H2/H1 Control of Singularly Perturbed Systems: The ... - Science Direct

European Journal of Control (2010)1:54–69 # 2010 EUCA DOI:10.3166/EJC.16.54–69

H2 =H1 Control of Singularly Perturbed Systems: The State Feedback Case Kanti B. Datta1,, Aparajita RaiChaudhuri2, 1 2

Department of Electrical Engineering, IIT Kharagpur, Kharagpur-721302, India; Department of Electrical Engineering, B.E.S.U, Howrah-711103, India

The design of a mixed H2 =H1 linear state variable feedback (LSVF) suboptimal controller for a continuoustime singularly perturbed system using reduced order slow and fast subsystems is described. It is shown that the designed controller based on reduced order models and the corresponding performance index (PI) both are OðÞ close to those synthesized using the full order system. Each of the three algebraic matrix equations to be sequentially solved in our study, contains three matrix variables coupled in them. Two of these equations can be made independent of the matrix gain variable under mild restrictions. Keywords: Singularly perturbed systems, H2 =H1 control, optimal control, robust control

1. Introduction In robust control problem with parameter uncertainties to have a reasonable performance level guaranteed, we define performance measures as H2 or H1 norm criteria. For parameter uncertainties one can at best minimize one of them keeping a constraint on the other. Such approaches are referred to as constrained optimal control problem (COCP). In H1 -COCP, an H2 performance is minimized simultaneously imposing a constraint on the H1 norm. These problems go by the name of mixed H2 =H1 control. There exists a number

*Correspondence to: K.B. Datta, E-mail: [email protected] **E-mail: [email protected]

of different formulations of these latter problems. To understand this, let Tzi wi , i ¼ 0; 1 denote the closed-loop transfer functions from the exogenous input wi to the controlled output zi . The H1 -COCP is to find an internally stabilizing controller which minimizes kTz0 w0 k2 while maintaining kTz1 w1 k1 < : A controller that solves this problem will ensure that the closed-loop system is robustly stable to all finite gain stable perturbations , interconnected to the system by w1 ¼ z1 , such that kk 1= where the kk is the induced operator norm obtained by taking L2 ½0; 1Þ norms on w1 and z1 . Currently no analytical solution to this problem is available. However, the convex programming approach of Boyd et al. [3] offers a feasible numerical alternative for the solvability of H1 -COCP. A closely related problem was studied by Bernstein and Haddad, in which they have considered w ¼ w0 ¼ w1 and established necessary conditions to minimize an upper bound of kTz0 w k2 subject to the constraint kTz1 w k1 < : These conditions are later found to be sufficient also by Yeh et al. [22], [2]. In the equalized case, z0 ¼ z1 , w0 ¼ w1 the problem of Bernstein and Haddad is equivalent to the maximum entropy problem investigated by Mustafa and Glover [16]. In this case the optimal controller is the so-called central controller. In [7], another version of mixed H2 =H1 problem is studied for a system in which the signal w0 is assumed to be white noise while w1 is a signal with bounded power. This problem is equivalent to the dual of the problem in [2]. Received 9 July 2008; Accepted 26 August 2009 Recommended by J.C. Ge´romel, E.F. Camacho

H2 =H1 Control of SPS with LSVF

Rotea and Khargonekar [20] parameterized the set of all internally stabilizing LSVF controllers K (in such cases we call K is admissible) for the unconstrained H2 -optimal control problem: inffkTz0 w0 ðKÞk2 : K is admissibleg, and derived necessary and sufficient conditions such that a controller in the above set also satisfies kTz1 w1 k1 < 1. These conditions involve two algebraic Riccati equations and a coupling condition. On the other hand, Khargonekar and Rotea [11] focused on the mixed H2 =H1 control problem as formulated by Bernstein and Haddad [2] and developed a design procedure for a static statefeedback gain controller by solving a finite-dimensional convex programming problem over a bounded set of real matrices. In [14], a state-feedback mixed H2 =H1 control problem is solved via the solution of an associated Nash game involving the two cost functions with w0 ¼ w1 ¼ w and leads to a central state-feedback controller as in minimum entropy problem of Mustafa and Glover [16]. Moreover, in [14], it is assumed that the worst case disturbance w ðt; xÞ can be generated by a linear, memoryless feedback strategy which is slightly restrictive but necessary for a unique Nash equilibrium. On the other hand, the H2 -COCP for continuous systems is excellently addressed in [8, 9, 19]. Based on a Riccati equation approach, Petersen and McFarlane [19] developed a construction procedure for an optimal state feedback quadratic guaranteed cost control. Using the quadratic stabilizability framework, Garcia et al. [8] studied the H2 guaranteed cost control for singularly perturbed uncertain systems and showed how to construct a quadratic stabilizing composite control minimizing an upper bound on the H2 norm of a certain transfer matrix. The high frequency phenomena and parameter variations in a dynamical system can be handled by considering a singularly perturbed uncertain model of Kokotovic et al. [12]. In this case, the slow subsystem takes into account low frequency uncertainties and the fast dynamics describes the neglected high-frequency part of the system. Important investigations into uncertain singularly perturbed systems are made in [4, 5, 15]. In [5], the control of uncertain systems which exhibit time-scale separation is studied by translating the problem into a two-frequency scale decomposition for H1 disk problems. A robust stabilizing composite control for singularly perturbed systems with timevarying uncertainties is developed in [4]. In [5], a composite controller is designed with the objective to guarantee stability of the uncertain system by the construction of an appropriate Lyapunov function. In these papers, no optimality properties have been associated with these controllers. In [17, 18], optim-

55

ality through an H1 criterion is proposed under perfect state measurements for a well-known model. In [1] it is established that the mixed H2 =H1 control problem is completely solved if a condition on the rank of the input matrix is assumed. In [6], H1 -COCP is studied for a discrete singularly perturbed system and a mixed H2 =H1 suboptimal linear state-feedback controller is designed based on reduced order models. From above it appears that H2 =H1 optimization for a continuous-time singularly perturbed system still remains an unsolved problem. The objective of the present paper is, therefore, to design a mixed H2 =H1 linear state variable feedback (LSVF) suboptimal controller for a continuous-time singularly perturbed system based on reduced order slow and fast subsystems which are defined independent of the small singular perturbation parameter . For the full-order case, our problem is to minimize an upper bound of kTz0 w k2 subject to the constraint kTz1 w k1 < as in [2]. In the discrete-time case, see [6], a weaker version of this problem is solved by assuming that z0 ¼ z1 , giving rise to a LSVF central controller. The slow and fast subsystems introduced for our present investigation are considerably different from those used in quadratic regulator theory of Kokotovic et al. [12]. The designed slow and fast LSVF gains are combined to produce a composite feedback for the fullorder system with an approach quite different from Kokotovic et al. [12]. The composite feedback gain and also the corresponding quadratic performance index (PI) are OðÞ close to the gain designed with full-order system and the corresponding value of PI.

2. The Full-Order Problem The singularly perturbed systems under consideration is described by, _ ¼ AxðtÞ þ B1 wðtÞ þ B2 uðtÞ; xðtÞ

ð1Þ

z1 ðtÞ ¼ C1 xðtÞ þ D1 uðtÞ

ð2Þ

z0 ðtÞ ¼ S1 xðtÞ þ S2 uðtÞ

ð3Þ

2

A11 xðtÞ ¼ ; A ¼ 4 A21 x2 ðtÞ 2 3 2 3 B11 B12 B1 ¼ 4 B21 5; B2 ¼ 4 B22 5; x1 ðtÞ

3 A12 A22 5; C1 ¼ ½C11

C12

S1 ¼ ½S11

S12 ;

and > 0 is a small parameter, x1 2 Rn1 , x2 2 Rn2 , w 2 Rm1 is the disturbance input with E½wðtÞw0 ðtÞ ¼ I,

56

K.B. Datta and A RaiChaudhuri

u 2 Rm2 is the controlled input, zi ðtÞ 2 Rri ði ¼ 0; 1Þ are the controlled outputs. Other matrices defined in (1)–(3) are of compatible dimensions with ðA; B2 Þ stabilizable, and ðC1 ; AÞ detectable. With a SVF control law x ð4Þ u ¼ Kx ¼ ½ K1 K2 1 x2 the full-order closed-loop system (1)–(3) becomes ~ ~ _ ¼ AxðtÞ xðtÞ þ BwðtÞ

