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Copyright IFAC Control Systems Design, Bratislava, Slovak Republic, 2003

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AFFI E ROB ST CO TROLLER DESIG FOR LI EAR PARAMETRIC U CERTAIN SYSTEMS WITH OUTPUT FEEDBACK M. Hypiusova, V. VeseIy

Department ofAutomatic Control Systems. Faculty ofElectrical Engineering and Information Technology, Slovak University ojTechnology, I/kovicova 3.81219 Bratislava. Slovakia. E-mail: hypiusov({j)[email protected].

Abstract: The paper deals with a new procedure for robust output feedback controller design providing affine quadratic stability. The developed procedure leads to a noniterative LMI based algorithm. The proposed affine robust controller applied to a plant consisting of two co-operating DC servomotors was evaluated using affine quadratic and quadratic stability test. Copyright ~ 2003 IFAC Keywords: Robust control. Linear matrix inequality. Affine quadratic stability. Output feedback.

I. I TRODUCTIO

the availability of many approaches and numerical algorithms.

For many real processes a controller design has to cope with the effect of uncertainties. which vel) often causes poor performance or even instability of the closed-loop systems. Considering these facts. the stability and performance robustness are of great practical importance and have attracted a lot of interest for several decades.

The theory of linear matrix inequalities (LMI) (Boyd. et al.. 1994) has been used in several design methods (Benton and Smith. 1999: Li Yu and Jian Chu. 1999). Most of above works present iterative algorithms in which a set of equations or a set of LMls is repeated until certain convergence criteria are met. The authors in (Kose and Jabbari. 1999) have studied conditions under which the designed output feedback controllers can be divided into two stages and the dynamic output feedback can be obtained. Authors in (Benton and Smith. 1999) have proposed an LMI based algorithm. which does not require iteration of LMI problems. The goal is to eliminate the need for iteration b) an appropriate choice of initializing state feedback matrix. The proposed algorithm can be used to robustly stabilize a pol)10pic system via static output feedback.

The output feedback problem is one of the most important open questions of control engineering. In a simple way. the problem can be formulated as follows: for a gi\en complex linear system a robust controller \\ ith output feedback is to be found. which would provide some desirable characteristics to the closed-loop system. or determine that such a feedback does not exist. During the last two decades numerous papers dealing with the design of robust output feedback control schemes to stabilize such systems have been published (Benton and Smith. 1999: Goh. et 0.1.. 1996: Iwasaki. et al.. 1994: Kose and Jabbari. 1999: Li Yu and Jian Chu. 1999: Vesel). 2000). In the above papers. the authors basically conclude that the problem of static output feedback is still open despite

In this paper one approach to the robust controller design has been developed. The new necessary and sufficient conditions to stabilize continuous time systems via static output feedback are used to design the robust controller. The proposed method is based

401

on the results of (Gahinet, el aI., 1996) to design a robust output feedback controller for linear continuous-time systems. which ensures affine quadratic closed loop stability. The proposed method for polytope systems leads to a non-iterative LMl based algorithm. For this method the designer can prescribe the structure of output feedback controller.

is stable for all admissible uncertainties described by (2) and simultaneously guaranteeing the suboptimal solution to the performance index (5)

where Q = QT ~ 0 and R = R T ~ 0 are matrices of compatible dimensions. The nominal model of the system ( I) is

The paper is organized as follows. In Section 2 the problem formulation and some preliminary results are presented. The main results are given in Section 3. In Section 4 the obtained theoretical results are applied to an example. We use a standard notation. A real symmetric positive (negative) definite matrix P is denoted by p > 0 (p < 0). Much of the notation and terminology complies with the references (Benton and Smith. 1999: Boyd. el aI., 1994; Kucera and De Souza.1995).

X(/) = AX(/) + BU(/) Y(/) = CX(/)

The following lemma is well known (Lankaster. 1969). Lemma I. Suppose P > 0 to be the solution of the following Lyapunov matrix equation

2. PROBLEM FORMULATIO A D PRELlMI ARIES

(7)

In the context of robustness analysis and synthesis of robust controllers for linear time invariant systems the following uncertain model is commonly used

X(/)= (A + 8A)x(/) + (B + 8B)u(/)

Y(/)= CX(/)

Then A is stable iff Q > 0 . If such P exists, we say that the matrix A is quadratically stable. A linear time invariant system is stable if and only if it is quadratically stable. It is possible. however. that e.g. linear polytopic systems can be stable without being quadratically stable (Boyd. et aI., 1994).

