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

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. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space; 𝑅^𝑛×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 𝑃≥0 for 𝑃∈𝑅^𝑛×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, ^{𝑋𝑌∗𝑍}=^{𝑌𝑋𝑌𝑇}_𝑍.

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)∈𝑅^𝑛 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 𝐹^𝑇(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=0𝑥^𝑇(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 𝐹^𝑇(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 𝐽≤𝑥^𝑇(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+∗𝜀𝐻𝐻^𝑇𝐴−𝑋𝑋𝑋𝐸𝟎_𝑇𝟎∗∗∗∗−∗𝜀𝐼−𝑄𝑋𝟎₋₁⎤⎥⎥⎥⎥⎦<0.(2.4) Moreover, the cost function (2.2) satisfies the bound 𝐽≤𝑥^𝑇(0)𝑋⁻¹𝑥(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 𝑀^𝑇(𝑡)𝑀(𝑡)≤𝐼. Then the following inequality holds: Σ₁𝑀(𝑡)Σ₂+Σ^𝑇2𝑀^𝑇(𝑡)Σ^𝑇1≤𝜀Σ₁Σ^𝑇1+𝜀⁻¹Σ^𝑇2Σ₂,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 −𝑃⁻¹∗𝐴+−𝐻𝑃+𝐹𝑄(𝑡)𝐸<0.(2.6) The inequality (2.6) can be further written as −𝑃⁻¹∗−𝑃𝐴+𝑄+𝐻𝟎𝐹(𝑡)𝟎𝐸+𝐸𝟎_𝑇𝐹^𝑇(𝑡)𝐻^𝑇𝟎<0.(2.7)
In light of Lemma 2.4, the inequality (2.7) holds for any 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 if and only if there exists a scalar 𝜀>0 such that −𝑃⁻¹+∗𝜀𝐻𝐻^𝑇−𝑃+𝑄+𝐴𝜀₋₁𝐸^𝑇𝐸<0,(2.8) which is further equivalent to ⎡⎢⎢⎣−𝑃⁻¹+∗𝜀𝐻𝐻^𝑇−𝑃𝐴+𝑄𝐸𝟎_𝑇∗∗−𝜀𝐼⎤⎥⎥⎦<0.(2.9)
Premultiplying and postmultiplying the inequality (2.9) by the matrix diag{𝐼,𝑃⁻¹,𝐼} yield ⎡⎢⎢⎣−𝑃⁻¹+∗𝜀𝐻𝐻^𝑇−𝑃₋₁𝐴𝑃⁻¹𝟎+𝑃⁻¹𝑄𝑃⁻¹𝑃⁻¹𝐸^𝑇∗∗−𝜀𝐼⎤⎥⎥⎦<0.(2.10)
By denoting X∶=𝑃⁻¹ 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 byΣ_𝑝∶𝑥^𝑝𝑦(𝑡+1)=𝐴_𝑝+Δ𝐴_𝑝𝑥_𝑝(𝑡)+𝐵_𝑝+Δ𝐵_𝑝𝑢_𝑝(𝑡),_𝑝(𝑡)=𝐶_𝑝+Δ𝐶_𝑝𝑥_𝑝(𝑡)+𝐷_𝑝+Δ𝐷_𝑝𝑢_𝑝(𝑡),(3.1) where 𝑥_𝑝(𝑡), 𝑢_𝑝(𝑡), and 𝑦_𝑝(𝑡) are the state vector, control input, and measured output, respectively; 𝐴_𝑝,𝐵_𝑝,𝐶_𝑝, and 𝐷_𝑝 are real constant matrices with appropriate dimensions; the pairs (𝐴_𝑝,𝐵_𝑝) and (𝐴_𝑝,𝐶_𝑝) are stabilizable and detectable, respectively; Δ𝐴_𝑝,Δ𝐵_𝑝,Δ𝐶_𝑝, and Δ𝐷_𝑝 are parameter uncertainties which are norm-bounded and can be described byΔ𝐴^𝑝Δ𝐵_𝑝Δ𝐶_𝑝Δ𝐷_𝑝=𝐻₁𝐻₂Δ(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 Δ^𝑇(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]Σ_𝑟=11−𝑧^−𝐿.(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 Σ_𝑟 can be written asΣ_𝑟∶𝑥^𝑟𝑢(𝑡+1)=𝐴_𝑟𝑥_𝑟(𝑡)+𝐵_𝑟𝑢(𝑡),_𝑝(𝑡)=𝐶_𝑟𝑥_𝑟(𝑡)+𝐷_𝑟𝑢(𝑡),(3.4) where𝐴_𝑟𝐵=⎡⎢⎢⎢⎢⎢⎣⋮0100⋱0010⋱0000⋱⋯⋯⋯⋯⋮0001⋮1000_𝑟⎤⎥⎥⎥⎥⎥⎦,𝐶_𝑟=100⋯00^𝑇,=0𝐷00⋯01_𝑟,=1.(3.5)

