Applied Nonlinear Control

Applied Nonlinear Control Nguyen Tan Tien - 2002.3 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Chapter 1 Introduction

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C.1 Introduction Why Nonlinear Control ? Nonlinear control is a mature subject with a variety of powerful methods and a long history of successful industrial applications ⇒ Why so many researchers have recently showed an active interest in the development and applications of nonlinear control methodologies ? • Improvement of existing control systems

Linear control methods rely on the key assumption of small range operation for the linear model to be valid. When the required operation range is large, a linear controller is likely to perform very poorly or to be unstable, because the nonlinearities in the system cannot be properly compensated for. Nonlinear controllers may handle the nonlinearities in large range operation directly. Ex: pendulum

• Analysis of hard nonlinearities One of the assumptions of linear control is that the system model is indeed linearizable. However, in control systems, there are many nonlinearities whose discontinuous nature does not allow linear approximation. Ex: Coulomb friction, backlash

• Dealing with model uncertainties In designing linear controllers, it is usually necessary to assume that the parameters of the system model are reasonably well known. However in many control problems involve uncertainties in the model parameters. Nonlinearities can be intentionally introduced into the control part of a control system so that model uncertainties can be tolerated. Two classes of nonlinear controllers for this purpose are robust controllers and adaptive controllers. Ex: parameter variations

• Design simplicity Good nonlinear control designs may be simpler and more intuitive than their linear counterparts. Ex: BuAxx +=& gufx +=&


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Chapter 2 Phase Plane Analysis

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2. Phase Plane Analysis Phase plane analysis is a graphical method for studying second-order systems. This chapter’s objective is to gain familiarity of the nonlinear systems through the simple graphical method. 2.1 Concepts of Phase Plane Analysis 2.1.1 Phase portraits The phase plane method is concerned with the graphical study of second-order autonomous systems described by

),( 2111 xxfx =& (2.1a) ),( 2122 xxfx =& (2.1b)

where

21, xx : states of the system

21, ff : nonlinear functions of the states Geometrically, the state space of this system is a plane having

21, xx as coordinates. This plane is called phase plane. The solution of (2.1) with time varies from zero to infinity can be represented as a curve in the phase plane. Such a curve is called a phase plane trajectory. A family of phase plane trajectories is called a phase portrait of a system. Example 2.1 Phase portrait of a mass-spring system_______

1=k

1=m

0

)(a )(b

x

x&

Fig. 2.1 A mass-spring system and its phase portrait

The governing equation of the mass-spring system in Fig. 2.1 is the familiar linear second-order differential equation

0=+ xx&& (2.2) Assume that the mass is initially at rest, at length 0x . Then the solution of this equation is

)cos()( 0 txtx = )sin()( 0 txtx −=&

Eliminating time t from the above equations, we obtain the equation of the trajectories

20

22 xxx =+ & This represents a circle in the phase plane. Its plot is given in Fig. 2.1.b. __________________________________________________________________________________________

The nature of the system response corresponding to various initial conditions is directly displayed on the phase plane. In the above example, we can easily see that the system trajectories neither converge to the origin nor diverge to infinity. They simply circle around the origin, indicating the marginal nature of the system’s stability. A major class of second-order systems can be described by the differential equations of the form

),( xxfx &&& = (2.3) In state space form, this dynamics can be represented with xx =1 and xx &=2 as follows

21 xx =& ),( 212 xxfx =&

2.1.2 Singular points A singular point is an equilibrium point in the phase plane. Since an equilibrium point is defined as a point where the system states can stay forever, this implies that 0x =& , and using (2.1)

==

0),(0),(

212

211xxfxxf

(2.4)

For a linear system, there is usually only one singular point although in some cases there can be a set of singular points. Example 2.2 A nonlinear second-order system____________

x

x&

3

6

9

3 6-6 -3

-3

-6

-9

convergencearea

divergencearea

to infinity

unstable

Fig. 2.2 A mass-spring system and its phase portrait

Consider the system 036.0 2 =+++ xxxx &&& whose phase portrait is plot in Fig. 2.2. The system has two singular points, one at )0,0( and the other at )0,3(− . The motion patterns of the system trajectories in the vicinity of the two singular points have different natures. The trajectories move towards the point 0=x while moving away from the point 3−=x . __________________________________________________________________________________________


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2

Why an equilibrium point of a second order system is called a singular point ? Let us examine the slope of the phase portrait. The slope of the phase trajectory passing through a point

),( 21 xx is determined by

),(),(

211

212

1

2xxfxxf

dxdx

= (2.5)

where 21, ff are assumed to be single valued functions. This implies that the phase trajectories will not intersect. At singular point, however, the value of the slope is 0/0, i.e., the slope is indeterminate. Many trajectories may intersect at such point, as seen from Fig. 2.2. This indeterminacy of the slope accounts for the adjective “singular”. Singular points are very important features in the phase plane. Examining the singular points can reveal a great deal of information about the properties of a system. In fact, the stability of linear systems is uniquely characterized by the nature of their singular points. Although the phase plane method is developed primarily for second-order systems, it can also be applied to the analysis of first-order systems of the form

0)( =+ xfx& The difference now is that the phase portrait is composed of a single trajectory. Example 2.3 A first-order system_______________________

Consider the system 34 xxx +−=& there are three singular

points, defined by 04 3 =+− xx , namely, 2,2,0 −=x . The phase portrait of the system consists of a single trajectory, and is shown in Fig. 2.3.

x

x&

stable unstableunstable

-2 0 2

Fig. 2.3 Phase trajectory of a first-order system

The arrows in the figure denote the direction of motion, and whether they point toward the left or the right at a particular point is determined by the sign of x& at that point. It is seen from the phase portrait of this system that the equilibrium point 0=x is stable, while the other two are unstable. __________________________________________________________________________________________

2.1.3 Symmetry in phase plane portrait Let us consider the second-order dynamics (2.3): ),( xxfx &&& = . The slope of trajectories in the phase plane is of the form

xxxf

dxdx

&

),( 21

1

2 −=

Since symmetry of the phase portraits also implies symmetry of the slopes (equal in absolute value but opposite in sign), we can identify the following situations:

),(),( 2121 xxfxxf −= ⇒ symmetry about the 1x axis. ),(),( 2121 xxfxxf −−= ⇒ symmetry about the 2x axis. ),(),( 2121 xxfxxf −−−= ⇒ symmetry about the origin. 2.2 Constructing Phase Portraits There are a number of methods for constructing phase plane trajectories for linear or nonlinear system, such that so-called analytical method, the method of isoclines, the delta method, Lienard’s method, and Pell’s method. Analytical method There are two techniques for generating phase plane portraits analytically. Both technique lead to a functional relation between the two phase variables 1x and 2x in the form

0),( 21 =xxg (2.6) where the constant c represents the effects of initial conditions (and, possibly, of external input signals). Plotting this relation in the phase plane for different initial conditions yields a phase portrait. The first technique involves solving (2.1) for 1x and 2x as a function of time t , i.e., )()( 11 tgtx = and )()( 22 tgtx = , and then, eliminating time t from these equations. This technique was already illustrated in example 2.1. The second technique, on the other hand, involves directly

eliminating the time variable, by noting that),(),(

211

212

1

2xxfxxf

dxdx

=

and then solving this equation for a functional relation between 1x and 2x . Let us use this technique to solve the mass-spring equation again. Example 2.4 Mass-spring system_______________________

By noting that )//()/( dtdxdxxdx &&& = , we can rewrite (2.2) as

0=+ xdxxdx&

& . Integration of this equation yields 20

22 xxx =+& . __________________________________________________________________________________________

Most nonlinear systems cannot be easily solved by either of the above two techniques. However, for piece-wise linear systems, an important class of nonlinear systems, this can be conveniently used, as the following example shows. Example 2.5 A satellite control system___________________

-U

U

p1

p1θ&u0=dθ

Jets Sattellite

θ

Fig. 2.4 Satellite control system

Fig. 2.4 shows the control system for a simple satellite model. The satellite, depicted in Fig. 2.5.a, is simply a rotational unit inertia controlled by a pair of thrusters, which can provide either a positive constant torqueU (positive firing) or negative torque (negative firing). The purpose of the control system is to maintain the satellite antenna at a zero angle by appropriately firing the thrusters.


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The mathematical model of the satellite is u=θ&& , where u is the torque provided by the thrusters andθ is the satellite angle. Let us examine on the phase plane the behavior of the control system when the thrusters are fired according to the control law

<>−

=00

)(θθ

ifUifU

tu (2.7)

which means that the thrusters push in the counterclockwise direction if θ is positive, and vice versa. As the first step of the phase portrait generation, let us consider the phase portrait when the thrusters provide a positive torque U . The dynamics of the system is U=θ&& , which implies that θθθ dUd =&& . Therefore, the phase portrait trajectories are a family of parabolas defined by

12 2 cU += θθ& , where 1c is constant. The corresponding

phase portrait of the system is shown in Fig. 2.5.b. When the thrusters provide a negative torque U− , the phase

trajectories are similarly found to be 12 2 cxU +−=θ& , with the

corresponding phase portrait as shown in Fig. 2.5.c.

Uu −=

x&

x

Uu =

x&

x

θantenna

u

Fig. 2.5 Satellite control using on-off thrusters The complete phase portrait of the closed-loop control system can be obtained simply by connecting the trajectories on the left half of the phase plane in 2.5.b with those on the right half of the phase plane in 2.5.c, as shown in Fig. 2.6.

parabolictrajectories

Uu += Uu −=switching line

x

x&

Fig.2.6 Complete phase portrait of the control system The vertical axis represents a switching line, because the control input and thus the phase trajectories are switched on that line. It is interesting to see that, starting from a nonzero initial angle, the satellite will oscillate in periodic motions under the action of the jets. One can concludes from this phase portrait that the system is marginally stable, similarly to the mass-spring system in Example 2.1. Convergence of the system to the zero angle can be obtained by adding rate feedback. __________________________________________________________________________________________

The method of isoclines (ñöôø ng ñaú ng khuynh) The basic idea in this method is that of isoclines. Consider the dynamics in (2.1): ),( 2111 xxfx =& and ),( 2122 xxfx =& . At a point ),( 21 xx in the phase plane, the slope of the tangent to the trajectory can be determined by (2.5). An isocline is defined to be the locus of the points with a given tangent slope. An isocline with slopeα is thus defined to be

α==),(),(

211

212

1

2xxfxxf

dxdx

This is to say that points on the curve

),(),( 211212 xxfxxf α= all have the same tangent slopeα . In the method of isoclines, the phase portrait of a system is generated in two steps. In the first step, a field of directions of tangents to the trajectories is obtained. In the second step, phase plane trajectories are formed from the field of directions. Let us explain the isocline method on the mass-spring system in (2.2): 0=+ xx&& . The slope of the trajectories is easily seen to be

2

1

1

2xx

dxdx

−=

Therefore, the isocline equation for a slopeα is

021 =+ xx α i.e., a straight line. Along the line, we can draw a lot of short line segments with slopeα . By takingα to be different values, a set of isoclines can be drawn, and a field of directions of tangents to trajectories are generated, as shown in Fig. 2.7. To obtain trajectories from the field of directions, we assume that the tangent slopes are locally constant. Therefore, a trajectory starting from any point in the plane can be found by connecting a sequence of line segments.

1=α 1−=α

∞=α

x&

x

Fig. 2.7 Isoclines for the mass-spring system

Example 2.6 The Van der Pol equation__________________

For the Van der Pol equation

0)1(2.0 2 =+−+ xxxx &&& an isocline of slopeα is defined by

α=+−−=

xxxx

dxxd

&

&& )1(2.0 2


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4

Therfore, the points on the curve

0)1(2.0 2 =++− xxxx && α all have the same slopeα . By takingα of different isoclines can be obtained, as plot in Fig. 2.8.

limit cycle

isoclines

trajectory

2-2

1x

2x

1=α

0=α 1−=α5−=α

Fig. 2.8 Phase portrait of the Van der Pol equation

Short line segments are drawn on the isoclines to generate a field of tangent directions. The phase portraits can be obtained, as shown in the plot. It is interesting to note that there exists a closed curved in the portrait, and the trajectories starting from both outside and inside converge to this curve. This closed curve corresponds to a limit cycle, as will be discussed further in section 2.5. __________________________________________________________________________________________

2.3 Determining Time from Phase Portraits Time t does not explicitly appear in the phase plane having

1x and 2x as coordinates. We now to describe two techniques for computing time history from phase portrait. Both of techniques involve a step-by step procedure for recovering time. Obtaining time from xxt &/∆≈∆ In a short time t∆ , the change of x is approximately

txx ∆≈∆ & (2.8) where x& is the velocity corresponding to the increment x∆ . From (2.8), the length of time corresponding to the increment x∆ is xxt &/∆≈∆ . This implies that, in order to obtain the time corresponding to the motion from one point to another point along the trajectory, we should divide the corresponding part of the trajectory into a number of small segments (not necessarily equally spaced), find the time associated with each segment, and then add up the results. To obtain the history of states corresponding to a certain initial condition, we simply compute the time t for each point on the phase trajectory, and then plots x with respects to t and x& with respects to t .

Obtaining time from dxxt ∫≈ )/1( &

Since dtdxx /=& , we can write xdxdt &/= . Therefore,

∫≈−x

xdxxtt

0

)/1(0 &

where x corresponding to time t and 0x corresponding to time 0t . This implies that, if we plot a phase plane portrait with new coordinates x and )/1( x& , then the area under the resulting curve is the corresponding time interval. 2.4 Phase Plane Analysis of Linear Systems The general form of a linear second-order system is

211 xbxax +=& (2.9a)

212 xdxcx +=& (2.9b) Transform these equations into a scalar second-order differential equation in the form )( 1112 xaxdxcbxb −+= && . Consequently, differentiation of (2.9a) and then substitution of (2.9b) leads to 111 )()( xdabcxdax −++= &&& . Therefore, we will simply consider the second-order linear system described by

0=++ xbxax &&& (2.10) To obtain the phase portrait of this linear system, we solve for the time history

tt ekektx 2121)( λλ += for 21 λλ ≠ (2.11a)

tt etkektx 2121)( λλ += for 21 λλ = (2.11b)

whre the constant 21,λλ are the solutions of the characteristic equation

0))(( 212 =−−=++ λλ ssbass

The roots 21,λλ can be explicitly represented as

242

1baa −+−

=λ and 2

42

2baa −−−

=λ

For linear systems described by (2.10), there is only one singular point )0( ≠b , namely the origin. However, the trajectories in the vicinity of this singularity point can display quite different characteristics, depending on the values of a and b . The following cases can occur • 21,λλ are both real and have the same sign (+ or -) • 21,λλ are both real and have opposite sign • 21,λλ are complex conjugates with non-zero real parts • 21,λλ are complex conjugates with real parts equal to 0

We now briefly discuss each of the above four cases Stable or unstable node (Fig. 2.9.a -b) The first case corresponds to a node. A node can be stable or unstable:

0, 21 <λλ : singularity point is called stable node. 0, 21 >λλ : singularity point is called unstable node.

There is no oscillation in the trajectories. Saddle point (Fig. 2.9.c) The second case ( 21 0 λλ << ) corresponds to a saddle point. Because of the unstable pole 2λ , almost all of the system trajectories diverge to infinity.


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ωj

σ

ωj

σ

ωj

σ

ωj

σ

ωj

σ

ωj

σ

center point

stable node

unstable node

saddle point

stable focus

unstable focus

x

x&

x

x&

x

x&

x

x&

x

x&

x

x&

)(a

)(b

)(c

)(d

)(e

)( f Fig. 2.9 Phase-portraits of linear systems

Stable or unstable locus (Fig. 2.9.d-e) The third case corresponds to a focus.

0),Re( 21 <λλ : stable focus 0),Re( 21 >λλ : unstable focus

Center point (Fig. 2.9.f) The last case corresponds to a certain point. All trajectories are ellipses and the singularity point is the centre of these ellipses.

⊗ Note that the stability characteristics of linear systems are uniquely determined by the nature of their singularity points. This, however, is not true for nonlinear systems. 2.5 Phase Plane Analysis of Nonlinear Systems In discussing the phase plane analysis of nonlinear system, two points should be kept in mind:

• Phase plane analysis of nonlinear systems is related to that of liner systems, because the local behavior of nonlinear systems can be approximated by the behavior of a linear system.

• Nonlinear systems can display much more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles.

Local behavior of nonlinear systems If the singular point of interest is not at the origin, by defining the difference between the original state and the singular point as a new set of state variables, we can shift the singular point to the origin. Therefore, without loss of generality, we may simply consider Eq.(2.1) with a singular point at 0. Using Taylor expansion, Eqs. (2.1) can be rewritten in the form

),( 211211 xxgxbxax ++=& ),( 212212 xxgxdxcx ++=&

where 21, gg contain higher order terms. In the vicinity of the origin, the higher order terms can be neglected, and therefore, the nonlinear system trajectories essentially satisfy the linearized equation

211 xbxax +=&

212 xdxcx +=& As a result, the local behavior of the nonlinear system can be approximated by the patterns shown in Fig. 2.9. Limit cycle In the phase plane, a limit cycle is defied as an isolated closed curve. The trajectory has to be both closed, indicating the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle (with near by trajectories converging or diverging from it). Depending on the motion patterns of the trajectories in the vicinity of the limit cycle, we can distinguish three kinds of limit cycles.

• Stable Limit Cycles: all trajectories in the vicinity of the limit cycle converge to it as ∞→t (Fig. 2.10.a).

• Unstable Limit Cycles: all trajectories in the vicinity of the limit cycle diverge to it as ∞→t (Fig. 2.10.b)

• Semi-Stable Limit Cycles: some of the trajectories in the vicinity of the limit cycle converge to it as

∞→t (Fig. 2.10.c)

2x

1x

convergingtrajectories 2x

1x

divergingtrajectories 2x

1x

convergingdiverging

limit cycle limit cycle limit cycle

)(a )(b )(c Fig. 2.10 Stable, unstable, and semi-stable limit cycles

Example 2.7 Stable, unstable, and semi-stable limit cycle___

Consider the following nonlinear systems

(a)

−+−−=

−+−=

)1(

)1(22

21212

22

21121

xxxxx

xxxxx

&

& (2.12)

(b)

−++−=

−++=

)1(

)1(22

21212

22

21121

xxxxx

xxxxx

&

& (2.13)

(c)

−+−−=

−+−=22

221212

222

21121

)1(

)1(

xxxxx

xxxxx

&

& (2.14)


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6

By introducing a polar coordinates

22

21 xxr +=

= −

1

21tan)(xxtθ

the dynamics of (2.12) are transformed as

)1( 2 −−= rrdtdr 1−=

dtdθ

When the state starts on the unicycle, the above equation shows that 0)( =tr& . Therefore, the state will circle around the origin with a period π2/1 . When 1<r , then 0>r& . This implies that the state tends to the circle from inside. When 1>r , then 0<r& . This implies that the states tend to the unit circle from outside. Therefore, the unit circle is a stable limit cycle. This can also be concluded by examining the analytical solution of (2.12)

tectr

201

1)(−+

= and tt −= 0)( θθ , where 1120

0 −=r

c

Similarly, we can find that the system (b) has an unstable limit cycle and system (c) has a semi-stable limit cycle. __________________________________________________________________________________________

2.6 Existence of Limit Cycles Theorem 2.1 (Pointcare) If a limit cycle exists in the second-order autonomous system (2.1), the N=S+1. Where, N represents the number of nodes, centers, and foci enclosed by a limit cycle, S represents the number of enclosed saddle points. This theorem is sometime called index theorem. Theorem 2.2 (Pointcare-Bendixson) If a trajectory of the second-order autonomous system remains in a finite region Ω , then one of the following is true:

(a) the trajectory goes to an equilibrium point (b) the trajectory tends to an asymptotically stable limit

cycle (c) the trajectory is itself a limit cycle

Theorem 2.3 (Bendixson) For a nonlinear system (2.1), no limit cycle can exist in the region Ω of the phase plane in which 2211 // xfxf ∂∂+∂∂ does not vanish and does not change sign. Example 2.8________________________________________

Consider the nonlinear system

22121 4)( xxxgx +=&

22112 4)( xxxhx +=&

Since )(4 22

21

2

2

1

1 xxxf

xf

+=∂∂+

∂∂ , which is always strictly

positive (except at the origin), the system does not have any limit cycles any where in the phase plane. __________________________________________________________________________________________


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Chapter 3 Fundamentals of Lyapunov Theory

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3. Fundamentals of Lyapunov Theory The objective of this chapter is to present Lyapunov stability theorem and illustrate its use in the analysis and the design of nonlinear systems. 3.1 Nonlinear Systems and Equilibrium Points Nonlinear systems A nonlinear dynamic system can usually be presented by the set of nonlinear differential equations in the form

),( txfx =& (3.1) where

nR∈f : nonlinear vector function nR∈x : state vectors

n : order of the system The form (3.1) can represent both closed-loop dynamics of a feedback control system and the dynamic systems where no control signals are involved. A special class of nonlinear systems is linear system. The dynamics of linear systems are of the from xAx )(t=& with

nnR ×∈A . Autonomous and non-autonomous systems Linear systems are classified as either time-varying or time-invariant. For nonlinear systems, these adjectives are replaced by autonomous and non-autonomous. Definition 3.1 The nonlinear system (3.1) is said to be autonomous if f does not depend explicitly on time, i.e., if the system’s state equation can be written

)(xfx =& (3.2) Otherwise, the system is called non-autonomous. Equilibrium points It is possible for a system trajectory to only a single point. Such a point is called an equilibrium point. As we shall see later, many stability problems are naturally formulated with respect to equilibrium points. Definition 3.2 A state *x is an equilibrium state (or equilibrium points) of the system if once )(tx is equal to *x , it

remains equal to *x for all future time.