(I)

where x E Rn. U E R m and yE RI are state. control and output vectors. respectively; A, Band Care known matrices of appropriate dimensions; 8A. 8B are unknown but norm bounded uncertainties. In the next development the matrix affine type uncertain structure will be used

8A

p

s

,=1

,=1

= LE,A, 8B = LE,B, f.,

$ E, $

3. OUTPUT FEEDBACK CONTROLLER DESIG In this paragraph we present a new procedure to design a static output feedback controller for continuous-time systems guaranteeing affine quadratic closed loop stability. This section is concerned with a class of uncertain polytope linear systems that can be described by state-space equations of the form

(2)

£,

where A,. B, are known matrices: £, are unknown parameters. In general. the polytope characterization of uncertainties results in less conservative controller designs than using other characterizations of uncertainty (Boyd. et al.. 1994).

X(/) = A(£ )x(t) + B(E: ~I(I )

= FCx

(8)

where

and £

(3)

such that the closed loop system

x = (.-1 + BFC)x + (8..-1 + 8BFC)x

X(o) = Xo

y(t) = C(£ )x (I)

The problem studied in this paper can be formulated as follows. For linear continuous-time invariant system described by (I) a robust static output feedback controller is to be designed with the control algorithm /I

(6)

A(E:)= Ao +A)£) +A 2£2 +

+Ap£p

B(£)= Bo+B)£) +B 2£2 +

+Bp£p

C(£) = Co +C)£I +C 2£2 +

+C p£p

=

l£).£2 ..... £pJ is a vector of uncertain and

possibly

time-varying rcal parameters; are known fixed matrices.

~.Bo.Co.A,.B),C",,,

Consider that either the matrix B(E:) or C(£) is varied and the other matrix B or C. respectively. is constant.

(4)

402

We also assume that lower and upper bounds for both the parameter values and rates of variation are available. Specifically. e, where

E(f,,£')

f,,£' ,~"r,

and E, E(~"r,), i=I,2, ... ,p

A system is affine quadratically stable if Ac(em ) is stable and if there exist p+ I symmetric matrices Po,···,Pp such that p(e)=PO+~el+P2e2+"'+

+Ppe p > 0 satisfies

(9)

are known constants. In the sequel

Ac(eY P(e) + P(e)Ac{e)+

f P''l", + fe, M , < 0 2

1=1

n = {(W I'W 2'.... wp ): w, E k,,£,}} T = {('l"I' 'l"2"'" 1): 'l", E k"r;}}

(10)

(w, 'l") E n

for all

The closed algorithm

=

FC(e)x

are

and some M i = Mr 2: 0 for the closed loop system analysis using affine quadratic stability test.

Definition J. The linear system (12) is affine quadratically stable if there exist p+ I symmetric matrices Po. PI. .... Pp such that

=

T

= M , 2: 0

The next theorem is the main result of our paper and gives sufficient conditions for output feedback stabilizability of the closed-loop system (12). We consider M I =... = Mp = 0 for the controller synthesis

The following definition has been introduced in (Gahinet. et al., 1996).

dt

Ai + BiFC, and M,

Using matrices M , in inequalities (16) and (17) the conservatism of the affine quadratic stability test we can be reduced.

is given as follows

dV

=

(17)

some nonnegative definite matrices.

(11)

v = x TP{e)x > 0

Ac,

where

loop system (8) with the control

u

x T and

A;'p, +p'A c, +M , 2:0 for i=I,2.... ,p

denote the set of 22p vertices or corners of this parameter box.

(16)

1=1

Theorem 2. The sufficient condition for the output feedback affine quadratic stabilizability of the closed loop system (12) is given as lollows. I. Ac(eJ is stable.

