Nonlinear Matrix Inequalities Research Articles

H∞ Control for Continuous‐Time Mean‐Field Stochastic Systems

AbstractAn H∞‐type control is considered for mean‐field stochastic differential equations (SDEs) in this paper. A stochastic bounded real lemma (SBRL) is proved for mean‐field stochastic continuous‐time systems with state‐ and disturbance‐dependent noise. Based on SBRL, a sufficient condition is given for the existence of a stabilizing H∞ controller in terms of coupled nonlinear matrix inequalities.

Generalized Differential Balancing for Variationally Symmetric Systems

Abstract: In this paper, we introduce the notion of variational symmetry for nonlinear prolonged systems (a variational extension of a nonlinear system) by using corresponding variational systems and by developing the recently introduced generalized differential balancing method for this type of system. Generalized differential balancing is proposed for model reduction of nonlinear systems in the contraction framework and is based on solutions to two nonlinear differential matrix inequalities. Our aim of this paper is to show that for variationally symmetric systems, these two inequalities reduce to one inequality. Finally, we show that variational symmetry is preserved under generalized balanced truncation if the variational Hankel singular values are distinct.

Robust state estimation and unknown inputs reconstruction for a class of nonlinear systems: Multiobjective approach

We consider a novel method to design H∞ observers for a class of uncertain nonlinear systems subject to unknown inputs. First, the main system dynamics are rewritten as an augmented system with state vector including both the state vector of the main system and the unknown inputs. Then, we design a H∞ reduced-order observer to estimate both state variables and unknown inputs simultaneously. Based on a Lyapunov functional, we derive a sufficient condition for existence of the designed observer which requires solving a nonlinear matrix inequality. To facilitate the observer design, the achieved condition is formulated in terms of a set of linear matrix inequalities (LMI). By extending the proposed method to a multiobjective optimization problem, the maximum bound of the uncertainty and the minimum value of the disturbance attenuation level are found. Finally, the proposed observer is illustrated with an example.

Synchronization of nonlinear master–slave systems under input delay and slope‐restricted input nonlinearity

This article addresses the synchronization of nonlinear master–slave systems under input time‐delay and slope‐restricted input nonlinearity. The input nonlinearity is transformed into linear time‐varying parameters belonging to a known range. Using the linear parameter varying (LPV) approach, applying the information of delay range, using the triple‐integral‐based Lyapunov–Krasovskii functional and utilizing the bounds on nonlinear dynamics of the nonlinear systems, nonlinear matrix inequalities for designing a simple delay‐range‐dependent state feedback control for synchronization of the drive and response systems is derived. The proposed controller synthesis condition is transformed into an equivalent but relatively simple criterion that can be solved through a recursive linear matrix inequality based approach by application of cone complementary linearization algorithm. In contrast to the conventional adaptive approaches, the proposed approach is simple in design and implementation and is capable to synchronize nonlinear oscillators under input delays in addition to the slope‐restricted nonlinearity. Further, time‐delays are treated using an advanced delay‐range‐dependent approach, which is adequate to synchronize nonlinear systems with either higher or lower delays. Furthermore, the resultant approach is applicable to the input nonlinearity, without using any adaptation law, owing to the utilization of LPV approach. A numerical example is worked out, demonstrating effectiveness of the proposed methodology in synchronization of two chaotic gyro systems. © 2015 Wiley Periodicals, Inc. Complexity 21: 220–233, 2016

Delay-range-dependent observer-based control of nonlinear systems under input and output time-delays

