Elsevier

Automatica

Volume 131, September 2021, 109728
Automatica

Technical communique
Finite-time output feedback stabilization of planar switched systems with/without an output constraint

https://doi.org/10.1016/j.automatica.2021.109728Get rights and content

Abstract

An output feedback control strategy is presented in this paper to make planar switched systems with or without an output constraint finite-time stable. By incorporating a common constructed tangent-type barrier Lyapunov function, the so-called adding a power integrator technique is revamped to design state feedback controllers systematically. Then, reduced-order switched observers are constructed deliberately so that the finite-time output feedback stabilization of planar switched systems can be realized with the output constraint guaranteed. The method proposed in this paper is in a unified form in the sense that, with no need of changing controllers and observers structures, it works for planar switched systems with or without an output constraint.

Introduction

In addition to achieving the established goal, control systems in practice also need to meet the constraints or various limits that are prevalent during operation (Blanchini, 1999, Gilbert and Tan, 1991, Hu and Lin, 2003, Kogiso and Hirata, 2006, Zhang and Liu, 2013). Constraints might come from performance or safety requirements, and equivalently represent physical restrictions imposed on the system, such as the flying speed of a spraying drone to be limited by the spraying effect and the restriction on maximum weight of bridges to be restricted for safety considerations.

Research on feedback stabilization for nonlinear systems with output constraints has attracted considerable attention in recent years. Approaches to coping with constraints include set invariance (Blanchini, 1999, Gilbert and Tan, 1991), model predictive control (Zhang & Liu, 2013), reference governors (Kogiso & Hirata, 2006) and composite quadratic function (Hu & Lin, 2003). However, the above-mentioned methods are usually numerical in nature, and the application of these methods is plagued by sophisticated calculations. Barrier Lyapunov functions (BLFs) are Lyapunov-like functions that grow to infinity as their arguments tend to some scheduled limits, and the approach based on BLFS is an alternative efficient method to handle constraints imposed on control systems (Ngo et al., 2005, Tee et al., 2009). The underlying principle of BLFs scheme is to make the value of BLFs finite during operation by control action so that the constraint requirements of dynamical systems can be achieved. Recently, a new tangent-type barrier Lyapunov function (T-BLF) method has been proposed to discuss asymptotic stability of nonlinear systems with output constraints (Chen and Sun, 2020a, Jin and Xu, 2013). However, nonlinear systems discussed in Chen and Sun (2020a) and Jin and Xu (2013) have only a single structure; nonlinear systems consisting of multiple subsystems with different structures along with switchings between them have not been considered.

Switched systems include a series of subsystems and a switching law that coordinates the switching between subsystems. Stability analysis and feedback stabilization of switched systems have been an active research topic for a long time (Liberzon and Morse, 1999, Lin and Antsaklis, 2009, Mancillar-Aguilar and Gracia, 2019, Rossa et al., 2020). It is well known that ensuring stability under arbitrary switchings is significantly pivotal of switched systems due to its practical importance; this issue has been widely studied by investigating the existence of a common Lyapunov function (CLF) (Liberzon and Morse, 1999, Lin and Antsaklis, 2009). Besides, another attractive problem of switched systems is finite-time stabilization task, which enables switched systems to enjoy desired features, including faster convergence rates and better disturbance rejection. In the literature, the finite-time stabilization issue of switched systems has been extensively discussed by state feedback (Fu et al., 2015, Huang and Xiang, 2016). Compared with finite-time stabilization by state feedback, finite-time output feedback stabilization in the large is much more challenging since the construction of observers is not a trivial work and separation principle usually does not hold for nonlinear switched systems, and thus stimulating a series of research works (Lin et al., 2017, Lin et al., 2020, Mazenc et al., 1994, Qian and Lin, 2002, Zhang et al., 2020).

In this paper, finite-time output feedback stabilization of planar switched systems with or without an output constraint is investigated. To the best of our knowledge, it is the first work concerning finite-time output feedback stabilization of switched systems while taking output constraints into consideration simultaneously. The superiorities of our method can be summarized as follows: (i) Reduced-order switched observers with novel structures, which are critical for state estimation and output feedback design, are delicately constructed. (ii) The control strategy, which is proposed by introducing a common T-BLF and revamping adding a power integrator technique with the implantation of the developed reduced-order switched observers, is a unified approach in the sense that, with no need of changing controllers and observers structures, it is workable for planar switched systems with or without an output constraint.

Section snippets

Preliminaries and technical lemmas

Consider the following planar switched systems ξ̇1=ξ2qδ(t),1+ϑδ(t),1(ξ1),ξ̇2=υδ(t)qδ(t),2+ϑδ(t),2(ξ1,ξ2),y=ξ1 with δ(t)S={1,2,,SN}, where ξ=(ξ1,ξ2)TR2, υδ(t)R, yR are state, input and output, respectively. For k=1,2, the parameter qδ(t),kRodd1 {qRq1 is a ratio of odd integers}. ϑδ(t),1(ξ1) and ϑδ(t),2(ξ1,ξ2) are smooth and satisfy ϑδ(t),1(0)=0,ϑδ(t),2(0,0)=0. S is the index set and SN denotes the subsystem number of switched system (1). Similarly to the works (Liberzon and Morse, 1999

Main results

Now, we present the main result of this paper.

Theorem 1

Planar switched system (1) can be finite-time stabilized under arbitrary switchings by the following output feedback controllers ξ̇z=L(ξ1)(ξz+Γ(ξ1))r2qδ(t),1+ϑδ(t),1(ξ1)υδ(t)=υ(ξ1,ξˆ2)withξˆ2=(Γ(ξ1)+ξz)r2 and the output constraint |y(t)|<bϵ for all t0 with bϵ>0 is guaranteed, where Γ(ξ1) is a continuously differentiable function with Γ(0)=0 and L(ξ1)=Γ(ξ1)ξ11.

Proof

The proof procedure can be divided into the following four steps.

Step 1: Finite-time

Simulations

Now, we provide simulation results to verify the main result of this paper. Consider system (1) with δ(t)S={1,2}, where q1,1=2321, q1,2=179, q2,1=1, q2,2=53, ϑ1,1(ξ1)=0, ϑ1,2(ξ1)=sin2(ξ1)ξ21721, ϑ2,1(ξ1)=ξ179cos(ξ1), ϑ2,2(ξ1)=sin(ξ2)(1+ξ12). It is not difficult to show that Assumption 1 holds with r1=1 r2=79, r3=13 and ω1=427, ω2=29. Moreover, |ϑ1,1(ξ1)|=0, |ϑ1,2(ξ1)||ξ2|1721, |ϑ2,1(ξ1)||ξ1|79, |ϑ2,2(ξ1)||ξ2|13 with ρ1,1(ξ1)=0,ρ2,1(ξ1)=1,ρ1,2(ξ1,ξ2)=ρ2,1(ξ1,ξ2)=1. By virtue

Conclusions

Finite-time output feedback stabilization problem for a class of planar switched nonlinear systems whose output is required to meet a predefined constraint has been investigated. By proposing reduced-order switched observers and introducing a common T-BLF, adding a power integrator technique was subtly revamped to establish a scheme that guides us in constructing an output feedback controller to perform the stabilization task and achieve the output constraint imposed on the system. An

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61773216 and the Ministry of Science and Technology (MOST), Taiwan , under Grant MOST 109-2221-E-006-089-.

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