Mathematically, this means that the constant vector *x satisfies

)( *xf0 = (3.3) Equilibrium points can be found using (3.3). A linear time-invariant system

xAx =& (3.4)

has a single equilibrium point (the origin 0) if A is nonsingular. If A is singular, it has an infinity of equilibrium points, which contained in the null-space of the matrix A, i.e., the subspace defined by Ax = 0. A nonlinear system can have several (or infinitely many) isolated equilibrium points. Example 3.1 The pendulum___________________________

R

θ

Fig. 3.1 Pendulum

Consider the pendulum of Fig. 3.1, whose dynamics is given by the following nonlinear autonomous equation

0sin2 =++ θθθ MgRbMR &&& (3.5) where R is the pendulum’s length, M its mass, b the friction coefficient at the hinge, and g the gravity constant. Leting

θ=1x , θ&=2x , the corresponding state-space equation is

21 xx =& (3.6a)

1222 sin xRgx

MRbx −−=& (3.6b)

Therefore the equilibrium points are given by ,02 =x

,0)sin( 1 =x which leads to the points )0],2[0( π and )0],2[( ππ . Physically, these points correspond to the

pendulum resting exactly at the vertical up and down points. __________________________________________________________________________________________

In linear system analysis and design, for notational and analytical simplicity, we often transform the linear system equations in such a way that the equilibrium point is the origin of the state-space. Nominal motion Let )(* tx be the solution of )(xfx =& , i.e., the nominal motion

trajectory, corresponding to initial condition 0* )0( xx = . Let

us now perturb the initial condition to be 00)0( xxx δ+= , and study the associated variation of the motion error

)()()( ttt *xxe −= as illustrated in Fig. 3.2.

nx

1x

2x

)(tx

)(t*x

)(te

Fig. 3.2 Nominal and perturbed motions


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8

Since both )(t*x and )(tx are solutions of (3.2): )(xfx =& , we have

)( ** xfx =& 0)0( xx* = )(xfx =& 00)0( xxx δ+=

then )(te satisfies the following non-autonomous differential equation

),(),(),( ** ttt egxfexfe =−+=& (3.8) with initial condition )0()0( xe δ= . Since 0),0( =tg , the new dynamic system, with e as state and g in place of f, has an equilibrium point at the origin of the state space. Therefore, instead of studying the deviation of )(tx from )(t*x for the original system, we may simply study the stability of the perturbation dynamics (3.8) with respect to the equilibrium point 0. However, the perturbation dynamics non-autonomous system, due to the presence of the nominal trajectory )(t*x on the right hand side. Example 3.2________________________________________ Consider the autonomous mass-spring system

0321 =++ xkxkxm &&&

which contains a nonlinear term reflecting the hardening effect of the spring. Let us study the stability of the motion

)(* tx which starts from initial point 0x . Assume that we slightly perturb the initial position to be 00)0( xxx δ+= . The resulting system trajectory is denoted as )(tx . Proceeding as before, the equivalent differential equation governing the motion error e is

0)](3)(3[ 2**2321 =++++ txetxeekekem &&

Clearly, this is a non-autonomous system. __________________________________________________________________________________________

3.2 Concepts of Stability Notation

RB : spherical region (or ball) defined by R≤x

RS : spherical itself defined by R=x

∀ : for any ∃ : there exist ∈ : in the set ⇒ : implies that ⇔ : equivalent

Stability and instability Definition 3.3 The equilibrium state 0x = is said to be stable if, for any 0>R , there exist 0>r , such that if r≤)0(x then

Rt ≤)(x for all 0≥t . Otherwise, the equilibrium point is unstable. Using the above symbols, Definition 3.3 can be written in the form

RttrrR <≥∀⇒<>∃>∀ )(,0)0(,0,0 xx or, equivalently

rr ttrR BxBx ∈≥∀⇒∈>∃>∀ )(,0)0(,0,0 Essentially, stability (also called stability in the sense of Lyapunov, or Lyapunov stability) means that the system trajectory can be kept arbitrarily close to the origin by starting sufficiently close to it. More formally, the definition states that the origin is stable, if, given that we do not want the state trajectory )(tx to get out of a ball of arbitrarily specified radius

RB . The geometrical implication of stability is indicated in Fig. 33.

12

3

0

)0(x rS

RS

curve 1 - asymptotically stablecurve 2 - marginally stable

curve 3 - unstable

Fig. 3.3 Concepts of stability

Asymptotic stability and exponential stability In many engineering applications, Lyapunov stability is not enough. For example, when a satellite’s attitude is disturbed from its nominal position, we not only want the satellite to maintain its attitude in a range determined by the magnitude of the disturbance, i.e., Lyapunov stability, but also required that the attitude gradually go back to its original value. This type of engineering requirement is captured by the concept of asymptotic stability. Definition 3.4 An equilibrium points 0 is asymptotically stable if it is stable, and if in addition there exist some 0>r such that r≤)0(x implies that 0x →)(t as ∞→t . Asymptotic stability means that the equilibrium is stable, and in addition, states start close to 0 actually converge to 0 as time goes to infinity. Fig. 3.3 shows that the system trajectories starting form within the ball rB converge to the origin. The ball rB is called a domain of attraction of the equilibrium point. In many engineering applications, it is still not sufficient to know that a system will converge to the equilibrium point after infinite time. There is a need to estimate how fast the system trajectory approaches 0. The concept of exponential stability can be used for this purpose. Definition 3.5 An equilibrium points 0 is exponential stable if there exist two strictly positive number α and λ such that

tett λα −≤>∀ )0()(,0 xx (3.9) in some ball rB around the origin. (3.9) means that the state vector of an exponentially stable system converges to the origin faster than an exponential


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9

function. The positive number λ is called the rate of exponential convergence.

For example, the system xxx )sin1( 2+−=& is exponentially convergent to 0=x with the rate 1=λ . Indeed, its solution is

ττ dxtextx ∫ +−−= 0

2 )](sin1[)0()( , and therefore textx −≤ )0()( . Note that exponential stability implies asymptotic stability. But asymptotic stability does not implies guarantee exponential stability, as can be seen from the system

1)0(,2 =−= xxx& (3.10) whose solution is )1/(1 tx += , a function slower than any

exponential function te λ− . Local and global stability Definition 3.6 If asymptotic (or exponential) stability holds for any initial states, the equilibrium point is said to be asymptotically (or exponentially) stable in the large. It is also called globally asymptotically (or exponentially) stable. 3.3 Linearization and Local Stability Lyapunov’s linearization method is concerned with the local stability of a nonlinear system. It is a formalization of the intuition that a nonlinear system should behave similarly to its linearized approximation for small range motions. Consider the autonomous system in (3.2), and assumed that f(x) is continuously differentiable. Then the system dynamics can be written as

)(... xfxxfx

0xtoh+

∂∂

==

& (3.11)

where ... tohf stands for higher-order terms in x. Let us use the constant matrix A denote the Jacobian matrix of f with respect

to x at x = 0: 0xx

fA=

∂∂

= . Then, the system

xAx =& (3.12) is called the linearization (or linear approximation) of the original system at the equilibrium point 0. In practice, finding a system’s linearization is often most easily done simply neglecting any term of order higher than 1 in the dynamics, as we now illustrate. Example 3.4________________________________________ Consider the nonlinear system

21221 cos xxxx +=&

211122 sin)1( xxxxxx +++=& Its linearized approximation about 0x = is

1.0 11 xx +=&

1221122 0 xxxxxxx +≈+++=&

The linearized system can thus be written xx

=

1101

& .

A similar procedure can be applied for a controlled system. Consider the system 0)1(4 25 =+++ uxxx &&& . The system can be linearly approximated about 0x = as 0)10(0 =+++ ux&& or

ux =&& . Assume that the control law for the original nonlinear

system has been selected to be xxxxu 23 cossin ++= , then the linearized closed-loop dynamics is 0=++ xxx &&& . __________________________________________________________________________________________

The following result makes precise the relationship between the stability of the linear system (3.2) and that of the original nonlinear system (3.2). Theorem 3.1 (Lyapunov’s linearization method)

• If the linearized system is strictly stable (i.e., if all eigenvalues of A are strictly in the left-half complex plane), then the equilibrium point is asymptotically stable (for the actual nonlinear system).

• If the linearizad system is un stable (i.e., if at least one eigenvalue of A is strictly in the right-half complex plane), then the equilibrium point is unstablle (for the nonlinear system).

• If the linearized system is marginally stable (i.e., if all eigenvalues of A are in the left-half complex plane but at least one of them is on the ωj axis), then one cannot conclude anything from the linear approximation (the equilibrium point may be stable, asymptotically stable, or unstable for the nonlinear system).

Example 3.5________________________________________ Consider the equilibrium point )0,( == θπθ & of the pendulum in the example 3.1. Since the neighborhood of πθ = , we can write

......)(cossinsin tohtoh +−=+−+= θππθππθ

thus letting πθθ −=~

, the system’s linearization about the equilibrium point )0,( == θπθ & is

0~~~

2 =−+ θθθRg

MRb &&&

Hence its linear approximation is unstable, and therefore so is the nonlinear system at this equilibrium point. __________________________________________________________________________________________

Example 3.5________________________________________ Consider the first-order system 5xbxax +=& . The origin 0 is one of the two equilibrium of this system. The linearization of this system around the origin is xax =& . The application of Lyapunov’s linearization method indicate the following stability properties of the nonlinear system

• 0<a : asymptotically stable • 0>a : unstable • 0=a : cannot tell from the linearization

In the third case, the nonlinear system is 5xbx =& . The linearization method fails while, as we shall see, the direct method to be described can easily solve this problem. __________________________________________________________________________________________


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10

3.4 Lyapunov’s Direct Method The basic philosophy of Lyapunov’s direct method is the mathematical extension of a fundamental physical observation: if the total energy of a mechanical (or electrical) system is continuous dissipated, then the system, whether linear or nonlinear, must eventually settle down to an equilibrium point. Thus, we may conclude the stability of a system by examining the variation of a single scalar function. Let consider the nonlinear mass-damper-spring system in Fig. 3.6, whose dynamic equation is

0310 =+++ xkxkxxbxm &&&& (3.13)

with

xxb && : nonlinear dissipation or damping 3

10 xkxk + : nonlinear spring term

mdamperandspringnonlinear

Fig. 3.6 A nonlinear mass-damper-spring system

Total mechanical energy = kinetic energy + potential energy

41

20

2

0

310

241

21

21)(

21)( xkxkxmdxxkxkxmV

x++=++= ∫ &&x

(3.14) Comparing the definitions of stability and mechanical energy, we can see some relations between the mechanical energy and the concepts described earlier:

• zero energy corresponds to the equilibrium point ),( 0x0x == &

• assymptotic stability implies the convergence of mechanical energy to zero

• instability is related to the growth of mechanical energy The relations indicate that the value of a scalar quantity, the mechanical energy, indirectly reflects the magnitude of the state vector, and furthermore, that the stability properties of the system can be characterized by the variation of the mechanical energy of the system. The rate of energy variation during the system’s motion is obtained by differentiating the first equality in (3.14) and using (3.13)

3310 )()()( xbxxbxxxkxkxxmV &&&&&&&&& −=−=++=x (3.15)

(3.15) implies that the energy of the system, starting from some initial value, is continuously dissipated by the damper until the mass is settled down, i.e., 0=x& . The direct method of Lyapunov is based on generalization of the concepts in the above mass-spring-damper system to more complex systems.

3.4.1. Positive definite functions and Lyapunov functions Definition 3.7 A scalar continuous function )(xV is said to be locally positive definite if 0)( =0V and, in a ball

0RB

0≠x ⇒ 0)( >xV

If 0)( =0V and the above property holds over the whole state space, then )(xV is said to be globally positive definite.

For instance, the function )cos1(21)( 1

22

2 xMRxMRV −+=x

which is the mechanical energy of the pendulum in Example 3.1, is locally positive definite. Let us describe the geometrical meaning of locally positive definite functions. Consider a positive definite function

)(xV of two state variables 1x and 2x . In 3-dimensional space, )(xV typically corresponds to a surface looking like an

upward cup as shown in Fig. 3.7. The lowest point of the cup is located at the origin.

1x

2x

V

0123 VVV >>

1VV =

2VV =

3VV =

Fig. 3.7 Typical shape of a positive definite function ),( 21 xxV The 2-dimesional geometrical representation can be made as follows. Taking 1x and 2x as Cartesian coordinates, the level curves αVxxV =),( 21 typically present a set of ovals surrounding the origin, with each oval corresponding to a positive value of αV .These ovals often called contour curves may be thought as the section of the cup by horizontal planes, projected on the ),( 21 xx plane as shown in Fig. 3.8.

1x

2x

0

123 VVV >>

1VV =2VV =

3VV =

Fig. 3.8 Interpreting positive definite functions using contour

curves Definition 3.8 If, in a ball

0RB , the function )(xV is positive

definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system (3.2) is negative semi-definite, i.e., 0)( ≤xV& then, )(xV is said to be a Lyapunov function for the system (3.2).


___________________________________________________________________________________________________________


11

A Lyapunov function can be given simple geometrical interpretations. In Fig. 3.9, the point denoting the value of

),( 21 xxV is seen always point down an inverted cup. In Fig. 3.10, the state point is seen to move across contour curves corresponding to lower and lower value ofV .

1x

2x

V

0

)(tx

V

Fig. 3.9 Illustrating Definition 3.8 for n=2

1x

2x

0

123 VVV >>

1VV =2VV =

3VV =

Fig. 3.10 Illustrating Definition 3.8 for n=2 using contour

curves 3.4.2 Equilibrium point theorems

Lyapunov’s theorem for local stability Theorem 3.2 (Local stability) If, in a ball

0RB , there exists a

scalar function )(xV with continuous first partial derivatives such that

• )(xV is positive definite (locally in0RB )

• )(xV& is negative semi-definite (locally in0RB )

then the equilibrium point 0 is stable. If, actually, the derivative )(xV& is locally negative definite in

0RB , then the

stability is asymptotic. In applying the above theorem for analysis of a nonlinear system, we must go through two steps: choosing a positive Lyapunov function, and then determining its derivative along the path of the nonlinear systems. Example 3.7 Local stability___________________________ A simple pendulum with viscous damping is described as

0sin =++ θθθ &&& Consider the following scalar function

221)cos1()( θθ &+−=xV

Obviously, this function is locally positive definite. As a mater of fact, this function represents the total energy of the pendulum, composed of the sum of the potential energy and the kinetic energy. Its time derivative yields

0sin)( 2 ≤−=+= θθθθθ &&&&&& xV Therefore, by involving the above theorem, we can conclude that the origin is a stable equilibrium point. In fact, using physical meaning, we can see the reason why 0)( ≤xV& , namely that the damping term absorbs energy. Actually,

)(xV& is precisely the power dissipated in the pendulum. However, with this Lyapunov function, we cannot draw conclusion on the asymptotic stability of the system, because )(xV& is only negative semi-definite. __________________________________________________________________________________________

Example 3.8 Asymptotic stability_______________________ Let us study the stability of the nonlinear system defined by

221

22

2111 4)2( xxxxxx −−+=&

)2(4 22

2122

212 −++= xxxxxx&

around its equilibrium point at the origin.

22

2121 ),( xxxxV +=

its derivativeV& along any system trajectory is

)2)((2 22

21

22

21 −++= xxxxV&

Thus, is locally negative definite in the 2-dimensional ball 2B ,

i.e., in the region defined by 2)( 22

21 <+ xx . Therefore, the

above theorem indicates that the origin is asymptotically stable. __________________________________________________________________________________________ Lyapunov theorem for global stability Theorem 3.3 (Global Stability) Assume that there exists a scalar function V of the state x, with continuous first order derivatives such that

• )(xV is positive definite

• )(xV& is negative definite • ∞→)(xV as ∞→x

then the equilibrium at the origin is globally asymptotically stable. Example 3.9 A class of first-order systems_______________ Consider the nonlinear system

0)( =+ xcx& where c is any continuous function of the same sign as its scalar argument x , i.e., such as 00)( ≠∀> xxcx . Intuitively, this condition indicates that )(xc− ’pushes’ the system back towards its rest position 0=x , but is otherwise arbitrary.


___________________________________________________________________________________________________________


12

Since c is continuous, it also implies that 0)0( =c (Fig. 3.13). Consider as the Lyapunov function candidate the square of distance to the origin 2xV = . The function V is radially unbounded, since it tends to infinity as ∞→x . Its derivative

is )(22 xcxxxV −== && . Thus 0<V& as long as 0≠x , so that 0=x is a globally asymptotically stable equilibrium point.

0

)(xc

x

Fig. 3.13 The function )(xc

For instance, the system xxx −= 2sin& is globally convergent

to 0=x , since for 0≠x , xxx ≤≤ sinsin2 . Similarly, the

system 3xx −=& is globally asymptotically convergent to 0=x . Notice that while this system’s linear approximation )0( ≈x& is inconclusive, even about local stability, the actual nonlinear system enjoys a strong stability property (global asymptotic stability). __________________________________________________________________________________________

Example 3 .10______________________________________ Consider the nonlinear system

)( 22

21121 xxxxx +−=&

)( 22

21212 xxxxx +−−=&

The origin of the state-space is an equilibrium point for this system. Let V be the positive definite function 2

221 xxV += .

Its derivative along any system trajectory is 222

21 )(2 xxV +−=&

which is negative definite. Therefore, the origin is a globally asymptotically stable equilibrium point. Note that the globalness of this stability result also implies that the origin is the only equilibrium point of the system. __________________________________________________________________________________________ ⊗ Note that:

- Many Lyapunov function may exist for the same system. - For a given system, specific choices of Lyapunov

functions may yield more precise results than others. - Along the same line, the theorems in Lyapunov analysis

are all sufficiency theorems. If for a particular choice of Lyapunov function candidate V , the condition on V& are not met, we cannot draw any conclusions on the stability or instability of the system – the only conclusion we should draw is that a different Lyapunov function candidate should be tried.

3.4.3 Invariant set theorem Definition 3.9 A set G is an invariant set for a dynamic system if every system trajectory which starts from a point in G remains in G for all future time.

Local invariant set theorem The invariant set theorem reflect the intuition that the decrease of a Lyapunov function V has to graduate vanish (i.e., ) V& has to converge to zero) because V is lower bounded. A precise statement of this result is as follows. Theorem 3.4 (Local Invariant Set Theorem) Consider an autonomous system of the form (3.2), with f continuous, and let )(xV be a scalar function with continuous first partial derivatives. Assume that

• for some 0>l , the region lΩ defined by lV <)(x is bounded

• 0)( ≤xV& for all x in lΩ

Let R be the set of all points within lΩ where 0)( =xV& , and M be the largest invariant set in R. Then, every solution )(tx originating in lΩ tends to M as ∞→t . ⊗ Note that:

- M is the union of all invariant sets (e.g., equilibrium points or limit cycles) within R

- In particular, if the set R is itself invariant (i.e., if once 0=V& , then 0≡ for all future time), then M=R

The geometrical meaning of the theorem is illustrated in Fig. 3.14, where a trajectory starting from within the bounded region lΩ is seen to converge to the largest invariant set M. Note that the set R is not necessarily connected, nor is the set M. The asymptotic stability result in the local Lyapunov theorem can be viewed a special case of the above invariant set theorem, where the set M consists only of the origin.

1x2x

V

lV =

lΩ

R

M0x

Fig. 3.14 Convergence to the largest invariant set M

Let us illustrate applications of the invariant set theorem using some examples. Example 3 .11______________________________________

Asymptotic stability of the mass-damper-spring system For the system (3.13), we can only draw conclusion of marginal stability using the energy function (3.14) in the local equilibrium point theorem, because V& is only negative semi-definite according to (3.15). Using the invariant set theorem, however, we can show that the system is actually asymptotically stable. TO do this, we only have to show that the set M contains only one point.


___________________________________________________________________________________________________________


13

The set R defined by 0=x& , i.e., the collection of states with zero velocity, or the whole horizontal axis in the phase plane ),( xx & . Let us show that the largest invariant set M in this set R contains only the origin. Assume that M contains a point with a non-zero position 1x , then, the acceleration at that

point is 0)/()/( 310 ≠−−= xmkxmkx&& . This implies that the

trajectory will immediately move out of the set R and thus also out of the set M, a contradiction to the definition. __________________________________________________________________________________________ Example 3 .12 Domain of attraction____________________

Consider again the system in Example 3.8. For 1=l , the

region lΩ , defined by 1),( 22

2121 <+= xxxxV , is bounded.

