Predictor-based Event-triggered Control of Non- linear Networked Systems with State and Input

3.3 Predictor-based Event-triggered Control of Nonlinear Networked Systems with State and Input Delays

In×n ∈R^n×n represent the zero and identity matrices, respectively. Then, we have ke˙^ETk ≤ kApkkXk+kΩpk (3.30) Since e^ET(tk) = 0 and _dt^dke^ETk ≤ ke˙^ETk, we get

ke^ETk ≤ Z t

t_k

e^kApk(t−τ)kΩpk dτ

= kΩpk

kApk((e^ET)^kApk(t−τ)−1), ∀t ∈[tk, tk+1) (3.31) Therefore,

t^∗ > 1 kApkln

kApk

kΩpk(ke^ETk+ 1)

≥0 (3.32)

Hence, the Zeno phenomenon is excluded. This completes the proof.

3.3 Predictor-based Event-triggered Control of Non-

3.3.2 Problem Formulation

Let us consider a similar class of MIMO nonlinear systems presented in Subsection 3.3.2. These systems are now considered to be controlled over communication network subject to state and input delays as

˙

x^τ₁^pc =x^τ₂^pc, x˙^τ₂^pc =ϕ^T₁(X^τpc)Θ +g11u^τ₁ +. . .+g1nu^τ_n

˙

x^τ₃^pc =x^τ₄^pc, x˙^τ₄^pc =ϕ^T₂(X^τpc)Θ +g21u^τ₁ +. . .+g2nu^τ_n ...

˙

x^τ2n−1^pc =x^τ_2n^pc, x˙^τ_2n^pc =ϕ^T_n(X^τpc)Θ +gn1u^τ₁+. . .+gnnu^τ_n (3.34) whereX^τpc :=X(t−τpc) = [x^τ₁^pc, x^τ₂^pc, . . . , x^τ_2n^pc]^T,u^τ :=u(t−τ) = [u^τ₁, u^τ₂, . . . , u^τ_n]^T and y= [x1 x3 . . . x2n−1] denote the delayed systems states, inputs and outputs, respectively.

Here, τ =τpc+τcp is the round-trip time delay in which τpc and τcp are the delays in the plant-to-controller and controller-to-plant channels, respectively. Further, ϕ1, . . . , ϕn ∈ R^dare known nonlinear functions. Different from previous Section 3.2, the vector Θ∈R^d and the matrixG∈R^n×nare assumed to be known in this section to further simplify the controller and predictor design. The system equations (3.34) can be written as

X˙₁^τpc =X₂^τpc

X˙₂^τpc = Φ^T(X^τpc)Θ +Gu^τ (3.35) where X₁^τpc = [x^τ₁^pc x^τ₃^pc. . . x^τ_2n−1^pc ]^T ∈Rⁿ, X₂^τpc = [x^τ₂^pc x^τ₄^pc. . . x^τ_2n^pc]^T ∈Rⁿ,

Φ(X_pc^τ ) = [ϕ1 ϕ2 . . . ϕn]∈R^d×n, and G =







g11 . . . g1n

... ...

gn1 . . . gnn





. Inspired by [46, 47], a state predictor is incorporated to deal with the network-induced time delays, and it designed as

X˙ˆ1 = ˆX2−K1 Xˆ₁^τ −X₁^τpc

(3.36) X˙ˆ2 = Φ^T( ˆX)Θ +Gu−K2 Xˆ₂^τ −X₂^τpc

(3.37) where K1 ∈R^n×n and K2 ∈R^n×n are gain matrices with positive diagonal elements. An event-triggered scheme is also proposed here to reduce the communication and computa- tion burden as follows

X₁^ET(t) =X1(tk), X₂^ET(t) = X2(tk), | ∀t∈[tk, tk+1) (3.38)

3.3 Predictor-based Event-triggered Control of Nonlinear Networked Systems with State and Input Delays

Plant

Controller

Event triggered

Predictor

Network channel

e^−τcp e^−τpc

e

^−τ

X(t) X(t_k)