ð5Þ

~ z1 ðtÞ ¼ CxðtÞ

ð6Þ

~ z0 ðtÞ ¼ SxðtÞ

ð7Þ

where

2

3 " # A~11 A~12 B11 A~ ¼ 4 A~21 A~22 5 ¼ A þ B2 K; B~ ¼ B21 ; ð8Þ ~ ~ ~ C ¼ ½C11 C12 ¼ C1 þ D1 K, S~ ¼ S1 þ S2 K, and for i; j ¼ 1; 2 A~ij ¼ Aij þ Bi2 Kj ; C~1i ¼ C1i þ D1 Ki : The transfer-function from wðtÞ to z1 ðtÞ is Tz1 w ðsÞ ¼ ~ AÞ ~ 1 B: ~ Here, symbol tilde denotes all fullCðsI order closed loop quantities containing K. The H2 performance functional to be minimized for the system (5)–(7) is chosen as n o JðKÞ ¼ lim E x0 ðtÞR1 xðtÞ þ 2x0 ðtÞR12 uðtÞ þ u0 ðtÞR2 uðtÞ t!1 ¼ lim E x0 ðS1 þ S2 KÞ0 ðS1 þ S2 KÞx t!1

~R~ ¼ trQ

ð9Þ

where Efg stands for expected value of fg. Above we defined, R1 :¼ S01 S1

R2 :¼ S02 S2

R12 :¼ S01 S2 ;

S1 ¼ ½S11

S12 ;

~ :¼ limt!1 E fxðtÞ R~ :¼ ðS1 þ S2 KÞ0 ðS1 þ S2 KÞ and Q 0 xðtÞ g is the closed-loop state covariance matrix, which satisfies the algebraic Lyapunov equation, [13], ~þQ ~A~0 þ B~B~0 ¼ 0: It can be easily shown that (9) A~Q is equal to the square of the H2 -norm of a transfer ~ AÞ ~ 1 B: ~ So an function given by Tz0 w ðsÞ ¼ SðsI H2 -optimization problem is to minimize (9). The aim of mixed H2 =H1 optimization problem is to minimize H2 PI (9) subject to the constraint kTz1 w k1 ;

> 0:

ð10Þ

This problem is not analytically solvable. Since the full-order algebraic Riccati equation (FOARE) ~ þ QA~0 þ 2 QC~0 CQ ~ þ B~B~0 0 ¼ AQ

ð11Þ

~ see Theorem 1, an upper implies (10), and Q Q, ~R~ is ~ of H2 cost JðKÞ ¼ trQ bound J ðK; QÞ :¼ tr ðQRÞ

minimized simultaneously satisfying (11). This is known as auxiliary minimization problem [2]. So our aim is now transformed to finding the control law (4) in such a way that (a) the closed-loop system (1)–(2) is asymptotically stable, (b) the closed-loop transfer function Tz1 w satisfies the bound (10), and (c) the PI J ðK; QÞ is minimized. With the structure of A~ as shown in (8), it is established by Kokotovic et al. [12] that the structure of Q is of the form given by (39), Q being the solution of the full-order algebraic Riccati equation (11). Theorem 1 [2]: Let K ¼ ½ K1 K2 be given and assume that 9Q 2 Rnn , a nonnegative-definite matrix satisfy~ BÞ ~ is stabilizable if and only if A~ ing (11). Then (a) ðA; is asymptotically stable. In this case, (b) kTz1 w k1 ~ Q: Consequently (d) JðKÞ J ðK; QÞ and (c) Q ~ where, J ðK; QÞ ¼ tr½QR. We now define S :¼ fðK; QÞ : Q > 0 and A~ þ ~ 0C ~ is asymptotically stable}. The task is to 2 QC optimize J ðK; QÞ subject to the constraint (11) over the open set S. The Lagrangian to be minimized is ~ þ QA~0 þ QR~1 Q þ VÞP ~ LðK; Q; PÞ ¼ tr QR~ þ ðAQ ~ V~ ¼ B~B~0 ; and P 2 Rnn is the where R~1 ¼ 2 C~0 C; symmetric Lagrange multiplier. The stationary condi@L @L ¼ 0 and ¼ 0 respectively result in tions @Q @K 0 ¼ ðA~ þ QR~1 Þ0 P þ PðA~ þ QR~1 Þ þ R~

ð12Þ

0 ¼ S02 ðS1 þ S2 KÞ þ fB02 þ 2 D01 ðC1 þ D1 KÞQgP Q: ð13Þ We call (12), the Lagrange multiplier equation, and (13), the gain equation for the full-order system. It follows from (12) that P is nonnegative definite. As the solution is sought in the asymptotic stability set S, a unique solution of P in (12) exists. Under the usual assumptions that D01 ½C1 D1 ¼ ½0 I we have: S02 S2 K þ 2 KQP ¼ B02 P S02 S1 ; which is the Sylvester equation. A full-order controller K can be found by solving coupled equations (11), (12) and ({13), see Section 4.

3. Design with Reduced Order Models Preliminary to any separation of slow and fast designs, the system (1)–(3) is decomposed into a slow system model with n1 small eigenvalues and a fast system model with n2 large eigenvalues. If xij and uj ,


57

i ¼ 1; 2 and j ¼ s; f are the slow and fast parts of the corresponding variables related by xi ¼ xis þ xif , i ¼ 1; 2 and u ¼ us þ uf , then the closed-loop slow and fast subsystems can be described by (14), (15), (17), and (18) given below where we have used ¼ 0 and us ¼ Ks xs and uf ¼ Kf xf in (1)–(3) to generate the closed-loop subsystems. As (71) implies kC~0 ðsI ~0 k ; the slow subsystem can be A~0 Þ1 B~0 þ D defined more logically with the help of (71). Similarly, the fast subsystem can be defined also with the help of (44). However, definition of (16) and (19) given below emerges from the study of the decomposition of full-order Lagrange multiplier equation into slow and fast components corresponding to the terms R~0 in (87) and R~22 in (80), respectively. For each of these subsystems suboptimal H2 =H1 -controller gains are designed separately. It is shown that the composite controller obtained from a combination of the suboptimal slow and fast gains generates the full-order suboptimal H2 =H1 controller’s gain to OðÞ approximation. Symbol bar is used to denote all closed loop quantities for slow and fast subsystems containing slow system gain Ks and fast system gain Kf . This difference in notation is used to emphasize the fact that the controller for a fullorder system and that of the reduced order models are designed by separate techniques, but finally it will be shown that Ks and Kf which form the composite controller are OðÞ close to components K1 and K2 , respectively, of the full-order controller. However, when we consider composite controller gain Kc :¼ ½Ks Kf in our discussion, we use the subscript c to all matrices.

A12 A1 22 A21 ; 1 C12 A22 A21 ;

Bs :¼ B11 A12 A1 22 B21 1 s :¼ C12 A B21 D 22

As :¼ A11 Cs :¼ C11 Af :¼ A22 þ B22 Kf ; Bf :¼ B21 ; Cf :¼ C12 þ D1 Kf Sf ¼ S12 þ S2 Kf ; Ss :¼ ½ðS11 þ S2 Ks Þ ðS12 þ S2 Kf ÞE01 E01 ¼ ½A22 þ 2 Qf C012 C12 1 ½A21 þ 2 Qf C012 C11 Ai1 :¼ Ai1 þ Bi2 Ks ; C11 :¼ C11 þ D1 Ks ; Ai2 :¼ Ai2 þ Bi2 Kf ; C12 :¼ C12 þ D1 Kf ;

for i ¼ 1; 2, where Ks and Kf are the slow and fast controller gains respectively, and A22 is invertible as the fast subsystem is stable in closed-loop. The design of the fast and slow controllers will be described separately in the following subsections. 3.2. Fast Subsystem Design For the system (17)–(19) the fast mixed H2 =H1 control problem is to find Kf such that (i) Af is asymptotically stable, (ii) the fast transfer-function from wðtÞ to zf1 ðtÞ, Tzf1 w ðpÞ ¼ Cf ðpI Af Þ1 Bf ;

p :¼ s

satisfies the constraint kTzf1 w k1 ;

ð20Þ

and (iii) the performance functional, Jf ðKf Þ ¼ lim E xf 0 ðS12 þ S2 Kf Þ0 ðS12 þ S2 Kf Þxf t!1

f Rf ; ¼ tr Q 3.1. Reduced Order Models

ð21Þ

We define the closed-loop slow subsystem by x_ s ¼ As xs þ Bs w

ð14Þ

is minimized, where f :¼ lim E xf ðtÞxf ðtÞ0 Q t!1

s w zs1 ¼ Cs xs þ D

ð15Þ

zs0 ¼ Ss xs

ð16Þ

and the closed-loop fast subsystem in fast time scale by x_ f ¼ Af xf þ Bf w

ð17Þ

zf1 ¼ Cf xf

ð18Þ

zf0 ¼ Sf xf

ð19Þ

where

is the closed-loop fast state covariance matrix, which satisfies the fast algebraic Lyapunov equation, f A0 þ Bf B0 ¼ 0; f þ Q Af Q f f

ð22Þ

Rf :¼ ðS12 þ S2 Kf Þ0 ðS12 þ S2 Kf Þ:

ð23Þ

and

It can be easily shown that (21) is equal to the square of the H2 -norm of the fast subsystem transfer function given by Tzf0 w ðpÞ ¼ Sf ðpI Af Þ1 Bf . To include the

58


disturbance attenuation constraint (20), as in the case of the full-order problem, the Lyapunov’s equation (22) is replaced with an algebraic Riccati equation, the solution of which overbounds the closed-loop steadystate covariance in view of the following theorem.