The set R is simply the origin 0, which is an invariant set (since it is an equilibrium point). All the conditions of the local invariant set theorem are satisfied and, therefore, any trajectory starting within the circle converges to the origin. Thus, a domain of attraction is explicitly determined by the invariant set theorem. __________________________________________________________________________________________

Example 3 .13 Attractive limit cycle_____________________

Consider again the system

)102( 22

41

7121 −+−= xxxxx&

)102(3 22

41

52

312 −+−−= xxxxx&

Note that the set defined by 102 2

241 =+ xx is invariant, since

)102)(124()102( 22

41

62

101

22

41 −++−=−+ xxxxxx

dtd

which is zero on the set. The motion on this invariant set is described (equivalently) by either of the equations

21 xx =& 312 xx −=&

Therefore, we see that the invariant set actually represents a limit circle, along which the state vector moves clockwise. Is this limit circle actually attractive ? Let us define a Luapunov function candidate 22

241 )102( −+= xxV which represents a

measure of the “distance” to the limit circle. For any arbitrary positive number l , the region lΩ , which surrounds the limit circle, is bounded. Its derivative

222

41

62

101 )102)(3(8 −++−= xxxxV&

Thus V& is strictly negative, except if 102 2

241 =+ xx or

03 62

101 =+ xx , in which cases 0=V& . The first equation is

simply that defining the limit cycle, while the second equation is verified only at the origin. Since both the limit circle and the origin are invariant sets, the set M simply consists of their union. Thus, all system trajectories starting in lΩ converge either to the limit cycle or the origin (Fig. 3.15)

1x0

2x

limit cycle

Fig. 3.15 Convergence to a limit circle

Moreover, the equilibrium point at the origin can actually be shown to be unstable. Any state trajectory starting from the region within the limit cycle, excluding the origin, actually converges to the limit cycle. __________________________________________________________________________________________

Example 3.11 actually represents a very common application of the invariant set theorem: conclude asymptotic stability of an equilibrium point for systems with negative semi-definite V& . The following corollary of the invariant set theorem is more specifically tailored to such applications. Corollary: Consider the autonomous system (3.2), with f continuous, and let )(xV be a scalar function with continuous partial derivatives. Assume that in a certain neighborhoodΩ of the origin

• is locally positive definite • )(xV& is negative semi-definite

• the set R defined by 0)( =xV& contains no trajectories of (3.2) other than the trivial trajectory 0≡x

Then, the equilibrium point 0 is asymptotically stable. Furthermore, the largest connected region of the form (defined by lV <)(x ) withinΩ is a domain of attraction of the equilibrium point. Indeed, the largest invariant set M in R then contains only the equilibrium point 0. ⊗ Note that:

- The above corollary replaces the negative definiteness condition on V& in Lyapunov’s local asymptotic stability theorem by a negative semi-definiteness condition on V& , combined with a third condition on the trajectories within R.

- The largest connected region of the form lΩ withinΩ is a domain of attraction of the equilibrium point, but not necessarily the whole domain of attraction, because the function V is not unique.

- The setΩ itself is not necessarily a domain of attraction. Actually, the above theorem does not guarantee thatΩ is invariant: some trajectories starting in Ω but outside of the largest lΩ may actually end up outsideΩ .

Global invariant set theorem The above invariant set theorem and its corollary can be simply extended to a global result, by enlarging the involved region to be the whole space and requiring the radial unboundedness of the scalar functionV .


___________________________________________________________________________________________________________


14

Theorem 3.5 (Global Invariant Set Theorem) Consider an autonomous system of the form (3.2), with f continuous, and let )(xV be a scalar function with continuous first partial derivatives. Assume that

• 0)( ≤xV& over the whole state space • ∞→)(xV as ∞→x

Let R be the set of all points where 0)( =xV& , and M be the largest invariant set in R. Then all solutions globally asymptotically converge to M as ∞→t For instance, the above theorem shows that the limit cycle convergence in Example 3.13 is actually global: all system trajectories converge to the limit cycle (unless they start exactly at the origin, which is an unstable equilibrium point). Because of the importance of this theorem, let us present an additional (and very useful) example. Example 3 .14 A class of second-order nonlinear systems___ Consider a second-order system of the form

0)()( =++ xcxbx &&& where b and c are continuous functions verifying the sign conditions 0)( >xbx && for 0≠x& and 0)( >xcx & for 0≠x . The dynamics of a mass-damper-spring system with nonlinear damper and spring can be described by the equation of this form, with the above sign conditions simply indicating that the otherwise arbitrary function b and c actually present “damping” and “spring” effects. A nonlinear R-L-C (resistor-inductor-capacitor) electrical circuit can also be represented by the above dynamic equation (Fig. 3.16)

)(xcvC = xvL &&= )(xbvR &=

Fig. 3.16 A nonlinear R-L-C circuit

Note that if the function b and c are actually linear

))(,)(( 1 xxcxxb αα == && , the above sign conditions are simply the necessary and sufficient conditions for the system’s stability (since they are equivalent to the conditions

0,0 01 >> αα ). Together with the continuity assumptions, the sign conditions b and c are simply that 0)0( =b and 0=c (Fig. 3.17). A positive definite function for this system is

∫+=x

dyycxV0

2 )(21& , which can be thought of as the sum of

the kinetic and potential energy of the system. Differentiating V , we obtain

0)()()()()( ≤−=+−−=+= xbxxxcxcxxbxxxcxxV &&&&&&&&&&& which can be thought of as representing the power dissipated in the system. Furthermore, by hypothesis, 0)( =xbx && only if

0=x& . Now 0=x& implies that )(xcx −=&& , which is non-zero

as long as 0≠x . Thus the system cannot get “stuck” at an equilibrium value other than 0=x ; in other words, with R being the set defined by 0=x& , the largest invariant set M in R contains only one point, namely ]0,0[ == xx & . Use of the local invariant set theorem indicates that the origin is a locally asymptotically stable point.

0

)(xb &

x& 0

)(xc

x

Fig. 3.17 The functions )(xb & and )(xc

Furthermore, if the integral ∫x

drrc0

)( is unbounded as ∞→x ,

then V is a radially unbounded function and the equilibrium point at the origin is globally asymptotically stable, according to the global invariant set theorem. __________________________________________________________________________________________

Example 3 .15 Multimodal Lyapunov Function___________

Consider the system

2sin1 32 xxxxx π

=+−+ &&&

Chose the Lyapunov function dyyyxVx

∫

−+=

0

22

sin21 π& .

This function has two minima, at 0,1 =±= xx & , and a local maximum in x (a saddle point in the state-space) at

0,0 == xx & . Its derivative 42 1 xxV && −−= , i.e., the virtual

power “dissipated” by the system. Now 00 =⇒= xV && or 1±=x . Let us consider each of cases:

0=x& ⇒ 02

sin ≠−= xxx π&& except if 0=x or 1±=x

1±=x ⇒ 0=x&& Thus the invariant set theorem indicates that the system converges globally to or )0,1( =−= xx & . The first two of these equilibrium points are stable, since they correspond to local minima of V (note again that linearization is inconclusive about their stability). By contrast, the equilibrium point

)0,0( == xx & is unstable, as can be shown from linearization ))12/(( xx −= π&& , or simply by noticing that because that point

is a local maximum of V along the x axis, any small deviation in the x direction will drive the trajectory away from it. __________________________________________________________________________________________

⊗ Note that: Several Lyapunov function may exist for a given system and therefore several associated invariant sets may be derived. 3.5 System Analysis Based on Lyapunov’s Direct Method How to find a Lyapunov function for a specific problem ? There is no general way of finding Lyapunov function for


___________________________________________________________________________________________________________


15

nonlinear system. Faced with specific systems, we have to use experience, intuition, and physical insights to search for an appropriate Lyapunov function. In this section, we discuss a number of techniques which can facilitate the otherwise blind of Lyapunov functions. 3.5.1 Lyapunov analysis of linear time-invariant systems Symmetric, skew-symmetric, and positive definite matrices Definition 3.10 A square matrix M is symmetric if M=MT (in other words, if jiij MMji =∀ , ). A square matrix M is skew-

symmetric if TMM −= (i.e., jiij MMji −=∀ , ). ⊗ Note that:

- Any square nn× matrix can be represented as the sum of a symmetric and a skew-symmetric matrix. This can be shown in the following decomposition

4342143421symmetricskew

T

symmetric

T

−

++

=22M-MMMM

- The quadratic function associated with a skew-symmetric matrix is always zero. Let M be a nn× skew-symmetric matrix and x is an arbitrary 1×n vector. The definition of

skew-symmetric matrix implies that xMxxMx TTT −= .

Since xMxT is a scalar, xMxxMx TT −= which yields

0xMxx =∀ T, (3.16) In the designing some tracking control systems for robot, this fact is very useful because it can simplify the control law.

- (3.16) is a necessary and sufficient condition for a matrix M to be skew-symmetric.

Definition 3.11 A square matrix M is positive definite (p.d.) if

0xMx0x >⇒≠ T . ⊗ Note that:

- A necessary condition for a square matrix M to be p.d. is that its diagonal elements be strictly positive.

- A necessary and sufficient condition for a symmetric matrix M to be p.d. is that all its eigenvalues be strictly positive.

- A p.d. matrix is invertible. - A .d. matrix M can always be decomposed as

ΛUUM T= (3.37)

where IUU =T , Λ is a diagonal matrix containing the eigenvalues of M

- There are some following facts

• 2max

2min )()( xMMxxxM λλ ≤≤ T

• ΛzzΛUxUxMxx TTTT == where zUx = • IMΛIM )()( maxmin λλ ≤≤

• 2xzz =T The concepts of positive semi-definite, negative definite, and negative semi-definite can be defined similarly. For instance, a square nn× matrix M is said to be positive semi-definite

(p.s.d.) if 0xMxx ≥∀ T, . A time-varying matrix M(t) is uniformly positive definite if IM αα ≥≥∀>∃ )(,0,0 tt .

Lyapunov functions for linear time-invariant systems Given a linear system of the form xAx =& , let us consider a

quadratic Lyapunov function candidate xPxTV =& , where P is a given symmetric positive definite matrix. Its derivative yields

xQ-xxPxxPx TTTV =+= &&& (3.18) where

-QAPPA =+T (3.19) (3.19) is so-called Lyapunov equation. Note that Q may be not p.d. even for stable systems. Example 3 .17 ______________________________________

Consider the second order linear system with

−−= 12840A .

If we take IP = , then

−−−=+= 244

40PAAPQ- T . The

matrix Q is not p.d.. Therefore, no conclusion can be draw from the Lyapunov function on whether the system is stable or not. __________________________________________________________________________________________

A more useful way of studying a given linear system using quadratic functions is, instead, to derive a p.d. matrix P from a given p.d. matrix Q, i.e.,

• choose a positive definite matrix Q • solve for P from the Lyapunov equation • check whether P id p.d.

If P is p.d., then xPxT)2/1( is a Lyapunov function for the linear system. And the global asymptotical stability is guaranteed. Theorem 3.6 A necessary and sufficient condition for a LTI system xAx =& to be strictly stable is that, for any symmetric p.d. matrix Q, the unique matrix P solution of the Lyapunov equation (3.19) be symmetric positive definite. Example 3 .18 ______________________________________

Consider again the second order linear system in Example

3.18. Let us take IQ = and denote P by

=

22211211

ppppP ,

where due to the symmetry of P, 1221 pp = . Then the Lyapunov equation is

−−=

−−+

−−

1001

12480

12840

22211211

22211211

pppp

pppp

whose solution is 511 =p , 12212 == pp . The corresponding

matrix

= 11

15P is p.d., and therefore the linear system is

globally asymptotically stable. __________________________________________________________________________________________

3.5.2 Krasovskii’s method Krasovskii’s method suggests a simplest form of Lyapunov function candidate for autonomous nonlinear systems of the


___________________________________________________________________________________________________________


16

form (3.2), namely, ff TV = . The basic idea of the method is simply to check whether this particular choice indeed leads to a Lyapunov function. Theorem 3.7 (Krasovkii) Consider the autonomous system defined by (3.2), with the equilibrium point of interest being the origin. Let A(x) denote the Jacobian matrix of the system, i.e.,

xfA

∂∂

=)(x

If the matrix TAAF += is negative definite in a neighborhood Ω , then the equilibrium point at the origin is asymptotically stable. A Lyapunov function for this system is

)()()( xfxfx TV = If Ω is the entire state space and, in addition, ∞→)(xV as

∞→x , then the equilibrium point is globally asymptotically stable. Example 3 .19 ______________________________________

Consider the nonlinear system

211 26 xxx +−=& 32212 262 xxxx −−=&

We have

−−−

∂∂

= 22662

26xx

fA

−−−=+= 2

212124412

xTAAF

The matrix F is easily shown to be negative definite. Therefore, the origin is asymptotically stable. According to the theorem, a Lyapunov function candidate is

23221

221 )262()26()( xxxxxV −−++−=x

Since ∞→)(xV as ∞→x , the equilibrium state at the origin is globally asymptotically stable. __________________________________________________________________________________________

The applicability of the above theorem is limited in practice, because the Jcobians of many systems do not satisfy the negative definiteness requirement. In addition, for systems of higher order, it is difficult to check the negative definiteness of the matrix F for all x. Theorem 3.7 (Generalized Krasovkii Theorem) Consider the autonomous system defined by (3.2), with the equilibrium point of interest being the origin, and let A(x) denote the Jacobian matrix of the system. Then a sufficient condition for the origin to be asymptotically stable is that there exist two symmetric positive definite matrices P and Q, such that

0≠∀x , the matrix

QPAPAxF ++= T)( is negative semi-definite in some neighborhood Ω of the origin. The function )()()( xfxfx TV = is then a Lyapunov

function for this system. If the region Ω is the whole state space, and if in addition, ∞→)(xV as ∞→x , then the system is globally asymptotically stable. 3.5.3 The Variable Gradient Method The variable gradient method is a formal approach to constructing Lyapunov functions. To start with, let us note that a scalar function )(xV is related to its gradient V∇ by the integral relation

∫∇=x

xx0

)( dVV

where T

nxVxVV /,,/ 1 ∂∂∂∂=∇ K . In order to recover a unique scalar function V from the gradient V∇ , the gradient function has to satisfy the so-called curl conditions

),,2,1,( njixV

xV

i

j

j

i K=∂

∂∇=

∂∂∇

Note that the ith component iV∇ is simply the directional derivative ixV ∂∂ / . For instance, in the case 2=n , the above simply means that

1

2

2

1xV

xV

∂∂∇

=∂∂∇

The principle of the variable gradient method is to assume a specific form for the gradient V∇ , instead of assuming a specific form for a Lyapunov function V itself. A simple way is to assume that the gradient function is of the form

∑=

=∇n

jjiji xaV

1

(3.21)

where the ija ’s are coefficients to be determined. This leads

to the following procedure for seeking a Lyapunov functionV

• assume that V∇ is given by (3.21) (or another form) • solve for the coefficients ija so as to sastify the curl

equations • assume restrict the coefficients in (3.21) so that V& is

negative semi-definite (at least locally) • computeV from V∇ by integration • check whetherV is positive definite

Since satisfaction of the curl conditions implies that the above integration result is independent of the integration path, it is usually convenient to obtain V by integrating along a path which is parallel to each axis in turn, i.e.,

++∇+∇= ∫∫ KKK21

0212

0111 )0,,0,()0,,0,()(

xxdxxVdxxVV x

∫ ∇nx

nn dxxV0

1 )0,,0,( K


___________________________________________________________________________________________________________


17

Example 3 .20 ______________________________________

Let us use the variable gradient method top find a Lyapunov function for the nonlinear system

11 2xx −=& 22122 22 xxxx +−=&

We assume that the gradient of the undetermined Lyapunov function has the following form

2121111 xaxaV +=∇

2221212 xaxaV +=∇ The curl equation is

1

2

2

1xV

xV

∂∂∇

=∂∂∇ ⇒

1

21121

2

12212 x

axaxaxa

∂∂

+=∂∂

+

If the coefficients are chosen to be 0,1 21122211 ==== aaaa

which leads to 11 xV =∇ , 22 xV =∇ then V& can be computed as

2)(

22

21

022

011

21 xxdxxdxxVxx +

=+= ∫∫x (3.22)

This is indeed p.d., and therefore, the asymptotic stability is guaranteed. If the coefficients are chosen to be ,,1 2

21211 xaa ==

3,3 222221 == axa , we obtain the p.d. function

321

22

21

23

2)( xxxxV ++=x (3.23)

whose derivative is )3(262 2

22121

22

22

21 xxxxxxxV −−−−=& .

We can verify that V& is a locally negative definite function (noting that the quadratic terms are dominant near the origin), and therefore, (3.23) represents another Lyapunov function for the system. __________________________________________________________________________________________

3.5.4 Physically motivated Lyapunov functions 3.5.5 Performance analysis Lyapunov analysis can be used to determine the convergence rates of linear and nonlinear systems. A simple convergence lemma Lemma: If a real function )(tW satisfies the inequality

0)()( ≤+ tWtW α& (3.26) where α is a real number. Then teWtW α−≤ )0()( The above Lemma implies that, if W is a non-negative function, the satisfaction of (3.26) guarantees the exponential convergence of W to zero.

Estimating convergence rates for linear system Let denote the largest eigenvalue of the matrix P by )(max Pλ , the smallest eigenvalue of the matrix Q by )(min Qλ , and their ratio )(/)( minmax QP λλ by γ . The p.d. of P and Q implies that these scalars are all strictly positive. Since matrix theory shows that IPP )(maxλ≤ and QIQ ≤)(minλ , we have

VxIPxPQxQx γλ

λλ

≥≥ ])([)()(

maxmax

min TT

This and (3.18) implies that VV γ−≤& .This, according to

lemma, means that .)0( tT e γ−≤ VxQx This together with the

fact 2min )()( tT xPxPx λ≥ , implies that the state x

converges to the origin with a rate of at least 2/γ . The convergence rate estimate is largest for IQ = . Indeed, let

0P be the solution of the Lyapunov equation corresponding to IQ = is

IAPPA −=+ 00T

and let P the solution corresponding to some other choice of Q

1QPAPA −=+T Without loss of generality, we can assume that 1)( 1min =Qλ since rescaling 1Q will rescale P by the same factor, and therefore will not affect the value of the corresponding γ . Subtract the above two equations yields

)()()( 100 I-QAP-PP-PA −=+T Now since )(1)( max1min IQ λλ == , the matrix )( 1 I-Q is positive semi-definite, and hence the above equation implies that )( 0P-P is positive semi-definite. Therefore

)()( 0maxmax PP λλ ≥ Since )(1)( min1min IQ λλ == , the convergence rate estimate

)(/)( maxmin PQ λλγ = corresponding to IQ = the larger than (or equal to) that corresponding to 1QQ = . Estimating convergence rates for nonlinear systems The estimation convergence rate for nonlinear systems also involves manipulating the expression of V& so as to obtain an explicit estimate of V . The difference lies in that, for nonlinear systems, V and V& are not necessarily quadratic function of the states. Example 3 .22 ______________________________________

Consider again the system in Example 3.8

221

22

2111 4)2( xxxxxx −−+=&

)2(4 22

2122

212 −++= xxxxxx&

Choose the Lyapunov function candidate 2x=V , its

derivative is )1(2 −= VVV& . That is dtVV

dV 2)1(

−=−

. The

solution of this equation is easily found to be


___________________________________________________________________________________________________________


18

dt

dt

eeV 2

2

1)(

−

−

+=

ααx , where

)0(1)0(

VV−

=α .

If 1)0()0( 2 <=Vx , i.e., if the trajectory starts inside the

unit circle, then 0>α , and tetV 2)( −<α . This implies that

the norm )(tx of the state vector converges to zero exponentially, with a rate of 1. However, if the trajectory starts outside the unit circle, i.e., if

1)0( >V , then 0<α , so that )(tV and therefore x tend to infinity in a finite time (the system is said to exhibit finite escape time, or “explosion”). __________________________________________________________________________________________

3.6 Control Design Based on Lyapunov’s Direct Method There are basically two ways of using Lyapunov’s direct method for control design, and both have a trial and error flavor:

• Hypothesize one form of control law and then finding a Lyapunov function to justify the choice

• Hypothesize a Lyapunov function candidate and then finding a control law to make this candidate a real Lyapunov function

Example 3 .23 Regulator design_______________________

Consider the problem of stabilizing the system uxxx =+− 23&&& . __________________________________________________________________________________________


___________________________________________________________________________________________________________

Chapter 4 Advanced Stability Theory

19

4. Advanced Stability Theory The objective of this chapter is to present stability analysis for non-autonomous systems. 4.1 Concepts of Stability for Non-Autonomous Systems Equilibrium points and invariant sets For non-autonomous systems, of the form

),( txfx =& (4.1) equilibrium points *x are defined by

00),( ttt* ≥∀≡xf (4.2) Note that this equation must be satisfied 0tt ≥∀ , implying that the system should be able to stay at the point *x all the time. For instance, we can easily see that the linear time-varying system

xAx )(t=& (4.3) has a unique equilibrium point at the origin 0 unless A(t) is always singular. Example 4.1________________________________________

The system

21)(xxtax

+−=& (4.4)

has an equilibrium point at 0=x . However, the system

)(1

)(2 tb

xxtax +

+−=& (4.5)

with 0)( ≠tb , does not have an equilibrium point. It can be regarded as a system under external input or disturbance )(tb . Since Lyapunov theory is mainly developed for the stability of nonlinear systems with respect to initial conditions, such problem of forced motion analysis are more appropriately treated by other methods, such as those in section 4.9. __________________________________________________________________________________________

Extensions of the previous stability concepts Definition 4.1 The equilibrium point 0 is stable at 0t if for any 0>R , there exists a positive scalar ),( 0tRr such that

rt <)( 0x ⇒ Rt <)(x 0tt ≥∀ (4.6) Otherwise, the equilibrium point 0 is unstable. The definition means that we can keep the state in ball of arbitrarily small radius R by starting the state trajectory in a ball of sufficiently small radius r .

