LMI-Based Model Predictive Control for a Class of Constrained Uncertain Fuzzy Markov Jump Systems

Abstract

An extended model predictive control algorithm is proposed to address constrained robust model predictive control. New upper bounds on arbitrarily long time intervals are derived by introducing two external parameters, which can relax the requirements for the increments of the Lyapunov function. The main merit of this new approach compared to other well-known techniques is the reduced conservativeness. The proposed method is proved to be effective for a class of uncertain fuzzy Markov jump systems with partially unknown transition probabilities. A single pendulum example is given to illustrate the advantages and effectiveness of the proposed controller design method.

1. Introduction

Model predictive control (MPC), a powerful strategy for dealing with input and state constraints for a control system, has first attracted notable attention in industrial applications [1]. Since stability analysis approach for MPC was proposed, many significant advances in understanding MPC from a control theoretician’s viewpoint have then been acquired [2–4]. Meanwhile, the MPC formulation for constrained linear systems has been naturally extended to not only nonlinear systems [5–7] but also some more complex systems, for example, stochastic systems [8, 9], time-delay systems [10], hybrid systems [11], uncertain systems [12], and so on. By employing the idea of control structure optimization, for example, distributed MPC [13], centralized MPC, and coordinated MPC, the theory of MPC has been further refined. As is well known, for a long time, efficient setting of the large set of tunable parameters has been a hard problem for MPC. Fortunately, many available methods have already been developed (see [14] and the references therein for more details). Nowadays, many scholars tend to develop “fast MPC” in order to ease the huge online computational burden [15–17]. However, it should be noticed that MPC algorithm may perform very poorly when model mismatch occurs in spite of the inherent robustness provided by the feedback strategy based on the plant measurement at the next sampling time.

Therefore, in the past decades, the issues of MPC algorithm for uncertain systems have been addressed much in the literature. A robust constrained MPC scheme considering two classes of system uncertainty, that is, polytopic paradigm and structured feedback uncertainty, has been analyzed by means of linear matrix inequalities (LMIs) by Kothare et al. [18]. As opposed to a single linear static state-feedback law in [18], Bloemen et al. [19] divided the input sequence into two parts, the first 𝐻_𝑆 inputs being computed by finite MPC method and the other inputs being calculated based on linear static state-feedback law. Because 𝐻_𝑆 was variable, the end-point state-weighting matrix and the invariant ellipsoid were transformed into variables in the online optimization, and thus this algorithm achieved the trade-off between feasibility and performance. This work has been improved by applying parameter-dependent Lyapunov function [20–22]. In [23–25], an efficient robust constrained MPC with a time-varying terminal constraint set was developed. The proposed algorithm can obtain a perfect control schema achieving lower online computation and larger stabilizable set of states while retaining the unconstrained optimal performance as much as possible. It is worth mentioning that in order to analyze the stability of the system and obtain an upper bound of cost function, the optimal cost function has been qualified as a constraint on the increments of Lyapunov function [26]. However, this constraint is too strict, since to guarantee the system stability, the increments of Lyapunov function are only required to be negative. Motivated by this, in the present paper, a modified MPC scheme in which two extra parameters are introduced is proposed based on the characteristics of convergent series in order to reduce the conservativeness.

Many real systems, such as solar thermal receivers, economic systems, and networked control systems, may experience random abrupt changes in system inputs, internal variables, and other system parameters. Uncertainties like these are best represented via stochastic models [27–29], such as fuzzy Markov jump system (MJS) in which the subsystems are modeled as fuzzy systems. Based on the approximation property of the fuzzy logic systems [30–32], fuzzy MJSs are developed for the nonlinear control systems with abrupt changes in their structure and parameters.

In this paper, the problem of MPC controller design for a class of uncertain fuzzy MJSs with partially unknown transition probabilities is addressed. This kind of system fits into a very wide range of practical dynamic systems combining nonlinear behaviors with changes or uncertainties of structure or parameters. Meanwhile, constraints, such as energy limitation, levels in tanks, flows in piping, and maximum of pH value, can be systematically included during the controller design process. The system concerned is much more complex than fuzzy systems with uncertainties or MJSs with partially unknown transition probabilities [33, 34], because here the uncertainties will consist of four levels (the system parameter uncertainty, the membership degree uncertainty, the mode uncertainty, and the transition probability uncertainty) and they are not mutually independent. Therefore, although the formulation seems similar, the results for fuzzy systems with uncertainties or MJSs with partially unknown transition probabilities cannot be directly used in this scenario. A comparison with the method in [20] (modified in [22]) is carried out by simulation on a single pendulum control problem. Due to LMIs’ prevalence in convex optimization problems, especially for the cases with high order matrices, and the availability of reliable general commercial solvers, the LMI algorithm is employed to deal with the underlying optimization problems in this study. The remainder of this paper is organized as follows. The mathematical model of the concerned system is formulated and some preliminaries are given in Section 2. Section 3 is devoted to deriving the results for the controller design. Numerical examples are provided in Section 4 and this paper is concluded in Section 5.

Notation. The notation used throughout the paper is fairly standard. The superscript “𝑇” stands for matrix transposition. ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space. The notation 𝑃>0 (≥0) means that 𝑃 is real symmetric and positive (semipositive) definite and 𝐴>𝐵 (≥𝐵) means 𝐴−𝐵>0 (≥0). In symmetric block matrices or complex matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry and diag{⋯} stands for a block-diagonal matrix. 𝐼 and 0 represent identity matrix and zero matrix, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. ‖⋅‖₂ stands for the usual Euclidean norm.

2. Problem Formulation and Preliminaries

2.1. Uncertain Fuzzy MJSs with Partially Unknown Transition Probabilities

Consider the following time-varying discrete-time fuzzy Markov jump system. For each mode 𝑟_𝑘, the fuzzy model is composed of 𝑠 plant rules that can be represented as follows.

