# Finite-Time Bounded Tracking Control for Linear Discrete-Time Systems

Authors: Fucheng Liao; Yingxue Wu; Xiao Yu; Jiamei Deng
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

A finite-time bounded tracking control problem for a class of linear discrete-time systems subject to disturbances is investigated. Firstly, by applying a difference method to constructing the error system, the problem is transformed into a finite-time boundedness problem of the output vector of the error system. In fact, this is a finite-time boundedness problem with respect to the partial variables. Secondly, based on the partial stability theory and the research methods of finite-time boundedness problem, a state feedback controller formulated in form of linear matrix inequality is proposed. Based on this, a finite-time bounded tracking controller of the original system is obtained. Finally, a numerical example is presented to illustrate the effectiveness of the controller.


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## Body

## 1. Introduction

In 1961, Dorato proposed the concept of finite-time stability (FTS) in [1]. The main concept is that if the bound of the initial condition is given, the state of the system does not exceed a certain bound over a given finite time-interval. Since then, many scholars have conducted indepth research on FTS. In the 1960s, Kushner investigated the FTS of stochastic systems in [2]. Weiss and Infante discussed the FTS of nonlinear systems in [3, 4]. However, due to the lack of effective mathematical tools at that time, the research progress is relatively slow.With the development of linear matrix inequality (LMI) theory, the research on FTS yielded fruitful results. In [5, 6], Amato et al. extended the concept of FTS to the linear continuous-time system with external disturbances and presented the concept of finite-time boundedness (FTB). The FTB of time-varying continuous-time systems was discussed in [7]. Subsequently, the discrete-time system was investigated in [8, 9] and further research was done in [10–12]. In [10], the state feedback controller and output feedback controller were designed to guarantee the FTB of the discrete-time system with disturbance. In [11], the FTS of discrete systems was analyzed by using polyhedral Lyapunov function. In [12], the sufficient conditions for FTS of time-varying discrete systems were given and an output feedback controller was developed.Following the pioneering work of Amato et al., many scholars extended the research of FTB of discrete-time systems. In [13], the finite-time control for linear discrete-time system with external disturbances was studied. The FTS of discrete-time stochastic systems with time-varying delays and its application to multiagent systems were considered in [14]. In [15], a finite-time optimal control method for a class of linear discrete-time systems with parameter variation was presented. By constructing the Lyapunov-Krasovskii functional, the FTS of discrete time-delay systems with nonlinear perturbations was studied in [16]. In [17], a robust controller was proposed to address the finite-time control problem of linear uncertain discrete systems by using an augmented LMI method. In [18], the FTS and H∞ control problem of discrete-time systems were discussed and a robust finite-time control scheme was provided.On the basis of [13], the tracking control problem of linear discrete-time systems with disturbances in a finite time-interval is considered in this paper. Firstly, the error system is constructed based on the preview control theory [19, 20], and the problem is turned into a FTB problem of the output vector of the error system. Then a state feedback controller is designed for the error system via the LMI approach. Finally, a finite-time state feedback controller of the original system is derived.Throughout this paper, the following notations are adopted. MatrixP>0 (or P<0) means that P is symmetric positive definite (or negative definite). P≥0 (P≤0) means that P is symmetric positive semidefinite (or negative semidefinite). P>Q (P<Q, P≥Q, and P≤Q) means that P-Q>0 (P-Q<0, P-Q≥0, and P-Q≤0). λmax(A) (λmin(A)) denotes the maximal (or minimal) eigenvalue of a real symmetric matrix A. diag⁡… denotes a block-diagonal matrix.


