# H ∞ Control for LPV Discrete Systems with Random Time-Varying Network Delay

Authors: Kewang Huang; Tao Ma; Feng Pan
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, we study theH∞ control problem for Linear Parameter Varying (LPV) discrete systems with random time-varying network delay. The state matrices of LPV discrete systems are deterministic functions and changed with parameters; the range of parameters is measurable. Considering the characteristics of networks with random time-varying delay, we proposed a new parameter-dependent H∞ performance criterion based on the Lyapunov stability theory. The coupling between Lyapunov functions and system matrices could be eliminated by introducing an additional matrix in this criterion, which made it easier for numerical implementation. On this basis, we designed a state feedback controller by virtue of linear matrix inequalities, which transformed the sufficient conditions into existence condition of solution of parametric linear matrix inequalities. The designed controller could keep the closed-loop system asymptotically stable under given time delay and probability and meet predefined performance metric. The validity of the proposed method is verified by numerical simulation.


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## Body

## 1. Introduction

In recent years, the gain scheduling control has become one of the most effective methods in nonlinear control domains, and the application of gain scheduling control based on Linear Parameter-Varying (LPV) system is boosting. The LPV system is a type of system with constantly changing parameters [1–4]. The state matrices of LPV systems are deterministic functions with time-varying parameters, and the range of these time-varying parameters associated with the functions can be online measured. However, most of the control research works for physical systems use a normal (or regular) model. In fact, LPV systems are of quite importance for the physical representation of some real systems [5–7]. Nowadays, the LPV system is widely used in aviation, aerospace, robotic, industrial control, and so on. The reports about the LPV systems can be found in much literature [7–10].The modern networked control systems are developing rapidly by virtue of such advantages as low cost, easy installation, high reliability/flexibility, strong fault tolerance/diagnosis ability, convenient remote manipulation/control, and so on. One of the most important requirements for a control system is the so-called robustness [11]. However, the combination of LPV system with network will cause problems as data quantization, network delay, and data loss, which will deteriorate the performance metric of the whole system and even lead to system instability [12]. Due to the network delay, system analysis will become much more complex and difficult in real closed-loop control system and the designed controller without considering network delay will always cause instability or the degradation of performance metric. Therefore, it is necessary to investigate the LPV system with network delay. To date, there are only a few reports about the networked LPV system. Wang [13] and Mehendale & Grigoriadis [14] investigated H∞ control of the LPV system with inherent delay. Hencey & Alleyne [15] analyzed the stability of LPV system with time delay. Wu & Su & Shi [16], Xiao & Jia & Matsuno [17], Shao [18], and Deuce & Sipahi [19] analyzed the time delay problems for networked control system. Faiz & Sing [20] and Luan & Shi & Liu [21] researched the stability problems of networked control system with random time delay. They proposed the design of controller with random time delay, which will keep the closed-loop system asymptotically stable. Yuan [22] investigated the stability of LPV system with state time delay and the controller design; he proposed the system model structure and designed an effective controller, which will keep the system asymptotically stable. However, H∞ control problem of LPV discrete systems with random time-varying network delay is still an open problem. The investigation of these systems has practical reference value and realistic signification. Therefore, we investigate the above-mentioned LPV discrete systems with random time-varying network delay in this paper.In summary, we investigate the control of LPV discrete system with time-varying network delay while there is very little literature on this aspect. The main contribution of this paper is as follows. First, it has lower conservative property in comparison with constant time delay system. We constructed a suitable Lyapunov function, and the occurrence of network-induced time delay was described by sequence generated from Bernoulli distribution. Secondly, the corresponding controller was designed to keep the quantized closed-loop LPV system asymptotically stable. Finally, the proposed controller design conditions were transformed into an optimization problem. The optimal solutions can be obtained by linear matrix inequalities based on stability theory and system can meet optimalH∞ performance metric [23–25].The following sections of this paper are as follows. Section2 outlines the problem formulation. Section 3 describes the main theorems with their proofs. Section 4 demonstrates numerical simulation results. Section 5 gives conclusion for this paper.


