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Output Linear Feedback for a Class of NonlinearSystems based on the Invariant Ellipsoid Method

S. Gonzalez-Garcia1, A. Polyakov1,2, A. Poznyak1

Department of Automatic Control. CINVESTAVAv. Instituto Politecnico Nacional 2508. 07300 Mexico D.F., Mexico

2 Post-doc position from Department of Applied Mathematics,Voronezh State University, Universitetskaja pl.1, Voronezh 394693, Russia.

E-mail: sgonzalez{apolyakov,apoznyak}@ctrl.cinvestav.mx

Abstract The robust output stabilization problemfor a class of uncertain nonlinear systems with dis-turbances is considered. The suggested control schemeas well as the observation algorithm are based on the

invariant ellipsoid method, which allows to obtain therobust linear feedback as a solution of the special linearoptimization problem with bilinear constraints. Themethods for solving this optimization problem involvesthe LMI technique. The stabilization of a single linkflexible joint manipulator is considered as an illustra-tive example.

Keywords: state estimation, uncertain systems, linearmatrix inequalities.

I. INTRODUCTION

The problem of persistent disturbance attenuation con-sists in the minimization of the perturbation effect in

system through an appropriate feedback control.The synthesis problem for a linear system with unknown

but bounded perturbations traces back to the work ofBertsekas and Schweppe [1] and [2] where an admissiblecontrol tube was found such that it maintained the state ina specified region and reached a given target set. This wasdone using a dynamic programming scheme and convexset operations.

A class of state feedback controls in order to guaran-tee uniform ultimate boundedness (UUB) for uncertaindynamic systems were presented in [3] and summarizedin [8] where additional results were presented for theanalysis of perturbed systems subjected to nonvanishing

perturbations.For the case of L2 perturbations, the explicit solutions

for the optimal controller with the H2 theory were ob-tained as it was shown in [4] and [5]. The restriction ofbounded perturbations was changed in [6] where only themaximum amplitude of the perturbations (l1 norm) wasconsidered, although the optimal controller designed withthe l1 scheme can be of high order.

An interesting methodology for the control of nonlinearsystems was presented in [7] that combines feedback lin-earization and linear compensation based on the stablefactorization approach to guarantee robustness againstparameter uncertainty and external disturbances.

In [9] the problem of peak to peak gain minimizationwith Linear Matrix Inequalities (LMI) was considered.

The notion of invariant sets and its close connection

with the Lyapunov theory was also exploited for the anal-ysis and control of dynamical systems such as constrainedcontrol, robustness analysis, and disturbance rejection aswas stated in [11].

The concept of invariance set states that any elementof this set for t = 0, will remain in the invariance setfor all t R. The most commonly used forms of invariantsets are ellipsoids and polytopes. The former are simpleto characterize the invariant set (a center and the matrixare only required) but it is rather conservative. The latterare the most natural and exact form to define constraintsalthough with a more complex representation.

In [12] the problem of synthesis of a static state-feedback

controller for an LTI system that minimizes the size of thecorresponding invariant ellipsoid was reduced to the opti-mization of a linear function under some LMI constraints.Although many robust control problems can be formulatedin terms of linear matrix inequalities (LMI) and be solvedwith semidefinite programming [10], a significantly widerclass of problems can be formulated in terms of BilinearMatrix Inequalities (BMI) as in [13]. As for their numer-ical solution, it can be mentioned that only in very fewcases (such as static state feedback and dynamic outputfeedback) it is possible to convexify the problem withthe appropriate change of variables and derive equivalentLMIs.

