RECURSIVE COMPUTATION OF PARETO OPTIMAL STRATEGY FOR MULTIPARAMETER SINGULARLY PERTURBED SYSTEMS

RECURSIVE COMPUTATION OF PARETO OPTIMAL STRATEGY FOR MULTIPARAMETER

SINGULARLY PERTURBED SYSTEMS

Hiroaki Mukaidani¹, Hua Xu² and Koichi Mizukami³

1Faculty of Information Sciences Hiroshima City University, 3-4-1, Ozuka-Higashi,

Asaminami-ku, Hiroshima, 731-3194Japan

2Graduate School of Business Sciences The University of Tsukuba, 3-29-1, Otsuka,

Bunkyou-ku, Tokyo, 112-0012 Japan

3Faculty of Engineering

Hiroshima Kokusai Gakuin University, 6-20-1, Nakano Aki-ku, Hiroshima, 739-0321 Japan

Abstract. In this paper, we study the Pareto optimal strategy for multiparameter singu- larly perturbed system (MSPS). In order to obtain the solution, we must solve the multipa- rameter algebraic Riccati equations (MARE). The main results in this paper are to propose a new recursive algorithm for solving the MARE and to find sufficient conditions regarding the convergence of our proposed algorithm. Using the recursive algorithm, we show that the solution of the MARE converges to a positive semi-definite stabilizing solution with the rate of convergence ofO(||µ||ⁱ⁺¹) under the sufficient conditions. Furthermore, we also show that the near-optimal strategy achieves the cost functionalJ_j^∗+O(||µ||ⁱ⁺¹).

Keywords. multiparameter singularly perturbed system (MSPS), multiparameter alge- braic Riccati equations (MARE), Pareto optimal strategy, recursive algorithm, Pareto near-optimal strategy.

AMS (MOS) subject classification: 34K26, 34K28

1 Introduction

Multimodeling stability, control and filtering problems have been investi- gated extensively (see e.g., [1-4,8-10,14]). The multimodeling problems arise in large scale dynamic systems. For example, these multimodel situations in practice are illustrated by the multi-area power system [8] and the passenger car model [2]. In order to obtain the optimal solution to the multimodel- ing problems, we must solve the multiparameter algebraic Riccati equation (MARE), which is parameterized by two small positive parameters ε₁and ε₂ of the same order. Various reliable approaches to the theory of the algebraic Riccati equation (ARE) have been well documented in the literature (see e.g., [6,11]). One of the approaches is the invariant subspace approach which

is based on the Hamiltonian matrix [11]. However, such an approach is not adequate to the multiparameter singularly perturbed systems (MSPS) since the dimension of the required workspace to carry out the calculations for the Hamiltonian matrix is twice the dimension of the original full-system. An- other disadvantage is that, for the computed solution there is no guarantee of symmetry when the ARE is ill-conditioned [11]. Note that it is very difficult to solve the MARE due to high dimension and numerical stiffness [1,2].

A popular approach to deal with the MSPS is the two-time-scale design method [3,4,8-10,14]. Recently, there has been interest in nonstandard sin- gularly perturbed systems [7,14] such that the fast state matrices may be singular. In [7], the linear-quadratic regulator problem for nonstandard sin- gularly perturbed systems has been studied. The results are then extended to the near-optimal control problem of nonstandard multiparameter/multitime scale singularly perturbed systems [14]. In [9,14], the resulting controllers for standard or nonstandard MSPS are proven to have the property of O(||µ||) (where µ =

ε₁ ε₂ 

) near-optimality. However, it is known from [1] that an O(||µ||) accuracy is very often not sufficient. More recently, the exact slow- fast decomposition method for solving the MARE of the MSPS has been proposed [1,2]. The solutions are obtained by solving the non-symmetric Sylvester equations of lower dimensions by means of the Newton method or the fixed point algorithm. However, these results are restricted to the MSPS such that the Hamiltonian matrices for the fast subsystems have no eigenval- ues in common ( Assumption 5, [2]). Hence, for example, we cannot apply the method proposed in [1,2] to the Pareto optimal strategy of a multi-area power system which was considered in [8]. Moreover, in order to obtain the exact solution, one needs the same workspace as the full-order MARE for calculating the inverse matrix. On the other hand, the recursive algorithm for the solution of the ARE of ordinary singularly perturbed systems has been developed in many literatures (see e.g., [5]). From a practical point of view, it has been shown that the recursive algorithm is very effective to solve the ARE when the system matrices are functions of a small perturbation parameter. However, the recursive algorithm for solving the MARE of the MSPS has not been investigated so far.

