Observer-Based Robust Controller Design for a Linear System with Time-Varying Perturbations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

213, 642]661 Ž1997.

AY975566

Observer-Based Robust Controller Design for a Linear System with Time-Varying Perturbations Shyi-Kae Yang Department of Mechanical Engineering, Wu-Feng Junior College of Technology and Commerce, Minshiung, Chiayi, Taiwan, Republic of China

and Chieh-Li Chen Institute of Aeronautics and Astronautics, National Cheng-Kung Uni¨ ersity, Tainan, Taiwan, Republic of China Submitted by Harold L. Stalford Received July 5, 1996

In this paper, an observer-based robust controller design problem for the linear uncertain system is considered, where the uncertain system is subject to the norm-bounded and the structured time-varying uncertainties. Using the Riccati approach, full-order and reduced-order observers are obtained to reconstruct the unmeasurable states so that the robust stability of the uncertain system can be guaranteed with the state feedback control law. Q 1997 Academic Press

NOTATION

l i Ž A.: eigenvalue of matrix A s  a i j 4 . Rew ax: real part of scalar a. < A <: absolute value of matrix A and < A < s < a i j <4 . A y B ) ŽG.0: a i j y bi j ) ŽG.0. 5 A 5 2 : matrix 2-norm and 5 A 5 2 s max i l i Ž ATA . . m 2 Ž A.: matrix measure of A and m 2 Ž A. s max i  l i wŽ A q AU .r2x4 .

'

642 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

OBSERVER-BASED ROBUST CONTROL

643

1. INTRODUCTION The robust controller design of uncertain systems has been widely studied these past years. Lots of design approaches have been proposed based on the consideration of different types of uncertainties. Among these approaches, the Lyapunov approach w2, 5, 10x and the Riccati approach w13, 15, 17, 18x are two main techniques to deal with the system with matchingrmismatching certainties. However, these two design approaches are based on the assumption that the system states are available such that the state feedback law can be applied to control the uncertain systems. In the case where system states cannot completely be measured, a state observer to reconstruct the states is usually introduced, and the influence of observer design on the stability of uncertain systems becomes an interesting issue in robust control research. The robustness of the observer-based control system has been studied in w3, 16x using the sensitivity analysis. The robustness designs for the fullorder and reduced-order state observers are also proposed in w1, 14x, respectively. However, both design techniques are based on the nominal system; the robust stability is then guaranteed by checking the sufficient conditions obtained by the Lyapunov function. In the case where the uncertain system is subject to matching uncertainties, the robust stability condition is proposed using the results of the constrained Lyapunov problem w9x. When the system uncertainty satisfies the rank 1 assumption, a full-order state observer is proposed in w12x by using the Riccati approach. The observer-based robust control problem subject to matching uncertainties is further studied in w11x using the Riccati design technique. In this paper, systems subject to incomplete matching uncertainties w6x described by the norm-bounded and structured manners are considered. Using the matrix 2-norm and the matrix measure, the robustness conditions corresponding to these two uncertainties are obtained in the same manner. Based on the Riccati approach, the robust stability conditions of the observed-based control system for both full-order and reduced-order state observers are proposed. Numerical examples are given to illustrate the proposed results.

2. PRELIMINARY In this paper, the matrix 2-norm and the matrix measure w7x are used to obtain the robust stability conditions for the system with normboundedrstructured uncertainties. Some properties of the matrix 2-norm and the matrix measure are listed in Appendix A. Based on their common

644

YANG AND CHEN

properties and for simplicity, the symbol GŽ . is used to represent 5 5 2 or m 2 Ž .. Thus, GŽ . satisfies the following properties. v

v

v

v

Property 1. Ž1. GŽ I . s 1 and GŽ ATA. s GŽ AAT . s 5 A 5 22 . Ž2. GŽ cA. s cGŽ A., ;c G 0. Ž3. yGŽyA. F Rew l i Ž A.x F GŽ A.. Ž4. GŽ A q B . F GŽ A. q GŽ B .. Ž5. If B G < A < G 0, then GŽ B . G GŽ< A <. G GŽ A.. Ž6. If A s AT , then GŽ A. s lmax Ž A. and yGŽyA. s l minŽ A.. LEMMA 1. Let A s AT , E g R n=n, and x TAx G 0, ; x g R n. Then x T Ž A q E . x G 0 if lmin Ž A . G G Ž yE . . Ž 1. Proof. For GŽ . ' 5 5 2 , it is clear that condition Ž1. is satisfied if lminŽ A. G 5 E 5 2 s 5 y E 5 2 . On the other hand, x T Ž A q E . x G 0 if lminŽ A q E . G 0. Therefore, using the property of matrix measure Žsee Appendix A., it is obtained that v

v

lmin Ž A q E . G ym 2 Ž yA y E . G y m 2 Ž yA . q m 2 Ž yE . . Therefore, if

lmin Ž A . s ym 2 Ž yA . G m 2 Ž yE . , then x T Ž A q E . x G 0.

