Abstract

The output feedback eigenvalue assignment problem for discrete-time systems is considered. The problem is formulated first as an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to find a local solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a logarithmic barrier method is proposed for finding the local solution. The conjugate gradient method is further extended to tackle the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. The performance of the methods is illustrated through various test examples.

1. Introduction

In this work, we consider the following unconstrained minimization problem: where is generally semismooth and nonconvex and is the spectral radius of the matrix function that will be defined later on.

It is desirable to have a local solution of (1) such that . Therefore, we include such a constraint as a cut to the objective function implying the following inequality constrained minimization problem: where is a given constant.

It is well known that, for any matrix norm, it holds that , where is any square matrix. Then, we can replace the eigenvalue constraint of problem (2) by a regular inequality constraint which yields the following relaxed minimization problem: where is as defined above. This problem can be tackled easily by any constrained optimization solver. However, there is no guarantee in general that a feasible solution exists for this problem.

The three problems (1)–(3) concern the well-known eigenvalue assignment problem (EAP) for discrete-time systems. These problems are generally nonconvex and semismooth optimization problems. Over the past decades, a considerable amount of attention has been given to the EAP where a lot of research can be found in systems and control literature in particular for continuous-time systems (see, e.g., [1–3] and the references therein). In the framework of linear discrete-time systems, a set of eigenvalue assignment algorithms have been developed (see, e.g., [2–11] and the references therein).

A related problem in output feedback control design is the linear-quadratic control problem in which the goal is to design an output feedback gain matrix that minimizes a certain performance cost function while all eigenvalues of the closed-loop system matrix must be within the unit circle (see, e.g., [12]). Such a controller can be calculated to this problem in case of continuous-time systems by available public software packages (e.g., HIFOO) [13]. However, for discrete-time systems, to the best of our knowledge, public software has not yet been developed.

In this work, we focus on the two problems (1) and (2) where the attempt is to minimize the spectral radius of the nonsymmetric real matrix . In this regard, we apply a three-term nonlinear conjugate gradient (CG) method [14] to find a local solution to the minimization problem (1) which is attempting to stabilize the associated control system; see the next section. In addition, a logarithmic barrier interior-point method is proposed to tackle the constrained minimization problem (2) for the same purpose.

Nonlinear conjugate gradient methods are widely studied and comprise a class of unconstrained optimization algorithms which are characterized by low memory requirements and strong global convergence properties (see the survey [15] and later references [3, 14]). We focus on a three-term nonlinear CG method which has a nice performance (see [14]). The CG methods are descent direction methods, which means that, starting from a given point , these methods generate a sequence according to the following relation: where the step size satisfies the line search rule and is a descent direction for at . The update of the new search direction varies from one CG method to another.

The logarithmic barrier interior-point method is one of the standard methods for solving constrained optimization problems (see, e.g., [16]). This method is employed to tackle the constrained problem (2) which converts it into a sequence of unconstrained minimization problems.

This article is organized as follows. In the next section, we state the formulation of the eigenvalue assignment problem and introduce some basic concepts which are needed in the subsequent analysis. In Section 3, we evaluate the required derivatives of the objective function. In Section 4, we introduce the proposed three-term CG method that finds a local solution of the unconstrained minimization problem. In Section 5, we extend the CG method to tackle the output feedback EAP problem for decentralized control systems. In Section 6, we reformulate the discrete-time control system with time delay as an augmented system without any delay so that the EAP can be tackled by the proposed CG method. In Section 7, we introduce a logarithmic barrier method for finding a local solution to problem (2). In Section 8, we test the proposed methods on different test examples from the literature. Then, we end with a conclusion.

Notations. For vectors, the symbol is the 2-norm, while for matrices denotes the Frobenius norm defined by , where is the inner product given by and is the trace operator. The eigenvalues of a matrix are denoted by ,  . The Greek letter denotes the spectral radius of a square matrix and is a diagonal matrix with eigenvalues on its main diagonal. Moreover, and denote the real and imaginary parts of a complex number, respectively. Sometimes and for the sake of simplicity, we skip the arguments of the considered functions; for example, we use to denote .

2. Problem Formulation and Preliminary

The static output feedback eigenvalue assignment problem for discrete-time systems can be stated as follows (see, e.g., the abovementioned citations and the references therein). Consider the linear time-invariant discrete-time system with the following state space realization: where , , and are the state, the control input, and the measured output vectors, respectively. Moreover, , , and are given constant matrices. Such a system is often closed by the control law which yields where and is the output feedback gain matrix which represents the unknown.

The following definitions are needed for later use.

Definition 1. The spectral radius of a matrix with eigenvalues is defined as

Definition 2. The discrete-time control system (6) is asymptotically stable (i.e., as for any initial ) if and only if .

The eigenvalue assignment problem is to design an output feedback gain matrix providing a closed-loop system in a satisfactory stage by shifting controllable eigenvalues to desirable locations in the complex plane. In particular, the EAP requires the spectral radius of the closed-loop matrix to be strictly within the unit circle in the complex plane.

A necessary condition for the EAP by constant output feedback is given in [9]. Fu [7] has also shown that the EAP via static output feedback is NP-hard. Systems with a symmetric state space realization, namely, , , occur in different applications such as RC networks [10]. The symmetric EAP might be stated as follows: find a matrix such that It is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable. A classical theorem (see, e.g., [17]) states that each eigenvalue of a symmetric matrix is the difference of two convex functions, which implies that the eigenvalues are semismooth functions. This fact allows us to use the theory of semismoothness to establish convergence results for the proposed methods.

Definition 3. A functional is said to be locally Lipschitz continuous at with a constant if and only if there exists a number such that where is an open ball with center at and radius .

