Non-linear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality

In a companion paper (Nonlinear Impulsive Dynamical Systems. Part I: Stability and Dissipativity) Lyapunov and invariant set stability theorems and dissipativity theory were developed for non-linear impulsive dynamical systems. In this paper we build on these results to develop general stability criteria for feedback interconnections of non-linear impulsive systems. In addition, a unified framework for hybrid feedback optimal and inverse optimal control involving a hybrid non-linear-non-quadratic performance functional is developed. It is shown that the hybrid cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the non-linear closed-loop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, hybrid Hamilton‐Jacobi‐Bellman equation.     

Vector dissipativity theory and stability of feedback interconnections for large-scale non-linear dynamical systems

Recent technological demands have required the analysis and control design of increasingly complex, large-scale non-linear dynamical systems. In analysing these large-scale systems, it is often desirable to treat the overall system as a collection of interconnected subsystems. Solution properties of the large-scale system are then deduced from the solution properties of the individual subsystems and the nature of the system interconnections. In this paper we develop an analysis framework for large-scale dynamical systems based on vector dissipativity notions. Specifically, using vector storage functions and vector supply rates, dissipativity properties of the composite large-scale system are shown to be determined from the dissipativity properties of the subsystems and their interconnections. Furthermore, extended Kalman–Yakubovich–Popov conditions, in terms of the subsystem dynamics and interconnection constraints, characterizing vector dissipativeness via vector system storage functions are derived. Finally, these results are used to develop feedback interconnection stability results for large-scale non-linear dynamical systems using vector Lyapunov functions.     

Limit cycle generation for a class of non-linear systems with jumps using a low dimensional predictive control

In this paper, it is shown that using a low dimensional non-linear predictive control scheme, provably stable limit cycles can be obtained for open-loop non-linear systems with unstable equilibrium point. A particular case, where the limit cycle may be reduced to a single point in the state space, can be obtained, which corresponds to asymptotic stabilization. The system may present hybrid nature in the sense that discontinuities (jump phenomena) on the state evolution may be handled. The proposed feedback scheme holds for classical jump-free systems as a particular case. The proposed strategy is illustrated through two examples: a jump-free system (the ball and beam) and a non-linear hybrid dynamical system including state jumps (the modified impulsive Lorenz chaotic system).     

Stability and dissipativity theory for discrete-time non-negative and compartmental dynamical systems

Non-negative and compartmental dynamical systems are derived from mass and energy balance considerations that involve dynamic states whose values are non-negative. These models are widespread in engineering, biomedicine and ecology. In this paper we develop several results on stability, dissipativity and stability of feedback interconnections of discrete-time linear and non-linear non-negative dynamical systems. Specifically, using linear Lyapunov functions we first develop necessary and sufficient conditions for Lyapunov stability and asymptotic stability for non-negative systems. In addition, using linear and non-linear storage functions with linear supply rates we develop new notions of dissipativity theory for non-negative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of non-negative dynamical systems.     

Optimal control and robust stability of uncertain impulsive dynamical systems

This paper studies the problem of optimal feedback control and robust stability for uncertain impulsive dynamical systems. By using algebraic inequalities, Riccati and Hamilton‐Jacobi inequalities, the conditions are derived under which not only the uncertain impulsive dynamical system has robust asymptotic stability but also the value of the optimal hybrid performance functional can be estimated. An example is also given to illustrate our results. © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

High-gain feedback in non-linear control systems†

High-gain state and output feedback are investigated for non-linear control systems with a single additive input by using singular perturbation techniques.

Classical approximation results (Tihonov-like theorems) in singular perturbation theory are extended to non-linear control systems by defining a composite additive control strategy, a control-dependent fast equilibrium manifold and non-linear change of coordinates.

Those tools and an appropriate change of coordinates show that high-gain state feedback and variable structure control systems can be equivalently used for approximate non-linearity compensation in feedback-linearizable systems.

Next the effect of high-gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility. For strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input-output feedback-linearizable transformation is applied.     

Dissipativity theory for nonnegative and compartmental dynamical systems with time delay

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences and typically involve the exchange of nonnegative quantities between subsystems or compartments wherein each compartment is assumed to be kinetically homogeneous. However, in many engineering and life science systems, transfers between compartments are not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. Including some information of past system states in the system model leads to infinite-dimensional delay nonnegative dynamical systems. In this note, we develop dissipativity theory for nonnegative dynamical systems with time delay using linear storage functionals with linear supply rates. These results are then used to develop general stability criteria for feedback interconnections of nonnegative dynamical systems with time delay.     

