# Distributed Optimal Control for a Class of Switched Nonlinear Systems with the State Time Delay

Authors: Yuxing Duan; Baili Su
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper is focused on a kind of distributed optimal control design for a class of switched nonlinear systems with the state time delay which have a prescribed switching sequence. Firstly, we design a bounded controller to make the system stable for each mode of the nominal system. Then, a distributed optimal controller which can satisfy input constraint is designed based on the bounded stabilization controller. A sufficient condition to guarantee ultimate boundedness of the system is given based on appropriate assumption. The significance of this paper is that distributed optimal control method is applied to switched nonlinear systems with the state time delay. Finally, a simulation example is given to verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched time-delay systems are an important kind of hybrid systems, which have attracted extensive attention in recent years. There are some results for switched time-delay systems [1–4]. However, the input of switched time-delay systems is often constrained by objective conditions, and some switched systems need to operate according to a prescribed switching sequence in many practical control systems. So, there are still many challenging control problems need to be solved for this class of nonlinear systems.Regression optimal control is an optimal control method that can deal with system constraints. It obtains the current control action by solving a finite time open loop optimal control problem at every sampling moment. It has well dynamic control effect and is helpful to improve the stability of the system. Therefore, its study has received considerable attention [5–8]. Moreover, the method of combining regression optimal control with the control Lyapunov function-based bounded control has made further development in the fields of single-objective control, multiobjective control, and so on. For example, the objective function in [9] adopts the nonquadratic cost function and studies a regression optimal control based on the Lyapunov economic model, which can directly solve economic problems. In [10], a fast feedback controller that stabilizes the fast dynamics and a regression optimal controller that stabilizes the slow dynamics and enforces desired performance objectives in the slow subsystem are designed for a kind of nonlinear singularly perturbed systems. He et al. [11] study an alternative utopia-tracking multiobjective economic regression optimal control scheme of constrained nonlinear systems with guaranteed asymptotic stability and convergence of average performance.In most existing results, centralized optimal control methods are adopted. When this kind of control method is applied to the system, the computational complexity reduces the performance of the system with the increase of variables and the expansion of model size. Distributed optimal control can reduce computational burden and fault tolerance of the system by using communication and cooperation among multiple controllers. Therefore, more and more scholars study distributed optimal control. The method of combining distributed optimal control with the control Lyapunov function-based bounded control also has made further development. Heidarinejad et al. [12] study a distributed optimal control of the switched nonlinear system. It redesigns the local control system (LCS) and the new control system (NCS) by using an optimal control method based on Lyapunov function to coordinate their respective operations. The distributed structure adopted in [12] enables LCS to stabilize the closed-loop system, and NCS can communicate and cooperate with LCS, which can further improve the closed-loop performance. A distributed optimal control scheme with such distributed structure is considered for switched nonlinear time-delay systems with delayed measurements and communication disruption in [13]. Liu et al. [14] propose a distributed optimal control strategy for nonlinear systems with asynchronous and time-delay measurements. Heidarinejad et al. [15] study a distributed optimal control scheme of nonlinear systems subject to communication disruptions—communication channel noise and data losses—between distributed controllers. It is noted that the distributed optimal control method is seldom applied to nonlinear time-delay systems. Hence, it is necessary to study the problem of distributed optimal control for switched nonlinear systems with the state time delay.In this paper, a distributed optimal control method is applied to switched nonlinear systems with the state time delay. The main ideas are as follows. A bounded controller to make system stable for each mode of the nominal system is designed. Then, a kind of distributed optimal control which can satisfy input constraint is designed based on the bounded stabilization controller. A sufficient condition to guarantee ultimate boundedness of the switched system is given. Finally, a simulation example is given to verify the effectiveness of the proposed distributed optimal control strategy.


