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.     

切换系统的不变性原理与不变集的状态反馈镇定

证明了一类切换系统的一个不变性原理,并将输入对状态稳定的概念推广到输入对系统某个非负能量函数稳定的情况.基于这个不变性原理以及输入对系统能量函数稳定的概念,利用多Lyapunov函数方法提出并证明了一类具有Lyapunov稳定子系统的切换系统的不变集可状态反馈镇定的条件.最后讨论了输入对系统能量函数稳定与输入对状态稳定的关系.仿真结果证明了该方法的可行性.     

A unifying framework for global regulation via nonlinear output feedback: from ISS to iISS

This paper presents a unifying framework for the problem of robust global regulation via output feedback for nonlinear systems with integral input-to-state stable inverse dynamics, subject to possibly unknown control direction. The contribution of the paper is two-fold. Firstly, we consider the problem of global regulation, instead of global asymptotic stabilization (GAS), for systems with generalized dynamic uncertainties. It is shown by an elementary example that GAS is not solvable using conventional smooth output feedback. Secondly, we reduce the stability requirements for the disturbance and demand relaxed assumptions for the system. Using our framework, most of the known classes of output feedback form systems are broadened in several directions: unmeasured states and unknown parameters can appear nonlinearly, restrictive matching and growth assumptions are removed, the dynamic uncertainty satisfies the weaker condition of Sontag's integral input-to-state stability, and the sign of high-frequency gain may be unknown. A constructive strategy is proposed to design a dynamic output feedback control law, that drives the state to the origin while keeping all other closed-loop signals bounded.     

Markov跳跃非线性系统逆最优增益设计

证明了一类严格反馈Markov跳跃系统是依概率输入–状态可稳定的.其次,证明了逆最优增益设计问题可解的一个充分条件是存在一组满足小控制量的依概率输入–状态稳定控制李雅普诺夫函数.最后,利用积分反推方法,给出了严格反馈Markov跳跃系统逆最优增益设计问题的一个构造性解.其中,为了克服由于Markov跳跃引起的耦合项所带来的困难,所设计的李雅普诺夫函数以及控制器是与模态无关的.     

Output feedback for stochastic nonlinear systems with unmeasurable inverse dynamics

This paper considers a concrete stochastic nonlinear system with stochastic unmeasurable inverse dynamics. Motivated by the concept of integral input-to-state stability （iISS） in deterministic systems and stochastic input-to-state stability （SISS） in stochastic systems, a concept of stochastic integral input-to-state stability （SiISS） using Lyapunov functions is first introduced. A constructive strategy is proposed to design a dynamic output feedback control law, which drives the state to the origin almost surely while keeping all other closed-loop signals almost surely bounded. At last, a simulation is given to verify the effectiveness of the control law.     

具有状态和控制约束的受扰离散线性切换系统的反馈控制

本文的主要贡献是针对一类具有重置函数及由外部不能控事件决定动态的离散时间线性切换系统,给出一些稳定性综合结论. 当系统受到外部有界扰动, 及状态和控制量约束时, 在输入到状态稳定性理论框架下, 研究使得系统镇定的线性状态反馈控制器设计方法. 针对这类混杂系统, 本文引入了受控D不变性的概念, 并给出检测某一混杂区域具有受控D不变性的充要条件. 进而, 提出一种能够使得受扰的线性切换系统镇定, 同时保证状态和控制量满足其约束的反馈矩阵的计算方法. 最后, 通过一个由两个子系统构成的数值例子来说明本文技术的应用性.     

A necessary and sufficient condition for input-to-state stability of quantised feedback systems

For zooming-out/in method used in the design of quantised feedback systems, the property of the duration of zoom-out mode (this duration is defined as capture time) is essential to input-to-state stability (ISS) of systems. This paper shows that a necessary and sufficient condition of achieving ISS with respect to external disturbances for quantised feedback systems is that capture time under the proposed coding scheme is uniformly bounded. It further shows that the coding scheme under which capture time is only bounded and not uniformly bounded cannot guarantee ISS of systems. A coding scheme is designed for uniformly bounded capture time and therefore achieves ISS of systems.     

