Parameter-dependent Lyapunov Function Research Articles

H∞ control of discrete-time uncertain linear systems with quantized feedback

This paper studies the problem of H∞ control for uncertain linear discrete-time systems with quantized state feedback. Consider that the uncertain parameters are supposed to reside in a polytope. The system state is quantized by a logarithmic static and time-invariant quantizer. Via giving a new control law and using parameter dependent Lyapunov function approach, new results on the quantized H∞ state feedback control are expressed in terms of linear matrix inequalities (LMIs). A numerical example is introduced to illustrate the effectiveness and applicability of the proposed methodology.

Stabilisation of asymmetrically structured back‐to‐back static synchronous compensator system with non‐linear damping control

This study describes a new back-to-back static synchronous compensator (BTB STATCOM) system with asymmetrically structured converters and develops its mathematical model. BTB STATCOM uses pulse-width modulation technology for a rectifier station along with multipulse technology for an inverter station. The authors propose a control strategy for the following three control variables: DC voltage and both active and reactive rectifier powers. In the rectifier station, input–output linearisation via the feedback control law is applied for independent control of the active and reactive powers. Conversely, the controller is not applicable for maintaining the DC voltage because of its irregularity. They design an alternative indirect damping controller for the reactive current within the inverter station. The proposed controller improves the stability margin and decreases the oscillation of DC voltage with a degraded reactive current tracking performance of the inverter. Using simulation, they validate the proposed method, comparing its effectiveness to that of a proportional-plus-integral-plus-derivative controller. They investigate the semi-globally exponential stability of the damped internal dynamics using a parameter-dependent Lyapunov (PDL) function. Using a composite Lyapunov function based on the PDL function, the uniform boundedness of the output regulation error as well as the closed-loop stability is established.

Gain-scheduled controller design for discrete-time Linear Parameter Varying systems with multiplicative noises

In this paper, two technologies to design gain-scheduled controller are proposed for discretetime Linear Parameter Varying (LPV) systems with multiplicative noises. In the considered systems, the time-varying parameters are proposed to describe the uncertainties. And, the added multiplicative noise term is applied to describe the stochastic behaviors of the system. Furthermore, the gainscheduled design scheme is applied to deal with the stabilization problem of the LPV systems with multiplicative noises. Through the parameter-independent and parameter-dependent Lyapunov functions, the corresponding sufficient conditions are derived to discuss the stability and stabilization problems of the system. In order to apply the convex optimization algorithm, the derived sufficient conditions are converted into Linear Matrix Inequality (LMI) problems. Based on the proposed design technologies, gain-scheduled controllers are designed such that the considered system is asymptotically stable in the mean square. Finally, two numerical examples are applied to demonstrate the application and effectiveness of the proposed design technologies of this paper.

Synchronization of fuzzy bidirectional associative memory neural networks with various time delays

The problem of synchronization for fuzzy bidirectional associative memory (BAM) neural networks (NNs) with various time delays is formulated and investigated. The various delays consist of discrete delays, unbounded distributed delays and constant delay in the leakage term (i.e. ‘leakage delay’) are considered. Then, some sufficient conditions are presented to guarantee the global asymptotic stability of the error dynamical system by using Lyapunov–Krasovskii functional and linear matrix inequality (LMI) method. As a result, we achieved synchronization of master-slave (drive-response) fuzzy BAM NNs systems. Even, if there is no leakage delay, the obtained results are less restrictive to some known sufficient conditions. Moreover, the proposed results do not require the boundedness, differentiability and monotonicity of the activation functions, which can also be easily checked via the LMI solver in robust control toolbox in Matlab. Furthermore, using a parameter-dependent Lyapunov function approach the synchronization problem for polytopic uncertain fuzzy BAM NNs with various time delays is considered. The parameter uncertainties under consideration are assumed to belong to a fixed convex polytope. Finally, two numerical examples and simulations are given to illustrate the effectiveness of the derived theoretical results.

A novel method for fault detection in singularly perturbed systems via the finite frequency strategy

This paper addresses the fault detection synthesis for a class of linear singularly perturbed systems with unknown disturbances. The observer is designed via a novel method, in which singularly perturbed systems could be processed within a uniform framework instead of using the classical slow–fast decomposition. The problem of robust fault detection can be converted into a standard H∞ model-matching. Based on the generalized Kalman–Yakubovich–Popov lemma and parameter-dependent Lyapunov functions, a full-order observer is designed such that the corresponding error dynamic system is asymptotically stable and satisfies a prescribed finite-frequency H_/H∞ performance index. A novel three-step design procedure is then proposed to extract the fault feature from strong background disturbances. An illustrative example is given to demonstrate the validity and applicability of the proposed approaches in the simulation part.