(26) has a unique solution as ðAf þ 2 Qf C0f Cf Þ is stable. Under the usual mild restriction: D01 ½C1 D1 ¼ ½0 I, Kf is the solution of the Sylvester’s equation S02 S2 Kf þ Kf 2 Qf P f ¼ S02 S12 B022 P f :

Theorem 2: Let Kf be given and assume that 9 Qf 2 Rnn , a nonnegative-definite matrix satisfying 0 ¼ Af Qf þ Qf A0f þ 2 Qf C0f Cf Qf þ Bf B0f

ð24Þ

Then ðaÞ ðAf ; Bf Þ is stabilizable if and only if Af is asymptotically stable. In this case, ðbÞ kTzf1 w k1 and f Qf : ðcÞ Q Consequently, ðdÞ

Jf ðKf Þ J f ðKf ; Qf Þ;

3.3. Slow Subsystem Design The solution of the slow suboptimal control problem for the system (14)–(16) differs from the fast one. In this case the structure of the ARE is more complex and similar to the one in [10]. For clarity the solution is described below in full. Consider the system (14)–(16). The slow mixed H2 =H1 -control problem is to find Ks such that (i) the As is asymptotically stable, (ii) the slow transfer-function from wðtÞ to zs1 ðtÞ, s ; Tzs1 w ðsÞ ¼ Cs ðsI As Þ1 Bs þ D

where J f ðKf ; Qf Þ ¼ tr½Qf Rf . The proof of this theorem follows the line of the proof of Theorem 1. Define the open set S fast :¼ fðKf ; Qf Þ : Qf > 0 and Af þ 2 Qf C0f Cf is asymptotically stable}. As discussed for the full-order problem the fast auxiliary minimization problem is to minimize J f ðKf ; Qf Þ ¼ tr½Qf Rf ;

h n o i Lf ¼ tr Qf Rf þ Af Qf þ Qf A0f þ 2 Qf C0f Cf Qf þ Bf B0f P f :

@Lf @Lf ¼ 0 and ¼ 0 give rise to @Qf @Kf

ð26Þ 0¼

S02 ðS12

þ S 2 Kf Þ þ

fB022

þ

2

D01 ðC12

þ D1 Kf ÞQf gP f Qf ð27Þ

respectively. We call (26), the Lagrange multiplier equation, and (27), the gain equation, for the fast subsystem. A solution of Qf , P f and the desired fast controller gain Kf can be found from the coupled equations (24), (26) and (27), see Section 4. As the solution of (24) is sought over the open set S fast ,

ð28Þ

and (iii) the performance functional, Js ðKs Þ ¼ lim E fxs 0 Rs xs g t!1

s Rs ; ¼ tr Q

ð29Þ

is minimized, where s :¼ lim E xs ðtÞxs ðtÞ0 Q t!1

is the closed-loop slow state covariance matrix, which satisfies the slow algebraic Lyapunov equation, s A0 þ Bs B0 ¼ 0; s þ Q As Q s s

0 ¼ Rf þ P f ðAf þ 2 Qf C0f Cf Þ þ ðAf þ 2 Qf C0f Cf Þ0 P f

kTzs1 w k1 ;

ð25Þ

an overbound of (21), where Qf is a solution to (24), subject to the constraint (24) over the open set S fast . This is equivalent to minimizing the Lagrangian Lf defined by

The conditions

satisfies the constraint

ð30Þ

and Rs : ¼ S0s Ss Ss : ¼ ½ðS11 þ S2 Ks Þ ðS12 þ S2 Kf ÞE01 E01 ¼ ½A22 þ 2 Qf C012 C12 1 ½A21 þ 2 Qf C012 C11 ð31Þ It can be easily shown that (29) is equal to the square of the H2 -norm of the slow subsystem transfer function given by Tzs0 w ðsÞ ¼ Ss ðsI As Þ1 Bs . To include the disturbance attenuation constraint (28), as in the


59

case of the full-order problem, the Lyapunov’s equation (30) is replaced with an algebraic Riccati equation, the solution of which overbounds the closedloop steady-state covariance in view of the following theorem. Theorem 3: Let Ks be given and assume that 9Qs 2 Rnn , a nonnegative-definite matrix satisfying s B0 þ Cs Qs Þ0 0 ¼ As Qs þ Qs A0s þ ðD s 2 0 1 0 ð I Ds D Þ ðDs B þ Cs Qs Þ þ Bs B0 s

s

s

ð32Þ Then ðaÞ ðAs ; Bs Þ is stabilizable if and only if As is asymptotically stable. In this case, ðbÞ kTzs1 w k1 and s Qs : ðcÞ Q Consequently, ðdÞ Js ðKs Þ J s ðKs ; Qs Þ; where J s ðKs ; Qs Þ ¼ tr½Qs Rs . The proof of this theorem follows the line of the proof of Theorem 1. Define the open set S slow :¼ fðKs ; Qs Þ : Qs > 0 and s D 0 þ Qs C0 Þð 2 I D 0 Þ1 Cs is asymptotic½As þ ðBs D s s s ally stable}. As discussed for the full-order problem the slow auxiliary minimization problem is to minimize J s ðKs ; Qs Þ ¼ tr ½Qs Rs ;

ð33Þ

an overbound of (29), where Qs is a solution to (32), subject to the constraint (32) over the open set S slow . This is equivalent to minimizing the Lagrangian Ls defined by n s B0 þ Cs Qs Þ0 Ls ðKs ; Qs ; P s Þ ¼ tr Qs Rs þ As Qs þ Qs A0s :ðD s o s D s B0 þ Cs Qs Þ þ Bs B0 P s : 0 Þ1 ðD ð 2 I D s s s

@Ls @Ls ¼ 0 and ¼ 0, we have @Qs @Ks 1 Cs 0 þ Qs C0 ÞM 0 ¼ P s As þ ðBs D s s s 1 Cs 0 P s þ Rs ; 0 þ Qs C0 ÞM þ As þ ðBs D s s s

s

s

Qs , P s and Ks can be solved iteratively from (32), (34) and (35) respectively, and Ms is assumed to be invertible. We call (34), the Lagrange multiplier equation, and (35), the gain equation, for the slow subsystem. As the solution of (32) is sought over the open set S slow , 0 þ (34) has a unique solution as As þ ðBs D s 0 1 Qs Cs ÞMs Cs is asymptotically stable. The slow problem is solved as done earlier for the fast case. However, the equations involved have more complex forms. As is evident from the equations involved here, the slow problem requires the knowledge of the fast gain Kf and also of Qf , which are already available from the solution of the fast problem. 3.4. Optimal Properties of H2 =H1 Composite Control A composite controller Kc is constructed from the fast controller Kf and the slow controller Ks as Kc :¼ ½Ks j Kf . Let J c denote the value of H2 -PI achieved by this controller Kc . The proposition below establishes near-optimal properties of the composite controller Kc and J c in comparison with the full-order controller K and value of full-order H2 -PI J ðK; QÞ. Proposition 1: If Kf and Ks are the solutions of matrix equations (27) and (35) respectively, then the composite controller defined by Kc :¼ ½Ks Kf ;

ð36Þ

and J c :¼ J c ðKc ; Qc Þ, the value of H2 -PI achieved by using this controller Kc are related to the full-order controller K in (4), respectively, by K ¼ Kc þ OðÞ; J ðK; QÞ ¼ J c ðKc ; Qc Þ þ OðÞ; where Qc satisfies the CARE given by ð37Þ

a subscript c denotes the quantities with a composite control and ð34Þ

s

s D 0 Þ; s :¼ ð 2 I D B2 :¼ B12 A12 A1 M s 22 B22 ; 1 1 :¼ D1 C12 A B22 ; S2 :¼ S12 þ S2 Kf ; D 22

0 ¼ A~c Qc þ Qc A~0c þ 2 Qc C~0c C~c Qc þ B~c B~0c ;

Putting

0 2 0 0 ¼ ½S2 S2 A1 22 ðB22 þ Qf C12 D1 Þ Ss Qs 1 D 0 þ Qs C0 ÞM 1 0 P s Qs ; þ ½B2 þ ðBs D

respectively, where

ð35Þ

~ A~c ¼ A þ B2 Kc ; C~c ¼ C1 þ D1 Kc ; B~c ¼ B1 ¼ B; J c ðKc ; Qc Þ ¼ tr Qc R~c ; R~c :¼ ðS1 þ S2 Kc Þ0 ðS1 þ S2 Kc Þ; Qc1 ¼ Qs þ OðÞ; Qc3 ¼ Qf þ OðÞ and Qc2 ¼ Q2 þ OðÞ: ð38Þ