线性多变量系统的稳定解耦

本文用Yokoyama标准形与多项式阵之间的对应关系及矩阵列的"零块指数"概念,给出 了用状态反馈实现解耦和稳定解耦的充分必要条件,并给出了确定实现解耦的状态反馈阵的 方法.     

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.     

Certainty-Equivalence Feedback Design With Polynomial-Type Feedbacks Which Guarantee ISS

The purpose of this note is to establish a certainty-equivalence feedback design for inverse optimally controlled affine systems. In particular, it is shown that a class of polynomial-type state feedbacks in conjunction with a globally asymptotically convergent observer leads to a globally asymptotically stable closed-loop. A key step in the proposed certainty-equivalence feedback design procedure is the identification of a new class of polynomial-type inverse optimal feedbacks which guarantees input-to-state stability (ISS) with respect to measurement errors. As a consequence, the proposed certainty-equivalence feedback design has the important feature that the state feedback is allowed to contain polynomial nonlinearities of arbitrarily high degree in the unmeasured states. This feature is illustrated on an example     

Robust ISS of uncertain discrete-time singularly perturbed systems with disturbances

This paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stability and stabilisation of uncertain discrete-time SPSs are also obtained as the particular cases of robust ISS and robust ISS by feedback. We first find a sufficient condition by using the fixed-point principle in terms of linear matrix inequalities (LMIs) to guarantee that the considered system is always standard discrete-time SPSs subject to uncertainty and disturbances. Then, the full systems could decompose into the continuous-time uncertain slow subsystem with disturbance and discrete-time uncertain fast subsystems with disturbance, respectively. Based on the two-time-scale decomposition technique, sufficient condition in terms of LMIs is given such that the full systems are uniformly standard and robust ISS simultaneously. In addition, a state feedback controller is constructed by using the LMI approach such that the resulting closed-loop systems are robust ISS. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.     

Output feedback distributed optimization algorithms of second-order Lipschitz nonlinear multi-agent systems

This paper introduces output feedback distributed optimization algorithms designed specifically for second-order nonlinear multi-agent systems. The agents are allowed to have heterogeneous dynamics, characterized by distinct nonlinearities, as long as they satisfy the Lipschitz continuity condition. For the case with unknown states, nonlinear state observers are designed first for each agent to reconstruct agents' unknown states. It is proven that the agents' unknown states are estimated accurately by the developed state observers. Then, based on the agents' state estimates and the gradient of each agent local cost function, a kind of output feedback distributed optimization algorithms are proposed for the considered multi-agent systems. Under the proposed distributed optimization algorithms, all the agents' outputs asymptotically approach the minimizer of the global cost function which is the sum of all the local cost functions. By using Lyapunov stability theory, convex analysis, and input-to-state stability theory, the asymptotical convergence of the output feedback distributed optimization closed-loop system is proven. Simulations are conducted to validate the efficacy of the proposed algorithms.     

Hybrid feedback stabilization of systems with quantized signals

This paper is concerned with global asymptotic stabilization of continuous-time systems subject to quantization. A hybrid control strategy originating in earlier work (Brockett and Liberzon, IEEE Trans. Automat. Control 45 (2000) 1279) relies on the possibility of making discrete on-line adjustments of quantizer parameters. We explore this method here for general nonlinear systems with general types of quantizers affecting the state of the system, the measured output, or the control input. The analysis involves merging tools from Lyapunov stability, hybrid systems, and input-to-state stability.     

