# Robust Preview Control for Uncertain Discrete Singular Systems

Authors: Jiang Wu; Fucheng Liao; Yifan Shao; Shujie Gao; Hao Wang; Zhengguang Xu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, the problem of robust preview control for uncertain discrete singular systems is considered. First of all, by employing the forward difference for uncertain discrete singular systems, the singular augmented error system with the state vector, the input control vector, and the previewable reference signal is derived. Since there is a singular matrix in the system, the existing method cannot be directly applied to this problem. By considering the stability of the transposition system with Linear Matrix Inequality (LMI) method, a new stability criterion for the transposition system is introduced. Then, the robust controller for the augmented error system is obtained, which is regarded as the robust preview controller for the original singular system. At last, the numerical simulation shows the correctness and effectiveness of the results.


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## Body

## 1. Introduction

Since the singular system model [1] is proposed by Rosenbrock in the early 1970s, the singular system has been widely studied, and many results are obtained [2, 3]. Especially after the 1980s, singular systems, as an appropriate tool to deal with large-scale complex systems with multiobjective and multidimensional, have been applied to the fields of singularly perturbed theory [4], large-scale systems theory [5], circuit system [6], and so on.In practical, due to the modeling error, the measurement error, the linear approximation, and the change of working environment, the uncertainty of the system is objective. And they are presented as uncertain model parameters, system perturbation, measurement noise, external interference, etc. There is no doubt that any controller without considering the uncertainties may be difficult to achieve an ideal actual effect or may even cause the collapse of the system. Therefore, robust control is still regarded as one of the hotspot research directions in control theory and the applications. And the study of robust control theory and methods has formed several important branches, such as Lyapunov-Razumikhin method [7], Riccati inequality [8], Linear Matrix Inequality method [9], etc.For singular systems, uncertain problems also exist [10]. Therefore, the modeling, analysis, and design of uncertain singular systems are particularly important. The related research of uncertain singular systems began in the 1980s. The main purpose of the study is to design a controller to make the corresponding closed-loop system stable and meet certain performance targets, when there are model uncertainty and external interference in the system.Preview control is a control technique to improve the performance of systems by making full use of known future reference or disturbance signals in advance. Since the preview control model was first proposed in 1960s, the theoretical study and the applications of preview control have never stopped. In recent years, preview control has been extended to multisampling system [11], singular system [12], time-varying system [13], and so on. And it has been applied to many fields, such as automobile control [14], robot control [15], electromechanical valve control system [16], etc.Based on many results on the robust preview control problem of nonsingular system, the robust preview control problem of singular system is further discussed. Because of the existence of the singular matrix, a proper Lyapunov function cannot be found by the current method to design the robust preview controller. In this paper, a singular augmented error system is constructed by using forward differences. Then, based on the method of [17], a Lyapunov function for the transposition system of the closed-loop system for the singular augmented error system is designed. And by the LMI method, a stability criterion for the transposition system is obtained. Since the two unforced systems before and after transposition have the same stability, the robust controller for the singular augmented error system can be obtained by the transposition system. As for the original singular system, this controller is the robust preview controller.Notation. P>0(P<0) denotes the notion that matrix P is a positive definite (negative definite) matrix; I denotes the unit matrix; A∈Rm×n denotes a m×n matrix; the symbol ∗ denotes the symmetric terms in a symmetric matrix.


## 2. Problem Statement

Consider the following linear uncertain singular system:(1)Exk+1=A+ΔAxk+B+ΔBukyk=Cxkwhere x(k)∈Rn is the state vector, u(k)∈Rm is the input vector, and y(k)∈Rp is the output vector. E∈Rn×n is the singular matrix with rank⁡(E)=q<n. The matrices A, B, and C are known real constant matrices with appropriate dimensions. The matrices ΔA and ΔB are uncertain matrices.First of all, the definition about admissible is needed.Definition 1 (see [17]).
The dynamic system(2)Exk+1=Axkis said to be admissible if it is regular, causal, and stable.Then, the following assumption is needed for system (1).Assumption 2.
The matrices pair(A,B) is stabilizable and the matrix A-EBC0 is of full row rank.By denoting the previewable reference signalr(k), the following assumption is needed.Assumption 3.
The reference signalr(k) is previewable. That is, at any time k, the future values r(k+1),r(k+2),…,r(k+Mr), are available. When σ>k+Mr, the reference signal r(σ) is assumed to be unchanged, namely,(3)rk+j=rk+Mr,j=Mr+1,Mr+2,…where Mr is the preview length [18].About the uncertain matricesΔA and ΔB, the following assumption is needed.Assumption 4.
There exist real constant matricesEi, Hi(i=1,2) with appropriate dimensions and uncertain matrices Σi(i=1,2) satisfying(4)ΔA=E1Σ1H1,ΔB=E2Σ2H2,(5)ΣiTΣi≤I,i=1,2.Remark 5.
Equation (4) shows that the uncertain matrices ΔA and ΔB in system (1) satisfy matching conditions, and (5) shows that they are norm bounded.The purpose in this paper is to design a controller for system (1) with preview compensation to make the output y(k) track the reference value r(k) accurately when there exist disturbances ΔA and ΔB in system (1).


