# Stabilization of Discrete-Time Delayed Systems in Presence of Actuator Saturation Based on Wirtinger Inequality

Authors: Vipin Chandra Pal; Richa Negi; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper examines the stability analysis of discrete-time control systems particularly during the event of actuator saturation and time-varying state delay. With the help of Wirtinger inequality along with Lyapunov-Krasovskii functional gain of state feedback controller is determined for stabilization of above system. The saturation nonlinearity is represented in the terms of convex hull. A new linear matrix inequality (LMI) criterion is settled with reciprocally convex combination based inequality which is dependent on delay. The proposed criterion is less conservative in concern to increase the delay bound and a controller is also simulated for real time problem of missile control system in this paper. It is also attained that projected stability criterion is less conservative compared to other outcomes. Furthermore, an optimization procedure together with LMI constraints has been proposed to maximize the attraction of domain.


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## Body

## 1. Introduction

Practically, every system is distressed due to common non-linearities like saturation, deadzone, backlash, etc. Among them actuator saturation is more general in industrial control systems since there is always limitation in inputs of control in the form of amplitude as well as rate saturation. Saturation nonlinearity leads to degradation of system stability and is responsible for poor performance of closed loop system. In most of the research works, sector based conditions [1] and convex hull approach [2] are selected to work on the actuator or input saturation nonlinearity. The problem of control system stability with actuator saturation has received extensive attention [3–10].Time delay is another phenomenon which exists in dynamical systems due to communication between data processing elements in computerised control systems, e.g., processes controlled via SCADA in chemical industries [11], optimal communication network systems [12], metrology systems [13], power systems, and neural network control system [14, 15]. Performance deterioration and instability are other issues created by this phenomenon. An adequate amount of work has been carried out on delayed systems [[16–21] and the references therein]. There are two common and popular ways to deal with time delayed systems, i.e., dependent on delay and not dependent on delay. The information about delay is utilized in delay-dependent method which yields less conservative results for limited delay range. Presently, many researchers are devoting their attention in stability analysis of time delayed processes using various approaches (such as model transformation, free weighting matrix, free matrix based integral inequality, Park’s inequality, Jensen’s inequality, Convex combination method and Wirtinger-based integral inequality, etc.) for achieving less conservative results.Numerous versions of Wirtinger-based integral inequality have been applied for stability of open loop system in presence of delay, which provides less conservative result as compared to well-known Jensen’s inequality [22–28]. In the same direction, the above cited new advancements are tested on delayed discrete systems [29–31]. In [28], an accurate integral inequality [32] is considered to reduce the conservatism and it is dependent on x(t),x(t-d) as well as on ∫t-dtx(s)ds. A new summation inequality is utilized for discrete-time delay systems in [30], which is extension of integral inequality given in [27]. The discrete version of Wirtinger’s inequality is derived in [30] and also applied for stabilization of delayed discrete systems along with interval like time-varying delay. A novel summation inequality has been projected for delayed discrete systems by considering Lyapunov-Krasovskii functional with two additional terms in difference of forward calculation in [31]. An advanced sequence of summation inequality has been obtained in [22], which includes Jensen, Wirtinger and Free matrix summation based inequality ones as specific case. Moreover, advancement in previous results has been obtained by Zhang et al. in 2018 [33] by extending the Wirtinger’s inequality. In [34], discrete inequalities for single summation and double summation have been proposed which are based on multiple auxiliary functions and include the Jensen discrete inequality and the discrete Wirtinger-based inequality as special cases. In [35], a new augmented Lyapunov-Krasovskii functional with single and double summation terms is used to derive the stability conditions for discrete-time neural networks by using Wirtinger-based inequality and decomposing the time-delay interval into two non-equidistant subintervals. Delay-dependent stability criteria are established by using Wirtinger inequality approach, and based upon the memoryless H∞ state feedback controller is designed for uncertain linear system with time-varying delay and disturbance [36]. They have utilized all possible configurations of the delay such as its, upper bounds, lower bounds and derivative of upper bounds. A new method of delay partitioning is introduced by [37], in which tractable conditions for stability are deduced for discrete time- delayed systems and the obtained conditions are also dependent on the partitioning part.Further, the problems reported in the above literature are related to stability analysis of considered system in the absence of controller design part, which is essentially required to stabilize the practical systems. For stabilization of closed loop systems, controller design is a very challenging task in industrial processes inflicted to different nonlinearities in which what is more challenging is the input saturation along with time-varying state delay. Very few results are obtained for controller design for time delayed systems [36]. Stability analysis and control synthesis problems incorporated with both nonlinearities (i.e., saturation and delay) have appeared in [8, 21, 38, 40] using LMI technique.Inspired by the above literature, especially [30], in this manuscript a controller based on feedback of state is invented to stabilize discrete systems along with input saturation and time-varying state delays through novel summation inequality in LKF. A new and less conservative stability criterion is established by using Lyapunov-Krasovskii functional based on reciprocal convex approach along with Wirtinger inequality. In this work, novelty is to synthesize a feedback controller for discrete-time systems in presence of nonlinearities like actuator saturation and time-varying state delays, and maximized domain of attraction is estimated. The anticipated criterion is simulated for stabilization of missile control system.The manuscript is arranged as follows. In Section2, problem is formalized and necessary lemmas are recalled. A delay-dependent approach is utilized for deriving asymptotic stability conditions in Section 3. To show the adequacy of the projected work numerical examples are demonstrated in Section 4. Finally, the paper ends with general remarks.Notations.Throughout this paper, the following representations are used:Rm×n-m×n real matrices;Rm- m×1 real matrices;I -identity matrix of suitable dimension;0 - null matrix or null vector;ΩT- transpose matrix of Ω;λmax(Ω) - maximum eigenvalue of any given matrix Ω;diaga1,a2,…,an- diagonal matrix with diagonal elements a1,a2,…,an;∗ - symmetricity in matrices;.- Euclidean normHe(M)-M+MT.


