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Abstract

This paper studies the output regulation problem of time-varying descriptor systems and the problem of designing state feedback and dynamic measurement output feedback control laws which asymptotically achieves output regulation and disturbance rejection is considered. New regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. Based on the solution of the regulator equations, the state feedback and dynamic measurement output feedback control laws are designed to solve the output regulation problem. The work extends the existing results of output regulation problem for time-varying linear systems to the time-varying descriptor systems. Numerical examples are given to show the effectiveness of our methodology.

Keywords

Time-varying descriptor systems output regulation state feedback dynamic measurement output feedback regulator equations

Introduction

The output regulation problem is an essential topic in control theory. It aims to achieve asymptotic tracking of the reference input and/or reject a class of exogenous disturbances while maintaining the internal stability of the closed-loop system.¹ This problem arises from many control problems in practice, such as cruise control of automobiles, aircraft taking-off and landing, manipulation of robot arms, orbiting of satellites, motor speed regulation, and so on.¹ It is shown in Francis² that studies the linear output regulation problem, and proposes that the solvability of a linear output regulation problem is equivalent to that of the given regulator equations of the associated linear time-invariant (LTI) system has a solution. In 1976, scholars invented the internal model principle to study the robust linear output regulation problem. In particular, a dynamic compensator called internal model is designed by using the internal model principle. The robust linear output regulation problem of uncertain linear time-invariant systems is transformed into the stabilization problem of augmented systems composed of original systems and internal models. Since then, scholars have been trying to extend the output regulation problem to various cases.^3–10 The perfect regulation problem under measurement output feedback is presented in Chen et al.,³ and a necessary and sufficient condition is derived. In Isidori et al.,⁴ the linear output regulation problem subject to input saturation is studied, where the sinusoidal or constant exogenous signals are considered. It is shown in Marino and Tomei⁵ that considers the case of the exosystem has unknown parameters and known order, and a dynamic output feedback control law is designed to achieve the object of output regulation problem based on the proposed adaptive internal model. Hu and Lin⁶ considers the output regulation problem of linear systems subject to control constraint which is described by a compact convex set. In Chen and Huang,⁷ the global robust output regulation problem for a class of nonlinear systems in output feedback form has been studied which assumes that the solution of the regulator equations is polynomial. It is shown in Zhang and Serrani⁸ that the output regulation problem of linear time-varying systems is studied. The regulator equations is given in the form of differential matrix equations. The output regulation problem of linear time-delay systems and the robust output regulation problem of linear time-delay systems are considered in Lu⁹ and Huang,¹⁰ respectively.

Descriptor systems arise in many aspects of our daily life such as cruise control of aircraft landing and taking-off, power systems, orbiting of satellites, motor speed regulation, and so forth. Since the late 1970s, descriptor systems have attracted the attention of many researchers.¹¹ They have studied the stability, solvability, controllability and observability, pole assignment, and other related problems of descriptor systems.^12–15 Descriptor systems are often used to describe systems with algebraic constraints on state variables. Therefore, it is of practical importance to study the ability to track targets and cope with disturbances for descriptor time-varying systems subject to external disturbances. It is shown in Lin and Dai¹⁶ that the output regulation problem of descriptor linear systems is studied, the regulator equations corresponding to the output regulation problem of descriptor linear systems is proposed. It is proved that the output regulation problem is solvable if and only if the corresponding regulator equations is solvable. In Ji and Huang,¹⁷ the necessary and sufficient conditions for the output regulation problem of descriptor nonlinear systems are given by using a generalized version of the center manifold theorem. Chen and Huang¹⁸ considers the robust output regulation problem for a class of descriptor nonlinear systems which contain uncertain parameters. It requires the solution of the regulator equations be a polynomial of the exogenous signal. The problems of semiglobal stabilization and output regulation of descriptor linear systems subject to input saturation is considered in Lan and Huang.¹⁹ It is shown in Wang and Huang²⁰ that a normal output feedback control law is designed to solve the output regulation problem of descriptor nonlinear systems without the normalizability assumption. In Pang et al.,²¹ the robust output regulation of descriptor nonlinear systems is solved via a nonlinear internal model. It is shown that the polynomial assumption in Chen and Huang¹⁸ can be replaced by a much milder condition. A new condition is proposed which can be applied to a larger class of non-polynomial nonlinear systems. It is shown in Jafari and Binazadeh²² that the robust output regulation of discrete-time singular systems with actuator saturation and uncertainties is solved by using composite nonlinear feedback control.

This paper considers the problem of output regulation for time-varying descriptor systems. It is of significance to study time-varying descriptor systems. The problem of controlling a mobile robot moving on an uneven surface can be formulated as stabilizing a time-varying system.^23,24 Generally, there are two methods to solve this problem, internal model method and regulator equation method, respectively. The major difference between the internal model method and the regulator equation method is that the former is usually used to solve output regulation problems in which the system contains uncertainty.^{7–10,16–18} The system studied in this paper does not contain uncertainty. Therefore, the regulator equation method is used to solve the output regulation problem of the time-varying descriptor systems. In this paper, new regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. In the stability analysis of the time-varying descriptor system, we extend the method in¹ for stability analysis to make the studied closed-loop system stable. Based on the given regulation equations and the conclusions of the stability analysis, the sufficient and necessary conditions for solving the output regulation problem of the time-varying descriptor system are given. The specific design methods of the state feedback controller and the dynamically measured output feedback controller were finally obtained, which expanded the research scope of the output regulation problem.

The contributions of this paper are as follows:

New regulator equations corresponding to the output regulation problem of time-varying descriptor systems are proposed in the form of differential-algebraic matrix equations. The unique solution of the new regulator equations is given in the Appendix as well which is used in the control design.

The state feedback and the dynamic measurement output feedback control are considered. We extend the results of output regulation problem from linear time-varying systems and descriptor systems to time-varying descriptor systems.

Notations: Throughout this paper, ∥·∥ stands for the Euclidean norm. $rank (E)$ represents the rank of matrix E. $\det (A)$ represents the determinant of matrix $A \in R^{n \times n}$ . $\deg (\cdot)$ represents the degree of polynomial (·).

Problem formulation and preliminaries

Consider the following time-varying descriptor system:

\begin{matrix} E \overset{\cdot}{x} (t) & = A (t) x (t) + B_{u} (t) u (t) + B_{v} (t) v (t), \\ \overset{\cdot}{v} (t) & = S (t) v (t), \\ e (t) & = C (t) x (t) + D_{u} (t) u (t) + D_{v} (t) v (t), \\ y_{m} (t) & = C_{m} (t) x (t) + D_{m} (t) u (t) + F_{m} (t) v (t), \end{matrix}

(1)

where $x (t) \in R^{n}$ is the state of the system, $u (t) \in R^{m}$ is the control input, $e (t) \in R^{p}$ is the tracking error, $y_{m} (t) \in R^{p_{m}}$ is the measurment output and $v (t) \in R^{q}$ is the exogenous signal representing the reference input and/or the disturbance. $A (t), B_{u} (t), C (t), D_{u} (t), B_{v} (t), D_{v} (t), C_{m} (t), D_{m} (t),$

$F_{m} (t), S (t)$ are matrices whose entries are the functions of time t, bounded and continuous on $[t_{0}, + \infty)$ . E is a singular matrix, i.e., $rank (E) < n$ .

