# Model Predictive Control for Continuous-Time Singular Jump Systems with Incomplete Transition Rates

Authors: Xinxin Gu; Jiwei Wen; Li Peng
Journal: Journal of Control Science and Engineering
 (2015)

---

## Abstract

This paper is concerned with model predictive control (MPC) problem for continuous-time Markov Jump Systems (MJSs) with incomplete transition rates and singular character. Sufficient conditions for the existence of a model predictive controller, which could optimize a quadratic cost function and guarantee that the system is piecewise regular, impulse-free, and mean square stable, are given in two cases at each sampling time. Since the MPC strategy is aggregated into continuous-time singular MJSs, a discrete-time controller is employed to deal with a continuous-time plant and the cost function not only refers to the singularity but also considers the sampling period. Moreover, the feasibility of the MPC scheme and the mean square admissibility of the closed-loop system are deeply discussed by using the invariant ellipsoid. Finally, a numerical example is given to illustrate the main results.


---

## Body

## 1. Introduction

As a class of stochastic hybrid systems, Markov Jump Systems (MJSs) are suitable to describe many practical systems whose structures and parameters suffer from random abrupt changes [1–5]. Furthermore, MJSs, which contain a non-full rank system matrix, are usually referred to as singular MJSs, and they have obtained much attention from mathematical and control community. It should be mentioned that a singular MJS can better describe some physical systems than standard MJS model and has extensive applications, such as microelectronic circuits and economics [6–9].In recent years, an ever increasing number of scholars have paid much attention in the study of singular MJSs [6, 10–12]. In [13], Boukas investigated the state feedback control for singular MJSs with norm-bounded uncertainties. By considering the full and partial knowledge of transition rates, sufficient conditions which guarantee that the closed-loop system is piecewise regular, impulse-free, and mean square stable are given in [14]. Due to the regularity and pulse phenomenon, the stability and stabilizability of singular MJSs are intensively studied [10]. In [11], the authors aimed to study the mean square stability of singular MJSs and extended the main criterion to frequency domain. The robust mean square stability problem for uncertain singular MJSs with actuator saturation is further considered [15], in which sufficient conditions for such systems to be regular, causal, and mean square stable are derived in LMI form.On another research frontline, model predictive control (MPC), which is also named as receding horizon control, has received tremendous attention in practical applications [16–19]. At each time instant, MPC requires the online solution of an optimization problem and then gets the optimized control inputs. We calculate a control sequence, but only the first element is implemented. At the next sampling time, we resolve the optimization problem again. Therefore, the control gain is updated at each sampling time and can naturally compensate the model uncertainties and exogenous disturbance to some extent. MPC has been successfully employed in industry since it can cope with hard constraints, control moves, system states, and outputs. As MJS is considered, MPC strategy has also received considerable research efforts on its analysis and synthesis from a theoretical point of view [20–23].Almost all of the above-mentioned studies are based on the assumption that the transition probabilities are completely known. However, in practice, obtaining the complete knowledge of transition probabilities is difficult and costly. Until very recently, some efforts start to focus on MJSs with incomplete transition probabilities. Zhang and Boukas firstly proposed an analysis and design method for MJSs with partly unknown transition probability, which does not require the information of unknown elements [24]. Subsequently, necessary and sufficient conditions for the stability analysis of MJSs with incomplete transition descriptions are further given in [25]. Based on the results of [24, 25], Zheng and Yang [26] investigated the robust stabilization problem with a sliding mode control approach. Shen et al. proposed a new method for the H∞ state feedback control [27] and established sufficient conditions for the existence of the H∞ static output feedback controller [28].It is worth noting that MPC for MJSs are usually based on discrete-time version. However, in the real world, we take the controlled objects as continuous-time models. To the best of our knowledge, few results about continuous-time singular MJSs are addressed due to the difficulty in stability analysis and controller design, especially further considering incomplete transition rates. In this paper, a new MPC method is developed for continuous-time singular MJSs with incomplete transitions rates. Our main contributions include the following three aspects.(1) A sampled-period MPC law is introduced to deal with the continuous-time plant. (2) A modified Lyapunov function approach is employed to deal with analysis and design problems for the underlying system concerning incomplete information. The approach not only refers to the knowledge of system state but also considers the sampling period and the singularity. (3) The feasibility of the MPC scheme for the continuous-time singular MJS and the mean square admissibility of the closed-loop system are discussed by using invariant ellipsoid.