Plant Rule 𝑖. If 𝜃₁(𝑘) is ϝ^𝑖₁,…, and 𝜃_𝑑(𝑘) is ϝ^𝑖_𝑑, then 𝑥(𝑘+1)=𝐴_𝑖𝑟_𝑘,𝑘𝑥(𝑘)+𝐵_𝑖𝑟_𝑘𝑢(𝑘),𝑦(𝑘),𝑘=𝐶_𝑖𝑟_𝑘,𝑘𝑥(𝑘),𝑖=1,2,…,𝑠, (1) where 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑢(𝑘)∈ℝ^𝑚 is the control input, 𝑦(𝑘)∈ℝ^𝑞 is the system output, 𝜃≜[𝜃₁,…,𝜃_𝑑]^𝑇 is the premise variable vector, 𝑑 is the number of premise variables, and {ϝ^𝑖_𝑗, 𝑖∈𝕊, 𝑗∈{1,2,…,𝑑}} is a fuzzy set. For simplicity, we denote {1,2,…,𝑠} by 𝕊. The Markov chain {𝑟_𝑘,𝑘≥0}, taking values in a finite set ℐ≜{1,…,𝑁}, governs switching among different system modes. The system matrices of the 𝑟_𝑘 mode are denoted by 𝐴_𝑖(𝑟_𝑘,𝑘)𝐵_𝑖(𝑟_𝑘,𝑘)𝐶_𝑖(𝑟_𝑘,𝑘)∈Ω(𝑟_𝑘,𝑖) and the set Ω(𝑟_𝑘,𝑖) is a set of polytopes Ω𝑟_𝑘,𝑖≜Co𝐴_𝑖1𝑟_𝑘,…,𝐶_𝑖1𝑟_𝑘,𝐴_𝑖2𝑟_𝑘,…,𝐶_𝑖2𝑟_𝑘,…,𝐴_𝑖𝐿𝑟_𝑘,…,𝐶_𝑖𝐿𝑟_𝑘, (2) where Co denotes the convex hull. In other words, if 𝐴_𝑖(𝑟_𝑘,𝑘)𝐵_𝑖(𝑟_𝑘,𝑘)𝐶_𝑖(𝑟_𝑘,𝑘)∈Ω(𝑟_𝑘,𝑖), then for some nonnegative 𝜆_𝑗(𝑖,𝑟_𝑘,𝑘),𝑗∈[1,𝐿] summing to one, the following holds: 𝐴_𝑖𝑟_𝑘,𝑘𝐵_𝑖𝑟_𝑘,𝑘𝐶_𝑖𝑟_𝑘=,𝑘_𝐿𝑗=1𝜆_𝑗𝑖,𝑟_𝑘,𝑘𝐴_𝑖𝑗𝑟_𝑘𝐵_𝑖𝑗𝑟_𝑘𝐶_𝑖𝑗𝑟_𝑘. (3) For the sake of simplicity, we denote {1,2,…,𝐿} by 𝕃 and 𝐴_𝑖(𝑟_𝑘,𝑘),𝐵_𝑖(𝑟_𝑘,𝑘),𝐶_𝑖(𝑟_𝑘,𝑘), and 𝜆_𝑗(𝑖,𝑟_𝑘,𝑘) by 𝐴^𝑘_𝑖𝑟𝑘,𝐵^𝑘_𝑖𝑟𝑘,𝐶^𝑘_𝑖𝑟𝑘, and 𝜆^𝑘_{𝑖𝑗𝑟𝑘}, respectively.

As commonly done in the literature, it is assumed that the premise variable vector 𝜃 does not depend on the control variables and the disturbance. The center-average defuzzification method is used as follows: ℎ_𝑖[𝜃(𝑘=∏)]^𝑑𝑗=1𝜇_𝑖𝑗𝜃_𝑗(𝑘)∑^𝑠_𝑙=1∏^𝑑_𝑗=1𝜇_𝑙𝑗𝜃_𝑗(𝑘)≥0,𝑖∈𝕊,_𝑠𝑖=1ℎ_𝑖[𝜃(𝑘)]=1, (4) where 𝜇_𝑖𝑗[𝜃_𝑗(𝑘)] is the grade of membership of 𝜃_𝑗(𝑘) in ϝ^𝑖_𝑗. In what follows, we use the following notation for simplicity: ℎ_𝑖^𝑘+𝑙=ℎ_𝑖[𝜃(𝑘+𝑙)].

A more compact presentation of system (1) is given by Σ∶𝑥(𝑘+1)=^𝑠_𝑖=1ℎ^𝑘_𝑖𝐴^𝑘_𝑖𝑟𝑘𝑥(𝑘)+𝐵^𝑘_𝑖𝑟𝑘𝑢(𝑘𝑦(𝑘)=_𝑠𝑖=1)ℎ^𝑘_𝑖𝐶^𝑘_𝑖𝑟𝑘𝑥(𝑘),𝑟_𝑘∈ℐ. (5) The process {𝑟_𝑘,𝑘≥0} is described by a discrete-time homogeneous Markov chain, which takes values in the finite set ℐ with mode transition probabilities Pr𝑟_𝑘+1=𝑗∣𝑟_𝑘=𝑖=𝜋_𝑖𝑗, (6) where 𝜋_𝑖𝑗≥0,∀(𝑖,𝑗)∈ℐ×ℐ, and ∑^𝑁_𝑗=1𝜋_𝑖𝑗=1. For example, the transition probability matrix (TPM) can be given by 𝜋Π=¹¹𝜋₁₂⋯𝜋_1𝑁𝜋₂₁𝜋₂₂⋯𝜋_2𝑁𝜋⋮⋮⋱⋮_𝑁1𝜋_𝑁2⋯𝜋_𝑁𝑁. (7) The transition probabilities described above are considered to be partially available and can be divided into two parts as ℐ_𝒦^(𝑖)≜𝑗∶𝜋_𝑖𝑗isknown,ℐ^(𝑖)_𝒰𝒦≜𝑗∶𝜋_𝑖𝑗isunknown. (8) In addition, if ℐ_𝒦^(𝑖)≠∅,ℐ_𝒦^(𝑖) is further described as ℐ_𝒦^(𝑖)=𝒦₁,𝒦₂,…,𝒦_𝑧𝑖,𝑧_𝑖∈{1,2,…,𝑁−2}, (9) where 𝒦_𝑠∈ℕ⁺,𝑠∈{1,2,…,𝑧_𝑖}, represents the index of the 𝑠th known element in the 𝑖th row of matrix Π. And we denote 𝜋_𝒦(𝑖)≜_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗. (10)

The following fuzzy control law is chosen: 𝑢(𝑘)=^𝑠_𝑝=1ℎ^𝑘_𝑝𝐹_𝑝𝑟_𝑘𝑥(𝑘),𝑟_𝑘∈ℐ. (11) Under the control law, the closed-loop system of Σ is given by Σ^𝐶∶𝑥(𝑘+1)=^𝑠𝑠_𝑝=1𝑞=1ℎ^𝑘_𝑝ℎ^𝑘_𝑞𝐴^𝑘_{𝑝𝑞𝑟𝑘}𝑥(𝑘),𝑦(𝑘)=^𝑠_𝑝=1ℎ^𝑘_𝑝𝐶^𝑘_𝑝𝑟𝑘𝑥(𝑘),𝑟_𝑘∈ℐ, (12) where 𝐴^𝑘_{𝑝𝑞𝑟𝑘}=𝐴_𝑝𝑞(𝑟_𝑘,𝑘)≜𝐴^𝑘_𝑝𝑟𝑘+𝐵^𝑘_𝑝𝑟𝑘𝐹_𝑞(𝑟_𝑘).

For the fuzzy MJS, the following definition will be adopted in the rest of this paper.

