# Observer-Based Bounded Control for Discrete Time-Delay Uncertain Nonlinear Systems

Authors: Bei Wu; Mou Chen; Xiaoming Chen
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

A bounded controller is proposed for a class of uncertain discrete time-delay systems with nonlinearity and disturbance based on state estimator and disturbance observer technique. A state estimator is developed to estimate the unmeasured system state vector. Suppose that the disturbance is generated by an exogenous system; a disturbance observer is designed to estimate the unknown disturbance. The parameters of the state estimator and the disturbance observer are calculated by solving linear matrix inequalities (LMIs). By applying the outputs of the state estimator and the disturbance observer, the sufficient condition for the existence of the bounded controller is derived based on an appropriate Lyapunov function candidate. Under the developed bounded controller, the stability of the closed-loop system can be guaranteed. Simulation examples are provided to show the effectiveness of the proposed bounded control scheme.


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## Body

## 1. Introduction

Time-delay phenomenon widely exists in most of industrial systems, such as biological systems, hydraulic systems, transmission systems, electrical networks, and chemical systems [1–5]. The existence of time-delay leads to performance degradation or even system instability and causes the complication to the analysis and design of the controller. Up to now, many control schemes have been studied for time-delay systems, such as guaranteed cost control [1], H∞ control [2, 4], adaptive neural network control [5, 6], sliding mode control [7, 8], and disturbance-observer-based control [9–14]. However, most of the above-mentioned research results are dealing with continuous-time systems. Due to the fast development of computers and digital signal processor (DSP) chips, considerable attention has been paid to the study of discrete time-delay systems [15, 16]. The stability control problem of discrete time-delay systems can be divided into delay-dependent stability control [17–19] and delay-independent stability control [20]. Since delay-dependent conditions are less conservative than delay-independent conditions, more concentrations have been given on the delay dependent stability analysis.Usually, the model errors, the measurement errors, and the external disturbances inevitably exist in dynamic systems, which can also further cause the degradation of system performance and even instability. The problem of robust stability analysis and robust controller design has been extensively studied for discrete-time systems with uncertainty and/or disturbance. A novel adaptive-critic-based neural network (NN) controller was proposed for nonlinear single-input-single-output (SISO) discrete-time systems in [21]. The novel adaptive control technique was proposed for discrete-time multi-input-multi-output (MIMO) systems in [22]. The improved dynamic surface control design was studied for constrained hypersonic flight models in [23]. In [24], based on the key ideas of “future outputs prediction” and “nearest-neighbour compensation,” the adaptive predictive control laws were developed for nonlinear autoregressive moving average systems. In [25], the problem of robust stabilization was investigated for discrete-time singular large-scale systems with time-invariant norm-bounded uncertainties. Based on the dynamic surface control technique, both indirect and direct global neural controllers were developed for the strict-feedback systems in [26]. In [27], the output feedback adaptive neural network controllers were developed for nonlinear discrete-time systems. The output feedback adaptive control technique was investigated for discrete-time systems with unknown control directions in [28]. In [29], a robust adaptive sliding mode controller was constructed for discrete time-delay systems with mismatched uncertainties and matched external disturbances. Since the disturbances widely exist in practical systems and most of them are difficult to be measured, many advanced control approaches, such as adaptive control and sliding mode control, have been proposed. However, the control approaches mentioned above are rejecting disturbances via feedback control based on the tracking error, which can not deal with strong disturbances directly and fast [30]. Then, the disturbance observer was proposed to estimate unmeasured disturbances for engineering application systems such as direct-current servomotor systems [31], robotic manipulators [32], and data storage systems [33]. A reduced order disturbance observer was studied for discrete-time linear systems in [33]. Based on the disturbance observer, an antidisturbance controller was investigated for discrete time-varying delay systems with multiple disturbances under actuator failures in [34]. In [35], disturbance-observer-based control and H∞ control were discussed for A4D aircraft at a flight condition of 15000 ft altitude and 0.9 Mach, the states were made up with the forward velocity, the angle of attack, the pitching velocity, and the pitching angle, and the state delay was considered. A composite fuzzy control was investigated for uncertain nonlinear systems based on the disturbance observer technique in [36]. However, the input time-delay was seldom considered in the research results mentioned above.Time delay in control inputs brings new challenges in the controller design. The problem ofH∞ control was investigated for discrete time-varying input-delay systems via Riccati difference equation in [37]. In [38], the state feedback prediction-based control law was developed for a discrete-time system with input time-varying delay. Applying Artstein’s reduction method and the scaled bounded real lemma, a sufficient condition was presented to stabilize linear discrete-time systems with time-varying input delay and model uncertainties in [39]. In [40, 41], the guaranteed cost control problem was studied for a class of discrete time-delay systems via state feedback. The problem of multiple input delays was discussed in [42]. A robust H∞ state feedback controller was designed for discrete time systems with interval time-varying delays in both states and control inputs in [43]. The state variables mentioned above were required as mensurable. However, not all of the state variables can be measured directly in practical applications. Thus, it is necessary to estimate the unmeasured state variables.State observer, the product of the development of modern control engineering with the characteristics of good stability and strong robustness, has been widely used in various fields, such as diesel engines [44], stirred tank reactor [45], and permanent magnet synchronous motors [46]. The problem of the state-observer-based finite-time H∞ control was studied for singular stochastic systems using the descriptor system approach in [47]. Based on the future states prediction, adaptive control technique was investigated for a class of discrete-time MIMO nonlinear systems in [48]. In [49], a robust delay-dependent state estimator was investigated for a class of discrete-time Bidirectional Associative Memory (BAM) neural networks. A reduced-order Kalman-type observer was designed for nonlinear discrete-time systems in [50]. In [51], a fuzzy observer was proposed for discrete time-delay systems with nonlinearity. An observer-based finite-time H∞ controller was developed for discrete-time Markovian jump nonlinear systems with time delays in [52]. In [53], a state observer was designed for discrete time-delay impulsive switched systems with nonlinearity. A nonlinear observer was proposed for discrete-time uncertain nonlinear systems in [20]. In [54], the problem of reliable observer-based H∞ control was studied for discrete-time Takagi-Sugeno (T-S) fuzzy systems with time-varying delays and stochastic actuator faults. To the best of author’s knowledge, the discrete-time system subject to input saturation, parameter uncertainties, nonlinearity, disturbance, and time delays in both the states and the control inputs has not been investigated.Motivated by the above discussion, we develop a bounded control scheme for uncertain discrete-time nonlinear system with disturbance and time delays in both the states and the control inputs in this paper. A state estimator and a disturbance observer are developed separately to estimate the unknown states and the disturbances generated by an exogenous system. Using the outputs of the state estimator and the disturbance observer, a bounded controller is designed. The Lyapunov function candidate is constructed so that the closed-loop system is asymptotically stable via LMI method. The paper is organized as follows: the problem formulation and preliminaries are given in Section2. In Section 3, a state estimator and a disturbance observer are presented, respectively, and the bounded controller is designed for the discrete time-delay uncertain systems. Simulation results are given in Section 4, followed by concluding remarks in Section 5.


