PASSIVITY–BASED CONTROL WITH

8/4/2019 PASSIVITYBASED CONTROL WITH

1/6

PASSIVITYBASED CONTROL WITH

SIMULTANEOUS ENERGYSHAPING AND

DAMPING INJECTION: THE INDUCTION MOTOR

CASE STUDY

R. Ortega G. Espinosa-Perez ,1

Laboratoire des Signaux et Systemes, SUPELEC, Plateau duMoulon, Gif-sur-Yvette 91192, FRANCE, [email protected] DEPFIUNAM, Apartado Postal 70-256, 04510 Mexico D.F.,

MEXICO, [email protected]

Abstract: We argue in this paper that the standard procedure used PassivityBasedController (PBC) designs of splitting the control action into energyshaping anddamping injection terms is not without loss of generality, and actually reduces theset of problems that can be solved with PBC. Instead, we suggest to carry outsimultaneously both stages. As a case in point, we show that the practically important

example of the induction motor cannot be solved with a PBC in two stages. It is,however, solvable carrying out simultaneously the energyshaping and the dampinginjection. The resulting controller is a simple output feedback scheme that ensuresglobal exponential convergence of the generated torque and the rotor flux norm totheir desired (constant) values. To the best of our knowledge, this is the first outputfeedback scheme that ensures such strong stability properties for this system.

Keywords: Nonlinear control, Passivity-based control, Energy-shaping, Inductionmotors.

1. INTRODUCTION

Passivitybased control (PBC) is a generic namegiven to a family of controller design techniquesthat achieve the control objective via the route ofpassivation, that is, rendering the closedloop sys-tem passive with a desired storage function (thatusually qualifies as a Lyapunov function for thestability analysis.) If the passivity property turnsout to be (output) strict, with an output signalwith respect to which the system is detectable,then asymptotic stability is ensured. See the fun-damental monograph (van der Schaft, 2000), and

(Ortega and Garcia-Canseco, 2004) for a recentsurvey.

1 This work has been partially supported by CONACyT(41298Y) and DGAPA-UNAM (IN119003).

As is wellknown, (van der Schaft, 2000), a pas-sive system can be rendered strictly passive sim-ply adding a negative feedback loop around thepassive outputan action sometimes called LgVcontrol (Sepulchre et al., 1997). For this reason, ithas been found convenient in some applications, inparticular for mechanical systems (Takegaki andArimoto, 1981), (Ortega et al., 2002b), to split thecontrol action into two terms, an energyshapingterm which, as indicated by its name, is responsi-ble of assigning the desired energy/storage func-tion to the passive map, and a second LgV termthat injects damping for asymptotic stability.

The purpose of this paper is to bring to the readersattention the fact that splitting the control actionin this way is not without loss of generality, andactually reduces the set of problems that can besolved via PBC. This assertion is, of course, not


2/6

surprising since it is clear that, to achieve strictpassivity, the procedure described above is justone of many other possible ways. The main objec-tive of the paper is to show that the practically

important example of the induction motor cannotbe solvedwith a PBC in two stages. It is, however,solvable carrying out simultaneously the energyshaping and the damping injection.

2. PBC WITH SIMULTANEOUS ENERGYSHAPING AND DAMPING INJECTION

To be more specific let us consider the Inter-connection and Damping Assignment (IDA) PBCproposed in (Ortega et al., 2002a) applied to non-linear systems of the form

x = f(x) + g(x)u (1)

where x Rn is the state vector and u Rm, m 0, which yields

the particular damping matrix

Rd(x) = g(x)Kdig(x).

In the next section we will show that, for theproblem of output feedback torque control of

2 All vectors in the paper are column vectors, even thegradient of a scalar function denoted () =

()

.

induction motors with quadratic in the incrementsdesired energy function, it is not possible to solve(4) with Rd(x) = 0. But the problem is solvable ifwe allow for a general damping matrix.

