1 Introduction When engineers analyze and design nonlinear dynamical systems in electrical circuits, mechanical systems, control systems, and other engineering disciplines, they need to be able to use a wide range of nonlinear analysis tools. Despite the fact that these tools have developed rapidly since the mid 1990’s, nonlinear control is still largely a tough challenge. In this course, we will present basic results for the analysis of nonlinear systems, emphasizing the differences to linear systems, and we will introduce the important nonlinear feedback control tools with the goal of giving an overview of the main possibilities available. Additionally, the lectures will aim to give the context on which each of these tools are to be used.

Requirements Difficulties Theoretical Results Modeling & Simulation Stabilization Tracking Disturbance Rejection Economic Optimization

Process Uncertainties Control Effort Quality of Measurements Nonlinearities

ODE Theory Lyapunov Theory Optimal Control Theory Model Predictive Control

1.1 Nonlinear Model and Nonlinear Phenomena We will deal with systems of the form:

( )( )

( )nnnn

nn

nn

uuxxtfx

uuxxtfxuuxxtfx

,...,,,...,,

,...,,,...,,,...,,,...,,

11

1122

1111

=

==

&

M

&

&

( )uxtfx ,,=&

( )uxthy ,,= where and . nx ℜ∈ pu ℜ∈ Often, we will neglect the time varying aspect. In the analysis phase, external inputs u are also often neglected, leaving system, ( )xtfx ,=& . Working with an unforced state equation does not necessarily mean that the input to the system is zero. It could be that the input has been specified as a given function of the state ( )xuu = .

Definition 1.1: A system is said to be autonomous or time invariant if the function does not depend explicitly on t ; that is

f( )xfx =& .

Definition 1.2: A point is called equilibrium point of *x ( )xfx =& if for some t implies for

( ) *xtx =( ) *xx =τ t≥τ .

The set of equilibrium points is equal to the set of real solutions of the equation ( ) 0=xf . Example 1.1:

2xx =& isolated equilibrium point ( )xx sin=& infinitely many equilibrium points ( )xx /1sin=& infinitely many equilibrium points in a finite region

For linear systems the state model takes the special form:

( ) ( )( ) ( )utBxtAy

utBxtAx+=+=&

As we move from linear to nonlinear systems, we face a more difficult situation. The superposition principle no longer holds, and analysis tools necessarily involve more advanced mathematics. Most importantly, as the superposition principle does not hold, we cannot assume that an analysis of the behaviour of the system – either analytically or via simulation – may be scaled up or down to tell us about the behaviour at large or small scales. These must be checked separately. The first step when analyzing a nonlinear system is usually to linearize it about some nominal operating point and analyze the resulting linear model. However, it is clear that linearization alone will not be sufficient. We must develop tools for the analysis of nonlinear systems. There are two basic limitation of linearization. First, since linearization is an approximation in the neighborhood of an operating point, it can only predict the local behavior of the nonlinear system in the vicinity of that point. Secondly, the dynamics of a nonlinear system are much richer than the dynamics of a linear system. There are essentially nonlinear phenomena that can take place only in the presence of nonlinearity; hence they cannot be described or predicted by linear models. The following are examples of nonlinear phenomena: Finite escape time: The state of an unstable linear system can go to infinity as time approaches infinity. A nonlinear system's state, however, can go to infinity in finite time. Multiple isolated equilibrium points: A linear system can have only one equilibrium point, and thus only one steady-state operating point that attracts or repels the state of the system irrespective of the initial state. A nonlinear system can have more than one equilibrium point.

Limit cycles: A linear system can have a stable oscillation if it has a pair of eigenvalues on the imaginary axis. The amplitude of the oscillation will then depend on the initial conditions. A nonlinear system can exhibit an oscillation of fixed amplitude and frequency which appears independently of the initial conditions. Chaos: A nonlinear system can have a more complicated steady-state behavior that is not equilibrium or periodic oscillation. Some of these chaotic motions exhibit randomness, despite the deterministic nature of the system.

1.2 Common Nonlinearities In the following subsections, various nonlinearities which commonly occur in practice are presented.

1.2.1 Memoryless nonlinearities:

They are called memoryless, zero memory or static because the output of the nonlinearity at any instant of time is determined uniquely by its input at that instant; it does not depend on the history of the input.

Relay Saturation

Dead Zone Quantization

1.2.2 Nonlinearity with memory Quite frequently, we encounter nonlinear elements whose input-output characteristics have memory; that is, the output at any instant of time may depend on the recent event or the entire history of the input.

Relay with hysteresis

1.3 Examples of Nonlinear Systems In this section we present some examples of nonlinear systems which demonstrate how nonlinearities may be present, and how they are then represented in the model equations. Example 1.2: (Chemical Reactor) This is an example of a strong nonlinear system

[ ] [ ]aaafa CrCCVqC 1])[]([ −−=&

The coefficients are an exponential function of the temperature and the concentration of the different reagents.

⎥⎦⎤

⎢⎣⎡

⋅−⋅= TREKr i

i exp

The reaction can be endothermic or exothermic.

[ ] ( )TTKCrKTTVqT caf −+⋅+−= 311)(&

The model has 2 states: the concentration of A and the temperature of the reaction vessel liquid. The manipulated variable is the jacket water temperature.

InputsStates

Tc

CA T

Cooling Jacket

Feed

Product

A BReaction

At a jacket temperature of 305K, the reactor model has an oscillatory response. The oscillations are characterized by reaction run-away with a temperature spike. When the concentration drops to a low value, the reactor cools until the concentration builds and there is another run-away reaction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9320

340

360

380

400

420

440

460

480

A

T

Example 1.3: (Tunnel Diode Circuit) We assume a time invariant linear capacitorC , inductor and resistorL R . The tunnel diode characteristic curve is plotted in the next figure.

( )RR vhi =

Choosing , and cvx =1 cix =2 Eu = we obtain the following system.

( )[ ]

[ ]ERxxL

x

xxhC

x

+−−=

+−=

212

211

1

1

&

&

The equilibrium points of the system are determined by setting 021 == xx .

( )[ ]

[ ]ERxxL

xxhC

+−−=

+−=

21

21

10

10

Therefore, the equilibrium points corresponds to the roots of the equation

( ) 111 xRR

Exh −=

The next figure shows graphically that, for certain values of E and R , this equation has three isolated roots which correspond to three isolated equilibrium points of the system. The number of equilibrium points might change as the values of E and R change. For example, if we increase E for the same value of R , we will reach a point beyond which only Q will exist. 3

Tunnel-diode circuit Tunnel-diode v characteristic RR i−

As we will see in the next chapter, the phase portrait in this case has two stable

equilibrium point and 1 unstable equilibrium point.

Phase-Portrait 21 xx −

Equilibrium points of the tunnel-diode circuit