Q.E.D.

Lemma 1 plays a key role in the robust analysis when E represents uncertainty. For example, if E is norm-bounded, 5 E 5 2 F q, then condition Ž1. becomes l minŽ A. G q. Alternatively, if E is absolute-value-bounded, < E < F qU g R n= n, then condition Ž1. becomes lminŽ A. G q m 2 ŽU .. Also, if the structured uncertainty, where E s Ý ris1 qi Ei , < qi < F q and Ei represents a perturbed structure, is considered, then condition Ž1. can be formulated as lminŽ A. G q m 2 ŽÝ ris1 < Ei <.. Therefore, using Property 1 and Lemma 1, the conditions of robust stability for uncertain systems with norm-bounded uncertainties or structured uncertainties can be obtained, respectively.

3. OBSERVER-BASED ROBUST CONTROLLER DESIGN Full-Order State Obser¨ er Consider the uncertain system

˙x Ž t . s Ž A q D A Ž t . . x Ž t . q Ž B q D B Ž t . . u Ž t .

Ž 2a .

y Ž t . s Ž C q DC Ž t . . x Ž t . ,

Ž 2b .

OBSERVER-BASED ROBUST CONTROL

645

where the incomplete matching condition is considered, that is, system uncertainties D B and DC satisfy the matching conditions D B Ž t . s BD b Ž t . ,

with G Ž DTb q D b . - 2,

Ž 3a .

and DC Ž t . s D c Ž t . C,

Ž 3b .

and D A relaxes the matching condition. Suppose that Ž A, B . is controllable and Ž A, C . is observable. There exist positive definite symmetric matrices Pc , Q c , Po , Q o and positive scalars sc , so , rc , r o such that the following AREs are satisfied, T Ž A q sc I . Pc q Pc Ž A q sc I . y 2 Ž 1 y a . rc Pc BB T Pc s yQc , Ž 4 . T Ž A q so I . Po q Po Ž A o q so I . y 2 r o Po C T CPo s yQ o ,

Ž 5.

where G Ž DTb q D b . 2

- a - 1.

The observer is described as

˙z s Az q Bu q L Ž y y Cz . ,

with L s Ž r o q k o . Po C T , k o ) 0, Ž 6 .

where the Po satisfy the ARE Ž5.. Let e s x y z. Then the error dynamic is

˙e s Ae q D Ax q BD b u y LCe y LD c Cx.

Ž 7.

Remark 1. In Eq. Ž4., the major purpose in introducing a is to overcome the input matching uncertainty. The parameter k o in Eq. Ž6. is applied to overcome the output matching uncertainty in an explicit manner. For simplicity, the form L s Ž r 0 q k 0 . P0 C T is applied instead of L s Ž P0 C Ž r 0 q k 0 .y1 .T . Apply the control u s yKz,

Ž 8.

where K s rc B T Pc and rc and Pc satisfy the ARE Ž4.. Then

˙e s Ž D A y rc BD b B T Pc y Ž r o q k o . Po C TD c C . x q Ž A q rc BD b B T Pc y Ž r o q k o . Po C T C . e.

Ž 9.

646

YANG AND CHEN

Thus, the Ž x, e . augmented system can be described as

˙x s A11 A 21 ˙e

A12 A 22

x , e

Ž 10 .

where A11 s A q D A y rc B Ž I q D b . B T Pc , A12 s rc B Ž I q D b . B T Pc , A 21 s D A y rc BD b B T Pc y Ž r o q k o . Po C TD c C, A 22 s A q rc BD b B T Pc y Ž r o q k o . Po C T C. Let the Lyapunov function candidate be V s w xT

eT x

Pc

0

0

Poy1

x . e

Ž 11 .