## 2. Problem Statement and Preliminaries

Consider a class of switched nonlinear systems with the state time delay as follows:(1)x˙t=fσtxt+g1σtxtu1σtt+g2σtxtu2σtt+qσtxt−τσt+kσtxtωσtt,where xt∈Rnx is the state, ujσtt∈Rnujj=1,2 is the input, ωσtt∈Rnω is external disturbance, respectively, fσt⋅, g1σt⋅, g2σt⋅, qσt⋅, and kσt⋅ are local Lipschitz functions satisfying fσt0=0, g1σt0=0, g2σt0=0, qσt0=0, and kσt0=0, and τσt denotes time delay. The initial value is xξ=φσtξ,∀ξ∈−τσt,0 with φσtξ being a continuous function and φσt=max−τσt≤ξ≤0φσtξ, ⋅ denotes the Euclidean norm. ujσtt is restricted on a nonempty set Ujσtt≔ujσtt∈Rnuj:ujσtt≤ujσtmax with ujσtmax being positive constant j=1,2. ωσtt∈Wσt≔ωσtt∈Rnω:ωσtt≤θσt with θσt being a positive constant. σt:0,∞⟶S=1,2,…,s is the right continuous switching symbol with s which denotes the number of switched modes. In other words, σti≜limt⟶ti+σt for all i, which means that a finite number of switches can only be switched in a finite time interval.In this paper,tirin and tirout denote the time of the rth switch in and out for the ith mode respectively, i.e., σtirin+=σtirout−=i. So, if tirin≤t<tirout, then system (1) can be described as(2)x˙t=fixt+g1ixtu1it+g2ixtu2it+qixt−τi+kixtωit.T
i
in
=
t
i
1
in
,
t
i
2
in
,
… and Tiout=ti1out,ti2out,… denote the time sets of switching in and out of the ith mode, respectively.The objective of this paper is to propose a distributed optimal control scheme for a class of switched nonlinear systems with the state time delay. System (1) adopts the distributed structure (see Figure 1), where u1 and u2 are designed by controller 1 and controller 2, respectively.Figure 1
The system structure.The following assumption is satisfied at modei i=1,2,…,s.Assumption 1.
For someai∈0,1, there holds(3)∂Vi∂xqixt−τi≤ΨiVixt+aiΨiVixt−τi,where Ψi⋅ is a continuous positive definite nondecreasing function.Remark 1.
Assumption1 illustrates that the decoupling between the delay state and the current state can be realized by the derivative of the Lyapunov function. This assumption is also used to realize the separation of time-delay state and nontime-delay state in [16].Remark 2.
The distributed optimal control can set up multiple part controllers to make the system stable and improve its performance. For simplicity, only two parts are considered in this paper. Distributed optimal controller designed in this paper is composed of controller 1 and controller 2. Controller 1 can stabilize the closed-loop system. Controller 2 can improve the performance of the closed-loop system by communicating with controller 1. If the distributed optimal controller is composed by multiple controllers, then controller 1 makes the closed-loop system stable, remaining controllers improve the performance of the closed-loop system through cooperation and communication with controller 1.


## 3. Main Results

### 3.1. Design of Bounded Controller

For each modei i=1,2,…,s, we can design bounded controllers acting on the following proposition.Proposition 1.
If the nominal system of system (2)(4)x˙t=fixt+g1ixtu1it+g2ixtu2it+qixt−τi,satisfies Assumption 1, we can choose bounded controllers u1it=hixt=−cix∂Vi/∂xg1ix and u2it=0 such that(5)∂Vi∂xfixt+g1ixtu1it≤−2ΨiVixt,where(6)cix=αix+αi2x+u1imaxβix4βi2x1+1+u1imaxβix2,βix≠0,0,βix=0,(7)αix=∂Vi∂xfixt+2ΨiVixt,βix=∂Vi∂xg1ixt.
Then, system (4) is asymptotically stable under the control of u1it=hixt and u2it=0.Proof.
Based on controller (6), from [17,18], the following stable set is given:(8)Φu1imax=xt∈Rnx:αix≤u1imaxβix.
The maximum estimation ofΦu1imax can be described by(9)Ωρi=xt∈Rnx:Vix≤ρi,where ρi>0 is the largest number for Ωρi⧹0⊆Φu1imax and Ωρi is invariant set of system (4).
Next, for all initial conditions inΩρ, we can prove that the controllers(10)u1it=hixt=−cix∂Vi∂xg1ixt,u2it=0,are bounded. From (6), we have(11)u1ix≤αix+αi2x+u1imaxβix4βix1+1+u1imaxβix2≤u1imaxβix+u1imaxβix2+u1imaxβix4βix1+1+u1imaxβix2=u1imax.
For system (4), Razumikhin’s original idea is that if Vixt≥Vixt−τi and dVi/dt≤0 are satisfied, then the system is stable. The derivative of function Vixt is(12)V˙ixt≤∂Vi∂xfixt+g1ixth1it+∂Vi∂xqixt−τi≤∂Vi∂xfixt−αix+αi2x+αi2xu1imaxβx21+1+u1imaxβx2+∂Vi∂xqixt−τi≤−2ΨiVxt+∂Vi∂xqixt−τi.
Because system (4) satisfies Assumption 1, we can obtain(13)∂Vi∂xqixt−τi≤−ΨiVixt+aiΨiVixt−τi.
IfVixt−τi≤Vixt, then ΨiVixt−τi≤ΨiVixt. We have(14)V˙ixt≤−1−aiΨiVixt.
Becauseai∈0,1, we have V˙ixt≤0. System (4) is asymptotically stable under control of u1it=hixt and u2it=0.