## 2. Problem Formulation and Preliminaries

Consider the following uncertain discrete-time nonlinear system with disturbance and time delays:(1)xk+1=G0+ΔG0xk+G0d+ΔG0dxk-T+F01f01xk,k+H0uk-T+dk,yk=C0xk,where x(k)∈Rn, u(k)∈Rm, y(k)∈Rp, and d(k)∈Rm are unmeasured state vector, control input, output, and disturbance, respectively. f01(x(k),k)∈Rq is the nonlinear function. G0, G0d, H0, and C0 are known constant matrices with appropriate dimensions. ΔG0 and ΔG0d are parameter uncertainties. F01 is a corresponding weighting matrix with appropriate dimensions. T is a known constant delay. The system parameter uncertainties are defined as(2)ΔG0ΔG0d=DΔkM1M2,where D, M1, and M2 are known constant matrices and Δ(k) is an unknown matrix satisfying(3)ΔTkΔk≤I.Here, I denotes the identity matrix with appropriate dimension.Assume thatsat(u(k))=[sat(u1(k)),…,sat(um(k))]T is a saturated nonlinear function. Each element sat(ui(k)) of sat(u(k)) is defined as(4)sat⁡uik=sign⁡uikmin⁡uimax,uik,where uimax represents the saturation level of ui(k) and is assumed as known, i=1,…,m.To design the bounded controller for the uncertain discrete time-delay system with nonlinearity and disturbance, the following assumptions and lemmas are needed.Assumption 1.
Assume that state vectorx(k) of the discrete time-delay system (1) is unmeasured.Assumption 2 (see Guo and Chen [55]).
Assume that the disturbanced(k) is generated by the following exogenous system:(5)wk+1=Wwk,dk=Vwk,where w(k)∈Rs, W, and V are matrices with corresponding dimensions. As shown in [9, 55, 56], a wide class of disturbances, such as harmonic disturbances, can be described by this model. This model has been extensively used in the disturbance observer design.Assumption 3 (see Wei et al. [35]).
Assume that there exists a constant matrixU01 such that(6)f01x1k,k-f01x2k,k≤U01x1k-x2k,f010,k=0.Assumption 4 (see Wei et al. [35]).
(
G
0
,
H
0
) is controllable; (W,H0V) and (G0,C0) are observable.Lemma 5 (see Shi et al. [57]).
Given symmetric constant matricesS11, S22 and constant matrix S12, then S22<0 and S11-S12S22-1S12T<0 hold if and only if(7)S11S12S12TS22<0.Lemma 6 (see Zhu et al. [58]).
Assume thatY, A, and B are real matrices with appropriate dimensions, Y is a symmetric matrix, and XTX≤I. Then(8)Y+AXB+BTXTAT<0holds if and only if there exists a scalar α>0 such that(9)Y+αAAT+α-1BTB<0.Lemma 7 (see Song and Wang [59]).
Assume thatD={D1,D2,…,D2m} is the set of m×m diagonal matrices with diagonal elements being either 0 or 1. If Dl∈D, we obtain that Dl-=I-Dl∈D, where l=1,…,2m.Lemma 8 (see Zheng and Wu [60]).
Assume thatu(k)∈Rm, v(k)=[v1,…,vm]T∈Rm. If |vi|≤uimax, the saturated input sat(u(k)) can be represented as sat(u(k))=∑l=12m‍ηl(Dlu(k)+Dl-v(k)), where i=1,…,m, 0≤ηl≤1, and ∑l=12m‍ηl=1.The control objective of this paper is to find a sufficient condition in order to guarantee the stability of the uncertain discrete time-delay system described by (1) under a bounded controller.


## 3. Design of Bounded Controller Based on the State Estimator and the Disturbance Observer

In this section, the state estimator and the disturbance observer are designed to estimate the unmeasured state vector and disturbance of the discrete time-delay uncertain system. The output signals of the disturbance observer and the state estimator are delivered to the bounded controller as its input signals. The stability condition of the closed-loop system is derived by the chosen Lyapunov function as well as LMIs method. The control diagram of the closed-loop system is shown in Figure1, where x^(k) and d^(k) are the estimations of x(k) and d(k).Figure 1
The control diagram of system.


### 3.1. Design of State Estimator

For the unmeasured state vector in the time-delay uncertain nonlinear system (1), a state estimator is needed to estimate x(k). The state estimator is designed as(10)x^k+1=G0x^k+G0dx^k-T+H0uk-T+H0d^k+F01f01x^k,k+L1yk-C0x^k,where L1 is an observer gain matrix which is needed to be designed by the comprehensive consideration between response speed and the sensitivity to disturbance and noise.Definex~(k)=x(k)-x^(k). Considering (1) and (10) yields(11)x~k+1=G0-L1C0x~k+ΔG0xk+G0dx~k-T+ΔG0dxk-T+F01τ01k+H0dk-d^k,where τ01(k)=(f01(x(k),k)-f01(x^(k),k)).