3. INDUCTION MOTOR: MODEL ANDEQUILIBRIA

In this section the basic motor model and theanalysis of its equilibria are presented. The latteris needed because, in contrast with the largemajority of controllers proposed for the inductionmotor, we are interested in this paper in thestabilization of a given equilibrium that generatesa desired torque and rotor flux amplitude.

3.1 Model

The standard three-phase induction motor rep-resented with a two-phase model defined in anarbitrary reference frame, which rotates at anarbitrary speed s R, is given by (Krause etal., 1995)

x12 = [I+ ( + u3)J] x12 +

+1 (I TrJ) x34 + 2u12 (6)

x34 = ( 1Tr

I + Ju3)x34 + Lsr

Trx12 (7)

= 3x12Jx34 L (8)

in which I R2 is the identity matrix,

J =

0 11 0

= J,

x12 R2 are the stator currents, x34 R

2 therotor fluxes, R the rotor speed, u12 R2 arethe stator voltages, L R is the load torque andu3 := s . The parameters, all positive, are

defined as

:=Rs

Lsr+

Lsr

LsLrTr; := 1

L2srLsLr

1 :=Lsr

LsLrTr; 2 :=

1

Ls

3 :=Lsr

Lr; Tr :=

Lr

Rr

with Ls, Lr the windings inductances, Rs, Rr thewindings resistances and Lsr the mutual induc-tance. Notice that, without loss of generality, therotor moment of inertia and the number of polepairs are assumed equal to one.

As first pointed out in the control literature in(Ortega and Espinosa, 1993), the signal u3 (equiv-alently s) effectively acts as an additional controlinput. Below, we will select u3 to transform theperiodic orbits of the system into constant equi-libria.


3/6

3.2 Controlled Outputs and Equilibria

We are interested in this paper in the problem ofregulation of the motor torque and the rotor flux

amplitude, that we denote,

y1 = h1(x) = 3x12Jx34

y2 = h2(x) = |x34|, (9)

respectively, to some constant desired values y =[y1, y2]

, where we defined x :=

x12

, x34

. To

solve this problem using IDAPBC it is necessaryto express the control objective in terms of adesired equilibrium. We make at this point thefollowing important observation:

From (8) we see that to operate the system in

equilibrium, y1 = Lhence, the load torque isassumed known. See, however, Remark 1.

As is wellknown (Marino et al., 1999), the zerodynamics of the induction motor is periodic, a factthat is clearly shown computing the angular speedof the rotor flux. Towards this end, we define therotor flux angle := arctan x4

x3, and evaluate 3

= Rry1

y22

u3,

from which have the following simple lemmawhose proof is obtained via direct substitution.

Lemma 1. Consider the induction motor model(6)(8) with u3 fixed to the constant

u3 = u3 := Rry1

y22

. (10)

Then, the set of assignable equilibrium points,denoted [x, ] R5, which are compatible withh(x) = y is defined by R and

x12 =1

Lsr

1 Lr

y1

y22

Lry1

y2

2

1

x34

|x34| = y2 (11)

Among the set of assignable equilibria definedabove we select, for the electrical coordinates,the one that ensures field orientation (Krause etal., 1995) and denote it

x := [LrLsr

y1

y2,

y2

Lsr, 0, y2]

. (12)

Remark 1. In practical applications an outer loopPI control around the velocity error is usually

3 From this relation it is clear that, if u3 is fixed to aconstant, say u3, and y = y, x34 is a vector of constantamplitude rotating at speed (t) = (Rr

y1y22

u3)t + (0).

added. The output of the integrator, on one hand,provides an estimate of L while, on the otherhand, ensures that speed also converges to thedesired value as shown via simulations in Section

6.

4. IDAPBC OF INDUCTION MOTOR

The following important aspects of the inductionmotor control problem are needed for its preciseformulation:

The only signals available for measurement arex12 and .