### 3.2. Distributed Optimal Control

For each modei i=1,2,…,s, a distributed optimal control scheme is designed.Firstly, we design controller 2 based on the bounded controller and the received measurementxtk at time tk. The input u2i can be calculated by the following optimal problem:(15)minu2i∈pΔ∫tktk+NΔx˜tTQix˜t+u1iTtR1iu1it+u2iTtR2iu2itdt,s.t.(16)x˜˙t=fix˜t+g1ix˜tu1it+g2ix˜tu2it+qx˜t−τi,∀t∈tk,tk+NΔ,(17)u1it=hix˜tk+mΔ,∀t∈tk+mΔ,tk+m+1Δ,m=0,1,…,N−1,(18)u2it∈U2i,∀t∈tk,tk+NΔ,(19)x^˙t=fix^t+g1ix^thixtk+mΔ+qix^t−τi,∀t∈tk,tk+NΔ,(20)x^tk=x˜tk,(21)∂Vi∂xg2ixtku2it≤0,where pΔ is a family of piecewise constants function with a sampling time Δ, NΔ is the predicted range, Qi, R1i, and R2i are the right of positive definite matrix, x˜t is the trajectory of system (4) with u1i=hixt, and u2 is the input trajectory computed by (15)–(21).The optimal solution is denoted byu2i∗ttk t∈tk,tk+NΔ.Onceu2i∗ttk is calculated, it is sent to controller 1 and the corresponding control actuator. Then, we design controller 1 based on the measurement xtk and u2i∗ttk. The input u1i is calculated by the following optimal problem:(22)minu1∈pΔ∫tktk+NΔx˜tTQix˜t+u1iTtR1iu1it+u2iTtR2iu2itdt,s.t.(23)x˜t=fix˜t+g1ix˜tu1it+g2ix˜tu2it+qix˜t−τi,∀t∈tk,tk+NΔ,(24)x˜˙t=fix˜t+g1ix˜thix˜tk+jΔ+g2ix˜tu2it+qix˜t−τi,∀t∈tk+mΔ,tk+m+1Δ,m=0,1,…,N−1,(25)u2it=u2i∗ttk,∀t∈tk,tk+NΔ,(26)u1it∈U1i,∀t∈tk,tk+NΔ,(27)x˜tk=x˜tk,(28)∂V∂xg1ixtku1it≤∂V∂xg1ixtkhixtk.The optimal solution is expressed asu1i∗ttk t∈tk,tk+NΔ.Remark 3.
We emphasize two points: (i) (18) and (26) ensure that system input satisfies the constraint. (ii) (21) and (28) guarantee the stability of mode i.