In the following, we introduce the definitions of uniform regularity and the impulse-freeness for time-varying descriptor systems.

Definition 1. (See Lin and Dai¹⁶) System (1) is said to be uniformly regular, if there exists a constant s such that

\det (sE - A (t)) \neq 0, \forall t \in [t_{0}, + \infty) .

(2)

Definition 2. Suppose the time-varying descriptor system (1) is uniformly regular. It is said to be impulse-free, if the pair $(E, A (t))$ satisfies $\deg (\det (sE - A (t)) = rank (E)$ , for all $t \geq t_{0}$ .

Condition (2) guarantees the unique solution of system (1). In this paper, we assume that system (1) is uniformly regular and impulse-free.

Now, we introduce an equivalent transformation of time-varying descriptor systems. Consider the time-varying descriptor system as follow:

\begin{matrix} E \overset{\cdot}{x} (t) = A (t) x (t) + B (t) u (t), \\ y (t) = C (t) x (t) . \end{matrix}

(3)

There always exist two invertible matrices $T_{1}, T_{2} \in R^{n \times n}$ , such that

\begin{matrix} T_{1} E T_{2} = [\begin{matrix} I & 0 \\ 0 & 0 \end{matrix}], T_{1} A (t) T_{2} = [\begin{matrix} A_{11} (t) & A_{12} (t) \\ A_{21} (t) & A_{22} (t) \end{matrix}], \\ T_{1} B (t) = {[\begin{matrix} \begin{matrix} B_{1}^{T} (t) & B_{2}^{T} (t) \end{matrix} \end{matrix}]}^{T}, \\ C (t) T_{2} = [\begin{matrix} \begin{matrix} C_{1} (t) & C_{2} (t) \end{matrix} \end{matrix}] . \end{matrix}

(4)

By taking linear coordinate transformation $z (t) = [\begin{matrix} z_{1}^{T} (t) & \begin{matrix} z_{2}^{T} (t) \end{matrix} \end{matrix}]^{T} = T_{2}^{- 1} x (t)$ and left multiplying $T_{1}$ on the first equation of (3), we have the following time-varying descriptor system

\begin{matrix} {\overset{\cdot}{z}}_{1} (t) = A_{11} z_{1} (t) + A_{12} (t) z_{2} (t) + B_{1} (t) u (t), \\ 0 = A_{21} (t) z_{1} (t) + A_{22} (t) z_{2} (t) + B_{2} (t) u (t), \\ y (t) = C_{1} (t) z_{1} (t) + C_{2} (t) z_{2} (t) . \end{matrix}

(5)

Clearly, system (3) and system (5) are equivalent, which implies that system (3) is uniformly regular and impulse-free if and only if system (5) is uniformly regular and impulse-free. It can be easily verified that system (3) is impulse-free if and only if $A_{22} (t)$ in the system (5) is invertible for all t.

We consider two classes of feedback control law in this paper.

State feedback control law:

u (t) = K_{x} (t) x (t) + K_{v} (t) v (t),

(6)

where $K_{x} (t)$ is the feedback gain and $K_{v} (t)$ is the feedforward gain. $K_{x} (t), K_{v} (t)$ are matrices in which the entries are the functions of time t, bounded and continuous on $[t_{0}, + \infty)$ .

2. Dynamic measurement output feedback control law:

\begin{matrix} \hat{E} \overset{\cdot}{z} (t) = G_{1} (t) z (t) + G_{2} (t) y_{m} (t), \\ u (t) = K (t) z (t), \end{matrix}

(7)

where $G_{1} (t), G_{2} (t)$ and $K (t)$ are matrices in which the entries are the functions of time t, bounded and continuous on $[t_{0}, + \infty)$ .

By combining control law (6) or (7) with (1), we can get the closed-loop system as follow:

\begin{matrix} \bar{E} {\overset{\cdot}{x}}_{c} (t) & = A_{c} (t) x_{c} (t) + B_{c} (t) v (t), \\ \overset{\cdot}{v} (t) & = S (t) v (t), \\ e (t) & = C_{c} (t) x_{c} (t) + D_{c} (t) v (t) . \end{matrix}

(8)

In the case of state feedback, let $x_{c} (t) = x (t)$ and

\begin{matrix} \bar{E} = E, \\ A_{c} (t) = A (t) + B_{u} (t) K_{x} (t), \\ B_{c} (t) = B_{u} (t) K_{v} (t) + B_{v} (t), \\ C_{c} (t) = C (t) + D_{u} (t) K_{x} (t), \\ D_{c} (t) = D_{u} (t) K_{v} (t) + D_{v} (t) . \end{matrix}

(9)

In the case of dynamic measurement output feedback, let $x_{c} (t) = {[\begin{matrix} x^{T} (t) & z^{T} (t) \end{matrix}]}^{T}$ and

\begin{matrix} \bar{E} = [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}], D_{c} (t) = D_{v} (t), \\ A_{c} (t) = [\begin{matrix} A (t) & B_{u} (t) K (t) \\ G_{2} (t) C_{m} (t) & G_{1} (t) + G_{2} (t) D_{m} (t) K (t) \end{matrix}], \\ B_{c} (t) = [\begin{matrix} B_{v} (t) \\ G_{2} (t) F_{m} (t) \end{matrix}], C_{c} (t) = [\begin{matrix} \begin{matrix} C (t) & D_{u} (t) K (t) \end{matrix} \end{matrix}] . \end{matrix}

(10)

Our object is to choose appropriate $K_{x} (t), K_{v} (t)$ and $K (t)$ , such that the closed-loop system (8) satisfies the following two properties:

Property 1. The pair $(\bar{E}, A_{c} (t))$ is stable.

Property 2. For all $x_{c} (t_{0})$ and $v (t_{0})$ , the tracking error $e (t)$ converges to $0$ , that is, $lim_{t \to + \infty} e (t) = 0$

Remark 1. Under control law (6) or (7), if the closed-loop system (8) satisfies Properties 1 and 2, then we said control law (6) or (7) solves the output regulation problem of system (1).

To solve the output regulation problem of the system (1), we need the following assumptions:

Assumption 1. The triple $(E, A (t), B_{u} (t))$ is stabilizable.

