分段仿射系统H∞事件触发控制策略

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"蒋永豪和吴炜在《基于观测器的分段仿射系统的H∞事件触发控制》一文中，研究了一种针对分段仿射系统的输出基H∞事件触发控制策略。他们考虑了系统状态不完全可测的情况，通过设计观测器来解决这一问题。随后，他们构建了事件触发机制和反馈控制器，以减少通信数据传输并确保系统渐近稳定性。文章利用退化椭球体描述分段仿射系统的输出空间，并借助分段Lyapunov函数将协同设计问题转化为线性矩阵不等式（LMI）形式的稳定性条件。通过求解这些LMI，可以得到观测器和控制器的设计方案。最后，通过仿真验证了该方法的有效性。" 这篇论文主要探讨了以下几个关键知识点： 1. **分段仿射系统**：这是一种非线性动态系统，其动态特性由多个线性子系统组成，它们在不同的系统状态下起作用。这种系统在工程应用中非常常见，如机器人、电力系统和交通流模型等。 2. **基于观测器的控制**：由于系统状态可能不可完全测量，观测器被设计用于估计未直接测量的状态变量。通过观测器，可以实现状态反馈控制，即使在部分信息可用的情况下也能进行有效的控制。 3. **H∞控制**：H∞控制理论是一种处理系统干扰和不确定性的方法，目标是保证系统性能的同时最小化输入到输出的传递函数的无穷范数，从而限制了系统对外部干扰的敏感性。 4. **事件触发控制**：与传统的周期性采样控制不同，事件触发控制仅在系统状态满足特定触发条件时进行更新，降低了系统的通信负担，提高了效率。 5. **事件触发机制**：论文设计了一种根据观测器输出和系统状态的触发机制，以决定何时更新控制器，减少了不必要的通信和计算。 6. **分段Lyapunov函数**：这是一种特殊的Lyapunov函数，适用于分段仿射系统，它由多个区域的Lyapunov函数组合而成，用于证明系统的稳定性。 7. **线性矩阵不等式（LMI）**：LMI是控制理论中的一种强大工具，用于求解优化问题，特别是稳定性分析和控制器设计。在文中，LMI被用来寻找观测器和控制器的参数，以满足系统的稳定性条件。通过以上知识点的综合运用，论文提供了一种有效的方法来处理分段仿射系统的控制问题，同时减少了通信开销，增强了系统的实际应用价值。

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

and event-triggering policy remains largely open, which motivates us for this study.

Motivated by the aforementioned discussions, in this paper, an observer-based H

∞

event-

triggered controller is designed for PWA systems. The piecewise quadratic Lyapunov function

technique is adopted which can derive less conservative results than the common Lyapunov

function technique. Based on piecewise quadratic Lyapunov function and the ellipsoidal ap-

proximation approach, the co-design of observer, event-triggering rule and H

∞

feedback con-

troller is formulated into LMIs. A numerical example is given to demonstrate the eﬀectiveness

of our approach.

Notation: ∥x∥ =

√

x denotes the Euclidean norm. l

[0, ∞) denotes the space of square

summable inﬁnite vector sequences over [0, ∞). A

and A

−1

represent the transposition and

the inverse of the matrix A, respectively. I and 0 are identity and zero matrices, respectively.

For a symmetric matrix P ∈ R

n×n

, P > 0 (P ≥ 0) means that P is positive deﬁnite (semi-

positive deﬁnite). He{A} means A + A

. diag(A, B) =



A 0

0 B





A ∗

B C





A B

B C



1 Problem Formulation

Considering the following discrete-time PWA system











x(k + 1) = A

x(k) + B

u(k) + D

ω(k) + a

y(k) = C

x(k),

z(k) = F

x(k) + G

ω(k),

for y(k) ∈ χ

, i ∈ I,

(1)

with x(k) ∈ R

the system state, u(k) ∈ R

the control input, y(k) ∈ R

the system

output, ω(k) ∈ R

the external disturbance which is assumed to be l

[0, ∞) and satisﬁes



∞

k=0

(k)ω(k) ≤ ω

max

< ∞, and z(k) ∈ R

the controlled output. The output space is

partitioned into a number of closed polyhedral regions, and χ

⊆ R

denotes these regions; I

is the index sets of χ

and I = I

∪I

, where I

is the index set of regions which contains the

origin, and I

is the index set of regions otherwise.

For piecewise aﬃne systems, the partition of the output/state space is typically done by

using closed polyhedral regions χ

= {y | T

y + l

≥ 0}

i∈I

∈ R

, with T

∈ R

p×p

, l

∈

. However, the polyhedral description of the output regions can not be cast into LMIs in

the synthesis problem. In order to establish some LMI-based conditions, we use degenerate

ellipsoids ε

to outer approximate the polyhedral regions. Assume that there exists matrices

∈ R

m×p

and f

∈ R

such that the polyhedral regions χ

, i ∈ I are included in the

ellipsoidal regions ε

, i ∈ I, that is, χ

⊆ ε

, i ∈ I, where

= {y |∥ E

y + f

∥≤ 1}, i ∈ I. (2)

- 3 -

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