Definition 1. The fuzzy MJS in (12) is said to be stochastically stable, if for any initial condition 𝑥(0)=𝑥₀,𝑟₀∈ℐ, the following inequality holds: lim_𝑁→∞𝐸^𝑁_𝑘=0𝑥𝑘,𝑥₀,𝑟₀²₂<∞, (13) where 𝐸{⋅} stands for the mathematical expectation.

The objective of this paper is to design a state-feedback model predictive controller such that the closed-loop system (12) is stochastically stable.

2.2. Model Predictive Control

Model predictive control (MPC) makes use of a receding horizon principle, which means that at each sampling time 𝑘, an optimization algorithm will be applied to compute a sequence of future control signals. The used performance index depends on the predicted future states of the plant 𝑥(𝑘+𝑖∣𝑘),𝑖≥0, which can be calculated through the newly obtained measurements 𝑥(𝑘) and the predictive model. Here, we use 𝑥(𝑘+𝑖∣𝑘),𝑦(𝑘+𝑖∣𝑘) as the state and output, respectively, of the plant at time 𝑘+𝑖 predicted by utilizing the measurements at time 𝑘; 𝑢(𝑘+𝑖∣𝑘) represents the control action moves at time 𝑘+𝑖 computed by the optimization problem (14); 𝑃 is the prediction horizon and 𝑀 is the control horizon. In what follows, let ℙ and 𝕄 represent {0,1,…,𝑃} and {0,1,…,𝑀}, respectively.

In this section, the problem formulation for MPC using the model (12) is discussed. The goal is to find an input sequence {𝑢(𝑘∣𝑘),𝑢(𝑘+1∣𝑘),…,𝑢(𝑘+𝑀−1∣𝑘)} at each sampling time 𝑘 which minimizes the following performance index 𝐽(𝑘): min_{𝑢()𝑘+𝑖∣𝑘,𝑖∈𝕄}𝐽(𝑘), (14) where 𝐽(𝑘)≜𝐸^𝑃_𝑖=0𝑥(𝑘+𝑖∣𝑘)^𝑇()+𝑄𝑥𝑘+𝑖∣𝑘_𝑀𝑗=0𝑢𝑘+𝑗∣𝑘^𝑇𝑅𝑢𝑘+𝑗∣𝑘, (15) and then it can be replaced by min_{𝑢()𝑘+𝑖∣𝑘,𝑖∈𝕄}max_{𝐴𝑘𝑘+𝑖𝑗𝑟𝐵𝑘𝑘+𝑖𝑗𝑟𝐶𝑘𝑘+𝑖𝑗𝑟𝑟∈Ω𝑘,𝑗,𝑖∈ℙ,𝑗∈𝕊}𝐽(𝑘). (16) The matrices 𝑄>0 and 𝑅>0 are symmetric weighting matrices. In particular, we will consider the case where 𝑃=𝑀, which means that the control horizon is equal to the prediction horizon. The input and output constraints focused on in this paper are Euclidean norm bounds and componentwise peak bounds, given, respectively, as ‖𝑢(𝑘+𝑖∣𝑘)‖₂≤𝑢_max𝑢,𝑘≥0,𝑖∈𝕄,_𝑗(𝑘+𝑖∣𝑘)≤𝑢_𝑗,max𝑦(,𝑘≥0,𝑖∈𝕄,𝑗=1,2,…,𝑚,𝑘+𝑖∣𝑘)₂≤𝑦_max,𝑘≥0,𝑖∈𝕄. (17)

3. Model Predictive Control Using Linear Matrix Inequalities

In this section, the problem is solved by deriving an upper bound on the objective function 𝐽(𝑘) based on the state-feedback control law 𝑢(𝑘+𝑖∣𝑘)=∑^𝑠_𝑝=1ℎ^𝑘_𝑝𝐹_𝑝(𝑟_𝑘)𝑥(𝑘+𝑖∣𝑘),𝑖∈ℙ. Therefore, via minimizing this upper bound at each sampling time 𝑘, we will obtain the sequence of control input signals.

3.1. Derivation of the Upper Bound

In this paper, the following objective function 𝐽(𝑘) is considered: min_{𝑢()𝑘+𝑖∣𝑘,𝑖∈ℙ}max_{𝐴𝑘𝑘+𝑖𝑗𝑟𝐵𝑘𝑘+𝑖𝑗𝑟𝐶𝑘𝑘+𝑖𝑗𝑟𝑟∈Ω𝑘,𝑗,𝑖∈ℙ,𝑗∈𝕊}𝐽(𝑘), (18) where 𝐽(𝑘)≜𝐸^𝑃_𝑖=0𝑥(𝑘+𝑖∣𝑘)^𝑇()()𝑄𝑥𝑘+𝑖∣𝑘+𝑢𝑘+𝑖∣𝑘^𝑇𝑅𝑢(𝑘+𝑖∣𝑘). (19)

Consider a parameter-dependent function 𝑉(𝑥(𝑘+𝑖∣𝑘))≜𝑥(𝑘+𝑖∣𝑘)^𝑇𝑠𝐿_𝑗=1𝑙=1ℎ_𝑗^𝑘+𝑖𝜆^𝑘+𝑖_{𝑗𝑙𝑟𝑘+𝑖}𝒫_𝑗𝑙𝑟_𝑘+𝑖×𝑥(𝑘+𝑖∣𝑘),𝑘≥0,𝑖∈ℙ, (20) where 𝒫_𝑗𝑙(𝑟_𝑘+𝑖)>0, 𝑉(𝑥(0))=0, and 𝑥(𝑘∣𝑘)=𝑥(𝑘) is the system state. Assume that there exists a convergent series 𝑎_𝑘, ∑^∞_𝑘=1𝑎_𝑘=𝑏<∞ such that, at each sampling time 𝑘, the following inequality holds for all 𝑥(𝑘+𝑖∣𝑘), 𝑢(𝑘+𝑖∣𝑘), 𝑖∈ℙ satisfying (12): ∀𝐴^𝑘+𝑖_𝑖𝑟𝑘𝐵^𝑘+𝑖_𝑖𝑟𝑘𝐶^𝑘+𝑖_𝑖𝑟𝑘∈Ω(𝑟_𝑘,𝑖),𝑖∈ℙ,𝑟_𝑘∈ℐ, 𝐸{𝑉(𝑥((𝑥(𝑥()𝑘+𝑖+1∣𝑘))−𝑉𝑘+𝑖∣𝑘))}≤𝐸−𝛽𝑘+𝑖∣𝑘^𝑇()()𝑄𝑥𝑘+𝑖∣𝑘+𝑢𝑘+𝑖∣𝑘^𝑇𝑅𝑢(𝑘+𝑖∣𝑘)−𝑎_𝑘+𝑖,0<𝛽≤1. (21) Summing (21) from 𝑖=0 to 𝑖=𝑃, we get 𝐸{𝑉(𝑥(𝑘+𝑃∣𝑘))−𝑉(𝑥(𝑘∣𝑘))}≤−𝛽(𝐽(𝑘)−𝑆(𝑘)), (22) where 𝑆(𝑘)=∑^𝑃_𝑖=0𝑎_𝑘+𝑖. Thus max_{𝐴𝑘𝑘+𝑖𝑗𝑟𝐵𝑘𝑘+𝑖𝑗𝑟𝐶𝑘𝑘+𝑖𝑗𝑟𝑟∈Ω𝑘,𝑗,𝑖∈ℙ,𝑗∈𝕊}𝐽(𝑘)1≤𝐸𝛽(𝑉(𝑥((𝑥((𝑘1𝑘∣𝑘))−𝑉𝑘+𝑃∣𝑘)))+𝑆)≤𝐸𝛽𝑉(𝑥(𝑘∣𝑘))+𝑏. (23) Specially, when 𝑃=𝑀=∞, we will have 𝑥(∞∣𝑘)=0 for the robust performance objective function to be finite and hence 𝑉(𝑥(∞,𝑘))=0. Summing (21) from 𝑖=0 to 𝑖=∞, we get 𝐸{−𝑉(𝑥(𝑘∣𝑘))}≤−𝛽(𝐽(𝑘)−𝑏). (24) Thus max_{𝐴𝑘𝑘+𝑖𝑗𝑟𝐵𝑘𝑘+𝑖𝑗𝑟𝐶𝑘𝑘+𝑖𝑗𝑟𝑟∈Ω𝑘,𝑗,𝑖∈ℙ,𝑗∈𝕊}𝐽(𝑘)1≤𝐸𝛽𝑉(𝑥(𝑘∣𝑘))+𝑏. (25)