## 3. Construction of the Augmented Error System

The error equation can be defined as follows:(6)ek=yk-rk.Employing the difference operator as Δv(k)=v(k+1)-v(k) to (1) and (6), we can obtain(7)EΔxk+1=A+ΔAΔxk+B+ΔBΔukand(8)Δek=Δyk-Δrk.Furthermore, (8) can be rewritten as(9)ek+1=ek+Δek=ek+CΔxk-Δrk.Putting (7) and (9) together, we have(10)E0X0k+1=A0+ΔA0X0k+B0+ΔB0Δuk+BrΔrk,where(11)X0k=ekΔxk,E0=Im00E,A0=ImC0A,ΔA0=000ΔA,B0=0B,ΔB0=0ΔB,Br=-Im0.DefineXr(k)=Δr(k)⋮Δr(k+Mr-1). Since the future Mr reference signal r(i)(i=k+1,k+2,…,k+Mr) is known at time k, it follows from the Assumption 3 that Xr(k) satisfies(12)Xrk+1=ArXrk,where(13)Ar=0Im⋯00⋱⋱⋮⋮⋱Im0⋯⋯0,BPr=Br0⋯0.With (10) and (12), the augmented error system can be written as follows:(14)E^X^k+1=A^+ΔA^X^k+B^+ΔB^Δukwhere(15)X^k=X0kXrk,E^=E000IMr,A^=A0BPr0Ar,ΔA^=ΔA0000,B^=B00,ΔB^=ΔB00.According to the properties ofΔA and ΔB in Assumption 4, we have(16)ΔA^=E11Σ11H11,ΔB^=E22Σ22H22where(17)E11=0E10,Σ11=Σ1,H11=0H10E22=0E20,Σ22=Σ2,H22=H2The matrices Σii(i=1,2) still satisfy ΣiiΣiiT≤I,(i=1,2). Therefore, ΔA^ and ΔB^ are normal bounded.