Remark 3.1. To enhance the robust stability, additional filtering is usually added to the repetitive controller. Selecting Σ_𝑟=1/(1−𝛾𝑧^−𝐿) 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 𝐴_𝑟 and 𝐶_𝑟 modified to include this scheme.

By using the augmented state vector 𝑥=[𝑥^𝑇𝑝,𝑥^𝑇𝑟]^𝑇, we combine (3.1) and (3.4) to yield the following system:𝑥(𝑦𝑡(+𝑡)1=)=𝐶𝐴+𝐻+₂𝐻₁Δ(𝑡)𝐸₁𝑥(𝑡)+𝐵+𝐻₁Δ(𝑡)𝐸₂𝑢(𝑡),Δ(𝑡)𝐸₁𝑥(𝑡)+𝐷+𝐻₂Δ(𝑡)𝐸₂𝑢(𝑡),(3.6) where𝐴=𝐴^𝑝𝐵_𝑝𝐶_𝑟𝟎𝐴_𝑟,𝐵=𝐵^𝑝𝐷_𝑟𝐶=𝐶_𝑝𝐵_𝑟,𝐻₁𝐷_𝑝𝐶_𝑟,𝐷=𝐷_𝑝𝐷_𝑟,=𝐻₁𝟎,𝐻²=𝐻₂,𝐸₁=𝐸₁𝐸₂𝐶_𝑟,𝐸₂=𝐸₂𝐷_𝑟.(3.7)

Associated with the system (3.6) is the quadratic cost function with given weighting matrices 𝑄>0 and 𝑅>0:𝐽=^∞_𝑡=0𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+𝑢^𝑇(𝑡)𝑅𝑢(𝑡).(3.8)

Remark 3.2. For square 𝑚×𝑚 MIMO linear systems, by selecting the repetitive controller as Σ_𝑟=11−𝑧^−𝐿×𝐼_𝑚×𝑚,(3.9) the design technique proposed in this paper is also applicable by just rewriting the state-space description of Σ_𝑟 to obtain the corresponding state-space matrices 𝐴_𝑟,𝐵_𝑟,𝐶_𝑟, and 𝐷_𝑟.

