An LMI Approach to Guaranteed Cost Control for Uncertain Delay Systems

Transcription

1 IEEE RASACIOS O CIRCUIS AD SYSEMS I: FUDAMEAL HEORY AD APPLICAIOS, VOL 5, O 6, JUE An LMI Approach to Guarantee Cost Control for Uncertain Delay Systems Hiroaki Mukaiani he notations use in this brief are fairly stanar he superscript enotes matrix transpose, I n 2 R n2n enote the ientity matrices, block-iag enotes the block iagonal matrix, kkenotes the Eucliean norm, an kk 2 enotes the largest singular value Abstract he guarantee cost-control problem for uncertain linear systems which have elay in both state an control input is consiere Sufficient conitions for the existence of guarantee cost controllers are given in terms of linear matrix inequality (LMI) It is shown that the state feeback controllers can be obtaine by solving the LMI Inex erms Delay, guarantee cost control, linear matrix inequality (LMI), uncertain linear systems I IRODUCIO Delay systems generally occur in moern society in the form of communication systems, transmission systems, chemical processing systems, power systems, an so on If the presence of elays, is not consiere in the controller esign, it may cause instability or serious eterioration in the performance of the resulting control systems [] he stuy of time-elay systems has receive ever greater attention in the past few ecaes In recent years, the problem of robust control of elay systems with parameter uncertainties has been wiely stuie in the literature [2] [4] Although there have been numerous results on robust control of uncertain elay systems, much effort has been mae towar fining a controller which guarantees robust stability However, when controlling such systems, it is also esirable to esign the control systems which is not only robustly stable, but also guarantees an aequate level of performance One approach to this problem is the so-calle guarantee cost-control approach [5] [] his approach has the avantage of proviing an upperboun on a given performance inex Recent avances in the theory of linear matrix inequality (LMI) have allowe a revisiting of the guarantee cost-control approach [6], [7] he LMI esign metho is a very well-known an powerful tool ot only can it efficiently fin feasible an global solutions, but also easily hanle various kins of aitional linear constraints he guarantee cost-control problem for a class of linear uncertain elay systems which is base on the LMI esign approach was solve [8] [] However, in [8] [], elay of the control input has not been consiere Furthermore, it shoul be pointe out that although the robust-control esign methos for parameter uncertain systems that have elay in both state an control input have been consiere (see for example [3], [4]), the guarantee cost control for such elay systems has not been iscusse so far In this brief, the guarantee cost-control problem of the robust control for uncertain system that has elay in both state an control input is consiere A sufficient conition for the existence of the robust feeback controllers is erive in terms of the LMI he main result of this brief shows that guarantee cost controllers can be constructe by solving the LMI he crucial ifference between the existing results [8] [] an our new one is that the controller which guarantees the stability an aequate level of performance for the time elay in both state an control input is given hus, the resulting controllers can be easily implemente for more practical elay system Manuscript receive May 8, 22; revise September 9, 22 his paper was recommene by Associate Eitor C Xiao he author is with the Grauate School of Eucation, Hiroshima University, Hiroshima , Japan ( mukaia@hiroshima-uacjp) Digital Object Ientifier 9/CSI II