## 4. Design of the Robust Preview Controller

In order to obtain the robust preview controller for system (1), the stability problem of the unforced discrete singular nominal system (2) should be considered first and the following lemmas are needed.Lemma 6 (see [17]).
The discrete singular system (2) is admissible if and only if there exists a matrix >0(P∈Rn×n), a symmetric matrix Φ∈R(n-q)×(n-q)such that(18)ATP-RTΦRA-ETPE<0,where RE=0. R∈R(n-q)×n is an annihilator basis of the range space of the matrix E and rank⁡(R)=n-q.If the matricesE and A are replaced with ET and ATin system (2), we have(19)ETxk+1=ATxkSince E is a singular matrix, there always exist nonsingular matrices P- and Q-, such that Q-EP-=Iq00N and Q-AP-=A100In-q according to the first restricted equivalent form in [19], where Nh=0 with a constant h. Substituting x(k)=P-x^(k) into (2) gives(20)EPx^k+1=AP-x^k.Premultiplying Q- on both sides of (20), we have(21)Q-EP-x^k+1=Q-AP-x^k. (21) can be rewritten as(22)Iq00Nx^qk+1x^n-qk+1=A100In-qx^qkx^n-qk.Concretely, we have that x^n-q(k)=0 is a stable solution for the second equation of (22). And the stability for the first equation of (22) is decided by the eigenvalues of A1.Similarly, substitutingx(k)=Q-Tx^(k) into (19), we have(23)ETQ-Tx^k+1=ATQ-Tx^k.Premultiplying P-T on both sides of (20), we have(24)P-TETQ-Tx^k+1=P-TATQ-Tx^k.Corresponding to (22), (24) can be rewritten as(25)Iq00NTx^qk+1x^n-qk+1=A1T00In-qx^qkx^n-qk.x^n-q(k)=0 is also a stable solution for the second equation of (25). And the stability for the first equation of (25) is decided by the eigenvalues of A1T, which are the same as A1. Therefore, system (19) is stable if and only if system (2) is stable.Through the definitions, it is obvious that system (19) is regular and casual if and only if system (2) is regular and casual. As a result, Lemma 7 can be obtained.Lemma 7.
The discrete singular system (2) is admissible if there exists a matrix P>0(P∈Rn×n) and a symmetric matrix Φ∈R(n-r)×(n-r) such that(26)AP-RΦRTAT-EPET<0where RE=0. R∈Rn×(n-q) is an annihilator basis of the range space of the matrix E and rank⁡(R)=n-q.Proof.
According to Lemma6, if (26) holds, we can get that system (19) is admissible. Because of the equality of the two systems, system (2) is also admissible and vice versa.Furthermore, Theorem8 can be obtained.Theorem 8.
The discrete singular system (2) is admissible if there exist a matrix P>0(P∈Rn×n), a symmetric matrix Φ∈R(n-q)×(n-q), and matrices Yi(i=1,2;Yi∈Rn×n) such that(27)Π=Π11∗Π21Π22<0where(28)Z=P-TΦTTΠ11=AY1+Y1TA-EPETΠ21=Y2TAT-Y1Π22=Z-Y2-Y2T,ET=0. T∈Rn×(n-q) is an annihilator basis of the range space of the matrix E and rank⁡(T)=n-q.Proof.
From (27) and (28), it is easy to see that(29)Π=M+ΛΥ+ΥTΛTwhere(30)M=-EPETZ,Λ=A-In,Υ=Y1Y2.Define H=InA. It is obvious that HΛ=0. Pre- and postmultiplying (29) by H and HT, from (27) we can get(31)HMHT=AZAT-EPET=HΠHT<0According to Lemma 7, it can be concluded that system (2) is admissible. This proves the theorem.In the following, the robust preview controller will be designed.Considering the state feedback input of system (14) described by(32)Δuk=LX^k,the closed-loop system is(33)E^X^k+1=A~+B~LX^kwhere A~=A^+ΔA^ and B~=B^+ΔB^.Next, the matrixL, which makes the closed-loop system (33) admissible, is to be obtained.Theorem 9.
The closed-loop discrete singular system (33) is admissible, if there exist a matrix P^>0(P^∈R(n+m+m×Mr)×(n+m+m×Mr)), a symmetric matrix Φ^(Φ^∈R(n-q+1)×(n-q+1)), a nonsingular matrix Y^(Y^∈R(n+m+m×Mr)×(n+m+m×Mr)), a matrix W∈Rp×(n+m+m×Mr), and some given scalars ρi(i=1,2), such that(34)Π^=Π^11∗Π^21Π^22<0where(35)Z^=P^-T^Φ^T^TΠ^11=-E^P^E^T+ρ1A~Y^+ρ1Y^TA~T+ρ1B~W+ρ1WTB~TΠ^21=-ρ1Y^+ρ2Y^TA~T+ρ2WTB~TΠ^22=Z^-ρ2Y^-ρ2Y^T,E^T^=0. T^∈R(n+m+m×Mr)×(n-q+1) is an annihilator basis of the range space of the matrix E^ and rank⁡(T^)=n-r+1. The state feedback controller (32) is Δu(k)=WY^-1X^(k).Proof.
