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Mathematical Problems in Engineering
Volume 2011, Article ID 904914, 16 pages
http://dx.doi.org/10.1155/2011/904914
Research Article

State Feedback Guaranteed Cost Repetitive Control for Uncertain Discrete-Time Systems

1Department of Marine Engineering, Shanghai Maritime University, Shanghai 200135, China
2Department of Automation, Shanghai Jiaotong University, Shanghai 200240, China
3Department of Electronic Information Engineering, Changzhou College of Information Technology, Changzhou 213164, China

Received 26 November 2010; Revised 18 February 2011; Accepted 2 April 2011

Academic Editor: Cristian Toma

Copyright © 2011 Wentao Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper considers the problem of guaranteed cost repetitive control for uncertain discrete-time systems. The uncertainty in the system is assumed to be norm-bounded and time-varying. The objective is to develop a novel design method so that the closed-loop repetitive control system is quadratically stable and a certain bound of performance index is guaranteed for all admissible uncertainties. The state feedback control technique is used in the paper. While for the case that the states are not measurable, an observer-based control scheme is adopted. Sufficient conditions for the existence of guaranteed cost control law are derived in terms of linear matrix inequality (LMI). The control and observer gains are characterized by the feasible solutions to these LMIs. The optimal guaranteed cost control law is obtained efficiently by solving an optimization problem with LMI constraints using existing convex optimization algorithms. A simulation example is provided to illustrate the validity of the proposed method.

1. Introduction

In practice, many tracking systems have to deal with periodic reference and/or disturbance signals, for example, industrial robots, computer disk drives, and rotating machine tools. Repetitive control, which is based on the internal model principle proposed by Francis and Wonham [1], has been proved to be a useful control strategy for this class of systems. Up to date, researchers have devoted considerable efforts to the analysis and design of repetitive control systems. For the continuous-time case, Weiss and Häfele [2] discussed the repetitive control of MIMO systems using 𝐻 design; Tsai and Yao [3] derived upper and lower bounds of the repetitive controller parameters that ensure stability and desired performance; Doh and Chung [4] presented a linear matrix inequality- (LMI-) based repetitive controller design method for systems with norm-bounded uncertainties, while for the discrete-time case, Osburn and Franchek [5] developed a method for designing repetitive controllers using nonparametric frequency response plant models; Freeman et al. [6] proposed an optimality-based repetitive control algorithm for time-invariant systems; Pipeleers et al. [7] proposed a novel design approach for SISO high-order repetitive controllers.

It is well known that in many practical systems, the system model always contains some uncertain elements due to poor plant knowledge, reduced-order models, and nonlinearities such as hysteresis or friction, slowly varying parameters, and so forth, and the uncertainties frequently lead to deterioration of system performance and instability of systems. Hence, robust stability and stabilization for uncertain systems have been the focus of much research in the recent years. However, for the repetitive control of uncertain discrete-time systems, to the best of our knowledge, there are no previous results reported in the literature. This motivates our research.

When controlling a system involving uncertainties, it is often desirable to design a robust controller that not only stabilizes the closed-loop system but also guarantees an ideal level of performance for all admissible uncertainties. One way to address this problem is the so-called guaranteed cost control technique (see, e.g., [8, 9]). Furthermore, LMI approach is a powerful tool in the control theory and applications and has been applied to a wide range of control problems, such as the output feedback control [9] and filter design of time-delayed systems [10]. In this paper, we will adopt these two useful methodologies (i.e., guaranteed cost control technique and LMI approach) to discuss the state feedback repetitive control for discrete-time systems with norm-bounded and time-varying uncertainties. The objective is to develop a novel design method that not only provides an ideal level of performance while preserving system stability but also can be efficiently implemented using existing software. The approach taken in this paper is as follows: we first combine the state vectors of the repetitive controller and the uncertain system and derive the sufficient condition in the form of LMI for the existence of guaranteed cost control law. Next, for the case that the states of a system are not available for measurement, we present an observer-based control scheme. The control and observer gains are characterized by the feasible solutions to some LMIs. Finally, a convex optimization problem with LMI constraints is introduced to solve the optimal guaranteed cost control law using existing LMI software [11].