### 3.2. Design of Disturbance Observer

In order to monitor the disturbance of the discrete time-delay uncertain system (1), a disturbance observer is proposed as follows [35]:(12)d^k=Vw^k,w^k=gk-Lx^k,gk+1=W+LH0Vw^k+LG0x^k+G0dx^k-T+F01f01x^k,k+H0uk-T,where w^(k)∈Rs is the estimation of w(k) and g(k)∈Rs is the auxiliary vector as the state of the disturbance observer.Define(13)w~k=wk-w^k.Considering (5), (13) and invoking (10), we obtain(14)w~k+1=Ww~k+L2C0x~k,where L2=LL1.Substituting (5), (12), and (13) into (11) gives(15)x~k+1=G0-L1C0x~k+ΔG0xk+G0dx~k-T+ΔG0dxk-T+F01τ01k+H0Vw~k.


### 3.3. Design of Bounded Controller

The state estimator and the disturbance observer have been designed to estimate the unmeasured state vector and the unknown disturbance. Based on the output signals ofx^(k) and d^(k), the bounded controller is designed as(16)uk=sat⁡K1x^k+K2d^k,where K1 and K2 are feedback gain matrices.The saturated linear region ofu(k) is defined as(17)LV1,V2=x^k,d^k∈Rn+m:Vi1x^k+Vi2d^k≤uimax,i=1,…,m,where V1=[V11,…,Vm1]T∈Rm×n, Vi1∈R1×n, V2=[V12,…,Vm2]T∈Rm×m, Vi2∈R1×m, and i=1,…,m. Using Lemma 8, (16) can be rewritten as(18)uk=∑l=12mηlDlK1x^k+K2d^k+Dl-V1x^k+V2d^k.Substituting (12) and (18) into (1), the closed-loop system is given by(19)xk+1=∑l=12mηlG0+ΔG0xk+G¯d+ΔG0dxk-T+H0Vwk+∑l=12mηlF01f01xk,k+W¯wk-T-K¯x~k-T-W¯w~k-T,where G¯d=G0d+H0(DlK1+Dl-V1), W¯=H0(DlK2+Dl-V2)V, and K¯=H0(DlK1+Dl-V1).Combining (14), (15), and (19), the whole closed-loop system is expressed as(20)x¯k+1=∑l=12mηlGx¯k+Gdx¯k-T+Ffx¯k,k+Hwk+Mwk-T+Eτk,where(21)x¯k=xkw~kx~k,x¯k-T=xk-Tw~k-Tx~k-T,F=F0100,G=G0+ΔG0000WL2C0ΔG0H0VG0-L1C0,H=H0V00,Gd=G¯d+ΔG0d-W¯-K¯000ΔG0d0G0d,M=W¯00,E=00F01,τk=τ01k,fx¯k,k=f01xk,k.According to Assumption3, we obtain(22)fx¯k,k=f01xk,k≤U01xk=U1x¯k,τk≤U01xk-x^k=U2x¯k,where U1=U0100∈Rq×(2n+s) and U2=00U01∈Rq×(2n+s).Disturbance gain matrixL, state estimator gain matrix L1, and feedback gain matrices K1, K2, V1, and V2 play an important role in closed-loop system stability and estimation error stability. The values of these matrices can be derived by solving LMIs. The stability condition of the closed-loop system (20) is given in the following theorem.Theorem 9.
For given scalarsɛ>0, λ1>0, and λ2>0, the closed-loop system (20) is asymptotically stable if there exist symmetric positive-defined matrices Q11=P11-1>0, P22>0, P33>0, N11=S11-1>0, S22>0, and S33>0 and matrices Z1, Z2, Z3, R1, R2, R3, R4, K1, K2, V1, and V2 such that the following matrix inequality holds: (23)N6=Π11∗∗Π21Π22∗Π31Π32Π33<0,where(24)Π11=-Q11∗∗∗∗∗∗0-P22+S22∗∗∗∗∗00-P33+S33∗∗∗∗000-N11∗∗∗0000-S22∗∗00000-S33∗000000-λ12I,Π21=0000000000000000000000000000G0Q1100G′-W¯-K¯F010P22WR2C000000P33H0VP33G0-R1C000P33G0d0,Π22=-λ22I∗∗∗∗∗∗0-Z1-Z1T∗∗∗∗∗0Z1-Z2TZ2+Z2T∗∗∗∗0Z1-Z3TZ2+Z3TZ3+Z3T∗∗∗0H0VW¯0-Q11∗∗00000-P22∗P33F0100000-P33,Π31=U01Q1100000000U0100000000000M1Q1100M2N11000Q11000000,Π32=000000000000000000DT0DTP3300000000000000,Π33=-λ1-2I∗∗∗∗0-λ2-2I∗∗∗00-ε-1I∗∗000-ɛI∗0000-N11,and G′=G0dN11+H0DlR3+H0Dl-R4, L1=P33-1R1, L2=P22-1R2, K1=R3N11-1, and V1=R4N11-1. ∗ denotes the transposed elements in the symmetric matrix.Proof.
Choose the Lyapunov function as(25)Vk=x¯TkP1x¯k+λ12∑i=1k-1U1x¯i2-fx¯i,i2+∑i=k-Tk-1x¯TiS1x¯i+λ22∑i=1k-1U2x¯i2-τi2,where(26)P1=P11000P22000P33=Q11-1000P22000P33>0,S1=S11000S22000S33=N11-1000S22000S33>0.
Definer(j)=w(j+1)-w(j). Then, we have(27)wk+1=wk+rk,wk-wk-T-∑j=k-Tk-1rj=0.