Then V˙ s w x T eT x

½

A11 A 21

A12 A 22

T

Pc

0

0

Poy1

q

Pc

0

0

Poy1

A11 A 21

A12 A 22

s Ux x q Ux e q Ue x q Ue e ,

5

x e

Ž 12 .

where Ux x s x T Ž AT Pc q Pc A y 2 rc Pc BB T Pc q D AT Pc q Pc D A yrc Pc B Ž DTb q D b . B T Pc . x s x T Ž yQc y 2 sc Pc y 2 a rc Pc BB T Pc q D AT Pc q Pc D A yrc Pc B Ž DTb q D b . B T Pc . x Ux e s x T Ž D AT Poy1 q rc Pc B Ž I q D b . B T Pc y rc Pc BDTb B T Poy1 y Ž r o q k o . C TDTc C . e Ue x s e T Ž Poy1D A q rc Pc B Ž I q D b . B T Pc y rc Poy1 BD b B T Pc

T

y Ž r o q k o . C TD c C . x Ue e s e T Ž AT Poy1 q Poy1A y 2 r o C T C y 2 k o C T C qrc Ž Pc BDTb B T Poy1 q Poy1 BD b B T Pc . . e s e T Ž yPoy1 Q o Poy1 y 2 so Poy1 y 2 k o C T C qrc Ž Pc BDTb B T Poy1 q Poy1 BD b B T Pc . . e.

OBSERVER-BASED ROBUST CONTROL

647

Using Lemma B1 Žsee Appendix B., T

Ux e q Ue x s rc x T Pc B Ž I q D b . B T Pc e q e T Pc B Ž I q D b . B T Pc x y rc x T Pc BDTb B T Poy1 e q eT Poy1 BD b B T Pc x y Ž r o q k o . x T C TDTc Ce q eT C TD c Cx q x TD AT Py1 0 e q eT P0y1D Ax F rc

1

T

x T Pc B Ž I q D b . Ž I q D b . B T Pc x q b 1 eT Pc BB T Pc e

b1

q

1

b2

x T Pc BDTb D b B T Pc x q b 2 eT Poy1 BB T Poy1 e

q Ž r o q k o . b4 x T x q q

1

b3

1

b4

eT C TD c CC TDTc Ce

x TD ATD Ax q b 3 eT Poy1 Poy1 e.

Ž 13 .

Then V˙ Fy x T Q c q2 sc Pc yD AT Pc yPc D Ay Ž r o qk o . b4 Iy

^

`

1

b3

D ATD A x

_

dV1

y x T Pc B 2 a rc I y rc Ž DTb q D b . yrc

ž

1

b1

T Ž I q Db. Ž I q Db. q

1

b2

DTb D b

/

B T Pc x

y eT Poy1 Q o Poy1 q 2 so Poy1 y b 3 Poy1 Poy1 < yrc Ž Pc BDTb B T Poy1 q Poy1 BD b B T Pc q b 1 Pc BB T Pc q b 2 Poy1 BB T Poy1 . e y eT C T 2 k o I y

ro q k o

b4

D c CC TDTc Ce.

Ž 14 .

THEOREM 1. The obser¨ er-based control system described abo¨ e is robustly stable if there exist positi¨ e definite symmetric matrices Pc , Q c , Po , Q o which satisfy AREs Ž4. and Ž5. with gi¨ en positi¨ e scalars a , sc , so , rc , r o , k o

648

YANG AND CHEN

and bi , i s 1, . . . , 4, such that the following conditions are simultaneously satisfied:

lmin Ž Q c q 2 sc Pc . G G Ž D AT Pc q Pc D A . q

Ž i.

1

b3

G Ž D ATD A .

q Ž r o q k o . b4 .

Ž 15a .

Ž ii . 2 a G G Ž DTb q D b . q

1

b1

GŽŽ I q Db . Ž I q Db .

T

1

. q b G Ž DTb D b . . 2

Ž 15b . lmin Ž Poy1 Q o Poy1 q 2 so Poy1 .

Ž iii .

G rc G Ž Pc BDTb B T Poy1 q Poy1 BD b B T Pc . 5 22 . q rc Ž b 1 5 B T Pc 5 22 q b 2 5 Poy1 B 5 22 . q b 3 5 Py1 0 ko G

Ž iv.

r o G Ž D c CC TDTc . 2 b4 y G Ž D c CC TDTc .

.

Ž 15c . Ž 15d.

Proof. From inequality Ž14. and Lemma 1, V˙ G 0 if

lmin Ž Q c q 2 sc Pc . G G D AT Pc q Pc D A q

ž

1

b3

D ATD A q Ž r o q k o . b4 I .

