#### Abstract

This paper is concerned with the singular LQ problem for irregular singular systems with persistent disturbances. The full information feedback control method is employed to achieve the optimal control. By restricted system equivalence transformation, the system state is decomposed into free state and restricted state and the input is decomposed into free input and forced input. Some sufficient conditions for the unique existence of optimal control-state pair are derived and these conditions are all described unitedly with matrix rank equalities. The optimal control-state pair can be explicitly formulated via solving an algebraic Riccati equation and a Sylvester equation. Moreover, under the optimal control-state pair, the resulting system has no free state.

#### 1. Introduction

In practical control systems there are unavoidably various external disturbances affecting the performances of systems. Many control problems involve designing a controller capable of stabilizing a given system while minimizing the worst-case response to some additive disturbances. The application areas of interest include, for example, the flight control through wind shear where disturbances arise from a model for wind shear based on harmonic oscillations [1], the active control for offshore structures where external disturbances are mainly from the wind or ocean wave forces [2], the noise reduction in vehicles and transformers, and the control of the linear course of ships [3, 4], and so forth. Therefore, the optimal control for systems with persistent disturbances is one of the very important control problems. Up to now, there has been much research work about the optimal control of linear standard state space systems with persistent disturbances (see, e.g., [5–7]).

Since, among all output feedback plants, it requires the weakest assumption, the full information feedback (FI, also called feedforward and feedback [8]) plant has received a great attention in the literature and has been extensively applied to control systems with disturbances. For example, Rusli et al. [9] design a low-order robust nonlinear FI controller for a multiscale system that dynamically couples Kinetic Monte Carlo and finite difference codes. Tang [5] proposes a FI control for linear systems affected by sinusoidal disturbances with known frequency and unknown magnitude and phase. Later, in his joint papers [6, 10–12] he brings forward a united description of disturbances including periodic disturbances as the special case and gives the FI controller designs for the continuous, discrete, time-delay, and continuous time-varying linear systems, respectively. Chiu [13] proposes a mixed FI based adaptive fuzzy control design for a class of MIMO uncertain nonlinear systems. Yu et al. [14] investigate the complete parametric approach for output regulation problems of matrix 2-order systems via FI.

Singular systems have comprehensive practical background [15–17]. Great progress has been made in the theory and its applications since 1970s [18–22]. However, there are few investigations on optimal control problems for irregular singular systems with disturbances, besides Chen [23, 24] where the singular LQ suboptimal and optimal control problems for irregular singular systems with disturbances are considered.

In this paper, the singular LQ problem for irregular singular systems with persistent disturbances is discussed. It is shown that the singular LQ problem for irregular singular systems with persistent disturbances can be transformed to the optimal problem for standard state space systems by restricted system equivalence transformation. The system state is decomposed into free state and restricted state and the input is decomposed into free input and forced input. Then, under some general conditions, FI optimal control-state pair and the optimal performance index are derived via solving an algebraic Riccati equation and a Sylvester equation.

The remainder of the paper is organized as follows. In Section 2, the singular LQ problem for irregular singular systems with persistent disturbances is transformed to the optimal problem for standard state space systems by restricted system equivalence. In Section 3, we deal with the FI optimal control problem for irregular singular systems and obtain the sufficient conditions for the unique existence of optimal control-state pair with regard to the cases when the disturbance is damped and not damped. A simulation example is exploited to demonstrate the effectiveness of the proposed results in Section 4. In Section 5, we give the brief conclusion of this paper.

*Notation 1. *Throughout the paper, the superscript “” stands for matrix transposition; denotes the -dimensional Euclidean space; is the set of real matrices; is the identity matrix; stands for the real part of ; for real symmetric matrix , means that is a definite-positive matrix and means that is a semidefinite-positive matrix. All of the matrices in the context, if not explicitly stated, are assumed to have compatible dimensions.

#### 2. Statement and Transformation of LQ Problem For Singular Systems

Consider the singular system with disturbances: where , , , , , and are state, disturbance, and input, respectively; . The disturbance is governed by the exosystem [12]: where is stable.

System (1) is said to be regular if and ; otherwise, it is irregular. For the regular singular system, Ishihara et al. [8] considered the FI and state feedback (SF) control problems. For the irregular singular system without disturbances, Zhu et al. [25] discussed the SF LQ control problem for system (1) with .

In this paper, the performance index is selected as follows.

In the case when is asymptotically stable, the quadratic performance index is and the corresponding admissible control-state pair set is where and . In the case when is stable but not asymptotically stable, the disturbance will have oscillation behaviour, the state and the control may not tend to zero at the same time, which may cause the quadratic performance index (3) tending to be infinite. So, in this case we adopt the quadratic average performance index The corresponding admissible control-state pair set is

The control objective of this paper is stated as follows: when is asymptotically stable, the control objective is to find an optimal control-state pair such that When is stable but not asymptotically stable, the control objective is to find an optimal control-state pair such that

First of all, it is necessary to discuss some properties of irregular singular systems.