In this paper, we study the Pareto optimal strategy for the MSPS. We first investigate the uniqueness and boundedness of the solution to the MARE and establish its asymptotic structure. The proof of the existence of the solution to the MARE with asymptotic expansion is obtained by an implicit function theorem [3,5]. This paper presents an improvement on some of the results of [3] in the sense that some assumptions are relaxed. We also investigate the stabilizability and detectability for the reduced-order ARE. The main contribution of this paper is to propose a new recursive algorithm for solving the MARE and to find the sufficient conditions regarding the convergence of the recursive algorithm by using the reduced-order ARE. It is important to note that the sufficient conditions derived here are independent of the small perturbation parameter µ. Furthermore, it is worth pointing out that

the numerical approach which is based on the recursive algorithm has never been studied. The resulting strategy is obtained by the solution to the MARE which is calculated by using the new recursive algorithm. We prove that the solution of the MARE converges to a positive semi-definite stabilizing solution with the rate of convergence of O(||µ||ⁱ⁺¹), where i is the iteration number. In addition, we show that the proposed strategy achieves the cost functional J_j^∗+O(||µ||ⁱ⁺¹), j = 1, 2, where J_j^∗is the optimal cost. As another important feature, we do not assume here that A_jj, j = 1, 2, is nonsingular nor that the Hamiltonian matrices T_jj, j = 1, 2 for the fast subsystems have no eigenvalues in common. Thus, our new results are applicable to more realistic MSPS. Moreover, it is easy to apply our analysis to the optimal regulator problem for the MSPS because the solution of such problem is a special case of the Pareto optimal strategy when the decision makers agree on a choice of a weighting factors.

2 Pareto Optimal Strategy

We consider the linear time–invariant MSPS

˙x₀(t) = A₀₀x₀(t) + A₀₁x₁(t) + A₀₂x₂(t) + B₀₁u₁(t) + B₀₂u₂(t),(1a) x₀(0) = x⁰₀,

ε₁˙x₁(t) = A₁₀x₀(t) + A₁₁x₁(t) + B₁₁u₁(t), x₁(0) = x⁰₁, (1b) ε₂˙x₂(t) = A₂₀x₀(t) + A₂₂x₂(t) + B₂₂u₂(t), x₂(0) = x⁰₂, (1c)

where x_j ∈ R^nj, j = 0, 1, 2 are the state vectors, u_j ∈ R^mj, j = 1, 2 are the control inputs. All the matrices are constant matrices of appropriate dimensions.

ε₁ and ε₂are two small positive singular perturbation parameters of the same order of magnitude such that

0 < k₁≤ α ≡ ε₁

ε₂ ≤ k₂<∞. (2)

That is, we assume that the ratio of ε₁ and ε₂ is bounded by some positive constants k_j, j = 1, 2. Note that the fast state matrices A_jj, j = 1, 2 may be singular. The system (1) is called the nonstandard MSPS if the matrix A_jj is singular. In the Pareto optimal strategy of the above MSPS, a quadratic cost functional is given by

J_j = 1 2

 _∞

0

[z_j^T(t)z_j(t) + u^T_j(t)R_ju_j(t)]dt, j = 1, 2, (3) where z_j(t) = C_j0x₀(t) + C_jjx_j(t)∈ R^rj, R_j> 0, j = 0, 1, 2.