Notation 1. 𝑅𝑛 denotes the 𝑛-dimensional Euclidean space; 𝑅𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices; 𝐼 is the identity matrix; null matrix or null vector of appropriate dimension is denoted by 0; the superscript “𝑇” stands for the transpose of a matrix; the notation 𝑃>0 and 𝑃0 for 𝑃𝑅𝑛×𝑛 means that the matrix 𝑃 is real symmetric positive definite or positive semidefinite, respectively; the symmetric terms in a symmetric matrix are denoted by *, for example, 𝑋𝑌𝑍=𝑌𝑋𝑌𝑇𝑍.

2. Preliminaries of Guaranteed Cost Control

Consider an uncertain discrete-time system described by the following state equation:𝑥(𝑡+1)=(𝐴+𝐻𝐹(𝑡)𝐸)𝑥(𝑡),(2.1) where 𝑥(𝑡)𝑅𝑛 is the state vector with initial condition 𝑥(0), 𝐴, 𝐻 and 𝐸 are known real constant matrices with appropriate dimensions, and 𝐹(𝑡) is a real uncertain matrix function satisfying 𝐹𝑇(𝑡)𝐹(𝑡)𝐼.

Associated with the uncertain system (2.1) is the following quadratic cost function with a given weighting matrix 𝑄>0:𝐽=𝑡=0𝑥𝑇(𝑡)𝑄𝑥(𝑡).(2.2)

Definition 2.1 (see [8]). A positive definite real matrix 𝑃 is said to be a quadratic cost matrix for the system (2.1) and cost function (2.2) if (𝐴+𝐻𝐹(𝑡)𝐸)𝑇𝑃(𝐴+𝐻𝐹(𝑡)𝐸)𝑃+𝑄<0(2.3) for all 𝐹(𝑡) satisfying the bound 𝐹𝑇(𝑡)𝐹(𝑡)𝐼.

Lemma 2.2 (see [8]). Suppose that 𝑃>0 is a quadratic cost matrix for the uncertain system (2.1) and cost function (2.2). Then the system is quadratically stable and the cost function satisfies the bound 𝐽𝑥𝑇(0)𝑃𝑥(0).

The following theorem shows that the existence of a quadratic matrix is equivalent to the feasibility of an LMI.

Theorem 2.3. Consider the system (2.1) and cost function (2.2). There exists a quadratic cost matrix if and only if there exist a scalar 𝜀>0 and matrix X>0 such that 𝑋+𝜀𝐻𝐻𝑇𝐴𝑋𝟎𝟎𝑋𝑋𝐸𝑇𝑋𝜀𝐼𝟎𝑄1<0.(2.4) Moreover, the cost function (2.2) satisfies the bound 𝐽𝑥𝑇(0)𝑋1𝑥(0).

To prove the theorem, we need the following lemma.

Lemma 2.4 (see [12]). Let Σ1 and Σ2 be real constant matrices of compatible dimensions and 𝑀(𝑡) a real matrix function satisfying 𝑀𝑇(𝑡)𝑀(𝑡)𝐼. Then the following inequality holds: Σ1𝑀(𝑡)Σ2+Σ𝑇2𝑀𝑇(𝑡)Σ𝑇1𝜀Σ1Σ𝑇1+𝜀1Σ𝑇2Σ2,forany𝜀>0.(2.5)

Proof of Theorem 2.3. By an obvious application of Schur’s complement formula [13], the inequality (2.3) is equivalent to 𝑃1𝐴+𝐻𝐹(𝑡)𝐸𝑃+𝑄<0.(2.6) The inequality (2.6) can be further written as 𝑃1𝐴+𝐻𝟎+𝟎𝐸𝑃+𝑄𝐹(𝑡)𝟎𝐸𝑇𝐹𝑇𝐻(𝑡)𝑇𝟎<0.(2.7)
In light of Lemma 2.4, the inequality (2.7) holds for any 𝐹(𝑡) satisfying 𝐹𝑇(𝑡)𝐹(𝑡)𝐼 if and only if there exists a scalar 𝜀>0 such that 𝑃1+𝜀𝐻𝐻𝑇𝐴𝑃+𝑄+𝜀1𝐸𝑇𝐸<0,(2.8) which is further equivalent to 𝑃1+𝜀𝐻𝐻𝑇𝐴𝟎𝑃+𝑄𝐸𝑇𝜀𝐼<0.(2.9)
Premultiplying and postmultiplying the inequality (2.9) by the matrix diag{𝐼,𝑃1,𝐼} yield 𝑃1+𝜀𝐻𝐻𝑇𝐴𝑃1𝟎𝑃1+𝑃1𝑄𝑃1𝑃1𝐸𝑇𝜀𝐼<0.(2.10)
By denoting X=𝑃1 and using Schur complements again, it is straightforward to verify that the inequality (2.10) is equivalent to (2.4). This completes the proof.

