Stability of sampled-data piecewise affine systems: A time-delay approach

This paper addresses the stability analysis of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant in feedback connection with a discrete-time emulation of a continuous-time state feedback controller. The sampled-data system is first considered as a continuous-time system with a variable time delay. Conditions under which the trajectories of the sampled-data closed-loop system will converge to an attracting invariant set are then presented. It is also shown that when the sampling period converges to zero, these conditions coincide with sufficient conditions for non-fragility of the stabilizing continuous-time PWA state feedback controller. The results are successfully applied to a helicopter example.     

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.     

Stability of sampled-data piecewise-affine systems under state feedback

This paper addresses stability of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant and a discrete-time emulation of a continuous-time state feedback controller. The paper presents conditions under which the trajectories of the sampled-data closed-loop system will exponentially converge to a neighborhood of the origin. Moreover, the size of this neighborhood will be related to bounds on perturbation parameters related to the sampling procedure, in particular, related to the sampling period. Finally, it will be shown that when the sampling period converges to zero the performance of the stabilizing continuous-time PWA state feedback controller can be recovered by the emulated controller.     

基于观测器的一类连续非线性系统的采样控制

首先使用反演方法分别设计了系统的连续时间状态反馈控制器、连续时间观测器和基于连续时间观测器的连续时间控制器. 接下来, 利用零阶保持法对连续时间状态反馈控制器进行离散化, 获得了状态反馈采样控制器; 利用零阶保持法对基于连续时间观测器 的连续时间控制器离散化, 获得了基于连续时间观测器的采样控制器; 利用Euler法对连续时间观测器离散化, 同时利用零阶保持法对控制器离散化, 从而获得了采样观测器和基于采样观测器的采样控制器. 本文论证了上述状态反馈采样控制器和基于连续时间观测器的采样控制器可以保证闭环系统渐近稳定, 而基于采样观测器的采样控制器可以保证被控对象的状态是有界的, 其最终边界依赖于设计参数与采样周期. 最后, 通过选择适当的采样周期, 完成了闭环采样控制系统的设计. 一个船舶航向控制的例子表明应用本文 所提方法设计出的三种采样控制器具有良好的控制效果.     

时滞不确定采样控制系统的鲁棒稳定性

本文给出了一种可定量分析采样控制系统的时滞鲁棒稳定性的方法.因为采样系统的对象是连续时间的,所以对象中的时滞也应该是按连续时间来处理.文中指出,一个整数倍时滞是稳定的采样系统,可能会因为有并不很大的连续时间时滞而失稳.定义了一个新的变量w(t),用来描述这个不确定连续时间时滞带来的动特性.将w(t)的反馈回路分成与时滞无关和有关的两个部分,并提出了一种用频率响应来确定是否存在由不确定时滞引起的周期解的方法.用修正z-变换法和仿真验证了这个由图解解析所求得的解.本方法既可用于采样系统,也可用于一般的连续时间系统.     

Optimal and control of hybrid multirate systems

The problem of using a synchronous multirate digital controller for a continuous-time plant is considered. The performance objectives considered are the H^∞ and the H² norms of the periodically time-varying continuous-time input-output behavior of the closed loop system. A continuous-time lifting technique is used to solve these hybrid sampled-data problems. This approach yields equivalent purely discrete-time problems while preserving the multirate causality of the systems. The later problems can then be solved using known techniques.     

Design and stabilization of sampled-data neural-network-based control systems.

This paper presents the design and stability analysis of a sampled-data neural-network-based control system. A continuous-time nonlinear plant and a sampled-data three-layer fully connected feedforward neural-network-based controller are connected in a closed loop to perform the control task. Stability conditions will be derived to guarantee the closed-loop system stability. Linear-matrix-inequality- and genetic-algorithm-based approaches will be employed to obtain the largest sampling period and the connection weights of the neural network subject to the considerations of the system stability and performance. An application example will be given to illustrate the design procedure and effectiveness of the proposed approach.     