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+𝐻₁Δ(𝑡)𝐸₁+𝐸₂𝐾𝑥(𝑡)(3.11) is not only stable, but also gives an upper bound for the closed-loop cost function𝐽=^∞_𝑡=0𝑥^𝑇(𝑡)𝑄+𝐾^𝑇𝑅𝐾𝑥(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+𝜀∗𝐻¹𝐻^𝑇1𝐴𝑋−+𝑋𝐵𝑌𝑌𝟎_𝑇𝟎𝟎𝑋𝑋𝐸^𝑇1+𝑌^𝑇𝐸^𝑇2∗∗∗∗−𝑅∗⁻¹−𝑄𝟎₋₁𝟎∗∗∗∗−𝟎𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0,(3.13) then 𝑢(𝑡)=𝑌𝑋⁻¹𝑥(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+𝜀∗𝐻¹𝐻^𝑇1−𝑋+𝑋(𝐴+𝑄𝐵+𝐾𝐾)_𝑇𝑋𝟎𝑅𝐾𝑋𝑋𝐸₁+𝐸₂𝐾^𝑇∗∗−𝜀𝐼⎤⎥⎥⎥⎦<0.(3.14) By using Schur complements, the inequality (3.14) is further equivalent to ⎡⎢⎢⎢⎢⎢⎢⎣−𝑋+𝜀∗𝐻¹𝐻^𝑇1𝐴𝑋−+𝑋𝐵𝐾𝑋𝑋𝐾𝟎_𝑇𝟎𝟎𝑋𝑋𝐸^𝑇1+𝑋𝐾^𝑇𝐸^𝑇2∗∗∗∗−𝑅∗⁻¹−𝑄𝟎₋₁𝟎∗∗∗∗−𝟎𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎦<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=𝑌𝑋⁻¹. 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 setΩ∶=𝑥(0)∈𝑅^𝑛∣𝑥(0)=Ψ𝜈,𝜈^𝑇𝜈≤1,(3.16) where Ψ is a given matrix. Then, the cost bound (3.8) leads to𝐽≤𝜆_maxΨ^𝑇𝑋⁻¹Ψ,(3.17) where 𝜆_max(⋅) denotes the maximum eigenvalue.

Furthermore, introduce a scalar 𝜆 satisfying𝜆⟺_max−Ψ𝜆^𝑇𝐼𝑋+⁻¹ΨΨ_𝑇<𝜆,⟺−𝜆𝐼Ψ_𝑇𝑋⁻¹Ψ<0,∗−𝑋<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:smubinjeimctiztoe_{𝜀>0,𝑋>0,𝑌}𝜆LMIs(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 asΣ_𝑜∶𝑥𝑦̂̂((𝑡𝑡)+=1𝐶)=𝑥̂(𝐴𝑡𝑢)𝑥+(̂𝑡()𝑡𝐷)=+𝑢𝐾(𝐵𝑡)𝑥𝑢,̂((𝑡𝑡)),+Γ(𝑦(𝑡)−𝑦̂(𝑡)),((44..12)) where 𝑥̂ is the estimation of 𝑥,𝑦̂ is the observer output, 𝐾 and Γ are the control gain and observer gain, respectively.

Define the state estimation error as𝑒(𝑡)=𝑥(𝑡)−𝑥̂(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 form𝑒(𝑡+𝑥̂1(𝑡)+=1)𝐻=₁𝐴+𝐵𝐾+Γ𝐻₂Δ𝐸₁+𝐸₂𝐾𝑥̂(𝑡)+Γ𝐶+Γ𝐻₂Δ𝐸₁𝑒(𝑡),−Γ𝐻₂Δ𝐸₁+𝐸₂𝐾𝑥̂(𝑡)+𝐴−Γ𝐶+𝐻₁−Γ𝐻₂Δ𝐸₁𝑒(𝑡),(4.4) which can be further written as𝑥𝑒̂((𝑡𝑡++11))=(Φ+𝑀Δ𝑁)𝑥𝑒̂((𝑡𝑡)),(4.5) whereΦ=𝐴+𝟎𝐵𝐾𝐴Γ−𝐶Γ𝐶,𝑀=Γ𝐻²𝐻₁−Γ𝐻₂,𝑁=𝐸¹+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=0𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)+𝑒^𝑇(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 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃¹∗−𝟎𝐼𝐴𝑃₁𝟎+𝐵𝑌𝐴Γ−𝐶Γ𝐶𝐻₁Γ𝐻₂𝟎𝟎𝟎∗∗−𝑃₁−Γ𝐻₂𝟎𝟎𝟎∗∗∗−𝟎𝐼𝟎𝟎𝑃₁𝐸^𝑇1𝐸+_𝑇1𝑌^𝑇𝐸^𝑇2𝑃₁𝑄^11/2^𝑇𝟎𝟎𝑄^21/2^𝑇∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−∗∗∗𝐼−∗∗𝟎𝐼−∗𝟎𝟎𝜀𝐼−𝟎𝟎𝟎𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(4.8) Moreover, the stabilizing observer and control gains are given by Γ and 𝐾=𝑌𝑃¹⁻¹, respectively.