3. State Feedback Repetitive Control

In this paper, we will consider the uncertain discrete-time SISO system described byΣ𝑝𝑥𝑝𝐴(𝑡+1)=𝑝+Δ𝐴𝑝𝑥𝑝𝐵(𝑡)+𝑝+Δ𝐵𝑝𝑢𝑝𝑦(𝑡),𝑝𝐶(𝑡)=𝑝+Δ𝐶𝑝𝑥𝑝𝐷(𝑡)+𝑝+Δ𝐷𝑝𝑢𝑝(𝑡),(3.1) where 𝑥𝑝(𝑡), 𝑢𝑝(𝑡), and 𝑦𝑝(𝑡) are the state vector, control input, and measured output, respectively; 𝐴𝑝,𝐵𝑝,𝐶𝑝, and 𝐷𝑝 are real constant matrices with appropriate dimensions; the pairs (𝐴𝑝,𝐵𝑝) and (𝐴𝑝,𝐶𝑝) are stabilizable and detectable, respectively; Δ𝐴𝑝,Δ𝐵𝑝,Δ𝐶𝑝, and Δ𝐷𝑝 are parameter uncertainties which are norm-bounded and can be described byΔ𝐴𝑝Δ𝐵𝑝Δ𝐶𝑝Δ𝐷𝑝=𝐻1𝐻2Δ(𝑡)𝐸1𝐸2,(3.2) where 𝐻1, 𝐻2, 𝐸1, and 𝐸2 are known constant matrices with appropriate dimensions, and Δ(𝑡) is an uncertain matrix satisfying the bound Δ𝑇(𝑡)Δ(𝑡)𝐼.

According to the internal model principle, in order to achieve zero tracking error in steady state, it is necessary to include in the loop the generator of periodic reference and/or disturbance signal, which is usually known as the repetitive controller. The transfer function of digital periodic signal generator with period 𝐿 is [14]Σ𝑟=11𝑧𝐿.(3.3)

As can be seen from (3.3), the periodic signal generator introduces 𝐿 open-loop poles uniformly distributed over a circumference of unit radius, which makes great differences between the design of repetitive control system and that of conventional feedback control system, and increases the difficulty of design work.

The state-space description of Σ𝑟 can be written asΣ𝑟𝑥𝑟(𝑡+1)=𝐴𝑟𝑥𝑟(𝑡)+𝐵𝑟𝑢𝑢(𝑡),𝑝(𝑡)=𝐶𝑟𝑥𝑟(𝑡)+𝐷𝑟𝑢(𝑡),(3.4) where𝐴𝑟=,𝐵00001100000100000010𝑟=10000𝑇,𝐶𝑟=,𝐷00001𝑟=1.(3.5)

Remark 3.1. To enhance the robust stability, additional filtering is usually added to the repetitive controller. Selecting Σ𝑟=1/(1𝛾𝑧𝐿) with 𝛾(0,1) yields a commonly used repetitive control scheme which sacrifices the high-frequency performance for system stability. All the results in this section can be extended with elements of 𝐴𝑟 and 𝐶𝑟 modified to include this scheme.

By using the augmented state vector 𝑥=[𝑥𝑇𝑝,𝑥𝑇𝑟]𝑇, we combine (3.1) and (3.4) to yield the following system:𝑥(𝑡+1)=𝐴+𝐻1Δ(𝑡)𝐸1𝑥(𝑡)+𝐵+𝐻1Δ(𝑡)𝐸2𝑢(𝑡),𝑦(𝑡)=𝐶+𝐻2Δ(𝑡)𝐸1𝑥(𝑡)+𝐷+𝐻2Δ(𝑡)𝐸2𝑢(𝑡),(3.6) where𝐴𝐴=𝑝𝐵𝑝𝐶𝑟𝟎𝐴𝑟𝐵,𝐵=𝑝𝐷𝑟𝐵𝑟,𝐶𝐶=𝑝𝐷𝑝𝐶𝑟,𝐷=𝐷𝑝𝐷𝑟,𝐻1=𝐻1𝟎,𝐻2=𝐻2,𝐸1=𝐸1𝐸2𝐶𝑟,𝐸2=𝐸2𝐷𝑟.(3.7)

