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基于浸入与不变原理的非线性切换系统的镇定研究
中文摘要

非线性切换系统由多个非线性子模态(或子系统)和一个切换规则构成。在切换规则的指导下,系统在非线性子模态(子系统)之间切换。这类系统具有重要的理论研究意义和广泛的工程应用背景,因此引起了众多研究人员的浓厚兴趣。近年,关于非线性切换系统的稳定性和镇定问题的研究取得了丰富的研究成果。但这些结果主要针对具有特殊结构的非线性切换系统而言,譬如下三角结构、上三角结构或级联结构的切换系统。对于一般形式的非线性切换系统,目前还没有统一有效的分析设计方法。基于标准型的非线性控制方法可以用来解决非线性切换系统镇定问题,但如何寻找一般系统的坐标变换,以及如何设计合适的切换律是很困难的。另外,基于标准型的方法以及现有研究工作中的其他控制方法基本都基于系统的Lyapunov函数,然而,对于许多复杂的实际系统,Lyapunov函数的构造几乎是不可能的。近年由Astolfi和Ortega提出的浸入与不变方法,是一种无需构造(控制)Lyapunov函数的非线性控制方法。但如何克服该方法本身的局限性,拓宽其应用范围,以及如何设计合适的切换律,从而将该方法推广到非线性切换系统情形,郁是非常具有挑战性的问题。目前,关于这方面的研究工作还非常有限。本文基于浸入与不变原理研究了非线性切换系统的镇定问题,主要工作包括以下几个方面: (一)基于标准型研究了一类非线性切换系统在具有平均驻留时间切换信号下的镇定问题。首先,针对源于航空发动机系统的一般非线性系统,通过求解偏微分方程,成功地找到了将系统化为非线性标准型的坐标变换,在所建立的非线性标准型基础上,构造了系统的状态反馈控制器和Lyapunov函数。另外,考虑到外部参数变化与发动机系统模型之间的关系,引入一种切换系统来描述具有变马赫数的发动机控制系统,并且进一步将基于标准型的控制方法推广到切换系统控制器设计中,使得非线性切换系统在具有平均驻留时间的切换信号下全局渐近稳定。 (二)研究了在任意切换信号下非线性切换系统的浸入与不变镇定问题。首先,给出一般仿射非线性切换系统浸入与不变可镇定的充分条件。然后,将相关结果应用于一类具有严格反馈型的非线性切换系统,为子系统构造浸入与不变状态反馈控制器,使得系统在任意切换信号下全局渐近稳定。值得一提的是,本文首次研究了非线性切换系统的浸入与不变镇定问题,不要求系统具有特殊结构,并且不需要被控系统Lyapunov函数的任何信息。 (三)研究了具有平均驻留时间切换信号下非线性切换系统的浸入与不变镇定问题。首先,受到航空发动机控制问题的启发,融合了两个相对较新的理论工具—浸入与不变理论和生存理论,提出了一种无需Lyapunov函数任何信息,且能克服浸入与不变定理中轨迹有界性验证困难的非线性控制方法,并将该方法应用于航空发动机控制系统设计中。其次,给出具有平均驻留时间切换信号下非线性切换系统浸入与不变可镇定的充分条件,在此基础上,引入生存理论进一步给出了状态受约非线性切换系统浸入与不变可镇定的充分条件,并应用于目标子系统全稳定和目标子系统不全稳定、状态约束由不等式表达的严格反馈型切换系统。值得强调的是,生存理论的引入既保证了浸入与不变理论中有界性条件又保证了系统的约束条件不被破坏。 (四)研究了一类线性参数化非线性切换系统的浸入与不变自适应镇定问题。首先,针对每个子系统均不可浸入与不变自适应镇定的情形,给出了线性参数化非线性切换系统浸入与不变自适应可镇定的充分条件;其次,采用广义多Lyapunov函数(GMLFs)方法和Backstepping方法,构造了子系统的公共虚拟镇定函数,并设计了状态依赖型切换律和各个子系统的浸入与不变自适应状念反馈控制器和误差估计器,使得闭环切换系统在所设计的状态依赖切换律下全局稳定。 关键词:非线性切换系统;标准型;浸入与不变原理;锁定;自适应控制;公共Lyapunov函数;多Lyapunov函数;生存理论;平均驻留时间