Energy- and Entropy-Based Stabilization for Lossless Dynamical Systems via Hybrid Controllers

A novel class of dynamic, energy-based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in lossless dynamical systems. These dynamic controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that the closed-loop system is consistent with basic thermodynamic principles. In particular, the existence of an entropy function for the closed-loop system is established that satisfies a hybrid Clausius-type inequality. Special cases of energy-based and entropy-based hybrid controllers involving state-dependent switching are described.     

Hybrid‐impulsive second‐order sliding mode control: Lyapunov approach

Dynamic system of relative degree two controlled by discontinuous‐hybrid‐impulsive feedback in the presence of bounded perturbations is considered. The state feedback impulsive‐twisting control exhibits a uniform exact finite time convergence to the second‐order sliding mode with zero convergence time. The output feedback discontinuous control augmented by a simplified hybrid‐impulsive functions provides uniform exact convergence with zero convergence time of the system's states to a real second‐order sliding mode in the presence of bounded perturbations. Only ‘snap’ knowledge of the output derivative, that is, the knowledge of the output derivative in isolated time instants, is required. The output feedback hybrid‐impulsive control with practically implemented impulsive actions asymptotically drives the system's states to the origin. The Lyapunov analysis of the considered hybrid‐impulsive‐discontinuous system proves the system's stability. The efficacy of the proposed control technique is illustrated via computer simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability theory of hybrid dynamical systems with time delay

By defining the bounded time lag space and other related concepts, this note formulates the notion of hybrid dynamical systems with time delay, which provides a unified framework for describing a large class of systems and is suitable for qualitative analysis. It also investigates stability properties of invariant sets by the method of Lyapunov functions and the Razumihin techniques. Criteria on uniform stability and uniform asymptotic stability are established for hybrid dynamical systems with time delay. Those criteria are then applied to conduct stability analysis for impulsive delay differential systems and nonlinear sampled-data feedback control systems with time delay.     

The Popov criterion and the inverse Nyquist method†

Kosonbrock's inverse Nyquist array (I.N.A.) theory for linear multivariable control Bystema with constant feedback elements is extended, to include systems lip to m nonlinear feedback elements, where the system has m inputs and m outputs. This extension is achieved by considering the Popov criterion for the most general case and through two further theorems. It shows that, as in the ease of Rosenbrock's I.N.A. theory, when certain auxiliary conditions are met with the help of suitable controllers, the design of multivariable controllers containing many non-linear feedbacks, can be based on the m frequency response loci corresponding to the diagonal entries of the open-loop inverse transfer function matrix. This leads to a simple design technique identical to the I.N.A. design method, suitable for use with a computer-aided design facility which permits a designer to use his intuitive understanding of transfer functions based on classical theory. The I.N.A. theory has been extended by Rosenbrock to cover systems having non-linear, time-varying feedback elements very recently.     

Robust H∞ control of uncertain linear impulsive stochastic systems

This paper develops robust stability theorems and robust H_∞ control theory for uncertain impulsive stochastic systems. The parametric uncertainties are assumed to be time varying and norm bounded. Impulsive stochastic systems can be divided into three cases, namely, the systems with stable/stabilizable continuous‐time stochastic dynamics and unstable/unstabilizable discrete‐time dynamics, the systems with unstable/unstabilizable continuous dynamics and stable/stabilizable discrete‐time dynamics, and the systems in which both the continuous‐time stochastic dynamics and the discrete‐time dynamics are stable/stabilizable. Sufficient conditions for robust exponential stability and robust stabilization for uncertain impulsive stochastic systems are derived in terms of an average dwell‐time condition. Then, a linear matrix inequality‐based approach to the design of a robust H_∞ controller for each system is presented. Finally, the numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Feedback passivation of switched nonlinear systems using storage-like functions

This paper addresses the issues of passivity and feedback passivation of switched nonlinear systems via multiple storage functions. The concept of storage-like functions for switched systems is presented. A sufficient condition for passivity of switched nonlinear systems is given via multiple storage functions under some switching signal. Then, the result is extended to find conditions under which a switched system is feedback equivalent to a passive switched system. Furthermore, passivity of switched cascade systems is investigated.     