A Pareto solution is a pair u₁, u₂which minimizes

J = γ₁J₁+ γ₂J₂, 0 < γ_j < 1, γ₁+ γ₂= 1 (4)

for some γ₁ and γ₂. It is well known that the solution of the Pareto optimal strategy is given by [8]

u^∗_j(t) =−1

γ_jR⁻¹_j B_je^TP_ex(t), j = 1, 2, (5) where P_e satisfies the MARE

A^T_eP_e+ P_eA_e− P_eS_eP_e+ Q = 0, (6) with

A_e=



 A₀₀ A₀₁ A₀₂

ε⁻¹₁ A₁₀ ε⁻¹₁ A₁₁ 0 ε⁻¹₂ A₂₀ 0 ε⁻¹₂ A₂₂



 ,

B_1e=



 B₀₁ ε⁻¹₁ B₁₁

0



 , B_2e=



 B₀₂ 0 ε⁻¹₂ B₂₂



 ,

Q₁=



 C₁₀^TC₁₀ C₁₀^TC₁₁ 0 C₁₁^TC₁₀ C₁₁^TC₁₁ 0

0 0 0



 , Q₂=



 C₂₀^TC₂₀ 0 C₂₀^TC₂₂

0 0 0

C₂₂^TC₂₀ 0 C₂₂^TC₂₂



 ,

S_je= B_jeR⁻¹_j B_je^T, j = 1, 2, S_e= 1

γ₁S_1e+ 1

γ₂S_2e, Q = γ₁Q₁+ γ₂Q₂. Since the matrices A_eand B_jecontain terms of order ε⁻¹_j , j = 1, 2, a solution P_e of (6), if it exists, must contain terms of order ε_j. Taking this fact into consideration, we look for a solutions P_eto the MARE (6) with the structure

P_e=



 P₀₀ ε₁P₁₀^T ε₂P₂₀^T ε₁P₁₀ ε₁P₁₁ √ε₁ε₂P₂₁^T ε₂P₂₀ √ε₁ε₂P₂₁ ε₂P₂₂



 ∈ R^N×N, where N = n₀+ n₁+ n₂, P₀₀= P₀₀^T, P₁₁= P₁₁^T, P₂₂= P₂₂^T.

A near–optimal Pareto strategy for the MSPS has been proposed in [8].

The algorithm consists of solving three separate subproblems, one in a slow time scale and two in fast time scale, and then combining the solutions of these problems to the specific form of the control law. However, in order to separate the MSPS, the nonsingularity of the matrices A_jj, j = 1, 2 are required. To avoid these assumptions we propose an iterative method which is different from the exact slow–fast decomposition method [1,2].

3 Multiparameter algebraic Riccati equation (MARE)

Before we present the high–order approximate Pareto strategy, we first study the asymptotic structure for the MARE (6). The MARE (6) can be parti-

tioned into

f₁= A^T₀₀P₀₀+ P₀₀A₀₀+ A^T₁₀P₁₀+ P₁₀^TA₁₀+ A^T₂₀P₂₀+ P₂₀^TA₂₀

−P00S₀₀P₀₀− P₁₀^TS^T₀₁P₀₀− P00S₀₁P₁₀

−P₂₀^TS₀₂^TP₀₀− P₀₀S₀₂P₂₀− P₁₀^TS₁₁P₁₀− P₂₀^TS₂₂P₂₀

+Q₀₀= 0, (7a)

f₂= P₀₀A₀₁+ P₁₀^TA₁₁+ ε₁A^T₀₀P₁₀^T + A^T₁₀P₁₁+√

αA^T₂₀P₂₁

−ε1(P₀₀S₀₀P₁₀^T + P₁₀^TS₀₁^TP₁₀^T + P₂₀^TS^T₀₂P₁₀^T)

−P00S₀₁P₁₁− P₁₀^TS₁₁P₁₁−√

α(P₀₀S₀₂P₂₁+ P₂₀^TS₂₂^TP₂₁)

+Q₀₁= 0, (7b)

f₃= P₀₀A₀₂+ P₂₀^TA₂₂+ ε₂A^T₀₀P₂₀^T + A^T₂₀P₂₂+ 1

√αA^T₁₀P₂₁^T

−ε2(P₀₀S₀₀P₂₀^T + P₁₀^TS₀₁^TP₂₀^T + P₂₀^TS^T₀₂P₂₀^T)

−P₀₀S₀₂P₂₂− P₂₀^TS₂₂P₂₂− 1

√α(P₀₀S₀₁P₂₁^T + P₁₀^TS₁₁P₂₁^T)

+Q₀₂= 0, (7c)

f₄= A^T₁₁P₁₁+ P₁₁A₁₁+ ε₁(A^T₀₁P₁₀^T + P₁₀A₀₁)