英文摘要

A nonlinear switched system is composed of several nonlinear modes (or subsystems) and a switching rule. Under the guidance of the switching rule, the system switches from one mode to another. Such a system is of great significance both in theory development and engineering applications. Therefore, it has attracted many researchers’ interest. In recent years, abundant research results on stability and stabilization of switched nonlinear systems have been reported. However, these results are mainly for switched nonlinear systems with special structures, for example, triangle structure, upper triangular structure or cascade structure. At present, there is no unified and effective analysis and design method for general nonlinear switched systems. The nonlinear control method based on normal norm is a method to solve the stabilization problem for nonlinear systems. But, the problem of how to obtain the proper coordinate transformation for a general subsystem and how to design a suitable switching law to solve the stabilization problem of switched nonlinear systems is very difficult. In addition, the method based on normal form and most of the other control methods in the current research are based on Lyapunov functions of plants. However, It is almost impossible to construct Lyapunov functions for many complex practical systems. In recent years, the Immersion and Invariance (I&I) approach proposed by Astolfi and Ortega is a kind of nonlinear control method, which does not require the Lyapunov function of the plant. But how to overcome the limitations of the method itself for a wider application range, and how to design a suitable switching law and thus extend this method to nonlinear switched systems case are really challenging. At present, research results on these problems are quite limited. Based on immersion and invariance principle, this dissertation focuses on the stabilization problem for nonlinear switched systems. The main contributions are as follows: 1.Based on the normal form, the stabilization problem for a class of nonlinear switched systems under switching signals with average dwell time is investigated. First of all, this dissertation investigates the design of a feedback controller via nonlinear normal form for nonlinear systems originated from the aero-engine control systems. By solving the partial differential equation, we successfully find the coordinate transformation, through which the plant is transformed into its nonlinear normal form. Then, based on the nonlinear normal form, a stabilizing state feedback control law and the Lyapunov function are obtained using a constructive design method. In addition, taking into account the relationship between the change of external parameters and the engine system model, a switching system is introduced to describe the engine control system with varying Mach number and the nonlinear normal form approach is further extended to design the switched controllers to stabilize the more general switched nonlinear systems with average dwell time. 2.The problem of immersion and invariant stabilization for nonlinear switched systems under arbitrary switchings is studied. First, a sufficient, condition for immersion and invariance stabilization of general affine nonlinear switched systems is obtained. Then, the relevant results are applied to the switched nonlinear system in strict feedback form and the immersion and invariance state feedback controllers for subsystems are constructed to globally asymptotically stabilize the closed-loop system under arbitrary switching. It is worth mentioning that the problem of immersion and stabilization for nonlinear switched systems is studied for the first time. The plant is not required to posses any special structure and the novel control method does not require the Lyapunov function of the plant. 3.The problem of immersed and invariant stabilization for nonlinear switched systems with average dwell time is investigated. First of all, inspired by the aeroengine control problem, a new approach based on the integration of immersion and invariance (I&I) theory and viability theory-two relatively new tools, is presented, and a sufficient condition for stabilization of nonlinear systems with state constraints is drived. Then, the proposed method is applied to the system in strict feedback form. We design a state feedback controller constructively to ensure the immersion and invariance stabilizability of the original system, and constraints are not destroyed simultaneously. Secondly, a sufficient condition for immersion and invariance stabilization of switched nonlinear systems with average dwell time is given. The viability theory is further introduced to drive a sufficient condition for stabilization of nonlinear switched systems with state constraints. Then, the relevant results are applied to the switched system in strict feedback form with state constraint represented by inequalities. It is worth pointing out that the introducing of viability theory provides another way to guarantee the trajectory boundedness of the closed-loop system and ensures that the constraints of the system are not destroyed. 4.The problem of immersion and invariance adaptive stabilization for a class of linear parametric switched nonlinear systems is investigated, where the solvability of the immersion and invariance adaptive stabilization problem for subsystems is not assumed. First, an adaptive law with an additional nonlinear term is chosen to ensure the uniform stability of the error switching dynamics under arbitrary switchings. Furthermore, a state-dependent switching law is designed by using the generalized multiple Lyapunov function (GMLFs) method to guarantee the stability of the closed-loop system. Then, along with the Backstepping technique, the proposed control method is applied to linearly parameterized nonlinear switched systems in strict-feedback form. The common stabilizing functions are successfully found so as to avoid the use of different coordinate transformation. Also, the immersion and invariant adaptive state feedback controllers and error estimators for subsystems together with a state-dependent switching law are designed simultaneously. Keywords: Switched nonlinear systems; Normal form; Immersion and invariance; Stabilization; Adaptive control; Common Lyapunov function (CLF); Multiple Lyapunov functions(MLFs); Viability theory; Average dwell time.

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