## 2. Problem Formulation and Prefaces

A Discrete-time system is considered in which the effect of actuator saturation and delay is present,(1a)xq+1=Axq+Adxq-dq+Bsatuq(1b)yq=Cpxq(1c)xq=ϕqq∈-d2,0where (1d)A=A11A12A21A22,Ad=Ad11Ad12Ad21Ad22,q∈z¯+ and z¯+ specify non-negative integers. State vector is represented by x(q)∈Rn, control vector as u(q)∈Rp whereas output vector is y(q)∈Rt. Matrices A∈Rn×n, Ad∈Rn×n, B∈Rn×p, Cp∈Rt×n are known value matrices, ϕ(q) is the initial condition and d(q) is time-varying delay.The saturation function is illustrated as below forn=1,2,…,2p as(2)satu¯n=1ifu¯n>1u¯0nif-1≤u¯n≤1-1ifu¯n<-1A d(q) is time-varying delay satisfying the following equation (3) as (3)d1≤dq≤d2for positive lower delay bounds d1 and upper delay bounds d2, respectively.A state feedback controller is considered asu(q)=Kx(q) for stabilization of closed loop system (1a), (1b), and (1d) (4a)xq+1=Axq+Adxq-dq+BsatKxq(4b)yq=CpxqIn this work the saturation nonlinearity is tackled by convex hull approach which is stated in Lemma 1.Lemma 1 (see [2]).
The set of convex hull is described as(5)satKxq∈coDmK+Dm-H¯xq,m∈1,M^for matrices K and H¯∈Rp×n so that all x(q)∈Rn satisfying H¯sx(q)≤1,s=1,2,…,p
whereH¯t is tth row of the matrix H¯ and co. represents convex hull. In this case there exist δ1≥0,…,δm≥0, with ∑m=1M^δ¯m=1 where M^=2p.Taking a setΨ∈Rp×p make up of elements either 1 or 0 which are diagonal, such as for p=2, Ψ=D¯1,D¯2,D¯3,D¯4(6a)=0000,0001,1000,1001For 0<P∈Rn×n, ellipsoid ε(P,1) is described as(6b)εP,1=xq∈Rn:xTqPxq≤1.A set l(H¯) is deduced as follows:(6c)lH¯=xq∈Rn:H¯sxq≤ρ,s=1,2,…,p When x(q)∈l(H), it follows from Lemma 1 that (7)satKxq=∑m=1M^δmDmK+Dm-H¯xqNext, using (5)–(7), equation (4a) and (4b) becomes as (8a)xq+1=∑m=1M^δmA~mxq+Adxq-dq=A⌢xq+Adxq-dq(8b)yq=Cpxqwith(9)A⌢=∑m=1M^δmA⌢m;A⌢m=A⌢11mA⌢12mA⌢21mA⌢22m;A⌢ijm=Aij+BiDmKj+BiDm-Hj. The initial condition for closed loop system (8a) and (8b) is given asx 0 = ϕ x ( q ), q=-d2,-d2+1,…,0 be ϕ(q,x0)and domain of attraction of the origin of system (8a) and (8b) is (10a)Γ≜ϕxq,q=-d2,-d2+1,…,0:limq→∞ϕxq,x0=0Domain of attraction is estimated as Eδ⊂Γ where(10b)Eδ≜ϕxq,-d2≤q≤0:maxϕxq≤δ.Lemma 2 (see [29, 30]).
For a given symmetric positive definite matrixW∈Rm×m, any sequence of discrete-time variable x(q) in -d0∩Z→Rm, the following inequality holds:(11)∑i=-d+10ωTiWωi≥1dθ0θ1TW003d+1d-1Wθ0θ1where d>1, ω(i)=x(i)-x(i-1) and(12)θ0=x0-x-h,θ1=x0+x-d-2d+1∑i=-d0xi. Since the factor (d+1)/(d-1) is difficult to handle in some practical systems with time-varying delay, the following Lemma 3 is given to make this term disappear from the inequality.Lemma 3 (see [30]).
For a given symmetric positive definite matrixW∈Rm×m, any sequence of discrete-time variable x in -d0∩Z→Rm, where d≥1, then the following inequality holds:(13)∑i=-d+10ωTiWωi≥1dθ0θ1TW003Wθ0θ1where ω(i)=x(i)-x(i-1) and(14)θ0=x0-x-h,θ1=x0+x-d-2d+1∑i=-d0xi.Lemma 4 (see [30, 41]).
For any vectorsσ1, σ2 matrices T, S and real numbers α1≥0, α2≥0 satisfying(15a)TS∗T≥0,α1+α2=1,(15b)σk=0ifαk=0k=1,2then(15c)-1α1σ1TTσ1-1α2σ2TTσ2≤-σ1σ2TTS∗Tσ1σ2