## 2. Problem Formulation

Consider the following polytypic LPV discrete control system:(1)xk+1=Aρkxk+B1ρkuk+B2ρkωkzk=Cρkxk+Dρkuk(2)xk=Φk,k∈-d,0where A(ρ(k))B1(ρ(k))B2(ρ(k))C(ρ(k))D(ρ(k))0=∑i=1rαiAiB1iB2iCiDi0,αi≥0,∑i=1rαi=1;x ( k ) ∈ R n represents the state vector; z(k)∈Rn represents the system output; u(k)∈Rp represents the system control input; ω(k)∈Rq represents the disturbance input. αi represents the coordinate of polytope. Ai, B1i, B2i, Ci, and Di represent each vertex of the polytope for system matrix A(ρ(k)), B1(ρ(k)), B2(ρ(k)), C(ρ(k)), and D(ρ(k)), respectively. {Φ(k),k=-d,-d+1,⋯,0} represents a sequence with known initial conditions. ρ(k)=ρ1(k)ρ2(k)⋯ρs(k)T represents a parameter vector and ρi(k) can be online measured and fall into the closed interval of [ρ_i,ρ-i]. For convenience, we use ρ and ρi to represent ρ(k) and ρi(k), respectively.Construct the following state feedback controller:(3)uk=1-δkKρxk+δkKρxk-dkwhere 1≤dm≤d(k)≤dM and K(ρ)∈Rp×n represents the unknown gain matrix of state feedback controller, which depends on parameter vector ρ(k), d(k) represents the random time-varying network delay. dm and dM represent the lower and upper bound of time delay, respectively. δk represents whether a delay will occur or not when the signal is transmitted from the sensor to the controller through network channel. The Bernoulli distribution sequence with a value of 0 or 1 is utilized to represent the occurrence of time delay. δk=0 means no random time delay while δk=1 means random time delay.Suppose the probability of random time-varying delay is(4)probδk=1=Eδk=αprobδk=0=E1-δk=1-αwhere 0≤α≤1 is a known constant.From (1) and (3), we can obtain the following closed-loop LPV control system:(5)xk+1=A¯ρxk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dkzk=C¯ρxk+D¯ρxk-dk+α-δkDρKρxk-α-δkDρKρxk-dkwhere A¯(ρ)=A(ρ)+(1-α)B1(ρ)K(ρ), Ad(ρ)=αB1(ρ)K(ρ), C¯(ρ)=C(ρ)+(1-α)D(ρ)K(ρ), and D¯(ρ)=αD(ρ)K(ρ).First, we investigate theH∞ performance criteria for (1). Secondly, we design the corresponding feedback controller, which should meet the following two metrics.(a) Equation (5) should be exponentially mean square stable.(b) Equation (5) has a certain level of disturbance attenuation for H∞; i.e., the gain from the disturbance input ω(k) to the system output z(k) is less than a predefine value. Moreover, (5) should meet the following criterion under zero initial condition (i.e., Φ(k)=0)(6)Ez22≤γ2ω22where performance metric γ>0, ω22=∑k=0∞ωT(k)ω(k) and z22=∑k=0∞zT(k)z(k).