Definition 4.2 The equilibrium point 0 is asymptotically stable at 0t if

• it is stable • 0)( 0 >∃ tr such that )()( 00 trt <x ⇒ 0x →)(t as

∞→t

The asymptotic stability requires that there exists an attractive region for every initial time 0t . Definition 4.3 The equilibrium point 0 is exponentially stable if there exist two positive numbers, α and λ , such that for sufficiently small )( 0tx ,

)(0

0)( ttet −−≤ λα xx 0tt ≥∀ Definition 4.4 The equilibrium point 0 is global stable )( 0tx∀ ,

0x →)(t as ∞→t Example 4.2 A first-order linear time-varying system_______ Consider the first-order system )()()( txtatx −=& . Its solution is

∫−=

t

tdrra

etxtx 0)(

0 )()( . Thus system is stable if

0,0)( ttta ≥∀≥ . It is asymptotically stable if ∫∞

+∞=0

)( drra .

It is exponentially stable if there exists a strictly positive

number T such that 0≥∀t , ∫+

≥Tt

tdrra γ)( , with γ being a

positive constant. For instance

• The system 2)1/( txx +−=& is stable (but not asymptotically stable)

• The system )1/( txx +−=& is asymptotically stable • The system xtx −=& is exponentially stable

Another interesting example is the system 2sin1)(

xxtx

+−=&

The solution can be expressed as∫ +

−=

t

tdr

rxx

etxtx 0 2 )(sin10 )()(

Since ∫−

≥+

t

t

ttdr

rxx

0 2)(sin10

2 , the system is exponentially

convergent with rate 2/1 . __________________________________________________________________________________________

Uniformity in stability concepts The previous concepts of Lyapunov stability and asymptotic stability for non-autonomous systems both indicate the importance effect of initial time. In practice, it is usually desirable for the systems to have a certain uniformity in its behavior regardless of when the operation starts. This motivates us to consider the definitions of uniform stability and uniform asymptotic stability. Non-autonomous systems with uniform properties have some desirable ability to


___________________________________________________________________________________________________________


20

withstand disturbances. The behavior of autonomous systems is dependent of the initial time, all the stability properties of an autonomous system are uniform. Definition 4.5 The equilibrium point 0 is locally uniformly stable if the scalar r in Definition 4.1 can be chosen independent of 0t , i.e., if )(Rrr = . Definition 4.6 The equilibrium point at the origin is locally uniformly asymptotically stable if

• it is uniformly stable • there exits a ball of attraction

0RB , whose radius is

independent of 0t , such that any trajectory with initial states in

0RB converges to 0 uniformly in 0t .

By uniform convergence in terms of 0t , we mean that for all 1R and 2R satisfying 0),(,0 21012 >∃≤<< RRTRRR such that, 0tt ≥∀

10 )( Rt <x ⇒ 2)( Rt <x ),( 210 RRTtt +≥∀ i.e., the trajectory, starting from within a ball

1RB , will

converges into a smaller ball 2RB after a time period T which

is independent of 0t . By definition, uniform asymptotic stability always implies asymptotic stability. The converse (ñaû o ñeà ) is generally not true, as illustrated by the following example. Example 4.3________________________________________

Consider the first-order system t

xx+

−=1

& . This system has

general solution ).(11

)( 00 txtt

tx++

= The solution asymptotically

converges to zero. But the convergence is not uniform. Intuitively, this is because a larger 0t requires a longer time to get close to the origin. __________________________________________________________________________________________

The concept of globally uniformly asymptotic stability can be defined be replacing the ball of attraction

0RB by the whole

state space. 4.2 Lyapunov Analysis of Non-Autonomous Systems In this section, we extend the Lyapunov analysis results of chapter 3 to the stability of non-autonomous systems. 4.2.1 Lyapunov’sdirect method for non-autonomous systems The basic idea of the direct method, i.e., concluding the stability of nonlinear systems using scalar Lyapunov functions, can be similarly applied to non-autonomous systems. Besides more mathematical complexity, a major difference in non-autonomous systems is that the powerful La Salle’s theorems do not apply. This drawback will partially be compensates by a simple result in section 4.5 called Barbalat’s lemma. Time-varying positive definite functions and decrescent functions

Definition 4.7 A scalar time-varying function ),( tV x is locally positive definite if 0),( =tV 0 and there exits a time-variant positive definite function )(0 xV such that

)(),(, 00 xx VtVtt ≥≥∀ (4.7) Thus, a time-variant function is locally positive definite if it dominates a time-variant locally positive definite function. Globally positive definite functions can be defined similarly. Definition 4.8 A scalar time-varying function ),( tV x is said to be decrescent if 0),( =tV 0 , and if there exits a time-variant positive definite function )(1 xV such that

)(),(,0 1 xx VtVt ≥≥∀ (4.7)

In other word, a scalar function ),( tV x is decrescent if it is dominated by a time-invariant p.d. function. Example 4.4________________________________________ Consider time-varying positive definite functions as follows

))(sin1(),( 22

21

2 xxttV ++=x 22

210 )( xxV +=x

)(2)( 22

211 xxV +=x

⇒ ),( tV x dominates )(0 xV and is dominated by )(1 xV because )(),()( 10 xxx VtVV ≤≤ . __________________________________________________________________________________________

Given a time-varying scalar function ),,( tV x its derivative along a system trajectory is

),( tfVtVV

tV

tdVd x

xx

x ∂∂

+∂∂

=∂∂

+∂∂

= & (4.8)

Lyapunov theorem for non-autonomous system stability The main Lyapunov stability results for non-autonomous systems can be summarized by the following theorem. Theorem 4.1 (Lyapunov theorem for non-autonomous systems)

Stability: If, in a ball 0RB around the equilibrium point 0,

there exits a scalar function ),( tV x with continuous partial derivatives such that

1. V is positive definite 2. V& is negative semi-definite

then the equilibrium point 0 is stable in the sense of Lyapunov.

Uniform stability and uniform asymptotic stability: If, furthermore

3. V is decrescent

then the origin is uniformly stable. If the condition 2 is strengthened by requiring thatV& be negative definite, then the equilibrium point is asymptotically stable.


___________________________________________________________________________________________________________


21

Global uniform asymptotic stability: If, the ball 0RB is

replaced by the whole state space, and condition 1, the strengthened condition 2, condition 3, and the condition

4. ),( tV x is radially unbounded

are all satisfied, then the equilibrium point at 0 is globally uniformly asymptotically stable.

Similarly to the case of autonomous systems, if in a certain neighborhood of the equilibrium point, V is positive definite andV& , its derivative along the system trajectories, is negative semi-definite, thenV is called Lyapunov function for the non-autonomous system. Example 4.5 Global asymptotic stability_________________ Consider the system defined by

)()()( 22

11 txetxtx t−−−=& )()()( 212 txtxtx −=&

Chose the Lyapunov function candidate

22

221 )1(),( xextV t−++=x

This function is p.d., because it dominates the time-invariant p.d. function 2

221 xx + . It is also decrescent, because it is

dominated by the time-invariant p.d. function 22

21 2xx + .

Furthermore,

)]1([2),( 22221

21

texxxxtV −++−−=x& This shows that

22

21

221

2221

21 )()(2),( xxxxxxxxtV −−−−=+−−≤x&

Thus, ),( tV x& is negative definite, and therefore, the point 0 is globally asymptotically stable. 4.2.2 Lyapunov analysis of linear time-varying systems Consider linear time-varying systems of the form

xAx )(t=& (4.17) Since LTI systems are asymptotically stable if their eigenvalues all have negative real parts ⇒ Will the system (4.17) be stable if any time 0≥t , the eigenvalues of )(tA all have negative parts ? Consider the system

−−=

212

21

101

xxe

xx t

&&

(4.18)

Both eigenvalues of )(tA equal to -1 at all times. The solution

of (4.18) can be rewritten as texx −= )0(22 , texxx )0(211 =+& . Hence, the system is unstable.

A simple result, however, is that the time-varying system (4.17) is asymptotically stable if the eigenvalues of the symmetric matrix )()( tt TAA + (all of which are real) remain strictly in the left-half complex plane

λλλ −≤+≥∀∀>∃ ))()((,0,,0 ttti Ti AA (4.19)

This can be readily shown using the Lyapunov function

xxTV = , since

VttV TTTTT λλ −=−≤+=+= xxAAxxxxx &&&& ))()(( so that tT eVtVt λ−≤=≤≥∀ )0()(0,0 xx and therefore x tends to zero exponentially. It is important to notice that the result provides a sufficient condition for any asymptotic stability. Perturbed linear systems Consider a linear time-varying system of the form

xAAx )]([ 21 t+=& (4.20) where 1A is constant and Hurwitz and the time-varying matrix

)(2 tA is such that 0A →)(2 t as ∞→t and

∫∞

∞<0

2 )( dttA (i.e., the integral exists and is finite)

Then the system (4.20) is globally stable exponentially stable. Example 4.8________________________________________

Consider the system defined by

183

521 )5( xxxx ++−=&

2322 4xxx +−=&

33 )sin2( xtx +−=& Since 3x tends to zero exponentially, so does 2

3x , and therefore, so does 2x . Applying the above result to the first equation, we conclude that the system is globally exponentially stable. __________________________________________________________________________________________

Sufficient smoothness conditions on the )(tA matrix Consider the linear system (4.17), and assume that at any time

0≥t , the eigenvalues of )(tA all have negative real parts

αλα −≤≥∀∀>∃ )]([,0,,0 tti i A (4.21) If, in addition, the matrix )(tA remains bounded, and

∞<∫∞

0)()( dtttT AA (i.e., the integral exists and is finite)

then the system is globally exponentially stable.


___________________________________________________________________________________________________________


22

4.2.3 The linearization method for non-autonomous systems Lyapunov’s linearization method can also be developed for non-autonomous systems. Let a non-autonomous system be described by (4.1) and 0 be an equilibrium point. Assume that f is continuously differentiable with respect to x. Let us denote

0xxfA

=

∂∂

=)(t (4.22)

The for any fixed time t (i.e., regarding t as a parameter), a Taylor expansion of f leads to

),()( ... txt toh xfAx +=& If f can be well approximated by xA )(t for any time t , i.e.,

00),(

suplim ... ≥∀=→

tttoh

xxf

0x (4.23)

then the system

xAx )(t=& (4.24) is said to be the linearization (or linear approximation) of the nonlinear non-autonomous system (4.1) around equilibrium point 0. ⊗ Note that:

- The Jacobian matrix A thus obtained from a non-autonomous nonlinear system is generally time-varying, contrary to what happens for autonomous nonlinear systems. But in some cases A is constant. For example, the nonlinear system txxx /2+−=& leads to the linearized system xx −=& .

- Our late results require that the uniform convergence condition (4.23) be satisfied. Some non-autonomous systems may not satisfy this condition, and Lyapunov’s linearization method cannot be used for such systems. For example, (4.23) is not satisfied for the system

2xtxx +−=& . Given a non-autonomous system satisfying condition (4.23), we can assert its (local) stability if its linear approximation is uniformly asymptotically stable, as stated in the following theorem: Therem 4.2 If the linearized system (with condition (4.23) satisfied) is uniformly asymptotically stable, then the equilibrium point 0 of the original non-autonomous system is also uniformly asymptotically stable. ⊗ Note that:

- The linearized time-varying system must be uniformly asymptotically stable in order to use this theorem. If the linearized system is only asymptotically stable, no conclusion can be draw about the stability of the original nonlinear system.

- Unlike Lyapunov’s linearization method for autonomous system, the above theorem does not relate the instability of the linearized time-varying system to that of the nonlinear system.

Therem 4.3 If the Jacobian matrix )(tA is constant,

0)( AA =t , and if (4.23) is satisfied, then the instability of the linearized system implies that of the original non-autonomous nonlinear system, i.e., (4.1) is unstable if one or more of the eigenvalues of 0A has a positive real part . 4.3 Instability Theorems 4.4 Existence of Lyapunov Functions 4.5 Lyapunov-Like Analysis Using Barbalat’s Lemma Asymptotic stability analysis of non-autonomous systems is generally much harder than that of autonomous systems, since it is usually very difficult to find Lyapunov functions with a negative definite derivative. An important and simple result which partially remedies (khaé c phuï c) this situation is Barbalat’s lemma. When properly used for dynamic systems, particularly for non-autonomous systems, it may lead to the satisfactory solution of many asymptotic stability problem. 4.5.1 Asymptotic properties of functions and their derivatives Before discussing Barbalat’s lemma itself, let us clarify a few points concerning the asymptotic properties of functions and their derivatives. Given a differentiable function f of time t , the following three facts are important to keep in mind

• ff ≠>→ 0& converges

The fact that 0→f& does not imply that )(tf has a limit as ∞→t .

• f converges 0→≠> f& The fact that )(tf has a limit as ∞→t does not imply

that 0→f& .

• If f is lower bounded and decreasing )0( ≤f& , then it converges to a limit.

4.5.2 Barbalat’s lemma Lemma 4.2 (Barbalat) If the differentiable function )(tf has

a finite limit as ∞→t , and if f& is uniformly continuous, then

0)( →tf& as ∞→t . ⊗ Note that:

- A function )(tg is continuous on ),0[ ∞ if

Rtgtg

ttttRRt

<−⇒

<−≥∀>∃>∀≥∀

)()(

,0,0),(,0,0

1

111 ηη

- A function )(tg is said to be uniformly continuous on ),0[ ∞ if

Rtgtg

ttttRR

<−⇒

<−≥∀≥∀>∃>∀

)()(

,0,0,0)(,0

1

11 ηη

or in other words, )(tg is uniformly continuous if we can always find anη which does not depend on the specific point 1t - and in particular, such thatη does not shrink as .∞→t t and 1t play a symmetric role in the definition of uniform continuity.


___________________________________________________________________________________________________________


23

- A simple sufficient condition for a differentiable function to be uniformly continuous is that its derivative be bound. This can be seen from the finite different theorem: ,, 1tt∀

)( 122 tttt ≤≤∃ such that ))(()()( 121 tttgtgtg −=− & . And therefore, if 01 >R is an upper bound on the

function g& , we can always use 1/ RR=η independently

of 1t to verify the definition of uniform continuity. Example 4.12_______________________________________ Consider a strictly stable linear system whose input is bounded. Then the system output is uniformly continuous. Indeed, write the system in the standard form

uBxAx +=& xCy =

Since u is bounded and the linear system is strictly stable, thus the state x is bounded. This in turn implies from the first equation that x& is bounded, and therefore from the second equation that xCy && = is bounded. Thus the system output y is uniformly continuous. __________________________________________________________________________________________

Using Barbalat’s lemma for stability analysis To apply Barbalat’s lemma to the analysis of dynamic systems, one typically uses the following immediate corollary, which looks very much like an invariant set theorem in Lyapunov analysis: Lemma 4.3 (Lyapunov-Like Lemma) If a scalar function

),( tV x satisfies the following conditions • ),( tV x is lower bounded

• ),( tV x& is negative semi-definite

• ),( tV x& is uniformly continuous in time

then 0),( →tV x& as ∞→t . Indeed, V the approaches a finite limiting value ∞V , such that

)0),0((xVV ≤∞ (this does not require uniform continuity). The above lemma then follows from Barbalat’s lemma. Example 4.13_______________________________________ Consider the closed-loop error dynamics of an adaptive control system for a first-order plant with unknown parameter

)(twee θ+−=&

)(twe−=θ& where e andθ are the two states of the closed-loop dynamics, representing tracking error and parameter error, and )(tw is a bounded continuous function. Consider the lower bounded function

22 θ+= eV Its derivative is

02)]([2)]([2 2 ≤−=−++−= etwetweeV θθ&

This implies that )0()( VtV ≤ , and therefore, that e andθ are bounded. But the invariant set cannot be used to conclude the convergence of e , because the dynamics is non-autonomous. To use Barbalat’s lemma, let us check the uniform continuity of V& . The derivative of V& is )(4 weeV θ+−−=&& . This shows

that V&& is bounded, since w is bounded by hypothesis, and e and θ were shown above to be bounded. Hence, V& is uniformly continuous. Application of Babarlat’s lemma then indicates that 0→e as ∞→t . Note that, although e converges to zero, the system is not asymptotically stable, because θ is only guaranteed to be bounded. __________________________________________________________________________________________

⊗ Note that: Such above analysis based on Barbalat’s lemma shall be called a Lyapunov-like analysis. There are two important differences with Lyapunov analysis:

- The function V can simply be a lower bounded function of x and t instead of a positive definite function.

- The derivative V& must be shown to be uniformly continuous, in addition to being negative or zero. This is typically done by proving that V&& is bounded.

4.6 Positive Linear Systems In the analysis and design of nonlinear systems, it is often possible and useful to decompose the system into a linear subsystem and a nonlinear subsystem. If the transfer function of the linear subsystem is so-called positive real, then it has important properties which may lead to the generation of a Lyapunov function for the whole system. In this section, we study linear systems with positive real transfer function and their properties. 4.6.1 PR and SPR transfer function Consider rational transfer function of nth-order SISO linear systems, represented in the form

01

1

01

1)(apap

bpbpbph nn

n

mm

mm

+++

+++=

−−

−−

K

K

The coefficients of the numerator and denominator polynomials are assumed to be real numbers and mn ≥ . The difference mn − between the order of the denominator and that of the numerator is called the relative degree of the system. Definition 4.10 A transfer function h(p) is positive real if

0)](Re[ ≥ph for all 0]Re[ ≥p (4.33) It is strictly positive real if )( ε−ph is positive real for some 0>ε Condition (4.33) is called the positive real condition, means that )( ph always has a positive (or zero) real part when p has positive (or zero) real part. Geometrically, it means that the rational function )( ph maps every point in the closed RHP (i.e., including the imaginary axis) into the closed RHP of )( ph .


___________________________________________________________________________________________________________


24

Example 4.14 A strictly positive real function_____________

Consider the rational functionλ+

=p

ph 1)( , which is the

transfer function of a first-order system, with 0>λ . Corresponding to the complex variable ωσ jp += ,

22)()(1)(

ωλσωλσ

λωσ ++

−+=

++=

jj

ph

Obviously, 0)](Re[ ≥ph if 0≥σ . Thus, )( ph is a positive real function. In fact, one can easily see that )( ph is strictly positive real, for example by choosing 2/λε = in Definition 4.9. __________________________________________________________________________________________

Theorem 4.10 A transfer function )( ph is strictly positive real (SPR) if and only if

i. )( ph is a strictly stable transfer function ii. the real part of )( ph is strictly positive along the ωj axis,

i.e., 0)](Re[0 >≥∀ ωω jh (4.34)

The above theorem implies necessary conditions for asserting whether a given transfer function )( ph is SPR:

• )( ph is strictly stable • The Nyquist plot of )( ωjh lies entirely in the RHP.