−ε₁(ε₁P₁₀S₀₀P₁₀^T + P₁₁S₀₁^TP₁₀^T +√

αP₂₁^TS^T₀₂P₁₀^T)

−ε1(P₁₀S₀₁P₁₁+√

αP₁₀S₀₂P₂₁)

−P₁₁S₁₁P₁₁− αP₂₁^TS₂₂P₂₁+ Q₁₁= 0, (7d) f₅= ε₁P₁₀A₀₂+ ε₂A^T₀₁P₂₀^T

−ε2(ε₁P₁₀S₀₀P₂₀^T + P₁₁S₀₁^TP₂₀^T +√

αP₂₁^TS^T₀₂P₂₀^T)

−ε₁(P₁₀S₀₂P₂₂+ 1

√αP₁₀S₀₁P₂₁^T)

+√

αP₂₁^T(A₂₂− S₂₂P₂₂) + 1

√α(A₁₁− S₁₁P₁₁)^TP₂₁^T = 0, (7e) f₆= A^T₂₂P₂₂+ P₂₂A₂₂+ ε₂(A^T₀₂P₂₀^T + P₂₀A₀₂)

−ε2(ε₂P₂₀S₀₀P₂₀^T + P₂₂S₀₂^TP₂₀^T + 1

√αP₂₁S₀₁^TP₂₀^T)

−ε₂(P₂₀S₀₂P₂₂+ 1

√αP₂₀S₀₁P₂₁^T)

−P22S₂₂P₂₂− 1

αP₂₁S₁₁P₂₁^T + Q₂₂= 0. (7f) It is assumed that the limit of α exists as ε₁and ε₂tend to zero [1-4,8-10,14], that is

¯

α = lim

ε1→+0 ε2→+0

α. (8)

Let us consider the following equations (9) which are associated to the

above equation (7) setting formally ε_j → +0, j = 1, 2.

A^T₀₀P¯₀₀+ ¯P₀₀A₀₀+ A^T₁₀P¯₁₀+ ¯P₁₀^TA₁₀+ A^T₂₀P¯₂₀+ ¯P₂₀^TA₂₀

− ¯P₀₀S₀₀P¯₀₀− ¯P₁₀^TS₀₁^TP¯₀₀− ¯P₀₀S₀₁P¯₁₀− ¯P₂₀^TS₀₂^TP¯₀₀− ¯P₀₀S₀₂P¯₂₀

− ¯P₁₀^TS₁₁P¯₁₀− ¯P₂₀^TS₂₂P¯₂₀+ Q₀₀= 0, (9a) P¯₀₀A₀₁+ ¯P₁₀^TA₁₁+ A^T₁₀P¯₁₁+√

¯

αA^T₂₀P¯₂₁− ¯P₀₀S₀₁P¯₁₁

− ¯P₁₀^TS₁₁P¯₁₁−√

¯

α( ¯P₀₀S₀₂P¯₂₁+ ¯P₂₀^TS₂₂P¯₂₁) + Q₀₁= 0, (9b) P¯₀₀A₀₂+ ¯P₂₀^TA₂₂+ A^T₂₀P¯₂₂+ 1

√α¯A^T₁₀P¯₂₁^T − ¯P₀₀S₀₂P¯₂₂

− ¯P₂₀^TS₂₂P¯₂₂− 1

√α¯( ¯P₀₀S₀₁P¯₂₁^T + ¯P₁₀^TS₁₁P¯₂₁^T) + Q₀₂= 0, (9c) A^T₁₁P¯₁₁+ ¯P₁₁A₁₁− ¯P₁₁S₁₁P¯₁₁− ¯α ¯P₂₁^TS₂₂P¯₂₁+ Q₁₁= 0, (9d)

√α ¯¯P₂₁^T(A₂₂− S22P¯₂₂) + 1

√α¯(A₁₁− S11P¯₁₁)^TP¯₂₁^T = 0, (9e)

A^T₂₂P¯₂₂+ ¯P₂₂A₂₂− ¯P₂₂S₂₂P¯₂₂−1

¯ α

P¯₂₁S₁₁P¯₂₁^T + Q₂₂= 0. (9f) The Pareto optimal strategy for the MSPS will be studied under the follow- ing control oriented assumptions [1,2,8,14]. In particular, note that we need to generalize the assumption about stabilizability and detectability, so that they can be applied to the nonstandard MSPS.