Denote that(28)Γ1=-wk+wk-T+∑j=k-Tk-1rj,Γ2=Z1wk+Z2wk-T+Z3∑j=k-Tk-1rj.
Taking the forward difference ofV(k) along the trajectories of system (20), we have(29)ΔVk=Vk+1-Vk=Vk+1-Vk+Γ1TΓ2+Γ2TΓ1=∑l=12mηlx¯Tk-P1+S1+λ12U1TU1+λ22U2TU2+GTP1Gx¯k+2x¯Tk-TGdTP1Gx¯k+x¯Tk-TGdTP1Gd-S1x¯k-T+2fTx¯k,kFTP1Gx¯k+2fTx¯k,kFTP1Gdx¯k-T+fTx¯k,kFTP1F-λ12Ifx¯k,k+2τTkETP1Gx¯k+2τTkETP1Gdx¯k-T+2τTkETP1Ffx¯k,k+τTkETP1E-λ22Iτk+2wTkHTP1Gx¯k+2wTkHTP1Gdx¯k-T+2wTkHTP1Ffx¯k,k+2wTkHTP1Eτk+wTkHTP1H-Z1-Z1Twk+2wTk-TMTP1Gx¯k+2wTk-TMTP1Gdx¯k-T+2wTk-TMTP1Ffx¯k,k+2wTk-TMTP1Eτk+2wTk-TMTP1H+Z1-Z2Twk+wTk-TMTP1M+Z2+Z2Twk-T+2∑j=k-Tk-1rTjZ1-Z3Twk+2∑j=k-Tk-1rTjZ2+Z3Twk-T+∑j=k-Tk-1rTjZ3+Z3T∑j=k-Tk-1rTj≤maxl∈1,2m⁡x¯Tk-P1+S1+λ12U1TU1+λ22U2TU2+GTP1Gx¯k+2x¯Tk-TGdTP1Gx¯k+x¯Tk-TGdTP1Gd-S1x¯k-T+2fTx¯k,kFTP1Gx¯k+2fTx¯k,kFTP1Gdx¯k-T+fTx¯k,kFTP1F-λ12Ifx¯k,k+2τTkETP1Gx¯k+2τTkETP1Gdx¯k-T+2τTkETP1Ffx¯k,k+τTkETP1E-λ22Iτk+2wTkHTP1Gx¯k+2wTkHTP1Gdx¯k-T+2wTkHTP1Ffx¯k,k+2wTkHTP1Eτk+wTkHTP1H-Z1-Z1Twk+2wTk-TMTP1Gx¯k+2wTk-TMTP1Gdx¯k-T+2wTk-TMTP1Ffx¯k,k+2wTk-TMTP1Eτk+2wTk-TMTP1H+Z1-Z2Twk+wTk-TMTP1M+Z2+Z2Twk-T+2∑j=k-Tk-1rTjZ1-Z3Twk+2∑j=k-Tk-1rTjZ2+Z3Twk-T+∑j=k-Tk-1rTjZ3+Z3T∑j=k-Tk-1rTj.
Then, (29) can be rewritten as(30)ΔVk≤maxl∈1,2m⁡x¯kx¯k-Tfx¯k,kτkwkwk-T∑j=k-Tk-1rjTN1x¯kx¯k-Tfx¯k,kτkwkwk-T∑j=k-Tk-1rj,where(31)N1=Π1∗∗∗∗∗∗GdTP1GΠ2∗∗∗∗∗FTP1GFTP1GdΠ3∗∗∗∗ETP1GETP1GdETP1FΠ4∗∗∗HTP1GHTP1GdHTP1FHTP1EΠ5∗∗MTP1GMTP1GdMTP1FMTP1EΠ8Π6∗0000Z1-Z3TZ2+Z3TΠ7,Π1=-P1+S1+λ12U1TU1+λ22U2TU2+GTP1G,Π2=GdTP1Gd-S1,Π3=FTP1F-λ12I,Π4=ETP1E-λ22I,Π5=HTP1H-Z1-Z1T,Π6=MTP1M+Z2+Z2T,Π7=Z3+Z3T,Π8=MTP1H+Z1-Z2T.
IfN1 can be proved to be less than 0, we have ΔV(k)<0. Then, the closed-loop system (20) is asymptotically stable. Applying Schur complement theorem, N1<0 if and only if(32)N2=Γ11∗Γ21Γ22<0,where(33)Γ11=-P1+S1∗∗∗∗∗∗0-S1∗∗∗∗∗00-λ12I∗∗∗∗000-λ22I∗∗∗0000-Z1-Z1T∗∗0000Z1-Z2TZ2+Z2T∗0000Z1-Z3TZ2+Z3TZ3+Z3T,Γ21=GGdFEHM0U1000000U2000000,Γ22=-P1-1∗∗0-λ1-2I∗00-λ2-2I.
Substituting (21) and (26) into (32) yields (34)N3=Λ11∗∗Λ21Λ22∗Λ31Λ32Λ33,where(35)Λ11=-P11+S11∗∗∗∗∗∗0-P22+S22∗∗∗∗∗00-P33+S33∗∗∗∗000-S11∗∗∗0000-S22∗∗00000-S33∗000000-λ12I,Λ21=0000000000000000000000000000G0+ΔG000G¯d+ΔG0d-W¯-K¯F010WL2C00000ΔG0H0VG0-L1C0ΔG0d0G0d0,Λ22=-λ22I∗∗∗∗∗∗0-Z1-Z1T∗∗∗∗∗0Z1-Z2TZ2+Z2T∗∗∗∗0Z1-Z3TZ2+Z3TZ3+Z3T∗∗∗0H0VW¯0-Q11∗∗00000-P22-1∗F0100000-P33-1,Λ31=U0100000000U010000,Λ32=00000000000000,Λ33=-λ1-2I∗0-λ2-2I.
Using Lemmas5 and 6, N3<0 if and only if(36)N4=Λ11∗∗Λ21′Λ22∗Λ31′Λ32′Λ33′<0,where(37)Λ21′=0000000000000000000000000000G000G¯d-W¯-K¯F010WL2C000000H0VG0-L1C000G0d0,Λ31′=U0100000000U0100000000000M100M2000,Λ32′=000000000000000000DT0DT0000000,Λ33′=-λ1-2I∗∗∗0-λ2-2I∗∗00-ɛ-1I∗000-ɛI.
Left and right multiplying both sides of (36) by diag⁡{Q11,I,I,I,I,I,I,I,I,I,I,I,I,I,I,I,I,I} and using Schur complement theorem, we have(38)N5=Ξ11∗∗Ξ21Λ22∗Ξ31Ξ32Ξ33,where(39)Ξ11=-Q11∗∗∗∗∗∗0-P22+S22∗∗∗∗∗00-P33+S33∗∗∗∗000-S11∗∗∗0000-S22∗∗00000-S33∗000000-λ12I,Ξ21=0000000000000000000000000000G0Q1100G¯d-W¯-K¯F010WL2C000000H0VG0-L1C000G0d0,Ξ31=U01Q1100000000U0100000000000M1Q1100M2000Q11000000,Ξ32=000000000000000000DT0DT00000000000000,Ξ33=-λ1-2I∗∗∗∗0-λ2-2I∗∗∗00-ɛ-1I∗∗000-ɛI∗0000-N11.
Left and right multiplying both sides of (38) by diag⁡{I,I,I,N11,I,I,I,I,I,I,I,I,P22,P33,I,I,I,I,I}, we obtain the matrix N6 of (23). Because N6<0, we have ΔV(k)<0, which implies that x(k)→0, w~(k)→0, and x~(k)→0 as k→∞. Namely, the closed-loop system (20) is asymptotically stable. This concludes the proof.