## 3. Main Results

The main outcomes of the research are as follows:Theorem 5.
For the scalarsd1, d2 fulfilling the condition 0<d1<d2 and any time-varying delay d(q)∈d1d2, if there exist symmetric matrices 0<P=diag(P1,P2,P3)∈R3n×3n, 0<Qq(q=1,2)∈Rn×n, 0<Zq(q=1,2)∈Rn×n, 0<Xq(q=1,…,3)∈Rn×n, controller gain matrix K∈Rp×n, H¯∈Rp×n, matrix Sq(q=1,…,4) with pertinent dimension fulfilling the following LMIs (16)–(18)(16)ϕ=Z~100∗Z~2S∗∗Z~2>0(17)P1HsT∗ρ2>0,s=1,2,…,p(18)ψ11-2Z1006Z100A⌢Td1A⌢-ITd12A⌢-IT∗ψ22ψ23ψ246Z16Z22S2+2S4000∗∗ψ33ψ340ψ36ψ37AdTd1AdTd12AdT∗∗∗ψ440-2S3T+2S4T6Z2000∗∗∗∗-12Z100000∗∗∗∗∗-12Z2-4S4000∗∗∗∗∗∗-12Z2000∗∗∗∗∗∗∗-P1-100∗∗∗∗∗∗∗∗-Z1-10∗∗∗∗∗∗∗∗∗-Z2-1<0where(19)Z~1=diagZ1,3Z1,Z~2=diagZ2,3Z2,d12=d2-d1,S=S1S2S3S4 then, the closed loop system defined by (8a) and (8b) is asymptotically stable using feedback controller of gain K. The predicted domain-attraction for (8a) and (8b) is represented by Γδ≤1
where(20)Γδ=δ2λmaxP1+d1λmaxP2+d2-d1λmaxP3+d1λmaxQ1+d2-d1λmaxQ2+2d1d1+1λmaxZ1+2d2-d1d2+d1+1λmaxZ2.Proof.
Following quadratic Lyapunov-Krasovskii functional is considered(21)V⌢xq=∑a=13V⌢axqwhere(22)V1xq=xTq∑i=q-d1q-1xTi∑i=k-d2q-d1-1xTiPxq∑i=q-d1q-1xi∑i=q-d2q-d1-1xi(23)V2xq=∑i=q-d1q-1xTiQ1xi+∑i=k-d2q-d1-1xTiQ2xi(24)V3xk=d1∑i=-d1+10∑j=q+iqηTjZ1ηj+d12∑i=-d2+1-d1∑j=q+iqηTjZ2ηjwhere η(i)=x(i)-x(i-1).
An augmented vectorξ(q) is defined as(25)ξq=xTqxTq-d1xTq-dkxTq-d2vT1qvT2qvT3qTwhere(26)v1q=1d1+1∑i=q-d1qxi(27)v2q=1dq-d1+1∑i=q-dqq-d1xi(28)v3q=1d2-dq+1∑i=q-d2q-dqxiFor system (8a) and (8b) the difference of above LKF will be(29)ΔVxq=∑a=13Vaxq+1-Vaxq=∑a=13ΔVaxqwhere(30)ΔV1xq=ξTqω1+βdqTPω1+βdq-ω2+βdqTPω2+βdqξq=ξTqω1TPω1-ω2TPω2+HeβTdqPω1-ω2ξqΔV1(x(q))is obtained by following relation(31)xq+1∑i=q-d1+1qxi∑i=q-d2+1q-d1xi=A⌢-Ixq+Adxq-dq-xq-d1+v1qv2q+v3q-xk-dq-xq-d2+xqd1v1qdq-d1v2q+d2-dqv3q=ω1+βdqξq(32)xq∑i=q-d1q-1xi∑i=q-d2q-d1-1xi=0-xq+v1qv2q+v3q-xq-d1-xq-dq+xqd1v1qdq-d1v2q+d2-dqv3q=ω2+βdqξqwhere (33)ω1=A⌢-I0Ad00000-I00I0000-I-I0II,ω2=0000000-I000I000-I-I00II,βdq=I0000000000d1I0000000dq-d1Id2-dqI(34)ΔV2xq=∑i=q-d1+1qxTiQ1xi+∑i=q-d2+1q-d1xTiQ2xi-∑i=q-d1q-1xTiQ1xi+∑i=q-d2q-d1-1xTiQ2xi=xTqQ1xq-xTq-d1Q1-Q2xq-d1-xTq-d2Q2xq-d2=ξTqQ^ξqwhere(35)Q^=diagQ1,-Q1+Q2,0,-Q2,0,0,0ΔV3xq=d1∑i=-d1+10∑j=q+i+1q+1ηTjZ1ηj+d12∑i=-d2+1-d1∑j=q+i+1q+1ηTjZ2ηj-d1∑i=-d1+10∑j=q+iqηTjZ1ηj-d12∑i=-d2+1-d1∑j=q+iqηTjZ2ηj=ηTq+1d12Z1ηq+1+ηTq+1d122Z2ηq+1-d1∑i=q-d1+1qηTiZ1ηi-d12∑i=q-d2+1q-d1ηTiZ1ηiThe summation limit in the last term of (35) can be split into two parts: first one containing the terms q-d1 to q-d(q)+1 and next one from q-d(q) to