### 3.3. Stability Analysis

In this section, the stability of distributed optimal control scheme is demonstrated.Theorem 1.
Considering that switched nonlinear system (1) satisfies Assumption 1, and under distributed optimal control of (15)–(21) and (23)–(28),(i)
When there is no switch, letΔi, εi>0, 0<ρsi<ρi satisfy the following inequality:(29)−1−aiΨiρsi+LxfiM1iΔ+Lxg1iM2iu1imaxΔ+Lxg2iM3iu2imaxΔ+Lxkθi+LxqiM4iΔ+2φi≤−εi,if initial state starts from the setΩρi, and ρmi<ρi, where ρmi=maxρsiVixt+Δ:Vixt≤Ωρsi, then xt is ultimately bounded in Ωρmi.(ii)
When there is switch, the following constraints need to be added to distributed optimal control of (15)–(21) and (23)–(28):(30)Vjxtjwin≤Vjxtjw−1in−ε∗,w>1,Vixtjw−1in−ρmj>0,ρmj,w>1,Vixtjw−1in−ρmj≤0,ρj,w=1,where Vjxtjwin denotes the value of Lyapunov functions of mode j for wth time and Vjxtjw−1in is the value of Lyapunov functions of mode j for w−1th time. If (29) is satisfied, then xt is ultimately bounded in Ωρmi.Proof.
The proof is divided into two parts.(i)
When there is no switch, ifxtk∈Ωρi, from (21) and (28), we have(31)V˙ixtk=∂Vi∂xfixtk+g1ixtku1i∗ttk+g2ixtku2i∗ttk+qixtk−τ≤∂Vi∂xfixtk+g1ixtkhixt+qixtk−τi.From (14), it can be deduced that(32)V˙ixtk≤−1−aiΨiVixtk.The derivative ofVixt along the actual state trajectory in t∈tk,tk+1 is(33)V˙ixt=∂Vi∂xfixt+g1ixtu1i∗ttk+g2ixtu2i∗ttk+qixt−τi+kixtωit.It follows from (31) and (33) that(34)V˙ixt≤−1−aiΨiVixtk+∂Vi∂xfixt+g1ixtu1i∗ttk+g2ixtu2i∗ttk+qixt−τ+kixtωit−∂Vi∂xfixtk+g1ixtku1i∗ttk+g2ixtu2i∗ttk+qixtk−τi.Vixt is acontinuous differentiable and fi⋅, g1i⋅, g2i⋅, qi⋅, and ki⋅ are local Lipschitz. So, there are positive numbers Lxfi, Lxg1i, Lxg2i, Lxqi, and Lxki which make the following equations true for ξ1,ξ2∈Ωρi:(35)∂Vi∂xfiξ1t−∂Vi∂xfiξ2t≤Lxfiξ1t−ξ2t,(36)∂Vi∂xg1iξ1t−∂Vi∂xg1iξ2t≤Lxg1iξ1t−ξ2t,(37)∂Vi∂xg2iξ1t−∂Vi∂xg2iξ2t≤Lxg2iξ1t−ξ2t,(38)∂Vi∂xqiξ1t−τi−∂Vi∂xqiξ2t−τi≤Lxqiξ1t−τi−ξ2t−τi,(39)∂Vi∂xkiξ1t−τi≤Lxki.Using (35)–(39), we can rewrite (34) as follows:(40)V˙ixt≤−1−aiΨiVixtk+Lxfixt−xtk+Lxg1ixt−xtku1imax+Lxg2ixt−xtku2imax+Lxqixt−τi−xtk−τi+Lxkiθi.For allxtk∈Ωρi/Ωρsi, it follows from (32) that(41)−1−aiΨiVixtk≤−1−aiΨiΩρsi.According to the smoothness of functionsfi⋅, g1i⋅, g2i⋅, and qi⋅, there exists positive constants Mjij=1,2,3,4 such that(42)fixt≤M1i,g1ixt≤M2i,g2ixt≤M3i,qixt≤M4i.Because of the continuity ofxt, for t∈tk,tk+1, there are(43)Lxfixt−xtk+Lxg1ixt−xtku1imax+Lxg2ixt−xtku2imax≤LxfiM1iΔ+Lxg1iu1imaxM2iΔ+Lxg2iu2imaxM3iΔ,Lxqixt−τi−xtk−τi≤LxqimaxM4iΔ,2φi,M4iΔ+2φi=LxqiM4iΔ+2φi.By (40) and all initial states xtk∈Ωi/Ωρsi, we have(44)V˙ixtk≤−1−aiΨiρsi+LxfiM1iΔ+Lxg1iM2iu1imaxΔ+Lxg2iM3iu2imaxΔ+Lxkiθi+LxqiM4iΔ+2φi.Forxtk∈Ωi/Ωρsi, if (29) is true, there exists ε>0 such that(45)V˙ixt≤−εi.Integrating this bound ont∈tk,tk+1, we have(46)Vixtk+1≤Vixtk−εiΔ,Vixt≤Vixtk,∀t∈tk,tk+1.By recursively applying (46), we can obtain that if the initial state starts from Ωi/Ωρsi, the state converges to Ωρsi. If the state converges to Ωρsi⊂Ωρmi (or starts at Ωρsi), the state remains in Ωρmi, determined by the definition of ρmi. So, xt is ultimately bounded in Ωρmi.(ii)
When there is switch, there are two cases.
In case oftirin≤t<tirout and tjwin=tirout<∞, we consider that the system has an infinite number of switches. At this point, ith mode is activated. If Vix>ρmi, then the constraints of (21) and (28) guarantee that Vixtirout<Vixtirin, and the constraints of (30) ensure that Vixtirin<Vixtir−1in<⋯<ρmi. So, Vix is always decrease. Even if ith mode is not activated, there is lth mode l∈S which is activated and guarantees that Vlx is decrease and Vlx<ρml. So, xt is ultimately bounded stable in Ωρmi.
In case oftirin≤t<tirout and tjwin=tirout=∞, we consider that the system has an finite number of switches. Similarly, Vixtirin<Vixtir−1in<⋯<ρmi. When switching to the ith mode, there is xtirin∈ρmi. From this point on, distributed optimal control strategy is not subject to any switching constraints. So, xt is ultimately bounded stable.
In conclusion, Theorem1 is proved.
The manipulation inputs of the distributed optimal control scheme are defined:(47)u1it=u1i∗ttk,u2it=u2i∗ttk,∀t∈tk,tk+1.
The executive strategy of optimal control in this paper is given as follows:Step 1: controller 2 receives the measurementxtk at time tk, and then controller 2 calculates the optimal trajectory of u2i.Step 2: the optimal trajectory ofu2i is transmitted to the actuator and controller 1.Step 3: once the optimal trajectory ofu2i is received by controller 1, the optimal trajectory of u1i is calculated.Step 4: controller 1 sends the optimal trajectory ofu1i to the actuator.Step 5: go to the next moment.