60


The proof is provided in Section 6.4. For this we need to decompose the full-order Riccati equation, fullorder Lagrange Multiplier equation and full-order gain equation into their slow and fast counterparts. They are included in Sections 6.1, 6.2 and 6.3. The controllers Ks and Kf are optimal only for the slow and the fast subsystems respectively. The above Proposition shows that the composite controller Kc even though does not contain explicitly and defined in terms of Ks and Kf , guarantees an OðÞ approximation of the optimal performance to the full order H2 =H1 control problem. If we apply the control uðtÞ ¼ Ks xs ðtÞ þ Kf xf ðtÞ, then there exists a solution Qc to (37). Indeed, following the procedure given in Section 6.1, the Riccati equation (37) can be decomposed to give us Qs þ OðÞ Qc2 Qc ¼ Q0c2 1 ðQf þ OðÞÞ where Qc2 can be determined with OðÞ approximation in terms of Qs and Qf , see (45). Since, ðKs ; Qs Þ 2 Sslow and ðKf ; Qf Þ 2 Sfast , as 7!0; Qc > 0, and by virtue of Theorem 1, when (37) is satisfied, A~c ¼ A þ B2 Kc , is asymptotically stable, where Kc ¼ ½Ks Kf . The following Proposition 2 is a direct consequence of Proposition 1. Proposition 2: Under the conditions of Proposition 1, kTcz1 w k1 ¼ kTz1 w k1 þ OðÞ; where ~ AÞ ~ B: ~ Tcz1 w ðsÞ ¼ C~c ðsI A~c ÞB~c ; and Tz1 w ðsÞ ¼ CðsI

4. Numerical Experimentation In this section a numerical result is presented to illustrate the effectiveness of the design procedure outlined so far. Consider the system 2 3 2 1 2 x_ 1 ðtÞ 6 7 x1 ðtÞ ¼ 4 3 1 25 x_ 2 ðtÞ x2 ðtÞ 2 3 2 2 3 2 3 1 1 1 6 7 6 7 þ 4 1 5wðtÞ þ 4 1 0 5uðtÞ 0 1 0:1000 0:4000 xðtÞ zðtÞ ¼ 0:4445 0 1:4845 0:10 0:40 uðtÞ þ 0:50 0:11

0 0:2000

For this system the H2 =H1 control problem described in Section 2 was solved using the Robust Control Toolbox in MATLAB. The full-order controller K for ¼ 0:1 and ¼ 1 was found to be 1:3401 1:6377 2:8185 K¼ 0:0906 0:9996 0:3402 The composite controller found using the procedure described in Section 3 was Kc ¼ ½Ks Kf 1:6149 1:6518 ¼ 0:2513 0:9146

2:9090

0:2728

It is evident that K and Kc are OðÞ close.

5. Conclusion In this paper it is shown how to design a mixed H2 =H1 LSVF controller for a singularly perturbed system based on reduced order models of slow and fast subsystems. The design of LSVF controller of the slow subsystem depends on Kf and Qf of the fast subsystem because the H1 design constraint is placed on the closed-loop system. An independent design of the slow and fast reduced order subsystems so as to satisfy the design requirements of the full-order system still remains an unsolved problem. The three design equations for the variables Q, P and K are coupled equations containing these variables which are to be solved recursively. With the assumption that z0 ¼ z1 and under some mild restrictions, it can be shown that the equations for Q, and P can be made independent of K for the full-order and also for the slow and fast subsystems. In this case we get a controller known as a central controller. The H2 =H1 theory of singularly perturbed system can be studied by representing it as a descriptor variable system [21], which remains a topic of future investigation.

6. Appendix 6.1. Decomposition of the Full-Order Riccati Equation The decomposition of full-order ARE (11) into slow and fast equations is done assuming, see [12], Q1 Q2 : ð39Þ Q¼ Q02 1 Q3


61

Substituting from (39), the terms of (11) are now simplified separately as shown below: " # ~12 Q1 ~11 Q2 A A ~ ¼ AQ 0 1 A~21 1 A~22 Q2 1 Q3 " # A~11 Q2 þ 1 A~12 Q3 A~11 Q1 þ A~12 Q02 ¼ 1 ðA~21 Q1 þ A~22 Q0 Þ 2 ðA~21 Q2 þ A~22 Q3 Þ 2

ð40Þ B~B~0 ¼

"

¼

B11 0 B11 B21 B11 B011

1

1 B11 B021

B21 B011

~0

~ ¼ QC CQ 2

1 B021

1

#

Q02

C~11 "

¼ 2

Q2

C~12

Q02

1 Q3 # Q1 C~011 þ Q2 C~012 Q0 C~0 þ 1 Q3 C~0 2

11

12

Writing the elements of (11) in the form of separate equations with OðÞ approximation we have ðiÞ 0 ¼ ðA~11 þ 2 Q2 C~012 C~11 ÞQ1 þ Q1 ðA~11 þ 2 Q2 C~0 C~11 Þ0 12

þ A~12 Q02 þ Q2 A~012 þ 2 ½Q1 C~011 C~11 Q1 þ Q2 C~0 C~12 Q0 þ B11 B0 12

0

ð42Þ

ð46Þ

1

E2 :¼ ðA~12 Q3 þ B11 B021 ÞA~22

ð47Þ

A~22 :¼ A~22 þ 2 Q3 C~012 C~12 ¼ ðI þ 2 Q3 C~0 C~12 A~1 ÞA~22 : 12

ð48Þ

22

Substituting (45) for Q2 in (42) and simplifying, we are led to an equation with Q2 removed as ^ 1 þ Q1 A^0 þ Q1 UQ1 þ B^0 B^ þ OðÞ 0 ¼ AQ

ð49Þ

where A^ :¼ A~11 A~12 E01 2 E2 C~012 C~11 þ 2 E2 C~0 C~12 E0

ð50Þ

U ¼ 2 C^0 C^

ð51Þ

1

B^0 B^ :¼ A~12 E02 E2 A~012 þ B11 B011 þ 2 E2 C~0 C~12 E0

ð52Þ

C^ :¼ C~11 C~12 E01

ð53Þ

12

2

Making use of (46), we have 0

0 1 1 ~0 C^0 ¼ C~011 ½I 2 C~12 Q3 A~22 C12 A~021 A~22 C~012 :

11

2

ðiiÞ 0 ¼ A~12 Q3 þ Q1 A~021 þ Q2 A~022 þ B11 B021 þ 2 ðQ1 C~0 þ Q2 C~0 ÞC~12 Q3 þ OðÞ 11

0 1

E1 :¼ ðA~021 þ 2 C~011 C~12 Q3 ÞA~22

12

½C~11 Q1 þ C~12 Q02 C~11 Q2 þ 1 C~12 Q3

ð45Þ

where

ð41Þ

" ~0 # C11 1 Q3 C~012 Q1 Q2

Q1

Q2 ¼ Q1 E1 þ E2 þ OðÞ;

2 B21 B021 2

the derivation of (71) for the benefit of the readers. From (43), solving for Q2 we get

ð43Þ

12

ðiiiÞ 0 ¼ A~22 Q3 þ Q3 A~022 þ B21 B021 þ 2 Q3 C~0 C~12 Q3 þ OðÞ

Defining the bracketed term by H and simplifying with the help of (48) we have 1 ~0 H :¼ I 2 C~12 Q3 A~22 C12 0

ð44Þ

12

Afterwards (44) will turn out to be the algebraic Riccati equation for designing suboptimal H2 =H1 controller for the fast subsystem to OðÞ approximation. However, in the process of decomposition of Lagrange’s multiplier equation and the gain equation into slow-fast components, we will need some of the intermediate steps in the process of derivation of (71). With this hindsight, we take pain to give a sketch of

0 0 1 1 1 1 ~0 ¼ I 2 C~12 Q3 A~22 ðI þ 2 C~012 C~12 Q3 A~22 Þ Þ C12 0 1 2 ~ 1 0 ¼ ðI þ C12 Q3 A~ C~ Þ

22

12

ð54Þ It is not difficult to show that 2 ~ ~0 ~ 1 A~1 22 :¼ ðA22 þ Q3 C12 C12 Þ 2 2 ~0 ~ ~ ~0 ~ 1 ¼ A~1 22 ½I Q3 C12 C12 ðA22 þ Q3 C12 C12 Þ ¼ A~1 ½I 2 Q3 C~0 H0 C~12 A~1 22

12

22

ð55Þ

62


^ 1 C~12 E0 C~11 Q1 þ C~12 Q02 ¼ CQ 2

ð56Þ

In view of this simplification in (54), C^0 becomes

2 ~ ~0 ~ A^ ¼ A~11 A~12 A~1 22 ðA21 þ Q3 C12 C11 2 Q3 C~0 C~12 E0 Þ 2 E2 C~0 ðC~11 C~12 E0 Þ 12

0 1 ~0 1 C12 C^0 ¼ C~011 ½I þ 2 C~12 Q3 A~22 0

0 1 1 1 ~ 0 A~021 A~22 ½I þ 2 C~012 C~12 Q3 A~22 C 12 0 ¼ C~ H;