Robustness With Respect to Sampling for Stabilization of Riesz Spectral Systems

We suppose that a continuous-time feedback is input–output stabilizing for an infinite-dimensional system. We address the question of whether the sampled-data controller obtained by applying idealized sample-and-hold to this continuous-time feedback is also input–output stabilizing if the sampling time is small enough. This question has been previously addressed for fairly general systems under various conditions. In this note, we restrict our attention to Riesz spectral systems, for which we generalize the existing results. Specifically, we give two relatively simple conditions which, combined, are sufficient for the sampled-data controller to be stabilizing. The first condition is a spectrum decomposition for the open-loop system generator, which by itself is necessary, but not sufficient, for the system to be stabilizable by sampled-data control. The second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators.     

A receding horizon control approach to sampled-data implementation of continuous-time controllers

We propose a novel way for sampled-data implementation (with the zero order hold assumption) of continuous-time controllers for general nonlinear systems. We assume that a continuous-time controller has been designed so that the continuous-time closed-loop satisfies all performance requirements. Then, we use this control law indirectly to compute numerically a sampled-data controller. Our approach exploits a model predictive control (MPC) strategy that minimizes the mismatch between the solutions of the sampled-data model and the continuous-time closed-loop model. We propose a control law and present conditions under which stability and sub-optimality of the closed loop can be proved. We only consider the case of unconstrained MPC. We show that the recent results in [G. Grimm, M.J. Messina, A.R. Teel, S. Tuna, Model predictive control: for want of a local control Lyapunov function, all is not lost, IEEE Trans. Automat. Control 2004, to appear] can be directly used for analysis of stability of our closed-loop system.     

Optimal fixed‐structure control for linear non‐negative dynamical systems

In this paper, we develop optimal output feedback controllers for set‐point regulation of linear non‐negative dynamical systems. Specifically, using a constrained fixed‐structure control framework we develop optimal output feedback control laws that guarantee that the trajectories of the closed‐loop system remain in the non‐negative orthant of the state space for non‐negative initial conditions. In addition, we characterize domains of attraction predicated on closed and open Lyapunov level surfaces contained in the non‐negative orthant for unconstrained optimal linear‐quadratic output feedback controllers. Output feedback controllers for compartmental systems with non‐negative inputs are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

Decentralized control and communication

In this paper, the past and current issues involved in the design of decentralized networked control systems are reviewed. The basic models of interconnected systems described as continuous-time linear time-invariant systems in the time domain serve as a framework for the inclusion of communication channels in the decentralized feedback loop. The I/O-oriented models and the interaction oriented models with disjoint subsystems and interactions are distinguished. The overview is focused on packet dropouts, transmission delays, and quantization effects which are included in the time-driven design of feedback loop components. Single- and multiple-packet transmissions are considered in this contents. The design of decentralized state feedback gain matrices with delayed feedback uses the methodology of sampled-data feedback design for continuous-time systems, while the decentralized H_∞ quantizer design is based on the static output controller. The Liapunov stability approach results in computationally efficient decentralized control design strategies described by using linear matrix inequalities.     

Asymptotic stabilisation of a class of nonlinear systems via sampled-data output feedback control

This article considers sampled-data output feedback control of a class of nonlinear systems in output feedback form. The underlying continuous-time controller is designed based on backstepping technique, employing a linear dynamic filter, and globally asymptotically stabilises the system. Rigorous analysis shows that when implemented with a sampling and zero-order hold device, the sampled-data version of the continuous-time controller semi-globally asymptotically stabilises the original system, given that the sampling period is smaller than a specific value, which depends on the initial values and nonlinearity of the system. Simulation results of a physical system are included in the end.     