This paper is concerned with the problem of stabiliza-tion of a nonlinear system in the presence of boundedperturbations and output measurements. The stabilizationscheme is based on the Lyapunov function approach andthe invariant ellipsoid method to construct a Luenbergerobserver and the observed state feedback controller. Dueto the nonlinearities of the system, the constraints of theoptimization problem are bilinear. However, with the useof some appropriate transformations, the obtained BMIconstraints are shown to be dependent only on two scalarparameters, i.e., for fixed scalar parameters, the BMIconstraints becomes LMI and the optimization problemcan be solved through standard semidefinite programming

2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)

IEEE Catalog Number: CFP08827-CDR

ISBN: 978-1-4244-2499-3

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technique. The outline of this paper is as follows. In section2, the problem statement and basic assumptions are intro-duced. Then, in section 3 the main results and its proofsare given; some numerical aspects of the robust feedback

synthesis are given in section 4. Next, section 5 presentsthe application of the obtained result to the stabilizationof a single link flexible joint robotic manipulator. Finally,some remarks conclude this paper.

II. PROBLEM STATEMENT

Consider the nonlinear dynamic system with linearstate-output mapping given by

x = f(x, t) + Buy = Cx + wy

(1)

where x Rn is the state vector, u Rm is the controlinput, y Rk is the system output, wy R

k is the output

perturbation, f : Rn

R Rn

is an unknown (from a classgiven below) nonlinear function, B Rnm, C Rkn arethe system matrices.

The following assumptions will be in force throughout:

the output disturbances wy are inside of a boundedellipsoid, that is,

wy2

K= wTy Kwy 1 (2)

where K > 0 is given; the nonlinear function f(x, t) is quasi-Lipschitz,

namely, it belongs to the class of functions satisfying

f(x, t)Ax2

Kf + x

2

Kx

where 0 < Kf Rnn, 0 < Kx R

nn, A Rnn are known matrices, 0; without loss ofgenerality, after the normalization of this inequalityit is sufficient to consider only two-valued case: =(0; 1).

The pair (A, B) is controllable and the pair (A, C) isobservable.

Using denotation wx := f(x, t)Ax the system (1) canbe rewritten in the form

x = Ax + Bu + wx (3)

y = Cx + wy (4)

with

wx2

Kf + x

2

Kx(5)

Here we only consider the linear feedback controls

u = Kx, K Rmn (6)

with respect to observed state x Rn which are obtainedby the classical Luenberger observer having the structure

x = Ax + Bu + F(y Cx), F Rnp (7)

The robust stabilization of the system (3),(4) will berealized using the method of Invariant Ellipsoids (see, for

example, [12]). Here we just present some basic ideas ofthis method.

The ellipsoid

(0, P) = x Rn : xTP1x 1 , P > 0with center in the origin and shape matrix P is saidto be state-invariant for the system (3),(4) under thedisturbances (2) and nonlinearities (5) if

1) the condition x(0) (0, P) implies x(t) (0, P)for all t 0;

2) if x(0) / (0, P) then x(t) (0, P) for t .In other words, any trajectory of the system starting

in the invariant ellipsoid stays in it for all t 0, but thetrajectory starting outside of the ellipsoid converges to thisellipsoid (asymptotically or in finite time). This is donewith the aid of the second method of Lyapunov.

This invariant ellipsoid can be considered as a charac-

teristic of the influence of the uncertainties in a system.So, the minimum (in some sense) invariant ellipsoid cor-responds to a robust feedback control. Hence, the mainproblem is to design the observer-based linear feedbackcontrol providing the convergence of any trajectory of thesystem (3),(4),(7) to the minimum invariant ellipsoid.Here we will use the trace as the criterion of the ellipsoidsminimality.

III. MAIN RESULTS

Define the state estimation error as e := x x. Then itstime derivative satisfies

e = (A F C)e + wx F wy

Introduce the extended vector z := ( x e )T where z R2n. Then it follows

z = Az + F w (8)

where A :=

A + BK F C

0 A F C

, F :=

0 FI F

and w :=

wxwy

.

Our aim here is to find the control gain matrix K andthe observer gain matrix F providing a good enoughstabilization as well as state estimation of the system (8),or more exactly, to design K and F such that the corre-sponding invariant ellipsoid, called below quasi-minimal,would contain the minimal one.