0

where

1

12

1

~0 ^ ¼ A~0 2 ðA~12 A~1 22 Q3 þ E2 gC12 C 0 ~ ¼ A~0 2 ½A~12 A~1 22 Q3 þ ðA12 Q3 þ B11 B21 Þ ðA~0 þ 2 C~0 C~12 Q3 Þ1 C~0 C^ 22

12

12

2 ~0 ~ ~0 ¼ A~0 2 ½A~12 A~1 22 fQ3 ðA22 þ C12 C12 Q3 Þ þ A~22 Q3 g þ B11 B021 ðA~022 þ 2 C~012 C~12 Q3 Þ1 C~012 C^ ~0 HC^ ¼ A~0 þ 2 B~0 D 0

~0 M ~ 1 ~ ¼ A~0 þ B~0 D 0 0 C0

~ C~0 :¼ C~11 C~12 A~1 22 A21

ð63Þ

With this simplification, U takes the form where U¼

2

C~00 HH0 C~0 :

~ A~0 :¼ A~11 A~12 A~1 22 A21 ; ~ ~ ~ ~1 ~ ~ B~0 :¼ B~11 A~12 A~1 22 B21 ; C0 :¼ C11 C12 A22 A21

Now, with the help of (44) we can write 2 ~ ~0 ~0 1 ~0 ðHH0 Þ1 ¼ ½I þ 2 C~12 A~1 22 Q3 C12 ½I þ C12 Q3 A22 C12 ¼ I þ 2 C~12 A~1 ½A~22 Q3 þ Q3 A~0 22

22

0 1 ~0 þ 2 Q3 C~012 C~12 Q3 A~22 C12 2 ~ ~1 0 ~0 1 ~0 ¼ I C12 A22 B21 B21 A22 C12 ; ~0 D ~0 ¼ I 2 D

ð64Þ ^ we write E2 with the help of (47) and To simplify B^0 B, (48) in the form 0 1 E2 ¼ ðA~12 Q3 þ B11 B021 2 E2 C~012 C~12 Q3 ÞA~22

ð65Þ

0

~0 ¼: 2 M

Inserting this expression for E2 in (52) ð57Þ

where

0 ~ B^0 B^ ¼ B11 B011 A~12 A~1 22 ðA12 Q3 þ B11 B21 2 E2 C~0 C~12 Q3 Þ0 ðA~12 Q3 þ B11 B0 12

~0 :¼ C~12 A~1 B21 ; D 22

~0 : ~ 0 :¼ 2 I D ~0 D M 0

ð58Þ

This gives us ~ 1 C~0 U ¼ 2 C^0 C^ ¼ C~00 M 0

ð59Þ

To simplify the expression for A^ we will use the following relations which are direct outcomes of (46), (48) and (54):

21

0 1 ~0 2 E2 C~012 C~12 Q3 ÞA~22 A12 þ 2 E2 C~012 C~12 E 02

ð66Þ With the help of (44), the second and fifth terms in (66) can be written as ~ ~0 1 ~0 ~0 A~12 A~1 22 ðQ3 A22 þ A22 Q3 ÞA22 A12 0 ¼ A~12 A~1 ðB21 B0 þ 2 Q3 C~0 C~12 Q3 ÞA~ 1 A~0 22

21

12

22

12

Hence, E1 ¼

ðA~021

þ

2

C~011 C~12 Q3

2

0 1 E1 C~012 C~12 Q3 ÞA~22

ð60Þ

0 0 0 ~0 1 ~0 B^0 B^ ¼ B11 B011 A~12 A~1 22 ðB11 B21 Þ ðB11 B21 ÞA22 A12 þ ðA~12 A~1 B21 ÞðA~12 A~1 B21 Þ0 22

22

0 ~0 ~ þ A~12 A~1 22 ðE2 C12 C12 Q3 Þ 0 þ 2 ðE2 C~0 C~12 Q3 ÞA~ 1 A~0 2

1 1 C~12 A~22 ¼ H0 C~12 A~22

ð61Þ

12

0

1 C~12 Q3 A~22

¼

0

1 HC~12 Q3 A~22

ð62Þ

22

Substituting the expressions from (47)–(48), (60) in (50) and simplifying with (44), (53) and (61), we obtain

22

12

~0 ~ ~0 1 ~0 þ 2 E2 C~012 C~12 E02 A~12 A~1 22 Q3 C12 C12 Q3 A22 A12 ¼ ðB11 A~12 A~1 B21 ÞðB11 A~12 A~1 B21 Þ0 22

~0 ~ ~ ~1 þ ðE2 þ A~12 A~1 22 Q3 ÞC12 C12 ðE2 þ A12 A22 Q3 Þ 2

ð67Þ


63

Now, E2 þ A~12 A~1 22 Q3 ¼ ðA~12 Q3 þ B11 B021 ÞðA~22 þ 2 C~012 C~12 Q3 Þ1 A~12 A~1 22 Q3 0 ¼ ½A~12 A~1 fA~22 Q3 þ Q3 ðA~ 1 þ 2 C~0 C~12 Q3 Þg 22 22 12 1 0 2 ~0 ~ ~ þ B11 B21 ðA22 þ C12 C12 Q3 Þ 1 0 0 2 ~0 ~ ~ A~12 A~1 22 B11 B21 þ B11 B21 ÞðA22 þ C12 C12 Q3 Þ B~0 B021 ðA~22 þ 2 C~012 C~12 Q3 Þ1

¼ ¼

ð68Þ

ðA~ þ QR~1 Þ ¼

A~11 þ 2 ðQ1 C~011 þ Q2 C~012 ÞC~11 1 ðA~21 þ 2 Q3 C~012 C~11 Þ þ Oð1Þ

So,

equation (11) in the last section. It should be noted that all through OðÞ approximation is used wherever necessary. The structure of P is assumed to be of the form, see [12], P¼

P1 P 02

P 2 : P 3

Proceeding with simplification of the terms in (72) we have successively,

A~12 þ 2 ðQ1 C~011 þ Q2 C~012 ÞC~12 : 1 ðA~22 þ 2 Q3 C~012 C~12 Þ þ Oð1Þ

ð73Þ

where we define, ~0 ðE2 þ A~12 A~1 22 Q3 ÞC12 0 0 1 ~0 1 ~0 1 ¼ B~0 B021 A~22 C12 ðI þ 2 C~12 Q3 A~22 C12 Þ 0 ~ H ¼ B~0 D

A11 :¼ A~11 þ 2 ðQ1 C~011 þ Q2 C~012 ÞC~11 A21 :¼ A~21 þ 2 Q3 C~0 C~11 ;

ð69Þ

12

A12

0

A22

Consequently, we have,

0

ð70Þ

0

~ ~0 M ~ 1 ~ ~ ~0 ~ 1 ~ 0 0 ¼ ðA~0 þ B~0 D 0 0 C0 ÞQ1 þ Q1 ðA0 þ B0 D0 M0 C0 Þ ~0 M ~ 1 C~0 Q1 þ B~0 ½I þ D ~ 1 D ~0 B~0 þ OðÞ; þ Q1 C~0 M 0

0

0

A11 A12 P 0 P 3 1 A21 1 A22 2 P 1 A11 þ P 2 A21 P 1 A12 þ P 2 A22 : ¼ P 3 A21 þ OðÞ P 3 A22 þ OðÞ

¼ ½ S11 þ S2 K1

¼

6.2. Decomposition of the Lagrange Multiplier Equation

ð72Þ

and has now to be decomposed into slow and fast subsystem components as done for algebraic Riccati

S12 þ S2 K2 0

½ S11 þ S2 K1 S12 þ S2 K2 " # S~0 1 S~1 S~0 1 S~2

ð76Þ

S~0 2 S~1 S~0 2 S~2 " # R~11 R~12 ¼: 0 R~12 R~22

0

~ 0 ¼ ðA~ þ QR~1 Þ0 P þ PðA~ þ QR~1 Þ þ R;

P 2

R~ ¼ ðS1 þ S2 KÞ0 ðS1 þ S2 KÞ

ð71Þ

The Lagrange multiplier equation (12), which is in fact a Lyapunov equation, is reproduced below

P1

Also,

0

~0 B~0 þ C~0 Q1 Þ0 M ~ 1 0 ¼ A~0 Q1 þ Q1 A~00 þ ðD 0 0 ~0 B~0 þ C~0 Q1 Þ þ B~0 B~0 þ OðÞ: ðD

ð75Þ

which after minor simplification finally reduces to

0

12

PðA~ þ QR~1 Þ ¼

We shall now replace (63)–(70) in (49) to obtain the following Riccati equation in Q1 which has now been made free from Q3 . However, it contains K1 and K2 .