Assumption 2. For any $t \geq t_{0}$ , $∥ W (t_{0}, t) ∥ \leq ϖ$ , where $ϖ > 0$ and $W (t, s)$ is the state transition matrix generated by $S (t)$ .

Assumption 3. The triple $([\begin{matrix} E & 0 \\ 0 & I \end{matrix}], [\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}], [\begin{matrix} \begin{matrix} C_{m} (t) & F_{m} (t) \end{matrix} \end{matrix}])$ is detectable.

Remark 2. Assumption 1 is made so that Property 1 can be achieved by a state feedback. Assumption 2 is general, because $ϖ$ can be any positive real number. One can replace Assumption 2 by $∥ W (t, t_{0}) v (t) ∥ \geq ∥ v (t) ∥$ for any $t \geq t_{0}$ . Clearly, this assumption is stronger than Assumption 2 (see Ichikawa and Katayama²⁵). It is analogous with Assumption 1.1 in Huang,¹in which the dynamic of the exosystem is anti-Hurwitz. Assumption 3, together with Assumption 1 guarantee the stability of the pair $(\bar{E}, A_{c} (t))$ by a measurement output feedback, where $\bar{E}$ and $A_{c} (t)$ are defined in (10).

Remark 3. More details about the stabilizability and detectability of time-varying descriptor systems can be found in Wang,²⁶Yao et al.²⁷

Main result

This section presents the solutions to the output regulation problem via state feedback and dynamic measurement output feedback.

Solution of output regulation via state feedback

Next, we present a necessity and sufficiency condition, which is the output regulation problem via state feedback is solvable if and only if the given regulator equations are solvable.

Theorem 1. Under Assumptions 1 and 2, a state feedback control law $(K_{x} (t), K_{v} (t))$ solves the output regulation problem of the time-varying descriptor system (1) if and only if there exist continuous bounded matrices $X (t)$ and $U (t)$ which satisfy the following regulator equations

E \overset{\cdot}{X} (t) = A (t) X (t) - EX (t) S (t) + B_{v} (t) + B_{u} (t) U (t),

(11)

0 = lim_{t \to + \infty} [C (t) X (t) + D_{v} (t) + D_{u} (t) U (t)] W (t, t_{0}) .

(12)

where $K_{x} (t)$ is a feedback gain such that the pair $(E, A (t) + B_{u} (t) K_{x} (t))$ is asymptotically stable and $K_{v} (t) = U (t) - K_{x} (t) X (t)$ .

Proof of Theorem 1. Necessity: Suppose control law $u (t) = K_{x} (t) x (t) + K_{v} (t) v (t)$ solves output regulation problem of system (1). Let $K_{x} (t)$ and $K_{v} (t)$ be bounded and continuous. According to Lemma 2 and Lemma 3 in Appendix and $U (t) = K_{x} (t) X (t) + K_{v} (t)$ , the following equation has unique solution which is bounded and continuous,

E \overset{\cdot}{X} (t) = A (t) X (t) - EX (t) S (t) + B_{v} (t) + B_{u} (t) U (t) .

(13)

Let $ξ (t) = x (t) - X (t) v (t)$ , we have

\begin{matrix} E \overset{\cdot}{ξ} (t) = E \overset{\cdot}{x} (t) - E \overset{\cdot}{X} (t) v (t) - EX (t) \overset{\cdot}{v} (t) \\ = (A (t) + B_{u} (t) K_{x} (t)) x (t) - EX (t) S (t) v (t) \\ + (B_{u} (t) K_{v} (t) + B_{v} (t)) v (t) - E \overset{\cdot}{X} (t) v (t) \\ = (A (t) + B_{u} (t) K_{x} (t)) x (t) - E \overset{\cdot}{X} (t) v (t) \\ + ((B_{u} (t) K_{v} (t) + B_{v} (t)) - EX (t) S (t)) v (t), \end{matrix}

by the definition of $K_{v} (t)$ and equation (13), we have

\begin{matrix} E \overset{\cdot}{ξ} (t) = (A (t) + B_{u} (t) K_{x} (t)) x (t) - E \overset{\cdot}{X} (t) v (t) \\ + (E \overset{\cdot}{X} (t) - (A (t) + B_{u} (t) K_{x} (t)) X (t)) v (t) \\ = (A (t) + B_{u} (t) K_{x} (t)) (x (t) - X (t) v (t)) \\ = (A (t) + B_{u} (t) K_{x} (t)) ξ (t) . \end{matrix}

Since the pair $(E, A (t) + B_{u} (t) K_{x} (t))$ is asymptotically stable, one has $lim_{t \to + \infty} ξ (t) = 0 .$

Note that the output regulation problem is solved, i.e., $lim_{t \to + \infty} e (t) = 0$ . Thus

\begin{matrix} lim_{t \to + \infty} e (t) = lim_{t \to + \infty} (C (t) x (t) + D_{u} (t) u (t) + D_{v} (t) v (t)) \\ = lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) x (t) \\ + (D_{u} (t) K_{v} (t) + D_{v} (t)) v (t)) \\ = lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) ξ (t) \\ + ((C (t) + D_{u} (t) K_{x} (t)) X (t) + D_{u} (t) K_{v} (t) \\ + D_{v} (t)) v (t)) . \end{matrix}

(14)

Since all the matrices are bounded and continuous, then we have

\begin{matrix} lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) X (t) + D_{u} (t) K_{v} (t) \\ + D_{v} (t)) v (t)) = 0 . \end{matrix}

\begin{matrix} With U (t) = K_{x} (t) X (t) + K_{v} (t), we have \\ lim_{t \to + \infty} (C (t) X (t) + D_{v} (t) + D_{u} (t) U (t)) v (t) = 0 . \end{matrix}

Due to $v (t) = W (t, t_{0}) v (t_{0})$ for any $v (t_{0}) \in R^{q}$ and $W (t, t_{0}) v (t_{0})$ does not necessarily decay to zero for all $v (t_{0}) \neq 0$ in general, one has $lim_{t \to + \infty} (C (t) X (t) + D_{v} (t) + D_{u} (t) U (t)) W (t, t_{0}) v (t_{0}) = 0$ which implies $lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) X (t) + (D_{u} (t) K_{v} (t) + D_{v} (t)) = 0$ . Proof of necessity is completed.