/

Since

ž

G D AT Pc q Pc D A q

1

b3

D ATD A q Ž r o q k o . b4 I

F G Ž D AT Pc q Pc D A . q

1

b3

/

G Ž D ATD A . q Ž r o q k o . b4 ,

therefore, if

lmin Ž Q c q 2 sc Pc . G G Ž D AT Pc q Pc D A . q

1

b3

G Ž D ATD A .

q Ž r o q k o . b4 then dV1 G 0. Similarly, conditions Žii. ] Živ. can be obtained from the other terms in inequality Ž14.. Q.E.D.

OBSERVER-BASED ROBUST CONTROL

649

Remark 2. It is obvious that the system uncertainty D A is overcome by Eq. Ž15. and the choices of a and k o in Eqs. Ž15b. and Ž15d. are used to overcome the effects due to the inputroutput matching conditions. The extra design parameters a , k o , and bi , i s 1, . . . , 4, in association with the LQG design technique are introduced and the remaining works are to choose suitable parameters mentioned above to satisfy the conditions in Theorem 1. In Theorem 1, the choice of the pair Ž b 1 , b 2 . will be based on a compromise between each other as well as the pair Ž k o , b4 . to guarantee the robust stability for the controller design. In the following corollary, optimal parameters are derived to obtain a simpler condition for the controller design. COROLLARY 1. The control system is robustly stable if there exist p.d. symmetric matrices Pc , Q c , Po , Q o which satisfy AREs Ž4. and Ž5. with positi¨ e scalars a , sc , so , rc , r o , k o , and b 3 such that

Ž i.

lmin Ž Q c q 2 sc Pc . G G Ž D AT Pc q Pc D A . q 2 r o G Ž D c CC TDTc . q

1

b3

G Ž D ATD A .

and

Ž 16a .

Ž ii . lmin Ž Poy1 Q o Poy1 q 2 so Poy1 . G rc G Ž Pc BDTb B T Poy1 q Poy1 BD b B T Pc . rc

5 22 q q b 3 5 Py1 0

2 a y G Ž DTb q D b . = GŽŽ I q Db. Ž I q Db.

ž

T 0.5

.

qG Ž DTb D b .

0.5

5 B T Pc 5 2

5 Poy1 B 5 2

2

/ Ž 16b .

with 1 ) a ) GŽ DTb q D b .r2. Proof. From condition Ži. in Theorem 1, to minimize d 1 ' b4 Ž r o q k o . and to satisfy Ž15d., k o is chosen as ko s

r o G Ž D c CC TDTc . 2 b4 y G Ž D c CC TDTc .

.

Substitute k o into d 1 ,

d1 s

r o b42 2 b4 y G Ž D c CC TDTc .

.

650

YANG AND CHEN

To minimize d 1 ,

­ ­b4

Ž d 1 . s 0 « b4 s G Ž D c CC TDTc .

k o s ro .

and

Then, the minimum of d 1 is

d 1 min s 2 r o G Ž D c CC TDTc . . Therefore, Ž15a. and Ž15d. can be reduced to

lmin Ž Q c q 2 sc Pc . 1

G G Ž D AT Pc q Pc D A . q 2 r o G Ž D c CC TDTc . q

G Ž D ATD A . .

b3

Similarly, to minimize d 2 ' b 1 5 Pc 5 22 q b 2 5 Poy1 5 22 and to satisfy Ž15b., the minimum of d 2 is Žsee Lemma B2 in Appendix B.

d 2 min s

1 2 a y GŽ

DTb

GŽŽ I q D . Ž I q D . qD . ž b

T 0.5

b

.

5 B T Pc 5 2

b

qG Ž DTb D b .

0.5

2

5 Poy1 B 5 2 .

/

Therefore, Ž15b. and Ž15c. can be reduced to

lmin Ž Poy1 Q o Poy1 q 2 so Poy1 . 5 22 G rc G Ž Pc BDTb B T Poy1 q Poy1 BD b B T Pc . 2 q b 3 5 Py1 0 q

rc 2 a y GŽ

DTb

GŽŽ I q D . Ž I q D . qD . ž b

b

T 0.5

.

qG Ž DTb D b . The proof is complete.

5 B T Pc 5 2

b

0.5

2

5 Poy1 B 5 2 .

/

Q.E.D.

Remark 3. Ži. The choice of parameter b 3 will affect the assessment of the robust stability. That is, a suitable choice of b 3 may reduce the conservatism in the determination of robust stability for the controller and observer design. In fact, let e1 and e2 represent the difference of the left hand side term and the sum of the right hand side terms of conditions Ž16a. and Ž16b., respectively. The determination of robust stability can be obtained from the plot of e1 and e2 with respect to b 3 .