BMI-Based Stabilization of Linear Uncertain Plants With Polynomially Time Varying Parameters

This technical note considers the robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics. Each mode is characterized by a theoretically unbounded dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. Using a parameter dependent Lyapunov function polynomially depending on time, the stabilizing controller for each single mode is directly obtained by the solution of some BMI's, which become LMI's by fixing two positive scalars. The stability conditions of the switching closed loop system are derived defining a switched Lyapunov function and involving the permanence time interval of the switching plant over each single mode. A salient feature of the technical note is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

FIR-type robust H 2 and H ∞ control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

In this paper, robust H 2 and H ∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.

Estimation of Attractive Stability Region via Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Impulsive Switched Linear System with Saturated Control Input

A class of homogeneous polynomial functions is used to estimate the attractive stability region for an impulsive switched linear system with saturated control input. Under the arbitrary switching rule and the dwell time switching rule, local stabilization conditions are obtained in terms of linear matrix inequalities. The corresponding optimization problems are formulated to obtain a larger attractive stability region. Using the simple ellipsoid method to estimate the attractive stability region produces very conservative results, because the attractive stability region is normally irregular. To solve this problem, a polyhedron constructed using a level set of the homogeneous parameter-dependent quadratic Lyapunov function is used to estimate the attractive stability region. The polyhedron is closer to the attractive stability region than the simple ellipsoid. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.

Robust stabilization of T–S fuzzy discrete systems with actuator saturation via PDC and non-PDC law

In this paper, the problem of robust stabilization of T–S fuzzy discrete systems with actuator saturation is studied. A set invariance condition is established as a null controllable region. The robust stabilization results of the closed-loop T–S fuzzy system have a domain of attraction which is arbitrarily close to the null controllable region. The problem of estimating the domain of attraction of the T–S fuzzy closed-loop systems is formulated and solved as an optimization problem. By using parameter-dependent Lyapunov function, both parallel-distributed compensation (PDC) control law and non-parallel-distributed compensation control law are designed. And there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T–S fuzzy system with actuator saturation. A numerical example is provided to illustrate the effectiveness of the proposed design techniques.

Parameter-Dependent Lyapunov Functions in the Robust Control of Discrete Linear Repetitive Processes Using Previous Pass-Windowed Information

Abstract The variables in multidimensional systems are functions of more than one indeterminate, and such systems cannot be controlled by standard systems theory. This paper considers a subclass of these systems that operate over a subset of the upper-right quadrant of the two-dimensional (2D) plane in the discrete domain with a specified recursive structure known as repetitive processes. Physical examples of such processes are known and also their representations can be used in the analysis of other classes of systems, such as iterative learning control. This paper gives new results on the use of the parameter-dependent Lyapunov functions for stability analysis and controls law design of a subclass of repetitive processes that arise in application areas. These results aim to eliminate or reduce the effects of model parameter uncertainty.

Asynchronous H∞ control for unmanned aerial vehicles: switched polytopic system approach

This study is concerned with the H∞ control for the full-envelope unmanned aerial vehicles (UAVs) in the presence of missing measurements and external disturbances. With the dramatic parameter variations in large flight envelope and the locally overlapped switching laws in flight, the system dynamics is modeled as a locally overlapped switched polytopic system to reduce designing conservatism and solving complexity. Then, considering updating lags of controller's switching signals and the weighted coefficients of the polytopic subsystems induced by missing measurements, an asynchronous H∞ control method is proposed such that the system is stable and a desired disturbance attenuation level is satisfied. Furthermore, the sufficient existing conditions of the desired switched parameter-dependent H∞ controller are derived in the form of linear matrix inequality (LMIs) by combining the switched parameter-dependent Lyapunov function method and average dwell time method. Finally, a numerical example based on a highly maneuverable technology (HiMAT) vehicle is given to verify the validity of the proposed method.

A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations

This paper provides a survey of results in linear parameter-varying (LPV) control that have been validated by experiments and/or high-fidelity simulations. The LPV controller synthesis techniques employed in the references of this survey are briefly reviewed and compared. The methods are classified into polytopic, linear fractional transformation, and gridding-based techniques and it is reviewed how in each of these approaches, synthesis can be carried out as a convex optimization problem via a finite number of linear matrix inequalities (LMIs) for both parameter-independent and parameter-dependent Lyapunov functions. The literature is categorized with regard to the application, the complexity induced by the controlled system’s dynamic and scheduling orders, as well as the synthesis method. Exemplary cases dealing with specific control design problems are presented in more detail to point control engineers to possible approaches that have been successfully applied. Furthermore, key publications in LPV control are related to application achievements on a timeline.

Adaptive Consensus Filters for Second-Order Distributed Parameter Systems Using Sensor Networks

Control and estimation of second-order distributed parameter systems are of importance in mechanical systems. In particular, flexible structures can be modeled as second-order distributed parameter systems. This paper investigates adaptive consensus filtering for a class of second-order distributed parameter systems under an abstract framework. We propose an adaptive consensus mechanism to minimize the disagreement among all local filters consisting of different sensor nodes and written in the natural setting of a second-order formulation with an additional coupling. A parameter-dependent Lyapunov function is presented to analyze the stability of the collective dynamics, that is, all filters agree with each other and converge to the true state of the second-order system. The performance is demonstrated on a numerical example of a second-order partial differential equation with point measurements.