Set 𝑎_𝑘+𝑖=𝑥(𝑘+𝑖∣𝑘)^𝑇𝑐_𝑘+𝑖𝐼𝑥(𝑘+𝑖∣𝑘), (26) where ∑^∞_𝑘=1𝑐_𝑘=𝑑. For the robust performance objective function to be finite, we will have 𝑥(𝑘+𝑖∣𝑘)∈𝑋_min𝑋_max and 𝑥(𝑘+𝑖∣𝑘)≤𝑋_max; hence, 𝑎_𝑘+𝑖≤𝑋^𝑇_max𝑐_𝑘+𝑖𝐼𝑋_max, and then ∑^∞_𝑘=1𝑎_𝑘<𝑋^𝑇_max∑^∞_𝑘=1𝑐_𝑘𝐼𝑋_max=𝑋^𝑇_max𝑑𝐼𝑋_max. Therefore, (21) can be written as 𝐸{𝑉(𝑥((𝑥(𝑥()𝑘+𝑖+1∣𝑘))−𝑉𝑘+𝑖∣𝑘))}≤𝐸−𝛽𝑘+𝑖∣𝑘^𝑇()()𝑄𝑥𝑘+𝑖∣𝑘+𝑢𝑘+𝑖∣𝑘^𝑇(()𝑅𝑢𝑘+𝑖∣𝑘)+𝛽𝑥𝑘+𝑖∣𝑘^𝑇𝑐_𝑘+𝑖𝐼𝑥(𝑘+𝑖∣𝑘). (27) In order to guarantee the increment Δ𝑉(𝑘)≤0, 𝑐_𝑘𝐼<𝑄 is required. This gives an upper bound on the robust performance objective. Thus the goal of our robust MPC algorithm has been redefined to deduce, at each time step 𝑘, a state-feedback control law 𝑢(𝑘+𝑖∣𝑘)=∑^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝐹_𝑝(𝑟_𝑘)𝑥(𝑘+𝑖∣𝑘) to minimize this upper bound 𝐸{(1/𝛽)𝑉(𝑥(𝑘∣𝑘))+𝑏}, that is to say, 𝐸{𝑉(𝑥(𝑘∣𝑘))}.

3.2. Minimization of the Upper Bound without Constraints

Theorem 2. Consider the closed-loop uncertain system (12) with the polytopic uncertainty set Ω(𝑟_𝑘,𝑖),𝑟_𝑘∈ℐ,𝑖∈𝕊. Let 𝑥(𝑘)=𝑥(𝑘∣𝑘) be the state measured at sampling time 𝑘. Assume that there are no constraints on the control input and plant output. Then the state-feedback matrix 𝐹_𝑝(𝑟_𝑘+𝑖) in the control law 𝑢(𝑘+𝑖∣𝑘)=∑^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝐹_𝑝(𝑟_𝑘+𝑖)𝑥(𝑘+𝑖𝑘),𝑖∈ℙ, that minimizes the upper bound 𝐸{(1/𝛽)𝑉(𝑥(𝑘∣𝑘))+𝑏} on the robust performance objective function at sampling time 𝑘 is given by 𝐹_𝑝𝑟_𝑘+𝑖=𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹, (28) where 𝐺>0 and 𝑌_𝑝(𝑟_𝑘+𝑖) are obtained from the solution (if it exists) to the following linear objective minimization problem: min_{𝛾,𝒬,𝐺,𝑌}𝛾()𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜1𝑥𝑘∣𝑘^𝑇𝑥(𝑘∣𝑘)𝒬_𝑝𝑗𝑟_𝑘≥0,∀𝑖∈ℙ,𝑝∈𝕊,𝑗∈𝕃,𝑟_𝑘∈ℐ,𝐺+𝐺^𝑇−𝒬_𝑝𝑗𝑟_𝑘+𝑖𝑀∗∗∗_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖𝑁_𝑤𝑣𝛽∗∗_1/2𝑄−𝑐̂_𝑘+𝑖𝐼^1/2𝛽𝐺0𝛾𝐼∗_1/2𝑅^1/2𝑌_𝑝𝑟_𝑘+𝑖00𝛾𝐼≥0,∀𝑖∈ℙ,𝑝≥𝑞,𝑤∈𝕊,𝑗,𝑣,𝑓≥𝑜∈𝕃,𝑟_𝑘+𝑖∈ℐ,𝑔∈ℐ^{(𝑟𝑘+𝑖)}_𝒰𝒦, (29) with 𝑀_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖≜𝜋_{𝑟1𝑘+𝑖𝒦}𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖,…,𝜋_{𝑟𝑧𝑟𝑘+𝑖𝒦𝑘+𝑖}𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖,1−𝜋_𝒦𝑟_𝑘+𝑖𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖^𝑇,𝑁_𝑤𝑣≜diag𝒬_𝑤𝑣𝒦₁,𝒬_𝑤𝑣𝒦₂𝒬,…,_𝑤𝑣𝒦_{𝑧𝑟𝑘+𝑖},𝒬_𝑤𝑣𝒦_𝑔,_∞𝑘=1𝑐_𝑘̂<∞,𝑐_𝑘̂𝐸𝐼<𝑄,_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖≜𝐴_{𝑝𝑓𝑟𝑘+𝑖}𝐺+𝐵_{𝑝𝑓𝑟𝑘+𝑖}𝑌_𝑞𝑟_𝑘+𝑖+𝐴_{𝑞𝑜𝑟𝑘+𝑖}𝐺+𝐵_{𝑞𝑜𝑟𝑘+𝑖}𝑌_𝑝𝑟_𝑘+𝑖×2⁻¹,𝒬_𝑝𝑗𝑟_𝑘>0. (30)