(13)

XT(Ac{eY P(e)+ P(e)Ac(e)+ dP(e))x < 0

2.

dt

There exist R

= R T > 0, Q = QT

> 0 such that

(14)

A(eY P(e)+ P(e )A(e)- P(e )B(e )R- I B(eY P(e)hold for all admissible trajectories of the parameter vector e with conditions (9) where

-C(eY FT RFC(e) + fPk'l"k +Q

(18)

k=1

(B{e Y P{e)+ RFC(e )}t>;1 (B(e YP(e)+ RFC(e))-R

hold for all

Next. we assume that A(e) affinely depends on the

=

[fl +£1 ..... f +£,,] 2 2 p

_(A(E)T P(E)+ P(E )A(E)- P(E )S(E )R- I S(E f P(E)-

Proof of this theorem is omitted. The system represented by (8) is a polytope of linear systems described by a set of its vertices. i.e. in the form

Theorem I. Consider a linear system governed by (12). Let nand T denote sets of vertices of the parameter box (10) and let

m

x T where

- c(Ef FT RFC(E)+ k~tk Tk + QJ

vector e as in (8) and that e and E range in the polytope defined by (9). The sufficient stability conditions for affine quadratic stability of the closed loop system (12) are given in the next Theorem (Gahinet. et al.. 1996).

e

(w. 'l") E n

( 15)

where

denote the average val ue of the parameter vector.

403

N

=2P

DI

=

Ao +flAI +f2 A2 + ...

4. EXAMPLE

El = Bo +flBI +fzB z + ...

Consider the mathematical model of a plant (I) and (2) where €, E (-1.1). i= 1,2

Equations (16), (17) and (18) have given impulse for the following algorithm for computation of the output feedback gain matrix F.

0 -0.215 0 I -1.014 0

Algorithm: I.

Compute ARJ

D,T Po + PoD, - PoB,R- B,T Po + Q ~ 0 1

i = 1.2..... N

2.

(20)

.40=

with respect to Po = P: > 0 . Compute the matrix P J from the following inequalities

D,T(pO+€I~)+(PO +€,P,)D, +'I~-(po +€I~)E,R-JE,T(pO+€IPJ)+ Q ~ 0 i= 1,2, ... ,N

AI

=

PO+€I~ ~O

3.

forall (€I"I)dhT and Ai~+~A, ~O Compute the matrix P2 from the following inequalities

D,T(PO+€JPJ +€zPz)+(po +€JPJ +€zPz)D,-(po +€JP1 +€zPZ)EiR-JE,T(PO+€I~ +€zpz )+ +TIP1 +'2P2 +Q~O

A1

for all (€Io '1'€2 '2)E n x T . i = 1.2, ... , Nand

=

0

A2T P2 +P2A2 ~O

(B{t:

(21 )

<1> < = -(A{t:

Yp(€) + P{t: )A(€)-

J - P{t: )B{t:)R- B(€

Y p(€)+ ktPk 'k + Q)

forall (€,')EnxT.

0

0

0

0

0 0

0

0

0

0

0

0

0

0 -0.164 0 I -0.814 0

0

0

0

0

0

0

0 0 -0.228

0

0

0

0

0

0

I -0.822

0

0 -0.0250 0

0

0

0

0

0 -0.1395 0

0

0

0

0

0

0

0

0

0 0 0 0.0188 0

0

0

0

0

0

0 -0.0938 0 0 -0.2911 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0.0208 0 0 0 0 0 -0.0333 0 0 0 -0.1173

0 0.0125 0

0

0

0

0

0

0 0.0594 0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0 0.0116 0 0 0.0308 0

0

0

0

0

0

0

0

0

0 -0.0156 0 0 -0.0565 0

0

0

0

0

0

0

0 0.0434

0

0

0

0

0

0

0 0.1258

o o

o o

-0.1028

-0.0841

0.0625 -0.0798

0

o o

o o

-0.0462

o o

0.0016

o

0.0072

o

o o

0.3676

o

0.0770

0.2448

o

-0.0050

-0.0094

o o

o

0.0019

o

0.0151

o o

ote that if the solutions to (20) and (21) are feasible for ~ = P2 = ... = 0 and the closed loop system (12) is quadratically stable 10 all 11' vertices of the pol~tope then the system is quadratically stable for all uncertainties given b) the vector € .