## 2. Preliminaries and Basic Concepts

This paper considers the following linear discrete-time system:(1)xk+1=Axk+Ewk,where x(k)∈Rn and w(k)∈Rp are the state vector and the disturbance vector of the system, respectively. A∈Rn×n and E∈Rn×p are known constant matrices.In [10–13], the FTB problem of system (1) was investigated and its basic definition was described as follows: system (1) is said to be finite-time bounded with respect to (δ,d,ε,R,N), where N≥1, d>0, ε>0, δ>0, and R>0, if(2)xT0Rx0≤δ2,∑k=0NwTkwk≤d2⟹xTkRxk≤ε2,∀k∈1,2,⋯,N.For convenience, hereinafter, the state vector of system (1) is also said to be finite-time bounded with respect to (δ,d,ε,R,N). The object of this paper is to generalize this concept and further study the finite-time bounded tracking problem of control systems. In the following, we first propose a definition of finite-time bounded tracking. Consider the discrete-time system(3)xk+1=Axk+Ewk,yk=Cxk,where x(k)∈Rn, w(k)∈Rp, y(k)∈Rq, A∈Rn×n, and E∈Rn×p.In some practical problems, it is hoped that the output of system (3) is always located in a ε neighborhood of a reference signal under some certain conditions. This kind of problem is referred to as “finite-time bounded tracking problem.” Let the reference signal be r(k)∈Rq. And the error signal e(k) is defined as(4)ek=yk-rk.The concept mentioned above can be described by the following definition.Definition 1.
System (3) achieves finite-time bounded tracking of the reference signal rk with respect to (δ,d,ε,R,N), where N≥1, δ>0, d>0, ε>0, and R>0, if(5)eT0Re⁡0≤δ2,∑k=0NwTkwk≤d2⟹eTkRe⁡k≤ε2,∀k∈1,2,⋯,N.Remark 2.
The conclusion of Definition1 is equivalent to the fact that the error signal e(k) is finite-time bounded with respect to (δ,d,ε,R,N); that is, the output y(k) of system (3) is always located in the ε neighborhood of the reference signal r(k) within a given time-interval 1,2,⋯,N.
In [13], the sufficient conditions for FTB of system (1) with respect to (δ,d,ε,R,N) were presented in terms of LMI. In this paper, the research methods in [13] will be modified and combined with the error system method in preview control theory to study the finite-time bounded tracking problem.
The Schur complement lemma is needed to deduce an LMI feasibility problem.Lemma 3 (see [21]).
Symmetric matrixS11S12S12TS22<0 if and only if one of the following two conditions is satisfied:(1)
S11<0,S22-S12TS11-1S12<0.(2)
S22<0,S11-S12S22-1S12T<0.


## 3. Problem Description

Let us consider the linear discrete-time system with disturbance(6)xk+1=Axk+Buk+Ewk,yk=Cxk,where x(k)∈Rn, u(k)∈Rm, w(k)∈Rp, and y(k)∈Rq are the state vector, the input vector, the disturbance vector, and the output vector of the system, respectively. A∈Rn×n, B∈Rn×m, E∈Rn×p, and C∈Rq×n are known constant matrices.The difference operatorΔ is defined as(7)Δvk=vk-vk-1.The reference signal isr(k)∈Rq, and the error signal is defined by (4). The assumptions on disturbance signal and reference signal of system (7) are presented as follows:A1:Assume that the disturbance vector satisfies the condition, ∑j=1NΔwT(j)Δw(j)≤d12, where d1>0.A2:Assume that the reference signal satisfies the condition, ∑j=1NΔrT(j)Δr(j)≤d22, where d2>0.The purpose of this paper is to design a controller with preview action for linear discrete-time system (6) so that the closed-loop system can achieve the finite-time bounded tracking of the reference signal r(k) with respect to (δ,d,ε,R,N).To achieve the above objective, an error system that includes the information of the error signale(k) will be first constructed. Then, the error signal is considered as the output vector of this system. By this means, the original problem is converted into a FTB problem of the output vector of the error system.


## 4. Derivation of the Error System

Taking the operatorΔ on both sides of the first equation of (6), it follows that(8)Δxk+1=AΔxk+BΔuk+EΔwk.Applying Δ to e(k+1)=y(k+1)-r(k+1) and noting that Δe(k+1)=e(k+1)-e(k), the following is obtained:(9)ek+1=ek+Δyk+1-Δrk+1=ek+CΔxk+1-Δrk+1=ek+CAΔxk+CBΔuk+CEΔwk-Δrk+1.Introduce the formal state vector X0k=e(k)Δx(k) and matrices Φ=ICA0A, G=CBB, Gr=-I0, and Gw=CEE. Combining (8) and (9) yields(10)X0k+1=ΦX0k+GΔuk+GrΔrk+1+GwΔwk.Define the new output(11)ek=C0X0k,where C0=I0. Thus, we can obtain the following error system:(12)X0k+1=ΦX0k+GΔuk+GrΔrk+1+GwΔwk,ek=C0X0k.Sincey(k)=Cx(k) is the output equation of system (6), y(k) is measurable. Moreover, the reference signal r(k) is known in advance. Thus, it is reasonable to consider e(k) as the output vector of system (10).Based on the above discussion, the finite-time bounded tracking problem of system (6) is transformed into the FTB problem of the output vector e(k) of the closed-loop system of error system (12).