q-d2+1(36)=xTq+1-xTqd12Z1+d122Z2xq+1-xq-d1∑i=q-d1+1qηTiZ1ηi-d12dq-d1dq-d1∑i=k-dq+1q-d1ηTiZ2ηi-d12d2-dqd2-dq∑i=q-d2+1q-dqηTiZ2ηiOn applying Lemma 2 on first term and Lemma 3 to the last three terms of (36), we get following relation:(37)ΔV3xq≤ξTqω1TZ~12ω1ξq-xq-xq-d1xq+xq-d1-2d1+1∑i=q-d1qxiTZ1003Z1×xq-xq-d1xq+xq-d1-2d1+1∑i=q-d1qxi-xq-d1-xq-dqxq-d1+xq-dq-2dq-d1+1∑i=q-dqq-d1xiT-d12dq-d1Z2003d12dq-d1Z2xq-d1-xq-dqxq-d1+xq-dq-2dq-d1+1∑i=q-dqq-d1xi-xq-dq-xq-d2xq-dq+xq-d2-2d2-dq+1∑i=q-d2q-dqxiTd12d2-dqZ2003d12d2-dqZ2×xk-dq-xq-d2xq-dq+xq-d2-2d2-dq+1∑i=q-d2q-dqxi where(38)Z~12=d12Z1+d122Z200000000,Z3=Z~1000d12dq-d1Z~2000d12d2-dqZ~2,Z~1=diagZ1,3Z1,Z~2=diagZ2,3Z2,θq=I-I00000II00-2I000I-I00000II00-2I000I-I00000II00-2IFurther, if there exists matrix S=S1S2S3S4∈R2n×2n such that (16) holds then based on Lemma 4, the above term can be modified as(39)ΔV3q≤ξTqω1TZ~12ω1ξq-θTqϕθqwhere(40)ϕ=Z~1000Z~2S0STZ~2,θq=I-I00000II00-2I000I-I00000II00-2I000I-I00000II00-2IEmploying (30)–(39), ΔV(x(q)) can be deduced as (41)ΔVxq≤ξTqQ^ξq+ξTqω1TPω1-ω2TPω2+HeβTdqPω1-ω2ξq+ξTqω1TZ~12ω1ξq-θTqϕθq(42)ΔVxq≤ξTqψξqwhere(43)ψ=ψ11+μ11-2Z1ψ1306Z100∗ψ22ψ23ψ246Z16Z22S2+2S4∗∗μ33+ψ33ψ340ψ36ψ37∗∗∗ψ440-2S3T+2S4T6Z2∗∗∗∗-12Z100∗∗∗∗∗-12Z2-4S4∗∗∗∗∗∗-12Z2(44a)μ11=A⌢-ITd12Z1A⌢-I+A⌢-ITd122Z2A⌢-I+A⌢TP1A⌢(44b)ψ11=-P1+P2+Q1-4Z1(44c)ψ13=A⌢TP1Ad+A⌢-ITd12Z1+d122Z2Ad(45a)ψ22=-P2+P3-Q1+Q2-4Z1-4Z2(45b)ψ23=-2Z2-S1-S2-S3-S4(45c)ψ24=S1-S2+S3-S4(46a)μ33=AdTP1AT+AdTd12Z1Ad+AdTd122Z2Ad(46b)ψ33=-8Z2+S1T+S1+S2T+S2-S3T-S3-S4T-S4(46c)ψ34=-2Z2-S1+S2+S3-S4(47a)ψ36=6Z2+2S3T+2S4T(47b)ψ37=6Z2-2S2+2S4(47c)ψ44=-4Z2-P3-Q2On applying Schur’s complement on (43), we get (48)ψ1=ψ11-2Z1006Z100A⌢Td1A⌢-ITd12A⌢-IT∗ψ22ψ23ψ246Z16Z22S2+2S4000∗∗ψ33ψ340ψ36ψ37AdTd1AdTd12AdT∗∗∗ψ440-2S3T+2S4T6Z2000∗∗∗∗-12Z100000∗∗∗∗∗-12Z2-4S4000∗∗∗∗∗∗-12Z2000∗∗∗∗∗∗∗-P1-100∗∗∗∗∗∗∗∗-Z1-10∗∗∗∗∗∗∗∗∗-Z2-1For symmetric matrices 0<Xk(k=1,…,3), we have(49)-P-1≤-2X1+X1PX1,-Z1-1≤-2X2+X2Z1X2,-Z2-1≤-2X3+X3Z2X3Remark 6.
For finding the valuesXk(k=1,2,3) the method discussed in [38, 42] is adopted. These values can be determined iteratively.
It may be perceived thatΔV(x(q))<0 for ξ(q)≠0 along with (16)-(17) are LMI conditions for asymptotic stability of closed system given by (8a) and (8b). The condition ΔV(x(q))<0 can be written as ψ1<0.
Equation (17) indicates that the set ε(P1)=x∈Rn;xTP1x≤1 is contained in polyhedral set l(H¯). It can be seen that ε(P1)=x∈Rn;xTP1x≤1 is equivalent to [43] (50)P1-HsTHsρ-2>0,s=1,2,…,pThe relation of x∈l(H¯) can be found by pre and post multiply of (50) with xT and x for all x∈ε(P1). Finally, (17) may be obtained by using Schur’s complement of (50).