Assumption 1. The triples (A_jj, B_jj, C_jj), j = 1, 2 are stabilizable and detectable.

Assumption 2.

rank



 sI_n0− A00 −A01 −A02 B₀₁ B₀₂

−A10 −A11 0 B₁₁ 0

−A20 0 −A22 0 B₂₂



 = N, (10a)

rank



 sI_n0− A^T₀₀ −A₁₀^T −A^T₂₀ C₁₀^T C₂₀^T

−A^T₀₁ −A^T₁₁ 0 C₁₁^T 0

−A^T₀₂ 0 −A^T₂₂ 0 C₂₂^T



 = N, (10b)

where∀s ∈ C with Re[s] ≥ 0.

If the assumption 1 holds, there exist the matrices ˜P_jj≥ 0, j = 1, 2 such that the matrices A_jj−S_jjP˜_jjare stable, where A^T_jjP˜_jj+ ˜P_jjA_jj− ˜P_jjS_jjP˜_jj+ Q_jj = 0. If we chose ¯P_jj to be ˜P_jj, the unique solution of (9e) is given by P¯₂₁ = 0 since the matrices A_jj− S_jjP¯_jj are stable. Thus the parameter ¯α does not appear in (9), that is, it does not affect the equation (9) in the limit when ε₁and ε₂tend to zero. Therefore, we obtain the following zeroth order equations

A^T_sP¯₀₀+ ¯P₀₀A_s− ¯P₀₀S_sP¯₀₀+ Q_s= 0, (11a) P¯_j0^T = ¯P₀₀N_0j− M_0j, j = 1, 2, (11b) A^T_jjP¯_jj+ ¯P_jjA_jj− ¯P_jjS_jjP¯_jj+ Q_jj= 0, j = 1, 2, (11c)

where

A_s= A₀₀+ N₀₁A₁₀+ N₀₂A₂₀+ S₀₁M₀₁^T + S₀₂M₀₂^T +N₀₁S₁₁M₀₁^T + N₀₂S₂₂M₀₂^T,

S_s= S₀₀+ N₀₁S^T₀₁+ S₀₁N₀₁^T + N₀₂S₀₂^T + S₀₂N₀₂^T +N₀₁S₁₁N₀₁^T + N₀₂S₂₂N₀₂^T,

Q_s= Q₀₀− M01A₁₀− A^T₁₀M₀₁^T − M02A₂₀− A^T₂₀M₀₂^T

−M₀₁S₁₁M₀₁^T − M₀₂S₂₂M₀₂^T,

N_0j=−D0jD_jj⁻¹, M_0j = ¯Q_0jD_jj⁻¹, ¯Q_0j= A^T_j0P¯_jj+ Q_0j, D₀₀= A₀₀− S00P¯₀₀− S01P¯₁₀− S02P¯₂₀,

D_0j= A_0j− S_0jP¯_jj, D_j0= A_j0− S_0j^TP¯₀₀− S_jjP¯_j0, D_jj= A_jj− S_jjP¯_jj, j = 1, 2.

The matrices A_s, S_s and Q_s do not depend on ¯P₁₁ and ¯P₂₂ because their matrices can be computed by using T_pq, p, q = 0, 1, 2 which is independent of ¯P₁₁and ¯P₂₂ [1,2], that is,

T_s= T₀₀− T₀₁T₁₁⁻¹T₁₀− T₀₂T₂₂⁻¹T₂₀=

 A_s −S_s

−Qs −A^T_s

,

T₀₀=

 A₀₀ −S₀₀

−Q00 −A^T₀₀

, T_0j=

 A_0j −S0j

−Q0j −A^T_j0

,

T_j0=

 A_j0 −S_0j^T

−Q^T_0j −A^T_0j

, T_jj =

 A_jj −Sjj

−Qjj −A^T_jj

, j = 1, 2.