## 2. Problem Statement

Consider a continuous-time singular MJS with uncertainties described as follows:(1)Ex˙t=Art+ΔArtxt+Brt+ΔBrtut,where x(t)∈Rn is the system state, u(t)∈Rm is the control input, E is a real constant matrix with rank(E)=r(r<n), A(r(t)) and B(r(t)) are known mode-dependent constant matrices with appropriate dimensions, and ΔAr(t) and ΔBr(t) are matrices with unknown uncertainties satisfying(2)ΔArt=DFrt,tH1,ΔBrt=DFrt,tH2,where D∈Rn×p, H1∈Rq×n, and H2∈Rq×m are known constant matrices and F(r(t),t)∈Rp×q is a time-varying matrix function satisfying FT(r(t),t)F(r(t),t)≤I. {r(t),t≥0} is a continuous-time Markov process which takes values in a finite set M={1,2,…,N} with the following transition rate:(3)Prrt+h=j∣rt=i=λijh+oh,i≠j1+λijh+oh,i=j,λii=-∑j∈M,i≠jλij,λij≥0,where h>0 and limh→0⁡o(h)/h→0 and λij is the switching rate from mode i at time t to mode j at time t+h.For the sake of convenience,A(r(t)), B(r(t)), ΔA(r(t)), and ΔB(r(t)) could be denoted by Ai, Bi, ΔAi, and ΔBi, respectively, where r(t)=i∈M.Moreover, in this paper, the transition rates are considered to be partly available; namely, some elements in matrixΛ are unknown. Taking singular MJS (1) with four operation modes as an example, the transition rate matrix could be expressed as(4)Λ⌢=λ11λ12λ^13λ^14λ^21λ^22λ23λ24λ31λ^32λ33λ^34λ^41λ42λ^43λ44,where the elements with “^” represent the unknown transition rate.In this paper, we denoteM=Mki+Muki, ∀i∈M, with(5)Mki≜j:λijis  known,Muki≜j:λijis  unknown.In addition, ifMki≠ϕ, it can be written as (6)Mki=K1i,…,Kmi,∀1≤m≤N,where Kmi∈N+ represent the mth known element in the ith row of matrix Λ. Also, we denote λki≜∑j∈Mkiλij.In the case thatλ^ii is unknown, providing a lower bound λdi with λdi≤-λki is necessary.MPC for (1) is studied over an infinite horizon. At each sampling time kT for k=0,1,…, we could obtain control gains by making use of the LMI techniques and we will only implement the first calculated control input u(kT)=Fix(kT), where T represents the fixed sampling interval. At the next sampling time, we will calculate the state x(k+1)T and then compute the feedback control gain Fi once again. Let x(kT,kT) denote the state measured at sampling time kT; the predicted state at time kT+τ is denoted by x(kT+τ,kT); u(kT+τ,kT) is the control action at time kT+τ. x(kT,kT) can be noted as x(kT) for simplicity.About the singular MJS (1), we give the following definitions.Definition 1 (see [10]).
The continuous-time singular MJSs (1) are said to be(1)
regular ifdet(sE-Ai) is not identically zero for each i∈M;(2)
impulse-free ifdegdet(sE-Ai)=rank(E) for each i∈M;(3)
mean square stable if there exists a scalarM(x0,r0)>0, for any x0∈Rn, r0∈M, such that(7)E∫0∞xt2dt∣x0,r0≤Mx0,r0;(4)
mean square admissible if it is regular, impulse-free, and mean square stable.Definition 2 (see [29] (invariant ellipsoid)).
Given a continuous-time dynamical systemx˙(t)=fx(t), a subset E=x∈Rn∣xTQ-1x≤1 of the state space Rn is said to be an invariant ellipsoid, if it has the property that whenever x(kT)∈E, then x(NT)∈E for all times NT>kT.Before giving the main results of the singular MJS (1), some useful lemmas are presented below.Lemma 3 (see [30]).
Given matricesG1 and G2 with appropriate dimensions and a symmetric matrix Ω, Ω+G1F(t)G2+G2TF(t)TG1T<0, for all F(t)TF(t)≤I, if and only if there exists a scalar δ>0 such that(8)Ω+δG1G1T+1δG2TG2<0.Lemma 4 (see [31]).
Given symmetric and positive definite matricesM and N, if M≥N>0, then N-1≥M-1>0.Lemma 5 (see [14]).
If a mode-dependent symmetric matrixPi and a real constant matrix E with rank(E)=r(r<n) are given a priori, then we will have ETPi≤(1/4)ε~i-1I+ε~iETPiPiTE for any εi>0, where I presents the identity matrices with compatible dimensions.In this paper, at each sampling timekT, we aim to design a state feedback controller(9)ukT+τ,kT=FixkT+τ,kTτ≥0,by solving the following optimization problem:(10)
min
F
i
⁡
J
k
,
J
k
≔
∫
0
∝
x
T
k
T
+
τ
,
k
T
Q
c
x
k
T
+
τ
,
k
T
+
u
T
k
T
+
τ
,
k
T
R
u
k
T
+
τ
,
k
T
d
τ
,where Qc>0 and R>0 are symmetric weighting matrices, Jk is the performance index of system (1) at time kT, and Fi is the state feedback control gain.