### 3.1. Maximization the Domain of Attraction

The domain of attraction forϕ(q;x), set of initial state x(q), is defined as limq→0ϕ(q;x)=0 [44].Next, theorem is illustrating the procedure for finding the estimate of domain of attraction.Theorem 7.
The maximized estimate of domain of attraction may be found by the following convex optimization problem.
minimizer,
with(51)r=r1+d1r2+d2-d1r3+d1r4+d2-d1r5+2d1d1+1r6+2d2-d1d2+d1+1r7subject to (16)–(18) and(52)r1I-P1≥0,r2I-P2≥0,r3I-P3≥0,r4I-Q1≥0,r5I-Q2≥0,r6I-Z1≥0,r7I-Z2≥0has a feasible solution for the weighting parameters ri>0,i=1,2,…,7, positive definite symmetric matrices P1∈Rn×n, P2∈Rn×n, P3∈Rn×n, Q1∈Rn×n, Q2∈Rn×n, Z1∈Rn×n, Z2∈Rn×n, Xk∈Rn×n,k=1,2,3, matrix K∈Rp×n, matrix H∈Rp×n and matrix S∈R2n×2n.
In this situation, state feedback controller gain matrixK provides a maximized estimation of domain of attraction given by δmax=1/τ, where (53)τ=λmaxP1+d1λmaxP2+d2-d1λmaxP3+d1λmaxQ1+d2-d1λmaxQ2+2d1d1+1λmaxZ1+2d2-d1d2+d1+1λmaxZ2.Proof.
If the conditions given by (52) hold true, then r1I≥λmax(P1), r2I≥λmax(P2), r3I≥λmax(P3), r4I≥λmax(Q1), r5I≥λmax(Q2),r6I≥λmax(Z1),r7I≥λmax(Z2).
From (20), we get δ=Γδ/τ. Thus, if we minimize the value of r which is given by (51), estimate of domain of attraction δ is being maximized. In other words, the optimization procedure in Theorem 7 orients the solutions of (16)–(18) in order to get the domain of attraction as large as possible.Remark 8.
By using LKF to deal with delay, conservative results are obtained because in this approach complication arises in model transformation and bound of some cross terms for finding the difference of above function.


