Abstract
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. The 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 well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is , where is a small positive singular perturbation parameter.
1. Introduction
The 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 [5–7]. On the other hand, several research studies for solving the Sylvester algebraic equation have gained attention in engineering problem [8–10]. 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.
The aim of our developed algorithm is to solve a large-scale Sylvester equation in order to overcome the numerical ill-conditioning problem of singularly perturbed systems presented in [11]. This 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.
The general Sylvester equation is defined as
Its unique solution exists under the assumption that matrices and have no eigenvalues in common [13].
Assumption 1. .
Without loss of generality, we will consider the Sylvester equation encountered in the control system design of linear systems:where is the state vector, is the control input vector, is the vector of system measurements, and , , and are constant matrices. Forms of the Sylvester algebraic equations that appear in the observer and controller designs are given bywhere stands for the observer feedback gain and is the system feedback gain. These Sylvester equations were studied in [15]. The 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 system-observer implicit singularly perturbed structure since the controller and the observer eigenvalue assignment problems are done independently using the separation principle. The 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 ill-conditioning.
1.1. Problem Statement
In this section, we study the Sylvester algebraic equation corresponding to singularly perturbed systems defined by [11] (Chapter 2):where , are, respectively, slow and fast state variables and 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). The following is the standard assumption used in theory of singular perturbation [11] (Chapter 2).
Assumption 2. The matrix is nonsingular.
We study, without loss of generality, a variant of observer design algebraic Sylvester equation (3) given byThe matrix is the matrix with the observer-desired closed-loop eigenvalues. The standard observer design assumptions are needed [13].
Assumption 3. .The pair (A, C) is observable, and the pair is controllable.
The general existence condition given in Assumption 1 and specialized to (5) leads to the following assumption.
Assumption 4. .
Having found an invertible solution of (5), then the observer gain is given by . 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 is obtained (see Section 5). The matrices in (4) and (5) are partitioned aswhere contains the desired observer closed-loop eigenvalues, that is, . 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 :which is consistent with the structures of matrices defined in (5) and (6). Namely, the right-hand side of (5) isWith the scaling chosen in (7), the left-hand side terms of (5), that is, and , are also both . Due to the structure of matrices and , the singularly perturbed algebraic Sylvester equation defined in (5) is numerically ill-conditioned. To overcome numerical ill-conditioning, we propose a new recursive algorithm for solving (5) in terms of reduced-order well-defined algebraic Sylvester equations. The dual version of (5) used for the system controller design is given byMultiplying matrices in (9), we can get four algebraic equations for the partitioned matrix as functions of the small parameter . It can be found from these equations [16, 17] that the structure of is given byAlgebraic Sylvester equations (9)-(10) will be solved numerically in terms of reduced-order numerically well-conditioned algebraic Sylvester equations under the standard controller design assumptions [13].
Assumption 5. The pair (A, B) is controllable and the pair is observable.
Moreover, the existence of a unique solution of (9) requires the assumption dual to Assumption 4.
Assumption 6. .
1.2. Parallel Algorithm for the Observer Sylvester Equation
The 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 and and the second one for and .
Setting in (11), the algebraic equations for zeroth-order approximations of solutions , , , and are obtained as
These equations can be solved independently as follows. Unique solution can be obtained from Sylvester equation (15) under the following assumption.
Assumption 7. .
Since , defined in (6), is chosen by the designer as an asymptotically stable matrix, this assumption is easily satisfied. Having obtained , from (13) and (14), we can obtain independently asSubstituting (17) into (12) results inThe unique solution of algebraic Sylvester equation (18) exists under the following assumption.
Assumption 8. .
The solutions , and are close to the exact solutions, that is,Now, we show that the values for , 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:The error equations can be solved iteratively using the fixed-point algorithm, in which the cross-coupling terms multiplied by are delayed by one iteration. This idea has been used in several algorithms that involve small parameters.
Algorithm 1. We first solve (22) asSubstituting (26) into (21) givesEquations (26) and (27) have nice forms since the quantity is multiplied by a small parameter . Similarly, equations for and can be iteratively solved asThe following theorem presents the main feature of Algorithm 1.
Theorem 1. Under Assumptions 7 and 8, Algorithm 1 converges to the exact solution with the rate of convergence of . The convergence is obtained for sufficiently small values of that makes the radius of convergence smaller than 1 in each iteration leading to a contraction mapping. Hence, after iterations, the solution is obtained with the accuracy of , that is,
Proof of Theorem. 1.
For , (27) impliesNote that . For , (27) producesSubtracting (30) from (31), we haveAt this point, we conclude thatIn a similar way, we can write the relationship between and aswhich implies thatContinuing the same procedure, we obtainNow, we work with using (26). For , we haveFor ,Using and the result in (33), we getConsidering (26) for and using (39), we obtainIf we keep repeating this process, we conclude thatIn addition, we have the following relationships. Subtracting (21)-(22) from (25) for producesFor , (21)-(22) and (25) produceContinuing the same procedure, we haveSimilar procedures applied to (23)-(24) producesResults established in (44)–(47) can be summarized inwhich completes the proof of the stated theorem.