Let be a locally Lipschitz continuous function. Radmacher’s theorem (see, e.g., [18]) implies that such a mapping is differentiable almost everywhere. Let be the set of all at which is differentiable and let be its Jacobian whenever it exists. Let be the set of all at which is differentiable. Such a set is open. Therefore, it is convenient to replace by the following level set: where is a given constant. This set is assumed to be bounded.

3. Derivative of the Objective Function

The eigenvalues of the matrix of (1) are not in general differentiable at a point where has repeated eigenvalues. Therefore, let us consider the following assumption on eigenvalues of .

Assumption 4. Assume that has no multiple eigenvalues for all .

Let be diagonalizable and let and be a couple of matrices whose columns are the left and right eigenvectors of ; that is, , , and satisfy

The columns of and satisfy

The following lemma provides the first-order derivatives of the objective function of the minimization problem (1) required by the CG methods.

Lemma 5. Suppose that satisfies Assumption 4 and is differentiable at . Let be the largest in magnitude eigenvalue of . Then, the entries of the gradient of the objective function are given by where and are the left and right eigenvectors associated with and is a matrix with zero entries except at the position where its value is one.

Proof (see [19, Lemma ]). The two eigenvectors are normalized such that which by differentiation yields where the dash denotes the first-order derivative with respect to the entries of . The eigenvalue also satisfies that which by differentiation gives Suppose that is the largest in magnitude eigenvalue of . Then, and by the chain rule we have Since , then the result follows.

4. Three-Term CG Method for the EAP

In this section, we analyze and study a three-term nonlinear CG method for computing the local solution of the minimization problem (1) (see [14]). For a given starting , the CG method generates a sequence of iterates according to the relation where is supposed to be a descent direction for at and is the step size. The new search direction for the CG method is given by the following relation: where is the gradient of at , , , and

In order to globalize this CG method, we recall Wolfe conditions (see, e.g., [16]) to update a suitable step size for the new iterate (20): where . The strong Wolfe conditions replace (24) by the following condition:

The following theorem shows that the search direction is a descent direction to the objective function.

Theorem 6. Let be generated by (20) and let the step size satisfy Wolfe conditions (23)-(24). Then, evaluated by (21)-(22) is a descent direction to for all .

Proof (see also [14, Proposition ]). From Wolfe’s second condition (24), we obtain the curvature condition . Moreover, from (21)-(22), it follows that

The three-term nonlinear CG algorithm is stated in the following lines.

Algorithm 7 (three-term CG method for the output feedback EAP). (0)Let , , be given constants and be the tolerance. Moreover, let be given constant matrices. Choose , compute , and set . If or , stop; otherwise, set and go to the next step.

While or , do(1)Compute that satisfies Wolfe conditions (23)-(24); set and then calculate the gradient .(2)If or , stop; otherwise, go to the next step.(3)Calculate a new search direction by (21)-(22).(4)Set and repeat.End (do)

Remark 8. For the considered linear control system, one of the major tasks is to achieve a stabilizing output feedback controller, that is. to calculate such that is strictly less than one. However, there is no relationship between achieving such a controller and finding , a local minimum of problem (1). Therefore, it is reasonable to stop the CG method as soon as a local solution is achieved or a stabilizing is reached with a sufficient stability margin.

To prove global convergence of the CG algorithm, we assume that for all ; otherwise, a stationary point is found.

Assumption 9. The following assumptions are assumed to hold: (a)The level set as defined in (10) is bounded; that is, there exists a constant such that, for all , .(b)In some neighborhood of the level set (10), the gradient of the objective function is Lipschitz continuous.

From Assumption 9(a), one has and from Assumption 9(b) we deduce that there exists a constant such that

The following lemma provides a lower bound to the step size ; see [14, Proposition ].

Lemma 10. Let be generated by Algorithm 7 and suppose that is a descent direction for . In addition, let Assumption 9 hold. Then,

Proof. By subtracting from both sides of (24) and using Lipschitz condition, Since is a descent direction for and is less than one, then (28) follows.

The result of the following lemma is used in proving the main global convergence theorem.

Lemma 11. Let be generated by Algorithm 7 and assume that is a descent direction. Furthermore, let Assumption 9 hold. Then,

Proof. From Wolfe condition (23) and Lemma 10, one has Then, Assumption 9 implies condition (30).

Next, we have the following result for the three-term CG method (see, e.g., [14, Proposition ]).

Lemma 12. Let Assumption 9 hold. Consider to be generated by Algorithm 7 where is a descent direction for and satisfies Wolfe conditions (23) and (24). If then

Proof (see also [20, Lemma ]). Suppose (33) is not true. Then, there exists a constant such that Therefore, from (32) and Lemma 11, we have This contradicts Lemma 12 which completes the proof.

Under condition (30), the following global convergence result is obtained.

Theorem 13. Let be generated by Algorithm 7. Assume that Assumption 9 holds. Assume further that there exists a constant such that for any . Then, it holds that

Proof (see also [14, Theorem ]). Since for all , then By direct computation, we have Consequently, one has Then, from Lemma 12, (36) follows.

5. The EAP for Decentralized Control Systems

Consider the linear time-invariant decentralized control system with control stations: where , , and are the state, the control input, and the measured output vectors, respectively. , , and are given constant matrices, .

The output feedback EAP for the decentralized system (40) is to find output feedback gain matrices that place, by using the control law the eigenvalues of the closed-loop system matrix to strictly lie within the unit disk.

By introducing the following augmented matrices, it is straightforward to rewrite the decentralized control system (40) in the original structure (5): The corresponding closed-loop system matrix is , where the output feedback gain matrix is given by the block-diagonal matrix and the corresponding unconstrained minimization problem takes the form