Since we are interested here in torque control,and this is only defined by the stator currents andthe rotor fluxes, its regulation can be achievedapplying IDAPBC to the electrical subsystemonly. Boundedness of will be established in asubsequent analysis.

Although with IDAPBC it is possible, in princi-ple, to assign an arbitrary energy function to theelectrical subsystem, we will consider here only aquadratic in errors form

Hd(x) =1

2xPx, (13)

with x := x x and P = P > 0 a matrix

to be determined. As first observed in (Fujimotoand Sugie, 2001), fixing Hd(x) transforms thematching equation (4) into a set of algebraicequationssee also (Rodriguez and Ortega, 2003)for application of this, socalled Algebraic IDAPBC, to general electromechanical systems.

The electrical subsystem (6)(7) with u3 = u3can be written in the form

x = f(x, ) +

I

022

u12.

Therefore, the matching equation (4) concerns

only the third and fourth rows of f(x, ) and ittakes the form

(1

TrI+Ju3)x34+

Lsr

Trx12 = [F3(x) F4(x)]Px,

(14)where, to simplify the notation, we define thematrix

F(x) := Jd(x) Rd(x),

that we partition into 2 2 sub-matrices as

F(x) =

F1(x) F2(x)F3(x) F4(x)

. (15)

The output feedback condition imposes an addi-tional constraint that involves now the first andsecond rows of f(x, ). Indeed, from (5) we seethat the control can be written as

u12 = u12(x12, ) + S(x, )x34


4/6

where u12(x12, ) is given in (24) and we havedefined

S(x, ) :=1

2(TrJ I)+

1

2[F1(x)P2 + F2(x)P3] ,

(16)with P partitioned as

P :=

P1 P2

P2 P3

; Pi R

22, i = 1, 2, 3.

It is clear that S(x, ) has to be set to zero tosatisfy the output feedback condition.

We thus have the following:

IDAPBC Problems. Find matrices F(x) andP = P > 0 satisfying (14) and S(x, ) = 0 withthe additional constraint that

(Energyshaping)F(x) + F(x) = 0, (17)

or the strictly weaker

(Simultaneous energyshaping and dampinginjection)

F(x) + F(x) 0. (18)

5. MAIN RESULTS

5.1 Solvability of the IDAPBC Problems

Proposition 1. The energyshaping problem isnot solvable. However, the simultaneous energyshaping and damping injection one is solvable.

Proof. First, we write the matching equation (14)in terms of the errors as

Lsr

Trx12

1

TrI+ u3J

x34 = [F3(x) F4(x)]Px,

which will be satisfied if and only if

[F3(x) F4(x)]P = [ LsrTr

I 1Tr

I+ u3J].(19)

Since P and the right hand side of the equationare constant, and P is full rank, we concludethat F3 and F4 should also be constant. (Tounderscore this fact we will omit in the sequeltheir argument.)

Let us consider first the energyshaping problem.From (15) and (17) we have that F2 = F3 .Then, setting (16) to zero and replacing the latterit is obtained that

F1(x)P2 F3 P3 = 1 (I TrJ) (20)

On the other hand, from the first two columns of(19) it follows that

F3 =

Lsr

TrI F4P

2

P11

(21)

Substitution of (21) into (20) leads to

F1(x)P2 P11

Lsr

TrI P2F

4

P3 =

= 1I 1TrJ(22)

Invoking again (17) we have that F1(x) must beskew-symmetric, that without loss of generalitywe can express in the form

F1(x) = 1(x)J + 2J,

where 2 R. Looking at the xdependent termswe get

1(x)JP2 + 1TrJ = 0,

which can be achieved only if P2 = I, with R, and 1(x) = 11Tr.