Associated with the system (3.6) is the quadratic cost function with given weighting matrices 𝑄>0 and 𝑅>0:𝐽=𝑡=0𝑥𝑇(𝑡)𝑄𝑥(𝑡)+𝑢𝑇(𝑡)𝑅𝑢(𝑡).(3.8)

Remark 3.2. For square 𝑚×𝑚 MIMO linear systems, by selecting the repetitive controller as Σ𝑟=11𝑧𝐿×𝐼𝑚×𝑚,(3.9) the design technique proposed in this paper is also applicable by just rewriting the state-space description of Σ𝑟 to obtain the corresponding state-space matrices 𝐴𝑟,𝐵𝑟,𝐶𝑟, and 𝐷𝑟.

The problem in this section is to design a memoryless state feedback control law𝑢(𝑡)=𝐾𝑥(𝑡)(3.10) such that for any admissible uncertain matrix Δ(𝑡), the resulting closed-loop system𝑥(𝑡+1)=𝐴+𝐵𝐾+𝐻1Δ𝐸(𝑡)1+𝐸2𝐾𝑥(𝑡)(3.11) is not only stable, but also gives an upper bound for the closed-loop cost function𝐽=𝑡=0𝑥𝑇(𝑡)𝑄+𝐾𝑇𝑅𝐾𝑥(𝑡).(3.12)

Remark 3.3. By combining the state vectors of the repetitive controller and the uncertain discrete-time system, the resulting closed-loop system with state feedback control law has a form similar to that of (2.1). Although similar problems have been investigated by some researchers for conventional uncertain systems without the repetitive controller, it is the merit of the paper that the simultaneous consideration of robust stability and performance for the repetitive control of uncertain discrete-time systems is achieved for the first time, and an optimal guaranteed cost control law, which not only preserves system stability but also ensures an adequate level of performance, can be obtained by the approach presented in the paper.

Definition 3.4. Consider the uncertain system (3.6) and cost function (3.8). The controller of the form (3.10) is said to be a state feedback guaranteed cost controller with cost matrix 𝑃>0 if the matrix 𝑃>0 is a quadratic cost matrix for the closed-loop system (3.11) and cost function (3.12).

Remark 3.5. Using the results of last section, it follows that if (3.10) is a guaranteed cost control law with cost matrix 𝑃>0, then the resulting closed-loop system will be quadratically stable. Furthermore, the closed-loop system guarantees an adequate level of performance.
The following theorem provides an efficient way to solve the guaranteed cost state feedback control law (3.10) by existing convex optimization algorithms.

Theorem 3.6. If there exist a scalar 𝜀>0 and matrices X>0,𝑌 such that the following LMI holds: 𝑋+𝜀𝐻1𝐻𝑇1𝐴𝑋+𝐵𝑌𝟎𝟎𝟎𝑋𝑌𝑇𝑋𝑋𝐸𝑇1+𝑌𝑇𝐸𝑇2𝑅1𝟎𝟎𝑄1𝟎𝜀𝐼<0,(3.13) then 𝑢(𝑡)=𝑌𝑋1𝑥(𝑡) is a guaranteed cost control law for the uncertain system (3.6).

Proof. According to Theorem 2.3, the existence of a quadratic cost matrix for the closed-loop system (3.11) and cost function (3.12) is equivalent to 𝑋+𝜀𝐻1𝐻𝑇1𝟎(𝐴+𝐵𝐾)𝑋𝑋+𝑋𝑄+𝐾𝑇𝐸𝑅𝐾𝑋𝑋1+𝐸2𝐾𝑇𝜀𝐼<0.(3.14) By using Schur complements, the inequality (3.14) is further equivalent to 𝑋+𝜀𝐻1𝐻𝑇1𝐴𝑋+𝐵𝐾𝑋𝟎𝟎𝟎𝑋𝑋𝐾𝑇𝑋𝑋𝐸𝑇1+𝑋𝐾𝑇𝐸𝑇2𝑅1𝟎𝟎𝑄1𝟎𝜀𝐼<0.(3.15) Now setting 𝑌=𝐾𝑋, it is ready to see that (3.15) yields (3.13). Moreover, the guaranteed cost control gain is 𝐾=𝑌𝑋1. This completes the proof.

