Control of Uncertain Bilinear Systems using Linear Controllers: Stability Region Estimation and Controller Design

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1 Control of Uncertan Blnear Systems usng Lnear Controllers: Stablty Regon Estmaton Controller Desgn Shoudong Huang Department of Engneerng Australan Natonal Unversty Canberra, ACT 2, Australa James Lam Department of Mechancal Engneerng Unversty of Hong Kong Pokfulam Road, Hong Kong Abstract Ths paper studes the problem of stablty regon estmaton controller desgn for uncertan blnear systems when lnear controllers are used. Iteratve lnear matrx nequalty (ILMI) algorthms are presented to estmate the closed-loop stablty regon desgn the controllers. No tunng of parameters s needed n the desgn methods. The desgn ams to optmze between the sze of the stablty regon, dampng of the state varables, the feedback gan.. Introducton Blnear system s a specal knd of nonlnear systems whch could represent a varety of mportant physcal processes [8]. A great amount of lterature related to the control problems of such systems has been developed over the past decades. Among them, some results were concerned wth blnear systems wth only multplcatve control [2, 9]. For blnear systems wth both addtve multplcatve control nputs, some control desgns, such as bang-bang control [7] or optmal control [], obtan global asymptotc stablty under the assumpton that the open-loop system s ether stable or neutrally stable. When the open-loop system s unstable, t s dffcult to obtan global asymptotc stablty except when ndependent addtve multplcatve control nputs exst [4]. Though much attenton has been pad to the study of blnear systems, lttle research work was devoted to the control problems for blnear systems wth uncertantes. Recently, the quadratc stablzaton problem of uncertan blnear systems was studed n [] a suffcent condton for the system to be quadratcally stablzable by quadratc feedback control was presented. In some practcal control system desgns, local closedloop stablty may be enough lnear controllers are preferred. In [5], Gutman suggested that a nonlnear controller be used frstly forced the trajectory of a blnear system suffcently near the orgn, then swtch to a lnear controller such that the closed-loop system s asymptotcally stable. Hence the study of local stablzaton usng lnear controller the estmaton of the closed-loop stablty regon are useful n practce. Such a problem had been studed n [3] but the results n there are not easy to be appled because t needs tunng of parameters. Moreover, [3] dd not consdered the robustness of the controller to model uncertantes. Ths paper consders the stablty regon estmaton controller desgn for uncertan blnear systems, where the controls act addtvely multplcatvely smultaneously. The uncertantes are assumed to have a polytopc form. No assumpton on the open-loop stablty s made lnear state feedback control s studed. As compared wth [3], the results n ths paper has the followng advantage. Frstly, we estmate the dervatve of the Lyapunov functon n a dfferent way, our results are less conservatve than that n [3]. Secondly, our results are presented n the form of teratve lnear matrx nequalty (ILMI) algorthms. It needs no tunng of parameters can be effcently solved numercally usng LMI toolbox n MATLAB. Thrdly, three desgn specfcatons, closed-loop system stablty regon, feedback gan, dampng of the state varables, are all taken nto account