## 3. Model Predictive Controller Design

In this section, we will design MPC controller for the singular MJSs with incomplete transition rates by presenting sufficient conditions, which could be efficiently solved by LMI toolbox.Theorem 6.
Consider the singular MJS (1) with incomplete transition rates and let x(kT) be the state of uncertain singular system (1) at sampling time kT. One will obtain Qi and Yi by solving the following optimization problem:(11)minγ,Qi,Yi,εi,δi⁡γsatisfying(12)1xTkTxTkTET∗4εiI0∗∗Qi+QiT-εiI≥0i∈M,(13)Ω1QiTYiTH1Qi+H2YiTΩ2Ω4-λkiQiT-λkiQiTET∗-γQc-1000000∗∗-γR-100000∗∗∗-δi0000∗∗∗∗-Ω3000∗∗∗∗∗-Ω500∗∗∗∗∗∗-4εjI0∗∗∗∗∗∗∗-Qj-QjT+εjI≤0∀j∈Muki,ifi∈Mki,(14)Ω1∗QiTYiTH1Qi+H2YiTΩ2Ω4-λdi-λkiQiT-λdi-λkiQiTET∗-γQc-1000000∗∗-γR-100000∗∗∗-δi0000∗∗∗∗-Ω3000∗∗∗∗∗-Ω500∗∗∗∗∗∗-4εjI0∗∗∗∗∗∗∗-Qj-QjT+εjI≤0∀j∈Muki,ifi∈Muki,where (15)Ω1=AiQi+BiYi+AiQi+BiYiT+δiDDT+λiiQiTET,Ω1∗=AiQi+BiYi+AiQi+BiYiT+δiDDT-λkiQiTET,Ω2=λiM1iQiT⋯λiMmiQiT,Ω3=diag⁡4εM1iI⋯4εMmiI,Ω4=λiM1iQiTET⋯λiMmiQiTET,Ω5=diag⁡QM1i+QM1iT-εM1iI⋯QMmi+QMmiT-εMmiI.The mode-dependent state feedback gain Fi is given by(16)Fi=YiQi-1.Proof.
Define a quadratic function(17)Vxt=xTtETPixt,ETPi=PiTE≥0.Here, we use Γ to denote the weak infinitesimal generator of random process x(t),i,t≥0 for each mode i∈M; it is defined as(18)
Γ
V
x
t
,
r
t
=
i
=
lim
δ
→
0
+
⁡
1
δ
E
V
x
t
+
δ
,
r
t
+
δ
∣
x
t
,
r
t
=
i
-
V
x
t
,
i
.At each sampling time kT, suppose that V satisfy the following inequality:(19)
Γ
V
x
k
T
+
τ
,
k
T
,
i
≤
-
x
T
k
T
+
τ
,
k
T
Q
c
x
k
T
+
τ
,
k
T
+
u
T
k
T
+
τ
,
k
T
R
u
k
T
+
τ
,
k
T
.Because the robust cost function will eventually become x(∝,kT)=0,  we will get(20)Jk≤VxkT,i=xTkTETPixkT,by integrating both the left and the right sides of inequality (19) with τ∈0∝.
Considering Lemma5, for any ε~i≥0, we will have(21)VxkT,i=xTkTETPixkT≤xTkT14ε~i-1I+ε~iETPiPiTExkT;let(22)xTkT14ε~i-1I+ε~iETPiPiTExkT≤γ,ε~i=γ-1εi,Pi=γQi-1;then, we have (23)1-xTkT14εi-1IxkT-xTkTεiETQi-1Qi-TExkT≥0.Using the fact that εi-1QiQiT≥Qi+QiT-εiI for any εi>0 and Lemma 4, we will obtain(24)εiQi-1Qi-T≤Qi+QiT-εiI-1.Obviously, via the well-known Schur complement, it is easy to get that conditions (12) and (24) imply (23).