OBSERVER-BASED ROBUST CONTROL

651

TABLE I Determination of GŽv. for Different Types of Uncertainty Type of uncertainty GŽv.

5 D A5 2 F q

< D A < F qU

D A s Ý ris1 qi Ei , < qi < F q

GŽ D AT Pc q Pc D A. GŽ D ATD A.

2 q 5 Pc 5 2 q2

q m 2 ŽU T < Pc < q < Pc < U . q 2m 2 ŽU T U .

q m 2 ŽÝ ris1 < EiT Pc q Pc Ei <. q 2m 2 ŽÝ ri, js1 < EiT Ej <.

Žii. In Corollary 1, the calculation of GŽ . depends on the type of uncertainty. Table I lists some results of GŽ . for different types of uncertainty Žsee Lemma A1 in Appendix A.. Different forms of GŽ . can be calculated in a similar way. v

v

v

Reduced-Order State Obser¨ er For the reduced-order state observer design, it is assumed that C in Ž3b. is C s Ip

0 ,

I p g R p= p

is an identity matrix.

Ž 17 .

That is, there are p states, represented as x a , which can be measured from the system output and n y p states, represented as x b , which are required to be reconstructed by the state observer. Let As

A aa Ab a

A ab , Abb

Bs

Ba , Bb

Ž 18a .

where A a a g R p= p , A a b g R p= Ž ny p . , A b a g R Ž ny p .= p , A b b g RŽ nyp.=Ž nyp., Ba g R p= m , Bb g RŽ nyp.=m , and uncertainty D A has the same partition described as DAs

D A aa D Ab a

D A ab . D Abb

Ž 18b .

Then the system can be rewritten as

˙x a s Ž A aa q D A aa . x a q Ž A ab q D A ab . x b q Ba Ž I q D b . Ž 19a . ˙x b s Ž A b a q D A b a . x a q Ž A b b q D A b b . x b q Bb Ž I q D b . Ž 19b . y s Ip

0

xa s xa. xb

Ž 19c .

To reconstruct state x b , let the observer be

˙ˆx b s A b b ˆx b q A b a x a q Bb u q L Ž ˙x a y A aa x a y Ba u y A ab ˆx b . , Ž 20.

652

YANG AND CHEN

where L is the observer gain. Substitute Ž19a. into Ž20.. Then

˙x b s A b b ˆx b q A b a x a q Bb u ˆ q L Ž D A aa x a q D A ab x b q Ba D b u q A ab x b y A ab ˆ xb . .

Ž 21 .

Let e s x b y ˆ x b . Then the error dynamics between x b and ˆ x b is

˙e s Ž A b b y LA ab . e q Ž D A b a y LD A aa . x a q Ž D A b b y LD A ab . x b q Ž Bb y LBa . D b u.

Ž 22 .

To simplify the representation, define L I ' w yL I x 0 Qb ' I ny p

and

Qa '

Ip 0

,

w Q a Q b x s In .

Ž 23 .

˙e s L I AQb e q L I D Ax q L I BD b u.

Ž 24 .

with

Then, Ž22. becomes

Apply the control law u s yK

xa

s y Ž Kx y KQb e . .

ˆx b

Ž 25 .

Then the augmented system similar to Ž11. has A11 s Ž A y BK . q Ž D A y BD b K . , A 21 s L I Ž D A y BD b K . ,

A12 s B Ž I q D b . KQb ,

A 22 s L I AQb q L I BD b KQb .

Ž 26 .

From the assumption that Ž A b b , A ab . is observable, there exist p.d. symmetric matrices Po , Q o and positive scalars so , r o such that the following ARE is satisfied: T Ž A b b q so I . Po q Po Ž A b b q so I . y 2 r o Po ATab A ab Po s yQ o . Ž 27 .

Since there is no output matching uncertainty, let L s r o Po ATab . Then Ž27. becomes Poy1 L I AQb q Ž L I AQb . Poy1 s yPoy1 Q o Poy1 y 2 so Poy1 .

T

Ž 28 .

OBSERVER-BASED ROBUST CONTROL

653

If the same linear state feedback gain K and Lyapunov function candidate is chosen as Ž8. and Ž11. Žbut Poy1 satisfies ARE Ž27.., then V˙ s Ux x q Ux e q Ue x q Ue e ,

Ž 29 .

where Ux x s x T Ž yQc y 2 sc Pc y 2 a rc Pc BB T Pc q D AT Pc q Pc D A yrc Pc B Ž DTb q D b . B T Pc . x Ux e s x T

ž Ž D A y r BD c

bB

T

T

Pc . LTI Poy1 q rc Pc B Ž I q D b . B T Pc Q b e T

Ue x s e T Poy1 L I Ž D A y rc BD b B T Pc . q rc QTb Pc B Ž I q D b .