X(tk−τpc)

X(tˆ −τ) X(t)ˆ

u(t−τ)

u(t)

ZOH

Figure 3.1: Block diagram of the system.

in which

tk+1 = inf{t|t > tk, J(e^ET, Z1, Z2)>0} (3.39) whereX₁^ET and X₂^ET denote the last-received states from the plant and J(e^ET, Z1, Z2) is the triggering function. Here, the measurement error is defined as e^ET = [e^ET₁ e^ET₂ ]^T, in which e^ET₁ (t) = X1(t)−X₁^ET(t) and e^ET₂ (t) =X2(t)−X₂^ET(t). The block diagram of the overall system with the predictor and triggering mechanism is illustrated in Fig. 3.1.

3.3.3 State Predictor Design

In this subsection, the error dynamics of the state predictor is analyzed. The conver- gence of the predicted states to the actual states is also proved.

Let us define the prediction error as

eX1 = ˆX₁^τ −X₁^τpc (3.40)

eX2 = ˆX₂^τ −X₂^τpc (3.41)

Then, we have

˙

eX1 =−K1e^τ_X1 + ˆX₂^τ −X₂^τpc

=−K1e^τ_X1 +eX2

≤ −K1e^τ_X1 +keX2k

≤ −K1e^τ_X1 +keXk (3.42)

where X = [X1 X2]^T and eX = ˆX^τ −X^τpc = [eX1 eX2]^T. We also have

˙

eX2 =−K2e^τ_X2 + Φ^T( ˆX^τ)Θ +Gu^τ−Φ^T(X^τpc)Θ−Gu^τ

=−K2e^τ_X2 + (Φ^T( ˆX^τ)−Φ^T(X^τpc))Θ

≤ −K2e^τ_X2 +kΦ^T( ˆX^τ)−Φ^T(X^τpc)kkΘk

≤ −K2e^τ_X2 +LkΘkkeXk (3.43)

The full error dynamics can be written in the following form

˙

eX ≤ −Ke^τ_X + Ψ (3.44)

where

K = K1 0 0 K2

!

and Ψ = keXk LkΘkkeXk

!

(3.45) Proposition 4. Let us consider the state predictor (3.36) of the system (3.34)along with the prediction error dynamics (3.44). If there exists a positive gain matrix (K)for which the following condition is satisfied

λmin(K)−λmin(K²)√qτ −λmin(K)Υτ −Υ>0 , (3.46) where λmin(.) denotes the smallest eigenvalues of (.) and Υ = max 1, LkΘk

in which L is the Lipschitz constant, then the prediction error e(t)→0 as t → ∞.

Proof. Let us choose the Lyapunov function candidate asVp = ¹₂e^T_XeX whose time deriva- tive is given as

V˙p =e^T_Xe˙X =−e^T_X(Ke^τ_X −Ψ) (3.47) From Leibnitz’s formula, we have

e^τ_X =eX − Z t

t−τ

˙

eX(ρ)dρ (3.48)

3.3 Predictor-based Event-triggered Control of Nonlinear Networked Systems with State and Input Delays

Substituting (3.48) in (3.47), one gets V˙p =−e^T_X

KeX +K² Z t

t−τ

eX(ρ−τ)dρ+K Z t

t−τ

Ψ(ρ)dρ−Ψ

(3.49)

≤ −λmin(K)keXk²−e^T_XK² Z t

t−τ

eX(ρ−τ)dρ−e^T_XK Z t

t−τkΨ(ρ)kdρ+keXkkΨk (3.50) The norm of the term Ψ can be written as

kΨk ≤ keXk+LkΘkkeXk ≤ΥkeXk (3.51) Using (3.51) and the relation that says Vp eX(t+γ)

≤qVp eX(t)

for−2τ ≤γ ≤0 and q >1, one gets

V˙p ≤ −λmin(K)keXk²+λmin(K²)√

qτkeXk²+λmin(K)ΥτkeXk²+ ΥkeXk²

=− λmin(K)−λmin(K²)√

qτ −λmin(K)Υτ −Υ

keXk² (3.52)

Herein, if the condition (3.46) is satisfied, ˙Vp becomes negative definite which implies that Vp → 0 as t → ∞. As a result, we can conclude that the prediction error e(t) → 0 as t→ ∞. This completes the proof.