Applying the control law (9) and the definition of weak infinitesimal generator, inequality (19) becomes(25)Ai+ΔAi+Bi+ΔBiFiTPi+PiTAi+ΔAi+Bi+ΔBiFi+∑j∈MλijETPj+Qc+FiTRFi≤0.Pre- and postmultiplying (25) by Pi-T and its transpose, replacing Pi and Yi, by γQi-1 and FiQi, respectively, one will get(26)AiQi+BiYi+AiQi+BiYiT+ΔAiQi+ΔBiYi+QiTΔAiT+YiTΔBiT+γ-1∑j∈MQiTλijETPjQi+γ-1QiTQcQi+γ-1YiTRYi≤0;considering the unknown uncertainty (2), (26) becomes(27)AiQi+BiYi+AiQi+BiYiT+DFitH1Qi+H2Yi+H1Qi+H2YiTFiTtDT+γ-1∑j∈MQiTλijETPjQi+γ-1QiTQcQi+γ-1YiTRYi≤0.According to Lemma 3, (27) holds if and only if there exists δi>0 satisfying (28)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1∑j∈MQiTλijETPjQi+γ-1QiTQcQi+γ-1YiTRYi≤0.According to Lemma 5, (28) holds if (29) is satisfied for any ε~j>0 satisfying(29)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1∑j∈MQiTλij14ε~j-1I+ε~jETPjPjTEQi+γ-1QiTQcQi+γ-1YiTRYi≤0.Replacing Pj and ε~j-1 by γQj-1 and γεj-1, respectively, we will have (30)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+∑j∈M14λijQiTεj-1IQi+∑j∈MλijQiTETQj-1Qj-TεjEQi+γ-1QiTQcQi+γ-1YiTRYi≤0.Subsequently, we take transition rate into account and separate it into two cases, i∈Mki and i∈Muki, respectively.
Case 1
(
i
∈
M
k
i
)
. Considering that ∑j∈Mλij=∑j∈Mkiλij+∑j∈Mukiλij, the left side of (30) is rewritten as(31)θi≜AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi+αki+14∑j∈Mukiλ^ijQiTεj-1IQi+βki+∑j∈Mukiλ^ijQiTETQj-1Qj-TεjEQi,where the elements λ^ij (∀j∈Muki) are unknown, αki≜(1/4)∑j∈MkiλijQiTεj-1IQi, and βki≜∑j∈MkiλijQiTET(Qj-1Qj-Tεj)EQi. Noting that, in this case, λii is known, λki=λii+∑j∈Mki,i≠j<0, and ∑j∈MQiTλijET(Qj-1)Qi=λiiQiTET, j=i, then(32)θi=AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi+λiiQiTET+αki∗+14∑j∈Mukiλ^ij-λki-λkiQiTεj-1IQi+βki∗+∑j∈Mukiλ^ij-λki-λkiQiTETQj-1Qj-TεjEQi,where αki∗=(1/4)∑j∈Mki,j≠iλijQiTεj-1IQi and βki∗=∑j∈Mki,j≠iλijQiTET(Qj-1Qj-Tεj)EQi, because of 0≤λ^ij/-λki≤1 and ∑j∈Mukiλ^ij/-λki=1; we will obtain that(33)