ž

Ue e s e

T

Ž

yPoy1 Q o Poy1

y

T

/ B P /x T

c

2 so Poy1

qrc Ž QTb Pc BDTb B T LTI Poy1 q Poy1 L I BD b B T Pc Q b . . e. Using Lemma B1, V˙ F yx T Q c q 2 sc Pc y D AT Pc y Pc D A y

1

b3

D ATD A x

y x T Pc B 2 a rc I y rc Ž DTb q D b . yrc

ž

1

b1

T Ž I q Db. Ž I q Db. q

1

b2

DTb D b

/

B T Pc x

y eT Poy1 Q o Poy1 q 2 so Poy1 yrc Ž QTb Pc BDTb B T LTI Poy1 q Poy1 L I BD b B T Pc Q b . yrc Ž b 1 QTb Pc BB T Pc Q b q b 2 Poy1 L I BB T LTI Poy1 . y b 3 Poy1 L I LTI Poy1 e. Similar to Corollary 1, the following result is obtained. COROLLARY 2. The control system is robustly stable if there exist p.d. symmetric matrices Pc , Q c , Po , Q o which satisfy AREs Ž4. and Ž27. with gi¨ en positi¨ e scalars a , sc , so , rc , r o , k o , and b 3 such that

Ž i . lmin Ž Qc q 2 sc Pc . G G Ž D AT Pc q Pc D A . q

1

b3

G Ž D ATD A . and

Ž 30a .

654

YANG AND CHEN

Ž ii . lmin Ž Poy1 Q o Poy1 q 2 so Poy1 . G rc G Ž QTb Pc BDTb B T LTI Poy1 q Poy1 L I BD b B T Pc Q b . 52 q b 3 5 Py1 0 LI 2 q

rc 2 a y G Ž DTb q D b . = GŽŽ I q Db. Ž I q Db.

ž

qG Ž DTb D b .

T 0.5

.

0.5

5 B T Pc Q b 5 2

5 Poy1 L I B 5 2

2

/ Ž 30b .

with 1 ) a ) GŽ DTb q D b .r2. Remark 4. In Eq. Ž20., the use of ˙ y is unacceptable because the differentiation will amplify the noise associated with the output. Therefore, a new state defined as z s ˆ x b y Ly is introduced when the observer is implemented w8x. However, this will not affect the results in Corollary 2.

4. NUMERICAL EXAMPLES EXAMPLE 1. Consider an uncertain system with its nominal system matrices described as As

1 4

2 , 4

Bs

1 , 2

Cs w3

2 x,

and the system uncertainties are D AŽ t . s

q1 Ž t .

2 q2 Ž t .

4 q3 Ž t .

4 q4 Ž t .

,

D B Ž t . s BD b Ž t . ,

DC Ž t . s D c Ž t . C, where < qi Ž t .< F 0.015, i s 1, . . . , 4, < D b Ž t .< F 0.015, and < D c Ž t .< F 0.015. For the observer design, choose Qo s

0.01 0

0 , 90.07

r o s 2.5,

and

so s 0.805,

and Q c s I, rc s 10, sc s 0.4, a s 0.5. The plot of e1 and e2 with respect to b 3 is shown in Fig. 1. It is found that there exists b 3 such that Corollary

OBSERVER-BASED ROBUST CONTROL

655

FIG. 1. The plot or e1 and e2 with respect to b 3 for Example 1.

1 is satisfied. Therefore, the resulting observer gain and state feedback gain are K s w 3.1197 6.2423x

and

Ls

1.0539 . 23.7791

EXAMPLE 2. The simple dynamic model w8x of a satellite can be considered as a system with two masses connected by a spring and a damper with torque constant k and viscous damping constant d; see Fig. 2. Choose the state vector x T s u 2 u 1 u˙2 u˙1 . Then the state space representation of the model is

˙x s Ž A q D A . x q Bu y s Cx,

656

YANG AND CHEN

FIG. 2. Physical model of a satellite.