## 3. Main Theorems

Lemma 1 (see [26] (Schur complement)).
Given real matricesL1,L2,L3 where L1=L1T and L2=L2T<0, then L1-L3TL2-1L3≤0 if and only if L1L3TL2L2≤0 or L2L3L3TL1≤0.Theorem 2.
For (5), given the probability of communication time delay occurrence 0<α≤1 and the performance metric γ>0, if there are matrices P(ρ)>0, Q(ρ)>0, Z>0, Y1, and Y2 such that the following linear matrix inequality equation holds for all parameters changing trajectories, then (5) meets the above-mentioned two performance metrics.(7)∏1∏2∗∏3<0where ∗ represents the symmetric matrices block transpose,(8)∏1=-Pρk+dM-dm+1Qρk+Y1+Y1TY2T-Y10∗-Qρk-dk-Y2-Y2T0∗∗-γ2I∏2=A¯TρPρk+1dMA¯Tρ-IZl·B1ρKρTPρk+1-dkY1C¯Tρβ·DρKρTAdTρPρk+1dMAdTρZ-l·B1ρKρTPρk+1-dkY2D¯Tρ-β·DρKρTB2TρPρk+1dMB2TρZ0000,∏3=-Pρk+100000∗-dMZ0000∗∗-Pρk+1000∗∗∗-dkZ00∗∗∗∗-I0∗∗∗∗∗-I,l=1+dMα1-α,β=α1-α.Proof.
Construct the following parameter-dependent Lyapunov function:(9)Vk,ρk=V1+V2+V3+V4where V1=xT(k)P(ρ(k))x(k), V2=∑i=k-d(k)k-1xT(i)Q(ρ(i))x(i), V3=∑j=-dM+2-dm+1∑i=k+j-1k-1xT(i)Q(ρ(i))x(i), V4=∑i=-dM-1∑m=k+ik-1δT(m)Zδ(m), δ(k)=x(k+1)-x(k).
Note thatE(α-δk)2=β2 and E(α-δk)=0; therefore we have(10)EVk+1,ρk+1-Vk,ρk=EΔV1+EΔV2+EΔV3+EΔV4ΔV1=xTk+1Pρk+1xk+1-xTkPρkxk=A¯ρxk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dkT·Pρk+1·A¯ρxk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dk-xTkPρkxkΔV2=xTkQρkxk-xTk-dkQρk-dkxk-dk+∑i=k-dm+1k-1xTiQρixi+∑i=k-dk+1+1k-dmxTiQρixi-∑i=k-dk+1k-1xTiQρixi≤xTkQρkxk-xTk-dkQρk-dkxk-dk+∑i=k-dM+1k-dmxTiQρixiΔV3=∑j=-dM+2-dm+1xTkQρkxk-xTk+j-1Qρk+j-1xk+j-1=dM-dmxTkQρkxk-∑i=k-dM+1k-dmxTiQρixiΔV4=∑i=-dM-1δTkZδk-δTk+iZδk+i≤dMxk+1-xkTZxk+1-xk-∑m=k-dkk-1δTmZδm=dMA¯ρ-Ixk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dkT·Z·A¯ρ-Ixk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dk-∑m=k-dkk-1δTmZδmThen we have(11)ΔVk,ρk=ΔV1+ΔV2+ΔV3+ΔV4≤1dk∑m=k-dkk-1Ξm,ρkwhere (12)Ξm,ρk=A¯ρxk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dkT·Pρk+1·A¯ρxk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dk-xTkPρkxk+dM-dm+1xTkQρkxk-xTk-dkQρk-dkxk-dk+dM·A¯ρ-Ixk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dkT·Z·A¯ρ-Ixk+Adρxk-dk+B2ρwk+α-δkB1ρKρxk-α-δkB1ρKρxk-dk-dkδTmZδm.
Based onxk-xk-dk=∑m=k-dkk-1δm, for any matrices Y1 and Y2,(13)Λ=1dk∑m=k-dkk-1xkxk-dkTY1Y2xk-xk-dk-dkδm=0
Then we have(14)EΔVk,ρk=EΔVk,ρk+2Λ≤1dk∑m=k-dkk-1ηTk,mψηk,mwhere(15)ηk,m=xkxk-dkwkδm,ψ=ψ11ψ12ψ13-dkY1∗ψ22ψ23-dkY2∗∗ψ330∗∗∗-dkZ,ψ11=A¯TρPρk+1A¯ρ-Pρk+dM-dm+1Qρk+Y1+Y1T+dMA¯ρ-ITZA¯ρ-I+l2B1ρKρTPρk+1B1ρKρψ12=Y2T-Y1+A¯TρPρk+1Adρ+dMA¯ρ-ITZΑdρ-l2B1ρKρTPρk+1B1ρKρψ22=AdTρPρk+1Adρ+dMAdTρZAdρ-Qρk-dk-Y2-Y2T+l2B1ρKρTPρk+1B1ρKρψ13=A¯TρPρk+1B2ρ+dMA¯ρ-ITZB2ρ,ψ23=AdTρPρk+1B2ρ+dMAdTρZB2ρψ33=B2TρPρk+1B2ρ+dMB2TρZB2ρ