In this paper, we are interested in designing a controller of the form (3.10) to minimize the upper bound of (3.8). However, this bound is dependent on the initial condition 𝑥(0). To remove this dependence on the initial condition, we adopt the approach proposed by Petersen et al. [8]. Suppose that the initial condition is arbitrary but belongs to the set𝑥Ω=(0)𝑅𝑛𝑥(0)=Ψ𝜈,𝜈𝑇𝜈1,(3.16) where Ψ is a given matrix. Then, the cost bound (3.8) leads to𝐽𝜆maxΨ𝑇𝑋1Ψ,(3.17) where 𝜆max() denotes the maximum eigenvalue.

Furthermore, introduce a scalar 𝜆 satisfying𝜆maxΨ𝑇𝑋1Ψ<𝜆,𝜆𝐼+Ψ𝑇𝑋1Ψ<0,𝜆𝐼Ψ𝑇𝑋<0.(3.18) Consequently, the design problem of the optimal guaranteed cost state feedback control law (3.10) can be formulated as the following optimization problem:minimize𝜀>0,𝑋>0,𝑌𝜆subjecttoLMIs(3.13),(3.18),(3.19) which is a convex optimization problem with LMI constraints and can be effectively solved by MATLAB LMI Toolbox.

4. Observer-Based Controller Design

In many practical control systems and applications, the states of a system are not always available for measurement. Hence, it is very necessary to introduce a state observer to reconstruct the states of the system. In the following work, we will focus on the design of an observer-based controller.

The dynamic observer-based control for the system (3.6) is constructed asΣ𝑜𝑢̂𝑥(𝑡+1)=𝐴̂𝑥(𝑡)+𝐵𝑢(𝑡)+Γ(𝑦(𝑡)̂𝑦(𝑡)),̂𝑦(𝑡)=𝐶̂𝑥(𝑡)+𝐷𝑢(𝑡),(4.1)(𝑡)=𝐾̂𝑥(𝑡),(4.2) where ̂𝑥 is the estimation of 𝑥,̂𝑦 is the observer output, 𝐾 and Γ are the control gain and observer gain, respectively.

Define the state estimation error as𝑒(𝑡)=𝑥(𝑡)̂𝑥(𝑡).(4.3) By applying the observer-based controller (4.1) and (4.2) to the system (3.6), we obtain the closed-loop system of the form̂𝑥(𝑡+1)=𝐴+𝐵𝐾+Γ𝐻2Δ𝐸1+𝐸2𝐾̂𝑥(𝑡)+Γ𝐶+Γ𝐻2Δ𝐸1𝑒𝐻(𝑡),𝑒(𝑡+1)=1Γ𝐻2Δ𝐸1+𝐸2𝐾𝐻̂𝑥(𝑡)+𝐴Γ𝐶+1Γ𝐻2Δ𝐸1𝑒(𝑡),(4.4) which can be further written aŝ𝑥(𝑡+1)𝑒(𝑡+1)=(Φ+𝑀Δ𝑁)̂𝑥(𝑡)𝑒(𝑡),(4.5) whereΦ=𝐴+𝐵𝐾Γ𝐶𝟎𝐴Γ𝐶,𝑀=Γ𝐻2𝐻1Γ𝐻2𝐸,𝑁=1+𝐸2𝐾𝐸1.(4.6)

As can be seen from (4.5), the expression of the closed-loop system with state observer is identical with that of (2.1). Therefore, we can utilize the results given in Theorem 2.3 to design the control gain 𝐾 and observer gain Γ. Associated with the system (4.5) is the following cost function with 𝑄1>0 and 𝑄2>0:𝐽=𝑡=0𝑥𝑇(𝑡)𝑄1𝑥(𝑡)+𝑒𝑇(𝑡)𝑄2𝑒(𝑡).(4.7)

The following theorem gives the main result on observer-based controller design by which the control gain 𝐾 and observer gain Γ could be solved.

