网络稳定性与控制的小增益原理:回顾与近期进展

小增益定理是现代控制理论中极为重要的基本工具之一,它在关联系统和不确定系统的鲁棒稳定性分析以及鲁棒控制器设计的许多工作中都发挥着极大的作用.基于输入到状态稳定性的概念,笔者于1994年首次提出了广义非线性小增益定理.与之前的小增益定理不同,这一结果为同时刻画关联系统的内部稳定性和外部稳定性提供了一个统一的框架.从镇定与鲁棒自适应控制到分散式或分布式控制以及输出调节(抗干扰渐近跟踪),基于非线性小增益定理已经发展出一系列的鲁棒非线性控制器设计新工具.在过去10年间,复杂非线性大系统已成为研究热点,驱动着小增益定理向更加完备的网络小增益定理方向发展,以期解决网络稳定性与控制中的新问题.对此,针对非线性小增益理论的一些最新研究进展及其在通讯和计算约束下的网络化控制和事件驱动控制应用结果进行综述,并对该理论的未来研究方向给出一些建议.     

iISS and ISS dissipation inequalities: preservation and interconnection by scaling

In analysis and design of nonlinear dynamical systems, (nonlinear) scaling of Lyapunov functions has been a central idea. This paper proposes a set of tools to make use of such scalings and illustrates their benefits in constructing Lyapunov functions for interconnected nonlinear systems. First, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre–Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.     

A Lyapunov-based small-gain theorem for interconnected switched systems

Stability of an interconnected system consisting of two switched systems is investigated in the scenario where in both switched systems there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signals neither switch too frequently nor activate non-ISS subsystems for too long, a small-gain theorem can be used to conclude global asymptotic stability (GAS) of the interconnected system. For each switched system, with the constraints on the switching signal being modeled by an auxiliary timer, a correspondent hybrid system is defined to enable the construction of a hybrid ISS Lyapunov function. Apart from justifying the ISS property of their corresponding switched systems, these hybrid ISS Lyapunov functions are then combined to establish a Lyapunov-type small-gain condition which guarantees that the interconnected system is globally asymptotically stable.     

A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability of interconnected systems

In this paper, a novel approach to constructing flexible Lyapunov inequalities is developed for establishing Input-to-State Stability (ISS) of interconnection of nonlinear time-varying systems. It aims at a useful tool for using nonlinear small-gain conditions by allowing some flexibility in Lyapunov inequalities each subsystem is to satisfy. In the application of the ISS small-gain “theorem”, achieving a Lyapunov inequality conforming to a nonlinear small-gain “condition” is not a straightforward task. The proposed technique provides us with many Lyapunov inequalities with which a single trade-off condition between subsystems gains can establish the ISS property of the interconnected system. Proofs are based on explicit construction of Lyapunov functions.     

A general boundedness result for interconnected nonlinear systems

A general boundedness result is given for the feedback interconnection of two nonlinear stable systems. Using the same input-output approach as in the standard small-gain theorem, the sufficient condition given here relaxes the finite-gain stability assumption and does not require a boundedness result for all possible bounded exogenous inputs. This condition has a simple graphical interpretation that utilizes graphs of bounding functions     

New results in global stabilization for stochastic nonlinear systems

This paper presents new results on the robust global stabilization and the gain assignment problems for stochastic nonlinear systems. Three stochastic nonlinear control design schemes are developed. Furthermore, a new stochastic gain assignment method is developed for a class of uncertain interconnected stochastic nonlinear systems. This method can be combined with the nonlinear small-gain theorem to design partial-state feedback controllers for stochastic nonlinear systems. Two numerical examples are given to illustrate the effectiveness of the proposed methodology.     

NN-adaptive output feedback tracking control for a class of discrete-time non-affine systems with a dynamic compensator

The problem of tracking control for a class of uncertain non-affine discrete-time nonlinear systems with internal dynamics is addressed. The fixed point theorem is first employed to ensure the control problem in question is solvable and well-defined. Based on it, an adaptive output feedback control scheme based on neural network (NN) is presented. The proposed control algorithm consists of two parts: a dynamic compensator is introduced to stabilise the linear portion of the tracking error system; a single-hidden-layer neural network (SHL NN) approximation mechanism is introduced to cancel the uncertainties resulting from the non-affine function, where the recursive weight update rules of NN estimation are derived from the discrete-time version of Lyapunov control theory. Ultimate boundedness of the error signals is shown through Lyapunov’s direct method and the discrete-time version of input-to-state stability (ISS) theory. Finally, a model of automatical underwater vehicle (AUV) is considered to show the effectiveness of the proposed control scheme.     

Quasi-ISS/ISDS observers for interconnected systems and applications

This paper considers interconnected nonlinear dynamical systems and studies observers for such systems. For single systems the notion of quasi-input-to-state dynamical stability (quasi-ISDS) for reduced-order observers is introduced and observers are investigated using error Lyapunov functions. It combines the main advantage of ISDS over input-to-state stability (ISS), namely the memory fading effect, with reduced-order observers to obtain quantitative information about the state estimate error. Considering interconnections quasi-ISS/ISDS reduced-order observers for each subsystem are derived, where suitable error Lyapunov functions for the subsystems are used. Furthermore, a quasi-ISS/ISDS reduced-order observer for the whole system is designed under a small-gain condition, where the observers for the subsystems are used. As an application, we prove that quantized output feedback stabilization for each subsystem and the overall system is achievable, when the systems possess a quasi-ISS/ISDS reduced-order observer and a state feedback law that yields ISS/ISDS for each subsystem and therefor the overall system with respect to measurement errors. Using dynamic quantizers it is shown that under the mentioned conditions asymptotic stability can be achieved for each subsystem and for the whole system.     