where the system matrices are 0 0

0 0 k

k

A s y J2

1 0 d y J2

J2

k

y

J1

k

d

J1

J1 1 0

Cs

0 1 d

y 0 1

0 0

Bs

,

J2 d

0 0 0 1

,

J1

J1 0 0

with J1 s 1, J 2 s 0.2, k s 0.15, d s 0.0153. Assume that k and d are subject to perturbation as the temperature fluctuates. Then the structured uncertainty can be described as D A s qk Ek q qd Ed , where 0 0 1 Ek s y J 2 1 J1

0 0 1 J2 y

1 J1

0 0

0 0

0

0 ,

0

0

0 0

0 0

Ed s 0

0

0

0

0 0 1 y J2 1 J1

0 0 1 J2 y

,

1 J1

< qk < F 0.012, < q d < F 0.0106. Using the reduced-order observer design technique in Corollary 2, Fig. 3 can be obtained, and the observer-based controller which guarantees the

OBSERVER-BASED ROBUST CONTROL

657

FIG. 3. The plot of e1 and e2 with respect to b 3 for Example 2.

robust stability is obtained with K s w y2.8148 16.9570 12.9901 11.6424x ,

Ls

28.2461 0.0229

0.0229 28.2766

by choosing Q c s 20 I, rc s 5, sc s 0, a s 0.5, Q o s 40 I, r o s 40, so s 0.

5. CONCLUSIONS Systems subject to incomplete matching uncertainties described by the norm-bounded and structured manners are considered in this paper. Using the Riccati design technique, an observer-based robust controller for uncertain systems where states cannot be completely measured is studied. The matrix 2-norm and the matrix measure are used as tools to obtain the conditions for robust system stability where a linear state feedback control law is applied and the states are reconstructed from the full-order or reduced-order observer. Numerical examples are given to illustrate the proposed results.

658

YANG AND CHEN

APPENDIX A Let A, B g C n= n. The properties of the matrix 2-norm and the matrix measure are listed as follows. Property A1. Ž1. 5 cA 5 2 s < c < 5 A 5 2 , ;c g C. Ž2. 5 A q B 5 2 F 5 A 5 2 q 5 B 5 2 . Ž3. For B G < A < G 0, 5 B 5 2 G 5 < A < 5 2 G 5 A 5 2 . Property A1 Ž3. can be derived from the property of spectral radius, r Ž . ' max i < l i Ž .<, that v

v

B T B G < A < T < A < G ATA « r Ž B T B . G r Ž < A < T < A < . G r Ž ATA . . Property A2 w4x. Ž1. m 2 Ž cA. s c m 2 Ž A., ;c G 0. Ž2. y5 A 5 F ym 2 ŽyA. F Rew l i Ž A.x F m 2 Ž A. F 5 A 5. Ž3. m 2 Ž A q B . F m 2 Ž A. q m 2 Ž B .. Ž4. For B G < A < G 0, m 2 Ž B . G m 2 Ž< A <. G m 2 Ž A.. LEMMA A1 ŽProof of Results of Table I.. Case Ži..

< D A < F qU.

G Ž D A Pc q Pc D A . s m 2 Ž D AT Pc q Pc D A . F m 2 Ž < D AT Pc q Pc D A < . T

F m 2 Ž < D A < T < Pc < q < Pc < < D A < . F q m 2 Ž U T < Pc < q < Pc < U . , G Ž D ATD A . s m 2 Ž D ATD A . F m 2 Ž < D ATD A < . F m 2 Ž < D A < T < D A < . F q 2m 2 Ž U T U . . Case Žii..

D A s Ý ris1 qi Ei , < qi < F q. r

G Ž D AT Pc q Pc D A . s m 2

qi Ž Ei Pc q Pc Ei .

žÝ žÝ žÝ is1 r

F m2

< qi < < Ei Pc q Pc Ei <

is1

r

/ /

< Ei Pc q Pc Ei < s q m 2

F m2 q

/

is1

r

G Ž D ATD A . s m 2

ž

r

i , js1 r

F q 2m 2

/ ž

qi q j EiT Ej F m 2

Ý

žÝ

i , js1

< EiT Ej < .

/

Ý i , js1

r

ž

Ý < Ei Pc q Pc Ei < is1

< qi q j < < EiT Ej <

/

/

,

OBSERVER-BASED ROBUST CONTROL

659

APPENDIX B Let M, N g R m= n and x g R n, y g R m. Then T

LEMMA B1.

"2 x T M T Ny F b x T M T Mx q

1

b

y T N T Ny, b ) 0.

Proof. The result is directly obtained from the inequality

ž

1

'b Mx " '

b

T

Ny

/ž

s b x T M T Mx q

1

'b Mx " '

1

b

b

Ny

/

y T N T Ny " 2 x T M T Ny G 0,

; x, y.