Theorem 4.1. The uncertain system (3.6) is quadratically stable by the observer-based control (4.1) and (4.2) provided that there exist a scalar 𝜀>0 and matrices 𝑃1>0, 𝑌, Γ, such that 𝑃1𝟎𝐴𝑃1+𝐵𝑌Γ𝐶Γ𝐻2𝟎𝟎𝟎𝐼𝟎𝐴Γ𝐶𝐻1Γ𝐻2𝟎𝟎𝟎𝑃1𝟎𝟎𝑃1𝐸𝑇1+𝑌𝑇𝐸𝑇2𝑃1𝑄11/2𝑇𝟎𝐼𝟎𝐸𝑇1𝟎𝑄21/2𝑇𝐼𝟎𝟎𝟎𝐼𝟎𝟎𝜀𝐼𝟎𝜀𝐼<0.(4.8) Moreover, the stabilizing observer and control gains are given by Γ and 𝐾=𝑌𝑃11, respectively.

Proof. Define the Lyapunov function as 𝑉(̂𝑥(𝑡),𝑒(𝑡))=̂𝑥𝑇(𝑡)𝑃1̂𝑥(𝑡)+𝑒𝑇(𝑡)𝑃2𝑒(𝑡), where 𝑃1>0 and 𝑃2>0. Then according to Lemma 2.2, the closed-loop system (4.5) is quadratically stable if the following inequality holds: (Φ+𝑀Δ(𝑡)𝑁)𝑇𝑃(Φ+𝑀Δ(𝑡)𝑁)𝑃+𝑄<0,(4.9) where 𝑃=diag{𝑃1,𝑃2},𝑄=diag{𝑄1,𝑄2}.
By applying Schur complements and some basic matrix manipulations to the LMI (2.4), the stability condition for system (4.5) can be equivalently written as 𝑃Φ𝑃𝑀𝟎𝟎𝑃𝟎𝑃𝑁𝑇𝑃𝑆𝑇𝐼𝟎𝟎𝐼𝟎𝜀𝐼<0,(4.10) where 𝑆=diag{𝑄11/2,𝑄21/2}.
Note that Φ𝑃=𝐴𝑃1+𝐵𝐾𝑃1Γ𝐶𝑃2𝟎𝐴𝑃2Γ𝐶𝑃2,𝑃𝑁𝑇=𝑃1𝐸𝑇1+𝑃1𝐾𝑇𝐸𝑇2𝑃2𝐸𝑇1,𝑃𝑆𝑇=𝑃1𝑄11/2𝑇𝟎𝟎𝑃2𝑄21/2𝑇.(4.11) Then it is straightforward to prove that (4.10) is equivalent to (4.8) with 𝑃2=𝐼, 𝐾=𝑌𝑃11. This completes the proof.

The optimal control gain 𝐾 and observer gain Γ can be obtained by solving the following optimization problem:minimize𝜀>0,𝑃1>0,𝑌,Γ𝜆,subjectto𝜆𝐼Ψ𝑇𝑃1𝟎𝟎𝐼<0,LMI(4.8).(4.12)

The LMI (4.8) of Theorem 4.1 provides an efficient way to solve the observer and control gains by existing LMI software. However, it will undoubtedly yield conservative results in view of the proof with 𝑃2=𝐼. As can be seen from the proof, since the entries in (4.10) occur in nonlinear fashion with respect to its arguments, it would be impossible to employ the standard LMI optimization approach to find the solutions if not letting 𝑃2=𝐼. The conservativeness brought by Theorem 4.1 may rest in the sense that in some cases it will fail to produce a feasible solution when one actually exists.

To reduce the conservativeness induced by setting 𝑃2=𝐼, in what follows, an alternative approach, which can be divided into two steps, will be presented. Firstly, the LMI result for the stability of closed-loop system (4.5), by which the suitable control gain 𝐾 and observer gain Γ could be obtained, is derived under the assumption that the original system described by (3.1) is with no perturbations in the output equation (i.e., 𝐻2=𝐻2=𝟎). Secondly, the observer gain Γ, which is solved in the first step, is supposed to be known a prior. Then sufficient condition for the existence of guaranteed cost control gain 𝐾 is derived in terms of LMI, and a convex optimization problem is formulated to solve the optimal control gain 𝐾 by minimizing the upper bound of the cost function.

First we present an LMI result for the stability of closed-loop system (4.5) with no perturbations in the output equation. Before proceeding, we need to introduce the following lemma.