Lemma 1. Under the assumptions 1 and 2, there exist a matrix Bs ∈ R^n0×M, M = m₁+ m₂and a matrix C_swith the same dimension as

 C₁₀ C₂₀

such that S_s = B_sR⁻¹B_s^T, R =

 R₁ 0 0 R₂

, Q_s = C_s^TC_s. Moreover, the triple (A_s, B_s, C_s) is stabilizable and detectable.

Proof. From (11a), it is easy to verify that S_s=
B¯₀₁+ N₀₁B¯₁₁ B¯₀₂+ N₀₂B¯₂₂   R⁻¹₁ 0

0 R⁻¹₂

 B¯₀₁^T + ¯B₁₁^TN₀₁^T B¯₀₂^T + ¯B₂₂^TN₀₂^T

,

where ¯B_0j= 1

√γ_jB_0j, ¯B_jj= 1

√γ_jB_jj, j = 1, 2.

Thus, we have

B_s=
B¯₀₁+ N₀₁B¯₁₁ B¯₀₂+ N₀₂B¯₂₂  .

However, it seems difficult to find C_s from (11a). In order to do that, we introduce a dual ARE

W˜_jjA^T_jj+ A_jjW˜_jj− ˜W_jjQ_jjW˜_jj+ S_jj= 0, j = 1, 2, (12)

which admits at least a symmetric positive semidefinite solution ˜W_jj under the assumption 1. Using (12), we find that

T_jj=

 I_j − ˜W_jj 0 I_j

·

 E_jj 0

−Qjj −E_jj^T

·

 I_j W˜_jj 0 I_j

, j = 1, 2,

where E_jj= A_jj− QjjW˜_jj, j = 1, 2 is stable under the assumption 1. After the calculation of T_s, we arrive at another expression for Q_s, that is,

Q_s = Q₀₀+ L^T₁₀Q^T₀₁+ Q₀₁L₁₀+ L^T₂₀Q^T₀₂+ Q₀₂L₂₀ +L^T₁₀Q₁₁L₁₀+ L^T₂₀Q₂₂L₂₀,

where

L_j0=−E_jj⁻¹E_j0, E_j0= A_j0− ˜W_jjQ^T_0j, j = 1, 2.

Hence, it is easy to find that

Q_s=
C¯₁₀^T + L₁₀^TC¯₁₁^T C¯₂₀^T + L^T₂₀C¯₂₂^T   ¯C₁₀+ ¯C₁₁L₁₀ C¯₂₀+ ¯C₂₂L₂₀

= C_s^TC_s, where ¯C_j0=√γ_jC_j0, ¯C_jj=√γ_jC_jj, j = 1, 2.

Let us now prove the second part of the lemma, namely stabilizability and detectability. Note the relation



 I_n0 −D₀₁D₁₁⁻¹ −D₀₂D⁻¹₂₂

0 D⁻¹₁₁ 0

0 0 D₂₂⁻¹





·



 sI_n0− A00 −A01 −A02 B¯₀₁ B¯₀₂

−A₁₀ −A₁₁ 0 B¯₁₁ 0

−A₂₀ 0 −A₂₂ 0 B¯₂₂





·









I_n0 0 0 0 0

−D⁻¹₁₁A₁₀ I_n1 0 π₂₁ 0

−D⁻¹₂₂A₂₀ 0 I_n2 0 π₂₂ π₁₁ π₃₁ 0 π₄₁ 0 π₁₂ 0 π₃₂ 0 π₄₂









=



 sI_n0− A₀₀− N₀₁A₁₀− N₀₂A₂₀ 0 0

0 I_n1 0

0 0 I_n2

B¯₀₁+ N₀₁B¯₁₁ B¯₀₂+ N₀₂B¯₂₂

0 0

0 0



 , (13)

where

π_1j=−R⁻¹_j B¯_jj^TP¯_jjD₁₁⁻¹A_j0, π_2j= D_jj⁻¹B¯_jj, π_3j = R⁻¹_j B¯_jj^TP¯_jj π_4j= I_nj+ R⁻¹_j B¯_jj^TP¯_jjD_jj⁻¹B¯_jj, j = 1, 2.

