Multiple Quadratic Lyapunov Functions Research Articles

Virtual-clock-dependent [formula omitted] controller design for discrete-time switched interval type-2 fuzzy systems with intermittent control inputs

This paper is devoted to the study of intermittent H∞ control problem for discrete-time switched interval type-2 fuzzy systems (DSIT2FSs) under minimum dwell time switching. A novel virtual-clock-dependent H∞ intermittent interval type-2 (IT2) fuzzy controller design scheme is proposed. Firstly, we transform the N-mode DSIT2FS with intermittent control inputs into a new one with N controller-busy modes and N controller-free modes. Then, by constructing a class of non-increasing virtual-clock-dependent multiple quadratic Lyapunov functions (VMQLFs), sufficient conditions on the existence of the H∞ intermittent IT2 fuzzy controllers are obtained for DSIT2FSs with dwell time constraints. The controller gains can be gotten by solving a set of strict linear matrix inequalities (LMIs). The obtained results are also applicable to non-switched IT2 fuzzy systems. It is shown that the designed IT2 fuzzy controllers work intermittently but still can guarantee stability and H∞ performance of the close-loop systems. Finally, a tunnel diode circuit system model and a single link robot arm model are provided to illustrate the effectiveness and advantages of the results.

Finite-time Annular Domain Stability and Stabilization of Itô-type Stochastic Time-Varying Systems with Wiener and Poisson noises

This paper investigates finite time annular domain (FTAD) stability and stabilization for Itô-type stochastic time-varying systems with continuous Wiener and discontinuous Poisson noises (STVSWPNs). First, using Itô-Levy formula and time-varying multiple quadratic Lyapunov functions, two less conservative FTAD-stability conditions based generalised differential Lyapunov equations (GDLEs) and differential linear matrix inequalities (DLMIs) are obtained. Second, the FTAD-stabilization is studied and some new sufficient conditions for the existence of state feedback and static output feedback controllers are presented by tractable differential linear matrix inequalities. Moreover, a new numerical algorithm is given. Finally, a numerical example and a real-world example are utilized to show the effectiveness of the proposed methods.

Unknown input interval observers for discrete-time linear switched systems

This paper deals with unknown input interval observer synthesis for discrete-time switched systems. First, a decomposition leading to obtain a subsystem not affected by the unknown input is presented. Second, an interval observer is designed based on the Input-to-State Stability (ISS). The gains are computed by solving Linear Matrix Inequalities (LMI) formulated based on multiple quadratic Lyapunov functions under average dwell time switching signals. In addition, a change of coordinates can be taken in order to ensure the positivity of the estimation errors. Finally, an explicit expression for the unknown input bounds is derived. Note that while the additive disturbances and the measurement noises are unknown but assumed to be bounded with known bounds, the unknown input signals are neither bounded nor stochastic.

Event-triggered [formula omitted] control of networked switched systems subject to denial-of-service attacks

This paper investigates the event-triggered H∞ control of networked switched systems in the presence of Denial-of-Service (DoS) attacks. DoS attacks prevent the sampled information of system state and switching signal from transmitting among the network. The transmission delay on the switching of controller leads to asynchronous switching. By a suitable event-triggered scheme, the bound of the actual DoS-induced transmission delay is obtained, which can be used to construct a merged switching signal combining the switching signal of system with DoS-induced switching signal of controller. The original networked switched systems under DoS attacks can be modeled as new switched systems with this merged switching signal. By constructing multiple quadratic Lyapunov functions with the merged switching signal, some sufficient conditions are provided for event-triggered H∞ controller. Finally, in order to demonstrate the effectiveness and feasibility of the proposed methods, a numerical simulation is performed.

Sensor fault detection for switched systems using interval observer with L∞ performance

This paper deals with sensor fault detection (FD) for discrete-time switched systems subject to bounded disturbances. A novel approach is investigated to construct residual framers using interval observer with L∞ performance. The proposed technique is used to provide more degrees of design freedom and to obtain accurate FD results. The design conditions of the FD observer are given in terms of Linear Matrix Inequalities (LMIs) adopting firstly a common quadratic Lyapunov function, under an arbitrary switching signal and secondly multiple quadratic Lyapunov functions, under an Average Dwell Time (ADT) switching signal. Furthermore, the FD decision is based on residual intervals generated by the proposed interval observer. The effectiveness of the proposed approach is highlighted through simulation results of an academic example.