0

ð74Þ

so that,

~0 HH0 D ~0 B~0 B^0 B^ ¼ B~0 B~00 þ 2 B~0 D 0 0 0 ~ 1 ~ ~0 ~ ~ ¼ B0 ½I þ D M D0 B 0

:¼ A~12 þ 2 ðQ1 C~011 þ Q2 C~012 ÞC~12 ; :¼ A~22 þ 2 Q3 C~0 C~12 ;

where ~ 1 :¼ ðS11 þ S2 K1 Þ; S ~ 2 :¼ ðS12 þ S2 K2 Þ; S ~ ~ ~0 S ~0 S R~11 :¼ S R~12 :¼ S 1 1; 1 2; 0 ~ ~ ~0 S ~0 S R~22 :¼ S R~12 :¼ S 2 1; 2 2: ð77Þ

64


Using (75) and (76) in (72), the elements in the resultant matrix are respectively: 0 ¼ R~11 þ P 1 A11 þ P 2 A21 þ A011 P 1 þ A021 P 02 þ OðÞ ð78Þ

ð79Þ 0 ¼ R~22 þ P 3 A22 þ

þ OðÞ

ð80Þ

It will be shown later that equation (80) is similar to the Lagrange multiplier equation for a properly defined fast subsystem of (1)–(3). We will now use (79) and (80) to eliminate P 2 and P 3 from (78). Solving (79) for P 2 we get, P 2 ¼ P 1 X1 þ X2 þ OðÞ;

ð81Þ

where A12 A1 22 ;

X1 ¼

X2 ¼ ðR~12 þ A021 P 3 ÞA1 22

ð82Þ

Substituting P 2 from (81) into (78) gives us 0 ¼ R~11 þ P 1 A11 þ A011 P 1 þ ðP 1 X1 þ X2 ÞA21 þ A021 ðP 1 X1 þ X2 Þ0 þ OðÞ ~ 1 þ P 1 A0 þ A0 P 1 þ X2 A21 þ ðX2 A21 Þ0 ~0 S ¼S

A0 ¼ A11 A12 A1 22 A21 2 ¼ A~11 þ ðQ1 C~011 þ Q2 C~012 ÞC~11 fA~12 þ 2 ðQ1 C~0 þ Q2 C~0 ÞC~12 gE0 11

12

1

¼ A~11 A~12 E01 þ 2 ðQ1 C~011 þ Q2 C~012 ÞðC~11 C~12 E01 Þ ¼ A~11 A~12 E0 2 E2 C~0 ðC~11 C~12 E0 Þ 1

12

1

þ 2 Q1 ðC~11 C~12 E01 Þ0 ðC~11 C~12 E01 Þ ^ ¼ A^ þ 2 Q1 C^0 C; ð85Þ

in which A^ and C^ are given in (63) and (53), see also (59). Putting (85) in (84), the equation for P 1 is ^ þ ðA^ þ 2 Q1 C^0 CÞ ^ 0P1 0 ¼ P 1 ðA^ þ 2 Q1 C^0 CÞ ~1 S ~ 2 A1 A21 Þ0 ðS ~1 S ~ 2 A1 A21 Þ þ OðÞ; þ ðS 22 22 ð86Þ

0

1

ð83Þ where A0 :¼ A11 þ X1 A21 : Equation (83) contains P 3 in X2 but P 2 is eliminated. The terms containing P 3 may be simplified as shown below. We have using (80), 0 0 ~0 S ~ S 1 1 þ X2 A21 þ A21 X2 ~0 S ~0 S ~ 1 ðS ~ 2 þ A0 P 3 ÞA1 A21 ¼S 1

E01 ¼ A1 22 A21 So, making substitutions from (45) and (50) we have

0 ¼ R~12 þ P 1 A12 þ P 2 A22 þ A021 P 3 þ OðÞ A022 P 3

using (43)–(44). Comparing (46) and (74), it is obvious that

1

21

22

0 1 ~ 0 ~ A021 A22 ðS 2 S 1 þ P 3 A21 Þ 0 0 ~ S ~ S ~1 S ~ 2 A1 A21 A0 A0 1 S ~0 S ~1 ¼S

~0 þ Q1 C~0 ÞM ~ 1 C~0 0 ¼ P 1 ½A~0 þ ðB~0 D 0 0 0 ~0 þ Q1 C~0 ÞM ~ 1 C~0 0 P 1 þ ½A~0 þ ðB~0 D 0

0

0

ð87Þ

þ R~0 þ OðÞ; ~1 S ~ 2 A1 A21 Þ0 ðS ~1 S ~ 2 A1 A21 Þ, which where R~0 ¼ ðS 22 22 is similar to Rs in which K1 ; K2 and Q3 are now replaced with Ks ; Kf and Qf respectively. The above equation will be shown in a subsequent section to represent the Lagrange multiplier equation of a properly defined slow subsystem of (1)–(3).

1

¼

1 2 22 21 22 0 0 1 ~ 0 ~ 1 þ A21 A22 S 2 S 2 A22 A21 ~1 S ~ 2 A1 A21 Þ0 ðS ~1 S ~ 2 A1 A21 Þ: ðS 22 22

which after substitution from (63) for A^ and from (59) for C^0 C^ finally reduces to

6.3. Decomposition of the Gain Equation The full-order controller gain K satisfies

With the above simplification (83) transforms to ~1 S ~ 2 A1 A21 Þ0 0 ¼ P 1 A 0 þ A0 P 1 þ ðS 22 ~1 S ~ 2 A1 A21 Þ þ OðÞ; ðS

ð84Þ

22

in which P 2 and P 3 have been eliminated. We will now remove Q2 and Q3 from A0 in (84)

0 ¼ S02 ðS1 þ S2 KÞQ þ ½B02 þ 2 D01 ðC1 þ D1 KÞQPQ; ð88Þ which will be decomposed into two parts, one approximating K1 , the controller’s gain for slow


65

subsystem and the other K2 , the controller’s gain for fast subsystem. Simplifying the individual terms of (88) we have, Q2 Q1 0 0 ðIÞ S2 S1 Q ¼ S2 ½ S11 S12 Q02 1 Q3 1 0 ¼ S2 S11 Q1 þ S12 Q02 S11 Q2 þ S12 Q3 ðIIÞ

K1 Q2 þ 1 K2 Q3

ðIIIÞ

¼ ¼

Q1

P1 1 0 B22 P 02 Q2

Q02

1 Q3

B012

P 2 ¼ P 1 X1 þ X2 þ OðÞ as defined in (45) and (81) in the following way to OðÞ approximation: ðAÞ ðBÞ

S02 S2 KQ ¼ S02 S2 K1 Q1 þ K2 Q02

B02 PQ

Q2 ¼ Q1 E1 þ E2 þ OðÞ;

P 2 P 3

S02 ðS11 Q1 þ S12 Q02 Þ þ S02 S2 ðK1 Q1 þ K2 Q02 Þ ðB022 P 3 þ 2 D01 C~12 Q3 P 3 ÞQ02 ¼ ðB0 P 3 þ 2 D0 C~12 Q3 P 3 ÞE0 Q1

ðCÞ

12

B012 P 1 Q1 þ B022 ðP 02 Q1 þ P 3 Q02 Þ þOðÞ 1 B022 P 3 Q3 þ Oð1Þ

ðDÞ

From the terms (I)–(IV) simplified above, assembling together ð1; 2Þ elements, we have

þ D1 K2 ÞQ3 gP 3 Q3 þ OðÞ;

ð89Þ

which turns out to be the controller’s gain for the fast subsystem described by (27) to OðÞ approximation. On the other hand, the (1,1)-elements lead to 0 ¼ S02 ðS11 Q1 þ S12 Q02 Þ þ S02 S2 ðK1 Q1 þ K2 Q02 Þ þ ðB0 P 3 þ 2 D0 C~12 Q3 P 3 ÞQ0 þ þ

1

ðB022 þ 2 D01 C~12 Q3 ÞP 02 Q1 ¼ ðB0 þ 2 D0 C~12 Q3 ÞX0 P 1 Q1 22

D01 ðC1 þ D1 KÞQPQ 2 D01 fC~11 Q1 P 1 Q1 þ C~12 ðQ02 P 1 Q1 þ Q3 P 02 Q1 þ Q3 P 3 Q02 Þg; 1 D01 C~12 Q3 P 3 Q3 þ Oð1Þ

0 ¼ ½S02 ðS12 þ S2 K2 Þ þ fB022 þ 2 D01 ðC12

1 1 0 ~ D1 C12 Q3 P 3 ÞE02

2 D01 C~12 E02 P 1 Q1

2

¼

þ

2

½B012 þ 2 D01 ðC~11 Q1 þ C~12 Q02 ÞP 1 Q1 ^ 1 P 1 Q1 ¼ B0 P 1 Q1 þ 2 D0 CQ

þ ðIVÞ

22 ðB022 P 3

22 1 2 ½B012 þ 2 D01 ðC~11 Q1 þ C~12 Q02 ÞP 1 Q1 ðB022 þ 2 D01 C~12 Q3 ÞP 02 Q1 þ OðÞ;