Based on the results of Theorem8, replacing A with A~+B~L in (27) and (28), we have (34), where Π^ij(i,j=1,2) are defined as(36)Z^=P^-T^Φ^T^TΠ^11=A~Y^1+Y^1TA~T-E^P^E^T+B~LY^1+Y^1TLTB~TΠ^21=Y^2TA~T+Y^2TLTB~T-Y^1Π^22=Z^-Y^2-Y^2T,and Y^i∈R(n+m+m×Mr)×(n+m+m×Mr)(i=1,2). Since there exist some quadratic matrices variables, such as Y^iTLT(i=1,2), (34) is not a strict LMI based on the description of (36). To tackle the problem effectively, matrices Y^, W and some scalars ρi(i=1,2) are introduced just like the condition in [17]. Let Y^i=ρiY^ and W=LY^. Substituting ρiY^ into Y^i(i=1,2) and substituting W into LY^ in (36), we have (34) and (35). Because Y^ is nonsingular, we have L=WY^-1 and Δu(k)=WY^-1X^(k). This proves the theorem.Since the uncertain termsΔA^ and ΔB^ are contained in A~ and B~ in (35), the robust controller cannot be directly obtained by computing (34).Next, the robust preview controller of system (1) is gained. First, some other lemmas are needed.Lemma 10 (see [20]).
LetF and G be matrices of appropriate dimensions. Let Ξ=diag⁡(Ξ1,Ξ2,…,Ξs), where Ξ1,Ξ2,…,Ξs are uncertain matrices that satisfy ΞiTΞi≤I,(i=1,2,…,s). Then, for arbitrary positive scalars ε1,ε2,…,εs, we have(37)FΞG+GTΞTFT≤FΛ~FT+GTΛ~-1G,where Λ~=diag⁡(ε1I,ε2I,…,εsI).Lemma 11 (see [21]).
Considering the matrixS=S11S12S12TS22, where S11 and S22 are symmetric matrices and invertible, the following three conditions are equivalent:(1)
S<0;(2)
S11<0,S22-S12TS11-1S12<0;(3)
S22<0,S11-S12S22-1S12T<0.Then, we can get Theorem12.Theorem 12.
If there exist a matrixP^>0(P^∈R(n+m+m×Mr)×(n+m+m×Mr)), a symmetric matrix Φ^(Φ^∈R(n-q+1)×(n-q+1)), a nonsingular matrix Y^(Y^∈R(n+m+m×Mr)×(n+m+m×Mr)), a matrix W∈Rp×(n+m+m×Mr), and some given scalars ρi,(i=1,2), ε1>0, ε2>0 such that(38)Λ1∗∗∗Λ2Z^-ρ2Y^-ρ2Y^T∗∗ρ1H11Y^ρ2H11Y^-ε1I0ρ1H22Wρ2H22W0-ε2I<0where(39)Z^=P^-T^Φ^T^TΛ1=ρ1A^Y^+ρ1Y^TA^T-E^P^E^T+ρ1B^W+ρ1WTB^TΛ2=ρ2Y^TA^T+ρ2WTB^T-ρ1Y^,the controller(40)uk=Ke∑j=1kej-1+Kxxk+∑j=0Mr-1∑i=1kKrjΔri+j-1+u0-Kxx0is the robust preview controller for the discrete singular system (1).Proof.
From (34) and (35), we can get(41)ρ1A^+ΔA^Y^+ρ1Y^TA^+ΔA^T-E^P^E^T+ρ1B^+ΔB^W+ρ1WTB^+ΔB^T∗ρ2Y^TA^+ΔA^T+ρ2WTB^+ΔB^T-ρ1Y^Z^-ρ2Y^-ρ2Y^T=ρ1A^Y^+ρ1Y^TA^T-E^P^E^T+ρ1B^W+ρ1WTB^T∗ρ2Y^TA^T+ρ2WTB^T-ρ1Y^Z^-ρ2Y^-ρ2Y^T+ρ1E11Σ11H11Y^+ρ1E22Σ22H22Wρ2E11Σ11H11Y^+ρ2E22Σ22H22W00+ρ1Y^TE11Σ11H11T+ρ1WTE22Σ22H22T0ρ2Y^TE11Σ11H11T+ρ2WTE22Σ22H22T0Denote(42)Ψ=ρ1A^+ΔA^Y^+ρ1Y^TA^+ΔA^T-E^P^E^T+ρ1B^+ΔB^W+ρ1WTB^+ΔB^T∗ρ2Y^TA^+ΔA^T+ρ2WTB^+ΔB^T-ρ1Y^Z^-ρ2Y^-ρ2Y^T,Ω=ρ1A^Y^+ρ1Y^TA^T-E^P^E^T+ρ1B^W+ρ1WTB^T∗ρ2Y^TA^T+ρ2WTB^T-ρ1Y^Z^-ρ2Y^-ρ2Y^TΞ1=E11E2200,Ξ2=Σ1100Σ22,Ξ3=ρ1H11Y^ρ2H11Y^ρ1H22Wρ2H22WThen (41) can be rewritten as(43)Ψ=Ω+Ξ1Ξ2Ξ3+Ξ3TΞ2TΞ1TAccording to the Assumption 4, Ξ2Ξ2T≤I. Applying Lemma 10 to (43), we have(44)Ψ<Ω+Ξ1Λ-Ξ1T+Ξ3TΛ--1Ξ3where Λ-=ε1Iε2I,(ε1,ε2>0).
Therefore, if there existε1,ε2>0 satisfying Ω+Ξ1Λ-Ξ1T+Ξ3TΛ--1Ξ3<0, according to Lemma 11, namely, if(45)Λ^1∗∗∗Λ^2Z-ρ2Y^-ρ2Y^T∗∗ρ1H11Y^ρ2H11Y^-ε1I0ρ1H22Wρ2H22W0-ε2I<0where(46)Z^=P^-T^Φ^T^TΛ^1=ρ1A^Y^+ρ1Y^TA^T-E^P^E^T+ρ1B^W+ρ1WTB^T+ε1E11E11T+ε2E22E22TΛ^2=ρ2Y^TA^T+ρ2WTB^T-ρ1Y^.The state feedback controller for system (14) is(47)Δuk=WY^-1X^k.
LetL=WY^-1 be blocked into L=KeKxKr(1)⋯Kr(Mr), then (47) can be expressed as(48)Δuk=Keek+KxΔxk+∑i=0Mr-1KriΔrk+i.With regard to the discrete singular system (1), since Δu(k)=u(k+1)-u(k), we have(49)u1-u0=Kee0+KxΔx0+∑j=0Mr-1KrjΔrj,u2-u1=Kee1+KxΔx1+∑j=0Mr-1KrjΔrj+1,⋮uk-uk-1=Keek-1+KxΔxk-1+∑j=0Mr-1KrjΔrj+k-1.Add the above k equations together on both sides, and move u(0) to the right side of the equation. When k≥1, we have(50)uk=Ke∑j=1kej-1+Kxxk+∑j=0Mr-1∑i=1kKrjΔri+j-1+u0-Kxx0. (50) is the robust preview controller for system (1). This proves the theorem.Remark 13.
Since there exists the term∑j=0Mr-1∑i=1kKr(j)Δr(i+j-1) in (50), the future know information about the reference signal r(t) is considered in the robust controller.Remark 14.
Unlike the free weighting matrix method for normal system in [22], the difficulty of designing proper Lyapunov function for singular systems is overcome by introducing the method of [17].