Theorem 1. If the following optimization problem

tr(X1) + tr(H) min (9)

subject to

R1 YT2 C 0 YT2

X2X1CTY2 X2 -Y

T2

I0 X2 -2Kf 0 0

Y2 -Y2 0 -3K 0X1X2 I 0 0 -

1

2K1x

0 (10)



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-R1-2X2 I

I X1AT+AX1+Y1BT+BYT1 +1X1

0

H II X2

0, 1 2 + 3, 2 0, 3 0

X1 0, X2 0, H 0

with := ATX2+X2A-Y

T2

C-CTY2+1X2

has a solution with respect to the matrix variablesH, X1, X2, R1, R2 Rnn, Y1 Rnm, Y2 Rkn andthe scalar variables 1, 2 and 3, then the ellipsoid withthe matrix

P =

X1 00 X1

2

(11)

is quasi-minimal invariant ellipsoid of the system (8) with(2), (5) with the feedback control gain matrix

K =

X1

1 Y1T

(12)and the observer gain matrix

F =

Y2X1

2

T(13)

Proof. Define the Lyapunov function as

V(z) : =(z, P1z), P1=

P1 00 P2

where P > 0 is the matrix of an invariant ellipsoid shouldbe minimized. Then

V = (z, P1z)+(z, P1z)

= (z, P1(Az+F w))+((Az+F w), P1

z)

= zT

[A

T

P1

+P1

A]z+wT

FT

P1

z+zT

P1

F wor expressing in its quadratic form

V=

zwx

wy

TATP1+P1A P1F

FTP1 0

zwxwy

0

(14)This ellipsoid (z, P1z) 1 will be invariant if and only

if outside of it we have V 0, that is, for z satisfying

zTP1z 1 (15)

Together with (14) and (15) we also have

wxKfwx + xT

Kxx and wy

2

K 1 (16)To fullfill all this constraints let us apply the, so-called,

S-procedure (see, for example, [17]). Define

A0 :=

ATP1 + P1A P1F

FTP1 0

, 0 := 0

Since

zTP1z 1

zwx

wy

T

A1

zwx

wy

1 := 1

A1 :=

P1 0 0

0 0 00 0 0

Representing (16) in the form

wxKfwx + zTKzz, Kz =

Kx KxKx Kx

one has zwx

wy

T

A2

zwx

wy

:= 2,

A2 :=

Kz 0 00 Kf 0

0 0 0

And analogously,

wy2

K 1

zwx

wy

T

A3

zwx

wy

1 := 3

A3 = 0 0 00 0 00 0 K

Finally, by the S-procedure if there exist 1 0 , 2 0and 3 0 such that

0 11 + 22 + 33A0 1A1 + 2A2 + 3A3

(17)

then the conditions (14), (15) and (16) hold and theellipsoid with the matrix P is an invariant ellipsoid of oursystem. The first condition in (17) can be represented as

1 2 + 3, i 0, i = 1, 2, 3 (18)

For the second one, define

Q := A0 1A1 2A2 3A3 0or

Q =

1 P1F C+2Kx 0 P1FCTFTP1+2Kx 2 P2 -P2F

0 P2 -2Kf 0FTP1 -FTP2 0 -3K

with

1 : =ATKP1+P1AK+2Kx, AK = A + BK +

12

I

2 : =ATFP2+P2AF+2Kx, AF = A F C+

12

I

Applying the quadratic non-singular transformation

T1 =

P11

0 0 00 I 0 00 0 I 00 0 0 I

to the matrix Q we get

Q1 = T1QTT1 =

1F C+

2P1

1Kx

0 F

CTFT+2KxP

1

1

2 P2 -P2F

0 P2 -2Kf 0FT -FTP2 0 -3K

0



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1 := P1

1ATK + AKP

1

1+ 1P

1

1+ 2P

1

1KxP

1

1

2 := ATFP2 + P2AF + 1P2 + 2Kx

Obviously that

Q1 = Q+

P11

I00

(2Kx) P11 I 0 0 0 (19)

