非线性切换广义系统的输入– 状态稳定性

针对一类非线性切换广义系统,分两种情况对其输入-状态稳定性问题进行了研究.采用平均驻留时间方法,通过设计适当的切换规则,给出了非线性切换广义系统整体输入-状态稳定的充分条件.相较于已有的控制方法,该方法无需构造输入-状态稳定的Lyapunov函数以及无需设计控制器的具体结构,这在一定程度上方便了控制器的设计.最后,两个仿真算例表明了所提方法的有效性.     

一类时滞切换系统的输入-状态稳定性分析

利用多Lyapunov函数方法、驻留时间法和Gronwall-Bellman不等式研究了一类时滞切换系统的输入-状态稳定性分析问题．从系统输入-状态稳定定义出发,给出了使得一类时滞切换系统输入-状态稳定的充分条件．与已有的方法相比,无需同时满足构造输入-状态稳定控制Lyapunov函数和所有子系统都是输入-状态稳定的条件,为控制器的设计提供了便利．最后,通过算例仿真验证了所提出方法的可行性．     

基于输入状态稳定的离散广义系统预测控制

针对一类具有持续扰动和输入约束的离散广义系统, 研究其鲁棒预测控制器的设计问题. 将输入状态稳定的概念引入广义系统预测控制, 在quasi-min-max 性能指标下, 提出了广义系统双模鲁棒预测控制器的设计方法, 证明了基于双模鲁棒预测控制器的闭环广义系统输入状态稳定, 且具有正则、因果性. 数值仿真结果验证了所提出方法的有效性.

     

Switched seesaw control for the stabilization of underactuated vehicles

This paper addresses the stabilization of a class of nonlinear systems in the presence of disturbances, using switching controllers. To this effect we introduce two new classes of switched systems and provide conditions under which they are input-to-state practically stable (ISpS). By exploiting these results, a methodology for control systems design—called switched seesaw control—is obtained that allows for the development of nonlinear control laws yielding input-to-state stability. The range of applicability and the efficacy of the methodology proposed are illustrated via two nontrivial design examples. Namely, stabilization of the extended nonholonomic double integrator (ENDI) and stabilization of an underactuated autonomous underwater vehicle (AUV) in the presence of input disturbances and measurement noise.     

一类前馈系统的鲁棒镇定

Two kinds of saturated controllers are designed for a class of feedforward systems and the closed-loop resulted is locally input-to-state stable and input-to-state stable, respectively. By the word "locally", it is meant that there are restrictions on the amplitude of inputs. At first, under the guidance of suitable energy functions, two kinds of saturated controllers are designed as locally input-to-state stabilizers for a class of perturbed linear systems, from which explicit gain estimations can be obtained for the subsequent design. Then under the conditions that two subsystems of the feedforward system are respectively of locally input-to-state stability and input-to-state stability, the small gain theory is used to determine saturated degrees for corresponding robust stabilizers. The stability proofs are given by using a new characterization of input-to-state stability that is based on the concept of ultimate boundedness. As an application, saturated controllers are designed for the partial dynamics of a certain inverted pendulum.     

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.     

Adaptive backstepping controller design using stochastic small-gain theorem

A more general class of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions are considered in this paper. With the concept of input-to-state practical stability (ISpS) and nonlinear small-gain theorem being extended to stochastic case, by combining stochastic small-gain theorem with backstepping design technique, an adaptive output-feedback controller is proposed. It is shown that the closed-loop system is practically stable in probability. A simulation example demonstrates the control scheme.     

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.     

Tracking control and robustness analysis for a nonlinear model of human heart rate during exercise

The control of human heart rate during exercise is an important problem that has implications for the development of protocols for athletics, assessing physical fitness, weight management, and the prevention of heart failure. Here we provide a new stabilization technique for a recently-proposed nonlinear model for human heart rate response that describes the central and peripheral local responses during and after treadmill exercise. The control input is the treadmill speed, and the control objective is to make the heart rate track a prescribed reference trajectory. We use a strict Lyapunov function analysis to design new state and output feedback tracking controllers that render the error dynamics globally exponentially stable. This allows us to prove input-to-state stability properties for our feedback stabilized systems under actuator errors. This robustness property quantifies the effects of variations of the treadmill speed from the controller values. We illustrate our feedback design through simulations.     

