Reduced-Order Algorithm for Eigenvalue Assignment of Singularly Perturbed Linear Systems

In this paper, we present an algorithm for eigenvalue assignment of linear singularly perturbed systems in terms of reduced-order slow and fast subproblem matrices. No similar algorithm exists in the literature. First, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation used for eigenvalue assignment. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. *e proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically wellconditioned algebraic Sylvester equations corresponding to slow and fast variables. *e convergence rate of the proposed algorithm is O(ε), where ε is a small positive singular perturbation parameter.


Introduction
e classical method for numerical solution of the Sylvester algebraic equation dates back to reference [1]. Solving the Sylvester algebraic equation numerically is not a simple task [2,3]. Namely, it was stated in [2,3] that the algorithm of [1] cannot produce a highly accurate solution. Another method for solving the large-scale Sylvester equations is introduced in [4]. In [4], the authors have shown that researchers have developed some methods for the solution of large-scale Sylvester equations [5][6][7]. On the other hand, several research studies for solving the Sylvester algebraic equation have gained attention in engineering problem [8][9][10]. In the image-processing problem, the presentation developed in [8] has shown that the image fusion method applied to a large-scale image facilitates reduction of computational complexity based on the explicit solution of large-scale Sylvester equations. Furthermore, many problems of control theory such as regulator problem [9] and particle swarm theory [10] lead to a Sylvester equation. e aim of our developed algorithm is to solve a largescale Sylvester equation in order to overcome the numerical ill-conditioning problem of singularly perturbed systems presented in [11]. is leads to reduced-order regular algebraic Sylvester equations [12], combined with the techniques presented in [13,14] which solves the eigenvalue assignment problem for singularly perturbed linear systems. e general Sylvester equation is defined as Its unique solution T exists under the assumption that matrices A and −M have no eigenvalues in common [13].

Assumption 1. λ(A) ≠ λ(−M) ≠ − λ(M).
Without loss of generality, we will consider the Sylvester equation encountered in the control system design of linear systems: where x(t) ∈ R n is the state vector, u(t) ∈ R m is the control input vector, y(t) ∈ R p is the vector of system measurements, and A, B, and C are constant matrices. Forms of the Sylvester algebraic equations that appear in the observer and controller designs are given by where K stands for the observer feedback gain and F is the system feedback gain.
ese Sylvester equations were studied in [15]. e system-observer configuration has slow and fast modes since the observer must be much faster than the system, and hence, it represents implicitly a singularly perturbed system. Note that the main difficulty in the numerical solution of algebraic Sylvester equation (3) will come from the fact that the system to be studied has a singularly perturbed structure and not from the systemobserver implicit singularly perturbed structure since the controller and the observer eigenvalue assignment problems are done independently using the separation principle. e main difficulty comes from the fact that since the system has slow and fast modes, then the observer must contain very fast modes, which leads to numerical illconditioning.

Problem Statement.
In this section, we study the Sylvester algebraic equation corresponding to singularly perturbed systems defined by [11] (Chapter 2): where x 1 (t) ∈ R n 1 and x 2 (t) ∈ R n 2 , n 1 + n 2 � n, are, respectively, slow and fast state variables and ε is a small positive singular perturbation parameter. Equation (4) can be obtained from (2) assuming that (2) has eigenvalues clustered into two groups: slow ones closer to the imaginary axis and fast ones farther from the imaginary axis (singularly perturbed structure). In such a case, a similarity transformation converts (2) into (4). e following is the standard assumption used in theory of singular perturbation [11] (Chapter 2).

Assumption 2.
e matrix A 4 is nonsingular. We study, without loss of generality, a variant of observer design algebraic Sylvester equation (3) given by e matrix A des is the matrix with the observer-desired closed-loop eigenvalues. e standard observer design assumptions are needed [13]. Assumption 3. . e pair (A, C) is observable, and the pair (A des , K) is controllable. e general existence condition given in Assumption 1 and specialized to (5) leads to the following assumption.
Having found an invertible solution of (5), then the observer gain is given by K � T −1 K. Note that Assumption 3 for single-input single-output systems is both sufficient and necessary condition for the existence of an invertible solution of (5). For multi-input multi-output systems, it is only a necessary condition [13], so that a repetitive design algorithm has to be performed until an invertible solution T is obtained (see Section 5). e matrices in (4) and (5) are partitioned as where A des contains the desired observer closed-loop eigenvalues, that is, λ(A des ) � λ(A − KC). Note that they are placed far to the left in the complex plane to make the observer asymptotically stable and much faster than the closed-loop system. We have found that the following scaling is appropriate for the solution matrix T: which is consistent with the structures of matrices defined in (5) and (6). Namely, the right-hand side of (5) is With the scaling chosen in (7), the left-hand side terms of (5), that is, TA and A des T, are also both O (1). Due to the structure of matrices A and A des , the singularly perturbed algebraic Sylvester equation defined in (5) is numerically illconditioned. To overcome numerical ill-conditioning, we propose a new recursive algorithm for solving (5) in terms of reduced-order well-defined algebraic Sylvester equations. e dual version of (5) used for the system controller design is given by 2 Mathematical Problems in Engineering Multiplying matrices in (9), we can get four algebraic equations for the partitioned matrix T c as functions of the small parameter ε. It can be found from these equations [16,17] that the structure of T c is given by Algebraic Sylvester equations (9)-(10) will be solved numerically in terms of reduced-order numerically wellconditioned algebraic Sylvester equations under the standard controller design assumptions [13].

Assumption 5.
e pair (A, B) is controllable and the pair (A c des , F) is observable. Moreover, the existence of a unique solution of (9) requires the assumption dual to Assumption 4.