## 5. Design of the Controller

Let us consider the following state feedback controller:(13)Δuk=KX0k,where K=KeKx will be determined later. Applying this controller to system (12) results in(14)X0k+1=Φ+GKX0k+GrΔrk+1+GwΔwk,ek=C0X0k.Compared with system (3), it can be seen that system (14) is exactly same as system (3) except GrΔrk+1. Hence, Δr(k+1) can be treated as the external disturbance. Putting GrΔrk+1 and GwΔw(k) together, a new disturbance vector Wk=Δw(k)Δr(k+1) is obtained. In this way, the closed-loop system (14) becomes(15)X0k+1=Φ+GKX0k+E-Wk,ek=C0X0k,where E-=GwGr.Remark 4.
System (15) is now fully in the form of system (3), which will facilitate the controller design. Since system (15) contains disturbances Δr(k+1) and Δw(k), the corresponding assumptions can be relaxed to A1 and A2. For w(k), it is easy to prove that A1 is much weaker than that of ∑k=0NwT(k)w(k)≤d12. In fact, if ∑k=0NwT(k)w(k)≤d12/4, then A1 is satisfied. This is because(16)∑k=1NΔwTkΔwk=∑k=1Nwk-wk-1Twk-wk-1≤∑k=1Nwk+wk-12=∑k=1Nwk2+wk-12+2wkwk-1≤∑k=1N2wk2+2wk-12=∑k=1N2wTkwk+2wTk-1wk-1=2wT0w0+2wTNwN+4∑k=1N-1wTkwk≤4∑k=0NwTkwk≤d12.
So far, the original problem has been converted into a FTB problem of partial variablee(k) of system (15). The conclusion of [13] cannot be directly applied to system (15). Therefore, it is necessary to combine relative ideas on partial stability with the proof methods in [13] to obtain the results of this paper. The following Theorem 5 is the first main result of this paper.Theorem 5.
The closed-loop system (15) achieves finite-time bounded tracking of the reference signal r(k) with respect to (δ,d,ε,R,N), if for a given scalar γ>1, there exist matrices P1>0, P2>0 and scalars λ1>0, λ2>0 such that(17)Φ+GKTC0TP1C0Φ+GK-γC0TP1C0Φ+GKTC0TP1C0E-E-TC0TP1C0Φ+GKE-TC0TP1C0E--γP2≤0,(18)R<P1<λ1R,(19)0<P2<λ2I,(20)λ1δ2+λ2d2<ε2γN-1.Moreover, the controller is Δu(k)=KX0(k).Proof.
Construct the following Lyapunov function:(21)Vek=eTkP1ek.Due to P1>0, V(e) is a positive-definite quadratic form with respect to e. Then, with some mathematical operations, we have(22)Vek+1=eTk+1P1ek+1=C0X0k+1TP1C0X0k+1=C0Φ+GKX0k+C0E-WkTP1C0Φ+GKX0k+C0E-Wk=X0TkWTkΦ+GKTC0TE-TC0TP1C0Φ+GKC0E-X0kWk=X0TkWTkΦ+GKTC0TP1C0Φ+GKΦ+GKTC0TP1C0E-E-TC0TP1C0Φ+GKE-TC0TP1C0E-X0kWk.If condition (17) holds, the following stands:(23)Vek+1≤γVek+γWTkP2Wk≤γVek+λmax⁡P2γWTkWk.Applying (23) iteratively leads to(24)Vek≤γVek-1+λmax⁡P2γWTk-1Wk-1≤γγVek-2+λmax⁡P2γWTk-2Wk-2+λmax⁡P2γWTk-1Wk-1=γ2Vek-2+λmax⁡P2γ2WTk-2Wk-2+γWTk-1Wk-1≤⋯≤γk-1Ve1+λmax⁡P2∑j=1k-1γjWTk-jWk-j.Taking γ>1 into account, it is easily obtained from (24) that(25)Vek≤γk-1Ve1+λmax⁡P2∑j=1k-1WTk-jWk-j.Thus for all k∈1,2,⋯,N, we get(26)Vek≤γN-1Ve1+λmax⁡P2∑j=1N-1WTN-jWN-j.By setting d2=d12+d22, the following is obtained:(27)∑j=1N-1WTN-jWN-j=∑j=1N-1ΔwTN-jΔwN-j+∑j=1N-1ΔrTN+1-jΔrN+1-j=∑j=1N-1ΔwTjΔwj+∑j=2NΔrTjΔrj≤∑j=1NΔwTjΔwj+∑j=1NΔrTjΔrj≤d12+d22=d2.Moreover,(28)Ve1=R1/2e1TR-1/2P1R-1/2R1/2e1.Denoting P~1=R-1/2P1R-1/2, it follows that(29)Ve1≤λmax⁡P~1eT1Re⁡1.Substituting (27) and (29) into (26) yields the further estimation:(30)Vek≤γN-1λmax⁡P~1eT1Re⁡1+λmax⁡P2d2.Since condition (18) is equivalent to I<R-1/2P1R-1/2<λ1I, i.e., I<P~1<λ1I, it can be obtained that(31)1<λmin⁡P~1≤λmax⁡P~1<λ1.In addition, condition (19) implies(32)0<λmin⁡P2≤λmax⁡P2<λ2.Hence, if (18) and (19) hold, it can be easily seen from (30) that(33)Vek≤γN-1λ1δ2+λ2d2.On the other hand, because of λmin(P~1)>1, then(34)Vek=eTkP1ek≥λmin⁡P~1eTkRe⁡k≥eTkRe⁡k.According to (33) and (34), the following is obtained:(35)eTkRe⁡k≤γN-1λ1δ2+λ2d2.Condition (20) implies that γN-1(λ1δ2+λ2d2)<ε2. Then, it can be concluded that eTkRe⁡(k)≤ε2 (k∈1,2,⋯,N). This completes the proof.By observing the inequality (17) carefully, it can be seen that (17) is not an LMI. Hence, it cannot be easily solved by Matlab LMI toolbox. To this end, a tractable LMI form will be presented in the following. This is the second main theorem of this paper.Theorem 6.
The closed-loop system (15) achieves finite-time bounded tracking of the reference signal r(k) with respect to (δ,d,ε,R,N), if for a given scalar γ>1, there exist matrices Q1>0, P2>0 and scalars λ1′>0, λ2>0 such that(36)-γQ100Q1+CBLT000CA+CBKxT00-γP2E~TQ1+CBLCA+CBKxE~-Q1≤0,(37)λ1′R-1<Q1<R-1,(38)0<P2<λ2I,(39)λ2d2-ε2γN-1δδ-λ1′<0,where E~=CE-I. In this case the controller is Δuk=Keek+KxΔxk with Ke=LQ1-1.Proof.
The key of the proof lies in that the conditions of Theorem5 are satisfied if the condition of this theorem holds.
To convert (17) to an LMI, let Q1=P1-1; then (17) can be equivalently written as(40)Φ+GKTC0TQ1-1C0Φ+GK-γC0TQ1-1C0Φ+GKTC0TQ1-1C0E-E-TC0TQ1-1C0Φ+GKE-TC0TQ1-1C0E--γP2≤0.Since the equivalent transformation of this inequality cannot yield the desired result, the matrix K=KeKx and the expressions of C0, Φ, G, K, E-, Gr, and Gw in the closed-loop system (15) are substituted into this inequality. Then the following can be obtained:(41)