Proof. Minimization of the upper bound means minimizing 𝑉(𝑥(𝑘∣𝑘))which is equivalent to min_𝛾,𝒫𝛾subjectto𝑥(𝑘∣𝑘)^𝑇𝑠𝐿_𝑝=1𝑗=1ℎ^𝑘_𝑝𝜆^𝑘_{𝑝𝑗𝑟𝑘}𝒫_𝑝𝑗𝑟_𝑘×𝑥(𝑘∣𝑘)≤𝛾. (31) Defining 𝒬_𝑝𝑗(𝑟_𝑘)≜𝛾𝒫⁻¹_𝑝𝑗(𝑟_𝑘)>0 and using Schur complements, this is equivalent to min_𝛾,𝒬𝛾()subjectto1𝑥𝑘∣𝑘^𝑇𝑥(𝑘∣𝑘)𝒬_𝑝𝑗𝑟_𝑘≥0,∀𝑖∈ℙ,𝑝∈𝕊,𝑗∈𝕃,𝑟_𝑘∈ℐ. (32)
The parameter-dependent function 𝑉 is required to satisfy (27). By substituting 𝑢(𝑘+𝑖∣𝑘)=^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝐹_𝑝𝑟_𝑘𝑥(𝑘+𝑖∣𝑘),𝑖∈ℙ, (33) and the state-space equations in (12), inequality (27) becomes𝐸𝑥(𝑘+𝑖∣𝑘)^𝑇𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1𝑗=1ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}⋅𝒫_𝑤𝑣𝑟_𝑘+𝑖+1𝐴_{𝑚𝑛𝑟𝑘+𝑖}−𝒫_𝑝𝑗𝑟_𝑘+𝑖+𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑞𝑟_𝑘+𝑖+𝛽𝑄−𝛽𝑐̂_𝑘+𝑖𝐼()()×𝑥𝑘+𝑖∣𝑘=𝑥𝑘+𝑖∣𝑘^𝑇𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1^𝑁_{𝑗=1𝑟𝑘+𝑖+1=1}ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}⋅𝜋_{𝑟𝑘+𝑖𝑟𝑘+𝑖+1}𝒫_𝑤𝑣𝑟_𝑘+𝑖+1𝐴_{𝑚𝑛𝑟𝑘+𝑖}−𝒫_𝑝𝑗𝑟_𝑘+𝑖+𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑞𝑟_𝑘+𝑖+𝛽𝑄−𝛽𝑐̂_𝑘+𝑖𝐼𝑥(𝑘+𝑖∣𝑘)≤0. (34)This is equivalent to^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1^𝑁_{𝑗=1𝑟𝑘+𝑖+1=1}ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}⋅𝜋_{𝑟𝑘+𝑖𝑟𝑘+𝑖+1}𝒫_𝑤𝑣𝑟_𝑘+𝑖+1𝐴_{𝑚𝑛𝑟𝑘+𝑖}−𝒫_𝑝𝑗𝑟_𝑘+𝑖+𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑞𝑟_𝑘+𝑖+𝛽𝑄−𝛽𝑐̂_𝑘+𝑖𝐼=^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1^𝑁_{𝑗=1𝑟𝑘+𝑖+1=1}14ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}⋅𝜋_{𝑟𝑘+𝑖𝑟𝑘+𝑖+1}𝒫_𝑤𝑣𝑟_𝑘+𝑖+1×𝐴_{𝑚𝑛𝑟𝑘+𝑖}+𝐴_{𝑛𝑚𝑟𝑘+𝑖}−4𝒫_𝑝𝑗𝑟_𝑘+𝑖+4𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑞𝑟_𝑘+𝑖+4𝛽𝑄−4𝛽𝑐̂_𝑘+𝑖𝐼=^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1^𝑁_{𝑗=1𝑟𝑘+𝑖+1=1}18ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}⋅𝜋_{𝑟𝑘+𝑖𝑟𝑘+𝑖+1}𝒫_𝑤𝑣𝑟_𝑘+𝑖+1×𝐴_{𝑚𝑛𝑟𝑘+𝑖}+𝐴_{𝑛𝑚𝑟𝑘+𝑖}+𝐴^𝑇_{𝑚𝑛𝑟𝑘+𝑖}+𝐴^𝑇_{𝑛𝑚𝑟𝑘+𝑖}×𝒫_𝑤𝑣𝑟_𝑘+𝑖+1𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}−8𝒫_𝑝𝑗𝑟_𝑘+𝑖+4𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑞𝑟_𝑘+𝑖+4𝛽𝐹_𝑞𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑝𝑟_𝑘+𝑖+8𝛽𝑄−8𝛽𝑐̂_𝑘+𝑖𝐼≤^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1^𝑁_{𝑗=1𝑟𝑘+𝑖+1=1}18ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}⋅𝜋_{𝑟𝑘+𝑖𝑟𝑘+𝑖+1}𝒫_𝑤𝑣𝑟_𝑘+𝑖+1×𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}+𝐴^𝑇_{𝑚𝑛𝑟𝑘+𝑖}+𝐴^𝑇_{𝑛𝑚𝑟𝑘+𝑖}×𝒫_𝑤𝑣𝑟_𝑘+𝑖+1𝐴_{𝑚𝑛𝑟𝑘+𝑖}+𝐴_{𝑛𝑚𝑟𝑘+𝑖}−8𝒫_𝑝𝑗𝑟_𝑘+𝑖+4𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑝𝑟_𝑘+𝑖+4𝛽𝐹_𝑞𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑞𝑟_𝑘+𝑖+8𝛽𝑄−8𝛽𝑐̂_𝑘+𝑖𝐼=^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1^𝑁_{𝑗=1𝑟𝑘+𝑖+1=1}14ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}⋅𝜋_{𝑟𝑘+𝑖𝑟𝑘+𝑖+1}𝒫_𝑤𝑣𝑟_𝑘+𝑖+1×𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}−4𝒫_𝑝𝑗𝑟_𝑘+𝑖+4𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑝𝑟_𝑘+𝑖+4𝛽𝑄−4𝛽𝑐̂_𝑘+𝑖𝐼=^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1𝑗=114ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}⋅_{𝑔∈ℐ)𝒦(𝑟𝑘+