where

Q =

1 F C 0 FCTFT 2 P2 P2F

0 P2 2Kf 0FT FTP2 0 3K

1 := P1

1ATK + AKP

1

1+ 1P

1

1

2 := ATFP2 + P2AF + 1P2

Using the Schur complement to (19) we obtain

Q2=

1 F C 0 F P1

1

CTFT 2 P2 -P2F I0 P2 -2Kf 0 0

FT FTP2 0 -3K 0P11

I 0 0 - 12

K1x

0

1 := P1

1ATK+AKP

1

1+1P

1

1

2 := ATFP2+P2AF+1P2

Analogously, apply the transformation

T2 =

P2 0 0 0 00 I 0 0 00 0 I 0 00 0 0 I 00 0 0 0 I

to Q2 we obtain

Q3 = T2Q2TT2

=

1 P2F C 0 P2F P2P1

1

CTFTP2 2 P2 -P2F I0 P2 -2Kf 0 0

FTP2 FTP2 0 -3K 0

P11

P2 I 0 0 -1

2K1x

0

(20)

1 : = P2(P1

1ATK+AKP

1

1+1P

1

1)P2

2 : = ATFP2+P2AF+1P2

By -inequality (see [17])

XYT + Y XT XXT + Y1YT (21)

valid for any X Rnk, Y Rnk and any 0 < =T Rkk being applied for X = P2 and Y = I it follows

X+ XT XXT + 1

that for

:= (P11

ATK

+ AKP1

1+ 1P

1

1)

implies

P2(P1

1ATK + AKP

1

1+ 1P

1

1)P

2

-P2-P2-(P1

1ATK+AKP

1

1+1P

1

1)1 (22)

Applying then the Schur complement to the matrix in-equality

2P2 (P1

1ATK + AKP

1

1+ 1P

1

1)1 R1 (23)

we get

R1 2P2 I

I P11

ATK + AKP1

1+ 1P

1

1

0 (24)

Defining

X1 := P1

1, Y1 := P

1

1KT, X2 := P2, Y2 := F

TP2

using (22), (23), (24) and the matrix inequality

X YT

Y Z

X YT

Y Z

valid for any X = XT, Z = ZT and X = XT X, then(20) can be restricted by

R1 YT2 C 0 YT2

X2X1CTY2 1 X2 -YT2 I

0 X2 -2Kf 0 0Y2 Y2 0 -3K 0

X1X2 I 0 0 -1

2 K1

x

0

1 := ATX2+X2A-Y

T2

C-CTY2+1X2

-R1-2X2 I

I X1AT+AX1+Y1BT+BYT1 +1X1

0

The minimization problem to be solved is

tr(X1) + tr(X1

2) min (25)

subject to the constraints (10). Introduce the followingadditional constraint

H X12

H II X2

0

This allows to reduce the optimization problem (25) to alinear one

tr(X1) + tr(H) min

Remark 1. Since the proof of the theorem 1 is basedon the S-procedure, Schur complement and -inequality[17], which give only the upper estimates of the matrixinequalities, we can guarantee that the obtained solutiongives only the quasi-minimal invariant ellipsoid.



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IV. ROBUST FEEDBACK SYNTHESIS:

NUMERICAL ASPECTS

The problem (9) is, in fact, a bilinear optimizationproblem because of the term X1X2 participating in the

matrix constraint (10). The numerical solution of suchproblems is usually a non trivial task. The feasible setfor a BMI is nonconvex in general case. Semi-definiterelaxations (as in [14] [15]) and the solution throughnonlinear programming methods (e.g. branch-bound al-gorithm) can be considered as two alternatives to solve agiven BMI. The efficiency of the Matlab-Tools PENBMIapplication is highly sensitive to the initial point selectionwhich is desired to be close to a solution [16]. It is basedon the penalty function approach. The following lemmasimplifies the problem of a feasible starting point finding.