## 3.1. Design of Bounded Controller

For each modei i=1,2,…,s, we can design bounded controllers acting on the following proposition.Proposition 1.
If the nominal system of system (2)(4)x˙t=fixt+g1ixtu1it+g2ixtu2it+qixt−τi,satisfies Assumption 1, we can choose bounded controllers u1it=hixt=−cix∂Vi/∂xg1ix and u2it=0 such that(5)∂Vi∂xfixt+g1ixtu1it≤−2ΨiVixt,where(6)cix=αix+αi2x+u1imaxβix4βi2x1+1+u1imaxβix2,βix≠0,0,βix=0,(7)αix=∂Vi∂xfixt+2ΨiVixt,βix=∂Vi∂xg1ixt.
Then, system (4) is asymptotically stable under the control of u1it=hixt and u2it=0.Proof.
Based on controller (6), from [17,18], the following stable set is given:(8)Φu1imax=xt∈Rnx:αix≤u1imaxβix.
The maximum estimation ofΦu1imax can be described by(9)Ωρi=xt∈Rnx:Vix≤ρi,where ρi>0 is the largest number for Ωρi⧹0⊆Φu1imax and Ωρi is invariant set of system (4).
Next, for all initial conditions inΩρ, we can prove that the controllers(10)u1it=hixt=−cix∂Vi∂xg1ixt,u2it=0,are bounded. From (6), we have(11)u1ix≤αix+αi2x+u1imaxβix4βix1+1+u1imaxβix2≤u1imaxβix+u1imaxβix2+u1imaxβix4βix1+1+u1imaxβix2=u1imax.
For system (4), Razumikhin’s original idea is that if Vixt≥Vixt−τi and dVi/dt≤0 are satisfied, then the system is stable. The derivative of function Vixt is(12)V˙ixt≤∂Vi∂xfixt+g1ixth1it+∂Vi∂xqixt−τi≤∂Vi∂xfixt−αix+αi2x+αi2xu1imaxβx21+1+u1imaxβx2+∂Vi∂xqixt−τi≤−2ΨiVxt+∂Vi∂xqixt−τi.
Because system (4) satisfies Assumption 1, we can obtain(13)∂Vi∂xqixt−τi≤−ΨiVixt+aiΨiVixt−τi.
IfVixt−τi≤Vixt, then ΨiVixt−τi≤ΨiVixt. We have(14)V˙ixt≤−1−aiΨiVixt.
Becauseai∈0,1, we have V˙ixt≤0. System (4) is asymptotically stable under control of u1it=hixt and u2it=0.