𝑖}𝜋_{𝑟𝑘+𝑖𝑔}𝒫_𝑤𝑣𝑔+_{𝑙∈ℐ)(𝑟𝑘+𝑖𝒰𝒦}𝜋_{𝑟𝑘+𝑖𝑙}𝒫_𝑤𝑣(𝑙)×𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}−4𝒫_𝑝𝑗𝑟_𝑘+𝑖+4𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑝𝑟_𝑘+𝑖+4𝛽𝑄−4𝛽𝑐̂_𝑘+𝑖𝐼=^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_𝑣=1𝑗=114ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}×𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}⋅_{𝑔∈ℐ)𝒦(𝑟𝑘+𝑖}𝜋_{𝑟𝑘+𝑖𝑔}𝒫_𝑤𝑣𝑔+1−𝜋_𝒦𝑟_{𝑘+𝑖𝑙∈ℐ)(𝑟𝑘+𝑖𝒰𝒦}𝜋_{𝑟𝑘+𝑖𝑙}1−𝜋_𝒦𝑟_𝑘+𝑖𝒫_𝑤𝑣(𝑙)×𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}−4𝒫_𝑝𝑗𝑟_𝑘+𝑖+4𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑝𝑟_𝑘+𝑖+4𝛽𝑄−4𝛽𝑐̂_𝑘+𝑖𝐼=^𝑠𝑠_𝑝=1^𝑠_𝑞=1^𝑠_𝑚=1^𝑠_𝑛=1^𝐿_𝑤=1^𝐿_{𝑣=1𝑗=1𝑙∈ℐ)(𝑟𝑘+𝑖𝒰𝒦}𝜋_{𝑟𝑘+𝑖𝑙}1−𝜋_𝒦𝑟_𝑘+𝑖×ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖ℎ_𝑚^𝑘+𝑖ℎ_𝑛^𝑘+𝑖ℎ_𝑤^𝑘+𝑖+1𝜆^𝑘+𝑖+1_{𝑤𝑣𝑟𝑘+𝑖+1}𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}⋅𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}2_{𝑔∈ℐ)𝒦(𝑟𝑘+𝑖}𝜋_{𝑟𝑘+𝑖𝑔}𝒫_𝑤𝑣𝑔+1−𝜋_𝒦𝑟_𝑘+𝑖𝒫_𝑤𝑣(𝑙)𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}2−𝒫_𝑝𝑗𝑟_𝑘+𝑖+𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇×𝑅𝐹_𝑝𝑟_𝑘+𝑖+𝛽𝑄−𝛽𝑐̂_𝑘+𝑖𝐼. (35)Then, (34) is satisfied for all 𝑖∈ℙ if 𝐴^𝑇_{𝑝𝑞𝑟𝑘+𝑖}+𝐴^𝑇_{𝑞𝑝𝑟𝑘+𝑖}2_{𝑔∈ℐ)𝒦(𝑟𝑘+𝑖}𝜋_{𝑟𝑘+𝑖𝑔}𝒫_𝑤𝑣𝑔+1−𝜋_𝒦𝑟_𝑘+𝑖𝒫_𝑤𝑣(𝑙)×𝐴_{𝑝𝑞𝑟𝑘+𝑖}+𝐴_{𝑞𝑝𝑟𝑘+𝑖}2−𝒫_𝑝𝑗𝑟_𝑘+𝑖+𝛽𝐹_𝑝𝑟_𝑘+𝑖^𝑇𝑅𝐹_𝑝𝑟_𝑘+𝑖+𝛽𝑄−𝛽𝑐̂_𝑘+𝑖𝐼≤0, (36) where 𝑐_𝑘̂ is the minimum term of 𝑐_𝑘+𝑖,𝑖∈0𝑃. Substituting 𝒫_𝑝𝑞(𝑟_𝑘)=𝛾𝒬⁻¹_𝑝𝑞(𝑟_𝑘),𝒬_𝑝𝑞(𝑟_𝑘)>0, using Schur complements, and then pre- and postmultiplying the above inequality by diag{𝒬_𝑝𝑗(𝑟_𝑘+𝑖),𝐼,𝐼,𝐼} and diag{𝒬_𝑝𝑗(𝑟_𝑘+𝑖),𝐼,𝐼,𝐼}, we can see that this is equivalent to 𝒬_𝑝𝑗𝑟_𝑘+𝑖𝑀∗∗∗_𝑝𝑞𝑟_𝑘+𝑖𝑁_𝑤𝑣𝛽∗∗_1/2𝑄−𝑐̂_𝑘+𝑖𝐼^1/2𝒬_𝑝𝑗𝑟_𝑘+𝑖𝛽0𝛾𝐼∗_1/2𝑅^1/2𝐹_𝑝𝑟_𝑘+𝑖𝒬_𝑝𝑗𝑟_𝑘+𝑖00𝛾𝐼≥0,∀𝑖∈ℙ,𝑝,𝑞,𝑤∈𝕊,𝑗,𝑣∈𝕃,𝑟_𝑘+𝑖∈ℐ,𝑔∈ℐ^{(𝑟𝑘+𝑖)}_𝒰𝒦, (37) with 𝑀_𝑝𝑞𝑟_𝑘+𝑖≜𝜋_{𝑟1𝑘+𝑖𝒦}𝐸_𝑝𝑞𝑟_𝑘+𝑖,…,𝜋_{𝑟𝑧𝑟𝑘+𝑖𝒦𝑘+𝑖}𝐸_𝑝𝑞𝑟_𝑘+𝑖,1−𝜋_𝒦𝑟_𝑘+𝑖𝐸_𝑝𝑞𝑟_𝑘+𝑖^𝑇,𝑁_𝑤𝑣≜diag𝒬_𝑤𝑣𝒦₁,𝒬_𝑤𝑣𝒦₂𝒬,…,_𝑤𝑣𝒦_{𝑧𝑟𝑘+𝑖},𝒬_𝑤𝑣𝒦_𝑔,𝐸_𝑝𝑞𝑟_𝑘+𝑖≜𝐴_{𝑝𝑟𝑘+𝑖}+𝐵_{𝑝𝑟𝑘+𝑖}𝐹_𝑞𝑟_𝑘+𝑖𝒬_𝑝𝑗𝑟_𝑘+𝑖+𝐴_{𝑞𝑟𝑘+𝑖}+𝐵_{𝑞𝑟𝑘+𝑖}𝐹_𝑝𝑟_𝑘+𝑖𝒬_𝑝𝑗𝑟_𝑘+𝑖×2⁻¹. (38) Inequality (37) is affine in 𝐴_𝑝(𝑘+𝑖,𝑟_𝑘)𝐵_𝑝(𝑘+𝑖,𝑟_𝑘). Following the logic of de Oliveira, Bernussou and Geromel [35], and Cuzzola et al. [20], (37) is satisfied for all 𝐴^𝑘+𝑖_𝑝𝑟𝑘𝐵^𝑘+𝑖_𝑝𝑟𝑘∈Ω_𝑝𝑟_𝑘=Co𝐴_𝑝1𝑟_𝑘𝐵_𝑝1𝑟_𝑘,𝐴_𝑝2𝑟_𝑘𝐵_𝑝2𝑟_𝑘𝐴,…,_𝑝𝐿𝑟_𝑘𝐵_𝑝𝐿𝑟_𝑘, (39) if and only if there exist 𝐺>0,𝑌_𝑝(𝑟_𝑘)=𝐹_𝑝(𝑟_𝑘)𝐺, and a positive 𝛾 such that 𝐺+𝐺^𝑇−𝒬_𝑝𝑗𝑟_𝑘+𝑖𝑀∗∗∗_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖𝑁_𝑤𝑣𝛽∗∗_1/2𝑄−𝑐̂_𝑘+𝑖𝐼^1/2𝛽𝐺0𝛾𝐼∗_1/2𝑅^1/2𝑌_𝑝𝑟_𝑘+𝑖𝑟00𝛾𝐼≥0,∀𝑖∈ℙ,𝑝≥𝑞,𝑤∈𝕊,𝑗,𝑣,𝑓≥𝑜∈𝕃,_𝑘+𝑖∈ℐ,𝑔∈ℐ^{(𝑟𝑘+𝑖)}_𝒰𝒦, (40) with 𝑀_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖≜𝜋_{𝑟1𝑘+𝑖𝒦}𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖,…,𝜋_{𝑟𝑧𝑟𝑘+𝑖𝒦𝑘+𝑖}𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖,1−𝜋_𝒦𝑟_𝑘+𝑖𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖^𝑇,𝑁_𝑤𝑣≜diag𝒬_𝑤𝑣𝒦₁,𝒬_𝑤𝑣𝒦₂𝒬,…,_𝑤𝑣𝒦_{𝑧𝑟𝑘+𝑖},𝒬_𝑤𝑣𝒦_𝑔,𝐸_{𝑝𝑞𝑓𝑜}𝑟_𝑘+𝑖≜𝐴_{𝑝𝑓𝑟𝑘+𝑖}𝐺+𝐵_{𝑝𝑓𝑟𝑘+𝑖}𝑌_𝑞𝑟_𝑘+𝑖+𝐴_{𝑞𝑜𝑟𝑘+𝑖}𝐺+𝐵_{𝑞𝑜𝑟𝑘+𝑖}𝑌_𝑝𝑟_𝑘+𝑖×2⁻¹. (41) The feedback matrix is then given by 𝐹_𝑝(𝑟_𝑘+𝑖)=𝑌_𝑝(𝑟_𝑘+𝑖)𝐺⁻¹.