### 3.2. Robust Stabilization

Consider the discrete time delayed system (1a), (1b), and (1d) in presence of actuator saturation and uncertainty as follows:(54a)xq+1=∑i=0NμiAixq+Adixq-dq+Bsatuq∀q>0(54b)yq=Cpxq(55)xq=ϕqq∈-d2,0 where ϕ(q) is the initial condition and time varying μi(q) belongs to unit simplex with μi(q)>0,∀i=0,…,Nand∑i=0Nμi(q)=1.By using Lemma1 and u(q)=Kx(q) the closed loop system (8a) and (8b) becomes (56a)xq+1=∑i=0NμiA~ixq+Adixq-dq(56b)yq=Cpxq The delay d(q) is pretended to gratify the restraint given in (3). The extension of Theorem 5 is given as follows:Theorem 9.
For given scalarsd1, d2 fulfilling the condition 0<d1<d2 and any time-varying delay d(q)∈d1d2, if there exist symmetric matrices 0<P=diag(P1,P2,P3)∈R3n×3n, 0<Qk(k=1,2)∈Rn×n, 0<Zk(k=1,2)∈Rn×n, 0<Xk(k=1,…,3)∈Rn×n, matrix K∈Rp×n, H¯∈Rp×n, matrix Sk(k=1,…,4) with pertinent dimension fulfilling the conditions (16), (17) and the following set of LMI (57)ψ11-2Z1006Z100A⌢iTd1A⌢i-ITd12A⌢i-IT∗ψ22ψ23ψ246Z16Z22S2+2S4000∗∗ψ33ψ340ψ36ψ37AdiTd1AdiTd12AdiT∗∗∗ψ440-2S3T+2S4T6Z2000∗∗∗∗-12Z100000∗∗∗∗∗-12Z2-4S4000∗∗∗∗∗∗-12Z2000∗∗∗∗∗∗∗-2X1+X1P1X100∗∗∗∗∗∗∗∗-2X2+X2Z1X20∗∗∗∗∗∗∗∗∗-2X3+X3Z2X3<0where(58)Z~1=diagZ1,3Z1,Z~2=diagZ2,3Z2,d12=d2-d1,S=S1S2S3S4 then, the closed loop system given by (8a) and (8b) is asymptotically stable by feedback controller gain K.Proof.
The proof of the above theorem is a direct use of Schur’s complement with ensuring the convexity of LMI (18) with respect to the system matrices A⌢ and Ad.


