Differential Linear Matrix Inequalities Research Articles

On finite time stability with guaranteed cost control of uncertain linear systems

This paper deals with the design of a robust state feedback control law for a class of uncertain linear time varying systems. Uncertainties are assumed to be time varying, in one-block norm bounded form. The proposed state feedback control law guarantees finite time stability and satisfies a given bound for an integral quadratic cost function. The contribution of this paper is to provide a sufficient condition in terms of differential linear matrix inequalities for the existence and the construction of the proposed robust control law. In particular, the construction of the feedback control law is brought back to a feasibility problem which can be solved inside the convex optimization framework. The effectiveness of the proposed approach is shown by means of the results obtained on a numerical and a physical example.

Differential linear matrix inequality in optimal sampled-data control

This paper addresses the design of optimal sampled-data output feedback full order controllers for linear continuous-time invariant systems. First, H2 and H∞ performance indices are determined and expressed through differential linear matrix inequalities (DLMIs). Second, in each scenario, the optimal sampled-data controller of the aforementioned class is determined from a convex programming problem expressed by DLMIs. Theoretical achievements are based on the direct application of the celebrated Bellman’s Principle of Optimality expressed in terms of the dynamic programming equation associated to the time interval corresponding to two successive sampling instants. The design problems to be dealt with are convex and are solved by converting all constraints to LMI and adopting a piecewise linear solution. The possibility of aperiodic sampling is considered and briefly discussed. Finally, academic examples are solved for illustration.

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

The goal of this paper is to assess the robustness of an uncertain linear time-varying (LTV) system on a finite time horizon. The uncertain system is modeled as an interconnection of a known LTV system and a perturbation. The input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Typical notions of robustness, e.g. nominal stability and gain/phase margins, can be insufficient for finite-horizon analysis. Instead, this paper focuses on robust induced gains and bounds on the reachable set of states. Sufficient conditions to compute robust performance bounds are formulated using dissipation inequalities and IQCs. The analysis conditions are provided in two equivalent forms as Riccati differential equations and differential linear matrix inequalities, and an algorithm is developed leveraging both forms.

Optimal and Robust Sampled-Data Control of Markov Jump Linear Systems: A Differential LMI Approach

This paper addresses the problem of designing optimal sampled-data state feedback control for continuous-time Markov jump linear systems. Stability and performance robustness against polytopic uncertainty acting on the system parameters including the transition rate matrix are analyzed. The main goal is to characterize the optimal solution of this class of problems in the context of ${\mathcal H}_2$ and ${\mathcal H}_\infty$ performances. The theoretical achievements are based on the direct application of the celebrated Bellman's Principle of Optimality expressed in terms of the dynamic programming equation applied to the time interval corresponding to two successive sampling instants. The design conditions are expressed through differential linear matrix inequalities. Examples are solved for illustration.

Synthesis of State Unknown Inputs Observers for Nonlinear Lipschitz Systems with Uncertain Disturbances

We propose methods to synthesize observers for the state and unknown input influences that ensure that estimation error is finite time bounded with respect to given sets of initial states and admissible trajectories or suppress initial deviations and uncertain bounded in L∞-norm external disturbances for time-varying continuous Lipschitz systems. Here gain coefficients of the observers depend on time and are determined based on numerical solutions of optimization problems with differential linear matrix inequalities or numerical solutions of the corresponding matrix comparison system. With the example of an electric drive system with elastic transmission of motion we show that their application for state estimation and unknown inputs for time-invariant systems proves to be more efficient (with respect to convergence time and accuracy of the resulting estimates) compared to observers with constant coefficients obtained based on numerical solutions of optimization problems with linear matrix inequalities.

Partial Sampled-Data State Feedback Control of Markov Jump Linear Systems

Abstract This paper aims at designing a partial sampled-data state feedback control law for Markov jump linear systems (MJLS). The interesting feature of the control structure is that only the state variable is sampled, while the stochastic parameter that defines the Markov mode of the system used for control purposes is free to change at any time between samples. The main goal is to provide sufficient convex conditions for the existence of a solution for this class of control design problems in the context of H∞ and H2 performances, which are expressed through Differential Linear Matrix Inequalities (DLMI). The proposed method is implemented using LMIs facilities and provides a minimum guaranteed cost control in only one shot. An example is solved for illustration and comparison.

