Robust H_∞ control for discrete-time polytopic uncertain systems with linear fractional vertices

The robust H∞ control problem for discrete-time uncertain systems is investigated in this paper. The uncertain systems are modelled as a polytopic type with linear fractional uncertainty in the vertices. A new linear matrix inequality (LMI) characterization of the H∞ performance for discrete systems is given by introducing a matrix slack variable which decouples the matrix of a Lyapunov function candidate and the parametric matrices of the system. This feature enables one to derive sufficient conditions for discrete uncertain systems by using parameter-dependent Lyapunov functions with less conservativeness. Based on the result, H∞ performance analysis and controller design are carried out. A numerical example is included to demonstrate the effectiveness of the proposed results.     

H-infinity control for switched and impulsive singular systems

A new model of dynamical systems is proposed which consists of singular systems with impulsive effects, i.e., switched and impulsive singular systems （SISS）. By using the switched Lyapunov functions method, a sufficient condition for the solvability of the H-infinity control problem for SISSs is given which generalizes the H-infinity control theory for singular systems to switched singular systems with impulsive effects. Then the sufficient condition of solvablity of the H-infinity control problem is presented in terms of linear matrix inequalities. Finally, the effectiveness of the developed aooroach for switched and imoulsive singular svstems is illustrated by a numerical example.     

Output Feedback Tracking Control for a Class of Switched Nonlinear Systems with Time-varying Delay

This paper studies the problem of tracking control for a class of switched nonlinear systems with time-varying delay. Based on the average dwell-time and piecewise Lyapunov functional methods, a new exponential stability criterion is obtained for the switched nonlinear systems. The designed output feedback H∞controller can be obtained by solving a set of linear matrix inequalities(LMIs).Moreover, the proposed method does not need that a common Lyapunov function exists for the switched systems, and the switching signal just depends on time. A simulation example is provided to demonstrate the effectiveness of the proposed design scheme.     

Control synthesis for a class of nonlinear systems based on partition of unity

A partition-of-unity-based approach is proposed to derive an approximate model for a class of nonlinear systems. The precision of the approximate model is analyzed by using the modulus of continuity of continuous functions. The system stability of the approximate model is analyzed by using Lyapunov stability theory. A design algorithm for constructing tracking controllers with tracking performance related to tracking error is given based on the approximate model and the partition of unity method.     

A computing capability test for a switched system control design using the Haris-Rogers method

The problem of finding stabilizing controllers for switched systems is an area of much research interest as conventional concepts from continuous time and discrete event dynamics do not hold true for these systems.Many solutions have been proposed,most of which are based on finding the existence of a common Lyapunov function(CLF) or a multiple Lyapunov function(MLF) where the key is to formulate the problem into a set of linear matrix inequalities(LMIs).An alternative method for finding the existence of a CLF by solving two sets of linear inequalities(LIs) has previously been presented.This method is seen to be less computationally taxing compared to methods based on solving LMIs.To substantiate this,the computational ability of three numerical computational solvers,LMI solver,cvx,and Yalmip,as well as the symbolic computational program Maple were tested.A specific switched system comprising four second-order subsystems was used as a test case.From the obtained solutions,the validity of the controllers and the corresponding CLF was verified.It was found that all tested solvers were able to correctly solve the LIs.The issue of rounding-off error in numerical computation based software is discussed in detail.The test revealed that the guarantee of stability became uncertain when the rounding off was at a different decimal precision.The use of different external solvers led to the same conclusion in terms of the stability of switched systems.As a result,a shift from using a conventional numerical computation based program to using computer algebra is suggested.     

Robust reliable H-infinity control for nonlinear uncertain stochastic time-delay systems with Markovian jumping parameters

This paper deals with the problems of robust reliable exponential stabilization and robust stochastic stabilization with H-infinity performance for a class of nonlinear uncertain time-delay stochastic systems with Markovian jumping parameters. The time delays are assumed to be dependent on the system modes. Delay-dependent conditions for the solvability of these problems are obtained via parameter-dependent Lyapunov functionals. Furthermore, it is shown that the desired state feedback controller can be designed by solving a set of linear matrix inequalities. Finally, the simulation is provided to demonstrate the effectiveness of the proposed methods.     