Further, the norm of LPV system output is(16)zk2=C¯ρxk+D¯ρxk-dk+α-δkDρKρxk-α-δkDρKρxk-dkT·C¯ρxk+D¯ρxk-dk+α-δkDρKρxk-α-δkDρKρxk-dk
Combining (14) and (16), we have(17)EΔVk,ρk+Ezk2-γ2wk2≤ηTkψηk-γ2wkTwk+ηTkC¯TρD¯Tρ00C¯TρD¯Tρ00Tηk+ηTkβ·DρKρT-β·DρKρT00β·DρKρT-β·DρKρT00TηkAccording to Schur complement theory, (7) can keep that EΔV(k,ρ(k))+Ez(k)2-γ2w(k)2<0; i.e., the closed-loop LPV system is exponential mean square stable and meets the predefined H∞ performance metrics. This completes the proof.We can conclude from Theorem2 that there exists a product term concerning with the Lyapunov function matrix and the closed-loop system matrix, which makes (7) change into bilinear matrix inequalities. To solve this problem, an additional matrix is introduced to eliminate the coupling between the system matrix and the parameter-dependent Lyapunov function [27]. A new H∞ performance metric is obtained for further design of gain scheduling feedback controller.Theorem 3.
Given the probability of communication time delay occurrence0<α≤1 and the performance metric γ>0, if there are symmetric positive definite matrices P¯, Q¯(ρ), Z, and matrices K¯(ρ), Y1¯,and Y2¯, such that the following linear matrix inequality equation holds for all parameters changing trajectories, then (5) is asymptotically stable and meets the predefined H∞ performance metrics.(18)Θ11Y¯2T-Y¯10Θ14dMΘ15lM-dkY¯1Θ18βN∗Θ220P¯AdTρdMP¯AdTρ-lM-dkY¯2P¯D¯Tρ-βN∗∗-γ2IB2TρdMB2Tρ0000∗∗∗-P¯00000∗∗∗∗dMZ-10000∗∗∗∗∗-P¯000∗∗∗∗∗∗-dkP¯ZP¯00∗∗∗∗∗∗∗-I0∗∗∗∗∗∗∗∗-I<0where(19)Θ11=-P¯+dM-dm+1Q¯ρk+Y¯1+Y¯1T,Θ14=P¯A¯Tρ+1-αB1ρKρT,Θ15=P¯A¯Tρ-I+1-αB1ρKρT,Θ18=P¯C¯Tρ+1-αDρKρT,Θ22=-Q¯ρk-dk-Y¯2-Y¯2T,Adρ=αB1ρKρ,D¯ρ=αDρKρ,M=P¯B1ρKρT,N=P¯DρKρT,
If (18) has a solution, then the gain matrix of controller is (20)Kρk=K¯ρkP¯-1Proof.
Suppose there are symmetric positive definite matricesP¯>0, Q¯(ρ)>0, Z>0, and matrices K¯(ρ), Y1¯, and Y2¯ meet (18). We do the equal transformation on (18) based on reversible matrix diagP¯-1,P¯-1,I,P¯-1,Z,P¯-1,P¯-1,I,I. Moreover, letting P=P¯-1, Q(ρ(k))=P¯-1Q¯(ρ(k))P¯-1, Y1=P¯-1Y1¯P¯-1, Y2=P¯-1Y2¯P¯-1, P(ρ(k))≡P, we can draw the conclusion that (18) is equivalent to (7). Therefore, the gain of controller designed by (18) can keep the closed-loop system stable, and system can meet the predefined performance metrics. This completes the proof.