## 3.2. Distributed Optimal Control

For each modei i=1,2,…,s, a distributed optimal control scheme is designed.Firstly, we design controller 2 based on the bounded controller and the received measurementxtk at time tk. The input u2i can be calculated by the following optimal problem:(15)minu2i∈pΔ∫tktk+NΔx˜tTQix˜t+u1iTtR1iu1it+u2iTtR2iu2itdt,s.t.(16)x˜˙t=fix˜t+g1ix˜tu1it+g2ix˜tu2it+qx˜t−τi,∀t∈tk,tk+NΔ,(17)u1it=hix˜tk+mΔ,∀t∈tk+mΔ,tk+m+1Δ,m=0,1,…,N−1,(18)u2it∈U2i,∀t∈tk,tk+NΔ,(19)x^˙t=fix^t+g1ix^thixtk+mΔ+qix^t−τi,∀t∈tk,tk+NΔ,(20)x^tk=x˜tk,(21)∂Vi∂xg2ixtku2it≤0,where pΔ is a family of piecewise constants function with a sampling time Δ, NΔ is the predicted range, Qi, R1i, and R2i are the right of positive definite matrix, x˜t is the trajectory of system (4) with u1i=hixt, and u2 is the input trajectory computed by (15)–(21).The optimal solution is denoted byu2i∗ttk t∈tk,tk+NΔ.Onceu2i∗ttk is calculated, it is sent to controller 1 and the corresponding control actuator. Then, we design controller 1 based on the measurement xtk and u2i∗ttk. The input u1i is calculated by the following optimal problem:(22)minu1∈pΔ∫tktk+NΔx˜tTQix˜t+u1iTtR1iu1it+u2iTtR2iu2itdt,s.t.(23)x˜t=fix˜t+g1ix˜tu1it+g2ix˜tu2it+qix˜t−τi,∀t∈tk,tk+NΔ,(24)x˜˙t=fix˜t+g1ix˜thix˜tk+jΔ+g2ix˜tu2it+qix˜t−τi,∀t∈tk+mΔ,tk+m+1Δ,m=0,1,…,N−1,(25)u2it=u2i∗ttk,∀t∈tk,tk+NΔ,(26)u1it∈U1i,∀t∈tk,tk+NΔ,(27)x˜tk=x˜tk,(28)∂V∂xg1ixtku1it≤∂V∂xg1ixtkhixtk.The optimal solution is expressed asu1i∗ttk t∈tk,tk+NΔ.Remark 3.
We emphasize two points: (i) (18) and (26) ensure that system input satisfies the constraint. (ii) (21) and (28) guarantee the stability of mode i.