Cooperative Intercept Guidance of Multiple Aircraft with a Lure Role Included

When threatened by a pursuer, an evading aircraft launches two defenders to accomplish a cooperative evasion, constituting a four-aircraft interception engagement. Under the assumption that the pursuing aircraft adopts the augmented proportional guidance law and first-order dynamics, a cooperative intercept mathematical model with an intercept angle constraint is established, allowing for the cooperative maneuvering of the evader. Based on the differential linear matrix inequality (DLMI), a controller design method of input and output finite time stability (IO-FTS) is proposed and applied to the aircraft’s cooperative intercept scenario. A cooperation performance analysis is carried out for two cases: (1) two defenders intercept the pursuer with various intercept angle constraints, and (2) the evader acts in a lure role to cooperate with two defenders. The simulation results indicate that the proposed method of controller design has the ability to guarantee that the two defenders intercept the pursuer at the preassigned intercept angles. The cooperative intercept scenario with a lure role is shown to be a very effective method for reducing the maximum required acceleration for defenders, which confirms the availability and advantage of cooperation. The strong adaptability and robustness of the cooperative guidance law with respect to various initial launch conditions is also verified.

Analysis of Adaptive Synchronization for Stochastic Neutral-Type Memristive Neural Networks with Mixed Time-Varying Delays

Linear feedback control and adaptive feedback control are proposed to achieve the synchronization of stochastic neutral-type memristive neural networks with mixed time-varying delays. By applying the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, we obtain some new adaptive synchronization criteria. A numerical example is given to illustrate the effectiveness of our results.

Guaranteed-Cost Finite-Time Fuzzy Control for Temperature-Constrained Nonlinear Coupled Heat-ODE Systems

This paper investigates the guaranteed-cost finite-time fuzzy control problem subject to a temperature constraint for a class of coupled systems represented by nonlinear ordinary differential equations (ODEs) and a scalar nonlinear heat equation. Initially, a finite-dimensional nonlinear coupled system is derived by combining the slow system of heat equation with the original ODE system, which can be exactly represented by the Takagi–Sugeno fuzzy model. Meanwhile, the temperature constraint is transformed into the state constraint performed on the finite-dimensional coupled system. Then, a guaranteed-cost finite-time constrained fuzzy control design is developed in terms of a set of time-dependent differential linear matrix inequalities (LMIs) to make the closed-loop of the original ODE system finite-time quasi-contractively stable with an upper bound of quadratic cost function, while the temperature constraint is respected. Furthermore, by utilizing the time-convexity and LMI optimization techniques, a suboptimal controller is obtained by means of minimizing the guaranteed-cost bound. Finally, the proposed design method is applied to the control of a temperature constrained hypersonic rocket car.

State and Unknown Inputs Finite Time Estimation for Time-Varying Nonlinear Lipschitz Systems with Uncertain Disturbances

The continuous time-varying systems with Lipschitz nonlinearities subject to bounded by L2 norm external disturbances and unknown inputs are considered. There suggested ways of observers synthesis for state and unknown inputs, providing H∞ rejection of initial deviations and bounded external disturbances simultaneously. Thus the observers gain depend on a time and is defined on the basis of the numerical solving of optimization problem with differential linear matrix inequalities or the numerical solving of corresponding matrix comparison system. The proposed observers are used to state and unknown inputs estimation of Rössler system.

State observer synthesis by measurement results for nonlinear Lipschitz systems with uncertain disturbances

We propose ways to synthesize state observers that ensure that the estimation error is bounded on a finite interval with respect to given sets of initial states and admissible trajectories and also simultaneous H?-suppression at every time moment of initial deviations and uncertain deviations bounded in L2-norm, external disturbances for non-autonomous continuous Lipschitz systems. Here the gain of the observers depend on the time and are defined based on a numerical solution of optimization problems with differential linear matrix inequalities or numerical solution of the corresponding matrix comparison system. With the example of a single-link manipulator we show that their application for the state estimating of autonomous systems proves to be more efficient (in terms of convergence time and accuracy of the resulting estimates) as compared to observers with constant coefficients obtained with numerical solutions of optimization problems with linear matrix inequalities.

Finite time control for one class of nonlinear systems with the H ∞ performance criterion

This work considers nonautonomous control systems with uncertain nonlinearities subjected to the impact of external disturbances which are continuous functions bounded by the L 2-norm. Based on the method of matrix comparison systems and technique of differential linear matrix inequalities, it is proposed to solve problems of finite time stability and boundedness with respect to the given sets, as well as suppress the disturbances and initial deviations using the state feedback with an estimation of the performance by the H ∞-criterion.

Anti-synchronization control for delayed memristor-based distributed parameter NNs with mixed boundary conditions

This paper addresses the anti-synchronization control problem of memristor-based distributed parameter neural networks (MDPNNs) with time-varying delays. An intermittent sampled-data controller is proposed such that the response system can be anti-synchronized with the drive system by using Lyapunov stability theory. The design approach of the desired intermittent sampled-data controller is first proposed. Based on a differential inequality and linear matrix inequality (LMI) technology, novel sufficient conditions for the exponential anti-synchronization are obtained in terms of line matrix inequalities. Numerical simulations are shown to demonstrate the effectiveness of the theoretical results.