Remark 3. Based on the above analysis, we can see that the choice of control horizon 𝑀 can amount to the choice of series 𝑐_𝑘̂. That is, given a constant control horizon 𝑀, we can construct some infinite convergent series whose corresponding series 𝑐_𝑘̂ possess some properties, such as 𝑐_𝑘̂=1/(𝑘+1)². Therefore, we will discuss neither the control horizon 𝑀 nor the predictive horizon 𝑃 which is supposed to be equal to 𝑀.

Remark 4. In particular, when the control horizon 𝑀=∞, the conditions for the above derivation can be satisfied if and only if the series 𝑐_𝑘̂=0,𝑘≥0. Then it follows that the method is equivalent to the approach in [20] (modified in [22]).

3.3. Minimization of the Upper Bound with Input and Output Constraints

Theorem 5. Consider the closed-loop uncertain system (12) with the polytopic uncertainty set Ω(𝑟_𝑘,𝑝),𝑝∈𝕊,𝑟_𝑘∈ℐ. Let 𝑥(𝑘)=𝑥(𝑘∣𝑘) be the state measured at sampling time 𝑘, and 𝒬_𝑝𝑗(𝑟_𝑘)>0, for all 𝑝∈𝕊,𝑗∈𝕃,𝑟_𝑘∈ℐ. The constraints on the control input and plant output in the form of (17) can be transferred into problems expressed by the following linear matrix inequalities, respectively: 𝑢²_max𝐼𝑌_𝑝𝑟_𝑘+𝑖𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝐺+𝐺^𝑇−𝒬_𝑝𝑗𝑟_𝑘+𝑖≥0,∀𝑝∈𝕊,𝑗∈𝕃,𝑟_𝑘+𝑖∈ℐ,𝑋𝑌_𝑝𝑟_𝑘+𝑖𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝐺+𝐺^𝑇−𝒬_𝑝𝑗𝑟_𝑘+𝑖≥0,∀𝑝∈𝕊,𝑗∈𝕃,𝑟_𝑘+𝑖∈ℐ, (42)(43) with 𝑋_𝑑𝑑≤𝑢²_𝑑,max,𝑑=1,2,…,𝑚, 𝐺+𝐺^𝑇−𝒬_𝑝𝑗𝑟_𝑘+𝑖∗𝐶_{𝑝𝑗𝑟𝑘+𝑖}𝐴_{𝑝𝑗𝑟𝑘+𝑖}𝐺+𝐵_{𝑝𝑗𝑟𝑘+𝑖}𝑌_𝑞𝑟_𝑘+𝑖𝑦²_max𝐼≥0,∀𝑝,𝑞∈𝕊,𝑗∈𝕃,𝑟_𝑘+𝑖∈ℐ. (44)

Proof. Following [36], we have max_𝑖≥0‖𝑢(𝑘+𝑖∣𝑘)‖²₂=max_𝑖≥0^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝐹_𝑝𝑟_𝑘+𝑖𝑥(𝑘+𝑖∣𝑘)²₂=max_𝑖≥0^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹𝑥(𝑘+𝑖∣𝑘)²₂≤max_𝑧∈𝜉^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹𝑧²₂=𝜎_𝑧∈𝜉^𝑠𝑠_𝑝=1𝑞=1ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖𝑧^𝑇𝐺^−𝑇×𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝑌_𝑞𝑟_𝑘+𝑖𝐺⁻¹𝑧=𝜎_𝑧∈𝜉^𝑠𝑠_𝑝=1𝑞=112ℎ_𝑝^𝑘+𝑖ℎ_𝑞^𝑘+𝑖𝑧^𝑇𝐺^−𝑇×𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝑌_𝑞𝑟_𝑘+𝑖+𝑌^𝑇_𝑞𝑟_𝑘+𝑖𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹𝑧≤𝜎_𝑧∈𝜉^𝑠_𝑝=1ℎ_𝑝^𝑘+𝑖𝑧^𝑇𝐺^−𝑇𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹𝑧=𝜎_𝑧∈𝜉^𝑠𝐿_𝑝=1𝑗=1ℎ_𝑝^𝑘+𝑖𝜆^𝑘+𝑖_{𝑝𝑗𝑟𝑘+𝑖}𝑧^𝑇𝒬^−1/2_𝑝𝑗𝒬^1/2_𝑝𝑗𝐺^−𝑇𝑌^𝑇_𝑝𝑟_𝑘+𝑖×𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹𝒬^1/2_𝑝𝑗𝒬^−1/2_𝑝𝑗𝑧≤𝜎𝒬^1/2_𝑝𝑗𝐺^−𝑇𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝑌_𝑝𝑟_𝑘+𝑖𝐺⁻¹𝒬^1/2_𝑝𝑗. (45) By virtue of Schur complements, we have ‖𝑢(𝑘+𝑖∣𝑘)‖₂≤𝑢_max,𝑘,𝑖≥0, if 𝑢²_max𝐼𝑌_𝑝𝑟_𝑘+𝑖𝑌^𝑇_𝑝𝑟_𝑘+𝑖𝐺𝒬_𝑝𝑗𝐺^𝑇≥0,∀𝑝∈𝕊,𝑗∈𝕃,𝑟_𝑘+𝑖∈ℐ. (46) Using the similar logic in [35], (46) is proved equivalent to (42). The proof of (43)-(44), which can be achieved by an analogous argument, is omitted for the sake of brevity.