Sufficiency: To show sufficiency, let $X (t)$ and $U (t)$ be solution of the regulator equations (11) and (12). Consider the stabilizing control law

\begin{matrix} u (t) = K_{x} (t) x (t) + K_{v} (t) v (t) \\ = [\begin{matrix} K_{x} (t) U (t) - K_{x} (t) X (t) \end{matrix}] [\begin{matrix} x (t) \\ v (t) \end{matrix}], \end{matrix}

where $K_{x} (t)$ is a matrix such that the pair $(E, A (t) + B_{u} (t) K_{x} (t))$ is asymptotically stable. Let $ξ (t) = x (t) - X (t) v (t)$ , and by the definitions of $u (t)$ and $U (t)$ , we have

\begin{matrix} lim_{t \to + \infty} e (t) = lim_{t \to + \infty} (C (t) x (t) + D_{u} (t) u (t) + D_{v} (t) v (t)) \\ = lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) ξ (t) \\ + ((C (t) + D_{u} (t) K_{x} (t)) X (t) \\ + D_{u} (t) K_{v} (t) + D_{v} (t)) v (t)) \\ = lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) ξ (t) \\ + (C (t) X (t) + D_{v} (t) + D_{u} (t) U (t)) v (t)) \\ = lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) ξ (t) \\ + (C (t) X (t) + D_{v} (t) + D_{u} (t) U (t)) W (t, t_{0}) v_{0}) . \end{matrix}

(15)

Because of the asymptotic stability of the pair $(E, A (t) + B_{u} (t) K_{x} (t))$ , one can get $lim_{t \to + \infty} ξ (t) = 0$ . By equations (12) and (15), it is easy to known that $lim_{t \to + \infty} e (t) = 0$ . Proof of sufficiency is completed.

Remark 4. The regulator equations in this paper are given in terms of differential-algebraic equations. We can first assume is known and compute the unique solution of the first equation (11), and then solve $U (t)$ by equation (12). The unique solution of the equation in the form of (11) is given by Lemma 2 in the Appendix.

Remark 5. If $∥ W (t, t_{0}) ∥ \leq ϖ$ , for any $t \geq t_{0}$ . It can be shown that

\begin{matrix} ∥ ((C (t) + D_{u} (t) K_{x} (t)) X (t) + (D_{u} (t) K_{v} (t) + D_{v} (t))) ∥ \\ = ∥ ((C (t) + D_{u} (t) K_{x} (t)) X (t) \\ + (D_{u} (t) K_{v} (t) + D_{v} (t))) W (t, t_{0}) W (t_{0}, t) ∥ \\ \leq ϖ ∥ ((C (t) + D_{u} (t) K_{x} (t)) X (t) \\ + (D_{u} (t) K_{v} (t) + D_{v} (t))) W (t, t_{0}) ∥ . \end{matrix}

Then the condition (12) implies the following condition:

\begin{matrix} lim_{t \to + \infty} ((C (t) + D_{u} (t) K_{x} (t)) X (t) + (D_{u} (t) K_{v} (t) + D_{v} (t)) \\ = lim_{t \to + \infty} (C (t) X (t) + D_{u} (t) U (t) + D_{v} (t)) = 0 . \end{matrix}

Remark 6. According to Lyapunov second method for a time-varying descriptor system,²⁸one can always find a $K_{x} (t)$ by solving a differential linear matrix inequality.

Solution of output regulation via dynamic measurement output feedback

Next, we present the necessity and sufficiency condition of the output regulation problem via dynamic measurement output feedback. Before giving the main theorem of this subsection, a technical lemma is given which will be used to proof the main result of this subsection.

Lemma 1. Suppose that Assumption 2 holds. Consider dynamic measurement output feedback control law in the form of (7) such that the pair $(\bar{E}, A_{c} (t))$ is stable. Then the following are equivalent:

(i) Output regulation problem of the system (1) can be solved via dynamic measurement output feedback control law $(G_{1} (t), G_{2} (t), K (t))$ .

(ii) There exists a matrix $X (t)$ satisfying the following differential-algebraic matrix equations:

\begin{array}{l} [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}] \overset{\cdot}{X} (t) = - [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}] X (t) S (t) \\ + [\begin{matrix} A (t) & B_{u} (t) K (t) \\ G_{2} (t) C_{m} (t) & Θ_{α} (t) \end{matrix}] X (t) \\ + [\begin{matrix} B_{v} (t) \\ G_{2} (t) F_{m} (t) \end{matrix}], \\ 0 = \lim_{t \to + \infty} ([\begin{matrix} \begin{matrix} C (t) & D_{u} (t) \end{matrix} K (t) \end{matrix}] X (t) \\ + D_{v} (t)) W (t, t_{0}) . \end{array}

(16)

where $\hat{E}$ is defined in (7) and $Θ_{α} (t) = G_{1} (t) + G_{2} (t) D_{m} (t) K (t)$ .

(iii) There exists matrices $(X_{1} (t), X_{2} (t), U (t))$ such that $X_{1} (t)$ and $U (t)$ satisfy the following regulator equations:

\begin{matrix} E {\overset{\cdot}{X}}_{1} (t) = A (t) X_{1} (t) - E X_{1} (t) S (t) \\ + B_{u} (t) U (t) + B_{v} (t), \\ 0 = lim_{t \to + \infty} (C (t) X_{1} (t) + D_{v} (t) \\ + D_{u} (t) U (t)) W (t, t_{0}), \end{matrix}

(17)

and $X_{2} (t)$ satisfies:

\begin{matrix} \hat{E} {\overset{\cdot}{X}}_{2} (t) = G_{1} (t) X_{2} (t) + G_{2} (t) (C_{m} (t) X_{1} (t) \\ + D_{m} (t) U (t) + F_{m} (t)) - \hat{E} X_{2} (t) S (t), \end{matrix}

(18)

where

\begin{matrix} U (t) & = K (t) X_{2} (t) . \end{matrix}

(19)

Proof of Lemma 1. $(i) \leftrightarrow (ii)$ The necessity and the sufficiency can be easily shown by the proof of Theorem 1 and the proof of Lemma 1.4 in,¹respectively. $(ii) \leftrightarrow (iii)$ Suppose $(ii)$ holds. Denote

\begin{matrix} X (t) = [\begin{matrix} X_{1} (t) \\ X_{2} (t) \end{matrix}], \end{matrix}

where $X_{1} (t)$ and $X_{2} (t)$ are bounded continuous matrices. Then (16) can be rewritten as

\begin{matrix} E {\overset{\cdot}{X}}_{1} (t) = A (t) X_{1} (t) - E X_{1} (t) S (t) \\ + B_{u} (t) K (t) X_{2} (t) + B_{v} (t), \\ \hat{E} {\overset{\cdot}{X}}_{2} (t) = G_{2} (t) C_{m} (t) X_{1} (t) + (G_{1} (t) \\ + G_{2} (t) D_{m} (t) K (t)) X_{2} (t) - \hat{E} X_{2} (t) S (t) \\ + G_{2} (t) F_{m} (t), \\ 0 = lim_{t \to + \infty} (C (t) X_{1} (t) + D_{u} (t) K (t) X_{2} (t) \\ + D_{v} (t)) W (t, t_{0}), \end{matrix}