θ
i
=
∑
j
∈
M
u
k
i
λ
^
i
j
-
λ
k
i
A
i
Q
i
+
B
i
Y
i
+
A
i
Q
i
+
B
i
Y
i
T
+
δ
i
D
D
T
+
1
δ
i
H
1
Q
i
+
H
2
Y
i
T
H
1
Q
i
+
H
2
Y
i
+
γ
-
1
Q
i
T
Q
c
Q
i
+
γ
-
1
Y
i
T
R
Y
i
+
λ
i
i
Q
i
T
E
T
+
α
k
i
∗
+
-
λ
k
i
Q
i
T
E
T
Q
j
-
1
Q
j
-
T
ε
j
E
Q
i
+
1
4
-
λ
k
i
Q
i
T
ε
j
-
1
I
Q
i
+
β
k
i
∗
.Since θi<0, we can have the following inequality:(34)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi+αki∗+14-λkiQiTεj-1IQi+βki∗+-λkiQiTETQj-1Qj-TεjEQi+λiiQiTET<0.Using (24), we can easily get (34), if(35)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+λiiQiTET+γ-1QiTQcQi+γ-1YiTRYi+αki∗+∑j∈Mki,j≠iλijQiTETQj+QjT-εjI-1EQi+14-λkiQiTεj-1IQi+-λkiQiTETQj+QjT-εjI-1EQi<0holds. Applying Schur complement to the above inequality, one can obtain (13).
Case 2
(
i
∈
M
u
k
i
)
. Likewise, considering ∑j∈Mλij=∑j∈Mkiλij+∑j∈Mukiλij, we rewrite the left side of (30) as(36)θi≜AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi+λ^iiQiTET+14∑j∈MkiλijQiTεj-1IQi+14∑j∈Muki,j≠iλ^ijQiTεj-1IQi+∑j∈MkiλijQiTETQj-1Qj-TεjEQi+∑j∈Muki,j≠iλ^ijQiTETQj-1Qj-TεjEQi.Noting that, in this case, λ^ii is unknown, λki=-λ^ii-∑j∈Muki,j≠iλij>0, then(37)θi=AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi+αki+14-λ^ii-λki·∑j∈Muki,j≠iλ^ij-λ^ii-λkiQiTεj-1IQi+-λ^ii-λki·∑j∈Muki,j≠iλ^ij-λ^ii-λkiQiTETQj-1Qj-TεjEQi+λ^iiQiTET+βki.Similarly, since we have 0≤λ^ij/(-λ^ii-λki)≤1 and ∑j∈Muki,j≠i(λ^ij/-λ^ii-λki)=1, we can easily obtain(38)
θ
i
=
∑
j
∈
M
u
k
i
,
j
≠
i
λ
^
i
j
-
λ
^
i
i
-
λ
k
i
A
i
Q
i
+
B
i
Y
i
+
A
i
Q
i
+
B
i
Y
i
T
+
δ
i
D
D
T
+
1
δ
i
H
1
Q
i
+
H
2
Y
i
T
H
1
Q
i
+
H
2
Y
i
+
γ
-
1
Q
i
T
Q
c
Q
i
+
γ
-
1
Y
i
T
R
Y
i
+
λ
^
i
i
Q
i
T
E
T
+
α
k
i
+
β
k
i
+
-
λ
^
i
i
-
λ
k
i
Q
i
T
E
T
Q
j
-
1
Q
j
-
T
ε
j
E
Q
i
+
1
4
-
λ
^
i
i
-
λ
k
i
Q
i
T
ε
j
-
1
I
Q
i