Lemma 4.2 (see [15]). For a given full row rank 𝐶𝑅𝑚×𝑛 with singular value decomposition 𝐶=𝑈[𝐶0𝟎]𝑉𝑇, where 𝑈𝑅𝑚×𝑚 and 𝑉𝑅𝑛×𝑛 are unitary matrices and 𝐶0𝑅𝑚×𝑚 is a diagonal matrix with positive diagonal elements in decreasing order, assume that 𝑋𝑅𝑛×𝑛 is a symmetric matrix, then there exists a matrix 𝑋𝑅𝑚×𝑚 such that 𝐶𝑋=𝑋𝐶 if and only if 𝑋𝑋=𝑉1𝟎𝟎𝑋2𝑉𝑇,(4.13) where 𝑋1𝑅𝑚×𝑚,𝑋2𝑅(𝑛𝑚)×(𝑛𝑚). Moreover, the matrix 𝑋 is given by 𝑋=𝑈𝐶0𝑋1𝐶01𝑈𝑇.

The suitable control gain 𝐾 and observer gain Γ for system (4.5) with 𝐻2=𝟎 could be solved by the following theorem.

Theorem 4.3. The uncertain system (3.6) with 𝐻2=𝟎 is quadratically stable by the observer-based control (4.1) and (4.2) provided that there exist a scalar 𝜀>0 and matrices 𝑃1>0,𝑃21>0,𝑃22>0,𝑌,𝑊, such that 𝑃1𝟎𝐴𝑃1+𝐵𝑌𝑊𝐶𝟎𝟎𝟎𝟎𝑃2𝟎𝐴𝑃2𝑊𝐶𝐻1𝟎𝟎𝟎𝑃1𝟎𝟎𝑃1𝐸𝑇1+𝑌𝑇𝐸𝑇2𝑃1𝑄11/2𝑇𝟎𝑃2𝟎𝑃2𝐸𝑇1𝟎𝑃2𝑄21/2𝑇𝐼𝟎𝟎𝟎𝐼𝟎𝟎𝜀𝐼𝟎𝜀𝐼<0,(4.14) where the singular value decomposition of full row rank matrix 𝐶 is 𝐶=𝑈[𝐶0𝟎]𝑉𝑇,𝑃2=𝑉𝑃21𝟎𝟎𝑃22𝑉𝑇. Moreover, the suitable control and observer gains are given by 𝐾=𝑌𝑃11 and Γ=𝑊𝑈𝐶0𝑃121𝐶01𝑈𝑇, respectively.

Proof. For SISO linear systems considered in this paper, it is obvious that the matrix 𝐶 is full row rank. For 𝑚×𝑚 MIMO systems, without loss of generality, we suppose that rank(𝐶𝑝)=𝑚, which implies rank(𝐶=[𝐶𝑝𝐷𝑝𝐶𝑟])=𝑚.
Since 𝑃2 can be expressed as 𝑃2=𝑉𝑃21𝟎𝟎𝑃22𝑉𝑇, then according to Lemma 4.2, there exists a matrix 𝑃2 such that the equality 𝐶𝑃2=𝑃2𝐶 holds. The matrix 𝑃2 and its inverse are given by 𝑃2=𝑈𝐶0𝑃21𝐶01𝑈𝑇 and 𝑃21=𝑈𝐶0𝑃121𝐶01𝑈𝑇, respectively.
Furthermore, noting that Φ𝑃=𝐴𝑃1+𝐵𝐾𝑃1Γ𝐶𝑃2𝟎𝐴𝑃2Γ𝐶𝑃2=𝐴𝑃1+𝐵𝐾𝑃1Γ𝑃2𝐶𝟎𝐴𝑃2Γ𝑃2𝐶,(4.15) then it is ready to see that (4.10) is equivalent to (4.14) with Γ=𝑊𝑃21,𝐾=𝑌𝑃11. This completes the proof.

Once the observer gain Γ has been yielded from Theorem 4.3, we may now proceed to design the optimal control gain 𝐾 which minimizes the upper bound of the cost function (4.7). The feasible control gain could be solved by the following theorem.