(20)

which is the same as:

\begin{matrix} E {\overset{\cdot}{X}}_{1} (t) = A (t) X_{1} (t) - E X_{1} (t) S (t) \\ + B_{u} (t) K (t) X_{2} (t) + B_{v} (t), \\ \hat{E} {\overset{\cdot}{X}}_{2} (t) = G_{1} (t) X_{2} (t) + G_{2} (t) (C_{m} (t) X_{1} (t) \\ + D_{m} (t) K (t) X_{2} (t) + F_{m} (t)) - \hat{E} X_{2} (t) S (t), \\ 0 = lim_{t \to + \infty} (C (t) X_{1} (t) + D_{u} (t) K (t) X_{2} (t) \\ + D_{v} (t)) W (t, t_{0}) . \end{matrix}

(21)

Let $U (t) = K (t) X_{2} (t)$ , then $(ii) \to (iii)$ . Similarly, supposing $(iii)$ holds, we need to show that $X_{1} (t)$ and $X_{2} (t)$ satisfy (20) or (21). Substituting $U (t) = K (t) X_{2} (t)$ into equation (17, one can conclude that $X_{1} (t)$ and $X_{2} (t)$ satisfy the first and third equations of (21), then by substituting $U (t) = K (t) X_{2} (t)$ into (18), it can be shown that satisfies the second equation of (21). The proof is completed.

Now, we present the necessity and sufficiency condition of the output regulation problem via dynamic measurement output feedback.

Theorem 2. Suppose Assumptions 1, 2 and 3 hold, the output regulation problem of the time-varying descriptor system (1) can be solved by the dynamic measurement output feedback control law $(G_{1} (t), G_{2} (t), K (t))$ defined in (7) if and only if there exists continuous bounded matrices $X (t)$ and $U (t)$ which satisfy the following regulator equations:

E \overset{\cdot}{X} (t) = A (t) X (t) - EX (t) S (t) + B_{v} (t) + B_{u} (t) U (t),

(22)

0 = lim_{t \to + \infty} (C (t) X (t) + D_{v} (t) + D_{u} (t) U (t)) W (t, t_{0}) .

(23)

Proof of Theorem 2. Necessity: Suppose the dynamic measurement output feedback control law $u (t) = K (t) z (t)$ solves the output regulation problem of system (1). The closed-loop system has the following form:

\begin{array}{l} [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}] [\begin{matrix} \overset{\cdot}{x} (t) \\ \overset{\cdot}{z} (t) \end{matrix}] = Λ (t) [\begin{matrix} x (t) \\ z (t) \end{matrix}] \\ + [\begin{matrix} B_{v} (t) \\ G_{2} (t) F_{m} (t) \end{matrix}] v (t), \\ \overset{\cdot}{v} (t) = S (t) v (t), \\ e (t) = [\begin{matrix} \begin{matrix} C (t) & D_{u} (t) K (t) \end{matrix} \end{matrix}] [\begin{matrix} x (t) \\ z (t) \end{matrix}] + D_{v} (t) v (t), \\ y_{m} (t) = [\begin{matrix} \begin{matrix} C_{m} (t) & D_{m} (t) K (t) \end{matrix} \end{matrix}] [\begin{matrix} x (t) \\ z (t) \end{matrix}] + F_{m} (t) v (t) . \end{array}

(24)

where

Λ (t) = [\begin{matrix} A (t) & B_{u} (t) K (t) \\ G_{2} (t) C_{m} (t) & G_{1} (t) + G_{2} (t) D_{m} (t) K (t) \end{matrix}] .

According to the proof in the necessity part of Theorem 1, it can be easily known that there exsit bounded continuous matrices $X_{1} (t)$ and $X_{2} (t)$ which satisfy the following matrix equations:

\begin{matrix} [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}] [\begin{matrix} {\overset{\cdot}{X}}_{1} (t) \\ {\overset{\cdot}{X}}_{2} (t) \end{matrix}] = Λ (t) [\begin{matrix} X_{1} (t) \\ X_{2} (t) \end{matrix}] \\ - [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}] [\begin{matrix} X_{1} (t) \\ X_{2} (t) \end{matrix}] S (t) + [\begin{matrix} B_{v} (t) \\ G_{2} (t) F_{m} (t) \end{matrix}], \\ 0 = lim_{t \to + \infty} ([\begin{matrix} C (t) & D_{u} (t) K (t) \end{matrix}] [\begin{matrix} X_{1} (t) \\ X_{2} (t) \end{matrix}] \\ + D_{v} (t)) W (t, t_{0}) . \end{matrix}

It can be easily obtained that

\begin{matrix} E {\overset{\cdot}{X}}_{1} (t) = A (t) X_{1} (t) + B_{u} (t) K (t) X_{2} (t) \\ - E X_{1} (t) S (t) + B_{v} (t), \end{matrix}

(25)

\begin{matrix} 0 = lim_{t \to + \infty} (C (t) X_{1} (t) + D_{u} (t) K (t) X_{2} (t) \\ + D_{v} (t)) W (t, t_{0}) . \end{matrix}

(26)

By defining $X (t) = X_{1} (t)$ , $U (t) = K (t) X_{2} (t)$ , the equations (25) and (26) become the regulator equations (22) and (23). The proof of necessity is completed.

Sufficiency: Suppose that there exist continuous bounded matrices $X (t)$ and $U (t)$ which satisfy the regulator equations (22) and (23). According to Theorem 1, the output regulation problem can be solved by state feedback control law

u (t) = [\begin{matrix} K_{x} (t) & U (t) - K_{x} (t) X (t) \end{matrix}] [\begin{matrix} x (t) \\ v (t) \end{matrix}] .

Under Assumption 3, that is

([\begin{matrix} E & 0 \\ 0 & I \end{matrix}], [\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}], [\begin{matrix} \begin{matrix} C_{m} (t) & F_{m} (t) \end{matrix} \end{matrix}])

is detectable, there exists a bounded and continuous matrix $L (t) = [\begin{matrix} L_{1} (t) \\ L_{2} (t) \end{matrix}]$ , such that the pair