Stabilization of constrained switched systems via multiple Lyapunov R-functions

This paper investigates the stabilization problem of switched linear systems subject to state constraints, input saturation and external disturbance. Within the set-theoretic framework of R-function, the multiple Lyapunov R-functions (MLRFs) are presented, and merge both the multiple polyhedral and the multiple quadratic Lyapunov functions. The advantages of the MLRFs are that the external level sets are designed in accordance to the state constraints and inner level sets arbitrary can be made as close as possible to the smooth one. Based on the MLRFs method, stabilization conditions for the constrained switched system under a gradient-based switching law are derived. The union of the invariant truncated ellipsoidal sets is provided as an alternative approximation to the domain of attraction of the constrained switched system. Each truncated ellipsoidal set is the intersection of a polyhedral set and an ellipsoidal set, which come from the polyhedral Lyapunov function and the quadratic Lyapunov function, respectively. The sub-optimality bounds are investigated by a policy iteration. Two examples are simulated to show the benefits of the proposed strategy.

Interval estimation for continuous-time LPV switched systems

This paper deals with interval state estimation for continuous-time linear parameter varying (LPV) switched systems with measured polytopic parameter dependence, assuming a known switching signal. Considering that the measurement noise and the state disturbance are unknown but bounded and that the dynamics of the system are described by a convex combination, lower and upper bounds of the state are therefore determined. An interval observer is designed in order to guarantee both stability and cooperativity of the upper and lower observation errors. For the proposed observer, sufficient input-to-state stability conditions are given in terms of linear matrix inequalities, adopting firstly common quadratic Lyapunov functions, under arbitrary switching signals and secondly multiple quadratic Lyapunov functions, under average dwell time (ADT) switching signals. The efficiency of the proposed interval observer is highlighted through simulation results on academic examples.

Asynchronous switching output feedback control of discrete-time switched linear systems

In this paper, the problem of dynamic output-feedback control synthesis is addressed for discrete-time switched linear systems under asynchronous switching. The proposed hybrid controller consists of a standard dynamic output-feedback switching control law and an impulsive reset law induced by controller state jumps. Using the average dwell time technique incorporating with multiple quadratic Lyapunov functions, the switching control synthesis conditions for asymptotic stability with guaranteed weighted ℓ2-gain performance are derived as a set of linear matrix inequalities (LMIs). The proposed hybrid synthesis scheme advances existing design methods for output-feedback asynchronous switching control of switched linear systems in two important aspects: LMI formulation of the synthesis problem; and arbitrary order of the controller state. A numerical example is used to illustrate the effectiveness and advantages of the proposed design technique.

A Switching Anti-windup Design Using Multiple Lyapunov Functions

This technical note proposes a switching anti-windup design, which aims to enlarge the domain of attraction of the closed-loop system. Multiple anti-windup gains along with an index function that orchestrates the switching among these anti-windup gains are designed based on the min function of multiple quadratic Lyapunov functions. In comparison with the design of a single anti-windup gain which maximizes a contractively invariant level set of a single quadratic Lyapunov function as a way to enlarge the domain of attraction, the use of multiple Lyapunov functions and switching in the proposed design allows the union of the level sets of the multiple Lyapunov functions, each of which is not necessarily contractively invariant, to be contractively invariant and within the domain of attraction. As a result, the resulting domain of attraction is expected to be significantly larger than the one resulting from a single anti-windup gain and a single Lyapunov function. Indeed, simulation results demonstrate such a significant improvement.

A New Fuzzy Lyapunov Approach to Non-Quadratic Stabilization of Takagi-Sugeno Fuzzy Models

A New Fuzzy Lyapunov Approach to Non-Quadratic Stabilization of Takagi-Sugeno Fuzzy ModelsIn this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.

SWITCHED STATE JUMP OBSERVERS FOR A CLASS OF NONLINEAR SWITCHED SYSTEMS

This paper addresses the problem of switched state ‘jump’ observer synthesis for a class of nonlinear switched systems for which the active mode is assumed to be unknown. The use of multiple quadratic Lyapunov functions and of the differential mean value theorem allows to transform the nonlinear error dynamics into a Linear Parameter Varying (LPV) system. Then we develop sufficient conditions for the observer synthesis that guarantee an upper bound on the estimation error. For a fixed value of a scalar parameter, the synthesis problem is brought to one of solving linear matrix inequalities (LMIs) that are easily tractable. A numerical example that demonstrates the applicability of our approach is also presented.

A new fuzzy Lyapunov function approach for a Takagi–Sugeno fuzzy control system design

In this paper, a new fuzzy Lyapunov function approach is presented for a class of continuous-time Takagi–Sugeno fuzzy control system. The proposed fuzzy Lyapunov function is formulated as a line-integral of a fuzzy vector which is a function of the state, and it can be regarded as the work done from the origin to the current state in the fuzzy vector field. Unlike the approaches using a fuzzy blending of multiple quadratic Lyapunov functions, the time-derivatives of membership functions do not appear in the proposed approach. The effectiveness of the proposed approach is shown through numerical examples.