2. Parallel Algorithm for Controller Sylvester Equation
The controller design algebraic Sylvester equation defined in (9)-(10) can be partitioned as
Setting in (49), the zeroth-order approximations , and can be obtained as follows:
The unique solution can be found from algebraic Sylvester equation (54) under the following assumption, which is easily satisfied since is an arbitrary stable closed-loop matrix.
Assumption 9. .
From (52)-(53), we can obtain and asSubstituting (56) into (51) results inThe solution can be obtained by solving algebraic Sylvester equation (57) under the following assumption.
Assumption 10. .
Since is chosen by the designer, this assumption is easily satisfied. We define the approximation errors asSubtracting (51)–(54) from (49) and using (59), we obtain the error equations in the following form:These error equations can be solved using the fixed-point algorithm, dual to Algorithm 1, as follows.
Algorithm 2. The 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 Theorem 1.
Theorem 2. Under Assumptions 5, 9, and 10, Algorithm 2 converges for sufficiently small values of with the rate of to the sought solution ; that is, after iterations, we haveThe proof of Theorem 2 parallels the one of Theorem 1.
3. Observer and Controller Designs
The 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. The observer design procedure for the system defined in (2) has the following steps [13].
The 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 . According to [13], the pair controllable and the pair observable are both the necessary and sufficient condition for invertibility of in the case of single-input single-output systems. For multiple-input multiple-output systems, this condition is only sufficient.
The feedback system is given by [13]
Comment 2. well-conditioning with respect to matrix inversion of matrices and can be established by using the formula for matrix inversion of partitioned matrices [13]. For matrix defined in (7), we obtainHence, the matrix is well conditioned with respect to matrix inversion if matrix is well conditioned with respect to matrix inversion. Similarly, for matrix inversion of , the matrix must be well conditioned with respect to the matrix inversion.
Comment 3. systems (63) and (64) have the structure of linear singularly perturbed systems.
4. Simulation Results
Consider a order system with the matrices , and taken from [11] (Chapter 3, 1999) as follows:
The pair is observable, and we can proceed with the observer design algorithm. The designer decides to place observer eigenvalues at the desired location by choosing matrices and . In the following, we will design a controller with the desired closed-loop eigenvalues placed at . Note that is controllable.
4.1. 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 as
We choose asso that is controllable, as required in Step 2 of Algorithm 1.
|
The matrix defined in (8) is given as
The zeroth-order approximations , and are obtained as
Performing iterations, we obtain the sixteen decimal digits accuracy after :
The corresponding iterative solution and the exact solution (obtained by using the MATLAB function lyap to solve the full-order Sylvester equation) are given by
The difference is
The solution is invertible in the first run of Algorithm 1 (see Section 5). The corresponding observer gain is
Checking the corresponding observer closed loop eigenvalues, we havewhich with the accuracy of is close to the chosen desired eigenvalues of the matrix . The general Sylvester equation for the observed case is given as in (63).
4.2. 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 havewherewhere and are the Schur transformation in order to construct a lower triangular form for and an upper triangular form for given in (5) and used in [1]. The formula for the solution 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
Partitioning (76) results inwhere dimensions of , , , , , , and , are, respectively, , , , , , , and , with the corresponding dimensions of , , , , , , and .
From equation (80) and the condition , we have
Solving reduced-order Sylvester equations (81)–(84), we have
The corresponding solution of is given by
Original solution using the inverse transformation of (78) is given by
The difference between and of our proposed Algorithm 1 is
Since they are different, we conclude that our proposed Algorithm 1 has a better accuracy regarding the solution of the general Sylvester equation.
Algorithm 2(Controller design). Similarly, we design a controller for the same system using the algorithm for solving the controller algebraic Sylvester equation from Section 4. We chooseThe zeroth-order approximations , and are obtained asUsing the proposed algorithm, we obtain after :The iterative solution and the exact solution are given bywhereTheir difference isThe solution is invertible in the first run of Algorithm 2 (see Section 5). The controller gain is given asChecking , we havewhich has produced the desired eigenvalues with the accuracy of .
5. Conclusions
It was shown that the numerically ill-conditioned Sylvester algebraic equation for singularly perturbed systems can be decomposed into four lower-order well-conditioned Sylvester equations. The recursive fixed-point-type methods were utilized in order to obtain numerical solutions for such lower-order algebraic Sylvester equations. The corresponding observer and controller design algorithms for assignment of observer and controller closed-loop eigenvalues in terms of reduced-order slow and fast subproblems were presented.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to thank a robot team in Agricultural Robotics and Automation Research Center for their assistance in this work. This work was supported by the Korea Institute of Planning and Evaluation for Technology in Food, Agriculture, Forestry (IPET) through the Agriculture, Food and Rural Affairs Research Center Support Program, funded by the Ministry of Agriculture, Food and Rural Affairs (MAFRA) (no. 714002-07).