## 3.1. Maximization the Domain of Attraction

The domain of attraction forϕ(q;x), set of initial state x(q), is defined as limq→0ϕ(q;x)=0 [44].Next, theorem is illustrating the procedure for finding the estimate of domain of attraction.Theorem 7.
The maximized estimate of domain of attraction may be found by the following convex optimization problem.
minimizer,
with(51)r=r1+d1r2+d2-d1r3+d1r4+d2-d1r5+2d1d1+1r6+2d2-d1d2+d1+1r7subject to (16)–(18) and(52)r1I-P1≥0,r2I-P2≥0,r3I-P3≥0,r4I-Q1≥0,r5I-Q2≥0,r6I-Z1≥0,r7I-Z2≥0has a feasible solution for the weighting parameters ri>0,i=1,2,…,7, positive definite symmetric matrices P1∈Rn×n, P2∈Rn×n, P3∈Rn×n, Q1∈Rn×n, Q2∈Rn×n, Z1∈Rn×n, Z2∈Rn×n, Xk∈Rn×n,k=1,2,3, matrix K∈Rp×n, matrix H∈Rp×n and matrix S∈R2n×2n.
In this situation, state feedback controller gain matrixK provides a maximized estimation of domain of attraction given by δmax=1/τ, where (53)τ=λmaxP1+d1λmaxP2+d2-d1λmaxP3+d1λmaxQ1+d2-d1λmaxQ2+2d1d1+1λmaxZ1+2d2-d1d2+d1+1λmaxZ2.Proof.
If the conditions given by (52) hold true, then r1I≥λmax(P1), r2I≥λmax(P2), r3I≥λmax(P3), r4I≥λmax(Q1), r5I≥λmax(Q2),r6I≥λmax(Z1),r7I≥λmax(Z2).
From (20), we get δ=Γδ/τ. Thus, if we minimize the value of r which is given by (51), estimate of domain of attraction δ is being maximized. In other words, the optimization procedure in Theorem 7 orients the solutions of (16)–(18) in order to get the domain of attraction as large as possible.Remark 8.
By using LKF to deal with delay, conservative results are obtained because in this approach complication arises in model transformation and bound of some cross terms for finding the difference of above function.


## 3.2. Robust Stabilization

Consider the discrete time delayed system (1a), (1b), and (1d) in presence of actuator saturation and uncertainty as follows:(54a)xq+1=∑i=0NμiAixq+Adixq-dq+Bsatuq∀q>0(54b)yq=Cpxq(55)xq=ϕqq∈-d2,0 where ϕ(q) is the initial condition and time varying μi(q) belongs to unit simplex with μi(q)>0,∀i=0,…,Nand∑i=0Nμi(q)=1.By using Lemma1 and u(q)=Kx(q) the closed loop system (8a) and (8b) becomes (56a)xq+1=∑i=0NμiA~ixq+Adixq-dq(56b)yq=Cpxq The delay d(q) is pretended to gratify the restraint given in (3). The extension of Theorem 5 is given as follows:Theorem 9.
For given scalarsd1, d2 fulfilling the condition 0<d1<d2 and any time-varying delay d(q)∈d1d2, if there exist symmetric matrices 0<P=diag(P1,P2,P3)∈R3n×3n, 0<Qk(k=1,2)∈Rn×n, 0<Zk(k=1,2)∈Rn×n, 0<Xk(k=1,…,3)∈Rn×n, matrix K∈Rp×n, H¯∈Rp×n, matrix Sk(k=1,…,4) with pertinent dimension fulfilling the conditions (16), (17) and the following set of LMI (57)ψ11-2Z1006Z100A⌢iTd1A⌢i-ITd12A⌢i-IT∗ψ22ψ23ψ246Z16Z22S2+2S4000∗∗ψ33ψ340ψ36ψ37AdiTd1AdiTd12AdiT∗∗∗ψ440-2S3T+2S4T6Z2000∗∗∗∗-12Z100000∗∗∗∗∗-12Z2-4S4000∗∗∗∗∗∗-12Z2000∗∗∗∗∗∗∗-2X1+X1P1X100∗∗∗∗∗∗∗∗-2X2+X2Z1X20∗∗∗∗∗∗∗∗∗-2X3+X3Z2X3<0where(58)Z~1=diagZ1,3Z1,Z~2=diagZ2,3Z2,d12=d2-d1,S=S1S2S3S4 then, the closed loop system given by (8a) and (8b) is asymptotically stable by feedback controller gain K.Proof.
The proof of the above theorem is a direct use of Schur’s complement with ensuring the convexity of LMI (18) with respect to the system matrices A⌢ and Ad.