Equivalently, the phase shift of the system in response to sinusoidal inputs is always less than 900

• )( ph has relative degree of 0 or 1 • )( ph is strictly minimum-phase (i.e., all its zeros are in

the LHP) Example 4.15 SPR and non-SPR functions______________ Consider the following systems

bapppph

++

−= 21

1)( 1

1)( 22+−

−=

pppph

bappph

++= 23

1)( 1

1)( 24++

+=

pppph

The transfer function 21,hh and 3h are not SPR, because 1h is non-minimum phase, 2h is unstable, and 3h has relative degree larger than 1. Is the (strictly stable, minimum-phase, and of relative degree 1) function 4h actually SPR ? We have

222

2

24)1(

)1)(1(1

1)(ωωωωω

ωωωω

+−

+−−+=

++−

+=

jjj

jjh

(where the second equality is obtained by multiplying numerator and denominator by the complex conjugate of the denominator) and thus

222222

22

4)1(1

)1(1)](Re[

ωωωωωωω

+−=

+−

++−=jh

which shows that 4h is SPR (since it is also strictly stable). Of course, condition (4.34) can also be checked directly on a computer. __________________________________________________________________________________________

⊗ The basic difference between PR and SPR transfer functions is that PR transfer functions may tolerate poles on the ωj axis, while SPR functions cannot. Example 4.16_______________________________________

Consider the transfer function of an integrator .1)(p

ph = Its

value corresponding to ωσ jp += is 22)(ωσωσ

+

−=

jph . We

can easily see from Definition 4.9 that )( ph is PR but not SPR. __________________________________________________________________________________________

Theorem 4.11 A transfer function )( ph is positive real if, and only if,

• )( ph is a stable transfer function • The poles of )( ph on the ωj axis are simple (i.e., distinct)

and the associated residues are real and non-negative • 0)](Re[ ≥ωjh for any 0≥ω such that ωj is not a pole of

)( ph The Kalman-Yakubovich lemma If a transfer function of a system is SPR, there is an important mathematical property associated with its state-space representation, which is summarized in the celebrated Kalman-Yakubovich (KY) lemma. Lemma 4.4 (Kalman-Yakubovich) Consider a controllable linear time-invariant system

ubxAx +=&

xcTy = The transfer function

bAIc 1][)( −−= pph T (4.35) is strictly positive real if, and only if, there exist positive matrices P and Q such that

-QPAPA =+T (4.36a) cbP = (4.36b)

In the KY lemma, the involved system is required to be asymptotically controllable. A modified version of the KY lemma, relaxing the controllability condition, can be stated as follows Lemma 4.5 (Meyer-Kalman-Yakubovich) Given a scalar

0≥γ , vector b and c , any asymptotically stable matrix A , and a symmetric positive definite matrix L , if the transfer function

bAIp 1][2

)( −−+= pcH Tγ


___________________________________________________________________________________________________________


25

is SPR, then there exist a scalar 0>ε , a vector q , and a symmetric positive definite matrix P such that

Lq-qAPPA ε−=+ TT

qcbP γ+= This lemma is different from Lemma 4.4 in two aspects.

• the involved system now has the output equation

uy T2γ

+= xc

• the system is only required to be stabilizable (but not necessary controllable)

4.6.3 Positive real transfer matrices The concept of positive real transfer function can be generalized to rational positive real matrices. Such generation is useful for the analysis and design of MIMO systems. Definition 4.11 An mm× matrix )( pH is call PR if

• )( pH has elements which are analytic for 0)Re( >p

• *)()( pp THH + is positive semi-definite for 0)Re( >p

where the asterisk * denote the complex conjugate transpose. )( pH is SPR if )( ε−pH is PR for some 0>ε .

4.7 The Passivity Formalism 4.8 Absolute Stability 4.9 Establishing Boundedness of Signal 4.10 Existence and Unicity of Solutions


___________________________________________________________________________________________________________

Chapter 6 Feedback linearization

26

6. Feedback Linearization Feedback linearization is an approach to nonlinear control design.

- The central idea of the approach is to algebraically transform a nonlinear system dynamics in to a fully or partly one, so that the linear control theory can be applied.

- This differs entirely from conventional linearization (such as Jacobian linearization) in that the feedback, rather than by linear approximations of the dynamics.

- Feedback linearization technique can be view as ways of transforming original system models into equivalent models of a simpler form.

6.1 Intuitive Concepts This section describes the basic concepts of feedback linearization intuitively, using simple examples. 6.1.1 Feedback linearization and the canonical form Example 6.1: Controlling the fluid level in a tank Consider the control of the level h of fluid in a tank to a specified level hd. The control input is the flow u into the tank and the initial value is h0.

houtputflow

u

Fig. 6.1 Fluid level control in a tank The dynamic model of the tank is

ghatudhhAdtd

h

o

2)()( −=

∫ (6.1)

where, A(h) is the cross section of the tank, a is the cross section of outlet pipe. The dynamics (6.1) can be rewritten as

ghauhhA 2)( −=& (6.2) If u(t) is chosen as

vhAghatu )(2)( += (6.3) with v being an “equivalent input” to be specified, the resulting dynamics is linear vh =& Choosing v as

hv~

α−= (6.4)

with dhthh −= )(~

is the level error, α is a strictly positive constant. Now, the close loop dynamics is

0~=+ hh α& (6.4)

This implies that 0)(~

→th as ∞→t . From (6.2) and (6.3), the actual input flow is determined by the nonlinear control law

)()(2)( hhAghatu α−= (6.5) Note that in the control law (6.5)

gha 2 : used provide the output flow )()( hhA α : used to rise the fluid level according to

the desired linear dynamics (6.4) If the desired level is a known time-varying function hd (t), the equivalent input v can be chosen as hthv d

~)( α−= & so as to

still yield 0)(~

→th when ∞→t . ⊗ The idea of feedback linearization is to cancel the nonlinearities and imposing the desired linear dynamics. Feedback linearization can be applied to a class of nonlinear system described by the so-called companion form, or controllability canonical form. Consider the system in companion form

+

=

uxbxf

xx

x

xx

n )()(

3

2

2

1

M

&

M

&

&

(6.6)

where

x : the state vector )(),( xbxf : nonlinear function of the state

u : scalar control input For this system, using the control input of the form

bfvu /)( −= (6.7) we can cancel the nonlinearities and obtain the simple input-output relation (multiple-integrator form) vx n =)( . Thus, the

control law )1(110

−−−−−−= n

n xkxkxkv K& with the ki chosen

so that the polynomial 01

1 kpkp nn

n +++ −− K has its roots

strictly in the left-half complex plane, lead to exponentially stable dynamics 00

)1(1

)( =+++ −− xkxkx n

nn K which

implies that 0)( →tx . For tasks involving the tracking of the desired output xd (t), the control law

)1(110

)( −−−−−−= n

nn

d ekekekxv K& (6.8) (where )()()( txtxte d−= is the tracking error) leads to exponentially convergent tracking.


___________________________________________________________________________________________________________


27

Example 6.2: Feedback linearization of a two-link robot Consider the two-link robot as in the Fig. 6.2

lc1

l1

l2

lc2

I2, m2

I1, m1

q2,τ2

q1,τ1

Fig. 6.2 A two-link robot

The dynamics of a two-link robot

=

+

−−−=

2

1

2

1

2

1

1

212

2

1

2221

12110 τ

τgg

qq

qhqhqhqh

qq

HHHH

&

&

&

&&&

&&

&&

(6.9) where,

[ ]Tqqq 21= : joint angles

[ ]T21 τττ = : joint inputs (torques)

222122

2121

21111 )cos2( IqllllmIlmH ccc +++++=

222222122112 cos IlmqclmHH c ++==

222222 IlmH c +=

2212 sin qllmh c= ]cos)cos([cos 1121221111 qlqqlgmqglmg cc +++=

)cos( 21222 qqglmg c +=

Control objective: to make the joint position 1q and 2q follows desired histories )(1 tqd and )(2 tqd To achieve tracking control tasks, one can use the follow control law

+

−−−=

=

2

1

2

1

1

212

2

1

2221

1211

2

10 g

gqq

qhqhqhqh

vv

HHHH

&

&

&

&&&

ττ

(6.10)

where, qqqv d~~2 2λλ −−= &&&&

[ ]Tvvv 21= : the equivalent input

dqqq −=~ : position tracking error λ : a positive number

The tracking error satisfies the equation 0~~2~ 2 =++ qqq λλ &&& and therefore converges to zeros exponentially. ⊗ When the nonlinear dynamics is not in a controllability canonical form, one may have to use algebraic transforms to

first put the dynamics into the controllability canonical form before using the above feedback linearization design. 6.1.2 Input-State Linearization Consider the problem of design the control input u for a single-input nonlinear system of the form

)( ,uxfx =& The technique of input-state linearization solves this problem into two steps:

- Find a state transformation )(xzz = and an input trans-formation )( vx,uu = , so that the nonlinear system dynamics is transformed into an equivalent linear time-invariant dynamics, in the familiar form vbzAz +=& .

- Use standard linear technique to design v . Example: Consider a simple second order system

1211 sin2 xxaxx ++−=& (6.11a) )2cos(cos 1122 xuxxx +−=& (6.11b)

Even though linear control design can stabilize the system in a small region around the equilibrium point (0,0), it is not obvious at all what controller can stabilize it in a large region. A specific difficulty is the nonlinearity in the first equation, which cannot be directly cancelled by the control input u. Consider the following state transformation

11 xz = (6.12a)

122 sin xxaz += (6.12b) which transforms (6.11) into

211 2 zzz +−=& (6.13b) )2cos(sincoscos2 111112 zuazzzzz ++−=& (6.13b)

The new state equations also have an equilibrium point at (0,0). Now the nolinearities can be canceled by the control law of the form

)cos2sincos()2cos(

11111

1zzzzv

zau +−= (6.14)

where v is equivalent input to be designed (equivalent in the sense that determining v amounts to determining u, and vise versa), leading to a linear input-state relation

11 2 zz −=& (6.15a) vz =2& (6.15b)

Thus,

the problem ofstabilizing the new

dynamics (6.15)using the new

input v

the problem ofstabilizing the originalnonlinear dynamics(6.11) using the originalcontrol input u

inputtransformation

(6.14)

statetransformation

(6.12)


___________________________________________________________________________________________________________


28

Now, consider the new dynamics (6.15). It is linear and controllable. Using the well known linear state feedback control law ,2211 zkzkv −−= one could chose 0,2 21 == kk or

22 zv −= (6.16) resulting in the stable closed-loop dynamics 211 2 zzz +−=& and 22 2 zz −=& . In term of the original state, this control law corresponds to the original input

)cos2sincossin22()2cos(

1111112

1xxxxxxa

xau +−−−=

(6.17) The original state x is given from z by

11 zx = (6.18a) azzx /)sin( 122 −= (6.18b)

The closed-loop system under the above control law is represented in the block diagram in Fig. 6.3.

v=-kTz u=u (x,v) = f(x,u)x&

z=z (x)

x0

pole-placement loop

linearization loop

z

Fig. 6.3 Input-State Linearization ⊗ To generalize the above method, there are two equations:

- What classes of nonlinear systems can be transformed into linear systems ?

- How to find the proper transformations for those which can ?

6.1.3 Input-Ouput Linearization Consider a tracking control problem with the following system

)( ,uxfx =& (6.19a) )(xhy = (6.19b)

Control objective: to make the output )(ty track a desired trajectory )(tyd while keeping the whole state bounded.

)(tyd and its time derivatives are assumed to be known and bounded. Consider the third-order system

3221 )1(sin xxxx ++=& (6.20a)

3512 xxx +=& (6.20b)

uxx += 213& (6.20c)

1xy = (6.20d) To generate a direct relationship between the output and input, let us differentiate the output 3221 )1(sin xxxxy ++== && . Since y& is still not directly relate to the input ,u let us differentiate again. We now obtain

)()1( 12 xfuxy ++=&& (6.21) 212233

511 )1()cos)(()( xxxxxxf ++++=x (6.22)

Clearly, (6.21) represents an explicit relationship between y and u . If we choose the control input to be in the form

)(1

11

2fv

xu −

+= (6.23)

where v is the new input to be determined, the nonlinearity in (6.21) is canceled, and we apply a simple linear double-integrator relationship between the output and the new input v,

vy =&& . The design of tracking controller for this double-integrator relation is simple using linear technique. For instance, letting )()( tytye d−= be the tracking error, and choosing the new input v such as

ekekyv d &&& 21 −−= (6.24)

where 21, kk are positive constant. The tracking error of the closed-loop system is given by

012 =++ ekeke &&& (6.25) which represents an exponentially stable error dynamics. Therefore, if initially 0)0()0( == ee & , then 0,0)( ≥∀≡ tte , i.e., perfect tracking is achieved; otherwise, )(te converge to zero exponentially. ⊗ Note that:

- The control law is defined anywhere, except at the singularity point such that 12 −=x .

- Full state measurement is necessary in implementing the control law.

- The above controller does not guarantee the stability of internal dynamics.

Example 6.3: Internal dynamics Consider the nonlinear control system

+=

uux

xx 3

2

2

1&

& (6.27a)

1xy = (6.27b) Control objective: to make y track to )(tyd

uxxy +== 321&& ⇒ )()(3

2 tytexu d&+−−= (6.28) yields exponential convergence of e to zero.

0=+ ee& (6.29) Apply the same control law to the second dynamic equation, leading to the internal dynamics

eyxx d −=+ && 322 (6.30)

which is non-autonomous and nonlinear. However, in view of the facts that e is guaranteed to be bound by (6.29) and dy& is


___________________________________________________________________________________________________________


29

assumed to be bounded, we have Detyd ≤−)(& , where D is positive constant. Thus we can conclude from (6.30) that

3/12 Dx ≤ , since 02 <x& when 3/1

2 Dx > , and 02 >x& when 3/1

2 Dx −< . Therefore, (6.28) does represent a satisfactory tracking control law for the system (6.27), given any trajectory

)(tyd whose derivative )(tyd& is bounded. ⊗ Note: if the second state equation in (6.27a) is replaced by

ux −=2& , the resulting internal dynamics is unstable. The internal dynamics of linear systems ⇒ refer the test book The zero-dynamics Definition: The zeros-dynamics is defined to be the internal dynamics of the systems when the system output is kept at zero by the input. For instance, for the system (6.27)

+=

uux

xx 3

2

2

1&

& (6.27a)

1xy = (6.27b) the out put 01 ≡= xy 01 ≡=→ xy && 3

2xu −≡→ , hence the zero-dynamics is

0322 =+ xx& (6.45)

This zero-dynamics is easily seen to be asymptotically stable by using Lyapunov function 2

2xV = . ⊗ The reason for defining and studying the zero-dynamics is that we want to find a simpler way of determining the stability of the internal dynamics.

- In linear systems, the stability of the zero-dynamics implies the global stability of the internal dynamics.

- In nonlinear systems, if the zero-dynamics is globally exponentially stable only local stability is guaranteed for the internal dynamics.

⊗ To summarize, control design based on input-output linearization can be made in three steps:

- differentiate the output y until the input u appears. - choose u to cancel the nonlinearities and guarantee

tracking convergence. - study the stability of the internal dynamics.

6.2 Mathematical Tools 6.3 Input-State Linearization of SISO Systems 6.4 Input-Output Linearization of SISO System


___________________________________________________________________________________________________________

Chapter 7 Sliding Control 30

7. Sliding Control In this chapter: - The nonlinear system with structured or unstructured

uncertainties (model imprecision) is considered. - A so-called sliding control methodology is introduced. 7.1 Sliding Surfaces Consider the SI dynamic system

ubfx n )()()( xx += (7.1) xy =

where, u : scalar control input

[ ]Tnxxx )1( −= L&x : state vector )(xf : unknown, bounded nonlinear function )(xb : control gain, unknown bounded but known sign

Control objective: To get the state x to track a specific time-

varying state [ ]Tndddd xxx )1( −= L&x in the presence of

model imprecision on )(xf and )(xb . Condition: For the tracking task to be achievable using a finite control u , the initial desired state must be such that

)0()0( xx =d (7.2)

7.1.1 A Notation Simplification - Tracking error in the variable x

dxxx −≡~ - Tracking error vector

[ ]Tnd xxx )1(~~~~ −=−≡ L&xxx

- Time-varying surface )(tS in the state-space )(nR by the scalar equation 0);( =ts x , where

λ

λ+=

−

,~);(1

xdtdts

nx is positive constant (7.3)

For example, xxxsnxxsn ~~2~3,~~2 2λ+λ+=→=λ+=→= &&&& . ⊗ Given initial condition (7.2), the problem of tracking

dxx ≡ is equivalent to that of remaining on the surfaces )(tS for all 0>t ; indeed 0≡s represents a linear differential

equation whose unit solution is 0~ ≡x , given initial condition (7.2). ⇒ The problem of tracking the n-dimensional vector

dx can be reduced to that of keeping the scalar quantity s at zero. Bounds on s can be directly translated into bounds on the tracking error vector x~ , and therefore the scalar s represents a true measure of tracking performance. Assume that

0)(~ =0x , we have

Φ≤≥∀ )(,0 tst ⇒ 1,,0,)2(~,0 )( −=ελ≤≥∀ nitxt ii L

(7.4) where 1/ −λ=ε nΦ .

By definition (7.3), the tracking error x~ is obtained from s through a sequence of first order low-pass filter as shown in Fig. 7.1.a, where )/( dtdp = is the Laplace operator.

sλ+p

1λ+p

1λ+p

1L

x~

4444444444 34444444444 21blocksn 1−

Fig. 7.1.a Computing bounds on x~

Let 1y be the output of the first filter ∫ −λ−=t

Tt dTTsey0

)(1 )( .

From Φ≤s we thus get ∫ −λ−=t

Tt dTTsety0

)(1 )()( Φ

λ≤−λ= λ− /)1)(/( ΦΦ te . Apply the same procedure, we get

ε=λ≤ λ− tx /~ Φ .

Similarly, )(~ ix can be thought of as obtained through the sequence of Fig. 7.1.b.

sLλ+p

1λ+p

1Lλ+p

1λ+p

1 )(~ ixz1

44444 344444 21blocksi

44444 344444 21blocksin 1−−

Fig. 7.1.a Computing bounds on )(~ ix From previous results, one has inz −−λ≤ 1

1 /φ , where 1z is

the output of the thin )1( −− filter. Furthermore, noting that

λ+λ

−=λ+ pp

p 1 ⇒ ελ=

λλ

+

λ

Φ≤

−−i

i

inix )2(1~

1)( is

bounded. In the case that 0~ ≠)0(x , bounds (7.4) are obtained asymptotically, i.e., within a short time-constant /λn )1( − . The simplified, 1st-order problem of keeping the scalar s at zero can now be achieved by choosing the control law u of (7.1) such that outside of )(tS

ssdtd

η−≤221 (7.5)

where η is a strictly positive constant. Essentially, (7.5) states

that the squared “distance” to the surface, as measured by 2s , decrease along system trajectories. Thus it constrains trajectories to point towards the surface )(tS , as illustrated in Fig. 7.2.

)(tS

Fig. 7.2 The sliding condition


___________________________________________________________________________________________________________


Condition (7.5) is called sliding condition. )(tS verifying (7.5) is referred to as sliding surface. The system‘s behavior once on the surface is called sliding regime or sliding mode. The other interesting aspect of the invariant set )(tS is that once on it, the system trajectories are defined by the equation of the set itself, namely ( ) 0~/ 1 =λ+ − xdtd n . Satisfying (7.5) guarantees that if condition (7.2) is not exactly verified, i.e., )0()0( dxx ≠ , the surface )(tS will be reach in a finite time smaller that η= /)0(ts . The typical system behavior implied by satisfying sliding condition (7.5) is illustrated in Fig. 7.3 for 2=n .

sliding modeexponential convergence

xd (t)

λslope -s = 0

finite -timereaching phase

x&

x

Fig. 7.3 Graphical interpretation of Eqs. (7.3) and (7.5),n=2

When the switching control is imperfect, there is chattering as shown in Fig. 7.4

sliding modeexponential convergence

xd (t)

λslope -s = 0

finite -timereaching phase

x&

x

Fig. 7.4 Chattering as a result of imperfect control switchings. 7.1.2 Filippov’s Construction of the Equivalent Dynamics The dynamics while in sliding mode can be written as 0=s& (7.6) By solving (7.6), we obtain an expression for u called the equivalent control, equ , which can be interpreted as the

continuous control law that would maintain 0=s& if the dynamics were exactly known. Fro instance, for a system of the form ufx +=&& , we have

)~( xxfxfuu deq&&&&&& ++−=+−=→

From (7.6)

0~~ =+= xxs λ&& ⇒ xx &&& ~~ λ−= Hence,

xxfu deq&&& ~λ−+−= (7.7)

And the system dynamics while in sliding mode is, of course,

xxufx deq&&&&& ~λ−=+= (7.8)

Geometrically, the equivalent control can be constructed as

−+ −+= uuueq )1( αα i.e., as a convex combination of the values of u on both side of the surface )(tS . The value of α can be obtained formally from (7.6), which corresponds to requiring that the system trajectories be tangent to the surface. This intuitive construction is summarized in Fig. 7.5

s = 0

s > 0

s < 0

f-

f+ feq

Fig. 7.5 Filippov’s construction of the equivalent

dynamics in sliding mode 7.1.3 Perfect Performance – At a Price A Basic Example: Consider the second-order system

uxxtax +−= 3cos)( 2&&& (7.10) where,

u : control input xy = : scalar output of interest

xxtaf 3cos)( 2&−= : unknown bounded nonlinear function

with 21 ≤≤ a . Let f be an estimation value of f , assume that the estimation error is bounded by some known function

),( xxFF &= as follows

Fff ≤−ˆ (7.9)

assume that xxf 3cos5.1ˆ 2&−= ⇒ xxF 3cos5.0 2&= . In

order to have the system track )()( txtx d= , we define a sliding surface 0=s according to (7.3), namely

xxxdtds ~~~ λ+=

λ+= & (7.11)

We then have

xxufxxxxxs dd&&&&&&&&&&&& ~~)(~~ λλλ +−+=+−=+= (7.12)

To achieve 0=s& , we choose control law as xxfu d

&&& ~λ−+−= .