## 5. Numerical Simulation

Consider the following system:(51)Exk+1=A+ΔAxk+B+ΔBukyk=Cxkwhere(52)A=01-1-1,B=0.21,C=0.50.1,ρ1=0.5,ρ2=1,E=1000,E1=0.150.30.30.15,E2=0.30.150.150.15,H1=0.150.150.150.15,H2=0.150.3,Σ1=10.5,Σ2=0.50.9.It is proved thatA-EBC0 is invertible and Σ1,Σ2 satisfy Assumption 4 for any k. So the system satisfies the fundamental assumptions.The reference signal is taken as(53)rk=1,k≥300,k<30The preview length isMr=0, Mr=1, and Mr=7, respectively. When Mr=0, Ke=0.8155, Kx=1.86653.4830. When Mr=1, Ke=0.8583, Kx=1.88903.5801, Kr=-0.7993-0.5449. When Mr=7, Ke=0.9344, Kx=1.93973.4174, Kr=[-0.9115-0.7297-0.3429-0.2242-0.1183-0.0682-0.0352-0.0175]. And we have Figures 1 and 2.Figure 1
The output response of the closed-loop system.Figure 2
The tracking error of the closed-loop system.Figure1 shows the output response for system (1). Figure 2 shows the tracking error. From the figures we can see that the output response with preview can track the reference signal much faster. And along with the preview length increasing, the overshoots reduced. Meanwhile, the maximum tracking errors decreased.As a comparison, the “nonrobust” controller is applied to the uncertain system (51). Similarly, the reference signal is also taken as(54)rk=1,k≥300,k<30and the preview length are Mr=0 and Mr=7, respectively. When Mr=0, Ke=2.8488, Kx=1.645610.8839. When Mr=7, Ke=1.2201, Kx=1.82653.7134, Kr=[-1.1122-0.8906-0.3268-0.1548-0.0588-0.0297-0.0111-0.0047]. Thus, we have Figures 3 and 4.Figure 3
The output response of the closed-loop system based on the robust controller and the “nonrobust” controller whenMr=0.Figure 4
The output response of the closed-loop system based on the robust preview controller and the “nonrobust” preview controller whenMr=7.Figures3 and 4 are the output response of the closed-loop system based on the robust controller and the “nonrobust” preview controller when Mr=0 and Mr=7, respectively. From these two figures, it can be easily seen that the overshoot of the output response based on the robust controller is smaller than that based on the nonrobust controller, which shows the effectiveness of the controller designed in this paper.


## 6. Conclusion

In this paper, the robust preview control problem for uncertain discrete singular systems is studied. At first, the singular augmented error system is constructed. Then, by considering the stability of the transposition system, some criterions are gained by using LMI method. At last, the robust controller for the singular augmented error system is obtained, which is the robust preview controller for the original uncertain discrete singular system. The numerical example shows the effectiveness of this paper.



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