PRELIMIARY We consier the autonomous uncertain elay system of the form _x(t) =[A A(t)]x(t) [A i A i (t)]x(t i ) [H h H h (t)]x(t h) x(t)= (t); t 2 [; ]; =maxf; ; ; hg where x(t) 2 R n is the state, i > are the elay constants, an (t) is the given continuous vector value initial function A, A i, an H h are the constant matrices of appropriate imensions he parameter uncertainties consiere here are assume to be of the following form: [A(t) H h (t)] =DF (t)[e E h 2 ] (2a) () A i (t) =D i F i (t)e i ; i =; ; (2b) where D, E, E h 2, Di, an Ei are known constant real matrices of appropriate imensions F (t) 2 R p2q an Fi (t) 2 R r 2s are unknown matrix functions with Lebesgue measurable elements an satisfying F (t)f (t) I q an F i (t)f i (t) I s : (3) Associate with () is the cost function J = x (t)qx(t) t (4) where Q is the given positive efinite symmetric matrices Definition : he matrix P > is sai to be the quaratic cost matrix for the uncertain elay systems () if the following inequality hols: t x (t)px(t) x (t)qx(t) < (5) for all nonzero x(t) 2 R n an all uncertainties (2) heorem : Suppose there exist the symmetric positive efinite matrices P > ; S i > ; ; ;;U> 2 R n2n such that for all uncertain matrices (2) the following matrix inequality (6) hols: 3 = 4 P ~ A P ~ A P ~ H h ~ P S ~ P S ~H h P U where 3 2 R 2, = n ( 2), an 4:= A ~ P P A ~ Q S i U ~A := A A(t) ~A i := A i A i (t) ~H h := H h H h (t): < (6) /3$7 23 IEEE

2 796 IEEE RASACIOS O CIRCUIS AD SYSEMS I: FUDAMEAL HEORY AD APPLICAIOS, VOL 5, O 6, JUE 23 hen, the autonomous uncertain elay system () is quaratically stable an the corresponing value of the cost function (4) satisfies the following inequality (7): J< ()P() he proof is given in Appenix I (s)s i(s) s (s)u(s) s: h III PROBLEM FORMULAIO In this section, we consier the problem of optimal guarantee cost control via the state feeback for a class of the elay system he uncertain elay system uner consieration are escribe by the following state equation: _x(t) =[AA(t)]x(t)[BB(t)]u(t) [A i A i (t)]x(t i)[b h B h (t)]u(t h); x(t)=(t); t2 [; ]; =maxf; ; ;hg where u(t) 2 R m is the control he parameter uncertainties satisfy (7) (8) [B(t) B h (t)] = DF (t)[e2 E h 2 ] : (9) B, B h, E2, an E2 h are the constant matrices of appropriate imensions he remainer constant real matrices an parameter uncertainties are the same as the elay systems () Associate with (8) is the cost function J = [x (t)qx(t) u (t)ru(t)] t () where Q an R are the given positive efinite symmetric matrices Definition 2: A control law u(t) =Kx(t) is sai to be a quaratic guarantee cost control with associate cost matrix P > for the elay system (8) an cost function () if the close-loop system is quaratically stable an the close-loop value of the cost function () satisfies the boun JJ 3 for all amissible uncertainties, that is t x (t)px(t) x (t)[q K RK]x(t) < () for all nonzero x 2 R n he objective of this brief is to esign a linear time-invariant feeback guarantee cost-control law u(t) =Kx(t) for the elay system (8) with uncertainties IV MAI RESULS We now give the LMI esign approach to the construction of a guarantee cost controller heorem 2: Suppose there exist the constant parameters >an " > such that the LMI shown in (2) at the bottom of the page, have the symmetric positive efinite matrices X > ; S i > ; i = ; ;;Z> 2 R n2n an a matrix Y 2 R m2n, where 8:= AX BY (AX BY ) Z DD H, H := "D i D If such conitions are met, the linear-state feeback-control law is the guarantee cost controller an J < ()X () i u(t) =Kx(t) =YX x(t) (3) h (s)s i (s) s (s)x ZX (s) s (4) is the guarantee cost for the close-loop uncertain elay systems he proof is given in Appenix II Since the LMI (2) consists of a convex solution set of (; "X; Y; S i ;Z), various efficient convex-optimization algorithms can be applie Moreover, its solutions represent the set of the guarantee cost controllers his parameterize representation can be exploite to esign the guarantee cost controllers which minimizes the value of the guarantee cost for the close-loop uncertain elay systems Consequently, solving the following optimization problem allows us to etermine the optimal boun: min X = min X race[m 2 i]c k k2 race[z] =J 3 ; X2 (; "X; Y; S i;z;;m i) (5) 8 B h Y (EX E2Y ) X Y A S A S X X Y B h Z Y E2 h EX E2Y E2 h Y I q X Q Y R S S SE E S "I s S S S E E S "I s X S < (2) X S

3 IEEE RASACIOS O CIRCUIS AD SYSEMS I: FUDAMEAL HEORY AD APPLICAIOS, VOL 5, O 6, JUE such that (2) an () () X < (6a) M i M i M i S i < ; i =; ; (6b) ci n I n I n X < (6c) where c is a given positive constant, M i M i := (s) (s) s, := h (s) (s) s hat is, the problem aresse in this brief is as follows Fin K = YX such that LMI (2) an (6) are satisfie an the cost J becomes as small as possible Finally, we are in a position to establish the main result of this section heorem 3: If the above optimization problem has the solution, ", X, Y, S i, Z,, an M i, then, the control law of the form (3) is the linear-state feeback-control law which ensures the minimization of the guarantee cost (4) for the uncertain elay systems Proof: By heorem 2, the control law (3) constructe from the feasible solutions, ", X, Y, S i, Z,, an M i is the guarantee cost controllers of the uncertain elay systems (8) Applying the Schur complement to the LMI (6) an using the following inequality [2]: race[xy] kxk2race[y]; Y = Y ; X = X we have (6a), ()X () < (6b) ) (s)s i (s) s = = race[m i S i M i] < race[m i] (6c) ) = h h (s)x ZX (s) s race[ (s)x ZX (s)] s race[ (s)s i (s)] s = race[ X ZX ]k k2 kx k2 race[z] <c 2 k k2 race[z]: It follows that J < ()X () (s)s i (s) s (s)x ZX (s) s h < race[m i ]c 2 k k2 race[z] min X J = J 3 : (7) hus, the minimization of J implies the minimum value J 3 of the guarantee cost for the uncertain elay systems (8) he optimality of the solution of the optimization problem follows from the convexity of the objective function uner the LMI constraints his is the require result Remark : he constant parameter c which is inclue in the inequality (6c) nee to be optimize as the LMI constraint In that case, it is har to obtain the optimum guarantee cost because the resulting problem is a nonconvex optimization problem Hence, we propose the above suboptimal guarantee cost control instea of solving the nonconvex optimization problem As a result, the robust suboptimal guarantee cost controller which minimizes the value of the guarantee cost for the close-loop uncertain elay system can be easily solve by using the LMI he constant parameter c nees to be chosen as small as possible However, if there exists no solution of the consiere optimization problem, then we nee to take the large parameter c On the other han, it shoul be note that the parameter c cannot become large because the matrix X is constraine by the inequality (6a) V UMERICAL EXAMPLE In orer to emonstrate the efficiency of our propose control, we have run a simple numerical example he system matrices with the uncertainties are given as follows: A = 2 A = A 2 = A 3 = B = B h = : : : : :2 : :3 :5 : :2 D = D = D2 = D3 = E = E = E2 = E3 =[ :] E2 = E2 h =[:2] = 2 =2 3 =3 h = (t) = 2 F (t) =cos(6t) F i (t) = exp(2t) i =; 2; 3 x = x : x2 ote that we cannot apply the technique propose in [8] [] to the above elay system, since the consiere system has elay in control input Moreover, compare with the multiple elay systems presente in [9], the uncertain matrices relate to state elays are more general forms ow, we choose as R =: an Q = iag [:2 :] Moreover, we take c =2 By applying heorem 3 an solving the corresponing optimization problem (5), we obtain the linear optimal state feebackcontrol law K =[: :634 ] : Consequently, the optimal guarantee cost of the uncertain close-loop elay system is J 3 =6:364 43