Proof. Define the Lyapunov function as 𝑉(𝑥̂(𝑡),𝑒(𝑡))=𝑥̂^𝑇(𝑡)𝑃₁𝑥̂(𝑡)+𝑒^𝑇(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 ⎡⎢⎢⎢⎢⎢⎣−∗𝑃Φ−𝑃𝑃𝑀𝟎𝑃𝑁𝟎_𝑇𝟎𝑃𝑆^𝑇∗∗∗∗∗∗−∗∗𝐼−∗𝟎𝐼−𝟎𝟎𝜀𝐼⎤⎥⎥⎥⎥⎥⎦<0,(4.10) where 𝑆=diag{𝑄^11/2,𝑄^21/2}.
Note that Φ𝑃=𝐴𝑃¹+𝐵𝐾𝑃₁Γ𝐶𝑃₂𝑃𝑁_𝑇𝟎𝐴𝑃₂−Γ𝐶𝑃₂,=𝑃¹𝐸^𝑇1+𝑃₁𝐾^𝑇𝐸^𝑇2𝑃𝑆_𝑇𝑃₂𝐸^𝑇1,=𝑃¹𝑄𝟎^11/2^𝑇𝑃₂𝟎𝑄^21/2^𝑇.(4.11) Then it is straightforward to prove that (4.10) is equivalent to (4.8) with 𝑃₂=𝐼, 𝐾=𝑌𝑃¹⁻¹. This completes the proof.

The optimal control gain 𝐾 and observer gain Γ can be obtained by solving the following optimization problem:smubinjeimctiztoe_{𝜀>0,𝑃1>0,𝑌,Γ}𝜆⎡⎢⎢⎣,−𝜆𝐼Ψ_𝑇∗−𝑃¹𝟎𝐼𝟎⎤⎥⎥⎦<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 𝐶∈𝑅^𝑚×𝑛 with singular value decomposition 𝐶=𝑈[𝐶₀𝟎]𝑉^𝑇, where 𝑈∈𝑅^𝑚×𝑚 and 𝑉∈𝑅^𝑛×𝑛 are unitary matrices and 𝐶₀∈𝑅^𝑚×𝑚 is a diagonal matrix with positive diagonal elements in decreasing order, assume that 𝑋∈𝑅^𝑛×𝑛 is a symmetric matrix, then there exists a matrix 𝑋∈𝑅^𝑚×𝑚 such that 𝐶𝑋=𝑋𝐶 if and only if 𝑋=𝑉𝑋¹𝟎𝑋𝟎₂𝑉^𝑇,(4.13) where 𝑋₁∈𝑅^𝑚×𝑚,𝑋₂∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}. Moreover, the matrix 𝑋 is given by 𝑋=𝑈𝐶₀𝑋₁𝐶⁰⁻¹𝑈^𝑇.