On hybrid control of a class of stochastic non-linear Markovian switching systems

In this paper, we address the problem of hybrid control for a class of stochastic non-linear Markovian switching systems. First, a hybrid controller is introduced for the systems. Then under some appropriate assumptions, the stabilization condition for the systems under pure impulsive control is given. Further under impulsive control, the output feedback stabilization problem of the systems is discussed and linear output feedback controllers are designed. Finally a numerical example is provided to illustrate the effectiveness of the proposed methods.     

Razumikhin stability theorems for a general class of stochastic impulsive switched time‐delay systems

This paper studies stability of a general class of impulsive switched systems under time delays and random disturbances using multiple Lyapunov functions and fixed dwell‐time. In the studied system model, the impulses and switches are allowed to occur asynchronously. As a result, the switching may occur in the impulsive intervals and the impulses can occur in the switching intervals, which have great effects on system stability. Since the switches do not bring about the change of the system state, we study two cases in terms of the impulses, ie, the stable continuous dynamics case and the stable impulsive dynamics case. According to multiple Lyapunov‐Razumikhin functions and the fixed dwell‐time, Razumikhin‐type stability conditions are established. Finally, the obtained results are illustrated via a numerical example from the synchronization problem of chaotic systems.     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

一类不确定脉冲型混杂系统的保代价控制

针对混合代价函数,研究了参数不确定脉冲型混杂系统的保代价控制问题,给出了混杂状态反馈保代价控制律的设计方法,由此得到的控制律既能使系统闭环鲁棒渐近稳定,又可使系统的闭环混合代价指标在对象参数摄动的范围内不超过确定的上界.本文提出的控制律不仅包含连续时间动态,也包含离散事件动态,而且其离散事件动态行为不需要与被控系统的离散事件动态行为一致,因此设计时不要求被控系统的每个连续时间子系统都具有可控性.仿真结果表明所提设计方法是可行有效的.     

Hybrid consensus protocols: an impulsive dynamical system approach

In this article, we develop a novel hybrid control framework to address fast consensus seeking problems for multiagent dynamical systems. Specifically, we present hybrid distributed controller architectures for multiagent coordination to improve the transient performance of coordination tasks. The proposed controller architectures are predicated on some novel hybrid dynamic compensation structures involving the exchange of information between agents. A unique feature of the proposed framework is that the proposed controller architectures are hybrid and appear to achieve finite-time coordination, and hence significantly improve the transient performance of the closed-loop system. The overall closed-loop dynamics under any of these controller algorithms achieving consensus possesses discontinuous flows since the controller algorithms combine logical switchings with continuous dynamics, leading to impulsive differential equations. Several simulation results are provided to validate the proposed consensus protocols.     

Passive stochastic feedback stability-Part II. Applications

In Part I of this paper it was shown that the concept of passivity, used extensively in deterministic feedback stability, could be extended to stochastic feedback systems described through Markov theory to produce stochastic feedback stability theorems. To apply the theory developed in Part I, one is required to do a passivity analysis of the sub-systems of the feedback system. In this paper, six particular types of stochastic systems (sub-systems) are considered and a passivity analysis of each is performed. These systems have the appearance of standardnth-order linear differential systems with a random process entering at some point. The randomness is either white noise, impulsive noise or a Markov jump process. Once the passivity analysis has been made, feedback stability theorems can be derived for many feedback systems containing one or more of these six systems as sub-systems. Two such feedback systems are considered in the paper.     

具有量化脉冲效应的时滞混杂动态网络同步分析

研究了具有量化脉冲效应的时滞混杂动态网络的同步动力学,利用偏收缩理论和矩阵测度概念,并结合量化器的特征,提出了网络节点实现同步的一般性条件.与以往大多数基于Lyapunov稳定性方法所提出的结果不同,利用本文的方法可以消除单个节点和网络耦合中的非线性函数的限制性假设,从而降低保守性.本文突出的特征就是综合了网络节点和耦合拓扑中的多重时滞影响和脉冲行为,并且考虑了脉冲效应在网络传输中的有限通信能力问题,具有实际意义.最后,通过对Lorenz混沌系统作为网络节点的混杂动态网络进行数值模拟,验证了所得结果的有效性.