## 4. Numerical Simulations

In this section, we will provide numerical simulation to show the effectiveness of the proposed method. The system matrix is originated from literature [22]. We consider the following polytypic LPV control system:(21)xk+1=0.11+0.2ρ1-1.2-1+0.1ρ2xk+0.20.3uk+0.20.2ωkzk=0100xk+01ukwhere ρ1 and ρ2 are online measurable time-varying parameters, and they are in the intervals ρ1∈-1,1 and ρ2∈1,2, respectively.The vertex of polytypicΘ is ϑ1,ϑ2,ϑ3,ϑ4=-1,1,-1,2,1,1,1,2. Let A(ρ)=∑i=1rαiAi and the initial state of the system is x0=1,-0.5, α1=0.1, α2=0.2, α3=0.3, α4=0.4. The disturbance is defined as follows:(22)ωk=10.1+k2.The state response of (21) is shown in Figure 1. As can be seen in Figure 1, the state response is divergent within the range of vector ρ(k), and the final state response will approximate infinity. Therefore, the original polytypic LPV system described by (21) is unstable.Figure 1
The state response of open-loop system.Consider theH∞ control, letting 1≤d(k)≤3, dm=1, dM=3, K(ρ)=∑i=1rαiKi, we substitute Θ into the system matrix. When d(k)=1 and α changes in the interval [0,1], we can obtain optimal performance metric under different time delay probabilities by solving (18), as shown in Table 1. The corresponding state responses are shown in Figures 2–4.Table 1
Performance metricγ under different time delay probabilities α.
time delay probabilityα performance metricγ 0 0.2079 0.2 0.2151 0.5 0.2271 0.6 0.2317 0.8 0.2429 1.0 0.2587Figure 2
The state response of closed-loop system whenα=0.2 and d(k)=1.Figure 3
The state response of closed-loop system whenα=0.5 and d(k)=1.Figure 4
The state response of closed-loop system whenα=0.6 and d(k)=1.Table1 implies that γ will gradually increase with the increase of α. This means the anti-interference ability will gradually deteriorate. As can be seen in Figures 2–4, the system is stable if d(k)=1 is fixed and α is less than or equal to 0.5; the system will become unstable gradually if d(k)=1 is fixed and α is greater than 0.6. In most cases, α is less than or equal to 0.5. Therefore, the controller designed in this paper can keep the closed-loop system stable and make system meet the predefined H∞ performance metric.Whenα=0.5 is fixed, the performance metric of the system will also change with the time delay, as shown in Table 2, and its corresponding state response is shown in Figure 5.Table 2
Performance metricγ under different time delay scale d(k).
time delay scaled(k) performance metricγ 1 0.2271 2 0.2551 3 0.2764Figure 5
The state response of the closed-loop system whenα=0.5 and d(k)=2.Table2 implies that γ will gradually increase with the increase of d(k). This means the anti-interference ability will gradually deteriorate. As can be seen in Figures 3 and 5, system stability and performance metric will be affected with the increase of d(k) when α=0.5 is fixed; the system is stable with good performance when d(k) is less than or equal to 2; the system is divergent, and it will gradually change into unstable system when d(k) is greater than or equal to 3.In summary, the stability and performance of the system are both affected by time delay probability and scale. The stability and performance of the system will deteriorate with the increasing of time delay probabilityα or time delay scale d(k). The controller designed in this paper can keep the closed-loop system asymptotically stable and meet predefined performance metric under given delay and probability.


## 5. Conclusion

In this paper, we investigateH∞ control of LPV polytypic system with random time-varying network delay. A method for the robust H∞ controller of gain scheduling LPV with random time-varying network delay is proposed based on the gain scheduling and H∞ theory. Utilizing LPV convex decomposition technique, each vertex of the polytypic LPV system is used to design the feedback gain so as to meet the H∞ performance metric and dynamic characteristics, respectively. Firstly, the feedback controller should be exponentially mean square stable under random delay distribution sequence conditions. Secondly, the designed system should meet predefined performance metric when taking anti-interference ability into account. Compared with the time delay in existing literature, we consider network-induced time delay; it will change stochastically, and it is a sequence under certain distribution. We can give the conclusion from equation derivations and numerical simulations that time delay is a key element for system stability. The stability and performance of the system will deteriorate if the time delay probability or the time delay scale is increasing. The proposed controller in this paper can keep closed-loop system asymptotically stable and meet expected performance under given delay and probability.



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