## 3.1. Design of State Estimator

For the unmeasured state vector in the time-delay uncertain nonlinear system (1), a state estimator is needed to estimate x(k). The state estimator is designed as(10)x^k+1=G0x^k+G0dx^k-T+H0uk-T+H0d^k+F01f01x^k,k+L1yk-C0x^k,where L1 is an observer gain matrix which is needed to be designed by the comprehensive consideration between response speed and the sensitivity to disturbance and noise.Definex~(k)=x(k)-x^(k). Considering (1) and (10) yields(11)x~k+1=G0-L1C0x~k+ΔG0xk+G0dx~k-T+ΔG0dxk-T+F01τ01k+H0dk-d^k,where τ01(k)=(f01(x(k),k)-f01(x^(k),k)).


## 3.2. Design of Disturbance Observer

In order to monitor the disturbance of the discrete time-delay uncertain system (1), a disturbance observer is proposed as follows [35]:(12)d^k=Vw^k,w^k=gk-Lx^k,gk+1=W+LH0Vw^k+LG0x^k+G0dx^k-T+F01f01x^k,k+H0uk-T,where w^(k)∈Rs is the estimation of w(k) and g(k)∈Rs is the auxiliary vector as the state of the disturbance observer.Define(13)w~k=wk-w^k.Considering (5), (13) and invoking (10), we obtain(14)w~k+1=Ww~k+L2C0x~k,where L2=LL1.Substituting (5), (12), and (13) into (11) gives(15)x~k+1=G0-L1C0x~k+ΔG0xk+G0dx~k-T+ΔG0dxk-T+F01τ01k+H0Vw~k.


