Linear Matrix Inequality Framework Research Articles

Design of a robust model predictive control for nonlinear switched systems: A low‐conservative and recursive feasibility strategy

AbstractThis article proposes a robust H∞ based model predictive control scheme for Lipschitz nonlinear switched systems with persistent dwell time. In the developed method, the constraints corresponding to the recursive feasibility are included and the recursive feasibility is guaranteed. In this scheme, the existence of unstabilizable sub‐systems and exogenous disturbances are included which makes it possible to avoid considering an ideal and simplistic assumption about switched systems. By simultaneously designing the controller and the switching signal and using multiple Lyapunov functions, the unacceptable conservatism in methods involving common Lyapunov functions and arbitrary switching signals is avoided. In the proposed solution, a time‐varying prediction horizon is used for finite horizon cost functions, which in turn will be effective in reducing conservatism. In the suggested design strategy, the non‐convex control scheme is formulated as an optimization problem in the framework of linear matrix inequality. Finally, a numerical example is developed and the performance of the proposed scheme is evaluated.

Secure Communication in Complex Dynamical Networks via Time-Delayed Feedback Control

In this article, a synchronization method of complex dynamical networks (CDNs) with time-varying delay feedback control is proposed to secure communication between the command system and each node of CDNs. To advance the security of the communication between the command system and each node of CDNs, the original information signal transmitted from the command system is encrypted with the techniques of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> -shift cipher and public key. Based on the Lyapunov stability sense and linear matrix inequality (LMI) framework, a new delay-dependent synchronization criterion is established to restore the original information signal on each node of the CDN as well as to ensure stable synchronization for secure communication of all nodes of the command system and CDN. With the support of numerical simulation, the validity of the proposed method is verified.

Hankel norm performance of 2-D digital filters described by Roesser model with overflow arithmetic

ABSTRACT This paper is concerned with the problem of asymptotic stability with Hankel norm performance (HNP) of two-dimensional (2-D) digital filter characterised by Roesser model, where the 2-D system includes overflow nonlinearities and external interference or excitation of finite duration. The underlying nonlinearities cover the usual types of overflow arithmetic utilised in practice such as saturation, two’s complement, zeroing and triangular. To solve this problem, new sufficient criterion is proposed by employing quadratic 2-D Lyapunov function. The proposed criterion guarantees that the addressed system has 2-D HNP bound. This criterion can examine the unwanted memory effect (ME) reduction of 2-D interfered digital filters for past excitations. In the absence of external inputs, the asymptotic convergence of 2-D digital filters is established under the proposed criterion. The developed HNP criterion is in linear matrix inequality framework and, therefore, computationally tractable. In addition, the optimisation problem is formulated to obtain the minimum HNP of the 2-D interfered digital filter. To illustrate the effectiveness of the proposed criterion, a numerical example is provided. The criteria addressed in current paper can give a whole analysis framework for the unwanted MEs of filters.

Practical Stabilization of Switched Affine Systems: Model and Data-Driven Conditions

An important trend about data-driven systems has recently emerged motivated for practical issues, that are the difficulties to identify a model of the plant. In this direction some recent results enhanced the robust control theory to get efficient and relevant solutions. To the best of or knowledge, the existing solutions are rather limited to the linear case, and only few addresses the stabilization of nonlinear systems, which still represents a major challenge. In this work, the objective is the design of switching control law for switched affine systems, ensuring that a neighborhood of a given operating point is stable. After proposing a model-based control design, we propose a systematic method to translate a model-based condition expressed using the framework of Linear Matrix Inequalities (LMI), into a data-driven condition. This is made possible thanks to a new formulation of the existing method from the recent literature on this topic. The method is then evaluated on an academic example.

Membership-Function-Dependent Design of Quantized Fuzzy Sampled-Data Controller for Semi-Markovian Jump Systems With Actuator Faults

This article concerns a fuzzy sampled-data controller for nonlinear semi-Markovian jump systems (SMJS) via the fuzzy dependent stochastic looped-Lyapunov functional (FSLF) approach. To do this, with the help of the Takagi–Sugeno (T–S) method, the stochastic jump parameters, actuator faults, and logarithmic quantizer are all considered in a unified framework. The stochastic parameters are generated by the semi-Markov process, whose transition rates depend upon the sojourn time. The nonlinear SMJS depicts real-time models subject to unpredictable structural changes, such as single-link robotic arm (S-LRA) systems. A key issue under consideration is how to design a fuzzy sampled-data controller to ensure stochastic stability for the nonlinear SMJS in the presence of actuator faults and state quantization. For this purpose, by using the weak infinitesimal operator of constructed FSLF, sufficient conditions are determined to realize the stochastic stability of the proposed nonlinear systems. Then, the stochastic stability conditions for the proposed sampled-data controller scheme can be derived under a linear matrix inequality framework. The validity of the theoretical findings is verified by Rossler’s chaotic systems. The S-LRA model is demonstrated to illustrate the applicability and efficacy of the proposed controller scheme.