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 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃¹∗−𝟎𝑃₂𝐴𝑃₁+𝐵𝑌𝑊𝐶𝟎𝟎𝟎𝟎∗∗−𝟎𝑃₁𝐴𝑃₂−𝑊𝐶𝐻₁𝟎𝟎𝟎∗∗∗−𝟎𝑃₂𝟎𝑃₁𝐸^𝑇1+𝑌^𝑇𝐸^𝑇2𝑃₁𝑄^11/2^𝑇𝟎𝟎𝑃₂𝐸^𝑇1𝟎𝑃₂𝑄^21/2^𝑇∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−∗∗∗𝐼−∗∗𝟎𝐼−∗𝟎𝟎𝜀𝐼−𝟎𝟎𝟎𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(4.14) where the singular value decomposition of full row rank matrix 𝐶 is 𝐶=𝑈[𝐶₀𝟎]𝑉^𝑇,𝑃₂=𝑉^{𝑃21𝟎𝟎𝑃}₂₂𝑉^𝑇. Moreover, the suitable control and observer gains are given by 𝐾=𝑌𝑃¹⁻¹ and Γ=𝑊𝑈𝐶₀𝑃⁻¹²¹𝐶⁰⁻¹𝑈^𝑇, 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(𝐶_𝑝)=𝑚, which implies rank(𝐶=[𝐶_𝑝𝐷_𝑝𝐶_𝑟])=𝑚.
Since 𝑃₂ can be expressed as 𝑃₂=𝑉^{𝑃21𝟎𝟎𝑃}₂₂𝑉^𝑇, 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=𝑈𝐶₀𝑃₂₁𝐶⁰⁻¹𝑈^𝑇 and 𝑃²⁻¹=𝑈𝐶₀𝑃⁻¹²¹𝐶⁰⁻¹𝑈^𝑇, respectively.
Furthermore, noting that Φ𝑃=𝐴𝑃¹+𝐵𝐾𝑃₁Γ𝐶𝑃₂𝟎𝐴𝑃₂−Γ𝐶𝑃₂=𝐴𝑃¹+𝟎𝐵𝐾𝑃₁𝐴𝑃₂Γ𝑃₂𝐶−Γ𝑃₂𝐶,(4.15) then it is ready to see that (4.10) is equivalent to (4.14) with Γ=𝑊𝑃²⁻¹,𝐾=𝑌𝑃¹⁻¹. 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=𝑌𝑃¹⁻¹⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃¹∗−𝟎𝑃₂𝐴𝑃₁+𝐵𝑌Γ𝐶𝑃₂Γ𝐻₂𝟎𝟎𝟎∗∗−𝟎𝑃₁𝐴𝑃₂−Γ𝐶𝑃₂𝐻₁−Γ𝐻₂𝟎𝟎𝟎∗∗∗−𝟎𝑃₂𝟎𝑃₁𝐸^𝑇1+𝑌^𝑇𝐸^𝑇2𝑃₁𝑄^11/2^𝑇𝟎𝟎𝑃₂𝐸^𝑇1𝟎𝑃₂𝑄^21/2^𝑇∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−∗∗∗𝐼−∗∗𝟎𝐼−∗𝟎𝟎𝜀𝐼−𝟎𝟎𝟎𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<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:msiunbimjeicztet_{𝜀>0,𝑃1>0,𝑃2>0,𝑌}o⎡⎢⎢⎣𝜆−,𝜆𝐼Ψ_𝑇∗−𝑃¹𝟎𝑃𝟎₂⎤⎥⎥⎦<0,LMI(4.17).(4.17)

5. Simulation Example

Consider the uncertain system (3.1) with the following parameters:𝐴_𝑝=01..75776−0.0776,𝐵_𝑝=10,𝐶_𝑝=1.056−1.105,𝐷_𝑝=4.4,𝐻₁=00.200..11,𝐻₂Δ=(𝑡0).1=0.s2in0,4𝜋𝑡si𝐸n0₁2=𝜋𝑡00...1200.1,𝐸₂=00..21,(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.

Figure 1 

Disturbance signal used in simulation.

Choose the weighting matrices as 𝑄₁=𝑄₂=𝐼_12×12 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) isΓ=[−0.03.103080,20,.00.200010,30,.02.008090,7−,00..00000097,,0−.000.00071,3−,0−.000.00050]_𝑇7,.(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 𝐽≤31.33,𝐾=[−−10..10507174,,0−.406.03060,10,.001.05080,20,.000,0−60,.−0000.010,207,−,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.

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.