## 3.3. Design of Bounded Controller

The state estimator and the disturbance observer have been designed to estimate the unmeasured state vector and the unknown disturbance. Based on the output signals ofx^(k) and d^(k), the bounded controller is designed as(16)uk=sat⁡K1x^k+K2d^k,where K1 and K2 are feedback gain matrices.The saturated linear region ofu(k) is defined as(17)LV1,V2=x^k,d^k∈Rn+m:Vi1x^k+Vi2d^k≤uimax,i=1,…,m,where V1=[V11,…,Vm1]T∈Rm×n, Vi1∈R1×n, V2=[V12,…,Vm2]T∈Rm×m, Vi2∈R1×m, and i=1,…,m. Using Lemma 8, (16) can be rewritten as(18)uk=∑l=12mηlDlK1x^k+K2d^k+Dl-V1x^k+V2d^k.Substituting (12) and (18) into (1), the closed-loop system is given by(19)xk+1=∑l=12mηlG0+ΔG0xk+G¯d+ΔG0dxk-T+H0Vwk+∑l=12mηlF01f01xk,k+W¯wk-T-K¯x~k-T-W¯w~k-T,where G¯d=G0d+H0(DlK1+Dl-V1), W¯=H0(DlK2+Dl-V2)V, and K¯=H0(DlK1+Dl-V1).Combining (14), (15), and (19), the whole closed-loop system is expressed as(20)x¯k+1=∑l=12mηlGx¯k+Gdx¯k-T+Ffx¯k,k+Hwk+Mwk-T+Eτk,where(21)x¯k=xkw~kx~k,x¯k-T=xk-Tw~k-Tx~k-T,F=F0100,G=G0+ΔG0000WL2C0ΔG0H0VG0-L1C0,H=H0V00,Gd=G¯d+ΔG0d-W¯-K¯000ΔG0d0G0d,M=W¯00,E=00F01,τk=τ01k,fx¯k,k=f01xk,k.According to Assumption3, we obtain(22)fx¯k,k=f01xk,k≤U01xk=U1x¯k,τk≤U01xk-x^k=U2x¯k,where U1=U0100∈Rq×(2n+s) and U2=00U01∈Rq×(2n+s).Disturbance gain matrixL, state estimator gain matrix L1, and feedback gain matrices K1, K2, V1, and V2 play an important role in closed-loop system stability and estimation error stability. The values of these matrices can be derived by solving LMIs. The stability condition of the closed-loop system (20) is given in the following theorem.Theorem 9.
For given scalarsɛ>0, λ1>0, and λ2>0, the closed-loop system (20) is asymptotically stable if there exist symmetric positive-defined matrices Q11=P11-1>0, P22>0, P33>0, N11=S11-1>0, S22>0, and S33>0 and matrices Z1, Z2, Z3, R1, R2, R3, R4, K1, K2, V1, and V2 such that the following matrix inequality holds: (23)N6=Π11∗∗Π21Π22∗Π31Π32Π33<0,where(24)Π11=-Q11∗∗∗∗∗∗0-P22+S22∗∗∗∗∗00-P33+S33∗∗∗∗000-N11∗∗∗0000-S22∗∗00000-S33∗000000-λ12I,Π21=0000000000000000000000000000G0Q1100G′-W¯-K¯F010P22WR2C000000P33H0VP33G0-R1C000P33G0d0,Π22=-λ22I∗∗∗∗∗∗0-Z1-Z1T∗∗∗∗∗0Z1-Z2TZ2+Z2T∗∗∗∗0Z1-Z3TZ2+Z3TZ3+Z3T∗∗∗0H0VW¯0-Q11∗∗00000-P22∗P33F0100000-P33,Π31=U01Q1100000000U0100000000000M1Q1100M2N11000Q11000000,Π32=000000000000000000DT0DTP3300000000000000,Π33=-λ1-2I∗∗∗∗0-λ2-2I∗∗∗00-ε-1I∗∗000-ɛI∗0000-N11,and G′=G0dN11+H0DlR3+H0Dl-R4, L1=P33-1R1, L2=P22-1R2, K1=R3N11-1, and V1=R4N11-1. ∗ denotes the transposed elements in the symmetric matrix.Proof.
Choose the Lyapunov function as(25)Vk=x¯TkP1x¯k+λ12∑i=1k-1U1x¯i2-fx¯i,i2+∑i=k-Tk-1x¯TiS1x¯i+λ22∑i=1k-1U2x¯i2-τi2,where(26)P1=P11000P22000P33=Q11-1000P22000P33>0,S1=S11000S22000S33=N11-1000S22000S33>0.
Definer(j)=w(j+1)-w(j). Then, we have(27)wk+1=wk+rk,wk-wk-T-∑j=k-Tk-1rj=0.
Denote that(28)Γ1=-wk+wk-T+∑j=k-Tk-1rj,Γ2=Z1wk+Z2wk-T+Z3∑j=k-Tk-1rj.
Taking the forward difference ofV(k) along the trajectories of system (20), we have(29)ΔVk=Vk+1-Vk=Vk+1-Vk+Γ1TΓ2+Γ2TΓ1=∑l=12mηlx¯Tk-P1+S1+λ12U1TU1+λ22U2TU2+GTP1Gx¯k+2x¯Tk-TGdTP1Gx¯k+x¯Tk-TGdTP1Gd-S1x¯k-T+2fTx¯k,kFTP1Gx¯k+2fTx¯k,kFTP1Gdx¯k-T+fTx¯k,kFTP1F-λ12Ifx¯k,k+2τTkETP1Gx¯k+2τTkETP1Gdx¯k-T+2τTkETP1Ffx¯k,k+τTkETP1E-λ22Iτk+2wTkHTP1Gx¯k+2wTkHTP1Gdx¯k-T+2wTkHTP1Ffx¯k,k+2wTkHTP1Eτk+wTkHTP1H-Z1-Z1Twk+2wTk-TMTP1Gx¯k+2wTk-TMTP1Gdx¯k-T+2wTk-TMTP1Ffx¯k,k+2wTk-TMTP1Eτk+2wTk-TMTP1H+Z1-Z2Twk+wTk-TMTP1M+Z2+Z2Twk-T+2∑j=k-Tk-1rTjZ1-Z3Twk+2∑j=k-Tk-1rTjZ2+Z3Twk-T+∑j=k-Tk-1rTjZ3+Z3T∑j=k-Tk-1rTj≤maxl∈1,2m⁡x¯Tk-P1+S1+λ12U1TU1+λ22U2TU2+GTP1Gx¯k+2x¯Tk-TGdTP1Gx¯k+x¯Tk-TGdTP1Gd-S1x¯k-T+2fTx¯k,kFTP1Gx¯k+2fTx¯k,kFTP1Gdx¯k-T+fTx¯k,kFTP1F-λ12Ifx¯k,k+2τTkETP1Gx¯k+2τTkETP1Gdx¯k-T+2τTkETP1Ffx¯k,k+τTkETP1E-λ22Iτk+2wTkHTP1Gx¯k+2wTkHTP1Gdx¯k-T+2wTkHTP1Ffx¯k,k+2wTkHTP1Eτk+wTkHTP1H-Z1-Z1Twk+2wTk-TMTP1Gx¯k+2wTk-TMTP1Gdx¯k-T+2wTk-TMTP1Ffx¯k,k+2wTk-TMTP1Eτk+2wTk-TMTP1H+Z1-Z2Twk+wTk-TMTP1M+Z2+Z2Twk-T+2∑j=k-Tk-1rTjZ1-Z3Twk+2∑j=k-Tk-1rTjZ2+Z3Twk-T+∑j=k-Tk-1rTjZ3+Z3T∑j=k-Tk-1rTj.
Then, (29) can be rewritten as(30)ΔVk≤maxl∈1,2m⁡x¯kx¯k-Tfx¯k,kτkwkwk-T∑j=k-Tk-1rjTN1x¯kx¯k-Tfx¯k,kτkwkwk-T∑j=k-Tk-1rj,where(31)N1=Π1∗∗∗∗∗∗GdTP1GΠ2∗∗∗∗∗FTP1GFTP1GdΠ3∗∗∗∗ETP1GETP1GdETP1FΠ4∗∗∗HTP1GHTP1GdHTP1FHTP1EΠ5∗∗MTP1GMTP1GdMTP1FMTP1EΠ8Π6∗0000Z1-Z3TZ2+Z3TΠ7,Π1=-P1+S1+λ12U1TU1+λ22U2TU2+GTP1G,Π2=GdTP1Gd-S1,Π3=FTP1F-λ12I,Π4=ETP1E-λ22I,Π5=HTP1H-Z1-Z1T,Π6=MTP1M+Z2+Z2T,Π7=Z3+Z3T,Π8=MTP1H+Z1-Z2T.
IfN1 can be proved to be less than 0, we have ΔV(k)<0. Then, the closed-loop system (20) is asymptotically stable. Applying Schur complement theorem, N1<0 if and only if(32)N2=Γ11∗Γ21Γ22<0,where(33)Γ11=-P1+S1∗∗∗∗∗∗0-S1∗∗∗∗∗00-λ12I∗∗∗∗000-λ22I∗∗∗0000-Z1-Z1T∗∗0000Z1-Z2TZ2+Z2T∗0000Z1-Z3TZ2+Z3TZ3+Z3T,Γ21=GGdFEHM0U1000000U2000000,Γ22=-P1-1∗∗0-λ1-2I∗00-λ2-2I.
Substituting (21) and (26) into (32) yields (34)N3=Λ11∗∗Λ21Λ22∗Λ31Λ32Λ33,where(35)Λ11=-P11+S11∗∗∗∗∗∗0-P22+S22∗∗∗∗∗00-P33+S33∗∗∗∗000-S11∗∗∗0000-S22∗∗00000-S33∗000000-λ12I,Λ21=0000000000000000000000000000G0+ΔG000G¯d+ΔG0d-W¯-K¯F010WL2C00000ΔG0H0VG0-L1C0ΔG0d0G0d0,Λ22=-λ22I∗∗∗∗∗∗0-Z1-Z1T∗∗∗∗∗0Z1-Z2TZ2+Z2T∗∗∗∗0Z1-Z3TZ2+Z3TZ3+Z3T∗∗∗0H0VW¯0-Q11∗∗00000-P22-1∗F0100000-P33-1,Λ31=U0100000000U010000,Λ32=00000000000000,Λ33=-λ1-2I∗0-λ2-2I.
Using Lemmas5 and 6, N3<0 if and only if(36)N4=Λ11∗∗Λ21′Λ22∗Λ31′Λ32′Λ33′<0,where(37)Λ21′=0000000000000000000000000000G000G¯d-W¯-K¯F010WL2C000000H0VG0-L1C000G0d0,Λ31′=U0100000000U0100000000000M100M2000,Λ32′=000000000000000000DT0DT0000000,Λ33′=-λ1-2I∗∗∗0-λ2-2I∗∗00-ɛ-1I∗000-ɛI.
Left and right multiplying both sides of (36) by diag⁡{Q11,I,I,I,I,I,I,I,I,I,I,I,I,I,I,I,I,I} and using Schur complement theorem, we have(38)N5=Ξ11∗∗Ξ21Λ22∗Ξ31Ξ32Ξ33,where(39)Ξ11=-Q11∗∗∗∗∗∗0-P22+S22∗∗∗∗∗00-P33+S33∗∗∗∗000-S11∗∗∗0000-S22∗∗00000-S33∗000000-λ12I,Ξ21=0000000000000000000000000000G0Q1100G¯d-W¯-K¯F010WL2C000000H0VG0-L1C000G0d0,Ξ31=U01Q1100000000U0100000000000M1Q1100M2000Q11000000,Ξ32=000000000000000000DT0DT00000000000000,Ξ33=-λ1-2I∗∗∗∗0-λ2-2I∗∗∗00-ɛ-1I∗∗000-ɛI∗0000-N11.
Left and right multiplying both sides of (38) by diag⁡{I,I,I,N11,I,I,I,I,I,I,I,I,P22,P33,I,I,I,I,I}, we obtain the matrix N6 of (23). Because N6<0, we have ΔV(k)<0, which implies that x(k)→0, w~(k)→0, and x~(k)→0 as k→∞. Namely, the closed-loop system (20) is asymptotically stable. This concludes the proof.


