Linear Matrix Inequality Approach Research Articles

LMI-based robust mixed-integer model predictive control for hybrid systems

ABSTRACT This paper proposes a linear matrix inequality (LMI) approach to mixed-integer model predictive control (MPC) of uncertain hybrid systems with binary and real valued control inputs. The stability condition of the hybrid system is obtained by using the Lyapunov function and then a sufficient state feedback control law is achieved so that guarantees the closed-loop stability and also minimises an infinite horizon performance index. The primal optimisation problem is convex, therefore, a convex relaxation is investigated by introducing the Lagrange dual function. The real and binary control inputs are obtained by solving the dual of the dual problem in the framework of LMIs. The performance and effectiveness of the proposed MPC approach and the duality gap of the convex relaxation are studied through simulation results of a hybrid system with mixed real and binary inputs.

Fractional controller based on a robust PIα observer for uncertain fractional systems

ABSTRACTIn this paper, we present a fractional sliding mode controller based on a robust observer for fractional systems with bounded disturbances. This controller is introduced to achieve closed-loop Mittag–Leffler stability for uncertain fractional systems; meanwhile, the robust observer is designed such that a stability is obtained via quadratic Lyapunov functions and the tuning is achieved with linear matrix inequality approach. The performance and stability of the proposed controller are analysed and demonstrated through a generalised -differentiator. Finally, to illustrate the effectiveness of the proposed approach a comparative analysis against the fractional Super-Twisting is carried out.

An AWID and AWIS X-By-Wire UGV: Design and Hierarchical Chassis Dynamics Control

In this paper, an all-wheel independently driven and all-wheel independently steered unmanned ground vehicle (UGV) is described. This paper investigates the hierarchical chassis yaw dynamics control (CYDC) and the tyre force control of the UGV in the remote control mode (RCM). The hierarchical CYDC scheme in RCM is proposed. As the key part in the control scheme, a yaw moment controller is proposed to deal with the oversteer problem of the UGV. Through the robust-based pole placement technique, the ideal poles’ zones of the lateral UGV dynamics system are able to be tuned to meet different dynamics behavior requirements in different UGV tasks. The robust state feedback yaw dynamics controller is investigated based on the linear matrix inequalities approach. It considers the unavoidable parametric disturbance and uncertainty, such as the variation of the UGV’s mass, yaw inertia, and tyre-road characteristics. In addition, in order to improve its performance in off-road conditions, the tyre traction force distribution algorithm and sliding mode wheel slip controller are designed to negotiate uneven terrains. The experiments in paved and off-road conditions are conducted to demonstrate the performance of the proposed controller.

Inertia Tensor Properties in Robot Dynamics Identification: A Linear Matrix Inequality Approach

Physical feasibility of robot dynamics identification is currently receiving renovated attention from the research community. Inertia tensor inequalities (namely the positive definite property) have been extensively used among other physical constraints to check physical consistency of identification methods. Recently, an extra inequality associated to inertia tensor eigenvalues has been included to check physical consistency, showing that the previous methods were incomplete. In this paper we show that this extra inequality incorporates the positive definite property, being more restrictive. We also include it in the linear matrix inequality framework for robot dynamics identification to obtain fully physical estimates. The relevance of the extra inequality is verified through real 7-DOF robot manipulator experiments.

Tracking Fault Tolerant Control for Hybrid System: Two-Link Human Arm Application

This paper proposes a novel fault tolerant control (FTC) scheme for a class of hybrid dynamical system (HDS) subject to sensor faults. The corresponding FTC architecture is designed around a reconfiguration mechanism. It aims to compensate the effects of the sensors degradation and maintain satisfactory performances including continuous stability. Moreover, by using the linear matrix inequalities (LMI) approach, a fault estimation algorithm is fulfilled and the compromise between robustness to disturbances and sensitivity to fault is guaranteed. For the sake of trajectory tracking, a combined robust state feedback and proportional-integral-derivative control system is proposed herein. Finally, extensive simulation results conducted on two-link arm system are included to illustrate the efficiency of the designed FTC scheme.