Stability analysis for discrete linear systems with state saturation by a saturation-dependent Lyapunov functional

This paper concerns the stability analysis problem of discrete linear systems with state saturation using a saturation-dependent Lyapunov functional.We introduce a free matrix characterized by the sum of the absolute value of each elements for each row less than 1,which makes the state with saturation constraint reside in a convex polyhedron.A saturation-dependent Lyapunov functional is then designed to obtain a sufficient condition for such systems to be globally asymptotically stable.Based on this stability criterion,the state feedback control law synthesis problem is also studied.The obtained results are formulated in terms of bilinear matrix inequalities that can be solved by the presented iterative linear matrix inequality algorithm.Two numerical examples are used to demonstrate the effectiveness of the proposed method.     

一类非线性系统控制Lyapunov函数的构造

The construction of control Lyapunov functions for a class of nonlinear systems is considered. We develop a method by which a control Lyapunov function for the feedback linearizable part can be constructed systematically via Lyapunov equation. Moreover, by a control Lyapunov function of the feedback linearizable part and a Lyapunov function of the zero dynamics, a control Lyapunov function for the overall nonlinear system is established.     

Robust control for a class of uncertain switched fuzzy systems

A model of uncertain switched fuzzy systems whose subsystems are uncertain fuzzy systems is presented. Robust controllers for a class of switched fuzzy systems are designed by using the Lyapunov function method. Stability conditions for global asymptotic stability are developed and a switching strategy is proposed. An example shows the effectiveness of the method.     

Practical stabilization of a class of uncertain time-varying nonlinear delay systems

In this paper we deal with a class of uncertain time-varying nonlinear systems with a state delay. Under some assumptions, we construct some stabilizing continuous feedback, i.e. linear and nonlinear in the state, which can guarantee global uniform exponential stability and global uniform practical convergence of the considered system. The quadratic Lyapunov function for the nominal stable system is used as a Lyapunov candidate function for the global system. The results developed in this note are applicable to a class of dynamical systems with uncertain time-delay. Our result is illustrated by a numerical example.     

控制李雅普诺夫承数的镇定应用

指出控制李雅普诺夫承数(源于松弛控制)正是零状态可检测性的一种刻面,并由此区别 两类镇定控制;无源性控制和Sontag型控制.然后考察一类典型的级联系统,基于一定条件下的控 制李雅普诺夫函数,给出了不同于无源性控制的Sontag型镇定设计;考察这类系统的扰动情形,基 于一定条件下的输入到状态稳定控制李雅普诺夫承数,给出了Sontag型输入到状态镇定设计.     

线性多变量系统的控制Lyapunov函数构造

提出线性多变量系统控制Lyapunov函数(CLF)构造的一般方法. 先证明可以通过解一类Lyapunov方程, 得到线性系统二次型的CLF. 接着证明了对于线性系统, 这种方法可以提供所有二次型的CLF. 最后证明了若线性系统存在CLF, 那么必存在二次型的CLF. 由此完全解决了线性系统的CLF构造问题.     

Constructive stabilization for quadratic input nonlinear systems

In this paper stabilization of nonlinear systems with quadratic multi-input is considered. With the help of control Lyapunov function (CLF), a constructive parameterization of controls that globally asymptotically stabilize the system is proposed. Two different cases are considered. Firstly, under certain regularity assumptions, the feasible control set is parameterized, and continuous feedback stabilizing controls are designed. Then for the general case, piecewise continuous stabilizing controls are proposed. The design procedure can also be used to verify whether a candidate CLF is indeed a CLF. Several illustrative examples are presented as well.     

CLF based designs with robustness to dynamic input uncertainties

The problem of robust stabilization of nonlinear systems in the presence of input uncertainties is of great importance in practical implementation. Stabilizing control laws may not be robust to this type of uncertainty, especially if cancellation of nonlinearities is used in the design. By exploiting a connection between robustness and optimality, “domination redesign” of the control Lyapunov function (CLF) based Sontag's formula has been shown to possess robustness to static and dynamic input uncertainties. In this paper we provide a sufficient condition for the domination redesign to apply. This condition relies on properties of local homogeneous approximations of the system and of the CLF. We show that an inverse optimal control law may not exist when these conditions are violated and illustrate how these conditions may guide the choice of a CLF which is suitable for domination redesign.     