State estimation and stabilization of continuous systems with uncertain nonlinearities and disturbances

Nonautonomous control systems with uncertain nonlinearities subjected to bounded exogenous disturbances are considered. Based on the method of matrix comparison systems and the framework of differential linear matrix inequalities, we suggest a solution approach to the problems of state estimation, stability and boundedness with respect to given sets, as well as suppression of disturbances and initial deviations using linear state-feedback control.

Cluster synchronisation in non‐linearly coupled impulsive networks with non‐identical nodes and time‐varying delays

This study investigates cluster synchronisation of non-linearly coupled impulsive networks with time-varying delay. Without loss of generality, it is assumed that the nodes in the networks to be non-identical and the coupling matrix to be asymmetric. Two kinds of topological structures are concerned, which correspond to positive weighted network, as well as cooperative and competitive network, respectively. Pinning controllers are designed in order to achieve cluster synchronisation. According to the Lyapunov stability theory, impulsive delay differential inequalities and linear matrix inequalities, some criteria for cluster synchronisation of non-linear impulsive dynamical systems are obtained. Two numerical examples are given to demonstrate the effectiveness of the authors' new results.

Adaptive sliding fault tolerant control for nonlinear uncertain active suspension systems

This paper is concerned with the fault tolerant control problem of nonlinear uncertain active suspension systems with constraint requirements. A novel adaptive sliding fault tolerant controller, which does depend on accurate models, is designed to stabilize the active suspension systems and thus to improve the ride comfort, without utilizing the bounds of actuator faults and parameter uncertainties. Furthermore, an H∞ optimization scheme based on differential evolution (DE) algorithm and linear matrix inequalities (LMIs) is introduced to design appropriate parameters of the sliding surface, which guarantees the constraint requirements of active suspension systems. Finally, simulation results are included to illustrate the effectiveness of the proposed strategy.

The Mixed Robust /FTS Control Problem Analysis and State Feedback Control

AbstractIn this paper we deal with the mixed /finite‐time stability control problem. More specifically, given an open loop uncertain linear system, we provide a necessary and sufficient condition for quadratic input‐output finite‐time stability with an bound. Exploiting this result we also give a sufficient condition to solve the related synthesis problem via state‐feedback. The property of quadratic input‐output finite‐time stability with an bound implies that the system under consideration satisfies an performance bound between the disturbance input and the controlled output and, at the same time, is input‐output finite‐time stable for all admissible uncertainties. This condition requires the solution of a feasibility problem constrained by a pair of differential linear matrix inequalities (LMIs) coupled with a time‐varying LMI. The proposed technique is illustrated by means of both a numerical and a physical example.

Robust Finite-TimeH∞Control for Linear Time-Varying Descriptor Systems with Jumps

The finite-timeH∞control problem is addressed for uncertain time-varying descriptor system with finite jumps and time-varying norm-bounded disturbance. Firstly, a sufficient condition of finite-time boundedness for the abovementioned class of system is obtained. Then the result is extended to finite-timeH∞for the system. Based on the condition, state feedback controller is designed such that the closed-loop system is finite-time boundedness and satisfiesL2gain. The conditions are given in terms of differential linear matrix inequalities (DLMIs) and linear matrix inequalities (LMIs), and such conditions require the solution of a feasibility problem involving DLMIs and LMIs, which can be solved by using existing linear algorithms. Finally, a numerical example is given to illustrate the effectiveness of the method.

State Estimation and Stabilization of Continuous Systems with Sector Nonlinearities and Bounded Disturbances

In this paper time-variant systems with sector nonlinearities tending to be affected by bounded external disturbances are considered. We present an approach to solving problems of state estimation, stability, boundedness relatively to given sets, rejection of bounded external disturbances and initial deviations by using static linear state feedback, based on matrix comparison systems method and differential linear matrix inequalities.

Robust Exponential Synchronization for a Class of Master-Slave Distributed Parameter Systems with Spatially Variable Coefficients and Nonlinear Perturbation

This paper addresses the exponential synchronization problem of a class of master-slave distributed parameter systems (DPSs) with spatially variable coefficients and spatiotemporally variable nonlinear perturbation, modeled by a couple of semilinear parabolic partial differential equations (PDEs). With a locally Lipschitz constraint, the perturbation is a continuous function of time, space, and system state. Firstly, a sufficient condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems is investigated for all admissible nonlinear perturbations. Secondly, a robust distributed proportional-spatial derivative (P-sD) state feedback controller is desired such that the closed-loop master-slave PDE systems achieve exponential synchronization. Using Lyapunov’s direct method and the technique of integration by parts, the main results of this paper are presented in terms of spatial differential linear matrix inequalities (SDLMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed methods applied to the robust exponential synchronization problem of master-slave PDE systems with nonlinear perturbation.