ð90Þ

ðB022

1 0 ~ D1 C12 Q3 ÞX02 Q1 ;

where in (C) we have used C^ ¼ C~11 C~12 E01 from (53). If Q is struck out from the right hand side of (88), then the coefficient of Q02 in (A) and all terms contained in (B) will not appear in the (1,1) term. These terms are, in fact, eliminated if (89) is valid after Q3 is removed. However, in the coefficient of Q02 in (A) and terms in (B) will cause a simplification while simplifying coefficients of Q1 , P 1 Q1 , and Q1 P 1 Q1 . The various terms in (A), (B), (C), and (D) are now simplified considering the coefficients of Q1 , P 1 Q1 , and Q1 P 1 Q1 successively. 6.3.1. (I) Coefficients of Q1 Stacking together the terms in (A), (B), and the coefficient of Q1 in (D), we have 0 0 ~ 1 Q1 S0 S ~ S02 S 2 2 ðE1 Q1 þ E2 Þ ðB0 P 3 þ 2 D0 C~12 Q3 P 3 ÞE0 Q1 22

1

1

ðB022 P 3 þ 2 D01 C~12 Q3 P 3 ÞE02 0 ~0 S ~ 1 þ P 3 A21 Q1 ðB0 þ 2 D0 C~12 Q3 ÞA~ 1 ½S 22

which will be shown to be the controller’s gain for the slow subsystem described by (35) to OðÞ approximation. Now, the various terms of (90) is rewritten by setting

þ

1 2

1

22

2

Taking into account (80) : 0 ¼ R~22 þ P 3 A22 þ A022 P 3 þ OðÞ we see that the coefficient of E02 is OðÞ, and hence the above expression becomes

66


~1 S ~ 2 E 0 ÞQ1 ðB 0 þ 2 D0 C~12 Q3 Þ S02 ðS 1 22 1 1 01 ~ 0 ~ ½P 3 A A21 þ A ðS S 1 þ P 3 A21 ÞQ1

So

2 22 22 ~1 S ~ 2 E 0 ÞQ1 ðB 0 þ 2 D0 C~12 Q3 Þ ¼ S02 ðS 1 22 1 0 1 ~0 ~ ½A01 ðA P þ P A ÞA A þ A01 3 22 22 22 3 22 21 22 S 2 S 1 Q1 0 ~ 0 ~0 ~ ¼ ½S20 ðB22 þ 2 D01 C~12 Q3 ÞA01 22 S 2 ðS 1 S 2 E1 ÞQ1

ð91Þ

2 D01 C~12 ðQ3 X10 E20 Þ coefficient of Q1 0 0 1 ~0 1 ~ 0 ~ A12 f 2 Q3 A~22 C12 C12 IgE20 ¼ 2 D01 C~12 ½Q3 A~22 0 0 1 ~0 1 ~ 0 ~ A12 f 2 HC~12 Q3 A~22 C12 C12 ¼ 2 D01 ½HC~12 Q3 A~22 0 1 0 0 ~ ~ ~ C12 gA ðQ3 A þ B21 B Þ

22

¼ ¼

6.3.2. (II) Coefficients of P 1 Q1 The terms in the coefficient of P 1 Q1 are 0 B12 2 D01 C~12 E20 þ ðB022 þ 2 D01 C~12 Q3 ÞX10 :

Now, observing that A~11 ¼ A22 , substituting for X1 from (82) and for Q2 from (45), simplifying with (53), (61), (62) and (55) we have, 0 0 B12 þ B22 X10 0 ~0 1 ~0 0 A22 ½A12 2 C~12 ¼ B012 B22 fC~11 Q1 C~12 ðE10 Q1 þ E20 Þg

¼

0 0 0 1 ~ 0 1 ~0 B22 ðI 2 A~22 C12 HC~12 Q3 ÞA~22 A12 0 2 0 ~ 1 ~ 0 0 ~0 0 ~ B22 A22 C12 ðH C0 Q1 C12 E2 Þ

B012

12

0 ~0 ~ ~1 A~012 þ C~12 A~1 22 Q3 A12 þ C12 A22 B21 B11 2 ~0 ~ ~ ¼ 2 D01 HH0 ½C~12 A~1 22 ðA22 Q3 þ Q3 C12 C12 Q3 0 þ Q3 A~0 ÞA~ 1 A~0 þ C~12 A~1 B21 B0 22

22

0 0 B12 þ B22 X10 coefficient of Q1 0 1 0 1 ¼ B~02 þ 2 B022 A~22 C~012 HC~12 Q3 A~22 A~012 0 1 ~0 0 ~0 þ 2 B022 A~22 C12 HH0 C~12 A~1 22 ðQ3 A12 þ B21 B11 Þ 0 1 ~0 ~ ~1 ¼ B~02 þ 2 B022 A~22 C12 M1 0 C12 A22 0 1 ~0 ½A~22 Q3 þ 2 Q3 C~012 C~12 Q3 þ Q3 A~022 A~22 A12 2 0 ~0 1 ~0 0 ~ ~1 0 þ B22 A22 C12 HH C12 A22 B21 B11 0 1 ~0 0 0 ~0 1 ~0 ~ ~1 C12 M1 ¼ B~02 þ B022 A~22 0 C12 A22 B21 ðB11 B11 A22 A12 Þ 0 1 ~0 ~ ~0 ¼ B~02 B022 A~22 C12 M1 0 D0 B0 ;

where B~2 ¼ B12 A~12 A~1 22 B22 . The remaining terms are 2 D01 C~12 ðQ3 X10 E20 Þ 0 1 ~0 ¼ 2 D10 C~12 ½Q3 A~22 fA12 þ 2 C~012 ðC~11 Q1 þ C~12 Q0 Þg þ E 0 2

¼

2 0 1 ~0 fA12 þ 2 D01 C~12 ½Q3 A~22 2 C~012 C~12 E02 g þ E20

2 C~012 ðC~11 C~12 E10 ÞQ1

12

22

11

0 0 ~0 1 ~ 0 ¼ 2 D01 HH0 C~12 A~1 22 B21 ðB11 B21 A22 A12 Þ ~0 B~0 ¼ D0 M1 D 1

0

0

ð92Þ

6.3.3. (III) Coefficients of Q1 P 1 Q1 Simplification of coefficients of Q1 P 1 Q1 appending those obtained from (II) gives us 1 ~0 1 ~0 2 ½D01 ðI C~12 Q3 A~22 C12 Þ B022 A~22 C12 C^ 0

0

0 0 1 ~0 1 ~0 ¼ 2 ½D01 ðI HC~12 Q3 A~22 C12 Þ B022 A~22 C12 HC^ 0 ¼ 2 ½D0 H B0 A~ 1 C~0 HC^

1

The coefficient of Q1 above is, in fact, a coefficient of Q1 P 1 Q1 , and so transferring it to be included in (III), we have

11

0 1 ~0 0 ~0 A12 þ HH0 A~1 D01 ½HC~12 Q3 A~22 22 ðQ3 A12 þ B21 B11 Þ 0 ~ ~0 ~ ~ 1 2 D01 HH0 ½ðI þ 2 C~12 A~1 22 Q3 C12 C12 ÞC12 Q3 A22

2

22

22

12

~0 M1 C~0 ¼D 1 0 ð93Þ Hence, combining the terms in (91)–(93), the (1,1)element in the full-order gain equation, with ~1 S ~ 2 E 0 Þ, becomes S~0 ¼ ðS 1 ~ 2 A1 ðB22 þ 2 Q3 C~0 D1 Þ0 S~0 Q1 0 ¼ ½S2 S 22 12 ~0 þ Q1 C~0 ÞM1 D ~1 0 P 1 Q1 þ OðÞ; þ ½B~2 þ ðB~0 D 0

0

0

ð94Þ which turns out to be the controller’s gain for the slow subsystem described by (35) to OðÞ approximation. 6.4. Proof of Proposition 1 The set of equations (44), (87) and (89) respectively closely approximate the fast system equations (24), (26) and (27). Hence it follows that K2 ¼ Kf þ OðÞ: Similarly the set of equations (71), (80) and (94) respectively are close approximations of (32), (34) and (35). So, K1 ¼ Ks þ OðÞ:


67

Hence, K ¼ Kc þ OðÞ. With the composite controller Kc , the closed-loop system becomes _ ¼ A~c xðtÞ þ B~c wðtÞ; xðtÞ z1 ðtÞ ¼ C~c xðtÞ;