Lemma 1. The set of variables satisfying (10) containsthe set of ones satisfying

R1 YT2 C 0 YT2

0CTY2 X2 -YT2 I

0 X2 -2Kf 0 0Y2 Y2 0 -3K 00 I 0 0 R2

0 (26)

-R1-2X2 I

I X1AT+AX1+Y1BT+BYT1 +1X1+

0

- 12

K1x -R2 X1X1 -

0,

H II X2

0

1 2 + 3, i 0, i = 1, 2, 3

with := ATX2+X2A-Y

T2

C-CTY2+1X2

Proof. By the -inequality (21) with

XT:=

0 0 0 0 X1

and Y:=

X2 0 0 0 0

the matrix inequality (20) containing Q3 can be estimatedas

Q3 Q

3 0

where

Q3 :=

1 P2F C 0 P2F 0CTFTP2 2 P2 P2F I

0 P2 2Kf 0 0FTP2 F

TP2 0 3K 00 I 0 0 3

0(27)

with:

1 : =P2(P1

1ATK + AKP

1

1+ 1P

1

1+ )P2

2 : =ATFP2 + P2AF + 1P2

3 : =1

2K1x + P

1

11P1

1

that, in view of (24),

-R1-2X2 I

I X1AT+AX1+Y1BT+BYT1

+1X1+ 0

The term 3 in (27) can be bounded by R2 as

P11

1P11

1

2K1x R2 (28)

Applying again the Schur complement we can express (28)as R2

1

2K1x P

1

1

P11

0

Remark 2. Notice that for the fixed scalar parameters1 and 2 the matrix inequalities (26) become LMIs.They can be solved using packages such as SeDuMi Tool-box, YALMIP Toolbox and the standard MATLAB LMI-toolbox.

Remark 3. The main optimization problem (9) can alsobe solved using the following two-steps procedure:

1) first, we fix the scalar parameters 1, 2 and solve

the problem (9) with respect to the matrix variableswhich satisfy LMI- constraints (26);

2) second, we solve the optimization problem (9) onlywith respect to scalar parameters 1, 2 (under thefound matrix variables).

V. NUMERICAL EXAMPLE

Consider the nonlinear model of a single link flexiblejoint manipulator. The equations of motion for this systemtaken from [18] are:

Iq1 + mgl sin(q1) + K(q1 q2) = 0

Jq2 + Bq2 K(q1 q2) = u

where q1and q2 are the link and motor angles, I is the linkinertia, J is the inertia of the motor, B is the actuatordamping, K is the spring stiffness, m and l are the massand length of the link respectively and u is the controlinput. Define the state variables x1 = q1, x2 = q1,x3 = q2 and x4 = q2 to obtain the following state spacerepresentation:

x=

0 1 0 0-0.2 0 0.2 0

0 0 0 19 0 -9 0

x+

0009

u+

0mgl sin(q

1)

00

with the following state-output mapping

y =

1 0 0 0

x1x2x3x4

+ wy

where wy R2 is an output noise which can be estimated

as follows

wTy Kwy 1, K = 500

Assume also that system nonlinearities are bounded as

f(x, t)Ax2 x12

1.5



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0 10 20 30 40 50 600.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time

q1

Real

Observed

Fig. 1: Angular position of the link.

One can rearrange them to the form (5) with

Kf =

1e-20 0 0 00 1 0 00 0 1e-20 00 0 0 1e-20

, =0

Kx =

1.5 0 0 00 1e 20 0 00 0 1e 20 00 0 0 1e-20

A linear feedback is designed according to the mainresults of the previous section. The obtained parametersK and F realizing the robust output linear controller are

as followsK =

28.27 946.11 616.23 488.42

F =

0.701 0.173 0.539 0.764

TThe obtained performance index (9) is 0.0287.

The figure 1 show the angular position of the controlledsingle link manipulator. As it can be seen from the abovenumerical simulation the closed loop system has a goodstabilization performance, but the convergence processesin general are slow. This situation seems to be usual forrobust linear controllers being applied to nonlinear modelscontaining external noise or disturbances.

Remark 4. For illustrative purposes the system param-eters of the single link flexible joint manipulator wereassumed to be known. If they were unknown but bounded,the parameters Kf and Kx must be changed according tothe corresponding uncertainties.