The constant part of (22), considering that P2 =I, reduces to

2J P11

Lsr

TrI F4

P3 = 1I

whichusing the fact that P3 is full rankcan beexpressed as F

4= GP1

3, where we have defined

the constant matrix

G :=1

Lsr

TrP3 + P1 (1I 2J)

Finally, since F4 must also be skewsymmetric, wehave that

G = P13 (G)P3,

i.e., G must be similar to G, and consequentlyboth have the same eigenvalues. A necessary con-dition for the latter is that trace(G) = 0, that isnot satisfied because

trace(G) =1

Lsr

Trtrace(P3)

>0

+1 trace(P1) >0

,

which is different from zero. This completes theproof of the first claim.

We will now prove that if we consider the largestclass of matrices (18) the problem is indeed solv-able, and actually give a very simple explicit ex-pression for F(x) and P. For, we set P2 = 0, andit is easy to see that

F2(x) = 1 (I TrJ) P13

F3(x) =Lsr

TrIP1

1

F4(x) =

1

TrI + u3J

P13

and F1(x) free provide a solution to (14) and make(16) equal to zero. It only remains to establish(18). For, we fix P1 = LsrTr I, P3 = 1I and

F1(x) = K(), with K() = K() > 0, then

F(x) =

K() I TrJ

I 11

1

TrI+ u3J

.


5/6

A simple Schur complement analysis shows thatF(x) + F(x) < 0 if and only if

K() >Lsr

4 (LsLr L2sr) T2r

2 + 4

I. (23)

5.2 Proposed Controller

Once the solvability of the problem with simulta-neous energyshaping and damping injection hasbeen established, the final part of the design isthe explicit definition of the resulting IDAPBCand the assessment of its stability properties. Thisis summarized in the proposition below whose

proof follows immediately from analysis of theclosedloop dynamics x = F(x)Hd, with F(x)+F(x) < 0, and (8).

Proposition 2. Consider the induction motor model(6)(8) with outputs to be regulated given by (9).Assume that

A.1 The measurable states are the stator cur-rents x12 and the rotor speed .

A.2 All the motor parameters are known.A.3 The load torque is constant and known.

Fix, the desired equilibrium to be stabilized as

(12), with y = [L, y2], y2 > 0, and setu3 = Rr

y1|y22|

and u12 = u12(x12, ) with

u12(x12, ) =1

2[I+ ( + u3)J] x12 + (24)

+1

2(I TrJ) x34 Jx12 +

Lsr

2TrK()x12]

with K() satisfying (23). Then, the xsubsystemadmits a Lyapunov function

Hd(x) =Lsr

2Tr

|x12|2 +

1

2|x34|

2.

that satisfies Hd Hd, for some > 0. Conse-quently, for all initial conditions,

limt

x(t) = x, limt

y(t) = y

exponentially fast. Furthermore limt (t) =.

6. SIMULATION RESULTS

The performance of the proposed controller was

investigated by simulations considering two ex-periments described below. The considered motorparameters, taken from (Ortega and Espinosa,1993), were Ls = 84mH, Lr = 85.2mH, Lsr =81.3mH, Rs = 0.687 and Rr = 0.842, withan unitary rotor moment of inertia. Regarding

the controller parameters, following field orientedideas, the rotor flux equilibrium value was set to

x34 =

0

with = 2, while x12 where computed accordingto (11). In order to satisfy condition (23), it wasdefined K = kI with

k =Lsr

(LsLr L2sr)

T2r

2 + 4

In a first experiment the motor was initially atstandstill with a zero load torque. At startup,the load torque was set to L = 20N m and thisvalue was maintained until t = 40sec when anew step in this variable was introduced changingto L = 40N m. Figure 1 shows the behaviorof the stator currents where it can be noticed

how, according to the field oriented approach, oneof the stator currents remains (almost) constantwhile the second one is dedicated to producethe required generated torque. In this sense, inFigure 2 it can be observed how the rotor fluxis aligned with the reference frame since one ofthe components equals while the other is zero.The internal stability of the closedloop systemis illustrated in Figure 3 where the rotor speed ispresented. As expected, besides its boundedness,it can be noticed that when the load torqueis increased, this variable decreses. In Figure 4

the main objective of the proposed controller isdepicted. Here it is shown how the generatedtorque regulation objective, both before and afterthe load torque change, is achieved. Figure 5shows the boundedness of the control (statorvoltages) inputs.