Because f is unknown and replaced by its estimation f , the control is chosen as

xxfuu d&&& ~ˆˆ λ−+−=→ (7.13)

u can be seen as our best estimate of the equivalent control. In order to stratify the sliding condition (7.5), despite the uncertainty on the dynamics f , we add to u a term discontinuous across the surface 0=s

)sgn(ˆ skuu −= (7.14)


___________________________________________________________________________________________________________


where “sgn” is the sign function:

<−=>+=

01sgn01sgn

sifsif

By choosing ),( xxkk &= in (7.14) to be large enough, we can now guarantee that (7.5) is verified. Indeed, from (7.12) and

(7.14), sksffsskffsssdtd

−−=−−== )ˆ()]sgn(ˆ[21 2 & .

So that, letting

η+= Fk (7.15)

we get from (7.9): ssdtd

η−≤221 as desired.

⊗ Note that: - from (7.15), the control discontinuity k across the

surface 0=s increases with the extent of parametric uncertainty.

- f and F need not depend only on x or x& . They may more generally be functions of any measured variables external to system (7.8), and may also depend explicitly on time.

- To the first order system, the sliding mode can be interpreted that “if the error is negative, push hard enough in the positive direction, and conversely”. It does not for higher-order systems.

Integral Control A similar result would be obtained by using integral control,

i.e., formally letting ∫t

drrx0

)(~ be the variable of interest. The

system (7.8) now third-order relative to this variable, and (7.3)

gives ∫∫ λ+λ+=

λ+=

ttdrxxxdrx

dtds

0

2

0

2~~2~~ & . Then, we

obtain, instead of (7.13), xxxfu d~~2ˆˆ 2λ−λ−+−= &&& with

(7.14) and (7.15) formally unchanged. Note that ∫t

drrx0

)(~ can

be replaced by ∫t

drrx )(~ , i.e., the integral can be defined

within a constant. The constant can be chosen to obtain 0)0( ==ts regardless of )0(dx , by letting

∫ −−++=t

xxdrxxxs0

2 )0(~2)0(~~~2~ λλλ &&

Gain Margin Assume now that (7.8) is replaced by

ubfx +=&& (7.16)

where the (possibly time-varying or state-dependent) control gain b is unknown but of known bounds

maxmin0 bbb ≤≤< (7.16) Since the control input enters multiplicatively in the dynamics, it is natural to choose our estimate b of gain b as the

geometric mean of the above bounds maxminˆ bbb = . Bound

(7.17) can then be written in the form

ββ ≤≤−

bb1 (7.18)

where, minmax / bb=β . Since the control law will be designed to be robust to the bounded multiplicative uncertainty (7.18), we shall call β the gain margin of our design. With s and u defined as before, one can then easily show that the control law

)]sgn(ˆ[ˆ 1 skubu −= − (7.19)

with

uFk ˆ)1()( −++≥ βηβ (7.20) satisfies the sliding condition. Indeed, using (7.19) in the expression of s& leads to

)sgn(ˆ)~)(ˆ1()ˆˆ( 111 skbbxxbbfbbfs d−−− −+−−+−= &&&& λ

Condition (7.5) can be rewritten as )sgn(sssss ηη −=−≤& . Hence we have ( )

)sgn()sgn(ˆ)~)(ˆ1()ˆˆ( 111

sssskbbxxbbfbbf dη

λ−≤

−+−−+− −−− &&&

or ( )

)sgn()~)(ˆ1()ˆˆ()sgn(ˆ 111

sssxxbbfbbfsskbb d

ηλ+

+−−+−≥ −−− &&&

⇒ ( ) ηλ 111 ˆ)sgn()~)(1ˆ()ˆˆ( −−− ++−−+−≥ bbsxxbbffbbk d&&&

so that k must verify

ηλ 111 ˆ)~)(1ˆ(ˆˆ −−− ++−−+−≥ bbxxbbffbbk d&&&

Since ),ˆ(ˆ ffff −+= where ,ˆ Fff ≤− this in turn leads to

xxfbbbbFbbk d&&& ~ˆ1ˆˆˆ 111 λη +−−++≥ −−−

and thus to (7.20). Note that the control discontinuity has been increased in order to account for the uncertainty on the control gain b . Example 7.1________________________________________

A simplified model of the motion of an under water vehicle can be written

uxxcxm =+ &&&& (7.21) where

x : position of vehicle u : control input (force provided by a propeller) m : mass of the vehicle c : drag coefficient

In practice, m and c are not known accurately, because they only describe loosely the complex hydrodynamic effects that govern the vehicle’s motion. From (7.3),

xxs ~~ λ+= &

⇒ xxxxxs d&&&&&&&&& ~)(~~ λλ +−=+=

⇒ xmxmuxxcxmxmxmsm dd&&&&&&&&&&& ~~ λλ +−+−=+−=

The estimated controller is chosen as


___________________________________________________________________________________________________________


xxcxxmu d &&&&& ˆ)~(ˆˆ +−= λ and a control law satisfying the sliding condition can be derived as

)sgn(ˆ)~(ˆ)sgn(ˆ skxxcxxmskuu d −+−=−= &&&&& λ (7.22)

where k is calculated from (7.20)

xxcxxmF

uFk

d &&&&& ˆ)~(ˆ)1()(

ˆ)1()(

+−−++≥

−++≥

λβηβ

βηβ

Hence k can be chosen as follows

)~()1(ˆ)( xxmFk d&&& λβηβ −−++= (7.23)

Note that the expression (7.23) is “tighter” than the general form (7.20), reflecting the simpler structure of parametric uncertainty: intuitively, u can compensate for xxc && directly, regardless of uncertainty on m . In general, for a given problem, it is a good idea to quickly rederive a control law satisfying the sliding condition, rather than apply some pre-packed formula. __________________________________________________________________________________________

7.1.4 Direct Implementations of Switching Control Laws The main direct applications of the above switching controller include the control of electric motors, and the use of artificial dither to reduce stiction effects.

- Switching control in place of pulse-width modulation - Switching control with linear observer - Switching control in place of dither 7.2 Continuous Approximations of Switching Control Laws In general, chattering must be eliminated for the controller to perform properly. This can be achieved by smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface

0);(,)( >≤= ΦΦxx tstB (7.25)

where, Φ is boundary layer thickness, and 1n-λΦ/=ε is the boundary layer width. Fig. 7.6.a illustrates boundary layer for the case 2=n .

φ

ε

ε

boundary layer

x&

x

Fig. 7.6.a The boundary layer

Fig. 7.6.b illustrates this concept: - Out side of )(tB , choose the control law u as before (7.5) - Inside of )(tB , interpolate to get u - for instance, replacing in the expression of u the term )sgn(s by Φ/s .

φ−

boundarylayer

u

sφ

u

Fig. 7.6.b Control interpolation in the boundary layer

Given the results of section 7.1.1, this leads to tracking to within a guaranteed precision ε , and more generally guarantees that for all trajectories starting inside )0( =tB

1,,0)2()(~,0 −=≤≥∀ nitxt ii Kελ

Example 7.2________________________________________

Consider again the system (7.10): uxxtax +−= 3cos)( 2&&& , and assume that the desired trajectory is )2/sin( txd π= . The constants are chosen as 1.0,20 == ηλ , sampling time

001.0=dt sec. Switching control law:

)~20~sgn()1.03cos5.0(

~203cos5.1

)sgn(ˆ

2

2

xxxx

xxxx

skuu

d

++−

−+=

−=

&&

&&&&

Smooth control law with a thin boundary layer 1.0=φ :

]/)~20~[()1.03cos5.0(

~203cos5.1

)/(ˆ

2

2

φ

φ

xxsatxx

xxxx

ssatkuu

d

++−

−+=

−=

&&

&&&&

The tracking performance with switching control law is given in Fig. 7.7 and with smooth control law is given in Fig. 7.8.

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5

Con

trol

inpu

t

Time (s)

-4

-3

-2

-1

0

1

2

3

4

6

5

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5

0.5

0.0

-0.5

-1.0

1.5

1.0

-1.5

Trac

king

Err

or

Time (s) Fig. 7.7 Switched control input and tracking performance

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5

Con

trol

Inpu

t

Time (s)

-4

-3

-2

-1

0

1

2

3

4

6

5

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5-5

-4

-3

-2

-1

0

1

2

3

5

4

Time (s)

Trac

king

Err

or (x

10-3 )

Fig. 7.8 Smooth control input and tracking performance

__________________________________________________________________________________________

⊗ Note that:

- The smoothing of control discontinuity inside )(tB essentially assigns a low-pass filter structure to the local


___________________________________________________________________________________________________________


dynamics of the variable s , thus eliminating chattering. Recognizing this filter-like structure then allows us, in essence, to tune up the control law so as to achieve a trade-off between tracking precision and robustness to un-modeled dynamics.

- Boundary layer thickness φ can be made time-varying, and can be monitored so as to well exploit the control “bandwidth” available.

Consider again the system (7.1): ubfx n )()()( xx += , with

1ˆ == bb . In order to maintain attractiveness of the boundary layer now that φ is allowed to vary with time, we must actually modify condition (7.5). Indeed, we now need to guarantee that the distance to the boundary layer always decreases.

φ≥s ⇒ η−≤− )( φsdtd

φ−≤s ⇒ η≥− )( φsdtd

Thus, instead of simply required that (7.5) be satisfy outside the boundary layer, we now required that

φ≥s ⇒ ssdtd )(

21 2 η-φ&≤ (7.26)

The additional term sφ& in (7.26) reflects the fact that the boundary layer attraction condition is more stringent during boundary layer contraction ( 0<φ& ) and less stringent during boundary layer expansion ( 0>φ& ).In order to satisfy (7.26), the quantity φ&− is added to control discontinuity gain )(xk , i.e., in our smooth implementation the term

)sgn()( sk x obtained from switched control law u is actually

replaced by )/sat()( φx sk , where

φxx &−= )()( kk (7.27)

and sat is the saturation function

=

≤=

otherwiseyy

yifyy

)sgn()sat(

1)sat(

and can be seen graphically as in the following figure

1−

)sat(y

y1

Accordingly, control law becomes )/sat()(ˆ φx skuu −= . Now, we consider the system trajectories inside the boundary layer. They can be expressed directly in terms of the variable s as

)()( xφ

x fsks ∆−−=& (7.28)

where fff −=∆ ˆ . Since k and f∆ are continuous in x , using (7.4) to rewrite (7.28) in the form

( ))()()( εOfsks +∆−+−= xφ

x& (7.29)

We can see from (7.29) that the variable s (which is a measure of the algebraic distance to the surface )(tS ) can be view as the output of the first order filter, whose dynamics only depend on the desired state dx , and whose input are, to the first order, “perturbations”, i.e., uncertainty )( df x∆ . Thus chattering can be eliminated, as long as high-frequency un-modeled dynamics are not excited. Conceptually, the perturbations are filtered according to (7.29) to give s , which in turn provides tracking error x~ by further low-pass filtering, according to definition (7.3)

s x~1storder filter(7.29) 1)(

1−+ np λ

)()( εOf d +∆− x

φofchoice sofdefinition

Fig. 7.9 Structure of the closed-loop error dynamics Control action is a function of x and dx . Since λ is break-frequency of filter (7.3), it must be chosen to be “small” with respect to high-frequency un-modeled dynamics (such as un-modeled structural modes or neglected time-delays). Furthermore, we can now turn the boundary layer thickness φ so that (7.29) also presents a first-order filter of bandwidth λ . It suffices to let

λ=φx )( dk

(7.30)

which can be written from (7.27) as

)( dk xφφ =+ λ& (7.31) (7.27) can be rewritten as

φxxx λ+−= )()()( dkkk (7.32) ⊗ Note that:

- The s-trajectory is a compact descriptor of the closed-loop behavior: control activity directly depends on s , while tracking error x~ is merely a filtered version of s

- The s-trajectory represents a time-varying measure of the validity of the assumptions on model uncertainty.

- The boundary layer thickness φ describes the evolution of dynamics model uncertainty with time. It is thus particularly informative to plot )(ts , )(tφ , and )(tφ− on a single diagram as illustrated in Fig. 7.11b.

Example 7.3________________________________________

Consider again the system described by (7.10): uxxtax +−= 3cos)( 2&&& . Assume that λη /)0( =φ with 1.0=η ,

20=λ . From (7.31) and (7.32)

( ) ( )φ

φ

&&

&&

−+=

++−+=

η

ληη

xx

xxxxxk dd

3cos5.0

3cos5.03cos5.0)(2

22

where, )3cos5.0( 2 ηλ ++−= xxd&& φφ . The control law is now


___________________________________________________________________________________________________________


]/)~20~[()3cos5.0(

~3cos5.1

)/()(ˆ

2

2

φφ

φ

xxsatxx

xxxx

ssatxkuu

d

+−+−

−+=

−=

&&&

&&&&

η

λ

⊗ Note that: - The arbitrary constant η (which formally, reflects the

time to reach the boundary layer starting from the outside) is chosen to be small as compared to the average value of )( dk x , so as to fully exploit our knowledge of the structure of parametric uncertainty.

- The value of λ is selected based on the frequency range of un-modeled dynamics.

The control input, tracking error, and s -trajectories are plotted in Fig. 7.11.

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5

Con

trol

Inpu

t

Time (s)

-4

-3

-2

-1

0

1

2

3

4

6

5

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5-5

-4

-3

-2

-1

0

1

2

3

5

4

Time (s)

Trac

king

Err

or (x

10-3 )

Fig. 7.11a Control input and resulting tracking performance

-8

-6

-4

-2

0

2

4

6

8

S-tr

ajec

tori

es (x

10-2 )

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5

Time (s)

φ

φ−

s

Fig. 7.11b s-trajectories with time-varying boundary layer

We see that while the maximum value of the time-varying boundary layer thickness φ is the same as that originally chosen (purposefully) as the constant value of φ in Example 7.2, the tracking error is consistently better (up to 4 times better) than that in Example 7.2, because varying the thickness of the boundary layer allow us to make better use of the available bandwidth. __________________________________________________________________________________________

In the case that 1≠β , one can easily show that (7.31) and (7.32) become (with )( dd xββ = )

ddk

βλ φ

x ≥)( ⇒ )( dd k xφφ βλ =+& (7.33)

ddk

βλ φ

x ≤)( ⇒ d

d

d

kββ

λ )(2

xφφ =+& (7.34)

ddkkk βλ φ/xxx +−= )()()( (7.35)

with initial condition )0(φ defined as:

λβ /))0(()0( dd k xφ = (7.36)

Example 7.4________________________________________

A simplified model of the motion of an under water vehicle can be written (7.21): uxxcxm =+ &&&& . The a priori bounds

on m and c are: 51 ≤≤ m and 5.15.0 ≤≤ c . Their estimate

values are 5ˆ =m and 1ˆ =c . 20=λ , 1.0=η . The smooth control input using time-varying boundary layer, as describe above is designed as follows:

xxs ~~ λ+= &

xxxs d&&&&&& ~λ+−=

)~( xxmuxxcsm d&&&&&& λ−−+−=

)~(ˆˆˆ xxmxxcuu d&&&&& λ−+=→

)sgn()~(ˆˆ

)sgn(ˆ

skxxmxxc

skuu

d −−+=

−=&&&&& λ

)sgn()~()ˆ()ˆ(

)~()sgn()~(ˆˆ

skxxmmxxcc

xxmskxxmxxcxxcsm

d

dd

−−−+−=

−−−−++−=&&&&&

&&&&&&&&&&&

λ

λλ

Condition (7.5): sss η−≤&

smssm η−≤&

( ) smsskxxmmxxcc d ηλ −≤−−−+− )sgn()~()ˆ()ˆ( &&&&&

( ) smsxxmmxxcxxccssk d ηλ +−−++−≥ )~()ˆ(ˆ)ˆ()sgn( &&&&&&&

( ) ηλ msxxmmxxcck d +−−+−≥ )sgn()~()ˆ()ˆ( &&&&&

ηλ msxxmmxxcck d +−−+−≥ )sgn()~()ˆ()ˆ( &&&&&

And the controller is

−=

+=

−=

−=

+−−+−=

−+=

)/sat(ˆ

~~)(

)(

)max(~ˆmaxˆmax)(

)~(ˆˆˆ

2

φ

φ

φφ

skuu

xxs

xkk

xk

mxxmmxccxk

xxmxxcu

d

d

d

λ

λ

ηλ

λ

&

&

&

&&&&

&&&&&

The results are given in Fig. 7.12

0 1.0 2.0 40.5 1.5 2.5-5

-4

-3

-2

-1

0

1

2

3

5

4

Des

ired

Tra

ject

orie

s

Time (s)

acceleration (m/s2 )velocity (m/s)distance (m)

-10

-5

0

5

10

15

20

25

30

35

0 1.0 2.0 40.5 1.5 2.5 3.0 3.5

Time (s)

Con

trol

Inpu

t

a. References b. Control input

0 2 4 61 3 5-25

-20

-15

-10

-5

0

5

10

15

Time (s)

Trac

king

Err

or (x

10-2 )

0 2 4 61 3 5-1.5

-1.0

-0.5

0

0.5

1.0

1.5

S-tr

ajec

tori

es (x

10-2 )

Time (s)

φ

s

φ−

c. Tracking error b. s- trajectories

Fig.12 __________________________________________________________________________________________

⊗ Remark:

- The desired trajectory dx must itself be chosen smooth enough not to excite the high frequency un-modeled dynamics.

- An argument similar to that of the above discussion shows that the choice of dynamics (7.3) used to define sliding surfaces is the “best-conditioned” among linear dynamics, in the sense that it guarantees the best tracking performance given the desired control bandwidth and the extent of parameter uncertainty.

- If the model or its bounds are so imprecise that F can only be chosen as a large constant, then φ from (7.31) is


___________________________________________________________________________________________________________


constant and large, so that the term )/sat( φsk simply equals βλ /s in the boundary layer.

- A well-designed controller should be capable of gracefully handling exceptional disturbances, i.e., disturbances of intensity higher than the predicted bounds which are used in the derivation of the control law.

- In the case that λ is time-varying, the term xu ~λ&−=′ should be added to the corresponding u , while

the augmenting gain )(xk according by the quantity )1( −′ βu . It will be discussed in next section.

7.3 The Modeling/Performance Trade-Offs The balance conditions (7.33)-(7.36) have practical implications in term of design/modeling/performance trade-offs. Neglecting time-constants of order λ/1 , condition (7.33) and (7.34) can be written

ddn kβελ ≈ (7.39)

Consider the control law (7.19): )]sgn(ˆ[ˆ 1 skubu −= − , we see that the effects of parameter uncertainty on f have been “dumped” in gain k . Conversely, better knowledge of f reduces k by a comparable quantity. Thus (7.39) is

particularly useful in an incremental mode, i.e., to evaluate the effects of model simplification on tracking performance:

)/( ndd k λβε ∆≈∆ (7.40)

In particular, margin gains in performance are critically dependent on control bandwidth λ : if large λ ’s are available, poor dynamic models may lead to respectable tracking performance, and conversely large modeling efforts produce only minor absolute improvements in tracking accuracy. And it is not overly surprising that system performance be very sensitive to control bandwidth. Thus, give system model (7.1), how large λ can be chosen ? In mechanical system, for instance, given clean measurements, λ typically limited by three factors:

i. structural resonant modes: λ must be smaller than the frequency Rν of the lowest un-modeled structural resonant mode; a reasonable interpretation of this constrain is, classically

RR νπλλ3

2≈≤ (7.41)

although in practice this bound may be modulated by engineering judgment, taking notably into account the natural damping of the structural modes. Furthermore, in certain case, it may account for the fact that Rλ may actually vary with the task. ii. neglected time delays: along the same lines, we have a condition for the form

AA T3

1≈≤ λλ (7.42)

where AT is the largest un-modeled time-delay (for instance in the actuators). iii. sampling rate: with a full-period processing delay, one gets a condition of the form

samplingS νλλ51

≈≤ (7.43)

where, samplingν is the sampling rate. The desired control bandwidth λ is the minimum of three bounds (7.41-43). Ideally, the most effective design corresponds to matching these limitations, i.e., having

λλλλ ≈≈≈ SAR (7.44) 7.4 Multi-Input System Consider a nonlinear multi-input system of the form

∑=

+=m

jjiji

ni ubfx i

1

)( )()( xx , mi ,,1 L= , mj ,,1 L=

where T

muuu ][ 21 L=u : the control input vector

[ ]Tnn xxx &L)2()1( −−=x : the state vector 7.5 Summary


___________________________________________________________________________________________________________

Chapter 8 Adaptive Control

37

8. Adaptive Control In this chapter: - The nonlinear system with structured or unstructured

uncertainties (model imprecision) is considered. - A so-called sliding control methodology is introduced. 8.1 Basic Concepts in Adaptive Control - Why we need adaptive control ? - What are the basic structures of adaptive control systems ? - How to go about designing adaptive control system ? 8.1.1 Why Adaptive Control ? 8.1.2 What is Adaptive Control ? An adaptive controller differs from an ordinary controller in that the controller parameters are variable, and there is a mechanism for adjusting these parameters on-line based on signals in the system. There are two main approaches for constructing adaptive controllers: so-called model-reference adaptive control method and so-called self-tuning method. Model-Reference Adaptive Control (MRAC)

reference model

plant

adaptation law

controller

my

y er u

a

Fig. 8.3 A model-reference adaptive control system

A MRAC can be schematically represented by Fig. 8.3. It is composed of four parts: a plant containing unknown parameters, a reference model for compactly specifying the desired output of the control system, a feedback control law containing adjustable parameters, and an adaptation mechanism for updating the adjustable parameters. The plant is assumed to have a known structure, although the parameters are unknown. - For linear plants, the numbers of poles and zeros are

assumed to be known, but their locations are not. - For nonlinear plants, this implies that the structure of the

dynamic equations is known, but that some parameters are not.