.Therefore, θi<0 is equivalent to ∀j∈Muki, j≠i,(39)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi+λ^iiQiTET+αki+14-λ^ii-λkiQiTεj-1IQi+-λ^ii-λkiQiTETQj-1Qj-TεjEQi+βki<0.As λ^ii is lower bounded by λdi, we know that λdi≤λ^ii<-λki. Thus, (39) holds if the following inequality holds:(40)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi-λkiQiTET+αki+14-λdi-λkiQiTεj-1IQi+-λdi-λkiQiTETQj-1Qj-TεjEQi+βki<0.Similarly, (40) is implied by (24) and(41)AiQi+BiYi+AiQi+BiYiT+δiDDT+1δiH1Qi+H2YiTH1Qi+H2Yi+γ-1QiTQcQi+γ-1YiTRYi-λkiQiTET+αki+14-λdi-λkiQiTεj-1IQi+-λdi-λkiQiTETQj+QjT-εjI-1EQi+∑j∈MkiλijQiTETQj+QjT-εjI-1EQi<0.Applying Schur complement to (41), we can obtain (14).
Certainly, MPC for singular MJSs without incomplete transition descriptions and the normal MJS with incomplete transition descriptions can be viewed as two special cases of singular MJS. Then, we have the following corollaries.Corollary 7.
Consider the singular MJS (1) without incomplete transition rate and let x(kT) be the state of singular system (1) at sampling time kT. One will obtain Qi and Yi by solving the following optimization problem:(42)minγ,Qi,Yi,εi,δi⁡γsatisfying (43)1xTkTxTkTET∗4εiI0∗∗Qi+QiT-εiI≥0i∈M,Ω1QiTYiH1Qi+H2YiTΩ2Ω4∗-γQc-10000∗∗-γR-1000∗∗∗-δi00∗∗∗∗-Ω30∗∗∗∗∗-Ω5≤0i∈M,where (44)Ω1=AiQi+BiYi+AiQi+BiYT+δiDDT+λiiQiTET,Ω2=λi1QiT⋯λi,i-1QiTλi,i+1QiT⋯λi,NQiT,Ω3=diag⁡4ε1I⋯4εi-1I4εi+1I⋯4εNI,Ω4=λi1QiTET⋯λi,i-1QiTETλi,i+1QiTET⋯λi,NQiTET,Ω5=diag⁡Q1+Q1T-ε1I⋯Qi-1+Qi-1T-εi-1IQi+1+Qi+1T-εi+1I⋯QN+QNT-εNI.Then, the mode-dependent state feedback gain Fi can be obtained by (16).Corollary 8.
WhenE is nonsingular and can be simplified as an identity matrix, system (1) becomes (45)x˙t=Art+ΔArtxt+Brt+ΔBrtut;let x(kT) be the state of uncertain MJS (45) at sampling time kT. One will obtain Qi and Yi by solving the following optimization problem:(46)minγ,Qi,Yi,δi⁡γsatisfying(47)1xTkT∗Qi≥0,i∈M,Ω1QiTYiTH1Qi+H2YiTΩ2-λkiQiT∗-γQc-10000∗∗-γR-1000∗∗∗-δi00∗∗∗∗-Ω30∗∗∗∗∗-Qj≤0∀j∈Muki,ifi∈Mki,Ω1∗QiTYiTH1Qi+H2YiTΩ2-λdi-λkiQiT∗-γQc-10000∗∗-γR-1000∗∗∗-δi00∗∗∗∗-Ω30∗∗∗∗∗-Qj≤0∀j∈Muki,ifi∈Muki,where (48)Ω1=AiQi+BiYi+AiQi+BiYT+δiDDT+λiiQiT,Ω1∗=AiQi+BiYi+AiQi+BiYT+δiDDT+λkiQiT,Ω2=λiM1iQiT⋯λiMmiQiT,Ω3=diag⁡QM1i⋯QMmi.
The mode-dependent state feedback gainFi is obtained by (16).