4 798 IEEE RASACIOS O CIRCUIS AD SYSEMS I: FUDAMEAL HEORY AD APPLICAIOS, VOL 5, O 6, JUE 23 ote that V (x(t)) > whenever x(t) 6= hen, the time erivative of V (x(t)) along any trajectory of the elay system (8) is given by Fig Response of the close-loop system with the propose control metho Finally, the result of the simulation of this example is epicte in Fig It is shown from Fig that the close-loop systems are asymptotically stable VI COCLUSIO In this brief, a solution of the guarantee cost-control problem for uncertain system that have elay in both state an control input has been presente he robust optimal guarantee cost controller which minimizes the value of the guarantee cost for the close-loop uncertain elay system can be solve by using software such as MALAB s LMI control oolbox [3] hus, the resulting linear feeback controller can guarantee the quaratic stability an the optimal cost boun for the uncertain elay systems On the other han, there is one rawback that we must not ignore for the propose controller esign It shoul be note that the uncertain system with the kin of uncertainty structure as in(2), usually results in very conservative controller esign However, since it is well known that there are many physical systems in which the uncertainty can be moele by this manner [4], the assumption that the uncertain system has the structure given by the uncertainty structure (2) is reliable After submitting this brief, we have notice that there exist the useful results for the guarantee cost-control problem for uncertain systems that have elay in both state an control [8] Although the propose esign approach is also base on the LMI technique, the controllers are not unique herefore, one nees to construct a guarantee cost controller through trial an error On the other han, our controllers are unique Moreover, since the consiere uncertain systems contain the multiple elay in the states, it is applicable to wier class Finally, it is expecte that the LMI approach is also applie to the output feeback case [5] Such a problem is more realistic than the state feeback case his problem will be aresse in future investigations APPEDIX I PROOF OF HEOREM Using the efinitions A, ~ A ~ i, an H ~ h, we can change the form () as _x(t) = Ax(t) ~ ~A i x(t i) ~ H h x(t h): (8) Suppose now there exist the symmetric positive efinite matrices P, S i an U such that the matrix inequality (6) hols for all amissible uncertainties (2) In orer to prove the asymptotic stability of the elay systems (8), let us efine the following Lyapunov function caniate: t V (x(t)) =x (t)( A ~ P P A)x(t) ~ 2x (t)p A ~ i x(t i) 2x (t)p H ~ h x(t h) [x (t)s ix(t) x (t i)six(t i)] x (t)ux(t) x (t h)ux(t h) 4 Q P A ~ P A ~ P H ~ h = z (t) ~ P S ~ P S ~H h P U = z (t)3z(t) x (t)qx(t) z(t) where z(t) = x (t) x (t ) x (t ) x (t h) 2 R an 4 an 3 are given in (6) aking the fact that the inequality (6) hols into account, it follows immeiately that t V (x(t)) < x (t)qx(t) < : (2) Hence, V (x(t)) is a Lyapunov function for the elay system (8) herefore, (8) is asymptotically stable Furthermore, by integrating both sies of the inequality (2) from to an using the initial conitions, we have V (x( )) V (x()) < x (t)qx(t) t: (2) Since (8) is asymptotically stable, that is, x( )!, when!, we obtain V (x( ))! hus, we get J = x (t)qx(t) t < V (x()) = ()P() he proof of heorem is complete (s)s i(s) s (s)u(s) s: h APPEDIX II PROOF OF HEOREM 2 Let us introuce the matrices X := P, Y := KP, S i := S i, an Z := P UP Premultiplying an postmultiplying both sies of the inequality (2) by block iag [P P I q I n I m S I s S I s I n I n] V (x(t)) = x (t)px(t) t t x (s)s ix(s) s t x (s)ux(s) s: (9) th yiels (22), shown at the top of the next page, where 9 := P P A U P DD P PHP, A := A BK, E := E E2K: Using the Schur complement [6], (22) hols if, an only if (23), shown at the top of the next page, hols,where := P P AR Si U P DD P PHP E E, R := QK RK