A reference model is used to specify the ideal response of the adaptive control system to external command. The choice of the reference model has to satisfy two requirements: - It should reflect the performance specification in the control

tasks such as rise time, settling time, overshoot or frequency domain characteristics.

- This ideal behavior should be achievable for the adaptive control system, i.e., there are some inherence constrains on the structure of reference model given the assumed structure of the plant model.

The controller is usually parameterized by a number of adjustable parameters. The controller should have perfect

tracking capacity in order to allow the possibility f tracking convergence. Existing adaptive control designs normally required linear parametrization of the controller in order to obtain adaptation mechanisms with guaranteed stability and tracking convergence. The adaptation mechanism is used to adjust the parameters in the control law. In MRAC systems, the adaptation law searches for parameters such that the response of the plant under adaptive control becomes the same as that of the reference model. The main difference from conventional control lies in the existence of this mechanism. Example 8.1 MRAC control of unknown mass____________

Consider the control of a mass on a frictionless surface by a motor force u , with the plant dynamics being

uxm =&& (8.1) Choose the following model reference

)(221 trxxx mmm λλλ =++ &&& (8.2)

where, 11,λλ : positive constants chosen to reflect the performance

specifications mx : the reference model output (ideal out put of the

controlled system) )(tr : reference position

* m is known exactly, we can choose the following control

law to achieve perfect tracking 0~~2~ 2 =++ xxx λλ &&& , with

mxxx −=~ representing the tracking error and λ is a strictly positive number. This control law leads to the exponentially convergent tracking error dynamics: )~~2(ˆ 2 xxxmu m λλ −−= &&& . * m is not known exactly, we may use the control law

)~~2(ˆ 2 xxxmu m λλ −−= &&& (8.3)

which contains the adjustable parameter m . Substitution this control law into the plant dynamics, yields

mmmxxxmm

xxxmxm

m

m

−≡−−+=

−−=

ˆ~with),~~2()~(

)~~2(ˆ2

2

λλ

λλ&&&

&&&&&

⇒ )~~2(~~~2~ 22 xxxmxmxmxm m λλλλ −−=++ &&&&&&

⇒ )~~2(~)~~()~~( 2 xxxmxxmxxm m λλλλλ −−=+++ &&&&&&&

Let the combined tracking error measure be

xxs ~~ λ+= & (8.5)

and the signal quantity v is defined as xxxv m~~2 2λλ −−= &&& .

The closed-loop zero dynamics

vmsmsm ~=+ λ& (8.4)

Consider Lyapunov function


___________________________________________________________________________________________________________


38

0~121

21 22 ≥+= msmV

γ (8.7)

Its derivative yields

( )vsmmsm

mmvmsms

mmsmsV

γγλ

γλ

γ

+−−=

−−−=

+=

−

−

−

&

&

&&&

ˆ~ˆ~)~(

~~

12

1

1

If the update law is chosen to satisfy

vsm γ−=& (8.6)

The derivative of Lyapunov function becomes

02 ≤−= smV λ& (8.8)

Using Barbalat’s lemma, it is easily to show that s converges to zero. The convergence of s to zero implies that of the

position tracking error x~ and the velocity tracking error x&~ . For illustration, the results of simulation for this example are given in Fig. 8.4 and 8.5. The numerical values are chosen as 2=m ,

,0)0(ˆ =m ,5.0=γ ,101 =λ ,252 =λ ,6=λ 0)0()0( == mxx && . In Fig. 8.4, 0)0()0( == mxx and 0)( =tr . In Fig. 8.5,

5.0)0()0( == mxx and )4sin()( ttr = .

-0.1

0.0

0.1

0.2

0.3

0.6

0.4

0.5

Trac

king

Per

form

ance

Time (s)0.0 1.0 2.0 3.00.5 1.5 2.5

0.0

0.5

1.0

1.5

2.0

2.5

Para

met

er E

stim

atio

n

Time (s)0.0 1.0 2.0 3.00.5 1.5 2.5

Fig. 8.4 Tracking performance and parameter estimation for

an unknown mass with reference path 0)( =tr

Trac

king

Per

form

ance

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Time (s)0.0 1.0 2.0 3.00.5 1.5 2.5

0.0

0.5

1.0

1.5

2.0

2.5

Para

met

er E

stim

atio

n

Time (s)0.0 1.0 2.0 3.00.5 1.5 2.5

Fig. 8.5 Tracking performance and parameter estimation for

an unknown mass with reference path )4sin()( ttr = __________________________________________________________________________________________

Self-Tuning Controller (STC)

plant

estimator

controlleryr u

a

Fig. 8.5 A self-tuning controller

A self-tuning controller is a controller which performs simultaneous identification of the unknown plant. Example 8.2 Self-tuning control of unknown mass________

Consider the control of a mass of Example 8.1. Let us still use the pole-placement (placing the poles of the tracking error dynamics) control laws (8.3) for generating the control input, but let us now generate the estimated mass parameter using a estimation law.

For simplicity, assume that the acceleration can be measured by an accelerometer. From (8.1), the simplest way of estimating m is

)()()(ˆ

txtutm

&&= (8.9)

However this is not good method because there may be considerable noise in the measurement x&& , and, furthermore, the acceleration may be close to zero. A better approach is to estimate the parameter using a least-squares approach, i.e., choosing the estimate in such a way that the total prediction error

∫t

drre0

2 )( (8.10)

is minimal, with the prediction error e defined as

)()()(ˆ)( tutxtmte −≡ && . The prediction error is simply the error

in fitting the known input u using the estimated parameter m . This total error minimization can potentially average out the effects of measurement noise. The resulting estimating is

∫∫

= t

t

drw

druwm

0

2

0ˆ (8.11)

with xw &&= . If actually, the unknown parameter m is slowly time-varying, the above estimate has to be recalculated at every new time instant. To increase computational efficiency, it is desirable to adopt a recursive formulation instead of repeatedly using (8.11). To do this, we define

∫≡ t

drwtP

0

2

1)( (8.12)

The function )(tP is called the estimation gain, its update can be directly obtained by using

( ) 21 wPdtd

=− (8.13)

Then differentiation of Eq. (8.11)(which can be written

∫=−t

druwmP0

1 ˆ ) leads to

ewtPm )(ˆ −=& (8.14)

In implementation, the parameter estimate m is obtained by numerically integrating Eqs. (8.13) and (8.14). __________________________________________________________________________________________

Relations between MRAC and ST methods 8.1.3 How to Design Adaptive Controllers ? The design of an adaptive controller usually involves the following three steps: - choose a control law containing variable parameters - Choose an adaptation law for adjusting those parameters - analyze the convergence properties of the resulting control system.


___________________________________________________________________________________________________________


39

Lemma 8.1: Consider two signals e and φ related by the following dynamic equation

)]()([)()( ttkpHte T vφ= (8.15) where )(te is a scalar output signal, )( pH is strictly positive real transfer function, k is an unknown constant with know sign, )(tφ is 1×m vector function of time, and )(tv is a measurable 1×m vector. If the vector )(tφ varies according to

)()sgn()( tekt vφ γ−=& (8.16) with γ being a positive constant, then )(te and )(tφ are globally bounded. Furthermore, if )(tv is bounded, then

0)( →te as ∞→t . 8.2 Adaptive Control of First-Order Systems Let us discuss the adaptive control of first-order plants using MRAC method. Consider the first-order differential equation

ubyay pp +−=& (8.20) where, y is the plant output, pa and pb are constant unknown plant parameters. Choice of reference model Let the desired performance of the adaptive control system be specified by a first-order reference model

)(trbyay mmmm +−=& (8.21) where 0,0 >≥ mm ba are constant parameters, and )(tr is bounded external reference signal. Choice of control law As first step in adaptive controller design, let us choose the control law to be

)()(ˆ)()(ˆ tytatrtau yr += (8.22) where yr aa ˆ,ˆ are variable feedback gains. The reason for the choice of control law (8.21) is clear: it allows the possibility of perfect model matching. With this control law, the closed-loop dynamics is

rbaybaay prpyp ˆ)ˆ( +−−=& (8.23) If the plant parameters were known, such as yyrr aaaa == ˆ,ˆ , comparing (8.21) and (8.23), we get

p

mr b

ba = p

mpy b

aaa

−= (8.24)

which lead to the closed-loop dynamics rbyay mm +−=& which is identical to the reference model dynamics, and yields zero tracking error. Choice of adaptation law

Now we choose the adaptation laws for ra and ya . Let the

tracking error be myye −= and the error of parameter estimation be

rrr aaa −= ˆ~ yyy aaa −= ˆ~ (8.25) The dynamics of tracking error can be found by subtracting (8.23) and (8.21)

rbabyabaayyae mrpyppmmm )ˆ()ˆ()( −++−+−−=&

)~~( yarabea yrpm −+−= (8.26)

The Lemma 8.1 suggests the following adaptation laws

reba pr γ)sgn(ˆ −=& (8.27)

yeba py γ)sgn(ˆ −=& (8.28) with γ being a positive constant representing the adaptation gain. The )sgn( pb in (8.27-28) determines the direction of the search for the proper controller parameters. Tracking convergence analysis We analyze the system’s stability and convergence behavior using Lyapunov theory. Choose the Lyapunov function candidate

0)~~(21

21 222 ≥++= yrp aabeV

γ (8.29)

Its derivative yields

( )( )yebaab

rebaabea

aaaabyarabbeea

aaaabyarabeaeV

pyyp

prrpm

yyrrpyrppm

yyrrpyrpm

γγ

γγ

γ

γ

)sgn(ˆ~1

)sgn(ˆ~1

)ˆ~ˆ~(1)~~()sgn(

)~~~~(1)]~~([

2

2

++

++−=

++−+−=

++−+−=

&

&

&&

&&&

With adaptation laws (8.27) and (8.28), the derivative of Lyapunov function becomes 02 ≤−= eaV m

& . Thus, the adaptive control system is globally stable, i.e., the signals e ,

ra~ and ya~ are bounded. Furthermore, the global asymptotic

convergence of the tracking error )(te is guaranteed by Barbalat’s lemma, because the boundedness of e , ra~ and ya~

imply the boundedness of e& and therefore the uniform continuity of V& . Example 8.3 A first-order plant________________________

Consider the control of the unstable plant uyy 3+=& using the previous designed adaptive controller. The numerical parameters are: 1−=pa , 3=pb , 4=ma , 4=mb , 2=γ , and

0)0()0( == myy . Two reference signals are used: * 4)( =tr ⇒ simulation results in Fig. 8.9 * )3sin(4)( ttr = ⇒ simulation results in Fig. 8.10


___________________________________________________________________________________________________________


40

Trac

king

Per

form

ance

0

1

2

3

4

5

6

0.0 1.5 3.5 5.01.0 2.5 4.53.00.5 2.0 4.0

Time (s)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Para

met

er E

stim

atio

n

Time (s)

ra

ya

0.0 1.5 3.5 5.01.0 2.5 4.53.00.5 2.0 4.0

Fig. 8.9 Tracking performance and parameter estimation with

reference path 4)( =tr

-4

-3

-2

-1

0

1

2

4

5

3

-5

Time (s)

Trac

king

Per

form

ance

0 3 7 102 5 961 4 8 0 3 7 102 5 961 4 8-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Para

met

er E

stim

atio

n

Time (s)

ra

ya

Fig. 8.10 Tracking performance and parameter estimation

with reference path )3sin(4)( ttr = __________________________________________________________________________________________

Parameter convergence analysis⇒ refer text book Extension to nonlinear plant The same method of adaptive control design can be used for the non-linear first-order plant describe by the differential equation

ubyfcyay ppp +−−= )(& (8.32) where f is any known nonlinear function. The nonlinear in these dynamics is characterized by its linear parametrization in terms of the unknown constant c . Instead of using (8.22), now we use the control law

rayfayau rfy ˆ)(ˆˆ ++= (8.33) where the second term in (8.33) is introduced with the intention of adaptively canceling the nonlinear term. Using the same procedure for the linear plant,

rabyfabcyaba

rayfayabyfcyay

rpfppypp

rfyppp

ˆ)()ˆ()ˆ(

]ˆ)(ˆˆ[)(

+−−−−=

+++−−=&

Comparing to (8.21) and define ppf bca /≡ and

fff aaa −≡ ˆ~ . The adaptation laws are

yeba py γ)sgn(ˆ −=& (8.34a)

feba pf γ)sgn(ˆ −=& (8.34b)

reba pr γ)sgn(ˆ −=& (8.34c) Example 8.4 A first-order non-linear plant_______________

Consider the control of the unstable plant uyyy 32 ++=& using the previous designed nonlinear adaptive controller. The numerical parameters are: 1−=pa , 3=pb , 4=ma , 4=mb ,

2=γ , and 0)0()0( == myy .

Two reference signals are used: * 4)( =tr ⇒ simulation results in Fig. 8.11 * )3sin(4)( ttr = ⇒ simulation results in Fig. 8.12

0 1.0 2.0 30.5 1.5 2.5

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.5

5.0

4.0

0.0

Time (s)

Trac

king

Per

form

ance

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Time (s)

Para

met

er E

stim

atio

n

ra

fa

ya

0.0 1.0 2.0 3.00.5 1.5 2.5


with reference path 4)( =tr

-4

-3

-2

-1

0

1

2

4

5

3

-5

Time (s)

Trac

king

Per

form

ance

0 3 7 102 5 961 4 8 0 3 7 102 5 961 4 8-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Para

met

er E

stim

atio

n

Time (s)

ra

ya

fa


with reference path )3sin(4)( ttr = __________________________________________________________________________________________

8.3 Adaptive Control of Linear Systems with Full States Feedback Consider the nth-order linear system in the canonical form

uyayaya nn

nn =+++ −

− 0)1(

1)( K (8.36)

where the state components )1(,,, −nyyy K& are measurable,

coefficient vector [ ]Tn aaaa 01L= is unknown, but their signs are known. The objective of the control system is to make y closely track the response of a stable reference model

)(0)1(

1)( tryyy m

nmn

nmn =+++ −

− ααα K (8.37) with )(tr being a bounded reference signal. Choice of control law Define a signal )(tz as follows

eeytz nn

nm 0

)1(1

)()( ββ −−−= −− K (8.38)

with nββ ,,1 K being positive constants chosen such that

01

1 ββ +++ −− Kn

nn pp is a stable (Hurwitz) polynomial.

Adding both side of (8.36) and rearranging, we can rewrite the plant dynamics as

yayazauzya nnn

nn 0

)1(1

)( ][ −−−−=− −− K

Let us choose the control law to be

)(ˆ)(ˆˆˆ 0)1(

1 ttyayazau Tnnn av=+++= −− K (3.39)

with [ ]Tn yyytzt &L1)()( −=v

[ ]Tnn aaaat 011 ˆˆˆˆ)(ˆ L−=a denoting the estimated parameter vector. This represents a pole-placement controller which places the poles at positions specified by the coefficients iβ . The tracking error

myye −= then satisfies the closed-loop dynamics

)(~)([ 0)1(

1)( tteeea Tn

nn

n av=+++ −− ββ K (3.40)

where aaa −= ˆ~


___________________________________________________________________________________________________________


41

Choice of adaptation law Rewrite the closed-loop system (3.40) in state space form

]~)/1[( avbxAx Tna+=& (3.41a)

xc=e (3.41b) where

−−−−

=

−1210

1000

01000010

n

A

ββββ L

L

MOMMM

L

L

,

=

10

00

Mb ,

=

00

01

MTc

Consider Lyapunov function candidate

aΓaxPxax ~~)~,( 1−+= TTV where both Γ and P are symmetric positive constant matrix, and P satisfies

QPAPA −=+ T 0>= TQQ for a chosen Q . The derivative V& can be computed easily as

aΓa2xPbvaxQx && ~~~2 1−++−= TTTV Therefore, the adaptation law

xPbvΓa T−=& (8.42) leads to 0xQx ≤−= TV& . 8.4 Adaptive Control of Linear Systems with Output Feedback Consider the linear time-invariant system presented bu the transfer function

nnn

mmm

pp

pp

ppapaappbpbbk

pRpZ

kpW++++

++++==

−−

−−

1110

1110

)()(

)(K

K(8.43)

where pk is called the high-frequency gain. The reason for this term is that the plant frequency response at high frequency

verifies mnpk

jW−

=ω

ω)( , i.e., the high frequency response is

essentially determined by pk . The relative degree r of this system is mnr −= . In our adaptive control problem, the coefficients ji ba , )1,,1,0;1,,1,0( −=−= mjni KK and the

high frequency gain pk are all assumed to be unknown. The desired performance is assumed to be described by a reference model with transfer function

m

mmm R

ZkpW =)( (8.44)

where mZ and mR are monic Hurwitz polynomials of degrees

mn and mm , and mk is positive. It is well known from linear

theory that the relative degree of the reference model has to be larger or equal to that the plant in order to allow the possibility of perfect tracking. Therefore, in our treatment, we will assume that mnmn mm −≥− . The objective of the design is to determine a control law, and an associated adaptation law, so that the plant output y asymptotically approaches my . We assume as follows

- the plant order n is known - the relative degree mn − is known - the sign of pk is known - the plant is minimum phase

8.4.1 Linear systems with relative degree one Choice of the control law To determine the appropriate control law for the adaptive controller, we must first know what control law can achieve perfect tracking when the plant parameters are perfect known. Many controller structures can be used for this purpose. The following one is particularly convenient for later adaptation design. Example 8.5 A controller for perfect tracking_____________

Consider the plant described by

uapap

bpky

pp

pp

212

)(

++

+= (8.45)

and the reference model

rapap

bpkymm

mmm

212

)(++

+= (8.46)

ke

mbpp++ 21 ββ

mbp +1α

)( pWm

)( pWp

)(tym

0u 1u)(tr

u

Fig. 8.13 Model-reference control system for relative degree 1 Let the controller be chosen as shown in Fig. 8.13, with the control law being

rkybp

pzum

+++

+= 211

ββα (8.47)

where )/( mbpuz += , i.e., z is the output of a first-order filter with input u , and k,,, 211 ββα are controller parameters. If we take these parameters to be mp bb −=1α ,

p

pm

kaa

111

−=β ,

p

pm

kaa

222

−=β ,

p

mkkk = , the transfer

function from the reference input r to the plant output is

)()(

212 pW

apapbpkW m

mm

mmry =

++

+=


___________________________________________________________________________________________________________


42

Therefore, perfect tracking is achieved with this control law, i.e., 0),()( ≥∀= ttyty m . Why the closed-loop transfer function can become exactly the same as that of the reference model ? To know this, note that the control input in (8.47) is composed of three parts: - The first part in effect replaces the plant zero by the reference model zero, since the transfer function from 1u to y is

2121

1 22,)()(

pp

mp

pp

pp

p

myu

apap

bpk

apap

bpkbpbp

W++

+=

++

+

++

=

- The second part places the closed-loop poles at locations of those of reference model. This is seen by noting that the transfer function from 0u to y is

)()(

)(1 21

2,

,,

211

10

pppp

mp

yuf

yuyu

kapkap

bpkWW

WW

ββ ++++

+=

+=

- The third part of the control law rkk pm )/( obviously

replaces pk , the high frequency gain of the plant, by mk . As a result of the above three parts, the closed-loop system has the desired transfer function. __________________________________________________________________________________________

The above controller in Fig. 8.13 can be extended to any plant with relative degree one. The resulting structure of the control system is shown in the Fig. 8.14, where *

2*1

* ,, θθk and *0θ

represents controller parameters which lead to perfect tracking when the plant parameters are known.

e

)( pWm

)( pWp

)(tym

0u 1u)(tr

u

*1θ hΛ,

hΛ,*2θ

*k1ω

2ω

*0θ

Fig. 8.14 A control system with perfect tracking The structure of this control system can be described as follows: - The block for generating the filter signal 1ω represent an

thn )1( − order dynamics, which can be described by huωΛω += 11& , where 1ω is an 1)1( ×−n vector, Λ is an )1()1( −×− nn matrix, and h is constant vector such that

),( hΛ is controllable. The poles of the matrix Λ are chosen to be the same as the roots of polynomial )( pZm , i.e.,

)(]det[ pZp m=−ΛI (8.48) - The block for generating the 1)1( ×−n vector 2ω has the same dynamics but with y as input, i.e., hyωΛω += 22&

- The scalar gain *k is defined to be pm kkk /* = and is intended to modulate the high frequency gain of the control system.