## 3.3. Stability Analysis

In this section, the stability of distributed optimal control scheme is demonstrated.Theorem 1.
Considering that switched nonlinear system (1) satisfies Assumption 1, and under distributed optimal control of (15)–(21) and (23)–(28),(i)
When there is no switch, letΔi, εi>0, 0<ρsi<ρi satisfy the following inequality:(29)−1−aiΨiρsi+LxfiM1iΔ+Lxg1iM2iu1imaxΔ+Lxg2iM3iu2imaxΔ+Lxkθi+LxqiM4iΔ+2φi≤−εi,if initial state starts from the setΩρi, and ρmi<ρi, where ρmi=maxρsiVixt+Δ:Vixt≤Ωρsi, then xt is ultimately bounded in Ωρmi.(ii)
When there is switch, the following constraints need to be added to distributed optimal control of (15)–(21) and (23)–(28):(30)Vjxtjwin≤Vjxtjw−1in−ε∗,w>1,Vixtjw−1in−ρmj>0,ρmj,w>1,Vixtjw−1in−ρmj≤0,ρj,w=1,where Vjxtjwin denotes the value of Lyapunov functions of mode j for wth time and Vjxtjw−1in is the value of Lyapunov functions of mode j for w−1th time. If (29) is satisfied, then xt is ultimately bounded in Ωρmi.Proof.
The proof is divided into two parts.(i)
When there is no switch, ifxtk∈Ωρi, from (21) and (28), we have(31)V˙ixtk=∂Vi∂xfixtk+g1ixtku1i∗ttk+g2ixtku2i∗ttk+qixtk−τ≤∂Vi∂xfixtk+g1ixtkhixt+qixtk−τi.From (14), it can be deduced that(32)V˙ixtk≤−1−aiΨiVixtk.The derivative ofVixt along the actual state trajectory in t∈tk,tk+1 is(33)V˙ixt=∂Vi∂xfixt+g1ixtu1i∗ttk+g2ixtu2i∗ttk+qixt−τi+kixtωit.It follows from (31) and (33) that(34)V˙ixt≤−1−aiΨiVixtk+∂Vi∂xfixt+g1ixtu1i∗ttk+g2ixtu2i∗ttk+qixt−τ+kixtωit−∂Vi∂xfixtk+g1ixtku1i∗ttk+g2ixtu2i∗ttk+qixtk−τi.Vixt is acontinuous differentiable and fi⋅, g1i⋅, g2i⋅, qi⋅, and ki⋅ are local Lipschitz. So, there are positive numbers Lxfi, Lxg1i, Lxg2i, Lxqi, and Lxki which make the following equations true for ξ1,ξ2∈Ωρi:(35)∂Vi∂xfiξ1t−∂Vi∂xfiξ2t≤Lxfiξ1t−ξ2t,(36)∂Vi∂xg1iξ1t−∂Vi∂xg1iξ2t≤Lxg1iξ1t−ξ2t,(37)∂Vi∂xg2iξ1t−∂Vi∂xg2iξ2t≤Lxg2iξ1t−ξ2t,(38)∂Vi∂xqiξ1t−τi−∂Vi∂xqiξ2t−τi≤Lxqiξ1t−τi−ξ2t−τi,(39)∂Vi∂xkiξ1t−τi≤Lxki.Using (35)–(39), we can rewrite (34) as follows:(40)V˙ixt≤−1−aiΨiVixtk+Lxfixt−xtk+Lxg1ixt−xtku1imax+Lxg2ixt−xtku2imax+Lxqixt−τi−xtk−τi+Lxkiθi.For allxtk∈Ωρi/Ωρsi, it follows from (32) that(41)−1−aiΨiVixtk≤−1−aiΨiΩρsi.According to the smoothness of functionsfi⋅, g1i⋅, g2i⋅, and qi⋅, there exists positive constants Mjij=1,2,3,4 such that(42)fixt≤M1i,g1ixt≤M2i,g2ixt≤M3i,qixt≤M4i.Because of the continuity ofxt, for t∈tk,tk+1, there are(43)Lxfixt−xtk+Lxg1ixt−xtku1imax+Lxg2ixt−xtku2imax≤LxfiM1iΔ+Lxg1iu1imaxM2iΔ+Lxg2iu2imaxM3iΔ,Lxqixt−τi−xtk−τi≤LxqimaxM4iΔ,2φi,M4iΔ+2φi=LxqiM4iΔ+2φi.By (40) and all initial states xtk∈Ωi/Ωρsi, we have(44)V˙ixtk≤−1−aiΨiρsi+LxfiM1iΔ+Lxg1iM2iu1imaxΔ+Lxg2iM3iu2imaxΔ+Lxkiθi+LxqiM4iΔ+2φi.Forxtk∈Ωi/Ωρsi, if (29) is true, there exists ε>0 such that(45)V˙ixt≤−εi.Integrating this bound ont∈tk,tk+1, we have(46)Vixtk+1≤Vixtk−εiΔ,Vixt≤Vixtk,∀t∈tk,tk+1.By recursively applying (46), we can obtain that if the initial state starts from Ωi/Ωρsi, the state converges to Ωρsi. If the state converges to Ωρsi⊂Ωρmi (or starts at Ωρsi), the state remains in Ωρmi, determined by the definition of ρmi. So, xt is ultimately bounded in Ωρmi.(ii)
When there is switch, there are two cases.
In case oftirin≤t<tirout and tjwin=tirout<∞, we consider that the system has an infinite number of switches. At this point, ith mode is activated. If Vix>ρmi, then the constraints of (21) and (28) guarantee that Vixtirout<Vixtirin, and the constraints of (30) ensure that Vixtirin<Vixtir−1in<⋯<ρmi. So, Vix is always decrease. Even if ith mode is not activated, there is lth mode l∈S which is activated and guarantees that Vlx is decrease and Vlx<ρml. So, xt is ultimately bounded stable in Ωρmi.
In case oftirin≤t<tirout and tjwin=tirout=∞, we consider that the system has an finite number of switches. Similarly, Vixtirin<Vixtir−1in<⋯<ρmi. When switching to the ith mode, there is xtirin∈ρmi. From this point on, distributed optimal control strategy is not subject to any switching constraints. So, xt is ultimately bounded stable.
In conclusion, Theorem1 is proved.
The manipulation inputs of the distributed optimal control scheme are defined:(47)u1it=u1i∗ttk,u2it=u2i∗ttk,∀t∈tk,tk+1.
The executive strategy of optimal control in this paper is given as follows:Step 1: controller 2 receives the measurementxtk at time tk, and then controller 2 calculates the optimal trajectory of u2i.Step 2: the optimal trajectory ofu2i is transmitted to the actuator and controller 1.Step 3: once the optimal trajectory ofu2i is received by controller 1, the optimal trajectory of u1i is calculated.Step 4: controller 1 sends the optimal trajectory ofu1i to the actuator.Step 5: go to the next moment.