Stability and stabilization of linear sampled-data systems with multi-rate samplers and time driven zero order holds

In multi-rate sampled-data systems, a continuous-time plant is controlled by a discrete-time controller which is located in the feedback loop between sensors with different sampling rates and actuators with different refresh rates. The main contribution of this paper is to propose sufficient Krasovskii-based stability and stabilization criteria for linear sampled-data systems, with multi-rate samplers and time driven zero order holds. For stability analysis, it is assumed that an exponentially stabilizing controller is already designed in continuous-time and is implemented as a discrete-time controller. For each sensor (or actuator), the problem of finding an upper bound on the lowest sampling frequency (or refresh rate) that guarantees exponential stability is cast as an optimization problem in terms of linear matrix inequalities (LMIs). Furthermore, sufficient conditions for controller synthesis are formulated as LMIs. It is shown through examples that choosing the right sensors (or actuators) with adequate sampling frequencies (or refresh rates) has a considerable impact on stability of the closed-loop system.     

Sufficient stability condition for a sampled-data system withdigital controller

A quantitative stability condition for a sampled-data feedback system is formulated. The system is obtained from a stable continuous-time feedback system by digitalization of the controller. The condition shows how to determine a sampling frequency f₀ such that for all f larger than f₀ the discrete-time system remains stable     

Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling

In practice, the system is often modeled as a continuous-time fuzzy system, while the control input is applied only at discrete instants. This system is called a sampled-data control system. In this paper, robust guaranteed cost control for uncertain sampled-data fuzzy systems is discussed. A guaranteed cost control where a quadratic cost function is bounded by a certain scalar, not only stabilizes a system but also considers a control performance. A typical sampled-data control is the zero-order input, which can be represented as a piecewise-continuous delay. Here we take a delay system approach to the sampled-data guaranteed cost control problem. The closed-loop system with a sampled-data state feedback controller becomes a system with time-varying delay. First, guaranteed cost control performance conditions for the closed-loop system are given in terms of linear matrix inequalities (LMIs). Such conditions are derived by using Leibniz–Newton formula and free weighting matrix method for fuzzy systems under the assumption that sampling time is not greater than some prescribed scalar. Then, a design method of robust guaranteed cost state feedback controller for uncertain sampled-data fuzzy systems is proposed. Examples are given to illustrate our robust sampled-data guaranteed cost control design.     

Performance recovery under output feedback sampled-data stabilization of a class of nonlinear systems

This paper studies sampled-data output feedback control of a class of nonlinear systems. It is shown that the performance of a stabilizing continuous-time state feedback controller can be recovered by a sampled-data output feedback controller when the sampling period is sufficiently small. The output feedback controller uses a deadbeat discrete-time observer to estimate the unmeasured states. Two schemes are proposed to overcome large initial transients when the controller is switched on.     

Spectral necessary and sufficient conditions for sampling-period-independent stabilisation of periodic and aperiodic sampled-data systems using a class of generalised sampled-data hold functions

Sampling-period-independent (SPI) stabilisation of both periodic and aperiodic sampled-data systems using a class of generalised sampled-data hold functions is addressed. It is proved that for this specific class of hold functions, a continuous-time linear system with rank-minimal input matrix and with non-defective eigenvalues on the imaginary axis is SPI-stabilisable if and only if the spectrum of the system is contained in the closed left-half plane. A systematic procedure for constructing suitable state-feedback controllers is also provided. Several examples are finally discussed for illustration.     

Indirect sampled-data control with sampling period adaptation

It is known that if a continuous-time feedback system is exponentially stable, then the corresponding sampled-data system obtained by sample-hold discretisation with constant sampling period is also exponentially stable, provided that the sampling period τ?>?0 is sufficiently small. In general, it is difficult to estimate how small the sampling period has to be in order to achieve the stability of the sampled-data system. In this article, we present an adaptive mechanism for adjusting the sampling period. This mechanism has the properties that, for every initial state, (i) the adaptation of the sampling period terminates after finitely many time steps and (ii) the state of the adaptive sampled-data system is integrable and converges to zero as time goes to infinity.     

Simple stability criteria for systems with time-varying delays

This paper considers the problem of checking stability of linear feedback systems with time-varying but bounded delays. Simple but powerful criteria of stability are presented for both continuous-time and discrete-time systems. Using these criteria, stability can be checked in a closed loop Bode plot. This makes it easy to design the system for robustness.