Integrated guidance and control for dual-control missiles based on small-gain theorem

An integrated guidance and control (IGC) design approach is proposed based on small-gain theorem for missiles steered by both canard and tail controls. The angle of attack and pitch rate commands, which are aimed at producing desired aerodynamic lift to achieve robust tracking of a maneuvering target, are generated by a guidance law that is designed using input-to-state stability (ISS) theory. An IGC law is developed utilizing generalized small-gain theorem to enforce the commands, and it can be shown that both the line-of-sight (LOS) rate and the tracking error are input-to-state practically stable (ISpS) with respect to target maneuvers and missile model uncertainties. The algorithm is tested using computer simulations against a maneuvering target.     

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.     

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.     

Input-to-state stability of networked control systems

A new class of Lyapunov uniformly globally asymptotically stable (UGAS) protocols in networked control systems (NCS) is considered. It is shown that if the controller is designed without taking into account the network so that it yields input-to-state stability (ISS) with respect to external disturbances (not necessarily with respect to the error that will come from the network implementation), then the same controller will achieve semi-global practical ISS for the NCS when implemented via the network with a Lyapunov UGAS protocol. Moreover, the ISS gain is preserved. The adjustable parameter with respect to which semi-global practical ISS is achieved is the maximal allowable transfer interval (MATI) between transmission times.     

Input-to-State Stabilization of Nonlinear Impulsive Delayed Systems: An Observer-Based Control Approach

This paper addresses the problems of input-to-state stabilization and integral input-to-state stabilization for a class of nonlinear impulsive delayed systems subject to exogenous dis- turbances. Since the information of plant’s states, time delays, and exogenous disturbances is often hard to be obtained, the key design challenge, which we resolve, is the construction of a state observer-based controller. For this purpose, we firstly propose a corresponding observer which is independent of time delays and exogenous disturbances to reconstruct (or estimate) the plant’s states. And then based on the observations, we establish an observer-based control design for the plant to achieve the input-to-state stability (ISS) and integral-ISS (iISS) properties. With the help of the comparison principle and average impulse interval approach, some sufficient conditions are presented, and moreover, two different linear matrix inequalities (LMIs) based criteria are proposed to design the gain matrices. Finally, two numerical examples and their simulations are given to show the effectiveness of our theoretical results.      

Preservation of input-to-state stability under sampling and emulation: multi-rate case

The stabilization problem of sampled-data non-linear systems is considered under the low measurement rate constraint. A multi-rate control scheme is proposed that utilizes a numerical integration scheme to approximately predict the current state. We show that if we design a continuous-time controller for a continuous-time plant so that the closed-loop continuous-time system is input-to-state stable and then discretize the controller and implement it using sample and zero order hold devices, then input-to-state stability property will be preserved for the sampled-data multi-rate closed loop system in a practical sense.     

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems.Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.     

Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities

The presence of friction in mechanical motion systems is a performance limiting factor as it induces stick-slip vibrations. To appropriately describe the stiction effect of friction, we adopt set-valued force laws. Then, the complete motion control system can be described by a Lur’e system with set-valued nonlinearities. In order to eliminate stick-slip vibrations for mechanical motion systems, a state-feedback control design is presented to stabilize the equilibrium. The proposed control design is based on an extension of a Popov-like criterion to systems with set-valued nonlinearities that guarantees input-to-state stability (ISS). The advantages of the presented controller is that it is robust to uncertainties in the friction and it is applicable to systems with non-collocation of actuation and friction where common control strategies such as direct friction compensation fail. Moreover, an observer-based output-feedback design is proposed for the case that not all the state variables are measured. The effectiveness of the proposed output-feedback control design is shown both in simulations and experiments for a typical motion control system.     

Quantized stabilization of strict-feedback nonlinear systems based on ISS cyclic-small-gain theorem

This paper proposes a new tool for quantized nonlinear control design of dynamic systems transformable into the dynamically perturbed strict-feedback form. To address the technical challenges arising from measurement and actuator quantization, a new approach based on set-valued maps is developed to transform the closed-loop quantized system into a large-scale system composed of input-to-state stable (ISS) subsystems. For each ISS subsystem, the inputs consist of quantization errors and interacting states, and moreover, the ISS gains can be assigned arbitrarily. Then, the recently developed cyclic-small-gain theorem is employed to guarantee input-to-state stability with respect to quantization errors and to construct an ISS-Lyapunov function for the closed-loop quantized system. Interestingly, it is shown that, under some realistic assumptions, any n-dimensional dynamically perturbed strict-feedback nonlinear system can be globally practically stabilized by a quantized control law using 2n three-level dynamic quantizers.