Linear Quadratic Differential Games based MIMO-PID design for Load Frequency Control of a 2-area interconnected power system: An Iterative LMI approach

In this paper a Multi-input-Multi-output (MIMO)-Proportional-Integral-Derivative (PID) control design method via sub-optimal solution of Linear Quadratic Differential Games (LQDG) for Load frequency Control (LFC) of a 2-area interconnected power system is proposed. The main motivation behind the proposed centralised LFC design scheme is to solve multi-objective optimization problem that can accomplish improvement in transient response & disturbance rejection simultaneously. The proposed iterative optimization algorithm in Linear matrix Inequality (LMI) framework yields appropriate sub-optimal PID gains. A 2-area LFC problem is solved using the proposed design and comparative results suggests that the proposed design can significantly attenuate the load disturbances as well as provide improved transient response specifications.

Proportional integral observer-based input–output finite-time stabilization for chaotic semi-Markov jump fuzzy systems

The goal of this article is to discuss the input–output finite-time (IO-FT) stabilization and state estimation problem for T-S fuzzy chaotic semi-Markov jump systems (TSFCS-MJSs) with uncertain transition rates, immeasurable states, quantization errors, gain fluctuations and external input signals by using quantized resilient proportional integral (PI) observer-based control. Precisely, to estimate the immeasurable states of the underlying TSFCS-MJSs, a mode-dependent fuzzy rule-based PI observer is implemented, wherein the inclusion of both proportional and integral loops provides more design freedom and finer steady-state accuracy. Additionally, the gain fluctuations that occur during the implementation of the addressed PI observer are taken into account and therefore, we focus on the design of resilient PI observer that provides satisfactory estimation results. Besides, by reason of the restricted capacity of the data transmission medium, the input signals are quantized prior to transmission. Following that, sufficient criteria are attained in the framework of linear matrix inequalities with the use of a mode-dependent Lyapunov function candidate can guarantee the resulting closed-loop system’s IO-FT stability. Thereafter, by solving the established linear matrix inequality-based criteria, the desired gain matrices of the controller and observer are computed. Finally, the simulation outcomes for standard Chua’s circuit system are displayed in accordance with the theoretical findings.

A New Method of Fault Estimation Observer Design for T-S Fuzzy Singular Systems With Disturbances Using Dissipativity Theory

This brief dedicates to addressing the robust fault estimation (FE) problem for a class of T-S fuzzy singular systems (TSFSSs) subject to faults and disturbances. A dissipativity-based FE observer design scheme is proposed to realize fault reconstruction, by which the FE performance can be improved in contrast to the existing one. With the help of the Lyapunov approach, dissipativity theory and matrix transformation technique, a brand-new condition is established to ensure observation error dynamics to be strictly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\mathscr {Q}, \mathscr {U}, \mathscr {R})-\iota -$ </tex-math></inline-formula> dissipative. It is worth mentioning that different from some previous studies, our method can solve observer gains simultaneously in a strict linear matrix inequality (LMI) framework rather than determine some of them artificially. Moreover, to bring extra degrees of freedom, slack scalars and matrices are added in FE observer design. The superiority of our method is fully illustrated via simulations.

Control-Oriented Physics-Based Modeling and Observer Design for State-of-Charge Estimation of Lithium-Ion Cells for High Current Applications

This article proposes a physics-based control-oriented model and observer design for the state-of-charge (SoC) estimation of lithium-ion cells for applications involving high magnitude fluctuating current profiles. The physics-based single-particle model (SPM) provides enhanced accuracy due to the inclusion of electrolyte dynamics and addresses the issue of nonobservability associated with it. The computationally efficient physics-based model is utilized to design a robust observer-based SoC estimator in the framework of linear matrix inequality to guarantee fast convergence despite parametric uncertainty in the state and output equations, and unknown initial conditions. The observer performance is validated using FTP75 and US06 dynamic tests at different temperatures, and the results are compared with the standard unscented Kalman filter (UKF). The mean SoC estimation error and the integral square error of the estimated SoC for the proposed observer are at least one order of magnitude smaller than that of UKF. Furthermore, robustness to ±30% parametric uncertainty, measurement noise, and unknown initial conditions is demonstrated through Monte Carlo simulations at different temperatures.

