Digital Stabilizer Design for a Switched Linear Control Delay System

The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.     

Stabilization analysis for discrete-time systems with time delay

The stabilization problem for a class of discrete-time systems with time-varying delay is investigated. By constructing an augmented Lyapunov function, some sufficient conditions guaranteeing exponential stabilization are established in forms of linear matrix inequality (LMI) technique. When norm-bounded parameter uncertainties appear in the delayed discrete-time system, a delay-dependent robust exponential stabilization criterion is also presented. All of the criteria obtained in this paper are strict linear matrix inequality conditions, which make the controller gain matrix can be solved directly. Three numerical examples are provided to demonstrate the effectiveness and improvement of the derived results.     

A new discrete mechanics approach to swing-up control of the cart-pendulum system

This paper develops a new swing-up control method for the cart-pendulum system via discrete mechanics. The swing-up control law consists of two parts: the swing-up stage and the stabilization one. In the swing-up stage, we use a controller based on a discrete Lyapunov function and it can swing up the pendulum. Then, in the stabilization stage, we utilize a stabilizing controller based on the linearized system and discrete-time optimal regulator theory. In addition, transformation methods from discrete control inputs into continuous zero-order hold inputs are introduced. From some simulation results, we can confirm that the cart-pendulum system is swung up and stabilized by our new method.     

T–S fuzzy-model-based robust stabilization for a class of nonlinear discrete-time networked control systems

In this paper, the robust stabilization problem is investigated for a class of nonlinear discrete-time networked control systems (NCSs). To study the system stability and facilitate the design of fuzzy controller, Takagi–Sugeno (T–S) fuzzy models are employed to represent the system dynamics of the nonlinear discrete-time NCSs with effects of the approximation errors taken into account, and a unified model of NCSs in the T–S fuzzy model is proposed by modeling the approximation errors as norm-bounded uncertainties in system metrics, where non-ideal network Quality of Services (QoS), such as data dropout and network-induced delay, are coupled in a unified framework. Then, based on the Lyapunov–Krasovskii functional, sufficient conditions are derived for the existence of a fuzzy controller. By these criteria, two approaches to design a fuzzy controller are developed in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are provided to show the effectiveness of the proposed methods.     

Stabilization and tracking controller for a class of nonlinear discrete-time systems

In this paper, stabilization and tracking control problem for parametric strict feedback class of discrete time systems is addressed. Recursive design of control function based on contraction theory framework is proposed instead of traditional Lyapunov based method. Explicit structure of controller is derived for the addressed class of nonlinear discrete-time systems. Conditions for exponential stability of system states are derived in terms of controller parameters. At each stage of recursive procedure a specific structure of Jacobian matrix is ensured so as to satisfy conditions of stability. The closed loop dynamics in this case remains nonlinear in nature. The proposed algorithm establishes global stability results in quite a simple manner as it does not require formulation of error dynamics. Problem of stabilization and output tracking control in case of single link manipulator system with actuator dynamics is analyzed using the proposed strategy. The proposed results are further extended to stabilization of discrete time chaotic systems. Numerical simulations presented in the end show the effectiveness of the proposed approach.     

Robust stabilization of discrete-time nonlinear Lur’e systems with sector and slope restricted nonlinearities

In this paper, we present a novel method for the robust control problem of uncertain nonlinear discrete-time linear systems with sector and slope restrictions. The nonlinear function considered in this paper is expressed as convex combinations of sector and slope bounds. Then the equality constraint is derived by using convex properties of the nonlinear function. A stabilization criterion for the existence of the state feedback controller is derived in terms of linear matrix inequalities (LMIs) by using Finsler’s lemma. The proposed method is demonstrated by a system with saturation nonlinearity.     

Quantized stabilization of discrete-time systems in a networked environment

This paper is concerned with the problem of output feedback stabilization for a class of discrete-time systems with sector nonlinearities and imperfect measurements. A unified control law model is proposed to take the network-induced delay, random packet dropout and measurement quantization into consideration simultaneously. By choosing appropriate Lyapunov functional, a new stability condition, which is dependent on multiple network status, is established for the resulting closed-loop system. Based on the result, a design criterion for the static output feedback controller is formulated in the form of nonconvex matrix inequalities, and the cone complementary linearization (CCL) procedure is exploited to solve the nonconvex feasibility problem. Incidentally, a less conservative synthesis method is also developed for the state feedback stabilization purpose. Finally, two illustrative examples are provided to illustrate the effectiveness and applicability of the proposed design method.     

具有量化误差的多步长非线性采样系统的镇定

该文研究一类非线性控制系统在采样器采样过程中产生量化误差的情况下多步长采样镇定问题. 运用近似DTD方法, 在非线性系统的近似离散时间模型上设计全局状态反馈镇定控制器. 当系统近似误差和采样量化误差被限制在一定的条件下, 可以得到含量化误差的多步长非线性采样系统是半全局实用渐近稳定. 最后, 仿真例子验证了所得结果的有效性.     