Global Asymptotic Stabilization of Cellular Neural Networks with Proportional Delay via Impulsive Control

This paper considers the global asymptotic stabilization of a class of cellular neural networks with proportional delay via impulsive control. First, from impulsive control point of view, some delay-dependent criteria are established to guarantee the asymptotic stability and global asymptotic stability of the zero solution of a general impulsive differential system with proportional delay by applying the Lyapunov–Razumikhin method. Second, based on the obtained criteria and LMI approach, several delay-dependent conditions are derived to ensure the uniqueness and global asymptotic stability of the equilibrium point of a class of cellular neural networks with impulses and proportional delay. It is shown that impulses can be used to globally asymptotically stabilize some unstable and even chaotic cellular neural networks with proportional delay. Moreover, the proposed stability conditions expressed in terms of linear matrix inequalities (LMIs) can be checked by the Matlab LMI toolbox, and so it is effective to implement in real problems. Finally, three numerical examples are provided to illustrate the effectiveness of the theoretical results.

Linear Matrix Inequality Approach to Stochastic Stabilization of Networked Control System with Markovian Jumping Parameters

This paper is concerned with the stochastic stabilization problem for a class of networked control system (NCS) with destabilizing transmission factors. By introducing the effective sampling instant to model random time delays and successive packet dropouts as two independent Markov chains, NCS is modeled as a discrete-time Markovian jump linear system with mixed integrated Markovian jumping parameters. In this way, a novel framework to analyze the stochastic stabilization problem of NCS is provided. The necessary and sufficient conditions for the stochastic stabilization of the NCS are obtained by the Lyapunov method and the state-feedback controller gain that depends on the delay modes is obtained in terms of the linear matrix inequalities (LMIs) formulation via the Schur complement theory. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.

Stability and Stabilization of 2D Singular Systems: A Strict LMI Approach

This paper deals with the stability and stabilization of 2D singular systems described by a Roesser model. The proposed results are presented in terms of a strict linear matrix inequality (LMI), which makes possible to elaborate a new sufficient admissibility condition. The design of a state feedback controller is then treated using this condition, deriving a sufficient condition for the admissibility of the closed-loop system. A numerical example is given at the end of the paper, which illustrates the effectiveness of the proposed methodology.

Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging

In this article, we analyze the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The Bocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation methods implemented in the HamPath software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. Symbolic computations techniques are used to handle the singularity analysis.

H∞ Guaranteed Cost Fault-Tolerant Control of Double-Fault Networked Control Systems: Piecewise Delay Method

The term double-fault networked control system means that sensor faults and actuator faults may occur simultaneously in networked control systems. The issues of modelling and an H∞ guaranteed cost fault-tolerant control in a piecewise delay method for double-fault networked control systems are investigated. The time-varying properties of sensor faults and actuator faults are modelled as two time-varying and bounded parameters. Based on the linear matrix inequality (LMI) approach, an H∞ guaranteed cost fault-tolerant controller in a piecewise delay method is proposed to guarantee the reliability and stability for the double-fault networked control systems. Simulations are included to demonstrate the theoretical results of the proposed method.

High-Performance Robust Controller Design of Plug-In Hybrid Electric Vehicle for Frequency Regulation of Smart Grid Using Linear Matrix Inequality Approach

This paper proposes a high-performance and robust linear quadratic regulator-proportional integral derivative (LQR-PID) controller for frequency regulation in a two-area interconnected smart grid with a population of plug-in hybrid electric vehicles. Controller robustness is achieved using a linear matrix inequality approach. The proposed control framework is tested in a simulated two-area interconnected smart grid integrated with plug-in hybrid electric vehicles under load disturbances and wind power fluctuations. The performance of the proposed controller is simulated using Matlab and compared with that of a conventional linear quadratic regulator controller. Simulation results show that the proposed controller provides reliable smart grid frequency control.