Output Feedback Regulation of Stochastic Nonlinear Systems With Stochastic iISS Inverse Dynamics

This paper develops a unifying framework for output feedback regulation of stochastic nonlinear systems with more general stochastic inverse dynamics. The contributions of this work are characterized by the following novel features: 1) Motivated by the concept of integral input-to-state stability (iISS) in deterministic systems and stochastic input-to-state stability (SISS) using Lyapunov function in stochastic systems, a concept of stochastic integral input-to-state stability (SiISS) using Lyapunov function is first introduced, two important properties of SiISS are obtained: (i) SiISS is strictly weaker than SISS using Lyapunov function; (ii) SiISS is stronger than the minimum-phase property. However, only under the minimum-phase assumption, there is no dynamic output feedback control law for global stabilization in probability. 2) Almost sure boundedness, a reasonable and stronger concept than boundedness in probability, is introduced. The purpose of introducing the concept is to prove the boundedness and convergence of some signals in the closed-loop control system. 3) Some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed-loop system are established. 4) A unifying framework is proposed to design a dynamic output feedback control law, which drives the states to the origin almost surely while maintaining all the closed-loop signals bounded almost surely.      

事件触发双模分布式预测控制

本文针对有界扰动作用下的线性离散大系统,提出了事件触发双模分布式预测控制设计方法.利用输入状态稳定性(input-to-state stability,ISS)理论建立了仅与子系统自身信息相关的事件触发条件.只有子系统满足相应的事件触发条件,才进行状态信息的传输和分布式预测控制优化问题的求解,并与邻域子系统交互最优解作用下的关联信息.当子系统进入不变集时,采用状态反馈控制律进行镇定,并与进入不变集的邻域子系统不再交互信息.分析了算法的递推可行性和系统的闭环稳定性,给出了扰动的上界.最后,通过车辆控制系统对算法进行仿真验证,结果表明,本文提出的方法能够有效降低优化问题的求解次数和关联信息的交互次数,节约计算资源和通信资源.     

A remark on partial-state feedback stabilization of cascade systems using small gain theorem

This paper points out that input-to-state stability of zero dynamics having a continuously differentiable (instead of locally Lipschitz continuous) gain function suffices to guarantee the existence of globally stabilizing, smooth partial-state feedback control laws for cascade systems, without imposing any extra condition. This conclusion is proved via the small gain theorem and a novel variable separation technique combined with feedback domination design.     

The disturbance decoupling problem with stability for switching dynamical systems

The disturbance decoupling problem with stability is dealt with by means of the geometric approach for switching systems. The existence of feedbacks which decouple the disturbance and, at the same time, assure stability is difficult to characterize, since the action of the feedback couples with that of the switching law. Under suitable conditions, it is shown that the above requirement can be dealt with in separate ways and this allows us to state a checkable necessary condition and, on that basis, also a sufficient condition for solvability of the problem.     

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).     

Stabilizability of linear systems using discrete feedback

The eigenvalue assignment problem of a controllable continuous linear system using continuous and discrete feedback loops is discussed. It is shown that for a given set of distinct self-conjugate complex numbers included inside of the unit circle, and for an arbitrarily given value of the sampling frequency (in the discrete feedback loop), it is possible to choose constant feedback matrices such that eigenvalues of the matrix of dynamics of the closed-loop system are located sufficiently near to appropriate numbers of the given set of complex numbers. The obtained result may immediately be applied in the case of discrete linear systems with sampling of the input signal and continuous state feedbacks.     

Smooth stabilization implies coprime factorization

It is shown that coprime right factorizations exist for the input-to-state mapping of a continuous-time nonlinear system provided that the smooth feedback stabilization problem is solvable for this system. It follows that feedback linearizable systems admit such fabrications. In order to establish the result, a Lyapunov-theoretic definition is proposed for bounded-input-bounded-output stability. The notion of stability studied in the state-space nonlinear control literature is related to a notion of stability under bounded control perturbations analogous to those studied in operator-theoretic approaches to systems; in particular it is proved that smooth stabilization implies smooth input-to-state stabilization