Hence, the pair (A_s, B_s) is stabilizable if and only if rank[sI_n0 − A₀₀− N₀₁A₁₀− N₀₂A₂₀ B_s] = n₀, ∀s ∈ C with Re[s] ≥ 0. In other words, the matrix pair (A₀₀+ N₀₁A₁₀+ N₀₂A₂₀, B_s) is stabilizable. Since

A_s = A₀₀+ N₀₁A₁₀+ N₀₂A₂₀+ B_sR⁻¹

 B¯₁₁^TM₀₁^T B¯₂₂^TM₀₂^T

= A₀₀+ N₀₁A₁₀+ N₀₂A₂₀+ B_sK

and the feedback K does not change the stabilizability property of (A00+ N₀₁A₁₀+ N₀₂A₂₀, B_s), we arrive at the conclusion that the matrix pair (A_s, B_s) is also stabilizable. Similarly, we can prove that (A^T_s, C_s^T) is de- tectable if and only if (10b) is satisfied. The detail is omitted for brevity.

Thereby, we have finished the proof of Lemma 1.

Thus, the required solution of the ARE (11a) exists under the assumptions 1 and 2.

It should be remarked that the solution P_e of (6) is a function of the multiparameters ε_j, j = 1, 2. But, the solutions ¯P₀₀, ¯P₁₁and ¯P₂₂ of (11a) and (11c) are independent of the multiparameters ε_j, respectively. Moreover, we do not assume here that A_jj, j = 1, 2 is nonsingular. Thus, our new results are applicable to more realistic MSPS.

The following theorem will establish the relation between P_eand reduced–

order solutions (11).

Theorem 1. Under the assumptions 1 and 2, there exist small ε^∗_j, j = 1, 2 such that for all ε_j∈ (0, ε^∗_j), the MARE (6) admits a symmetric positive semidefinite stabilizing solution P_e which can be written as

P_e=





P¯₀₀+F₀₀(||µ||) ε₁( ¯P₁₀+F₁₀(||µ||))^T ε₂( ¯P₂₀+F₂₀(||µ||))^T ε₁( ¯P₁₀+F₁₀(||µ||)) ε₁( ¯P₁₁+F₁₁(||µ||)) √ε₁ε₂F₂₁(||µ||)^T ε₂( ¯P₂₀+F₂₀(||µ||)) √ε₁ε₂F₂₁(||µ||) ε₂( ¯P₂₂+F₂₂(||µ||))



 ,

(14) whereF_pq(||µ||) is defined as follows

F_pq(||µ||) = O(||µ||) < ∞, pq = 00, 10, 20, 11, 21, 22, µ =

ε₁ ε₂  . Proof. We apply the implicit function theorem [3,5] to (7). To do so, it is enough to show that the corresponding Jacobian is nonsingular at ε_j= 0, j = 1, 2. It can be shown, after some algebra, that the Jacobian of (7) in the limit as µ→ µ0 is given by

J_P = ∇F = ∂vec(f₁, f₂, f₃, f₄, f₅, f₆)

∂vec(P₀₀, P₁₀, P₂₀, P₁₁, P₂₁, P₂₂)^T 

µ=µ0, P=P0

=











J₀₀ J₀₁ J₀₂ 0 0 0 J₁₀ J₁₁ 0 J₁₃ J₁₄ 0 J₂₀ 0 J₂₂ 0 J₂₄ J₂₅

0 0 0 J₃₃ 0 0

0 0 0 0 J₄₄ 0

0 0 0 0 0 J₅₅











, (15)

where vec denotes an ordered stack of the columns of its matrix [12] and µ₀=

0 0 

, P = (P00, P₁₀, P₂₀, P₁₁, P₂₁, P₂₂), P₀= ( ¯P₀₀, ¯P₁₀, ¯P₂₀, ¯P₁₁, 0, ¯P₂₂),

J₀₀= (I_n0⊗ D^T₀₀)U_n0n0+ D₀₀^T ⊗ In0, J_0j= (I_n0⊗ D^T_j0)U_n0nj+ D_j0^T ⊗ I_n0, J_j0= D^T_0j⊗ I_n0, J_jj= D_jj^T ⊗ I_n0, J₁₃= I_n1⊗ D₁₀, J₁₄=√

¯

α(I_n1⊗ D₂₀)U_n1n2, J₂₄= 1

√α¯I_n2⊗ D10, J₂₅= I_n2⊗ D20, J₃₃= (I_n1⊗ D^T₁₁)U_n1n1+ D₁₁^T ⊗ I_n1, J₄₄=√

¯

αD₂₂^T ⊗ In1+ 1

√α¯I_n2⊗ D₁₁^T,

J₅₅= (I_n2⊗ D^T₂₂)U_n2n2+ D₂₂^T ⊗ In2, j = 1, 2,

where ⊗ denotes Kronecker products and U_njnj, j = 0, 1, 2 is the permu- tation matrix in the Kronecker matrix sense [12].