LMI-Based Stability of Nonlinear Non-Autonomous Fractional-Order Systems With Multiple Time Delays

Linear matrix inequality (LMI), as a class of stability conditions for non-autonomous fractional-order (NAFO) systems, is proposed in this paper. Based on the fractional-order Lyapunov direct method, the Mittag-Leffler stability of fractional-order systems without time delay is analyzed under LMI conditions first. For fractional-order systems with multiple time delays, the LMI conditions of Lyapunov asymptotical stability are studied by using the fractional-order comparison principle. Besides, the stability of linear NAFO systems is analyzed by using the LMI approach. What is more, the LMI-based stability method of nonlinear NAFO systems with multiple time delays is presented. Especially, for nonlinear fractional-order systems, a sufficient condition of the existence and uniqueness of the equilibrium point is given in the LMI form. In addition, two examples are provided to verify the effectiveness of the obtained theoretical results.

Design of <inline-formula> <tex-math notation="LaTeX">$H_\infty$ </tex-math> </inline-formula> Output Feedback Controller for Gas Turbine Engine Distributed Control With Random Packet Dropouts

The distributed control architecture is expected to dominate the gas turbine engine (GTE) control systems in future, in which the sensors and actuators are connected to the controllers via a network. Hence, the control problem of network-enabled high-performance distributed engine control (DEC) becomes important in modern gas turbine control systems. Due to the properties of the network, the packet dropouts must be considered. This study introduces a distributed control system architecture based on a networked cascade control system (NCCS). Typical turboshaft engine-distributed controllers are designed based on the NCCS framework with a $H_\infty $ output feedback under random packet-dropouts. The sufficient robust mean-stable conditions are derived via the Lyapunov stability theory and linear matrix inequality approach. In addition, the effectiveness of the presented method is illustrated through a series of simulations.

Global Synchronization of Complex-Valued Neural Networks With Stochastic Disturbances and Time-Varying Delay

This paper discusses synchronization problem of two delayed stochastic neural networks. By constructing the Lyapunov functions, using stochastic analysis technique, some sufficient conditions guaranteeing the synchronization in mean square for the drive system with the response system have been derived in terms of linear matrix inequalities (LMIs) approach. Moreover, the main feature of this paper lies in that the present results are applicable for the drive system and the response system which all are considered not only in complex-valued domain but also with stochastic noise case. Two numerical examples are given to illustrate the effectiveness and merits of the present results.

A New Observer Design for Systems in Presence of Time-varying Delayed Output Measurements

This paper presents a state observer for linear systems and Lipschitz nonlinear systems with delayed output measurements, which are affected by a known and bounded time-varying delay. The structure of the proposed observer is based on a proportional-integral term, which allows to compensate the time-varying delay. The observer gain depends on the maximum bounded delay. This gain is computed by a Linear Matrix Inequality (LMI) approach. The observer exhibits good performance for state estimation of the system despite the presence of significantly long delay. A Lyapunov-Krasovskii functional is used to prove the asymptotical convergence to zero of the observation error. This observer is applied to the case of systems with time-varying delay whose dynamic is described by a piecewise differentiable function. Examples and numerical simulations are provided in order to support the validity of the main results.

Dynamic Output Feedback Finite-Horizon Control for Markov Jump Systems With Actuator Saturations

This paper considers the problem of asynchronous observer-based finite-horizon control of Markov jump systems (MJSs) with actuator saturations. The hidden Markov model is employed to describe asynchronous phenomenon between observer-based controller and the plant, where the observer designed has its own jumping mode that is different from that of the controlled plant. The purpose of this paper is to develop an asynchronous observer-based controller to ensure that an H ∞ performance index, over a given finite-horizon, can be satisfied for MJSs with actuator saturation. A sufficient condition is derived to guarantee that the H ∞ performance index can be achieved by using the stochastic Lyapunov function theory and S-Procedure lemma. Then, a recursive linear matrix inequality (RLMI) approach is applied to design the gains of the controller and observer. Finally, an example is given to verify the proposed algorithm.