3.3.4 Predictor-based Event-triggered Control Design

The predictor-based event-triggered controller is now designed based on the predicted states. The event-triggered condition is consequently derived in the last step of the design procedure by ensuring the negative semi-definiteness of the derivative of Lyapunov function. Let us now redefine the error variables based on the predicted states as

Z1 = ˆX1

Z2 = ˆX2 −α1 (3.53)

where ˆX1 and ˆX2 are the predicted states, α1 is the virtual control law to be designed in the following steps. The event-triggered error variables are accordingly defined as Z₁^ET = ˆX₁^ET, Z₂^ET = ˆX₂^ET −α1.

Step 1: The dynamics of the first error subsystem is given as

Z˙1 =X˙ˆ1 = ˆX2 =Z2+α1 (3.54) Similar to Section 3.2.2, the virtual control is chosen as

α1 =−C1Z₁^ET (3.55)

where C1 ∈ R^n×n is a gain matrix with positive diagonal elements. However, it is to be noted here that, unlike Section 3.2.2, the virtual control law (3.55) is designed based on the predicted states. Differentiating V1 = ¹₂Z₁^TZ1 with respect time along the trajectories of the system (3.34) is

V˙1 ≤Z₁^TZ2−Z₁^TC1Z1+kC1kkZ1kke^ET₁ k (3.56) Step 2: In this step, the actual control law is designed to stabilize the second error variable Z2. The Lyapunov function candidate is therefore chosen as V2 =V1 + ¹₂Z₂^TZ2. Differentiating V2 with respect time is

V˙2 ≤Z₁^TZ2−Z₁^TC1Z1+kC1kkZ1kke^ET₁ k+Z₂^TZ˙2 (3.57) It is to be noted that the control signal is constant between inter-event periods. Thus

˙

α1 = 0. The second error dynamics ˙Z2 is then expressed as

Z˙2 = Φ^T( ˆX)Θ + Φ^T( ˆX^ET)Θ−Φ^T( ˆX^ET)Θ +Gu (3.58) Now, the actual control law is chosen as

u=−G⁻¹(C2Z₂^ET +Z₁^ET + Φ^T( ˆX^ET)Θ), (3.59) where C2 ∈R^n×n is a gain matrix with positive diagonal elements. The time derivative of Lyapunov function V2 becomes

V˙2 ≤Z₁^TZ2−Z₁^TC1Z1+kC1kkZ1kke^ET₁ k −Z₂^TC2Z₂^ET −Z₂^TZ₁^ET +Z₂^T

Φ^T( ˆX)−Φ^T( ˆX^ET)

Θ (3.60)

Adding and subtracting the term Z₂^TC2Z2+Z₁^TZ2 to (3.60) and using Assumption (2), one gets

V˙2 ≤ −Z₁^TC1Z1−Z₂^TC2Z2+kC1kkZ1kke^ET₁ k+kC2kkZ2kke^ET₂ k+kZ2kke^ET₁ k+LΘkZ2kke^ET₂ k

≤ −Z₁^TC1Z1−Z₂^TC2Z2+kC1kkZ1kke^ETk+kC2kkZ2kke^ETk+kZ2kke^ETk+LΘkZ2kke^ETk

≤ −Z₁^TC1Z1−Z₂^TC2Z2+γke^ETk (3.61)

where e^ET = [e^ET₁ , e^ET₂ ]^T and γ =kC1kkZ1k+kC2kkZ2k+kZ2k+LΘkZ2k.

Based on the results obtained in Subsection 3.2.2, the triggering condition can be directly designed as

J(e^ET, Z1, Z2) =ke^ETk² −2ζ Z₁^TC1Z1+Z₂^TC2Z2

+kγk² (3.62)