The problem of designing observer-based controller for a class of nonlinear time-delay systems subjected to input and output time-varying delays is addressed in this paper. More incisively, a delay-range-dependent criterion, incorporating variation of delays between known lower and upper bounds, is established for control of nonlinear time-delay systems. The proposed methodology ensures asymptotic stability of the closed-loop system's state and the estimation error between states of the system and observer. By application of a Lyapunov–Krasovskii functional, Jensen's inequality, standard matrix inequality procedures, Luenberger-type observer, delay-interval bounds and delay-derivative limits, observer-based controller synthesis approach using nonlinear matrix inequalities for the nonlinear time-delay systems is provided. Further, a decoupling approach is employed to render a simpler condition for the observer and the controller synthesis. A novel observer-based control scheme for the linear time-delay systems with interval time-varying input as well as output lags is derived from the proposed control strategy. The traditional delay-dependent controller design, incorporating zero lower bound of the delay, for the time-delay systems is a particular scenario of the projected delay-range-dependent approach. The proposed decoupled observer-based controller synthesis condition is cast into linear matrix inequalities by means of the cone complementary linearization approach. Finally, a numerical simulation example of control of one-link flexible joint robotic arm is provided to verify the proposed design methodology.

Output Feedback H∞ Control for Discrete‐time Mean‐field Stochastic Systems

AbstractThis paper is to consider dynamic output feedback H∞ control of mean‐field type for stochastic discrete‐time systems with state‐ and disturbance‐dependent noise. A stochastic bounded real lemma (SBRL) of mean‐field type is derived. Based on the SBRL, a sufficient condition with the form of coupled nonlinear matrix inequalities is derived for the existence of a stabilizing H∞ controller. Moreover, a numerical example is given to examine the effectiveness of the theoretical results.

Multi-objective and reliable output feedback control for spacecraft hovering

A multi-objective and reliable dynamic output feedback controller design method is proposed in this paper for spacecraft hovering around targets in circular orbits. The hovering process contains two steps namely, orbital transfer and hovering maintenance. Firstly, the dynamical model of spacecraft hovering is built and multi-objective output feedback control problems for the both steps are formulated. For orbital transfer control, we focus on fuel optimization and pole placement. And H2 performance, H∞ performance and pole placement are considered in hovering maintenance control. Then, sufficient conditions for guaranteed cost and pole placement are provided in the forms of nonlinear matrix inequalities. Using some changes of variables and Schur complement, the obtained sufficient conditions can be rewritten as linear matrix inequality constraints. Likewise, sufficient conditions for H2 control, H∞ control and pole placement are presented and transformed into linear matrix inequalities. Thus, the multi-objective controller design problems for both orbital transfer control and hovering maintenance are described as convex optimizations subject to linear matrix inequalities, respectively. Finally, numerical simulations are presented to illustrate the effectiveness and validity of the proposed dynamic output feedback controller design algorithms.

Static output feedback stabilization for systems with time-varying delay based on a matrix transformation method

This paper is concerned with the problem of static output feedback stabilization for linear systems with time-varying delay. A novel controller design is proposed based on a matrix transformation method with a new equation condition, which can solve the controller gain more easily and avoid complicated calculations of the non-linear matrix inequality. Then, the corresponding criteria can be obtained by combining the matrix transformation method and the ideas of non-uniformly dividing delay interval, which can guarantee the asymptotic stability of the closed-loop systems and calculate the satisfactory controller gain. Finally, two numerical examples are provided to verify the effectiveness of the proposed design scheme.

Modelling and observer‐based H ∞ controller design for networked control systems

This study is concerned with modelling and observer-based H∞ controller design for a continuous-time networked control system with network-induced delays and packet dropouts. A new model for an observer-based networked control system is first established by proposing a linear estimation-based delay compensation method. Then some controller design criteria are obtained by constructing an interval time-varying delay decomposition-based Lyapunov functional. A new bounding inequality is introduced to transfer non-linear matrix inequalities into a solvable optimisation problem. A numerical example is given to illustrate the merits and effectiveness of the obtained results.