## 4. Simulation

In order to verify the correctness of the proposed method, simulation examples are given in the following.


### 4.1. The Case for Two-Dimension System

Consider the uncertain nonlinear discrete time system (1) with disturbance and time delays in both the states and the control inputs with(40)G0=0.60.800.5,G0d=0.20.100.1,D=0.10.2,H0=0.01000.01,M1=0.20.3T,M2=00.1T,F01=0.150,f01xk,k=sin⁡3x2k.The parameter matrices of exogenous system (5) are chosen as(41)W=0.99980.02-0.020.9998,V=1001.Suppose that the initial values of state variable arex(0)=-22T, the initial values of state estimator are x^(0)=-2.0012.001T, the initial values of exogenous inputs are w(0)=0.21-0.22T, and w^(0)=0.2-0.2T. Choosing U01=diag⁡00.2, T=1, u1max=0.03, u2max=0.1, λ1=0.1, λ2=0.5, and ɛ=10, by solving (23), we obtain(42)K1=-0.00380.00060.00060.1235,K2=-0.0502-0.1198-0.1198-0.3544,L1=0.51710.46200.08020.5118,L=10-5×-0.16390.27910.12620.0229.Under the observer-based bounded controller, simulation results are obtained for the two-dimension uncertain discrete time-delay system with nonlinearity and disturbance, as shown in Figures2–7. It is shown in Figures 2 and 3 that the stability of the closed-loop system can be obtained under the proposed bounded controller. Figure 4 shows that the state estimation errors of x~1(k) and x~2(k) are small, and the outputs of the state estimator can estimate the system state effectively. The simulation results shown in Figures 5 and 6 indicate that the outputs of the disturbance observer can effectively and quickly approximate the unknown external disturbance. Figure 7 shows that the control input is bounded under the designed controller. From these simulation results of the example, we can know that the state estimator and the disturbance observer can well approximate the system state and the disturbance, and the designed bounded control scheme based on the state estimator and the disturbance observer is valid for the discrete-time system subject to input saturation, parameter uncertainties, nonlinearity, disturbance, and state and input time delays.Figure 2
The system statex1(k).Figure 3
The system statex2(k).Figure 4
The estimation error outputs ofx~1(k) and x~2(k).Figure 5
The disturbanced1(k) and the estimation output of d^1(k).Figure 6
The disturbanced2(k) and the estimation output of d^2(k).Figure 7
The control inputsu1(k) and u2(k).


### 4.2. The Case for Three-Dimension System

Considering the following discrete-time uncertain time-delay system with disturbance and nonlinearity, originally given by [41], the parameter matrices of system (1) are(43)G0=0.70-0.50.050.8000.30.6,G0d=-0.02000-0.010.0100-0.02,D=0.100.2,H0=0.01000.01-0.0120.12,M1=0.200.3T,M2=00.10.01T,F01=0.100,f01xk,k=sin⁡2x2k.The parameter matrices of exogenous system (5) are chosen as(44)W=0.99980.02000-0.02000.99980000.9802,V=100010.Suppose that the initial values of state variable arex(0)=2-22.5T, the initial values of state estimator are x^(0)=2.001-2.0012.499T, the initial values of exogenous inputs are w(0)=0.110.220T, and w^(0)=0.10.210.01T. Choosing U01=diag⁡00.20, T=1, u1max=0.01, u2max=0.08, λ1=0.1, and λ2=ɛ=10, by solving (23), we have(45)K1=-0.0029-0.06010.0482-0.0009-0.01910.0154,K2=-0.00870.02420.0242-0.3696,L1=0.6584-0.2042-0.3962-0.04300.77100.0020-0.93303.37980.3040,L=10-3×-0.0124-0.0111-0.00660.0844-0.45820.1109-0.00000.0001-0.0000.Using the developed bounded controller, simulation results for the three-dimension uncertain discrete time-delay system with nonlinearity and disturbance are shown in Figures8–14. From Figures 8–10, we can observe that the stability of the closed-loop system can be guaranteed under the observer-based bounded controller. According to Figure 11, we can see that the outputs of the state estimator can estimate the system state effectively. The disturbance estimate outputs d^1(k), d^2(k) and the disturbance d1(k), d2(k) are presented in Figures 12 and 13 which illustrate the effectiveness of the proposed disturbance observer. Figure 14 indicates that the control input is bounded under the designed controller. Based on these simulation results, we can obtain that the state estimator and the disturbance observer can effectively approximate the system state and the disturbance, and the proposed observer-based bounded control is valid for the discrete-time uncertain nonlinear system with time delays and disturbance.Figure 8
The system statex1(k).Figure 9
The system statex2(k).Figure 10
The system statex3(k).Figure 11
The estimation error outputs ofx~1(k), x~2(k), and x~3(k).Figure 12
The disturbanced1(k) and the estimation output of d^1(k).Figure 13
The disturbanced2(k) and the estimation output of d^2(k).Figure 14
The control inputsu1(k) and u2(k).