5 IEEE RASACIOS O CIRCUIS AD SYSEMS I: FUDAMEAL HEORY AD APPLICAIOS, VOL 5, O 6, JUE PB h K E I n K PA PA I n I n K B h P U K E h 2 E E h 2 K I q I n Q K R P S E E "I s P S E E "I s < (22) I n S I n S F := PA PA PB h K E E h 2 K P " E E S P " E E S K B h P K E2 h E K E2 h E2 h K U < (23) Using a stanar matrix inequality [7], for all amissible uncertainties (2) an (9), the following matrix inequality hols: > F P P AR PD S i U PA PA PB h K P S P S K B h P U F (t) E K E2 h PD PD E F (t) PD K E2 h F F E E F E E PD PD F = L: (24) Let A DF (t)e! ~ A = A A(t), [B h B h (t)]k! ~ H h = H h H h (t) an Q K RK = R! Q hen, L =3 Hence, the close-loop systems are asymptotically stable uner heorem On the other han, since the results of the cost boun (4) can be prove by using the similar argument for the proof of heorem, it is omitte REFERECES [] Jalili an E Esmailzaeh, Optimum active vehicle suspensions with actuator time elay, rans ASME, J Dyn Syst Meas Control, vol 23, pp 54 6, 2

6 8 IEEE RASACIOS O CIRCUIS AD SYSEMS I: FUDAMEAL HEORY AD APPLICAIOS, VOL 5, O 6, JUE 23 [2] S O R Moheimani, A V Savkin, an I R Petersen, Synthesis of minimax optimal controllers for uncertain time-elay systems with structure uncertainty, Int J Syst Sci, vol 3, no 2, pp 37 47, 2 [3] Y S Moon, P Park, an W H Kwon, Robust stabilization of uncertain input-elaye systems using reuction metho, Automatica, vol 37, pp 37 32, 2 [4] J H Kim, E Jeung, an H B Park, Robust control for parameter uncertain elay systems in state an control input, Automatica, vol 32, pp , 996 [5] I R Petersen an D C McFarlane, Optimal guarantee cost control an filtering for uncertain linear systems, IEEE rans Automat Contr, vol 39, pp , Sept 994 [6] L Yu, G Chen, an J Chu, Optimal guarantee cost control of linear systems: LMI approach, in Proc 4th IFAC Worl Congress, vol G, Beijing, China, 999, pp [7] H Mukaiani, Y akato, Y anaka, an K Mizukami, he guarantee cost control for uncertain large-scale interconnecte systems, in Proc 5th IFAC Worl Congress, Barcelona, Spain, 22, CD-ROM [8] L Yu an J Chu, An LMI approach to guarantee cost control of linear uncertain time elay systems, Automatica, vol 35, pp 55 59, 999 [9] S O R Moheimani an I R Petersen, Optimal quaratic guarantee cost control of a class of uncertain time-elay systems, Proc Inst Elect Eng Control heory Applicat, vol 44, no 2, pp 83 88, 997 [] S H Esfahani, S O R Moheimani, an I R Petersen, An LMI approach to sub-optimal guarantee cost control for uncertain time-elay systems, Proc IEE Control heory Applicat, vol 45, no 6, pp , 998 [] S H Esfahani an I R Petersen, An LMI approach to output-feebackguarantee cost control for uncertain time-elay systems, Int J Robust onlinear Control, vol, pp 57 74, 2 [2] Z Gajic an M J Qureshi, Lyapunov Matrix Equation in System Stability an Control San Diego, CA: Acaemic, 995 [3] S Boy, L E Ghaoui, E Feron, an V Balakrishnan, Linear Matrix Inequalities is System an Control heory Philaelphia, PA: SIAM, 994 [4] P P Khargoneka, I R Petersen, an K Zhou, Robust stabilization of uncertain linear system: Quaratic stabilizability an H control theory, IEEE rans Automat Contr, vol 35, pp , Mar 99 [5] J C Geromel, J Bernussou, an M C e Oliveira, H -norm optimization with constraine