([\begin{matrix} E & 0 \\ 0 & I \end{matrix}], [\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}] - L (t) [\begin{matrix} \begin{matrix} C_{m} (t) & F_{m} (t) \end{matrix} \end{matrix}])

is asymptotically stable. Because the informations of $x (t)$ and $v (t)$ is not available, we design an observer in the following form to estimate the state $x (t)$ and the exogenous signal $v (t)$ ,

\begin{array}{l} [\begin{matrix} E & 0 \\ 0 & I \end{matrix}] \overset{\cdot}{z} (t) = [\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}] z (t) + L (t) y_{m} (t) \\ + [\begin{matrix} B_{u} (t) \\ 0 \end{matrix}] [\begin{matrix} \begin{matrix} K_{x} (t) & U (t) - K_{x} (t) X (t) \end{matrix} \end{matrix}] z (t) \\ - [\begin{matrix} L_{1} (t) \\ L_{2} (t) \end{matrix}] ([\begin{matrix} C_{m} (t) & F_{m} (t) \end{matrix}] + D_{m} (t) [K_{x} (t) \\ U (t) - K_{x} (t) X (t)]) z (t) \\ = ([\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}] \\ + [\begin{matrix} B_{u} (t) K_{x} (t) & B_{u} (t) (U (t) - K_{x} (t) X (t)) \\ 0 & 0 \end{matrix}] \\ - [\begin{matrix} L_{1} (t) (C_{m} (t) + D_{m} (t) K_{x} (t)) \\ L_{2} (t) (C_{m} (t) + D_{m} (t) K_{x} (t)) \end{matrix} \\ \begin{matrix} L_{1} (t) (F_{m} (t) + D_{m} (t) U (t) - D_{m} (t) K_{x} (t) X (t)) \\ L_{2} (t) (F_{m} (t) + D_{m} (t) U (t) - D_{m} (t) K_{x} (t) X (t)) \end{matrix}]) z (t) \\ + L (t) y_{m} (t) . \end{array}

(27)

Define

\begin{matrix} \hat{E} = [\begin{matrix} E & 0 \\ 0 & I \end{matrix}], G_{2} (t) = L (t) = [\begin{matrix} L_{1} (t) \\ L_{2} (t) \end{matrix}], \\ G_{1} (t) = [\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}] + [\begin{matrix} B_{u} (t) \\ 0 \end{matrix}] [K_{x} (t) \\ U (t) - K_{x} (t) X (t)] - L (t) ([\begin{matrix} C_{m} (t) F_{m} (t) \end{matrix}] \\ + D_{m} (t) [\begin{matrix} K_{x} (t) & U (t) - K_{x} (t) X (t) \end{matrix}]), \\ K (t) = [\begin{matrix} K_{x} (t) U (t) - K_{x} (t) X (t) \end{matrix}] . \end{matrix}

(28)

Then, $\bar{E}$ and $A_{c} (t)$ in (10) become

\begin{matrix} [\begin{matrix} E & 0 \\ 0 & \hat{E} \end{matrix}] = [\begin{matrix} E & 0 & 0 \\ 0 & E & 0 \\ 0 & 0 & I \end{matrix}], \\ [\begin{matrix} A (t) & B_{u} (t) K (t) \\ G_{2} (t) C_{m} (t) & G_{1} (t) + G_{2} (t) D_{m} (t) K (t) \end{matrix}] = \\ [\begin{matrix} A (t) & B_{u} (t) K_{x} (t) \\ L_{1} (t) C_{m} (t) & A (t) + B_{u} (t) K_{x} (t) - L_{1} (t) C_{m} (t) \\ L_{2} (t) C_{m} (t) & - L_{2} (t) C_{m} (t) \end{matrix} \\ \begin{matrix} B_{u} (t) Λ (t) \\ B_{v} (t) + B_{u} (t) Λ (t) - L_{1} (t) F_{m} (t) \\ S (t) - L_{2} (t) F_{m} (t) \end{matrix}], \end{matrix}

(29)

where $Λ (t) = U (t) - K_{x} (t) X (t)$ .

After appropriate elementary transformation, (29) becomes

[\begin{matrix} ϒ_{1} (t) & B_{u} (t) K_{x} (t) & ϒ_{2} (t) \\ 0 & ϒ_{3} (t) & ϒ_{4} (t) \\ 0 & - L_{2} (t) C_{m} (t) & ϒ_{5} (t) \end{matrix}] .

where

\begin{matrix} ϒ_{1} (t) = A (t) + B_{u} (t) K_{x} (t), \\ ϒ_{2} (t) = B_{u} (t) (U (t) - K_{x} (t) X (t)), \\ ϒ_{3} (t) = A (t) - L_{1} (t) C_{m} (t), \\ ϒ_{4} (t) = B_{v} (t) - L_{1} (t) F_{m} (t), \\ ϒ_{5} (t) = S (t) - L_{2} (t) F_{m} (t) . \end{matrix}

Because the pairs $(E, A (t) + B_{u} (t) K_{x} (t))$ and

([\begin{matrix} E & 0 \\ 0 & I \end{matrix}], Ω_{α} (t))

are stable,where

Ω_{α} (t) = [\begin{matrix} A (t) - L_{1} (t) C_{m} (t) & B_{v} (t) - L_{1} (t) F_{m} (t) \\ - L_{2} (t) C_{m} (t) & S (t) - L_{2} (t) F_{m} (t) \end{matrix}] .

Then the closed-loop system (24) is stable (see Wang²⁹), that is, Property 1 is satisfied. Now we show that Property 2 is satisfied. Define $X_{1} (t) = X (t)$ , and $X_{2} (t) = [\begin{matrix} X_{1} (t) \\ I \end{matrix}] .$ According to the solvability of equation (22) and (23), $(X_{1} (t), U (t))$ also satisfies (17) in Lemma 1. Now we show that $X_{2} (t)$ satisfies (18). Note that $U (t) = K (t) X_{2} (t)$ where $K (t) = [K_{x} (t) K_{v} (t)]$ and $K_{v} (t) = U (t) - K_{x} (t) X_{1} (t)$ . By the definition of $G_{1} (t)$ in (28) and equation (22), we have

\begin{matrix} G_{1} (t) X_{2} (t) = [\begin{matrix} A (t) & B_{v} (t) \\ 0 & S (t) \end{matrix}] [\begin{matrix} X_{1} (t) \\ I \end{matrix}] \\ + [\begin{matrix} B_{u} (t) K_{x} (t) & ϒ_{a} (t) \\ 0 & 0 \end{matrix}] [\begin{matrix} X_{1} (t) \\ I \end{matrix}] \\ - L (t) ϒ_{b} (t) [\begin{matrix} X_{1} (t) \\ I \end{matrix}] \\ = [\begin{matrix} ϒ_{c} (t) \\ S (t) \end{matrix}] - L (t) (ϒ_{d} (t) + F_{m} (t)) \\ = [\begin{matrix} A (t) X_{1} (t) + B_{u} (t) U (t) + B_{v} (t) \\ S (t) \end{matrix}] \\ - L (t) (C_{m} (t) X_{1} (t) + D_{m} (t) U (t) + F_{m} (t)) \\ = [\begin{matrix} E {\overset{\cdot}{X}}_{1} (t) + E X_{1} (t) S (t) \\ S (t) \end{matrix}] \\ - L (t) (C_{m} (t) X_{1} (t) + D_{m} (t) U (t) + F_{m} (t)) \\ = [\begin{matrix} E & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\overset{\cdot}{X}}_{1} (t) \\ 0 \end{matrix}] \\ + [\begin{matrix} E & 0 \\ 0 & I \end{matrix}] [\begin{matrix} X_{1} (t) \\ I \end{matrix}] S (t) \\ - L (t) (C_{m} (t) X_{1} (t) + D_{m} (t) U (t) + F_{m} (t)) \\ = \hat{E} X_{2} (t) + \hat{E} X_{2} (t) S (t) \\ - L (t) (C_{m} (t) X_{1} (t) + D_{m} (t) U (t) + F_{m} (t)), \end{matrix}