LEMMA B2. Suppose that a, b, c, d, e are positi¨ e scalars. Then the optimization problem minimize

d s d b1 q e b 2

subject to

b 1 ) 0, b 2 ) 0, and

a

b1

q

b

b2

Fc

has the minimal ¨ alue dmin s Ž1rc .Ž'ad q 'be . 2 with b 1 s Ž1rc .Ž a q aberd . and b 2 s Ž1rc .Ž b q abdre ..

'

'

Proof. The problem is equivalent to finding the line d s d b 1 q e b 2 which is tangent to the hyperbolic H: arb1 q brb 2 s c. The slope of the tangent line T to the hyperbolic H is slope T s y

ab

Ž c b1 y a.

2

for a given b 1 .

,

To find the line d s d b 1 q e b 2 tangent to the hyperbolic H, it follows that slope T s y

ab

Ž c b1 y a.

2

sy

d e

.

Then, the tangent points are

b1 s

1 c

abe

ž ( / a"

d

and

b2 s

1 c

abd

ž ( / b"

e

.

660

YANG AND CHEN

Since b 1 s Ž1rc .Ž a y aberd . and b 2 s Ž1rc .Ž b y abdre . cannot satisfy the constraints b 1 , b 2 ) 0, therefore, the only feasible solution is

'

b1 s

1 c

'

abe

ž ( / aq

and

d

b2 s

1 c

abd

ž ( / bq

e

and the corresponding minimal value of d is

dmin s

1 c

Ž 'ad q 'be .

2

.

REFERENCES 1. B. R. Barmish and A. R. Galimidi, Robustness of Luenberger observers: Linear systems stabilized via non-linear control, Automatica 22 Ž1986., 413]423. 2. B. R. Barmish and G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions, IEEE Trans. Automat. Control 27 Ž1982., 153]158. 3. S. P. Bhattacharyya, The structure of robust observer, IEEE Trans. Automat. Control 22 Ž1976., 581]588. 4. C. L. Chen and S. K. Yang, Robust performance analysis for linear continuousrdiscrete time systems with linear dependent and time-varying perturbations, Internat. J. Systems Sci. 26 Ž1995., 1583]1591. 5. Y. H. Chen, On the robustness of mismatched uncertain dynamical systems, ASME J. Dynam. System Measure. Control 109 Ž1987., 29]35. 6. L. Cong and P. H. Landers, Robust control design for nonlinear uncertain systems with incomplete matching conditions, Trans. Inst. Measure. Control 15 Ž1993., 46]52. 7. C. A. Desoer and M. Vidyasagar, ‘‘Feedback Systems: Input-Output Properties,’’ Academic Press, New York, 1975. 8. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, ‘‘Feedback Control of Dynamic System,’’ Addison]Wesley, Reading, MA, 1986. 9. A. R. Galimidi and B. R. Barmish, The constrained Lyapunov problem and its application to robust output feedback stabilization, IEEE Trans. Automat. Control 31 Ž1986., 410]419. 10. S. Gutman, Uncertain dynamical systems}A Lyapunov max-min approach, IEEE Trans. Automat. Control 24 Ž1979., 437]443. 11. F. Jabbari and W. E. Schmitendorf, Robust linear controllers using observers, IEEE Trans. Automat. Control 36 Ž1991., 1509]1514. 12. I. R. Petersen, A Riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems, IEEE Trans. Automat. Control 30 Ž1985., 904]907. 13. Z. Qu and J. Dorsey, Robust control of generalized dynamic systems without the matching conditions, ASME J. Dynam. System Measure. Control 113 Ž1991., 582]589. 14. W. E. Schmitendorf, Design of observer-based robust stabilizing controllers, Automatica 24 Ž1988., 693]696.

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15. W. E. Schmitendorf, Designing stabilizing controllers for uncertain systems using the Riccati equation approach, IEEE Trans. Automat. Control 33 Ž1991., 376]379. 16. R. T. Stefani, Reducing the sensitivity to parameter variations of a minimum-order reduced-order observer, Internat. J. Control 35 Ž1982., 983]995. 17. H. Wu and K. Mizukami, Robust stabilization of uncertain linear dynamical systems, Internat. J. Systems Sci. 24 Ž1993., 265]276. 18. S. K. Yang and C. L. Chen,Robust controller design for systems with mismatching uncertainty, Systems Anal. Modelling Simulation, in press.