n the controller desgn. Three desgn algorthms are presented to optmze one specfcaton when the other two specfcatons are gven. Fnally, our results are sutable for blnear systems wth polytopc uncertantes. Ths paper s organzed as follows. Secton 2 gves the problems statement. The stablty regon estmaton method controller desgn methods are presented n Sectons 3 4, respectvely. Two examples are gven n Secton 5 to llustrate our methods. Secton 6 concludes the paper. *ThsworkssupportednpartbytheRGCGrantHKU 73/P the Wllam Mong Post-doctoral Research Fellowshp

2 2. Problem Statement The followng notatons are used n ths paper. Ω denotes the boundary of a set Ω R n. e denotes the th stard bass of R n. (P ) denotes the element n the th column the th row of a matrx P. σ(a) denotes the spectrum of matrx A. For a real symmetrc matrx M, M >( ) means that M s postve (sem-)defnte. If not explctly stated, I denote the dentty matrx the zero matrx of approprate dmensons, respectvely. Besdes, all matrces are assumed to have compatble dmensons. Consder the blnear system S : ẋ = A(t)x + x N (t)u + B(t)u () where x(t) R n s the state vector, x (t) R s the -th element of x(t), u(t) R m s the control nput. Suppose the system under consderaton has polytopc uncertantes. That s, the system matrces belong to matrx polytopes A(t) Co {A,...,A na }, (2) B(t) Co {B,...,B nb }, (3) N (t) Co ª N,...,N nn,,...,n. (4) where Co denotes the convex hull. Consder a lnear state feedback u = Kx, the closed-loop system s S c : ẋ =[A(t)+B(t)K] x + x N (t)kx (5) The frst problem consdered n ths paper s to estmate the stablty regon of S c for a gven K. The second problem s to desgn K such that S c satsfes certan specfcatons. The specfcatons consdered n ths paper nclude the sze of the stablty regon, the dampng of the state varables, the magntude of the feedback gan. 3. Estmaton of Stablty Regon Snce there are uncertantes n system (5), we wll consder ts local quadratc stablty. Consder the Lyapunov functon V (x) =x T Px wth P>. The dervatve of V (x) along the trajectores of S c s V n o (x) = x T P [A(t)+B(t)K]+[A(t)+B(t)K] T P x à n T X +x T P x N (t)k + K T x N (t)! P x = x T {P [A(t)+B(t)K]+[A(t)+B(t)K] T P + x PN (t)k + K T N T (t)p }x. Defne a set Ω R n as follows. For any x R n, x Ω f only f P [A(t)+B(t)K]+[A(t)+B(t)K] T P + x PN (t)k + K T N T (t)p < for all A(t),B(t) N (t) ( =,...,n) that satsfy (2), (3), (4), respectvely, where x s the -th element of x. Further defne a set D R n as D = {x R n : V (x) <V } (6) where V =mn x Ω xt Px. From Lyapunov stablty theorem, D s contaned n the stablty regon of S c. The followng lemma gves a method to compute D for gven K P. Lemma Suppose K P>are gven, r R > s the soluton of LMI problem max r subject to P (A la + B lb K)+(A la + B lb K) T P p ³ +r (P ) PN ln K + K T Nl T N P < x j for all l A =,...,n A, l B =,...,n B, l N =,...,n N (7) ( =,...n), j =,...,2 n, where x j = hx j,...,xj n (j =,...,2 n ) are the 2 n vectors whose elements x j canonlytakethevalueof or, then an estmate of the stablty regon can be obtaned by D = x R n : x T Px < r 2ª. (8) Proof. For any r>, snce x = r p (P ) ( =,...,n) are tangental to the ellpsod x T Px = r 2, we consder the polyhedron q ¾ Ω r = x R n : x <r (P ), =,...,n (9) that contans the ellpsod x R n : x T Px < r 2ª.Snce Ω r s convex, P [A(t)+B(t)K]+[A(t)+B(t)K] T P + x PN (t)k + K T N T (t)p () < for all x n Ω r f only f () holds for all the 2 n vertces of Ω r,thats, P [A(t)+B(t)K]+[A(t)+B(t)K] T P +r x j p (P ) PN (t)k + K T N T (t)p <, () for all j =,...,2 n. Snce A(t),B(t) N (t) ( =,...,n) satsfy (2), (3) (4), () s further equvalent to (7). After the largest r that satsfes (7) s obtaned, from the fact about quadratc functons (see e.g. Lemma n T