## 4.1. The Case for Two-Dimension System

Consider the uncertain nonlinear discrete time system (1) with disturbance and time delays in both the states and the control inputs with(40)G0=0.60.800.5,G0d=0.20.100.1,D=0.10.2,H0=0.01000.01,M1=0.20.3T,M2=00.1T,F01=0.150,f01xk,k=sin⁡3x2k.The parameter matrices of exogenous system (5) are chosen as(41)W=0.99980.02-0.020.9998,V=1001.Suppose that the initial values of state variable arex(0)=-22T, the initial values of state estimator are x^(0)=-2.0012.001T, the initial values of exogenous inputs are w(0)=0.21-0.22T, and w^(0)=0.2-0.2T. Choosing U01=diag⁡00.2, T=1, u1max=0.03, u2max=0.1, λ1=0.1, λ2=0.5, and ɛ=10, by solving (23), we obtain(42)K1=-0.00380.00060.00060.1235,K2=-0.0502-0.1198-0.1198-0.3544,L1=0.51710.46200.08020.5118,L=10-5×-0.16390.27910.12620.0229.Under the observer-based bounded controller, simulation results are obtained for the two-dimension uncertain discrete time-delay system with nonlinearity and disturbance, as shown in Figures2–7. It is shown in Figures 2 and 3 that the stability of the closed-loop system can be obtained under the proposed bounded controller. Figure 4 shows that the state estimation errors of x~1(k) and x~2(k) are small, and the outputs of the state estimator can estimate the system state effectively. The simulation results shown in Figures 5 and 6 indicate that the outputs of the disturbance observer can effectively and quickly approximate the unknown external disturbance. Figure 7 shows that the control input is bounded under the designed controller. From these simulation results of the example, we can know that the state estimator and the disturbance observer can well approximate the system state and the disturbance, and the designed bounded control scheme based on the state estimator and the disturbance observer is valid for the discrete-time system subject to input saturation, parameter uncertainties, nonlinearity, disturbance, and state and input time delays.Figure 2
The system statex1(k).Figure 3
The system statex2(k).Figure 4
The estimation error outputs ofx~1(k) and x~2(k).Figure 5
The disturbanced1(k) and the estimation output of d^1(k).Figure 6
The disturbanced2(k) and the estimation output of d^2(k).Figure 7
The control inputsu1(k) and u2(k).


## 4.2. The Case for Three-Dimension System

Considering the following discrete-time uncertain time-delay system with disturbance and nonlinearity, originally given by [41], the parameter matrices of system (1) are(43)G0=0.70-0.50.050.8000.30.6,G0d=-0.02000-0.010.0100-0.02,D=0.100.2,H0=0.01000.01-0.0120.12,M1=0.200.3T,M2=00.10.01T,F01=0.100,f01xk,k=sin⁡2x2k.The parameter matrices of exogenous system (5) are chosen as(44)W=0.99980.02000-0.02000.99980000.9802,V=100010.Suppose that the initial values of state variable arex(0)=2-22.5T, the initial values of state estimator are x^(0)=2.001-2.0012.499T, the initial values of exogenous inputs are w(0)=0.110.220T, and w^(0)=0.10.210.01T. Choosing U01=diag⁡00.20, T=1, u1max=0.01, u2max=0.08, λ1=0.1, and λ2=ɛ=10, by solving (23), we have(45)K1=-0.0029-0.06010.0482-0.0009-0.01910.0154,K2=-0.00870.02420.0242-0.3696,L1=0.6584-0.2042-0.3962-0.04300.77100.0020-0.93303.37980.3040,L=10-3×-0.0124-0.0111-0.00660.0844-0.45820.1109-0.00000.0001-0.0000.Using the developed bounded controller, simulation results for the three-dimension uncertain discrete time-delay system with nonlinearity and disturbance are shown in Figures8–14. From Figures 8–10, we can observe that the stability of the closed-loop system can be guaranteed under the observer-based bounded controller. According to Figure 11, we can see that the outputs of the state estimator can estimate the system state effectively. The disturbance estimate outputs d^1(k), d^2(k) and the disturbance d1(k), d2(k) are presented in Figures 12 and 13 which illustrate the effectiveness of the proposed disturbance observer. Figure 14 indicates that the control input is bounded under the designed controller. Based on these simulation results, we can obtain that the state estimator and the disturbance observer can effectively approximate the system state and the disturbance, and the proposed observer-based bounded control is valid for the discrete-time uncertain nonlinear system with time delays and disturbance.Figure 8
The system statex1(k).Figure 9
The system statex2(k).Figure 10
The system statex3(k).Figure 11
The estimation error outputs ofx~1(k), x~2(k), and x~3(k).Figure 12
The disturbanced1(k) and the estimation output of d^1(k).Figure 13
The disturbanced2(k) and the estimation output of d^2(k).Figure 14
The control inputsu1(k) and u2(k).


## 5. Conclusion

Stability of the uncertain discrete time-delay systems with disturbance and nonlinearity has been studied in this paper. The state estimator and the disturbance observer have been designed to estimate the unmeasured state vector and the unknown disturbance generated by the exogenous system. A bounded controller has been developed by the outputs of the state estimator and the disturbance observer. The stability of the closed-loop system has been proved via LMIs. Simulation examples have been provided to show the effectiveness of the proposed scheme. In the future, the time-varying state delay and input delay can be further considered for the discrete-time uncertain nonlinear systems with disturbance and input saturation.



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*Source: 135248-2015-11-16.xml*