模糊Delta算子系统的鲁棒镇定

研究一类基于Delta算子描述的T-S模糊模型状态反馈镇定设计问题。首先将全局模糊模型按隶属函数划分成若干子空间,并被表示成不确定系统的形式；采用分段Lyapunov函数法，得到鲁棒稳定化控制律存在的充分条件．该条件被进一步等价表示成一组线性矩阵不等式的可解性问题。克服了以往设计法中需要求解一公共正定矩阵P的不足，也无需求解繁琐的Riccati方程。所得结果可将连续和离散模糊系统的有关结论统一到Delta算子框架内。     

Quantized stabilization of stochastic systems with multiplicative noise under Markovian switching

A problem of quantized state feedback quadratic mean-square stabilization of discrete-time stochastic processes under Markovian switching and multiplicative noise is considered. A static quantizer is used in the feedback channel and the jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. It is shown that the coarsest quantization density that permits quadratic mean-square stabilization of this system is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special linear stochastic Markovian switching control system. Also, sufficient conditions for exponential mean-square stabilization of such systems are also explored. An example is given to demonstrate the obtained results.     

Dynamic output feedback fault tolerant controller design for discrete-time switched systems with actuator fault

This paper investigates the problem of dynamic output feedback fault tolerant controller design for discrete-time switched systems with actuator fault. By using reduced-order observer method and switched Lyapunov function technique, a fault estimation algorithm is achieved for the discrete-time switched system with actuator fault. Then based on the obtained online fault estimation information, a switched dynamic output feedback fault tolerant controller is employed to compensate for the effect of faults by stabilizing the closed-loop systems. Finally, an example is proposed to illustrate the obtained results.     

Multirate sampling and input-to-state stable receding horizon control for nonlinear systems

In this paper, a robust receding horizon control for multirate sampled-data nonlinear systems with bounded disturbances is presented. The proposed receding horizon control is based on the solution of Bolza-type optimal control problems for the approximate discrete-time model of the nominal system. “Low measurement rate” is assumed. It is shown that the multistep receding horizon controller that stabilizes the nominal approximate discrete-time model also practically input-to-state stabilizes the exact discrete-time system with disturbances.     

Stabilization of planar discrete-time switched systems: Switched Lyapunov functional approach

In this paper, we investigate the problem of stabilization for single-input planar discrete-time switched systems by establishing necessary and/or sufficient conditions for the existence of switched quadratic Lyapunov functions of the closed-loop system. The results given in terms of a series of matrix inequalities generalize those results in our recent paper [Y.G. Sun, L. Wang, G. Xie, Necessary and sufficient conditions for stabilization of discrete-time planar switched systems, Nonlinear Anal.: Theory and Methods 65 (2006) 1039–1049] and clearly describe the set of switched quadratic Lyapunov functions for the system.     

Necessary and sufficient conditions for robust global asymptotic stabilization of discrete-time systems

We give Lyapunov-like conditions for non-uniform in time output stabilization of discrete-time systems. Particularly, it is proved that for a discrete-time control system there exists a (continuous) output stabilizing feedback if and only if there exists a (strong) output control Lyapunov function (OCLF). Moreover, strategies for the construction of continuous robust feedback stabilizers are presented.     

Dynamic feedback control and exponential stabilization of a compound system

We studied the exponential stabilization problem of a compounded system composed of a flow equation and an Euler–Bernoulli beam, which is equivalent to a cantilever Euler–Bernoulli beam with a delay controller. We designed a dynamic feedback controller that stabilizes exponentially the system provided that the eigenvalues of the free system are not the zeros of controller. In this paper we described the design detail of the dynamic feedback controller and proved its stabilization property.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Event-triggered feedback stabilization of switched linear systems using dynamic quantized input

This paper is concerned with the co-design of event-triggered sampling, dynamic input quantization and constrained switching for a switched linear system. The mismatch between the plant and its corresponding controller is considered. This behavior is raised by switching within the event-triggered sampling interval. Accordingly, novel update laws of dynamic quantization parameter are designed separately for matched sampling intervals (without switching) and mismatched sampling intervals (with a switch). We technically transform the total variation (increment or decrement) of Lyapunov functions in one sampling interval into the discrete-time update of quantization parameter. Based on this transformation, a hybrid quantized control policy is developed. This policy, in conjunction with the average dwell-time switching law and the constructed event-triggered condition, can ensure the exponential stabilization of the switched system with finite-level quantized input. Besides, the event-triggered scheme is proved to be Zeno-free. The effectiveness of the developed method is verified by a simulation example.     

Stability and stabilization of switched linear discrete-time systems with interval time-varying delay

This paper deals with stability and stabilization of a class of switched discrete-time delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the asymptotic stability and stabilization for the system is designed via linear matrix inequalities. Numerical examples are included to illustrate the effectiveness of the results.