3 [6]), the estmated stablty regon D can be computed by (6) where V = mn n V ¾ V = mn x T Px : x R n, x = r q(p ) ¾ = mn x T Px : x R n, e T x = r q(p ) r p (P ) 2 = e T P e = r 2. Generally speakng, t s not easy to solve the above optmzaton problem drectly. However, when p (P ) Hence (8) holds. The proof s completed. ( =,...,n) are fxed, the problem becomes a generalzed Now suppose K s gven, we consder the problem to egenvalue mnmzaton problem whch can be choose P such that D s the largest. The problem s to solved by the LMI toolbox. The followng teratve LMI choose P r satsfyng (7) such that D n (8) s the (ILMI) algorthm s provded. largest. Algorthm ESTIMATION: For any scalar α >, ˆP = αp mples ˆP = α r P ³ Step Choose a small tolerance scalar η >. Let P ˆP p = α (P ). Snce (7) s equvalent ε I R n n for a suffcently small ε ( > ε > ). q P to Step 2 Compute ( =,...,n). αp (A la + B lb K)+(A la + B lb K) T αp Step 3 Solve LMI problem + αr x j p α (P ) ³αPN ln K + K T Nl T mn λ subject to N αp < P I for all l A =,...,n A, l B =,...,n B, l N =,...,n N P P ( =,...n), j =,...,2 n, or equvalently, q P x j ³ ˆP (A la + B lb K)+(A la + B lb K) T PN ln K + K T Nl T N P ˆP h + ³ ³ αr xr j ˆP ˆPNlN K + < λ P (A KT Nl T la + B lb K) (A la + B lb K) T P, N ˆP <, for all l A =,...,n A,l B =,...,n B,l N =,...,n N ( =,...n), j =,...,2 n,obtanp λ. for all l A =,...,n A, l B =,...,n B, l N =,...,n N ( =,...n), j =,...,2 n. Hence ˆP kp Pk = αp ˆr = Step 4 If kp k < η, go to Step 6, else go to Step 5. αr also satsfy (7). Notce that D s kept unchanged Step 5 Let P = P, go to Step 2. when substtutng P r by ˆP ˆr, re- Step 6 The estmated stablty regon s gven by (3). spectvely. Wthout loss of generalty, we can suppose P I. In order to obtan a larger D, we can maxmze r (maxmze the largest ball contaned n D ). The Remark 2 TheLMIproblemnStep3sfeasbleprovded problem becomes that there exsts a P>such that max r subject to (7) P (A la + B lb K)+(A la + B lb K) T P< P I wth varables r P. Denote λ = r, then we have the followng theorem. Theorem Suppose P R n n > λ R > are the solutons of the followng optmzaton problem mn λ subject to P I, x j p ³ (P ) PN ln K + K T Nl T N P h (2) < λ P (A la + B lb K) (A la + B lb K) T P for all l A =,...,n A, l B =,...,n B,l N =,...,n N ( =,...n), j =,...,2 n, then an estmate of the closed-loop stablty regon s D = x R n : x T Px < λ ¾ 2, (3) whch contans the largest ball x R n : x T x< λ 2 ª. Remark Dependng on applcatons, we can also consder maxmzng the volume of D nstead of maxmzng the largest ball contaned n D a smlar result can be obtaned. for all l A =,...,n A l B =,...,n B (P can be chosen suffcently q small such that P P ). Snce P P P mples p (P ), the P λ obtaned n Step 3 satsfy (2). Moreover, once the LMI problem n Step 3 s feasble, t s also feasble for the next terate defned n Step 5 because at least the new P tself s a soluton. Furthermore, the postve defnte matrx sequences {P } wth P I obtaned n Algorthm ESTIMATION s non-decreasng (n the sense of postve sem-defnteness), the postve sequence {λ} obtaned n Algorthm ESTIMATION s non-ncreasng, thus {P } {λ} wll converge the algorthm wll converge.