Robust Position Control of a Two‐Sided 1‐DoF Impacting Mechanical Oscillator Subject to an External Persistent Disturbance by Means of a State‐Feedback Controller

This paper proposes a state‐feedback controller using the linear matrix inequality (LMI) approach for the robust position control of a 1‐DoF, periodically forced, impact mechanical oscillator subject to asymmetric two‐sided rigid end‐stops. The periodic forcing input is considered as a persistent external disturbance. The motion of the impacting oscillator is modeled by an impulsive hybrid dynamics. Thus, the control problem of the impact oscillator is recast as a problem of the robust control of such disturbed impulsive hybrid system. To synthesize stability conditions, we introduce the S‐procedure and the Finsler lemmas by only considering the region within which the state evolves. We show that the stability conditions are first expressed in terms of bilinear matrix inequalities (BMIs). Using some technical lemmas, we convert these BMIs into LMIs. Finally, some numerical results and simulations are given. We show the effectiveness of the designed state‐feedback controller in the robust stabilization of the position of the impact mechanical oscillator under the disturbance.

Stabilization for Markovian Jump Distributed Parameter Systems With Time Delay

The problems of stochastic stability and stabilization for a class of Markovian jump distributed parameter systems with time delay are researched in this paper. First, taking advantage of a combination of Poincare inequality and Green formula, a stochastic stability criterion is presented by a linear matrix inequality (LMI) approach. Then, a state feedback controller is designed. Based on the proposed results, the sufficient conditions of the close-loop systems' stochastic stability are given in terms of a set of LMIs by constructing the appropriate Lyapunov functionals, calculating the weak infinitesimal generator, and using the Schur complement lemma. The sufficient conditions could be solved directly and applied to engineering practice conveniently. The obtained results generalize and enrich the theory of distributed parameter systems with time delay. The model of Markovian jump distributed parameter systems is more fitting the actual systems' requirements and has wider application scope. Finally, numerical examples are used to demonstrate the validity of the method.

A State Feedback Controller Based on GCC Algorithm against Wind‐Induced Motion for High‐Rise Buildings with Parametric Uncertainties

High‐rise buildings with an active mass damper/driver (AMD) system generally use a simplified mathematical model. The result leads to parametric uncertainties including the stiffness and mass variations exist, and the accuracy of its controller is seriously affected. In view of the above uncertainties, based on a Lyapunov stability theory and linear matrix inequality (LMI) approach, a state feedback controller based on the guaranteed cost control (GCC) algorithm is proposed in this paper. A ten‐storey frame with an AMD system is taken as a numerical example, and its control effect and AMD parameters are viewed as the control indexes. The performance of the proposed robust controller is compared with a conventional controller based on the classical LQR algorithm. The analysis results show that the new controller effectively suppresses the dynamic responses of high‐rise buildings with parametric uncertainties and keeps the AMD parameters stable. Finally, a four‐storey steel experimental frame with an AMD system is used to verify the correctness of the conclusions.

Non‐fragile sampled‐data guaranteed cost control for bio‐economic fuzzy singular Markovian jump systems

This study investigates the non-fragile sampled-data guaranteed cost control problem for a bio-economic singular Markovian jump system that is represented by the Takagi–Sugeno fuzzy model. The main intention of this study is to design a non-fragile sampled-data controller for the considered model to handle the issue of tax fluctuations by means of showing that the closed-loop system is regular, impulse free and stochastically finite-time bounded. Sampled-data controller is the one where the continuous system is controlled by the digital control algorithms. By introducing a proper Lyapunov–Krasovskii functional and using linear matrix inequality (LMI) approach, a new set of criteria is obtained in terms of LMIs for achieving the required result. More precisely, by solving LMIs, an upper bound for the cost function can be obtained. Finally, a simulation result is given to illustrate the effectiveness of the proposed control design.