随机马尔可夫切换系统：时变延迟与饱和非线性的抗饱和设计

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本文探讨了"具有模式相关时变时滞和饱和非线性的随机马尔可夫切换系统"的抗饱和设计问题。在随机马尔可夫切换系统的研究背景下，这种系统模型结合了时变延迟和饱和非线性特性，这是现代控制理论中的一个重要课题，尤其是在处理工业过程、通信网络和自动化系统等复杂动态环境中，这些特性经常出现。马尔可夫切换系统是一种多模态系统，其状态根据一个不可观测的马尔可夫链随机地在多个行为模式之间切换。在这种系统中，模式依赖的时间变时滞反映了系统内部或外部环境变化对切换事件的影响，而饱和非线性则可能出现在系统输入或输出的限制条件下，如电机驱动器、电力电子设备等。抗饱和设计旨在防止由于系统受到饱和限制导致性能退化或控制失灵。设计的目标是确保在面对饱和时，系统仍然能够维持稳定性和鲁棒性，同时保持期望的行为。这通常涉及到寻找合适的控制器结构，例如预估控制器或自适应控制器，以补偿饱和效应并减小其负面影响。作者Wenhai Qia、Ju H. Park（共同作者）、Jun Cheng、Yonggui Kao和Xianwen Gao分别来自中国曲阜师范大学、韩国延世大学、湖北省民族大学和哈尔滨工业大学威海分校以及东北大学的信息科学与工程学院，他们合作研究了这一问题，并于2017年在《非线性分析：混合系统》期刊上发表了这篇研究成果。该文章于2016年9月13日接收，6月6日被接受，并于同年7月4日在线发布。论文的关键字包括“抗饱和设计”、“马尔可夫切换系统”、“模式依赖的时间变时滞”，表明研究的重点在于如何在这些复杂的系统特征下，提出有效的设计策略来解决实际应用中的挑战。这篇文章深入分析了如何通过理论建模和控制器设计，对抗饱和非线性和模式相关时变时滞对随机马尔可夫切换系统性能的影响，为这类系统的设计和控制提供了理论支持和技术指导。对于从事控制系统设计、优化和故障应对的工程师和研究人员来说，这篇论文是一个有价值的参考资源。

W. Qi et al. / Nonlinear Analysis: Hybrid Systems 26 (2017) 201–211 203

Fig. 1. Anti-windup compensation scheme.

For system (1), we design the following n

-order dynamic output stabilizing controllers:

(t) = [A

(t) + A

(t − τ (g

, t)) + G

(t)]dt + [W

(t) + W

(t − τ (g

, t))]dω(t),

(t) = C

)x(t) + C

(t),

+ θ ) = ϕ

(θ), ∀θ ∈ [−τ , 0], (4)

where x

(t) ∈ R

, u

(t) = y(t), and u(t) = v

(t) are, respectively, the state of the controller, the input of the controller, and

the output of the controller. An anti-windup term E

)(sat(v

(t)) − v

(t)) can be added to mitigate the undesirable effects

of windup caused by actuator saturation. Thus, the system (4) can be written as

(t) = [A

(t) + A

(t − τ (g

, t)) + G

(t) + E

)(sat(v

(t)) − v

(t))]dt

+ [W

(t) + W

(t − τ (g

, t))]dω(t),

(t) = C

)x(t) + C

(t),

+ θ ) = ϕ

(θ), ∀θ ∈ [−τ , 0]. (5)

Remark 1. The ‘‘windup’’ term is historically related to the actuator saturation effects in control loops containing integral

action with unnecessary charge producing undesirable effects on the transient response of the system, such as oscillations

and slow convergence to the equilibrium point. In order to avoid this performance degradation generated by the saturation,

an additional control-loop, called anti-windup compensation can be added to the closed-loop system.

Remark 2. The general principle of the anti-windup scheme can be depicted in Fig. 1. In this figure, the (unconstrained)

signal produced by the controller is compared to that which is actually fed into the plant (the constrained signal). In Fig. 1,

the linear closed-loop system is ‘‘disturbed’’ by the dead-zone signal corresponding to sat(v

(t)) −v

(t) and this same signal

is exploited to activate the anti-windup filter action whenever necessary. From Fig. 1, we can easily get Eq. (5).

When g

= i, i ∈ S, A(g

), A

), B(g

), W (g

), W

), C(g

), A

), G

), W

), C

), τ (g

, t),

τ (g

), ˙τ (g

, t), and h(g

) are denoted as A

, A

, B

, W

, C

, A

d1i

, G

, W

d1i

, C

, τ

(t), τ

, ˙τ

(t), and h

Define an extended state vector

ξ(t) =



(t) x

(t)



∈ R

n+n

, (6)

and the following augmented matrices



+ B





0 A

d1i





−B



R =



−I





0 W





0 W

d1i



, K





As a result, the closed-loop system is rewritten as follows:

dξ(t) = [

ξ(t) +

ξ(t − τ

(t)) + (

)ψ(v

)]dt + [

ξ(t) +

ξ(t − τ

(t))]dω(t),

(t) = K

ξ(t),

ξ(t

+ θ ) = ϕ(θ), ∀θ ∈ [−τ , 0], (7)

where ψ(v

) = v

(t) − sat(v

(t)).

For any matrices P

> 0 , an ellipsoid is defined as

η(P

) = {ξ (t) ∈ R

n+n

: ξ

(t)P

ξ(t) ≤ 1}. (8)

Here, a definition and a lemma are given for further analysis.

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