## 4. Numerical Examples

The numerical example are given to demonstrate the main results.Example 1.
Consider a discrete-time delayed systems(59)A=0.8000.97,Ad=-0.1-0.10-0.1,B=10,Cp=10.5. The LMI conditions (16)–(18) and (52) in Theorem 7 are found to be feasible for delay range 1≤d(k)≤11 by using LMI Toolbox [45], and maximized estimated domain of attraction is 0.0647.
The other unknown parameters and controllerK which stabilizes the above system are obtained as-(60)K=0.0185-0.0679,P1=9.31591.08391.083915.4223,P2=0.59460.21220.21220.0826,P3=0.34420.13440.13440.5680In Table 1, the obtained result is compared with the existing one.Table 1
Maximized Delay bounds.
Method d 1 ≤ d ( q ) ≤ d 2 Theorem 2 [38] 1 ≤ d ( q ) ≤ 5  Theorem 2 [proposed work] 1 ≤ d ( q ) ≤ 11It is seen that the delay bounds in [38] are obtained as 1≤d(q)≤5 but in the proposed method the delay bounds are obtained as 1≤d(q)≤11.Example 2.
Consider discrete-time delayed systems represented by (1a), (1b), (1d), (2), (3), (4a), (4b), (5), (6a), (6b), (6c), (7), (8a), and (8b) with the parameters as follows:(61)A=1.20.50.80.7,Ad=-0.015-0.010-0.013,B=10.45,Cp=-0.0019-0.0017 Since 1.6301 and 0.2699 are calculated as the eigen values ofA, so it is unstable model. The considered system is stabilized through proposed criterion given in Theorem 7. On using LMI toolbox, conditions (16)–(18) and (52) stated in Theorem 7 are found to be feasible for delay range 1≤d(q)≤19 and maximized estimated domain of attraction is 0.0737.
The controller gainK along with other unknown parameters are found as(62)K=-1.1593-0.5452,P1=130.465824.735724.735791.3822,P2=0.4365-0.1219-0.12190.3462,P3=0.28000.00770.00770.3048In Figure 2(a), state trajectories are sketched for preliminary state of above system as x0=1.2-2.5T.
The control effort is shown in Figure2(b). From Figure 2, it can be concluded that the above unstable system is asymptotically stable under the effect of controller u(q)=Kx(q) with saturation limit of ±1 (Figure 1).Figure 1
Characteristic of saturation.Figure 2
State trajectories of system and control effort.
(a) (b)Example 3.
A missile control system given in [46] is considered in this example.
Overall closed loop transfer functionT(z) can be obtained as(63)Tz=1.31z+0.5566z2+0.2428z+0.6238 with the sampling interval T=0.1 sec, amplifier gain Ka=20 and airframe dynamics is given by(64)Dm=27ss+27. In state space the above system is represented by (65)x˘n+1=-0.2428-0.623810xn+10u˘n;y˘n=1.310.5566x˘n. The model of missile control system must be robust and have an ability to adapt significant changes in atmosphere/environment. In this framework, model of block diagram is given by the Figure 3, digital computer operated as a controller by building up the proper controlling input. The impact of delay likewise is considered for is missile control system because of consuming time in calculation/computation and transmission of flag/data.
In this way, the above system may be expressed in terms of (1a), (1b), (1d), (2), (3), (4a), (4b), (5), (6a), (6b), (6c), (7), (8a), and (8b) with the accompanying parameters(66)A=-0.2428-0.623810,Ad=-0.016-0.0120-0.013,B=10,Cp=1.310.5566. By using LMI Toolbox (Gahinet et al., 1995), the LMI conditions (16)–(18) stated in Theorem 5 are found feasible for delay range 1≤d(q)≤15.
The controller gainK and other unknown parameters are obtained as(67)K=0.55550.6290,P1=103×1.5853-0.0930-0.09300.0912,P2=398.2891-3.9665-3.96656.4803,P3=270.6726-2.8491-2.84914.8553In Table 2, the obtained result is compared with the existing one.
Forx0=0.4-0.2T (initial condition), the system trajectories are presented in the Figure 4(a) whereas control effort is predicted by Figure 4(b) for saturation limit of ±1. So, from the application of above approach, the overall system is asymptotically stable.Table 2
Maximized Delay bounds.
Method d 1 ≤ d ( q ) ≤ d 2 Theorem 1 [39] 1 ≤ d ( q ) ≤ 10  Theorem 1 [proposed work] 1 ≤ d ( q ) ≤ 15Figure 3
Control system of missile.Figure 4
State trajectories of system and control effort.
(a) (b)Remark 4.
From Table2, it is clear that when stability criteria given in Theorem 1 of [39] are applied, it is found that the system is asymptotic stable for delay range 1≤d(k)≤10 in presence of actuator saturation and time-varying delay, whereas in the present approach using Wirtinger based inequality the system is found to be asymptotically stable for delay range 1≤d(k)≤15.


## 5. Conclusion

The stability investigation of time delay control system using writinger’s inequality has been carried out in this paper for optimized performance and improvement of discrete systems used in various industries and practical systems. A qualitative analysis of the delay dependent LKF is considered and LMI based stability criterion is derived. The state feedback technique is utilized for designing of controller which stabilizes the closed loop delayed practical systems. The set of convex hull is considered to deal with input saturation. The potency of the projected work has been exposed in numerical examples section.



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