New approach to [formula omitted] filtering for discrete-time systems with polytopic uncertainties

This paper is concerned with the problem of H∞ filtering for polytopic uncertain discrete-time systems. The parameter-dependent Lyapunov function approach and linear matrix inequality (LMI) technique are applied to solve the filtering problem. By considering auxiliary slack variables with free structure and introducing additional scalar parameters, new sufficient conditions for H∞ filter design are presented in terms of linear matrix inequalities (LMIs), which guarantee the filtering error systems to be asymptotically stable with prescribed H∞ performances. In contrast to the existing results for H∞ filter design, the new proposed design method can provide less conservative results. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Extended dissipativity-based synchronization of uncertain chaotic neural networks with actuator failures

This paper addresses the problem of extended dissipativity-based synchronization for chaotic neural networks with polytopic-type uncertainties and actuator failures. By using a parameter-dependent Lyapunov function and a novel extended dissipation inequality, a criterion is proposed to guarantee that the synchronization error system is extended (X,Y,Z)-dissipative for all admissible uncertainties and actuator failures. Based on this criterion, a desired fault-tolerant controller is designed, which takes the possible actuator failures into account. A numerical example is presented to show the effectiveness of our proposed design method.

Robust consensus for linear multi-agent systems with mixed uncertainties

This paper studies the robust consensus problem for a class of linear multi-agent systems (MASs) with polytopic uncertainties and external disturbances. The uncertainties appear in both the dynamics of each agent and the topology of the overall network. Based on Lyapunov stability theory, the robust consensus problem is converted to the robust H∞ stabilization of the error system. With the aid of the homogeneous parameter-dependent (HPD) Lyapunov functions and sum of squares (SOS) technique, the robust consensus problem is further formulated as an SOS programming. Sufficient conditions to achieve robust consensus with disturbance rejection capability are presented in terms of SOS constraints. As an extension, the robust consensus problem for discrete-time MASs is studied in a similar way. Numerical examples are provided to illustrate the effectiveness of the approach.

On Homogeneous Parameter-Dependent Quadratic Lyapunov Function for RobustH∞Filtering Design in Switched Linear Discrete-Time Systems with Polytopic Uncertainties

This paper is concerned with the problem of robustH∞filter design for switched linear discrete-time systems with polytopic uncertainties. The condition of being robustly asymptotically stable for uncertain switched system and less conservativeH∞noise-attenuation level bounds are obtained by homogeneous parameter-dependent quadratic Lyapunov function. Moreover, a more feasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the given arbitrary switching signal. Lastly, simulation results are used to illustrate the effectiveness of our method.

A Parameter-Dependent Approach to Robust H<sub>∞</sub> Control of Norm Bounded Uncertain Systems

A new approach to analyze and synthesize linear uncertain systems with time varying norm bounded uncertainty is presented in this paper. First, the uncertainty set is separated into several different sets according to the maximum singular value of uncertainty.Based on the new classification, the original uncertain system is transformed into a Markovian jump system. Second, via a parameter-dependent Lyapunov function, the problems of robust H∞ analysis and synthesis is studied.Based on this, two sufficient conditions for designing parameter-dependent controllers are established. Finally, numerical simulations are used to illustrate the developed method.

Robust MPC of an unstable chemical reactor using the nominal system optimization

Abstract The continuous stirred-tank reactor with uncertain parameters was stabilized in the open-loop unstable steady state using the robust model predictive control. The gain matrices of the robust state-feedback controller were designed using the nominal system optimization and the quadratic parameter-dependent Lyapunov functions. The controller was verified by simulations using the non-linear model of the reactor and compared with the robust model predictive controller designed using the worst-case system optimization. The values of the quadratic cost function and the consumption of coolant were observed. Both robust model predictive controllers stabilized the reactor despite constrained control inputs and states. The robust model predictive control based on the nominal system optimization improved control responses and decreased the consumption of coolant.

Intermittent observer-based consensus control for multi-agent systems with switching topologies

In this paper, we focus on the consensus problem for leaderless and leader–followers multi-agent systems with periodically intermittent control. The dynamics of each agent in the system is a linear system, and the interconnection topology among the agents is assumed to be switching. We assume that each agent can only share the outputs with its neighbours. Therefore, a class of distributed intermittent observer-based consensus protocols are proposed for each agent. First, in order to solve this problem, a parameter-dependent common Lyapunov function is constructed. Using this function, we prove that all agents can access a prescribed value, under the designed intermittent controller and observer, if there are suitable conditions on communication. Second, based on the investigation of the leader-following consensus problem, we design a new distributed intermittent observer-based protocol for each following agent. Finally, we provide an illustrative example to verify the effectiveness of the proposed approach.