*Definition 1. *The irregular singular system with disturbances
is said to be restricted system equivalent (r.s.e.) to the system (1) if there exist two nonsingular matrices and such that , , , , , and .

Obviously, restricted system equivalence is an equivalent relationship and it is consistent with the definition in [25] for the systems without disturbances.

Denote and , where and , and then .

Since , there exist nonsingular matrices and such that . Accordingly, let where .

*Definition 2. *System (1) is impulse controllable if, for every initial condition and disturbance governed by the system (2), there exists a smooth (impulse-free) control-state pair of system (1).

Definition 2 is consistent with the definition in [26] for the systems without disturbances.

Obviously, it is necessary for the solvability of the problem and that system (1) is impulse controllable. The following lemma establishes two necessary and sufficient conditions for the impulse controllability of system (1).

Lemma 3. *System (1) is impulse controllable if and only if
**
or
*

The proof is similar to that of Theorem 13 in [26], and it is presented in the Appendix.

In the following discussions, we always assume that (10) holds.

Denote and ; then there exist matrices , , , , , and and nonsingular matrices , , , and such that where , , , , , , , and

Let where , , , , , and and then the system (1) is r.s.e to the following system where .

The second and third equations in system (15) can be written as

Therefore, in the system (15), the state variables , and input variable are determined uniquely by , , and . Thus the state variable is free and the input variable is not free. We call the free state and the forced input. Accordingly, and are called restricted state and is called free input.

*Remark 4. *System (1) has no free state if and only if

*Remark 5. *System (1) has no forced input if and only if

Let and ; then the dynamic equation of , , and is

From (16), where is a symmetric matrix, and

Let and then the LQ problem is substantially transformed to the optimal problem, denoted by , of finding an optimal solution at which the performance index achieves the minimum, and the LQ problem is substantially transformed to the optimal problem, denoted by , of finding an optimal solution at which the performance index achieves the minimum.

Obviously, and are optimal problems of the standard state space system, and the singularity of and is determined by the property of . Now, we give a necessary and sufficient condition for .

Theorem 6. * if and only if
*

*Proof. * is equivalent to , which is , where

From (8) and (11),
which finishes the proof.

*Remark 7. *It is a routine matter to show that if is regular and , then (25) is equivalent to that is impulse observable.

#### 3. Design of the FI Controller

In this section, we solve the problem and via solving and , respectively.

##### 3.1. Solution of the Problem

Denoting , (20) can be written as where , , and where is a symmetric matrix, and

Let and then the optimal problem is transformed to the optimal problem, denoted by , of finding an optimal solution at which the performance index achieves the minimum.

Denote . According to the Maximum Principle, if is stabilizable and is detectable, then has a unique solution , and the optimal value is , where satisfies the equation is the unique positive semidefinite solution of the Riccati equation and the matrix is asymptotically stable.

Before further discussion, we first give two lemmas.

Lemma 8. *When is asymptotically stable, is stabilizable if and only if
*

*Proof. *From (8) and (11),

When is asymptotically stable, , the matrix is nonsingular, so
Since , (36) holds if and only if

Lemma 9. *When is asymptotically stable, is detectable if and only if
*

*Proof. *From (8) and (11),

When is asymptotically stable, , we have

Since is nonsingular, an easy computation shows that (40) is equivalent to

*Remark 10. *In the case when is regular and , (40) equivalently implies that (, , ) is -detectable.

*Definition 11 (see [19]). *The finite ’s satisfying are called finite poles for the singular system .

In the following, we give the conclusion concerning the problem .

Theorem 12. *Assume that is asymptotically stable and the rank equalities (10), (25), (36), and (40) hold, then there exists a unique FI optimal control-state pair for . Under the optimal control-state pair, the finite poles of resulting system all are located on the left-half complex plane and the optimal value is
**
where is the unique positive semidefinite solution of the Riccati equation (35) and .*

*Proof. *From Lemmas 8 and 9, the problem has a unique solution given by (33) and (34) and every eigenvalue of has negative real part.

In accord with (13), (14), (17), (33), and (34), it follows that the unique optimal control-state pair of is
and satisfies the equation
where , .

Since the matrix is asymptotically stable, the matrix is asymptotically stable, too. Substituting (45) into (1) leads to (46), so, under the control-state pair (), the finite poles of resulting system all are the eigenvalues of the matrix . Hence, the finite poles of resulting system all are located on the left-half complex plane.

It is obvious that (44) holds from .