Robust Nonfragile Stabilizing Controller for an Uncertain Stochastic Nonlinear System With Interval Time-Varying State-Delays: Prescribed H∞ Performance Level

In this technical brief, delay-dependent nonfragile H∞ control problem of a class of stochastic nonlinear systems with interval time-varying state-delays has been considered using Lyapunov–Krasovskii (LK) functional approach. By exploiting a candidate LK functional and using free-weighting matrix technique, a less conservative delay-dependent stabilization criterion is presented for the existence of a nonfragile memoryless state-feedback controller, which ensures stochastic stability as well as a prescribed H∞ performance level of the closed-loop system in the presence admissible parametric uncertainties in the system as well as in the controller gains and exogenous input signal. Since the resulting stabilization criterion is in terms of nonlinear matrix inequalities (NLMIs), it is solved using cone complementarity algorithm (CCA) to obtain a stabilizing controller. A numerical example is presented to illustrate the effectiveness of the proposed result.

시변지연 및 임의 발생 외란이 존재하는 선형 동적 시스템의 신뢰성 제어

In this paper, the problem of reliable control of linear systems with time-varying delays, randomly occurring disturbances, and actuator failures is investigated. It is assumed that actuator failures occur when disturbances affect to the systems. Firstly, by using a suitable Lyapunov-Krasovskii functional and some recent techniques such as Wirtinger-based integral inequality and reciprocally convex approach, stabilization criterion for nominal systems with time-varying delays is derived. Secondly, the proposed method is extended to the reliable H∞ controller design for linear dynamic systems with time-varying delays, randomly occurring disturbances, and actuator failures. Since nonlinear matrix inequalities (NLMIs) are involved in proposed results, the cone complementarity algorithm will be introduced. Finally, two numerical examples are included to show the effectiveness of the proposed criteria.

Quantised and sampled output feedback for nonlinear systems

In this paper we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantised. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimisation problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented in order to illustrate the applicability of the proposed design method.

A feasible direction interior point algorithm for nonlinear semidefinite programming

We present a new algorithm for nonlinear semidefinite programming, based on the iterative solution in the primal and dual variables of Karush-Kuhn-Tucker optimality conditions, which generates a feasible decreasing sequence. At each iteration, two linear systems with the same matrix are solved to compute a feasible descent direction and then an inexact line search is performed in order to determinate the new iterate. Feasible iterates are essential in applications where feasibility is required to compute some of the involved functions. A proof of global convergence to a stationary point is given. Several numerical tests involving nonlinear programming problems with linear or nonlinear matrix inequality constraints are described. We also solve structural topology optimization problems employing a mathematical model based on semidefinite programming. The results suggest efficiency and high robustness of the proposed method.

Input-to-State Stability for Nonlinear Systems With Large Delay Periods Based on Switching Techniques

The paper is concerned with the problem of input-to-state stability (ISS) for nonlinear delay systems with large delay periods (LDP). The concepts of the length rate of LDP and the frequency of LDP are introduced. First the considered system is converted into a switched delay system which may include an unstable subsystem. Then based on a piecewise Lyapunov functional, the ISS properties of the systems are developed under the constraints of the length rate and the frequency of LDP. ISS for a special kind of nonlinear delay systems is also considered and the conditions of nonlinear matrix inequalities (NLMIs) are proposed to guarantee the ISS properties of such a system. A numerical example and an example of a series DC motor are given to show the effectiveness of the proposed methods.

Smooth switching LPV controller design for LPV systems

This paper presents a method to design a smooth switching gain-scheduled linear parameter varying (LPV) controller for LPV systems. The moving region of the gain-scheduling variables is divided into a specified number of local subregions as well as subregions for the smooth controller switching, and one gain-scheduled LPV controller is assigned to each of the local subregions. For each switching subregion, a function interpolating two local LPV controllers associated with its neighborhood subregions is designed to satisfy the constraint of smooth transition of controller system matrices. The smooth switching controller design problem amounts to solving a feasibility problem which involves nonlinear matrix inequalities. To find a solution to the feasibility problem, an iterative descent algorithm which solves a series of convex optimization problems is proposed. The usefulness of the proposed controller design method is demonstrated with a control example of a flexible ball-screw drive system.