Input-to-state stability and gain computation for networked control systems

This paper studies the stability problem for networked control systems. A general result, called network gain theorem, is introduced to determine the input-to-state stability (ISS) for interconnected nonlinear systems. We show how this result generalises the previously known small gain theorem and cyclic small gain theorem for ISS. For the case of linear networked systems, a complete characterisation of the stability condition is provided, together with two distributed algorithms for computing the network gain: the classical Jacobi iterations and a message-passing algorithm. For the case of nonlinear networked systems, characterisation of the ISS condition can be done using M-functions, and Jacobi iterations can be used to compute the network gain.     

Quasi-exponential input-to-state stability for discrete-time impulsive hybrid systems

This paper investigates the problem of input-to-state stability (ISS) for the general discrete-time impulsive hybrid systems. Quasi-exponential input-to-state stability criteria are established for non-linear discrete-time impulsive hybrid systems, based on the Lyapunov function. The results are then specialized to the linear discrete-time impulsive hybrid systems and fairly simple algebraic conditions are derived. Moreover, ISS small-gain properties are also investigated for a class of interconnected discrete-time impulsive systems. Several examples are given to illustrate results.     

Input-to-state stability of infinite-dimensional control systems

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.     

Input-to-state stability analysis for interconnected difference equations with delay

Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.     

动态不确定非线性系统直接自适应模糊backstepping控制

对一类单输入单输出动态不确定非线性系统,提出一种直接自适应模糊backstepping和小增益相结合的控制方法.设计中,首先用模糊逻辑系统逼近虚拟控制器:其次把自适应模糊控制和backstepping控制设计技术相结合.给出了直接自适应模糊控制设计方法.最后基于Lyapunov函数和小增益方法证明了整个闭环系统的稳定性.仿真实例进一步验证了所提方法的有效性.     

输入饱和及输出受限的纯反馈非线性系统控制

针对具有输入饱和和输出受限的纯反馈非线性系统,设计了神经网络自适应控制器.首先利用隐函数定理和中值定理将非仿射形式的纯反馈非线性系统转换成有显式输入的非线性系统,基于李雅普诺夫第二方法以及反推法并采用障碍型李雅普诺夫函数进行控制器的设计,最后通过稳定性分析证明了闭环控制系统是半全局一致最终有界的,利用仿真例子验证了控制...     

Stable adaptive fuzzy control of nonlinear systems using small-gain theorem and LMI approach

A new design scheme of stable adaptive fuzzy control for a class of nonlinear systems is proposed in this paper. The T-S fuzzy model is employed to represent the systems. First, the concept of the so-called parallel distributed compensation (PDC) and linear matrix inequality (LMI) approach are employed to design the state feedback controller without considering the error caused by fuzzy modeling. Sufficient conditions with respect to decay rate α are derived in the sense of Lyapunov asymptotic stability. Finally, the error caused by fuzzy modeling is considered and the input-tostate stable (ISS) method is used to design the adaptive compensation term to reduce the effect of the modeling error. By the small-gain theorem, the resulting closed-loop system is proved to be input-to-state stable. Theoretical analysis verifies that the state converges to zero and all signals of the closed-loop systems are bounded. The effectiveness of the proposed controller design methodology is demonstrated through numerical simulation on the chaotic Henon system.     

Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids

Motivated by the problem of phase-locking in droop-controlled inverter-based microgrids with delays, the recently developed theory of input-to-state stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using Lyapunov–Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum. For this case, corresponding delay-dependent conditions for asymptotic phase-locking are given.     

Mean-square small gain theorem for stochastic control:discrete-time case

This paper presents a small gain theorem in the mean square sense for multiple (interconnected) linear systems with multiplicative noises. The small-gain theorem is proposed in terms of the spectral radius of a matrix, whose elements are the squares of H₂ norms of the involved transfer functions. Both robust stability and performance conditions are characterized by the new small-gain theorem     

Nonlinear small-gain theorems for discrete-time feedback systems and applications

We derive in this work a local nonlinear small-gain theorem in the framework of input-to-state stability for discrete time systems. Our primary objective is to show that, as in the continuous-time context, these discrete-time nonlinear small-gain theorems are very effective in stability analysis and synthesis for various classes of discrete-time control systems. Two converse Lyapunov theorems for discrete exponential stability are developed to assist these applications. New results in stability and stabilization presented in this paper are significant extensions of previous work by other authors (IEEE Trans. Automat. Control 38 (1993) 1398; 39 (1994) 2340; 33 (1988) 1082).     

Further results on input-to-state stability for nonlinear systems with delayed feedbacks

We consider a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions for the closed-loop systems are available. In the presence of feedback delays and actuator errors, we explicitly construct input-to-state stability (ISS) Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the available Lyapunov functions for the original undelayed dynamics, which establishes that the closed-loop systems are input-to-state stable (ISS) with respect to actuator errors. We illustrate our results using a generalized system from identification theory and other examples.     

An ISS-modular approach for adaptive neural control of pure-feedback systems

Controlling non-affine non-linear systems is a challenging problem in control theory. In this paper, we consider adaptive neural control of a completely non-affine pure-feedback system using radial basis function (RBF) neural networks (NN). An ISS-modular approach is presented by combining adaptive neural design with the backstepping method, input-to-state stability (ISS) analysis and the small-gain theorem. The difficulty in controlling the non-affine pure-feedback system is overcome by achieving the so-called “ISS-modularity” of the controller-estimator. Specifically, a neural controller is designed to achieve ISS for the state error subsystem with respect to the neural weight estimation errors, and a neural weight estimator is designed to achieve ISS for the weight estimation subsystem with respect to the system state errors. The stability of the entire closed-loop system is guaranteed by the small-gain theorem. The ISS-modular approach provides an effective way for controlling non-affine non-linear systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.