The second experiment was aimed to illustrate theclaim stated in Remark 1. In this sense, the controlinput u3 was set to

u3 = u3 = Rry1

y22

where the estimate of the load torque is obtained

as the output of a PI controller, defined over thespeed error between the actual and the desiredvelocities, of the form

y1 = kp ( d) + ki

( d) dt

Figure 6 shows the rotor speed behavior when thedesired velocity is (initially) d = 100rpm andat t = 50sec it is changed to d = 150rpm. Inthis simulation it was considered L = 10N m,ki = .1 and kp = 1. All the other parameterswere the same than in the first experiment.

7. REFERENCES

Fujimoto, K. and T. Sugie (2001). Canonicaltransformations and stabilization of general-ized hamiltonian systems. Systems and Con-trol Letters 42(3), 217227.


6/6

0 10 20 30 40 50 60 700

5

10

15

20

25

30

time [sec]

statorcurrents[amp]

Fig. 1. Stator currents

0 10 20 30 40 50 60 700.5

0

0.5

1

1.5

2

2.5

3

time [sec]

rotorf

luxes[wb]

Fig. 2. Rotor fluxes

0 10 20 30 40 50 60 7010

0

10

20

30

40

50

60

time [sec]

rotor

speed[rpm]

Fig. 3. Rotor speed

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40

time [sec]

generatedandloadtorque[Nm]

Fig. 4. Generated and load torque

Krause, P.C., O. Wasynczuk and S.D. Sudhoff(1995). Analysis of Electric Machinery. IEEEPress. USA.

Marino, R., S. Peresada and P. Tomei (1999).Global adaptive output feedback control of

induction motors with uncertain rotor resis-tance. IEEE Transactions on Automatic Con-trol 44(5), 967983.

Ortega, R., A. van der Schaft, B. Maschkeand G. Escobar (2002a). Interconnection anddamping assignment passivitybased control

0 10 20 30 40 50 60 700

100

200

300

400

500

600

time [sec]

statorvo

ltages[v]

Fig. 5. stator voltages

0 10 20 30 40 50 60 70 80 90 10020

0

20

40

60

80

100

120

140

160

time [sec]

S

peed[rpm]

Fig. 6. Speed behavior

of port controlled hamiltonian systems. AU-TOMATICA.

Ortega, R. and E. Garcia-Canseco (2004).Interconnection and damping assignmentpassivitybased control: Towards a construc-

tive procedurepart i and ii. In: Proc. 43thIEEE Conference on Decision and Control(CDC03). Bahamas.

Ortega, R. and G. Espinosa (1993). Torque reg-ulation of induction motors. AUTOMATICA(Regular Paper) 29(3), 621633.

Ortega, R., M. Spong, F. Gomez and G. Blanken-stein (2002b). Stabilization of underactuatedmechanical systems via interconnection anddamping assignment. IEEE Trans. AutomaticControl 47(8), 12181233.

Rodriguez, H. and R. Ortega (2003). Intercon-nection and damping assignment control ofelectromechanical systems. Int. J. of Robustand Nonlinear Control 13(12), 10951111.

Sepulchre, R., M. Jankovic and P. Koko-tovic (1997). Constructive Nonlinear Control.Springer-Verlag. London.

Takegaki, M. and S. Arimoto (1981). A newfeedback for dynamic control of manipulators.Trans. of the ASME: Journal of DynamicSystems, Measurement and Control102, 119125.

van der Schaft, A. (2000). L2Gain and PassivityTechniques in Nonlinear Control. second ed..

Springer Verlag. London.

Documents

PASSIVITY–BASED CONTROL WITH