Common Lyapunov Function Approach Research Articles

Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks

We consider adaptive compensation for infinite number of actuator failures in the tracking control of uncertain nonlinear systems. We construct an adaptive controller by combining the common Lyapunov function approach and the structural characteristic of neural networks. The proposed control strategy is feasible under the presupposition that the systems have a nonstrict-feedback structure. We prove that the states of the closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin under the designed controllers, even though there are an infinite number of actuator failures. At last, the validity of the proposed control scheme is demonstrated by two examples.

Output Reachable Set Estimation for Switched Linear Systems and Its Application in Safety Verification

This paper addresses the output reachable set estimation problem for continuous-time switched linear systems consisting of Hurwtiz stable subsystems. Based on a common Lyapunov function approach, the output reachable set is estimated by a union of bounding ellipsoids. Then, multiple Lyapunov functions with time-scheduled structure are employed to estimate the output reachable set for switched systems under dwell-time constraint. Furthermore, the safety verification problem of uncertain switched systems is investigated based on the result of output reachable set estimation. First, a sufficient condition ensuring the existence of an approximate bisimulation relation between two switched linear systems with a prescribed precision is proposed. Then, the safety verification for an uncertain switched system can be performed through an alternative safety verification for a switched system with exact parameters. Numerical examples are provided to illustrate our results.

Adaptive finite-time tracking control of switched nonlinear systems

The finite-time tracking issue of switched nonlinear systems is taken into consideration in this paper. By combining the common Lyapunov function approach and the structural characteristic of neural networks, a simplified adaptive controller is constructed. Under arbitrary switching, the constructed common controller ensures that the system output is located in a small neighborhood of the reference signal in finite time. Different from the existing literature on the finite-time stabilizing issue, the linear growth conditions for the nonlinear functions of the controlled systems can be removed; besides, the tracking problem is considered, which covers the finite-time stabilization issue as a special case.

Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model

This paper deals with the problem of switching design for guaranteed cost control of discrete-time two-dimensional (2-D) nonlinear switched systems described by the Roesser model. The switching signal, which determines the active mode of the system, is subject to a state-dependent process whose values belong to a finite index set. By using 2-D common Lyapunov function approach, a sufficient condition expressed in terms of tractable matrix inequalities is first established to design a min-projection switching rule that makes the 2-D switched system asymptotically stable. The obtained result on stability analysis is then utilized to synthesize a suboptimal state feedback controller that minimizes the upper bound of a given infinite-horizon cost function. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.

Robust Stabilization for Discrete Uncertain Takagi–Sugeno Fuzzy Systems Based on a Piecewise Lyapunov Function

This article investigates the robust H∞ fuzzy output-feedback control problem for a discrete Takagi–Sugeno (T–S) fuzzy system with uncertainties. By constructing a discrete piecewise Lyapunov function in each maximal overlapped-rule group, the systematic design of the fuzzy robust H∞ controller is studied through the parallel distributed compensation control scheme, and a new sufficient condition to check the robust asymptotic stabilization of the closed-loop discrete uncertain T–S fuzzy system is proposed and confirmed. This sufficient condition requires only finding common matrices in each maximal overlapped-rule group. Compared with the common Lyapunov function approach and the fuzzy Lyapunov function approach, the proposed sufficient condition can not only overcome the defect of finding common matrices in the whole feasible region but also significantly reduce the number of linear matrix inequalities to be solved. Therefore, the proposed condition is less conservative and difficult than the other two approaches. Finally, the simulation results for a class of two-echelon uncertain nonlinear supply chain systems show the effectiveness of the proposed approach.

New approaches to finite-time stability and stabilization for nonlinear system

This paper is concerned with the problems of finite-time stability (FTS) and finite-time stabilization for both continuous and discrete nonlinear systems, which can be represented by affine fuzzy system. Some new FTS conditions are provided and applied to the design problem of finite-time fuzzy regulators. The common Lyapunov function (CLF) approach is first used to analyze the FTS of affine fuzzy system. Then, a piecewise Lyapunov function (PLF), which is less conservative than CLF, is adopted to analyze the FTS of affine fuzzy systems and design corresponding finite-time fuzzy controller. In analysis, the FTS conditions are formulated in terms of linear matrix inequalities (LMIs). In synthesis, the conditions of finite-time stabilization turn out to be in the formulation of nonconvex matrix inequalities for discrete affine fuzzy system (DAFS) and bilinear matrix inequalities (BMIs) for continuous affine fuzzy system (CAFS). Thus, iterative LMI (ILMI) approach is applied to obtain the feasible solutions. Two examples are provided to illustrate the validity of the proposed results.

Absolute exponential stability and stabilization of switched nonlinear systems

This paper is concerned with the problems of absolute exponential stability and stabilization for a class of switched nonlinear systems whose system matrices are Metzler. Nonlinearity of the systems is constrained in a sector field, which is bounded by two odd symmetric piecewise linear functions. Multiple Lyapunov functions are introduced to deal with the stability of such nonlinear systems. Compared with some existing results obtained by the common Lyapunov function approach in the literature, the conservatism of our results is reduced. All present conditions can be solved by linear programming. Furthermore, the absolute exponential stabilization for the considered systems is designed by the state-feedback and average dwell time switching strategy. Two examples are also given to illustrate the validity of the theoretical findings.