where $\hat{E}$ is defined in (28), and

\begin{matrix} ϒ_{a} (t) = B_{u} (t) (U (t) - K_{x} (t) X_{1} (t)), \\ ϒ_{b} (t) = D_{m} (t) [\begin{matrix} K_{x} (t) U (t) - K_{x} (t) X_{1} (t) \end{matrix}] \\ + [\begin{matrix} C_{m} (t) F_{m} (t) \end{matrix}], \\ ϒ_{c} (t) = B_{u} (t) (K_{x} (t) X_{1} (t) + U (t) - K_{x} (t) X_{1} (t)) \\ + A (t) X_{1} (t) + B_{v} (t), \\ ϒ_{d} (t) = D_{m} (t) (K_{x} (t) X_{1} (t) + U (t) - K_{x} (t) X_{1} (t)) \\ + C_{m} (t) X_{1} (t) . \end{matrix}

We proved that $X_{2} (t)$ satisfies the equation (18) in Lemma 1. According Lemma 1 $(i) \leftrightarrow (iii)$ , the proof of sufficiency is completed.

A numerical example

In this section, we employ two numerical examples to verify the effectiveness of Theorem 1 and Theorem 2 proposed in this paper. Firstly, consider output regulation problem of the following time-varying descriptor system:

\begin{matrix} E \overset{\cdot}{x} (t) = A (t) x (t) + B_{u} (t) u (t), \\ \overset{\cdot}{v} (t) = S (t) v (t), \\ e (t) = C (t) x (t) + D_{v} (t) v (t), \\ y_{m} (t) = C_{m} (t) x (t) + F_{m} (t) v (t), \end{matrix}

(30)

where

E = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], A (t) = [\begin{matrix} 0 & 1 & 0 \\ \sin 2 π t & \sin t & 0 \\ 0 & 1 & 1 \end{matrix}],

B_{u} (t) = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], C (t) = [\begin{matrix} \begin{matrix} 1 & 0 & 0 \end{matrix} \end{matrix}], D_{v} (t) = [\begin{matrix} \begin{matrix} - 1 & 0 & 0 \end{matrix} \end{matrix}],

(31)

S (t) = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] .

The reference signal to be tracked is $v_{1} (t)$ $(v (t) = (v_{1} (t), v_{2} (t), v_{3} (t {))}^{T})$ . By solving the regulator equations (11) and (12) in Theorem 1. $X (t)$ and $U (t)$ are given by

\begin{array}{l} X (t) = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & - 1 & 0 \end{matrix}], U (t) = [\begin{matrix} \begin{matrix} - \sin 2 π t & - \sin t & 0 \end{matrix} \end{matrix}] . \end{array}

We choose feedback gain as follows:

\begin{matrix} K (t) = [\begin{matrix} - 1 - \sin 2 π t & - 2 - \sin t & 0 \end{matrix}] . \end{matrix}

The initial states of the original system and the exogenous system are given as $x (0) = [\begin{matrix} - 1 & 1 & - 1 \end{matrix}]^{T}$ and $v (0) = [\begin{matrix} 0 & 1 & 0 \end{matrix}]^{T}$ . The simulation result of system (30) with the designed state-feedback control law is shown in Figure 1. The output $y (t) = C (t) x (t) = x_{1} (t)$ asymptotically converges to the reference signal $v_{1} (t)$ . The state feedback control law solves the output regulation problem.

Figure 1.

The states and the reference signal.

Next, the output regulation problem of the system (30) via dynamic measurement output feedback is considered, where $E, A (t), B_{u} (t)$ and $D_{v} (t)$ are defined in (31) and

C (t) = [\begin{matrix} \begin{matrix} 1 & 0 & - 1 \end{matrix} \end{matrix}], D_{v} (t) = [\begin{matrix} \begin{matrix} - 1 & - 1 & 0 \end{matrix} \end{matrix}],

C_{m} (t) = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{matrix}], F_{m} (t) = [\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}],

S (t) = [\begin{matrix} 0 & 1 & 0 \\ - {(\frac{π}{2})}^{2} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] .

The reference signal to be tracked is $v_{1} (t)$ and $v_{2} (t)$ and the disturbance signal to be rejected is $v_{3} (t)$ . Solving the regulator equations (22) and (23) in Theorem 2. The solution $X (t)$ and $U (t)$ are given by

\begin{array}{l} X (t) & = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & - 1 & 0 \end{matrix}], \\ U (t) & = [\begin{matrix} \begin{matrix} - s i n 2 π t - {(\frac{π}{2})}^{2} & - s i n t & 0 \end{matrix} \end{matrix}] . \end{array}

We choose the feedback gain and the observer gain as follow:

\begin{array}{l} K (t) = [\begin{matrix} \begin{matrix} - 1 - \sin 2 π t & - 2 - \sin t & 0 \end{matrix} \end{matrix}], \\ L (t) = [\begin{matrix} 2 + \sin t & 0 & 0 \\ 1 + \sin 2 π t + (2 + \sin t) \sin t & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 1 - {(\frac{π}{2})}^{2} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] . \end{array}

The simulation results with $x (0) = [\begin{matrix} - 1 & 1 & - 1 \end{matrix}]^{T}$ and $v (0) = [\begin{matrix} 5 & 0.5 & - 1 \end{matrix}]^{T}$ are shown in Figures 2 and 3. The tracking error asymptotically converges to zero. The dynamic output feedback law solves the output regulation problem of the system (30).

Figure 2.

The states and the reference signals.

Figure 3.

The tracking error signal.

The simulation results verified the effectiveness of our methodology.

Conclusions

In this paper, the output regulation problem of time-varying descriptor systems is studied. The state feedback control and dynamic measurement output feedback control are considered. New regulator equations and its unique solution are given for the output regulation problem of time-varying descriptor systems. The necessary and sufficient conditions for solving the output regulation problem (using regulator equations) are presented. In the future, we will consider the robust output regulation problem of time-varying descriptor systems by using the internal model principle, where the exosystem is time-varying.