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,that is,(42)I+CBKeTQ1-1I+CBKe-γQ1-1I+CBKeTQ1-1CA+CBKxI+CBKeTQ1-1CE-I+CBKeTQ1-1CA+CBKxTQ1-1I+CBKeCA+CBKxTQ1-1CA+CBKxCA+CBKxTQ1-1CE-CA+CBKxTQ1-1CETQ1-1I+CBKe-Q1-1I+CBKeCETQ1-1CA+CBKx-Q1-1CA+CBKxCETQ1-1CE-CETQ1-1-Q1-1CEQ1-1-γP2≤0.Rewriting the left side of (42) yields(43)I+CBKeTQ1-1I+CBKe-γQ1-1I+CBKeTQ1-1CA+CBKxI+CBKeTQ1-1CE-ICA+CBKxTQ1-1I+CBKeCA+CBKxTQ1-1CA+CBKxCA+CBKxTQ1-1CE-ICET-IQ1-1I+CBKeCET-IQ1-1CA+CBKxCET-IQ1-1CE-I-γP2≤0.By denoting E~=CE-I, the above inequality becomes(44)I+CBKeTQ1-1I+CBKe-γQ1-1I+CBKeTQ1-1CA+CBKxI+CBKeTQ1-1E~CA+CBKxTQ1-1I+CBKeCA+CBKxTQ1-1CA+CBKxCA+CBKxTQ1-1E~E~TQ1-1I+CBKeE~TQ1-1CA+CBKxE~TQ1-1E~-γP2≤0.
Pre- and postmultiplying (44) by the symmetric matrix diag⁡Q1,I,I and its transpose, respectively, we obtain(45)Q1I+CBKeTQ1-1I+CBKeQ1-γQ1Q1I+CBKeTQ1-1CA+CBKxQ1I+CBKeTQ1-1E~CA+CBKxTQ1-1I+CBKeQ1CA+CBKxTQ1-1CA+CBKxCA+CBKxTQ1-1E~E~TQ1-1I+CBKeQ1E~TQ1-1CA+CBKxE~TQ1-1E~-γP2≤0.In fact, (45) can be rewritten as(46)-γQ10000000-γP2-Q1I+CBKeTCA+CBKxTE~T-Q1-1I+CBKeQ1CA+CBKxE~≤0.From -Q1<0 and Lemma 3 (2), (45) is equivalent to(47)-γQ100Q1I+CBKeT000CA+CBKxT00-γP2E~TI+CBKeQ1CA+CBKxE~-Q1≤0.By setting L=KeQ1, it can be seen that (47) becomes (36) and (17) is finally converted to an equivalent LMI (36).
Because ofQ1=P1-1, (18) becomes(48)R<Q1-1<λ1R.Then the above equation leads to(49)1λ1R-1<Q1<R-1,which implies that Q1 needs to meet the condition: 1/λ1R-1-Q1<0. Since λ1>0 and Q1>0 are unknown, this inequality cannot be solved by the LMI toolbox in Matlab. Thus, let λ1′=1/λ1; then (49) is converted into a computationally tractable condition (37).
Takingλ1′=1/λ1 into account, (20) can be written as(50)1λ1′δ2+λ2d2<ε2γN-1.It is also necessary to convert (50) to an LMI. For this purpose, we rewrite (50) as(51)λ2d2-ε2γN-1-δ-λ1′-1δ<0.Because of -λ1′<0, it can be obtained by Lemma 3 (2) that conditions (50) and (39) are equivalent. Besides, (19) in Theorem 5 does not require any change and it is (38). This completes the proof.Note thatΔu(k)=u(k)-u(k-1). By solving u(k), the following result is derived instantly.Theorem 7.
Assume that A1-A2 are satisfied. The control input of linear discrete-time system (6) is given by(52)uk=u0+Ke∑j=1kej+Kxxk-x0,k∈1,2,⋯,N,where Ke and Kx are determined by LMIs (36)-(39). Under the controller, system (6) achieves finite-time bounded tracking of the reference signal r(k) with respect to (δ,d,ε,R,N).Remark 8.
The results in this paper can be readily extended to linear discrete-time systems with state delay. In this case, we can construct a delay-free error system by applying the difference method and the discrete lifting technique [22]. Furthermore, the error vector is still taken as the output vector of the error system. Then, applying the controller design method in this paper, a finite-time bounded tracking controller of discrete time-delay systems can be obtained.