A New Finite-Time Bounded Control of Stochastic Itô Systems with (x,u,v)-Dependent Noise: Different Quadratic Function Approach

This paper addresses the finite-time bounded control problem of linear stochastic systems with state, control input, and external disturbance-dependent noise ((x,u,v)-dependent noise for short). The notion of finite-time boundedness of linear stochastic systems is first introduced. Then a different quadratic function approach is proposed to give a sufficient condition for finite-time boundedness of such a class of systems, and its superiority to common quadratic approach is shown. Moreover, the finite-time bounded controller design problem is studied and two sufficient conditions for the existence of state and output feedback controllers are presented in terms of nonlinear matrix inequalities. An algorithm is given for solving the obtained nonlinear matrix inequalities. Finally, an example is employed to illustrate the effectiveness of our obtained results.

Low Conservative Criteria for Robust Consensus of Multiagent Systems with Delays, Disturbances, and Topologies Uncertainties

Considering the limited communications conditions such as delays, disturbances, and topologies uncertainties, the stability criteria for robust consensus of multiagent systems are proposed in this paper. Firstly, by using the idea of state decomposition and space transformation, the condition for guaranteeing consensus is converted into verifying the robust stability of the disagreement system. In order to deal with multiple time-varying delays and switching topologies, jointly quadratic common Lyapunov-Krasovskii (JQCLK) functional is built to analyze the robust stability. Then, the numerical criterion can be obtained through solving the corresponding feasible nonlinear matrix inequality (NLMI); at last, nonlinear minimization is used like solving cone complementarity problem. Therefore, the linear matrix inequality (LMI) criterion is obtained, which can be solved by mathematical toolbox conveniently. In order to relax the conservativeness, free-weighting matrices (FWM) method is employed. Further, the conclusion is extended to the case of strongly connected topologies. Numerical examples and simulation results are given to demonstrate the effectiveness and the benefit on reducing conservativeness of the proposed criteria.

H∞ Filter Design for State Estimation and Unknown Inputs Reconstruction of a Class of Nonlinear systems

We consider a novel method to design a H∞ filter for a class of nonlinear systems subject to unknown inputs. First, we rewrite the system dynamics as a descriptor system. Then, we design a robust H∞ reduced-order filter to estimate both state variables and unknown inputs at the same time. Based on a Lyapunov functional, we derive a sufficient condition for existence of the designed filter which requires solving a nonlinear matrix inequality. The achieved condition is further formulated in terms of a linear matrix inequality (LMI) that is straightforward to solve by popular methods. Finally, the proposed filter is illustrated with an example.

A novel stabilization approach for small signal disturbance of power system with time-varying delay

Small signal instability may cause severe accidents for power system if it can not be dealt correctly and timely. How to maintain power system stable under small signal disturbance is a big challenge for power system operators and dispatchers. Time delay existing in signal transmission process makes the problem more complex. Conventional eigenvalue analysis method neglects time delay influence and can not precisely describe power system dynamic behaviors. In this work, a modified small signal stability model considering time varying delay influence was constructed and a new time delay controller was proposed to stabilize power system under disturbance. By Lyapunov-Krasovskii function, the control law in the form of nonlinear matrix inequality (NLMI) was derived. Considering synthesis method limitation for time delay controller at present, both parameter adjustment method by using linear matrix inequality (LMI) solver and iteration searching method by solving nonlinear minimization problem were suggested to design the controller. Simulation tests were carried out on synchronous-machine infinite-bus power system. Satisfactory test results verify the correctness of the proposed model and the feasibility of the stabilization approach.

Exponential Stabilization of Neutral-Type Neural Networks with Mixed Interval Time-Varying Delays by Intermittent Control: A CCL Approach

For a class of neutral-type neural networks with discrete and distributed delays, the problem of exponential stabilization is studied in this paper by using periodically intermittent control. By introducing an appropriate Lyapunov–Krasovskii functional and using matrix inequality techniques to deal with its derivative, exponential stabilization criteria are presented in the form of nonlinear matrix inequalities. In order to solve the nonlinear problem, a cone complementarity liberalization (CCL) algorithm is proposed. In addition, numerical examples are provided to illustrate the applicability of the proposed approach.