0

0

-0.0287

where

0 0

0

0.0478

Y p(€)+ RFC }D;' (B(€ Yp(€) + RFC)- R < 0

0 0

0

0.3148

Last Step: Compute the output feedback gain matrix F

0 0

0

0

Po +€I~ +€2 P2 >0,

0

0 0 -0.261 0 0 I -0.911 0

-0.0121 -0.0030

o o

-0.0091

-0.0030

o o

0.0449

0

I 0

0

I 0

-0.0640

o o o

0

0.0189

o

I

0 I

In case of a PI controller design the extended state space model is to be used. This extension is carried out as follows

404

Table I. Controllers designed for various robustness and performance requirements q

[1.~10

100 In case of decentralized control with 2 inputs and 4 outputs, application of the static output feedback yields the matrix F in the following form

5

o 0.5

and the ith obtained transfer function of the PI subcontroller is as follows

G RI) =

F,H2 F" +

0.676 0

1.~72]

-0.27±0.57i

0.402

0~86]

-0.13±0.37i

0~77]

-0.12±O.55i

[0~75

0 0.491

[0~62

0 0.935 0.696 0

0

s

F

controllers stabilize the polytope vertices and one point inside the polytope, they ensure the afline quadratic stability for the closed loop system with uncertainties. Table 2 summarizes robust analysis results of closed loop systems under affine quadratic controllers. Quadratic stability test (QS) and affine quadratic stability test (AQS) were applied. Presented QS and AQS results indicate percentages by which it is possible to extend the maximal value of the parameters

0

4559

Table 2. Robust analysis of affine controllers by quadratic stability test and affine quadratic stability test

Some of the calculated controller parameters for various settings of the Q and R matrices along with the maximum close loop eigenvalues are listed in the Table I, where e, = I. Q = qI and R = rI. As the

uncertainty

I

Max. close loop eigenvalues

F

I'

elm

= maxl~,11

[1.~10

0.676 0 4.559 0

1.~72]

[0~75

0.402 0 0.491 0

0~86]

[O~62

0 0.935 0.696 0

0~77 ]

QS

AQS

[%]

[%]

173.14

246.19

102.15

245.31

82.13

246.09

REFERENCES

still

Benton. I.E. and D. Smith (1999). A non iterative LMI based algorithm for robust static output feedback stabilization. IJc. Vot. 72. pp. 13221330.

preserving either closed loop quadratic stability or closed loop affine quadratic stability. respectively. For the third controller QS is less than 100 %. which means that the closed loop quadratic stability is satisfied only for e/II/ = 0.8213.

Boyd. S.. L. El Ghaoui. E. Feron and V. Balakrishnam (1994). Linear Matrix Inequalities in System and Control Theory. SIAM. 15. Gahinet. P.. P. Apkarian and M. Chilali (1996). Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Trans. on AC. Vol. 42. pp. 436-442.

The results show that the affine controllers give good performance and larger region of affine quadratic stabi I ity. Further it is possible to see that the affine quadratic stabi I ity test is less conservative than the quadratic stability test.

Goh. K.Coo M.G. Safonov and J.H. Ly (1996). Robust synthesis via bilinear matrix inequalities. 1nl. Journal of Robust and Nonlinear Control. Vot. 6. pp. 1079-1095.

5. CONCLUSIO

Iwasaki. Too R.E. Skelton and J.C Geromel (1994). Linear quadratic suboptimal control with static output feedback. Systems and Control Lellers. Vot. 23. pp. 421-430.

The main aim of this paper has been to present a new method for solving the problem of designing robust controllers with output feedback via non-iterative LMI approach guaranteeing the affine quadratic stability of the uncertain closed loop system. Application of this approach to a real laboratory model has resulted in a high robustness measure.

Kose. I.E. and F. Jabbari (1999). Robust control of linear systems with real parametric uncertainty. Automatica. Vol. 35, pp. 679-687. Kucera. V. and C.E. De Souza (1995). A necessary and sufficient conditions for output feedback

405

stabilizability. Automatica. Vol. 31. pp. 13571359. Lankaster, P. (1969). Theory of matrices. Academic press. ew-York. London. Li Yu and Jian Chu (1999). An LMI approach to guaranteed cost control of linear uncertain timedelay systems. Automatica. Vol. 35, pp. 11551159. Vesely, V. (2000). Output robust controller design for linear parametric uncertain systems. In: The 3rd IFAC Symposium on Robust Contra! Design. Praha. Czech Republic.

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