## 6. Simulation Example

The effectiveness of the proposed method will be shown by a numerical example, in which two different reference signals are considered.Example 1.
Consider system (6) with the following system matrices:(53)A=12-1-2,B=-0.51,E=-0.10.5,C=0.10.3.
TakeR=I, δ=0.1, d=6/2, ε=5, N=100, and γ=1.01. By using the LMI toolbox in Matlab to solve the LMIs (36)-(39) in Theorem 6, the feedback gain matrices are given by(54)Ke=-4,Kx=0.81.6.Then let x0=00 and u(0)=0; we obtain(55)uk=u0+Ke∑j=1kej+Kxxk-x0=-4∑j=1kej+0.8x1k+1.6x2k.
The disturbance is taken as(56)wk=3sin⁡0.07πk0.5+k0.6.By calculation, we have(57)∑j=1NΔwTjΔwj≈0.3917≤0.5≝d12.Below, two different reference signals are considered to do the numerical simulation.
(1) The reference signal is taken as(58)rk=0,0≤k<10,0.2,10≤k<20,0.4,20≤k<30,0.6,30≤k<40,0.8,40≤k<50,1,50≤k<60,1.2,60≤k<70,1.4,70≤k<80,1.6,80≤k≤100.In this case, r(k) satisfies ∑j=1NΔrT(j)Δr(j)=0.32≤1≝d22. Note that(59)d12+d22=32=d2.In addition, from x0=00 and u(0)=0, eT1Re⁡(1)=eT(1)e(1)≈0.0052 can be obtained. Then the following condition is guaranteed:(60)eT1Re⁡1≤0.01=δ2.Therefore, the tracking error between the closed-loop output and the reference signal (58) should satisfy(61)eTkRe⁡k≤ε2k∈1,2,⋯,100.
Figure1 shows the output response of the closed-loop system, and Figure 2 shows the tracking error between the closed-loop output and the reference signal.
As shown in Figures1-2, the proposed controller guarantees that the closed-loop output signal is always in the ε neighborhood of the reference signal r(k) within a given time-interval 1,2,⋯,100 and the error signal is always in a given range. That is to say, the closed-loop system achieves finite-time bounded tracking of the reference signal r(k) with respect to (0.1,6/2,5,I,100). It needs to be emphasized that the tracking error is very small even if a strong disturbance signal exists in the system.
Note that from Definition1, if eT0Re⁡(0)≤δ2 and other conditions are satisfied, eTkRe⁡(k)≤ε2 holds. This result has nothing to do with the selection of initial state x(0). But in fact, due to e(0)=y(0)-r(0)=Cx(0)-r(0), the initial state x(0) is still limited. In this example, If we let x0=-0.00083-0.07 and u(0)=0, this results in eT1Re⁡(1)=eT(1)e(1)=0.01=δ2. γ=1.01 is still taken to solve the corresponding LMIs. In this case, the simulation results are completely consistent with the theoretical results, and they are omitted here.Figure 1
The output response of the closed-loop system to reference signal (58).Figure 2
The tracking error of the closed-loop system to reference signal (58).Note that the reference signal (58) is very valuable in practice. In fact, the desired trajectory of a biped robot in the upslope process is usually in the form of function (58) [23].(2) The reference signal is taken as the periodic function given by(62)rk=0,80<k≤100,0.5sin⁡π10k-10,20≤k≤80,0,0≤k<20.In this case, the following can be obtained:(63)∑j=1NΔrTjΔrj≈0.7819≤1≝d22,which implies ∑j=1NΔwT(j)Δw(j)+∑j=1NΔrT(j)Δr(j)≤d2. Thus, the condition of Theorem 6 holds.Figure3 shows the output response of the closed-loop system, and Figure 4 shows the tracking error between the actual output and the desired output. It can be seen that the closed-loop system achieves finite-time bounded tracking of the reference signal r(k) with respect to (0.1,6/2,5,I,100).Figure 3
The output response of the closed-loop system to reference signal (62).Figure 4
The tracking error of the closed-loop system to reference signal (62).


## 7. Conclusion

In this paper, the concept of finite-time bounded tracking control for linear discrete-time systems is proposed. Using the difference method, we construct an error system where the tracking error is only a part of the augmented state vector. Then, by constructing a Lyapunov function with respect to the tracking error, a sufficient condition guaranteeing that the norm of tracking error is finite-time bounded is presented in terms of a set of LMIs. Based on this criterion, a feedback controller of the original system is derived, under which the closed-loop output achieves finite-time bounded tracking of the reference signal. Numerical simulation shows the effectiveness of the proposed controller.



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