## 4. Simulation Example

We consider the following switched system to verify the effectiveness of the proposed distributed optimal control scheme:(48)x˙t=fixt+g1ixtu1it+g2ixtu2it+qixt−τi+kixtωit,where f1x=3x23t,0T and f2x=x23,0T; g11x=0,1T and g12x=0,1T; g21x=1,0T and g22x=1,0T; q1xt−1=2x22t−1,2x12t−1T and q2xt−1=x22t−1,x12t−1T; k1xt=0.2,0Tandk2xt=0.23,0T; and ω1t=sinx2t,0T and ω2t=sinx2t,0T. Our goal is to adjust the initial state x0 to the stable state xsT=0,0 for the system. The control constraints are u1i∈u1i∈R1u1i≤20 and u2i∈u2i∈R1u2i≤15, xt=φit=1.2,−0.9, t∈−1,0.We choose the Lyapunov functionV1x1,x2=0.25x12+x22, Ψ1V=V1+22V12, and a1=0.85, Lyapunov function V2x1,x2=0.25x12+x22, Ψ2V=V2+20V22, and a2=0.79 to satisfy Assumption 1. We use Proposition 1 to construct the bounded controller u1i=hix and u2i=0. We apply the distributed optimal control and the centralized optimal control, respectively. Both control methods use the same parameters. The prediction horizon is N=6, the sampling interval is Δ=0.01, and the weighted matrices are chosen as Qi=diag10,10, R1i=0.50, and R2i=0.60(i=1,2). Select the initial state as x0T=−2.6,1.9.Set the system switching mode 2 from mode 1 at timet=3.50s (the system state enters the stable region of mode 2). The simulation results are shown in Figures 2–5:Figure 2
The trajectories ofx1.Figure 3
The trajectories ofx2.Figure 4
The trajectories ofu1.Figure 5
The trajectories ofu2.The solid lines and the dotted lines represent the state trajectories under distributed optimal control and centralized optimal control, respectively. Simulation results indicate that optimal control via distributed optimal control method can ensure that the time-delay system state is ultimately bounded and stable. It also shows that both control designs give similar closed-loop performance and drive the state and the state close to the stable state in about 7.5 s and 8.5 s, respectively. To further illustrate that the two control designs have similar closed-loop performance, we illustrate this point from the perspective of the performance index. A series of simulation comparisons are made between the distributed optimal control and the centralized optimal control with the same parameters by using ten sets of different initial values. We calculated the total performance cost along the closed-loop system trajectories in 10s at different initial values. The comparison results are shown in Table1.Table 1
Total performance cost along the closed-loop system trajectories.
Number
1
2
3
4
5
6
7
8
9
10
Initial value
[−1.1, 0.9]
[1.2, −0.9]
[−0.7, 1.0]
[0.8, 1.05]
[−0.97, −1.3]
[0.86, −1]
[−0.5, 1.4]
[1.5, −1.7]
[−1.4, 0.97]
[1.6, −1.8]
Distributed optimal control
1778
1693
1860
1792
2120
1673
1732
3170
1973
2970
Centralized optimal control
1687
1801
1778
1933
1946
1854
1973
3457
1669
2532It can be seen from Table1 that the cost of distributed optimal control method is lower than that of the centralized optimal control method in 6 out of 10 simulations. This shows that the closed-loop performance of the distributed optimal control method is comparable to that of the centralized optimal control method. Among the 10 simulation results, we also compared the computation time of the distributed optimal control method and centralized optimal control method. The average time to calculate the centralized optimal control is 2.64 s. The average time to calculate controller 1 and controller 2 is 0.9 s and 1.05 s, respectively. Obviously, the computation time of the distributed optimal control method is less than that of centralized optimal control. This is because centralized optimization control must optimize two inputs simultaneously in an optimization problem, while distributed optimization control is to solve two optimization problems which only have one decision variable.


## 5. Conclusions

This paper solves the problem of distributed optimal control for switched nonlinear systems with the state time delay. For each mode, a bounded controller is designed to stabilize the system. Then, a distributed optimal control scheme which can satisfy the input constraint is designed based on the bounded stabilization controller. Finally, it is proved that the system state is ultimately bounded stable.



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