## 4. Mean Square Admissibility Analysis

Previous section gives sufficient conditions, which could guarantee the existence of model predictive controller. In this section, we mainly study the mean square stability of the closed-loop singular MJS (1).Lemma 9 (feasibility).
The solution of the optimization problem in Theorem6 is feasible over any time interval NT,(N+1)T with N>k, if it has the property that the solution is feasible over the time interval kT,(k+1)T.Proof.
At sampling timekT, from inequality (12), we limit the measured state x(kT) in the ellipsoid region xT[(1/4)εi-1I+ET(Qi+QiT-εiI)-1E]x≤1. According to the proof of Theorem 6, we can get(49)ΓVxt,rt=i<0.Then, for any sampling time NT with N>k, the future state xNT remains in the ellipsoid set. Therefore, by referring to Definition 2, xT[(1/4)εi-1I+ET(Qi+QiT-εiI)-1E]x≤1 is an invariant ellipsoid.
The solution of the optimization problem in Theorem6 is feasible over the time interval kT,(k+1)T. By observing (12), (13), and (14), it could be concluded that the feasibility of the MPC strategy only depends on (12). Thus, to prove Lemma 3, we only need to prove that the LMI (12) is feasible for all future states x(NT), N>k. Considering the invariance of the ellipsoid, we obtain(50)xTk+1T14εi-1I+ETQi+QiT-εiI-1E·xk+1T≤1,which means LMI (12) is feasible with state x(k+1)T and then the feasibility of optimization problem (11) could be achieved over the time interval (k+1)T,(k+2)T. Similarly, we can complete the proof on the time intervals (k+2)T,(k+3)T, (k+3)T,(k+4)T,….Theorem 10.
The feedback predictive control gain computed from Theorem6 guarantees that the closed-loop singular MJS (1) is mean square admissible.Proof.
Firstly, we show that the singular MJS (1) is regular and impulse-free. There exist two orthogonal matrices U=U1U2 and V=V1V2 such that (51)E-=I000=UTEV-T.
Based on singular value decomposition, system (1) can be written as the following differential and algebraic form:(52)x˙1t=Aic1∗x1t+Aic2∗x2t,0=Aic3∗x1t+Aic4∗x2t,where Aic=(Ai+ΔAi)+(Bi+ΔBi)Fi, and it can be decomposed as Aic=Aic1Aic2Aic3Aic4.
Assuming the optimal solution of optimization problem (11) at sampling time kT can be denoted as Pi∗kT and Fi∗(kT), from the proof of Theorem 6, we can obtain that there exists Pi∗(kT) satisfying (25) and(53)ETPi∗kT=PiT∗kTE≥0.
Let(54)A-i=Aic1∗Aic2∗Aic3∗Aic4∗=UTAic∗V-T,P-i∗kT=Pi1∗kTPi2∗kTPi3∗kTPi4∗kT=UTPi∗kTV-T,Q-c=Qc1Qc2Qc2TQc3=V-1QcV-T,F-i∗kT=Fi1∗kTFi2∗kT=Fi∗kTV-T.By (53), we have Pi1∗(kT)>0 (because Pi∗(kT) is nonsingular) and Pi2∗(kT)=0. Pre- and postmultiplying the left and right sides of (25) by V-1 and V-T, we obtain(55)Aic4T∗Pi4∗kT+Pi4T∗kTAic4∗+Qc3+Fi2T∗kTRFi2∗kT<0.
From (55), we can easily get that Aic4T∗ is invertible. Hence, for any t≥0,(56)det⁡sE-Aic∗=det⁡U-Tdet⁡VTdet⁡Aic4∗·det⁡sI-Aic1∗-Aic2∗Aic4-1∗Aic3∗,deg⁡det⁡sE-Aic∗=rankE,
from which we can see thatdetsE-Aic∗ is not identically zero; by Definition 1, the closed-loop system is regular and impulse-free.
What is more, we consider the mean square stability of the singular MJS (1). Here, the piecewise Lyapunov function is described as (57)V∗xt,i=xTtETPi∗kTxt,t∈kT,k+1T.
From (25), we have(58)Ai+ΔAi+Bi+ΔBiFiTPi∗+PiT∗Ai+ΔAi+Bi+ΔBiFi+∑j∈MλijETPj∗≤0which yields(59)ΓV∗xt,i<0,t∈kT,k+1T.Therefore, Vx(t),i is strictly decreasing over each time interval kT,(k+1)T.
Note thatPi∗(kT) represents a feasible solution at time (k+1)T while Pi∗(k+1)T is an optimal solution at time (k+1)T. Since V∗x(t),i is a piecewise function, we further consider(60)limt→k+1T-⁡V∗xt,i=xTk+1TEPi∗kTxk+1T≥xTk+1TEPi∗k+1Txk+1T=V∗xk+1T,for all k≥0. Inequalities (59) and (60) guarantee that Vx(t),i is strictly decreasing when t∈0,∝; system (1) is mean square stable.
Therefore, according to the property of being regular and impulse-free, the closed-loop singular MJS is mean square admissible.