Reachable Set Estimation for T–S Fuzzy Markov Jump Systems With Time-Varying Delays via Membership Function Dependent H∞ Performance

This article considers the reachable set estimation problem and membership function dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance analysis for a class of fuzzy Markov jump systems (FMJSs) with mode-dependent time-varying delays and bounded external disturbances via sampled-data control. First, mode-dependent sampled-data control for the FMJS is designed using the Takagi–Sugeno (T–S) fuzzy method. Then, a novel stochastic Lyapunov–Krasovskii functional (LKF) is constructed in mode-dependent augmented form by taking full advantage of the variable characteristics related to the actual sampling pattern. At the same time, a membership function dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance index is introduced for the first time to attenuate the impact of disturbances on the closed-loop FMJS. Based on the novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance index and LKF, new delay-dependent conditions are derived in the framework of linear matrix inequalities to ensure stochastic stability of the closed-loop system and its reachable set is bounded by an ellipsoid in the presence of bounded disturbances. Finally, two illustrated application problems validate theoretical results with less conservatism in the sense of enlarging the sampling period and minimizing the disturbance attenuation level.

Subpredictor-Based Fuzzy Control Design for a Reaction–Diffusion Equation

This article addresses the stabilization of the nonlinear reaction–diffusion equation with an unknown but bounded delay. Based on Takagi–Sugeno (T–S) fuzzy rules, a T–S fuzzy partial differential equation (PDE) model is constructed to describe the nonlinear distributed parameter system. More importantly, we contribute to the large delay compensation and the future state prediction by introducing a chain of subpredictors for the T–S fuzzy PDE model, and the convergence of observation errors is guaranteed as well. Different from the conventional approaches, the proposed observer can compensate a larger delay. To stabilize the system, for the first time, a new subpredictor-based T–S fuzzy control law is suggested by using the proposed observer, which is composed of elementary observers connected in series. The implementation of the subpredictor feedback is simpler than the classical methods that only one of the elementary observers is employed. Via Lyapunov–Krasovskii approach, sufficient exponential stability conditions are established in the framework of linear matrix inequalities. A numerical example is provided to illustrate the effectiveness of theoretical results.

Stabilization Criteria for T–S Fuzzy Systems With Multiplicative Sampled-Data Control Gain Uncertainties

This study concerns the stabilization criteria of Takagi–Sugeno (T–S) fuzzy systems under the memory-based sampled-data control. Contrasted to the traditional sampled-data control strategy, a more general control scheme that admits the signal transmission delay and the multiplicative control gain uncertainties is designed. In order to indicate the perturbations of control gain, the Bernoulli sequence is applied to the designed control scheme. At the same time, the membership function dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance index is introduced to attenuate the disturbances of continuous-time T–S fuzzy systems for the first time. By utilizing the novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance index together with the appropriate Lyapunov–Krasovskii functional, including the information of sampling period and signal transmission delay, the sufficient conditions are derived in the framework of linear matrix inequalities. These conditions ensure the stabilization of concerned T–S fuzzy systems and enlarging the maximum sampling period. At last, the advantages of the derived results are shown by the numerical examples.

Gain-Scheduled Steering and Braking Coordinated Control in Path Tracking of Intelligent Heavy Vehicles

Abstract In the context of intelligent transportation system and advanced vehicle control system, higher requirements for intelligent control and coordinated control of heavy vehicles (HVs) have been proposed. Automatic path tracking control is essential for automatic driving of heavy vehicles. However, uncertain lateral disturbances and time-varying characteristics of system parameters make it difficult to guarantee roll stability and path tracking accuracy during automatic path tracking. This paper investigates the coordination of active front steering (AFS) and direct yaw moment control (DYC) in the automatic path tracking system for intelligent heavy vehicles. The main idea is to adjust the braking action according to the rollover risk evaluated by a specific rollover index (RI). The coordination of steering and braking systems and the robustness of the system against time-varying parameters are achieved by a gain-scheduled linear parameter varying (LPV) controller. Based on the linear matrix inequality (LMI) framework, the LPV controller is synthesized to ensure the robust H∞ performance against external disturbances. Simulation results show that the proposed gain-scheduled LPV/H∞ control strategy can enhance roll stability and path tracking accuracy in the path tracking process of intelligent heavy vehicles.

Event‐triggered stabilization for large‐scale interconnected systems with time delays

AbstractThis article addresses the event‐triggered stabilization for large‐scale interconnected systems in presence of time delays. Two types of event‐triggering mechanisms (static event‐triggering mechanism and dynamic event‐triggering mechanism) are constructed. For each case, event‐triggered predictor‐based controller is designed to compensate time delays by using backstepping method. Via Lyapunov approach, constructive exponential stability conditions are established in the framework of linear matrix inequalities (LMIs). A numerical simulation example is given to demonstrate the effectiveness of the proposed methods.