##### 3.2. Solution of the Problem

When has an eigenvalue locating on imaginary axis, the Riccati equation (35) has no unique positive semidefinite solution, thereby solving the problem cannot use the same method as the problem .

Theorem 13. *Assume that is stable but not asymptotically stable and the rank equalities (10), (25), (36), and (40) hold; then there exists a unique FI optimal control-state pair for . Under the optimal control-state pair, the finite poles of resulting system all are located on the left-half complex plane and the optimal value is
**
where , is the unique solution of the Sylvester equation
**
and is the unique positive semidefinite solution of the Riccati equation
**
with , , , and .*

*Proof. *Consider the problem . It follows that from .

Construct the Hamilton function
and then from ,

Since
the Hamilton function achieves minimum at .

By (20) and (51), the two-point boundary value problem is

Let
Then
where .

On the other hand, by the second equation of (53),

Combine (55), (56), and the randomicity of and ; it follows that and satisfy (49) and (48), respectively.

It is obvious that is stabilizable if and only if (36) holds, and is detectable if and only if (40) holds. So when (36) and (40) hold, the Riccati equation (49) has a unique positive semidefinite solution such that is asymptotically stable. Since is stable,
where and are eigenvalues of and , respectively. Therefore, the Sylvester equation (48) has a unique solution ([27]).

In accord with (13), (14), (17), (51), (53), and (54), it follows that the unique optimal control-state pair of is
and satisfies the equation

Obviously, under the control-state pair (), the finite poles of resulting system all are the eigenvalues of the matrix . Hence, the finite poles of resulting system all are located on the left-half complex plane.

By (48), (49), (51), (54), and (59), one can obtain that
and then
which indicates that (47) holds.

#### 4. A Simulation Example

In this section, we give a simple example to illuminate the design method of FI control law and demonstrate its feasibility.

Consider the irregular singular system (1) and exosystem (2), where In the performance indices (3) and (5), we choose

Obviously, , , and we can choose and . Let and then the system (1) is r.s.e to the following system: under the transformation (13).

(i) Let ; then is damped. We consider the performance index (3). The unique positive semidefinite solution of the Riccati equation (35) is The simulation results are displayed in Figure 1 and the optimal performance index is .

(ii) Let ; then is sinusoidal perturbation. We consider the performance index (5). The unique positive semidefinite solution of the Riccati equation (49) is , and the unique solution of the Sylvester equation (48) is . The simulation results are displayed in Figure 2 and the optimal performance index is .

#### 5. Conclusion

In this paper the singular LQ problem for irregular singular systems with persistent disturbances has been investigated. By restricted system equivalence transformation, we transformed the singular LQ problem for irregular singular systems with persistent disturbances to the optimal problem for standard state space systems. Consequently, based on optimization theory for standard state space systems, we have derived FI optimal control-state pair under some matrix rank equality conditions by solving an algebraic Riccati equation and a Sylvester equation. Under the optimal control-state pair, the finite poles of resulting system are all located on the left-half complex plane.

The significance of the paper, we think, can be summarized as follows: (1) to our knowledge, it seems that the present paper is the first to apply restricted system equivalence transformation to decompose system state into free state and restricted state and decompose input into free input and forced input; (2) it seems that the present paper is the first to apply the full information feedback (FI) method to the singular LQ problem for irregular singular systems with persistent disturbances; (3) all the conditions adopted in this paper are unitedly described by matrix rank equalities; (4) the optimal performance indices are formulated explicitly.

However, there are many problems unsolved about the singular LQ problem of the irregular singular systems. For example, in this paper only the case where the performance index is nonnegative was treated, and the indefinite LQ problem for irregular singular systems with persistent disturbances has not been involved. More importantly, in the controller design, we need to solve an algebraic Riccati equation, which is still a challenge. Therefore, we think that the significance of the paper exists in theory more than in practice.

#### Appendix

#### The Proof of Lemma 3

First of all, we prove that system (1) is impulse controllable if and only if (9) holds.

*Necessity*. By the transformation , the system (1) is r.s.e to the following system:
where .

Premultiplying the second equation of (A.1) by , where , we have that From the randomicity of and , it follows that , which implies that , so (9) holds.

*Sufficiency*. When (9) holds, there exists a matrix such that . Denote and ; then first equation of (A.1) is changed to

Obviously, for every initial condition and disturbance governed by the system (2) there exists a smooth solution of (A.3). Then () is the smooth control-state pair of system (A.1), which finishes the proof of sufficiency.

From (8), we can prove that (9) is equivalent to (10) by easy computation.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by Natural Science Foundation of Zhejiang Province under Grant Y1110036 and Research Project of Zhejiang Province Education Department under Grant Y201018827.