Parallel Algorithm for the Observer Sylvester Equation.
e partitioned form of the Sylvester equation given in (5) subject to (6)-(8) is given by It can be seen from (11) that this system has two independent sets of linear algebraic equations: the first one for T 1 and T 2 and the second one for T 3 and T 4 .
Setting ε � 0 in (11), the algebraic equations for zerothorder approximations of solutions are obtained as ese equations can be solved independently as follows. Unique solution T (0) 4 can be obtained from Sylvester equation (15) under the following assumption.
Since A f , defined in (6), is chosen by the designer as an asymptotically stable matrix, this assumption is easily satisfied. Having obtained T (0) 4 , from (13) and (14), we can obtain T (0) 3 and T (0) 2 independently as Substituting (17) into (12) results in e unique solution T (0) 1 of algebraic Sylvester equation (18) exists under the following assumption.
close to the exact solutions, that is, Now, we show that the values for E i , i � 1, 2, 3, 4, can be obtained by running iterations on independent linear reduced-order algebraic equations. Subtracting (12)-(15) from (11) and using (20), we obtain the error equations (after some algebra) in the following form: e error equations can be solved iteratively using the fixed-point algorithm, in which the cross-coupling terms multiplied by ε are delayed by one iteration. is idea has been used in several algorithms that involve small parameters.

Mathematical Problems in Engineering
Algorithm 1.
We first solve (22) as Substituting (26) into (21) gives Equations (26) and (27) have nice forms since the quantity E 2 is multiplied by a small parameter ε. Similarly, equations for E 3 and E 4 can be iteratively solved as Proof of eorem 1.
Note that Subtracting (30) from (31), we have At this point, we conclude that In a similar way, we can write the relationship between E (3) 1 and E (2) 1 as which implies that Continuing the same procedure, we obtain Now, we work with E 2 using (26). For i � 0, we have For i � 1, Using E (0) 2 � 0 and the result in (33), we get Considering (26) for i � 2 and using (39), we obtain If we keep repeating this process, we conclude that In addition, we have the following relationships. Subtracting (21)-(22) from (25) for i � 0 produces For i � 1, (21)-(22) and (25) produce Continuing the same procedure, we have 4 Mathematical Problems in Engineering Similar procedures applied to (23)-(24) produces

Results established in (44)-(47) can be summarized in
which completes the proof of the stated theorem.

Parallel Algorithm for Controller Sylvester Equation
e controller design algebraic Sylvester equation defined in (9)-(10) can be partitioned as Setting ε � 0 in (49), the zeroth-order approximations 1c , T (0) 2c , T (0) 3c , and T (0) 4c can be obtained as follows: e unique solution T (0) 4c can be found from algebraic Sylvester equation (54) under the following assumption, which is easily satisfied since A fc is an arbitrary stable closed-loop matrix.
Since A sc is chosen by the designer, this assumption is easily satisfied. We define the approximation errors as Subtracting (51)-(54) from (49) and using (59), we obtain the error equations in the following form: (60) ese error equations can be solved using the fixed-point algorithm, dual to Algorithm 1, as follows.

Algorithm 2.
e convergence proof of Algorithm 2 can be done via the dual arguments used in Algorithm 1. Similarly, we can state the corresponding theorem dual to eorem 1.
e proof of eorem 2 parallels the one of eorem 1.

Observer and Controller Designs
e general design of an observer and a controller using the Sylvester approach is presented in [13]. We will exploit two-time scale property so that the design is done in terms of reduced-order problems. e observer design procedure for the system defined in (2) has the following steps [13].
e state feedback controller for the system defined in (2) can be obtained using the steps. Comment 1. according to our experience, we need only one repetition to obtain invertible T c . According to [13], the pair (A, B) controllable and the pair (A des , F) observable are both the necessary and sufficient condition for invertibility of T c in the case of single-input single-output systems. For multiple-input multiple-output systems, this condition is only sufficient.
e feedback system is given by [13] _ Comment 2. well-conditioning with respect to matrix inversion of matrices T and T c can be established by using the formula for matrix inversion of partitioned matrices [13]. For matrix T defined in (7), we obtain Hence, the T matrix is well conditioned with respect to matrix inversion if matrix T 1 is well conditioned with respect to matrix inversion. Similarly, for matrix inversion of T c , the matrix T 4c must be well conditioned with respect to the matrix inversion.

Simulation Results
Consider a 4 th −order system with the matrices A, B, and C taken from [11] (Chapter 3, 1999) as follows: e pair (A, C) is observable, and we can proceed with the observer design algorithm. e designer decides to place observer eigenvalues at the desired location by choosing matrices A s and A f . In the following, we will design a controller with the desired closed-loop eigenvalues placed at −0.

Observer Design Algorithm 1.
We choose the observer eigenvalues such that it is roughly ten times faster than the closed-loop system. Consequently, we choose A des as We choose K as so that (A des , K) is controllable, as required in Step 2 of Algorithm 1. e matrix Q defined in (8) is given as Checking the corresponding observer closed loop eigenvalues, we have which with the accuracy of O(10 − 12 ) is close to the chosen desired eigenvalues of the matrix A des . e general Sylvester equation for the observed case is given as in (63).

Result of Comparison Algorithm 1 with Algorithm 432.
To compare our proposed Algorithm 1 against the existing design algorithm, we implement the Algorithm 432 presented in [1]. Applying the Schur transformation to equation (5), we have where where U and V are the Schur transformation in order to construct a lower triangular form for A des and an upper triangular form for A given in (5) and used in [1]. e formula for the solution T in (5), based on the relationship in (77), is given in Using the MATLAB Schur function and the similarity transformation given in (77), we have