Fault estimation in discrete-time linear systems with mixed uncertainties using proportional integral observer

In this article, the problem of fault estimation is addressed for the class of discrete-time linear systems subject to mixed uncertainties (norm bounded and stochastic uncertainty) and unknown input disturbances. A robust proportional integral (PI) observer is proposed based on H∞ norm minimization for dealing with uncertainties and disturbances. An augmented observer is constructed first where the sensor fault is treated as a state variable and then, H∞ norm is minimized in the linear matrix inequalities (LMI) framework with the application of the Lyapunov function and Young inequalities. The application of the DC motor system and well-known three tank system is given to demonstrate the performance of the proposed observer. Through simulation results, a proposed PI observer has shown the precise estimation of a sensor fault.

Finite-time reliable nonfragile control for fractionalorder nonlinear systems with asymmetrical saturation and structured uncertainties

This paper investigates the finite-time stabilization problem of fractional-order nonlinear differential systems via an asymmetrically saturated reliable control in the sense of Caputo’s fractional derivative. In particular, an asymmetrical saturation control problem is converted to a symmetrical saturation control problem by using a linear matrix inequality framework criterion to achieve the essential results. Specifically, in this paper, we obtain two sets of sufficient conditions under different scenarios of structured uncertainty, namely, norm-bounded parametric uncertainty and linear fractional transformation uncertainty. The uncertainty considered in this paper is a combination of polytopic form and structured form. With the help of control theories of fractional-order system and linear matrix inequality technique, some sufficient criteria to ensure reliable finite-time stability of fractional-order differential systems by using the indirect Lyapunov approach are derived. As a final point, the derived criteria are numerically validated by means of examples based on financial fractional-order differential system and permanent magnet synchronous motor chaotic fractional-order differential system.

High-gain interval observer for continuous–discrete-time systems using an LMI design approach

In this paper, a high-gain interval observer (HGIO) for a class of partially linear continuous-time systems with sampled measured outputs in the presence of bounded noise and additive disturbances is proposed. The design of the HGIO is formulated in the Linear Matrix Inequality (LMI) framework. The gain of the HGIO is designed to satisfy the cooperative property using a time-varying change of coordinates based on pole placement in separate LMI regions. Moreover, a procedure for designing the HGIO gain to minimise the effect of the noise and disturbance in the estimation is provided. The stability of the proposed HGIO is also guaranteed. The proposed approach is assessed in simulation using a numerical example.

H∞-based control of multi-agent systems: Time-delayed signals, unknown leader states and switching graph topologies

The paper investigates a leader-following scheme for nonlinear multi-agent systems (MASs). The network of agents involves time-delay, unknown leader's states, external perturbations, and switching graph topologies. Two distributed protocols including a consensus protocol and an observer are utilized to reconstruct the unavailable states of the leader in a network of agents. The H∞-based stability conditions for estimation and consensus problems are obtained in the framework of linear-matrix inequalities (LMIs) and the Lyapunov-Krasovskii approach. It is ensured that each agent achieves the leader-following agreement asymptotically. Moreover, the robustness of the control policy concerning a gain perturbation is guaranteed. Simulation results are performed to assess the suggested schemes. It is shown that the suggested approach gives a remarkable accuracy in the consensus problem and leader's states estimation in the presence of time-varying gain perturbations, time-delay, switching topology and disturbances. The H∞ and LMIs conditions are well satisfied and the error trajectories are well converged to the origin.

Robust stabilization of power systems subject to a series of lightning strokes modeled by Markov jumps: Attracting ellipsoids approach

Power systems are subject to stochastic faults and process random noise. The faults (due to e.g. lightning strokes) can be temporary or permanent. When a lightning stroke hits a transmission line, the circuit breakers open to clear the fault and reclose. The resulting parameters changes are modelled as a discrete-time Markov jumps in this paper using the practical statistical failure rates. System parameters are also subject to random noise e.g. line reactance’s depend on the conductors spacing which in turn depends on stochastic wind speeds. This paper presents a novel power system excitation control robust against such stochastic uncertainties. The design is based on a derived sufficient condition in the framework of linear matrix inequalities (LMI), and the attracting ellipsoid approach. The effectiveness of the proposed control is tested on a multi-machine power system.

Consensus in discrete-time one-sided Lipschitz nonlinear multi-agent systems with time-varying communication delay

This paper discusses consensus in discrete-time nonlinear multi-agent systems with time-varying delay in a directed communication topology. Since each agent can obtain information only from its neighbouring agents, for the purpose of analysis, a consensus protocol is proposed that considers the relative position information between the leader and followers and between neighbouring followers under time-varying communication delay. This nonlinear problem is solved by employing nonlinear behaviour based on the one-sided Lipschitz method. By constructing an appropriate Lyapunov-Krasovskii function, the consensus criterion for the leader-following problem is established in terms of the linear matrix inequality (LMI) framework. Furthermore, the solution of gain matrix is addressed by utilizing the cone complementarity linearization (CCL) algorithm. The results of a numerical simulation indicate that this method can be used to effectively solve this problem.