VI. CONCLUSIONS

Here the stabilization scheme for a class of nonlinear sys-tems under the presence of uncertainties and disturbancesis presented. As a principle part it contains the linearoutput controller using the current state estimates anddesigned based on the invariant ellipsoid technique. It min-imizes the effect of the perturbations and nonlinearities in

the system. Due to nonlinearities and perturbations theconstraints involved into this optimization problem turnsout to be bilinear matrix inequalities. Using appropriatetransformations and fixing the scalar parameters (1 and

2), the bilinear constraints become LMI. The robustnessproperty of this scheme is confirmed by the application tothe single link flexible joint robotic manipulator.

References

[1] D.P. Bertsekas and I.B. Rhodes, On the minmax reachabilityof target set and target tubes, Automatica, vol. 7, pp. 233-247,1971.

[2] D. Glover and F. Schweppe, Control of linear dynamic systemswith set constrained disturbances. IEEE Trans. Automat Con-trol, vol. 16, no. 5, pp. 411-423, 1971.

[3] M.J. Corless and G. Leitmann, Continuous state-feedbackguaranteeing uniform ultimate boundedness for uncertain dy-namic systems, IEEE Trans. Automat. Contr., vol. 26, no. 5,pp. 1139-1144, 1981.

[4] J.C. Doyle, Synthesis of robust controllers and filters, Pro-

ceedings of the 22th IEEE Conference on Decision and Control,pp. 109-114, 1983.

[5] B.A. Francis, J.W. Helton and G. Zames, H-optimal feed-back controllers for linear multivariable systems, IEEE Trans.Automat. Contr., vol. 29, pp. 888-900, 1984.

[6] M.A. Dahleh and J.B. Pearson, l1 optimal feedback con-trollers for MIMO discrete-time systems, IEEE Trans. Au-tomat. Contr., vol. 32, pp. 314-322, 1987.

[7] M.W. Spong and M. Vidyasagar, Robust Linear Compen-sator Design for Nonlinear Robotic Control, IEEE Journal ofRobotics and Automation, vol RA-3, no. 4, 1987.

[8] H.K. Khalil, Nonlinear Systems, Prentice-Hall, EnglewoodCliffs, NJ, 1992.

[9] J. Abedor, K. Nagpal and K. Poola, A Linear Matrix InequalityApproach to Peak-to-Peak Gain Minimization, Int. J. RobustNonlinear Control, vol. 6, pp. 899-927, 1996.

[10] S. Boyd, E.G. Ghaoui, E. Feron, and V. Balakrishnan, Lin-

ear Matrix Inequalities in System and Control Theory, SIAM,Philadelphia, 1994.[11] F. Blanchini, Set invariance in control-a survey, Automatica,

vol. 35, no. 11, pp.1747-1767, 1999.[12] B.T. Polyak and M.V. Topunov, Suppression of bounded ex-

ogenous disturbances: output feedback, Automation and Re-mote Control, vol. 69, no. 5, pp. 801-818, 2008.

[13] M.G. Safonov, K.C. Goh and J.H. Ly, Control system synthesisvia bilinear matrix inequalities, in Proceedings of the 1994American Control Conference, Baltimore, MD, USA.

[14] J. Loefberg, Minimax Approaches to Robust Model PredictiveControl, Ph.D. Thesis, Linkoeping University, 2003.

[15] S. Boyd and L. Vandenberghe, Semidefinite programming re-laxations of non-convex problems in control and combinatorialoptimization, In: Communications, Computation, Control andSignal Processing: A Tribute to Thomas Kailath, Kluwer, 1997.

[16] D. Henrion, J. Loefberg, M. Kocvara and M. Stingl, Solving

polynomial static output feedback problems with PENBMI,In Proceedings of the 44th IEEE Conference on Decision andControl and the European Control Conference, pp. 7581-7586,Seville, Spain, 2005.

[17] A.S. Poznyak, Advanced mathematical tools for automatic con-trol engineers: Deterministic techniques, vol. 1, Elsevier, NewYork, NY, 2008.

[18] M.W. Spong and M. Vidyasagar, Robot Dynamics and Control,John Wiley, New York, 1989.



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