A Survey on the Control Lyapunov Function and Control Barrier Function for Nonlinear-Affine Control Systems

This survey provides a brief overview on the control Lyapunov function (CLF) and control barrier function (CBF) for general nonlinear-affine control systems. The problem of control is formulated as an optimization problem where the optimal control policy is derived by solving a constrained quadratic programming (QP) problem. The CLF and CBF respectively characterize the stability objective and the safety objective for the nonlinear control systems. These objectives imply important properties including controllability, convergence, and robustness of control problems. Under this framework, optimal control corresponds to the minimal solution to a constrained QP problem. When uncertainties are explicitly considered, the setting of the CLF and CBF is proposed to study the input-to-state stability and input-to-state safety and to analyze the effect of disturbances. The recent theoretic progress and novel applications of CLF and CBF are systematically reviewed and discussed in this paper. Finally, we provide research directions that are significant for the advance of knowledge in this area.      

Robust switching adaptive control of multi-input nonlinear systems

During the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters. New design tools such as adaptive feedback linearization, adaptive back-stepping, control Lyapunov functions (CLFs) and robust control Lyapunov functions (RCLFs), nonlinear damping and switching adaptive control have been introduced. Most of the results developed are applicable to single-input feedback-linearizable systems and parametric-strict-feedback systems. These results, however, cannot be applied to multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. In this paper, we develop a general procedure for designing robust adaptive controllers for a large class of multi-input nonlinear systems. This class of nonlinear systems includes as a special case multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. The proposed approach uses tools from the theory of RCLF and the switching adaptive controllers proposed by the authors for overcoming the problem of computing the feedback control law when the estimation model becomes uncontrollable. The proposed control approach has also been shown to be robust with respect to exogenous bounded input disturbances     

Prescribed performance adaptive neural output control for a class of switched nonstrict‐feedback nonlinear time‐delay systems: State‐dependent switching law approach

This paper investigates the tracking problem for a class of uncertain switched nonlinear delayed systems with nonstrict‐feedback form. To address this problem, by introducing a new common Lyapunov function (CLF), an adaptive neural network dynamic surface control is proposed. The state‐dependent switching rule is designed to orchestrate which subsystem is active at each time instance. In order to compensate unknown delay terms, an appropriate Lyapunov‐Krasovskii functional is considered in the constructing of the CLF. In addition, a novel switched neural network–based observer is constructed to estimate system states through the output signal. To maintain the tracking error performance within a predefined bound, a prescribed performance bound approach is employed. It is proved that by the proposed output‐feedback control, all the signals of the closed‐loop system are bounded under the switching law. Moreover, the transient and steady‐state tracking performance is guaranteed by the prescribed performance bound. Finally, the effectiveness of the proposed method is illustrated by two numerical and practical examples.     

Multilayer Minimum Projection Method with Singular Point Assignment for Nonsmooth Control Lyapunov Function Design

Control Lyapunov function (CLF) design on a manifold is a difficult problem in control theory. To address this problem, we have proposed the multilayer minimum projection method. The method requires CLFs on different manifolds from the manifold where the control problem is defined. In this paper, we relax the requirement by desingularization of the functions on the manifolds. The paper focuses on the problem of desingularization in the multilayer minimum projection method. We show that the functions on other manifolds need not be CLFs by consideration of desingularization. Moreover, we propose a CLF design method by singular point assignment based on the advantage of desingularization. The method enables us to merge local CLFs into the global CLF. This paper proposes two CLF design methods: desingularization and singular point assignment. A CLF design example is provided for each method; the advantages of the proposed methods are confirmed by those two examples.     

Stabilization of nonlinear diffusion Ito processes with Markov switchings

Consideration is given to a class of systems that are described by the finite set of affine-controlled diffusion Ito processes with jump-like passages between them, definable by the evolution of the uniform Markov chain (Markov switchings). The stochastic version of the notion of the control Lyapunov function (CLF) is introduced. On the assumption of existence of the CLF, the explicit expression is found for the smooth stabilizing scalar control with the state feedback. In the case when a system is under the action of only noises depending on the state, the vector stabilizing control with the state feedback is found under the same assumptions. In the latter case, formulas for the stabilizing control represent the extension of the “universal” Sontag formula for the stabilizing control of a determinate system.