4 4. Controller Desgn Now we consder the controller desgn problems. Three desgn specfcatons wll be consdered: closed-loop system stablty regon, feedback gan dampng of the state varables. Certanly, we want to desgn a controller to obtan a larger stablty regon, a better dampng of the state varables, a smaller feedback gan. However, a desgn should compromse the above three specfcatons. In ths secton, we wll consder the problems of optmzng one specfcaton when the other two specfcatons are gven. Frst consder the problem of maxmzng the stablty regon. Suppose a lnear feedback controller u = Kx should be desgned such that () for all l A =,...,n A l B =,...,n B,thereal parts of the egenvalues of A la + B lb K s less than δ for a gven δ >, thats,σ (A la + B lb K) C δ where C δ = {s C :Re(s) < δ } s a subset of the complex plane; () the feedback gan s constraned by KK T k I for agvenk >. From Theorem, the problem of obtanng the largest stablty regon can be stated as mn λ subject to x j p ³ (P ) PN ln K + K T Nl T N P h < λ P (A la + B lb K) (A la + B lb K) T P, for all l A =,...,n A, l B =,...,n B,l N =,...,n N ( =,...n), j =,...,2 n, P (A la + B lb K)+(A la + B lb K) T P +2δ P<, (4) for all l A =,...,n A l B =,...,n B, KK T k I, P I where (4) mples that σ (A la + B lb K) C δ, for all l A =,...,n A l B =,...,n B. In the above problem, varables are K, P λ. Denote P = P, the problem s converted nto mn λ subject to ³ xr j P ³N ln K P + PK T N T ln h < λ (A la + B lb K) P P (A la + B lb K) T, for all l A =,...,n A, l B =,...,n B,l N =,...,n N ( =,...n), j =,...,2 n, (A la + B lb K) P + P (A la + B lb K) T +2δ P <, for all l A =,...,n A l B =,...,n B, KK T k I, P I. Further denote W = K P, then K = W P. Snce P I, the constrant KK T k I can be substtuted by WW T k I. Hence the followng theorem holds. Theorem 2 Suppose P R n n >,W R m n λ R are the solutons of the followng optmzaton problem mn λ subject to ³ xr j P ³N ln W + W T N T ln < λ h A la P T PAlA B lb W W T B T, lb for all l A =,...,n A, l B =,...,n B,l N =,...,n N ( =,...n), j =,...,2 n, A la P + T PA la + B lb W + W T Bl T B +2δ P <, (5) for all l A =,...,n A l B =,...,n B, k I W W T (6) I P I (7) then the controller whch can obtan the largest closedloop system stablty regon s u = Kx = W P x. (8) Moreover, an estmate of the closed-loop system stablty regon s D = x R n : x T P x< λ ¾ 2, (9) whch contans the ball x R n : x T x< λ 2 ª. The optmzaton problem n Theorem 2 can be solved usng the followng algorthm, whch s smlar to Algorthm ESTIMATION. Algorthm DESIGN: Step Choose a small tolerance scalar η >. Let P = ρ I R n n for a suffcently large ρ >. r ³ Step 2 Compute P ( =,...,n). Step 3 Solve LMI problem mn λ subject to (5), (6), (7) r ³ P ³N ln W + W T N T ln x j < λ h A la P T PAlA B lb W W T B T, lb for all l A =,...,n A,l B =,...,n B,l N =,...,n N ( =,...n), j =,...,2 n, P P to obtan P,W λ. Step 4 If k P P k k P < η, go to Step 6, else go to Step 5. k Step 5 Let P = P, go to Step 2. Step 6 The controller the estmated stablty regon are obtaned by (8) (9). Smlar to Theorem 2, when consderng the problems of maxmzng the dampng rate of the state varables mnmzng the feedback gan, the followng results

5 hold. Theorem 3 Among all the controllers satsfyng KK T k I n such that the closed-loop o stablty regon contans x R n : x T x<, the controller n (8) has the maxmal dampng rate of the state varables, λ 2 σ (A la + B lb K) C δ for all l A =,...,n A l B =,...,n B,where P R n n >,W R m n δ R > are the solutons of the followng optmzaton problem mn δ subject to ³ xr j P ³N ln W + W T N T ln h < λ A la P T PAlA B lb W W T Bl T B, for all l A =,...,n A, l B =,...,n B, l N =,...,n N ( =,...n), j =,...,2 n, 2 P < δ ³ A la P T PA la B lb W W T BlB T, for all l A =,...,n A l B =,...,n B, k I W W T, I P I. Theorem 4 Among all the controllers satsfyng σ (A la + B lb K) C δ for all l A =,...,n A l B =,...,n B, such that the closed-loop stablty regon contans x R n : x T x< ¾ λ 2, the controller