ynamic output feeback controllers: Decentralize an reliable control, IEEE rans Automat Contr, vol 44, pp , July 999 [6] K Zhou, Essentials of Robust Control Englewoo Cliffs, J: Prentice-Hall, 998 [7] Y Wang, D J Hill, an G Guo, Robust ecentralize control for multimachine power systems, IEEE rans Circuits Syst I, vol 45, pp , Mar 998 [8] H Li, S-I iculescu, L Dugar, an J-M Dion, Robust guarantee cost control for uncertain linear time-elay systems, in Stability an Control of ime-delay Systems Lonon, UK: Springer Verlag, 998, pp Pursley s Aperioic Cross-Correlation Functions Revisite ohru Koha an Hiroshi Fujisaki Abstract Pursley s aperioic cross-correlation function of one elay parameter, which plays an important role in the quasi-synchronous state, is revisite Using sequences up-sample by a factor of, we generalize this function to the one with two iscrete elay parameters which play an important role in asynchronous state Furthermore, Markov spreaing sequences are shown to be simply generate by a two-state Markov chain Applying the central limit theorems, in particular, the Fortet Kac theorem to the aperioic cross-correlation function of spreaing sequences with Markovity, we can get theoretical estimate of the variance of multiple-access interference Inex erms Asynchronous irect-sequence coe-ivision-multiple-access (DS/CDMA) system, average interference parameter (AIP), multipleaccess interference (MAI), Markov chains, up-sample sequences I IRODUCIO Consier baseban irect-sequence sprea-spectrum (DS/SS) communications of J users as shown in Fig We efine the ata signal of the jth user an its assigne SS coe signal (j = ; 2; ;J) respectively, by (j) (t) = X (j) (t) = q= X(j) q p= (j) p u (t q c ), an X (j) q u (t p ), (j) p 2 f; g, 2f; g, where ; for t D u D(t) = ; otherwise Since the jth user s coe sequence has perio = = c, it is enote by X (j) = fx (j) q g q= Without loss of generality, we assume c = Let s (j) (t) = X (j) (t) (j) (t) be the jth user s SS moulate signal, an t j be its time elay In an asynchronous DS coe-ivision-multiple-access (DS/CDMA) system, values of the aperioic cross-correlation function R A (`), introuce by Pursley [] etermine the magnitue of multiple-access interference (MAI) from other channels he average interference parameter (AIP), a quaratic form of R A (`), has often been iscusse as a measure of bit-error probabilities in asynchronous DS/CDMA systems [] [8] or as a measure of correlational properties of spreaing sequences [9], [] Recently, by iscussing a variance of MAI with respect to coe symbols, it has been confirme that sequences generate by some Markov chains are superior/inferior [] [4] to sequences of inepenent an ientically istribute (ii) ranom variables in an asynchronous/quasi-synchronous (or chip-synchronous) state [5], [6] In this brief, we show that from the viewpoint of symmetry, it woul be better to evaluate the MAI in terms of the even an o cross-correlation functions rather than in terms of the aperioic cross-correlation function Generalization of the aperioic cross-correlation function /3$7 23 IEEE Manuscript receive September 3, 2; revise April 4, 2, an April 8, 22 his work was presente in part at 2 International Symposium on Circuits an Systems, Geneva, Switzerlan, May 28 3, 2, an at the 2 IEEE International Symposium on Sprea Spectrum echnologies an Applications, Parsippany, J, September 6 8, 2 his paper was recommene by Associate Eitor G Setti Koha is with the Department of Computer Science an Communication Engineering, Kyushu University, Fukuoka , Japan ( koha@cscekyushu-uacjp) H Fujisaki is with the Grauate School of atural Science an echnology, Kanazawa University, Kanazawa , Japan ( Digital Object Ientifier 9/CSI238265