Driving State Adaptive Control of an Active Vehicle Suspension System

A new adaptive vehicle suspension control method is presented that adjusts the controller parametrization to the current driving state and thereby enables to significantly enhance ride comfort while the dynamic wheel load and the suspension deflection remain within safety critical bounds. To this end, the adaptive controller structure dynamically interpolates between differently tuned linear quadratic regulators governed by the dynamic wheel load and the suspension deflection. The stability of the adaptive controller structure is analyzed by means of a common Lyapunov function approach taking into account the nonlinear damper characteristic of the suspension system. In order to provide a realistic framework for the controller design and the performance analysis, a quarter-car test rig based on an all-terrain vehicle suspension that has been equipped with an electrical linear motor to realize an active suspension system, is employed as testbed for the study. On this test rig, the significant performance of the adaptive control concept is successfully validated in a comparison to benchmark suspension controllers.

State feedback H∞ control for discrete 2D switched systems

This paper investigates the state feedback H∞ stabilization problem for a class of discrete-time 2D (two-dimensional) switched systems represented by a model of Roesser type. First, sufficient conditions for the asymptotical stability and H∞ disturbance attenuation performance of the underlying system are derived via common Lyapunov function approach and multiple Lyapunov function approach, respectively. Then, based on the obtained stability results, two state feedback controller design methods are proposed to guarantee that the resulting closed-loop system is asymptotically stable and achieves a prescribed disturbance attenuation level γ. Finally, two numerical examples are provided to verify the effectiveness of the proposed methods.

Robust Switching Rule Design for Boost Converters with Uncertain Parameters and Disturbances

This paper is concerned with the design problem of robust switching rule for Boost converters with uncertain parameters and disturbances. Firstly, the Boost converter is modeled as a switched affine linear system with uncertain parameters and disturbances. Then, using common Lyapunov function approach and linear matrix inequality (LMI) technique, a novel switching rule is proposed such that the model reference tracking performance is satisfied. Finally, a simulation result is provided to show the validity of the proposed method.

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.

Robust Systems with Discontinuous Parameters

Abstract In this paper we consider robust SISO systems with nonlinear controller. The plant dynamics are described by a switched system model. The control law is defined with help of the approach of adaptive control system design. Output derivatives are used to form control. Control systems are robust to discontinuous parameters disturbances. The system stability is studied by common Lyapunov function approach.

Robust H ∞ control for a class of switched non-linear systems with neutral uncertainties

This paper focuses on the problem of robust H∞ control for a class of switched non-linear systems with neutral uncertainties. The multiple Lyapunov function approach and the common Lyapunov function approach are exploited respectively for the cases whether states are measurable or not. Uncertainties are allowed to appear in channels of state, control input and disturbance input. Conditions for the solvability of the robust H∞ control problem and design of both hybrid state feedback and dynamic output feedback controllers are presented. As an application, a hybrid state feedback strategy is proposed to solve the standard robust H∞ control problem for non-linear systems when no single continuous controller is effective.

Decentralized Control of Swarms with Collision Avoidance Implications

The paper reviews some approaches to the decentralized control of a swarm of unmanned vehicles, and then proposes a new algorithm capable of managing collisions between vehicles and with obstacles. The swarm goal is to achieve a desired shape and position in space, formalized using an abstraction based approach. Formation statistics are defined in analogy with phisical bodies: center of mass position and inertia moments. Each agent elaborates its own estimate of these variables using a consensus algorithm capable of tracking a ramp reference, in order to reduce tracking errors. Gyroscopic and damping terms are added to the control law in order to avoid collision between vehicles and with obstacles. The obstacle avoidance terms appear in the control law only in the presence of obstacles or nearby vehicles; thus system dynamics change during the system evolution. This behavior was modeled as a hybrid system and proof of stability is given, under mild conditions, using the Common-Lyapunov function approach. The proposed methodology is validated through extensive numerical simulation.

STABILIZING DYNAMIC CONTROLLER OF SWITCHED LINEAR SYSTEMS

This paper is devoted to the problem of design a dynamic controller of switched linear systems. In the first part of this paper, we give some remarks about the influences of switching signal on the asymptotic stability of switched systems. We give a practical example to generate a large class of switching signals. The second part is devoted to the switched dynamic controller design, based on a common Lyapunov function approach. A sufficient condition are formulated as an LMI problem for the switched controller design under arbitrary switching. A stabilizing switched controller with regional pole placements is also formulated as a convex problem, an LMI approach is used to derive the switched dynamic controller with performance limitations.

Output-feedback control of systems over communication networks using a deterministic switching system approach*1

This paper proposes a new control problem of systems over asymmetric path-delay configurations on arbitrary communication networks, where the criterion is the H ∞ norm minimization via dynamic output-feedback controllers. Network-delay-dependent switching controllers are designed via a piecewise Lyapunov function approach as well as a common Lyapunov function approach. The controllers are found through convex optimization over linear matrix inequalities.

Output-feedback [formula omitted] control of systems over communication networks using a deterministic switching system approach

This paper proposes a new control problem of systems over asymmetric path-delay configurations on arbitrary communication networks, where the criterion is the H ∞ norm minimization via dynamic output-feedback controllers. Network-delay-dependent switching controllers are designed via a piecewise Lyapunov function approach as well as a common Lyapunov function approach. The controllers are found through convex optimization over linear matrix inequalities.

Common Lyapunov function approach to matrix root clustering

This paper proposes an alternative route to the matrix root clustering problem, which enables to use a result on the common Lyapunov function for solving the problem efficiently. A necessary and sufficient condition is obtained in terms of the existence of a common positive definite solution to a set of Lyapunov inequalities for eigenvalues of a matrix to lie in a prescribed subregion of the complex plane. Applications to root clustering in sector regions are shown for illustration.