Writing (11) in the form 0 ¼ ðA~c þ 2 Qc C~0c C~c ÞQ þ QðA~c þ 2 Qc C~0c C~c Þ0 þ 2 QC~0 C~c Q þ B~c B~0 2 Qc C~0 C~c Q c

c

c

2 QðQc C~0c C~c Þ0 þ OðÞ;

where " A~c ¼

A~c11 A~c21 1

A~c12 A~c22

ð98Þ

# we have, subtracting (98) from (97)

1

A~cij ¼ Aij þ Bi2 Kcj ; B~c ¼ B1 ; C~c ¼ ½C~c11 C~c12 ¼ C1 þ D1 Kc ; C~c1i ¼ C1i þ D1 Kci Kc1 ¼ Ks ; Kc2 ¼ Kf ; A~c22 ¼ Af ; B21 ¼ Bf ; C~c12 ¼ Cf : Since Qc satisfies (37), proceeding as in Section 6.1 for the decomposition of ARE we have ~cs B~0 þ C~cs Qc1 Þ0 0 ¼ A~cs Qc1 þ Qc1 A~0cs þ ðD cs ~ 1 ðD ~cs B~0 þ C~cs Qc1 Þ þ B~cs B~0 þ OðÞ; M cs

cs

ð95Þ

cs

0 ¼ A~c22 Qc3 þ Qc3 A~0c22 þ 2 Qc3 C~0c12 C~c12 Qc3 þ B21 B021 þ OðÞ;

0 ¼ ðA~c þ 2 Qc C~0c C~c ÞðQc QÞ þ ðQc QÞðA~c þ 2 Qc C~0 C~c Þ0 c

2 ðQc QÞC~0c C~c ðQc QÞ þ OðÞ: ð99Þ By implicit function theorem, it can be shown that Qc in (37) possesses a power series at ¼ 0 represented by " # X ðiÞ 1 i QðiÞ Qc2 Qc1 Qc2 c1 Qc ¼ þ 0 ðiÞ Q0c2 1 Qc3 i! QðiÞ 1 Qc3 i¼1 c2 ð100Þ

ð96Þ

and so also Q. The representation of Q is the same as Qc but replacing Qc with Q in (100). Thus V :¼ Qc Q can be expanded as " # X ðiÞ 1 i VðiÞ V2 V1 V2 1 þ V¼ ; 0 V02 1 V3 i! VðiÞ 1 VðiÞ i¼1

where ~ A~cs :¼ A~c11 A~c12 A~1 c22 Ac21 ; ~ B~cs :¼ B~11 A~c12 A~1 c22 B21 ; C~cs :¼ C~c11 C~c12 A~1 A~c21

2

c22

3

ð101Þ

~cs :¼ C~c12 A~1 B21 ; D c22 ~0 ~ cs :¼ 2 D ~cs D M cs

which satisfies in view of (99),

As A~c22 ¼ Af ; B21 ¼ Bf ; C~c12 ¼ Cf , it follows from (24) and (96) that Qc3 ¼ Qf þ OðÞ. Similarly, comparing (32) and (95) we have Qc1 ¼ Qs þ OðÞ. Also as in Section 6.1, see (45), we can show

0 ¼ ðA~c þ 2 Qc C~0c C~c ÞV þ VðA~c þ 2 Qc C~0c C~c Þ0 2 VC~0 C~c V þ OðÞ:

Qc2 ¼ Qc1 E1 ðKf ; Qc3 Þ þ E2 ðKs ; Kf ; Qc3 Þ þ OðÞ;

Putting (101) into (102) and simplifying, the zero order term for the (2,2) block element in the resulting matrix equation obtained by putting ¼ 0 becomes

where E1 ðKf ; Qc3 Þ is obtained replacing K2 and Q3 in E1 with Kf and Qc3 and E2 ðKs , Kf ; Qc3 Þ is obtained replacing K1 ; K2 ; Q3 in E2 with Ks ; Kf , and Qc3 using (46) and (47) respectively. This argument shows that Qc2 ¼ Q2 þ OðÞ. To show that J ðK; QÞ ¼ J c ðKc ; Qc Þþ OðÞ, we note that, since K ¼ Kc þ OðÞ, we have C~ ¼ C~c þ OðÞ, and A~ ¼ A~c þ OðÞ in view of (5), (6) and (38). Now, (37) can be written as, 0 ¼ ðA~c þ 2 Qc C~0c C~c ÞQc þ Qc ðA~c þ 2 Qc C~0c C~c Þ0 2 Qc ðQc C~0 C~c Þ0 þ B~c B~0 : c

c

ð97Þ

c

ð102Þ

0 ¼ ðA~c22 þ 2 Qc3 C~0c12 C~c12 ÞV3 þ V3 ðA~c22 þ 2 Qc3 C~0 C~c12 Þ0 c12

ð103Þ

V3 C~0c12 C~c12 V3 : 2

Assuming that ðAf ; Bf Þ is stabilizable and ðAf ; Cf Þ is detectable, it follows from (24) that Af þ 2 Qf C0f Cf is stable. Since, Af ¼ A~c22 ; Cf ¼ C~c12 and Qc3 ¼ Qf , so A~c22 þ 2 Qc3 C~0c12 C~c12 is stable and ðA~c22 ; C~c12 Þ is detectable, which implies that the solution of the above equation (103) gives rise to V3 ¼ 0. In view of

68


this, the (1,2) and (1,1) elements in (102) at ¼ 0 are respectively given by 0 ¼V2 ðA~c22 þ 2 Qc3 C~0c12 C~c12 Þ0 þ V1 ðA~c21 þ 2 Qc3 C~ 0 C~c11 Þ0

J ðK; QÞ J c ðKc ; Qc Þ ¼ tr½ðQ Qc ÞB~B~0 ¼ OðÞ:

c12

0 0 0 ¼ ½A~c11 þ 2 ðQc1 C~c11 þ Qc2 C~c12 ÞC~c11 V1 2 0 þ ½A~c12 þ ðQc1 C~ þ Qc2 C~ 0 ÞC~c12 V 0 c11

c11

c12

0 0 ½ðV1 C~c11 þ V2 C~c12 ÞC~c11 V1 þ ðV1 C~ 0 þ V2 C~ 0 ÞC~c12 V 0 : 2

2

c12

With the help of first equation eliminating V2 from the second equation, left coefficient of V1 becomes, 0 0 0 0 2 Ec2 C~c12 ðC~c11 C~c12 Ec1 Þ Cc ¼ A~c11 A~c12 Ec1 0 þ 2 Qc1 ðC~c11 C~c12 E 0 Þ ðC~c11 C~c12 E 0 Þ c1

c1

¼ A^c þ 2 Qc1 C^c C^c0 ~0 þ Qc1 C~ 0 ÞM ~ 1 C~cs ¼ A~cs þ ðB~cs D cs cs cs where we have used the notations, A^c :¼ A~c11 A~c12 E0c1 2 Ec2 C~0c12 ðC~c11 C~c12 E0c1 Þ0 C^c :¼ C~c11 C~c12 E0c1 ; Qc2 ¼ Qc1 Ec1 þ Ec2 þ OðÞ; 0 Ec1 :¼ ðA~0c21 þ 2 C~0c11 C~c12 Qc3 ÞðA~c22 þ 2 Qc3 C~0c12 C~c12 Þ 1 ; 0 Ec2 :¼ ðA~c12 Qc3 þ B11 B021 ÞðA~c22 þ 2 Qc3 C~0c12 C~c12 Þ 1 ;

which follows from the observation that the decomposition of (37) will generate similar equations as given in Section 6.1 inserting only an additional subscript c in the variables. Simplifying other terms, the (1,1) element finally becomes ~ 1 C~cs V1 : 0 ¼ Cc V1 þ V1 C0c V1 C~0cs M cs Since Cc is stable, and ðA~cs ; C~cs Þ is detectable, which follow from the property of the slow-subsystem, the solution of the above equation gives rise to V1 ¼ 0, and consequently, V2 ¼ 0. Now, the first order term for the (2,2) block element obtained from (101) and (102) with V1 ¼ 0 ; V2 ¼ 0 and V3 ¼ 0 is ð1Þ 0 ¼ ðA~c22 þ 2 Qc3 C~0c12 C~c12 ÞV3

þ V3 ðA~c22 þ 2 Qc3 C~0c12 C~c12 Þ0 ð1Þ

ð1Þ ð1Þ 2 V3 C~0c12 C~c12 V3 þ OðÞ;

References

2

c12

0 0 þ V1 ½A~c11 þ 2 ðQc1 C~c11 þ Qc2 C~c12 ÞC~c11 0 þ V2 ½A~c12 þ 2 ðQc1 C~ 0 þ Qc2 C~ 0 ÞC~c12 0

c11

which is of the same form as (103) except for the OðÞ ð1Þ term. Arguing similarly we have, V3 ¼ OðÞ: Thus it is evident that V ¼ Qc Q ¼ OðÞ: Hence,

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