Remark 6. The obtained theorems can be easily reduced to simple situations, for example, fuzzy systems with uncertainties and Markov jump systems, and can be solved via commercial solvers such as LMITOOL, YALMIP, and GloptiPoly.

4. Numerical Simulation

In this section, we present a numerical example that clearly illustrates the improvement obtained with Theorem 2. We will compare under different situations the improved method with the approach in [20] (modified in [22]) which can be seen as 𝑐_𝑘̂=0,𝑘>0 and 𝛽=1. Consider the single pendulum system: 𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)=−𝑐_1,𝑙sin𝑥₁(𝑡)+𝑐_2,𝑙𝑥₂(𝑡)+𝑢(𝑡)+0.1𝑤(𝑡), (47) where 𝑥₁(𝑡) is the angular displacement, 𝑥₂(𝑡) is the angular velocity, 𝑢(𝑡) is the control torque, 𝑤(𝑡) is the disturbance, and 𝑐_1,𝑙 and 𝑐_2,𝑙 are jump parameters with values 𝑐_1,1=1,𝑐_1,2=2, 𝑐_1,3=0.5, 𝑐_2,1=1, 𝑐_2,2=0.5, and 𝑐_2,3=2. The angular displacement 𝑥₁(𝑡) is assumed to vary in the intervals [−𝜋/2,𝜋/2]. This system can be represented as the following discrete-time fuzzy model with partly unknown transition probabilities: Π=∗0.4∗0.9∗∗0.250.40.35. (48) Set 𝑥(𝑘)=𝑥₁(𝑘)𝑥₂(𝑘)^𝑇, and choose the membership functions as ℎ₁𝑥₁(𝑘)=4𝑥²₁(𝑘)𝜋²,ℎ₂𝑥₁(𝑘)=1−ℎ₁𝑥₁(𝑘). (49) For the sake of simplicity, we use two T-S fuzzy rules to approximate this system. Plant Rule  1. If 𝑥₁(𝑘) is about 𝜋/2, then 𝑥(𝑘+1)=𝐴₁𝑟_𝑘𝑥(𝑘)+𝐵₁𝑟_𝑘𝑢(𝑘),𝑦(𝑘)=𝐶₁𝑟_𝑘𝑥(𝑘). (50) Plant Rule  2. If 𝑥₁(𝑘) is about −𝜋/2, then 𝑥(𝑘+1)=𝐴₂𝑟_𝑘𝑥(𝑘)+𝐵₂𝑟_𝑘𝑢(𝑘),𝑦(𝑘)=𝐶₂𝑟_𝑘𝑥(𝑘). (51)where 𝐴_𝑙𝑟_𝑘∈Ω𝑙,𝑟_𝑘,𝐵_𝑙𝑟_𝑘=0𝑇^𝑇,𝐶_𝑙𝑟_𝑘=11,𝑙=1,2,𝑟_𝑘Ω()=,,Ω()=,,Ω()=,,Ω()=,,Ω()=,,Ω()=,.=1,2,3,1,11𝑇−0.4530𝑇1+0.7116𝑇1+0.2𝑇𝑇−0.3530𝑇1+0.7116𝑇2,11𝑇−𝑇1+𝑇1+0.2𝑇𝑇−0.9𝑇1+𝑇1,21𝑇−0.9060𝑇1+0.0768𝑇1+0.2𝑇𝑇−0.8060𝑇1+0.0768𝑇2,21𝑇−2𝑇1+0.5𝑇1+0.2𝑇𝑇−1.9𝑇1+0.5𝑇1,31𝑇−0.2265𝑇1+1.8558𝑇1+0.2𝑇𝑇−0.1265𝑇1+1.8558𝑇2,31𝑇−0.5𝑇1+2𝑇1+0.2𝑇𝑇−0.4𝑇1+2𝑇 (52)

Our purpose here is to illustrate the advantages of the proposed method by comparing the optimal parameter 𝛾 for different situations. First of all, supposing 𝑐_𝑘̂=1/(𝑘+1)², 𝛽=0.5, and the initial states 𝑥(0)=[3𝜋/8;0.5], the steady-state responses of the closed-loop fuzzy MJS with input constraints ‖𝑢(𝑘+𝑖∣𝑘)‖₂≤3,𝑘>0,𝑖∈𝕄, are shown in Figure 1. Meanwhile, the optimal results of 𝛾 compared for different initial conditions are shown in Table 1. One may note that the average values of 𝛾 become much smaller when the additional parameters 𝑐_𝑘̂ and 𝛽 are introduced. Then, assuming initial states 𝑥(0)=[𝜋/4;0.5], 𝛽=0.5, we can obtain the corresponding values of 𝛾 for different 𝑐_𝑘̂ listed in Table 2. In the same way, by assuming initial states 𝑥(0)=[𝜋/4;0.5], 𝑐_𝑘̂=1/(𝑘+1)², the information about 𝛾 for different values of 𝛽 is given in Table 3. It is easy to observe from Tables 2 and 3 that the optimal performance is closely related to the two parameters.

Table 1 

The value of 𝛾 for different initial conditions with constraints.

Table 2 

The value of 𝛾 for different 𝑐_𝑘̂,𝑘>0 with constraints.

Table 3 

The value of 𝛾 for different 𝛽 with constraints.

(a)

(b)

Figure 1 

(a) State response of MJS system with 𝑥(0)=[3𝜋/8;0.5] and 𝑐𝑘̂=1/(𝑘+1)2, and 𝛽=0.5 for case 1, 𝑐𝑘̂=0, 𝛽=1 for case 2. (b) Input signals obtained based on Theorem 2.

5. Conclusion

In this paper, the problem of controller design based on MPC algorithm for uncertain systems is discussed. A relaxed scheme which has less conservativeness than traditional approaches is derived through introducing two additional parameters. Based on this scheme, a new set of criteria for model predictive controller design is obtained based on the fuzzy Markov jump system with partially unknown TPMs in an arbitrarily large horizon. A practical example is presented to show the effectiveness and applicability of the developed method. It is expected that the methods and ideas behind the paper could be extended to other systems or issues, such as filter design for the underlying system.

Conflict of Interests

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