n (8) has the smallest feedback gan, KK T ki, where P R n n >,W R m n k R > are the solutons of the followng optmzaton problem mn k subject to ³ xr j P ³N ln W + W T N T ln h < λ A la P T PAlA B lb W W T Bl T B, for all l A =,...,n A, l B =,...,n B, l N =,...,n N ( =,...n), j =,...,2 n, A la P + T PA la + B lb W + W T Bl T B +2δ P <, for all l A =,...,n A l B =,...,n B, ki W W T, I P I. r ³ Remark 3 When P ( =,...,n) are fxed, the optmzaton problems n Theorems 3 4 become a generalzed egenvalue mnmzaton problem a mnmzaton of a lnear objectve under LMI constrants, respectvely, whch can be solved by the LMI toolbox n MATLAB. Hence, two ILMI algorthms smlar to Algorthm DESIGN can be establshed to desgn a controller such that a better dampng of state varables or a smaller feedback gan can be obtaned. 5. Examples 5. Example Consder the example used n [3]. The system s expressed by () where A(t) =A, B(t) =B, N (t) =N,N 2 (t) =N 2 A = 5 2,B=,N = 2,N 2 = We frst estmate the closed-loop stablty regon for the followng three controllers gven by [3]: K =[ 3.9,.4],K 2 =[ 4.35,.44],K 3 =[2.87,.47]. Usng Algorthm ESTIMATION, we obtan the followng stablty regon estmates: D = D 2 = D 3 = x R n : x T x R n : x T x R n : x T ¾ x<.69, ¾ x<.627, ¾ x<.59 respectvely. The regons are compared wth those obtaned n [3] n the three fgures. Itcanbeseenthatour estmates are much better than those n [3]. 5.2 Example 2 Consder an open-loop unstable system wth two-dmensonal controlnput(examplen[]). Thesystemsexpressed by () where A(t) =Co {A,A 2 },B(t) =B, N (t) =N,N 2 (t) =N 2 A = 6,A 2 = 2, B =,N 2 2 =,N =. 4 We consder the controller desgn problems. For δ = k =5, usng Algorthm DESIGN we obtan λ =.323 the controller u = K x = x, the estmaton of stablty regon s ¾ D = x R n : x T x< For λ = k =5,usngTheorem3anILMI.

6 algorthm, we obtan δ =.9632 the controller u = K x = x For λ = δ =,usngtheorem4anilmi algorthm, we obtan k = the controller u = K x = x The results are summarzed n the followng table. controller λ λ δ k K K K Concluson Ths paper studed the stablty regon estmaton lnear controller desgn problems for uncertan blnear systems. Iteratve lnear matrx nequalty (ILMI) algorthms were presented to estmate the stablty regon desgn the controllers. Three closed-loop specfcatons, closed-loop system stablty regon, feedback gan dampng of the state varables were consdered n the controller desgn problems. The controllers desgned are such that one specfcaton s enhanced whle the other two specfcatons are constraned at a certan level. Examples show that our stablty regon estmaton method s less conservatve than that n [3], our controller desgn methods are more effectve. References [] A. Benallou, D. A. Mellchamp, D. E. Seborg. Optmal stablzng controllers for blnear systems. Int. J. Control, 48(4):487 5, 988. [2] M. S. Chen S. T. Tsao. Exponental stablzaton of a class of unstable blnear systems. IEEE Transactons on Automatc Control, 45(5): , 2. [3] I. Derese E. Noldus. Desgn of lnear feedback laws for blnear systems. Int. J. Control, 3(2):29 237, 98. [4] I. Derese E. Noldus. Optmzaton of blnear control systems. Int. J. Systems Scence, 3(3): , 982. [5] P. O. Gutman. Stablzng controllers for blnear systems. IEEE Transactons on Automatc Control, 26(4):97 922, 98. [6] H. Hnd S. Boyd. Analyss of lnear systems wth saturaton usng convex optmzaton. In Proc. of the 37th IEEE Conference on Decson Control, pages 93 98, Tampa, 998. [7] R. Longchamp. Controller desgn for blnear systems. IEEE Transactons on Automatc Control, 25(3): , 98. [8] R.R.Mohler.Nonlnear Systems,volume2.Prentce- Hall, Inc., Englewood Clffs, N.J., 99. [9] C. D. Rahn. Stablzablty condtons for strctly blnear systems wth purely magnary spectra. IEEE Transactons on Automatc Control, 4(9): , 996. [] H. D. Tuan S. Hosoe. On robust H controls for a class of lnear blnear systems wth nonlnear uncertanty. Automatca, 33(7): , ths paper [DN8] K =[ 3.9,.4] ths paper [DN8] K 2 =[ 4.35,.44] ths paper [DN8] K 3 =[2.87,.47]