Footnotes

Appendix

In this section, the unique solution of the following differential-algebraic matrix equation is proposed,

(A.1)

E \overset{\cdot}{X} (t) = A (t) X (t) - EX (t) S (t) + R (t) .

By using the equivalent transformation of the time-varying descriptor system defined in (4), there exist invertible matrices $T_{1}$ and $T_{2}$ , such that

(A.2)

\begin{matrix} T_{1} E T_{2} T_{2}^{- 1} \overset{\cdot}{X} (t) = T_{1} A (t) T_{2} T_{2}^{- 1} X (t) \\ - T_{1} E T_{2} T_{2}^{- 1} X (t) S (t) + T_{1} R (t) \end{matrix}

(A.1) is transformed into the following equation:

(A.3)

\begin{matrix} [\begin{matrix} I_{q} & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} {\overset{\cdot}{M}}_{1} (t) \\ {\overset{\cdot}{M}}_{2} (t) \end{matrix}] = [\begin{matrix} A_{11} (t) & A_{12} (t) \\ A_{21} (t) & A_{22} (t) \end{matrix}] [\begin{matrix} M_{1} (t) \\ M_{2} (t) \end{matrix}] \\ - [\begin{matrix} I_{q} & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} M_{1} (t) \\ M_{2} (t) \end{matrix}] S (t) + [\begin{matrix} R_{1} (t) \\ R_{2} (t) \end{matrix}] \end{matrix}

where

(A.4)

M (t) = [M_{1}^{T} (t) M_{2}^{T} (t)]^{T} = T_{2}^{- 1} X (t)

(A.5)

{[R_{1}^{T} (t) R_{2}^{T} (t)]}^{T} = T_{1} R (t)

It is obvious that system (1) is uniformly regular and impulse-free if and only if equations (A.1) and (A.3) are uniformly regular and impulse-free. Due to the invertibility of $T_{1}$ and $T_{2}$ , equation (A.1) has a unique solution if and only if equation (A.3) has a unique solution.

The following technical lemma plays a vital role in proving the main result.

(A.6)

\begin{matrix} E \overset{\cdot}{X} (t) = A (t) X (t) - EX (t) S (t) + R (t), \\ X (t_{0}) = X_{0}, \end{matrix}

which is given by

(A.7)

\begin{matrix} X (t) = [\begin{matrix} Δ_{a} (t) \\ Θ (t) - T_{2} A_{22}^{- 1} (t) R_{2} (t) \end{matrix}], \end{matrix}

where $T_{1}$ and $T_{2}$ are invertible matrices defined in (4), $X (t) = [X_{1}^{T} (t) X_{2}^{T} (t)]^{T}$ , and

\begin{matrix} Δ_{a} (t) = T_{2} Ξ (t, t_{0}) T_{2}^{- 1} X_{1 (0)} W (t_{0}, t) \\ + T_{2} \int_{t_{o}}^{t} Ξ (t, r) R_{t} (r) W (r, t) dr \\ Θ (t) = - T_{2} A_{22}^{- 1} (t) A_{21} (t) (T_{2} Ξ (t, t_{0}) T_{2}^{- 1} X_{1 (0)} W (t_{0}, t) \\ + T_{2} \int_{t_{o}}^{t} Ξ (t, r) R_{t} (r) W (r, t) dr) . \end{matrix}

$Ξ (t, s)$ and $W (t, s)$ are the state transition matrices generated by $Φ (t)$ and $S (t)$ , respectively. $Φ (t)$ and $R_{t} (t)$ are defined as follow:

(A.8)

Φ (t) = A_{11} (t) - A_{12} (t) A_{22}^{- 1} (t) A_{21} (t),

(A.9)

R_{t} (t) = R_{1} (t) - A_{12} (t) A_{22}^{- 1} (t) R_{2} (t),

(A.10)

\begin{matrix} {\overset{\cdot}{M}}_{1} (t) = A_{11} (t) M_{1} (t) - M_{1} (t) S (t) + R_{1} (t) \\ + A_{12} (t) M_{2} (t), \end{matrix}

(A.11)

0 = A_{21} (t) M_{1} (t) + A_{22} (t) M_{2} (t) + R_{2} (t) .

According to the fundamental theory of the ordinary differential equation, the matrix equation (A.10) has a unique solution. By the impulse-free property of the system (1), $A_{22} (t)$ is invertible. Then we can solve equation (A.11), that is

(A.12)

M_{2} (t) = - A_{22}^{- 1} (t) A_{21} (t) M_{1} (t) - A_{22}^{- 1} (t) R_{2} (t)

Substituting (A.12) into (A.10), we have

(A.13)

\begin{matrix} {\overset{\cdot}{M}}_{1} (t) = (A_{11} (t) - A_{12} (t) A_{22}^{- 1} (t) A_{21} (t)) M_{1} (t) \\ - M_{1} (t) S (t) + R_{1} (t) - A_{12} (t) A_{22}^{- 1} (t) R_{2} (t) \\ = Φ (t) M_{1} (t) - M_{1} (t) S (t) + R_{t} (t) . \end{matrix}

According to Lemma 2.1 in Wang²⁹and Theorem 1 in Pinzoni,²⁸where $M_{1} (t_{0}) = M_{1 (0)}$ , the unique solution of equation (A.10) is

(A.14)

\begin{matrix} M_{1} (t) = Ξ (t, t_{0}) M_{1 (0)} W (t_{0}, t) \\ + \int_{t_{o}}^{t} Ξ (t, r) R_{t} (r) W (r, t) dr . \end{matrix}

Then we have

(A.15)

\begin{matrix} M_{2} (t) = - A_{22}^{- 1} (t) A_{21} (t) (Ξ (t, t_{0}) M_{1 (0)} W (t_{0}, t) \\ + \int_{t_{o}}^{t} Ξ (t, r) R_{t} (r) W (r, t) dr) - A_{22}^{- 1} (t) R_{2} (t) . \end{matrix}

By the definitions of $M (t)$ in (A.4) and $R_{t} (t)$ in (A.5), we can conclude that (A.7) is the solution of equation (A.6).

Acknowledgements

The authors are grateful to the editor and anonymous reviewers for their valuable suggestions that helped in improving the initial version of the manuscript. The authors would like to express their appreciation to the referees for their helpful comments and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Project supported by the National Natural Science Foundation of China under Grant (No.61074005) and the Talent Project of the High Education of Liaoning Province (LR2012005).

ORCID iD

Wenai Liu

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Solutions to the output regulation problem of time-varying descriptor systems

Abstract

Keywords

Introduction

Problem formulation and preliminaries

Main result

Solution of output regulation via state feedback

Solution of output regulation via dynamic measurement output feedback

A numerical example

Conclusions

Footnotes

Appendix

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References