The Jacobian (15) can be expressed as

detJ_P = detJ₃₃· detJ44· detJ55· det



 J₀₀ J₀₁ J₀₂ J₁₀ J₁₁ 0 J₂₀ 0 J₂₂





= detJ₃₃· detJ₄₄· detJ₅₅· detJ₁₁· detJ₂₂

·det(J00− J01J₁₁⁻¹J₁₀− J02J₂₂⁻¹J₂₀)

= detJ₁₁· detJ22· detJ33· detJ44· detJ55

·det[I_n0⊗ D^T₀U_n0n0+ D₀^T ⊗ I_n0], (16) where D₀≡ D00− D01D⁻¹₁₁D₁₀− D02D₂₂⁻¹D₂₀. Obviously, J_jj, j = 1,· · · , 5 are nonsingular because the matrices D₁₁ = A₁₁− S11P¯₁₁ and D₂₂= A₂₂− S₂₂P¯₂₂ are stable under the assumption 1. After some straightforward but tedious algebra, we see that A_s−SsP¯₀₀= D₀₀−D01D⁻¹₁₁D₁₀−D02D⁻¹₂₂D₂₀= D₀. Therefore, the matrix D₀ is stable if the assumption 2 holds. Thus, detJ_P = 0, i.e., JP is nonsingular at (µ, P) = (µ0, P0). The conclusion of Theorem 1 is obtained directly by using the implicit function theorem.

The remainder of the proof is to show that P_eis the positive semidefinite stabilizing solution. Firstly, from (14), we get

P_e=





P¯₀₀ 0 0

0 0 0

0 0 0



 + O(||µ||),

Taking into consideration the fact that the solution ¯P₀₀is positive semidefi-

nite, we have P_e≥ 0. Secondly, using (14), we obtain

A_e− S_eP_e= Φ⁻¹_e







 D₀₀ D₀₁ D₀₂ D₁₀ D₁₁ 0 D₂₀ 0 D₂₂



 + O(||µ||)



 ,

where

Φ_e=



 I_n0 0 0

0 ε₁I_n1 0 0 0 ε₂I_n2



 .

The matrix D₁₁, D₂₂and D₀ are stable since the assumptions 1 and 2 hold.

Therefore, if parameter ε_j is very small, A_e− S_eP_e is stable by applying Theorem 1 in [8].

In contrast with the results of [3], we do not require singularity of A_jj, j = 1, 2. Moreover, we believe that our proof is very direct.

Remark. The assumptions 1 and 2 guarantee the existence of the solu- tions ¯P_pqof the equations (9). Then after obtaining the asymptotic structure of the stabilizing solution of MARE (6) via the implicit function theorem, it can be verified that the limits of P_pqin the equations (7) are ¯P_pq.

4 The Recursive Algorithm for the MARE

Now, let us define||µ|| ≡ E. The solution (14) can be changed as follows.

P_e=





P¯₀₀+EE00 ε₁( ¯P₁₀+EE10)^T ε₂( ¯P₂₀+EE20)^T ε₁( ¯P₁₀+EE₁₀) ε₁( ¯P₁₁+EE₁₁) E²E₂₁^T ε₂( ¯P₂₀+EE₂₀) E²E₂₁ ε₂( ¯P₂₂+EE₂₂)



 , (17)

whereE :=√ε₁ε₂, E₀₀= E₀₀^T, E₁₁= E₁₁^T, E₂₂= E^T₂₂.

The O(||µ||ⁱ) approximation of the error terms E_pqwill result in O(||µ||ⁱ⁺¹) approximation of the required matrix P_pq. That is why we are interested in finding equations of the error terms and a convenient algorithm to find their solutions. Substituting (17) into (7) and subtracting (9) from (7), we arrive