## 5. Numerical Example

To illustrate the efficiency of the proposed MPC scheme for continuous-time MJSs, a numerical example is presented in the following.Consider a system with the form of (1), where r(t)=1,2,3,4. The system matrices are A1=0.75-0.750-1.5, A2=0.15-0.490-2.1, A3=0.3-0.150-1.8, A4=0.9-0.340-1.65, B1=5-1, B2=2-1, B3=1-1, B4=3-1, D=0.10, E=1000, H1=0.10, and H2=0.1. The weighting matrices are Qc=1001 and R=1.The TRM is given as follows:(61)Λ=-1.30.2λ^13λ^14λ^21λ^220.30.30.1λ^32-2.5λ^34λ^410.2λ^43-1.2,where the diagonal element λ^22 is unknown and its lower bound λd2=-5 is a priori given value. The initial value of the state is x0=-1.20.6T and the initial mode is r0=1.The simulation step is taken as 150 time units and each unit length is taken as 0.01. We get the jump mode (Figure1) and apply the designed MPC algorithm and normal state feedback control to system (1), separately. Then, we get the simulation results in Figures 2–5.Figure 1
Jump mode.Figure 2
State responses under MPC.Figure 3
State responses under normal state feedback control.Figure 4
Control input under MPC.Figure 5
Control input under normal state feedback control.Obviously, the unstable system (1) becomes more easily stable by using MPC strategy.


## 6. Conclusions

In this paper, MPC strategy is presented for continuous-time singular MJSs with incomplete transition rates. The controller design problem is formulated as LMI optimization algorithm. The predictive control strategy is proved to be feasible at every sampling time and it also can guarantee the mean square admissibility of the closed-loop system.



---

*Source: 140527-2015-08-17.xml*