- The vector *1θ contains )1( −n parameters which intend to

cancel the zeros of plant. - The vector *

2θ contains )1( −n parameters which, together

with the scalar gain *0θ can move the poles of the closed-loop

control system to the locations of the reference model poles. As before, the control input in this system is a linear combination of: - the reference signal )(tr - the vector signal 1ω obtained by filtering the control input u - the vector signal 2ω obtained by filtering the plant output y and the output itself. The control input can be rewritten in terms of the adjustable parameters and the various signals, as

yθωθωθ *02

*21

*1

** )( +++= rktu (8.49) Corresponding to this control law and any reference input )(tr , the output of the plant is

)()()()()( * trWtu

pApBty m== (8.50)

since these parameters result in perfect tracking. At this point, we can see the reason for assuming the plant to be minimum-phase: this allows the plant zeros to be cancelled by the controller poles. In adaptive control problem, the plant parameters are unknown, and the ideal control parameters described above are also unknown. Instead (8.49), the control law is chosen to be

yθωθωθ 02211)( +++= rktu (8.49) where, 2,, θθ1k and 0θ are controller parameters to be provided by the adaptation law. Choice of adaptation law For the sake of simplicity, define as follows

[ ]Tttttkt )()()()()( 321 θθθθ =

[ ]Tttttrt )()()()()( 321 ωωωω = Then the control law (8.51) becomes

)()()( tttu T ωθ= (8.52) Let the ideal value of θ be *θ and the error *)()( θθφ −= tt ,

then )()( * tt φθθ += . Therefore, the control law (8.52) can

also be written as ωφωθ )()( ttu TT* += . With the control law (8.52), the control system with variable gains can be equivalently represented as shown in Fig. 8.15, with */)( ktT ωφ regarded as an external signal. The output here must be

]/)[()()( *kpWrpWty Tmm ωφ+= (8.53)


___________________________________________________________________________________________________________


43

)( pWp

0u 1u)(tr u

*1θ hΛ,

hΛ,*2θ

*k1ω

2ω

*0θ

y

*

T

kωφ

Fig. 8.15 An equivalent control system for time-varying gains

Since rpWty mm )()( = , the tracking error is seen to be related to the parameter error by the simple equation

]/)()([)()( *kttpWte Tm ωφ= (8.54)

Since this is the familiar equation seen in Lemma 8.1, the following adaptation law is chosen

)()()sgn( ttek p ωθ γ−=& (8.55) where γ is positive number representing the adaptation gain

and we have used the fact that the sign of *k is the same as that of pk , due to the assumed positiveness of mk . Based on Lemma 8.1 and through a straightforward procedure for establishing signal boundedness, we can show that the tracking error in the above adaptive control system converges to zero asymptotically. 8.4.2. Linear system with higher relative degree The design of adaptive controller for plants with relative degree larger than 1 is both similar to, and different from, that for plants with relative degree 1. Specifically, the choice of control law is quite similar but the choice of adaptation law is very different. Choice of control law Let us start from a simple example. Example 8.6 _______________________________________

Consider the second-order plant described by the transfer function

uapap

ky

pp

p

212 ++

=

and the reference model

rapap

ky

mm

mm

212 ++

=

ke

)( pWm

)( pWp

)(tym

0u 1u)(tr

u

0p +1αλ

λ 0pp++ 21 ββ

Fig. 8.16 Model-reference control system for relative degree 2

Let the controller be chosen as shown in Fig. 8.16. Noting that

mb in the filter in Fig. 8.13 has been replaced by a positive number .λ The closed-loop transfer function from the reference signal r to the plant output y is

)())((

)(

1

212

10

0

20

21

10

0

210

0

21

21

21

ββαλ

λ

λββ

αλλ

αλλ

++++++

+=

++++

+++

+

+++++

=

pkapapp

pkk

apap

k

pp

pp

apap

k

pp

kW

ppp

p

pp

p

pp

p

ry

Therefore, if the controller parameters 211 ,, ββα , and k are chosen such that

))((

)())((

21

21

20

212

10

mm

ppp

apapp

pkapapp

+++

=++++++

λ

ββαλ

and pm kkk /= , then the closed-loop transfer function

ryW becomes identically the same as that of the reference model. Clearly, such choice of parameters exists and is unique. __________________________________________________________________________________________

For a general plants of relative degree larger than 1, the same control structure as given in Fig. 8.14 is chosen. Note that the order of the filters in the control law is still )1( −n . However, since the model numerator polynomial )( pZ m is of degree smaller than )1( −n , it is no longer possible to choose the poles of the filters in the controller so that )(]det[ pZp m=−ΛI as in (8.48). Instead, we now choose

)()()( 1 ppZp m λλ = (8.57) where ]det[)( ΛI −= ppλ and )(1 pλ is a Hurwitz polynomial of degree )1( mn −− . With this choice, the desired zeros of the reference model can be imposed. Let us define the transfer function of the feed-forward part 1/ uu of the controller by ))()(/()( pCpp +λλ , and that of the feedback part by )(/)( ppD λ , where the polynomial )( pC contains the parameter in the vector 1θ , and the polynomial )( pD contains the parameter in the vector 2θ . Then the closed-loop transfer function is easily found to be

)()]()()[()()(1

pDZkpCppRpZpZkk

Wppp

mppry ++=

λλ

(8.58)

The question now is whether in this general case, there exists choice of values for 2,, θθ1k and 0θ such that the above transfer function becomes exactly the same as )( pWm , or equivalently

)()())()(( 1 pRZpDZkpCpR mpppp λλ =++ (8.59) The answer to this question can be obtained from the following lemma


___________________________________________________________________________________________________________


44

Lemma 8.2: Let )( pA and )( pB be polynomials of degree 1n and 2n , respectively. If )( pA and )( pB are relative prime, then there exist polynomials )( pM and )( pN such that

)()()()()( * pApNpBpMpA =+ (5.60)

where )(* pA is an arbitrary polynomial. This lemma can be used straight forward to answer our question regarding to (8.59). Choice of adaptation law When the plant parameters are unknown, the controller (8.52) is used again

)()()( tttu T ωθ= (8.61) and the tracking error from (8.54)

]/)()([)()( *kttpWte Tm ωφ= (8.62)

However, the adaptation law (8.55) cannot be used. A famous technique called error augmentation can be used to avoid the difficulty in finding an adaptation law for (8.62). The basic idea of the technique is to consider a so-called augmented error )(tε which correlates to the parameter errorφ in a more desirable way than the tracking error ).(te

)( pWp *1k

Tφ

)( pWp

)( pWp

)(tα

)(te )(tε

)(tη

)(tω

Tθ

Tθ

Fig. 8.17 The augmented error Let define an auxiliary error )(tη by

)]()()[()]()[()()( ttpWtpWtt Tmm

T ωθωθ −=η (8.63) as shown in Fig. 8.17. It is useful to note two features about this error - Firstly, )(tη can be computed on-line, since the estimated

parameter vector )(tθ and the signal vector )(tω are both available.

- Secondly, )(tη is caused by time-varying nature of the estimated parameters )(tθ , in the sense that when )(tθ is

replaced by the true (constant) parameter vector *θ , we

have .0)]()()[()]()[( ** =− ttpWtpW mm ωθωθ This also

implies thatη can be written: )()()( ωφωφ Tmm

T WWt −=η Define an augmented error )(tε

)()()()( tttet ηαε += (8.64) where )(tα is a time-varying parameter to be determined by adaptation. For convenience, let us write )(tα in the form

)(/1)( * tkt αφα += . From (8.62)-(8.64) we obtain

)()(1)( * ttk

t T ηφε α+= ωφ (8.65)

where

])[()( ωω pWt m= (8.66) This implies that the augmented error can be linearly parameterized by the parameter error )(tφ and αφ . Using the gradient method with normilazation, the controller parameters

)(tθ and the parameter )(tα for forming the augmented error are updated by

ωω

ωθ T

pk

+−=

1

)sgn( εγ& (8.67a)

ωωT+−=

1ηεγα& (8.67b)

With the control law (8.61) and adaptation law (8.67), global convergence of the tracking error can be shown. 8.5 Adaptive Control of Nonlinear System Consider a class of nonlinear system satisfying the following conditions:

1. the nonlinear plant dynamics can be linearly parameterized.

2. the full state is measurable 3. nonlinearities can be cancelled stably (i.e., without

unstable hidden modes or dynamics) by the control input if the parameters are known.

In this section, we consider the case of SISO system. Problem statement Consider nth-order nonlinear systems in companion form

ubtfyn

iii

n =+∑=

),(1

)( xα (8.68)

where

[ ]Tnyyyx )1( −= L& : the state vector ),( tfi x : known nonlinear functions

bi ,α : unknown constant The control objective is track a desired output )(tyd despite the parameter uncertainty. (8.68) can be rewritten in the form

utfayhn

iii

n =+∑=

),(1

)( x (8.70)

where, bh /1= and ba ii /α= . Choice of control law Similarly to the sliding control approach, define a combined error epeees n

nn )(0

)2(2

)1( ∆=+++= −−

− λλ K , where the output tracking error is dyye −= and a stable (Hurwitz)

polynomial is 02

21)( λλ +++=∆ −

−− Kn

nn ppp . Note that s

can be rewritten as )1()1( −− −= nr

n yys with )1( −nry is defined as

eeyy nn

nd

nr 0

)2(2

)1()1( λλ −−−= −−

−− K .


___________________________________________________________________________________________________________


45

Consider the control law

∑=

+−=n

iii

nr tfaskyhu

1

)( ),(x (8.71)

where k is constant of the same sign as h , and )(nry is the

derivative of )1( −nry , i.e., eeyy n

nn

dn

r &K 0)1(

2)()( λλ −−−= −

− .

Noting that )(nry , the so-called “reference” value of )(ny , is

obtained by modifying )(ndy according to the tracking errors. If

the parameters are all known, this choice leads to the tracking error dynamics 0=+ sksh & and therefore gives exponential convergence of s , which in turn, guarantees the convergence of e .

Choice of adaptation law For our adaptive control, the control law (8.71) is replaced by

∑=

+−=n

iii

nr tfaskyhu

1

)( ),(ˆˆ x (8.72)

where iah, have been replaced by their estimated values. The tracking error yields

∑=

+=+n

iii

nr tfayhsksh

1

)( ),(~~x& (8.73)

where, iii hhh −= ˆ~, iii aaa −= ˆ~ . (8.73) can be written in the

form

[ ]

+=

+

+= ∑

=

),(

),(~~~)/(

/1

),(~~)/(

/1

1

)(

1

1

)(

tf

tfy

aahhkp

h

tfayhhkp

hs

n

nr

n

n

iii

nr

x

x

x

ML

(8.74)

Lemma 8.1 suggests the following adaptation law

)()sgn(ˆ nryshh γ−=

&

ii fsha )sgn(ˆ γ−=& Specially, using the Lyapunov function candidate

01

2212 ≥

++= ∑

=

−n

iiahshV γ

it is straight forward to verify that 02 2 ≤−= skV& and therefore the global tracking convergence of the adaptive control system can be easily shown. Example___________________________________________

Consider a mass-spring-damper system with nonlinear friction and nonlinear damping described by the equation

uxfkxfcxm =++ )()( 21 &&& (8.69) Apply the above analysis procedure. Define the error

rd xxexxees &&&&& −≡−−=+≡ )( 00 λλ Chose the control law

21 fkfcsxmu r ++−= α&& (→ 8.71) which yields 2121 fkfcsxmfkfcxm r ++−=++ α&&&& or

0)( =+− sxxm r α&&&& . Because the unknown parameters, the controller is

21ˆˆˆˆ fkfcsxmuu r ++−=→ α&& (→ 8.72)

which leads to the tracking error

2121ˆˆˆ fkfcsxmfkfcxm r ++−=++ α&&&&

0)ˆ()ˆ()ˆ()( 21 =+−−−−−−− sfkkfccxmxmxmxm rrr α&&&&&&&&

0~~~)( 21 =+−−−− sfkfcxmxxm rr α&&&&&&

[ ]

=+

2

1~~~

ffx

kcmssmr&&

& α (→ 8.74)

Adaptation laws: rxsm &&& γ−=ˆ , 1ˆ fsc γ−=& , 2ˆ fsk γ−=&

Lyapunov function: )

~~~( 22212 kcmsmV +++= −γ and its derivative with the above adaptation laws yields

02

)ˆ(~

)ˆ(~)ˆ(~22

)ˆ~ˆ~ˆ~(2)

~~~(2

)ˆ~ˆ~ˆ~(22

2

2112

121

1

≤−=

++++++−=

+++−++=

+++=

−

−

−

s

fskkfsccxsmms

kkccmmsfkfcxms

kkccmmssmV

r

r

α

γγγγα

γα

γ

&&&&&

&&&&&

&&&&&

__________________________________________________________________________________________ 8.6 Robustness of Adaptive Control System The above tracking and parameter convergence analysis has provided us with considerable insight into the behavior of the adaptive control system. The analysis has been carried out assuming that no other uncertainties exist in the control system besides parametric uncertainties. However, in practice, many types of non-parametric uncertainties can be present. These include

- high-frequency un-modeled dynamics, such as actuator dynamics or structural vibrations

- low-frequency un-modeled dynamics, such as Coulomb friction and stiction

- measurement- noise - computation round-off error and sampling delay

Since adaptive controllers are designed to control real physical systems and such non-parametric uncertainties are unavoidable, it is important to ask the following questions concerning the non-parametric uncertainties:

- what effects can they have on adaptive control systems ? - when are adaptive control systems sensitive to them ? - how can adaptive control systems be made insensitive to

them ?

While precise answers to such questions are difficult to obtain, because adaptive control systems are nonlinear systems, some


___________________________________________________________________________________________________________


46

qualitative answers can improve our understanding of adaptive control system behavior in practical applications. Parameter drift When the signal v is persistently exciting, both simulations and analysis indicate that the adaptive control systems have some robustness with respect to non-parametric uncertainties. However, when the signals are not persistently exciting, even small uncertainties may lead to severe problems for adaptive controllers. The following example illustrates this situation. Example 8.7 Rohrs’s example_________________________

Consider the plant described by the following nominal model

p

p

ap

kpH

+=)(0

The reference model has the following SPR function

33)(+

=+

=pap

kpM

m

m

The real plant, however, is assumed to have the transfer function relation

uppp

y22930

2291

22 +++

=

This means that the real plant is of third order while the nominal plant is of only first order. The un-modeled dynamics are thus to seen to be )22930/(229 2 ++ pp , which are high-frequency but lightly-damped poles at )15( j±− . Beside the un-modeled dynamics, it is assumed that there is some measurement noise )(tm in the adaptive system. The whole adaptive control system is shown in Fig. 8.18. The measurement noise is assumed to be )1.16sin(5.0)( ttn = .

)(tym)(tr

u

22930229

12

2 +++ ppp

33+p )(te

1y

reference model

nominal un-modeled

ya

ra

)(tn

Fig. 8.18 Adaptive control with un-modeled dynamics and

measurement noise Corresponding to the reference input 2)( =tr , the results of adaptive control system are shown in Fig. 8.19. It is seen that the output )(ty initially converges to the vicinity of 2=y , then operates with a small oscillatory error related to the measurement noise, and finally diverges to infinity.

Fig. 8.19 Instability and parameter drift

_________________________________________________________________________________________ In view of the global tracking convergence proven in the absence of non-parametric uncertainties and the small amount

of non-parametric uncertainties present in the above example, the observed instability can seem quite surprising. Dead-zone 8.7 On-line Parameter Estimation Few basic methods of on-line estimation are studied. Continuous-time formulation is used. 8.7.1 Linear parameterization model The essence of parameter estimation is to extract parameter information from available data concerning the system. The quite general model for parameter estimation applications is in the linear parameterization form

aWy )()( tt = (8.77) where

nR∈y : known “output” of the system mR∈a : unknown parameters to be estimated

mnRt ×∈)(W : known signal matrix (8.77) is simply a linear equation in terms of the unknown a. Model (8.77), although simple, is actually quite general. Any linear system can be rewritten in this form after filtering both side of the system dynamics equation through an exponentially stable filter of proper order, as seen in the following example. Example 8.9 Filtering linear dynamics__________________

Consider the first-order dynamics

ubyay 11 +−=& (8.78) Assume that 11,ba in model are unknown, and that the output y and the input u are available. The above model cannot be

directly used for estimation, because the derivative of y appears in the above equation (noting that numerically

differentiating y is usually undesirable because of noise consideration). To eliminate y& in the above equation, let us filter (multiply) both side of the equation by )/(1 fp λ+

(where p is the Laplace operator and fλ is a known positive constant). Rearranging, this leads to the form

11)()( buayty fff +−= λ (8.78) where )/( ff pyy λ+= and )/( ff puu λ+= with subscript

f denoting filtered quantities. Note that, as a result of the filtering operation, the only unknown quantities in (8.79) are the parameters )( 1af −λ and 1b . The above filtering introduces a d.c. gain of fλ/1 , i.e., the

magnitudes of fy and fu are smaller than those of y and

u by a factor of fλ at low frequencies. Since smaller signals may lead to slower estimation, we may multiply both side of (8.79) by a constant number, i.e., fλ .


___________________________________________________________________________________________________________


47

Generally, for a linear SISO system, its dynamics can be described by

upBypA )()( = (8.80) where

nnn ppapaapA ++++= −−

1110)( K

1110)( −−+++= n

n papbbpB K Divide both sides of (8.80) by a known monic polynomial of order n, leading to

upApB

pApApA

y)()(

)()()(

00

0 −= (8.81)

where nn

n pppA ++++= −−

11100 ααα K

has known coefficients. In view of the fact that

11111000 )()()()()( −−− −++−+−=− n

nn papaapApA ααα K

(8.81) can be rewritten in the form

)(ty T wθ= (8.82) where

[ ]Tnnn bbaaa 10111100 −−− −−−= LL αααθ Tnn

Aup

Au

Ayp

Apy

Ay

=

−−

0

1

00

1

00LLw

Note that w can be computed on-line based on the available values of y and u . _________________________________________________________________________________________ Example 8.9 Linear parametrization of robot dynamics_____

lc1

l1

l2

lc2

I2, m2

I1, m1

q2,τ2

q1,τ1

Fig. 6.2 A two-link robot

Consider the nonlinear dynamics of a two-link robot

=

+

−−−=

2

1

2

1

2

1

1

212

2

1

2221

12110 τ

τgg

qq

qhqhqhqh

qq

HHHH

&

&

&

&&&

&&

&&

(6.9) where,

[ ]Tqqq 21= : joint angles

[ ]T21 τττ = : joint inputs (torques)

222122

2121

21111 )cos2( IqllllmIlmH ccc +++++=

222222122112 cos IlmqclmHH c ++==

222222 IlmH c +=

2212 sin qllmh c= ]cos)cos([cos 1121221111 qlqqlgmqglmg cc +++=

)cos( 21222 qqglmg c += Let us define ,21 ma = ,222 clma = ,2

1113 clmIa += and 22224 clmIa += . Then each term on the left-hand side of (6.9)

is linear terms of the equivalent inertia parameters [ ]Taaaa 4321=a . Specially,

2122114311 cos2 qlalaaaH +++=

422 aH =

42122112 cos aqlaHH +== Thus we can write

aqqqYτ ),,(1 &&&= (8.83) This linear parametrization property actually applies to any mechanical system, including multiple-link robots. Relation (8.83) cannot be directly used for parameter estimation, because of the present of the un-measurable joint acceleration q&& . To avoid this, we can use the above filtering technique. Specially, let )(tw be the impulse response of a stable, proper filter. Then, convolving both sides of (6.9), yields

∫∫ ++−=−tt

drrtwdrrrtw00

])[()()( GqCqHτ &&& (8.84)

Using partial integration, the first term on the right hand side of (8.84) can be rewritten as

∫

∫∫

−−

−−=

−−=−

t

ttt

drrtwrtw

ww

drwdrdrtwdrrtw

0

000

])()([

)0(])0([)0()()0(

][)()(

qH-qH

qqHqqH

qHqHqH

&&&&

&&

&&&&

This means that the equation (8.48) can be rewritten as

aqqWy ),()( &=t (8.85) where y is the filtered torque and W is the filtered version of

1Y . Thus the matrix W can be computed from available measurements of q and q& . The filtered torque y can also be computed because the torque signals issued by the computer are known. _________________________________________________________________________________________ 8.7.2 Prediction-error-based estimation model 8.7.3 The gradient estimator 8.7.4 The standard least-squares estimator 8.7.5 Least-squares with exponential forgetting


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48

8.7.6 Bounded gain forgetting 8.7.7 Concluding remarks and implementation issues 8.8 Composite Adaptation

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Applied Nonlinear Control