Theorem 4.4. Suppose that the observer gain Γ in (4.1) is solved a priori by Theorem 4.3. Then the closed-loop system (4.5) is quadratically stable provided that there exist a scalar 𝜀>0 and matrices 𝑃1>0,𝑃2>0, 𝑌 satisfying the following LMI. Moreover, if this condition holds, then the control gain is given by 𝐾=𝑌𝑃11𝑃1𝟎𝐴𝑃1+𝐵𝑌Γ𝐶𝑃2Γ𝐻2𝟎𝟎𝟎𝑃2𝟎𝐴𝑃2Γ𝐶𝑃2𝐻1Γ𝐻2𝟎𝟎𝟎𝑃1𝟎𝟎𝑃1𝐸𝑇1+𝑌𝑇𝐸𝑇2𝑃1𝑄11/2𝑇𝟎𝑃2𝟎𝑃2𝐸𝑇1𝟎𝑃2𝑄21/2𝑇𝐼𝟎𝟎𝟎𝐼𝟎𝟎𝜀𝐼𝟎𝜀𝐼<0.(4.16)

Proof. It can be completed immediately from (4.10) by setting 𝑌=𝐾𝑃1.

Hence, the optimal guaranteed cost control gain 𝐾 which minimizes the upper bound of the cost function (4.7) can be obtained by solving the following LMI-constrained optimization problem:minimize𝜀>0,𝑃1>0,𝑃2>0,𝑌𝜆,subjectto𝜆𝐼Ψ𝑇𝑃1𝟎𝟎𝑃2<0,LMI(4.17).(4.17)

5. Simulation Example

Consider the uncertain system (3.1) with the following parameters:𝐴𝑝=1.570.7760.7760,𝐵𝑝=10,𝐶𝑝=1.0561.105,𝐷𝑝=4.4,𝐻1=,0.20.100.1𝐻2=,0.10.2𝐸1=,0.100.20.1𝐸2=,.0.20.1Δ(𝑡)=sin4𝜋𝑡00sin2𝜋𝑡(5.1) The control performance specification is to design an observer-based state feedback controller which stabilizes the closed-loop system and rejects a disturbance signal defined by 𝑑(𝑡)=0.7+0.5sin(𝜔𝑡)+0.3sin(2𝜔𝑡)+0.2sin(3𝜔𝑡)+0.1sin(4𝜔𝑡), as shown in Figure 1, where 𝜔=2𝜋/𝐿 and 𝐿=10.

904914.fig.001
Figure 1: Disturbance signal used in simulation.

Choose the weighting matrices as 𝑄1=𝑄2=𝐼12×12 and 𝑅=0.2. Now, we are in a position to solve the control gain 𝐾 and observer gain Γ by the approach presented in Section 4. Firstly, we consider utilizing the result given in Theorem 4.1. However, it is found that the LMI (4.8) is infeasible, which, in some sense, validates the conservativeness induced by the proof with 𝑃2=𝐼. Next, we turn to the results given in Theorems 4.3 and 4.4. The feasible solution of the observer gain Γ obtained by solving the LMI (4.14) is[]Γ=0.3138,0.0201,0.2089,0.0007,0.0013,0.0007,0.0002,0.0003,0.0007,0.0009,0.0007,0.0005𝑇.(5.2) Then, with the observer gain Γ known a prior, the optimal guaranteed cost control gain 𝐾 is obtained as follows by solving the optimization problem (4.17), and the corresponding cost bound is 𝐽31.33,[].𝐾=1.1577,0.4636,0.0158,0.0006,0.0027,0.0014,0.0001,0.0002,0,0.0001,0,1(5.3)

Figure 2 shows the response of the system output. It can be seen that the disturbance is attenuated to about 2.76 percent in four-sample periods when considering the amplitude of the disturbance and the output, although biggish amplitude oscillations occur as the output tends to steady state.

904914.fig.002
Figure 2: Time response of system output.

6. Conclusion

In this paper, a solution to the problem of repetitive control for uncertain discrete-time systems is presented. The state feedback control and guaranteed cost control techniques are adopted. Sufficient conditions for the existence of guaranteed cost control law are derived in terms of LMI, and it is shown that the control and observer gains can be characterized by the feasible solutions to the LMIs. A convex optimization problem is introduced to solve the optimal guaranteed cost control law. The validity of the proposed method is verified by a simulation example.

Acknowledgment

This paper is supported by the Science and Technology Program of Shanghai Maritime University under Grant no. 20110023.

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