Solution Of Linear Matrix Inequalities Research Articles

Tensor Product Alternatives for Nonlinear Field-Oriented Control of Induction Machines

The paper presents a nonlinear field-oriented control technique based on the tensor product representation of the nonlinear induction machine model and the solvability of linear matrix inequalities. The nonlinear model has 32 quasi linear parameter-varying equivalent variants, and it is shown that only half of the models result in feasible controller. Two control goals are realized: torque control and speed control. The controller is a nonlinear state feedback controller completed by integral action. A new block diagram is investigated for speed control. The controller gains are designed by the solution of linear matrix inequalities to solve the Lyapunov inequality to obtain a stable and fast response and constraints on the control signal. The presented methods are verified and compared by simulations.

Memory sampled-data control design for attitude stabilization of uncertain spacecraft with randomly missing measurements

This paper focuses on stabilizing the attitude of the rigid spacecraft under the sampled-data control design technique with constant communication delay. The system dynamics with model uncertainty (perturbation) and missing measurements are considered. The aim is to design a desirable attitude control that ensures the stabilization of the rigid spacecraft with an optimal H∞ performance level. A novel Lyapunov–Krasovskii functional is developed, incorporating looped characteristics, for the proposed study on the rigid spacecraft model by developing relevant new terms. Making use of the Lyapunov function as well as free-matrix-based integral inequality, sufficient stability criteria are established to assure the stability of the rigid spacecraft model. Furthermore, the desired sampled-data control gain matrices are acquired through the solution of linear matrix inequalities, which guarantees the asymptotic stability of the rigid spacecraft model. Finally, the numerical simulation demonstrates the efficacy of the theoretical study proposed for the rigid spacecraft model.

Fuzzy control of singular fractional order multi-agent systems with actuator saturation

In this paper, a T-S fuzzy model is established to describe nonlinear fuzzy singular fractional order multi-agent systems (SFOMASs) exhibiting actuator saturation when the order is between 0 and 2. We focus on the case where the system state is difficult to measure or cannot be directly measured, so it needs to be estimated based on other measurable signals. To this end, a fuzzy observer is designed, which can process fuzzy or imprecise input and output signals through fuzzy processing. Then the observer-based leader-following consensus problem of fuzzy SFOMASs with actuator saturation is addressed. To address input saturation, we use some mathematical tools to represent the saturated linear feedback on the convex hull, which is then transformed into seeking the optimal solution of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed approach is verified through the adoption of a numerical example.

Robust formation control of uncertain linear multi‐agent systems in presence of time‐delay using output information

AbstractA major challenge in multi‐agent formation is the issue of delay and uncertainty. This article investigates a robust formation control problem for linear multi‐agent systems with input delay and model uncertainty. The strategy adopted by an agent in the system aims to predict the delayed state using output information from neighboring agents over a fixed communication network and generate the control input from the predictor output. The predictor is employed using the finite spectrum assignment (FSA) technique. The overall strategy leads to a unified framework representation, and the multi‐agent formation control problem is simplified to a closed‐loop stability problem for an agent. The overall problem comes down to deciding two feedback gains: predictor gain and controller gain. For a perfect plant model, any choice of stabilizing gains can achieve formation. But when agent models are uncertain, the gains need to be derived from the solutions of linear matrix inequalities (LMIs) containing network information and delay. LMIs are obtained from the bounded real lemma and the gains obtained from the solution guarantee to maximize the norm bound of allowable perturbation, modeled as additive uncertainty, for a known delay size. Further, since the FSA technique is sensitive to discretization, a digital implementation of the overall scheme is elaborated, which may help to debug stability issues in the implementation process. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed ideas.

Sliding Mode Control Against False Data Injection Attacks in DC Microgrid Systems

In this letter, the problem of false data injection (FDI) attacks is investigated for dc microgrid systems, which involves Markovian jumping parameters and Lévy noises. Moreover, an adaptive sliding mode controller is constructed to assure the stability of such dc microgrid systems. First, the dynamical equations of dc microgrid systems are proposed, which consider Lévy noises and Markovian jumping parameters. Meanwhile, the impact of FDI attacks is taken into account in dc microgrid systems. Next, the sliding mode motion dynamic of dc microgrid systems can be attained by designing a sliding mode surface (SMS) in the form of integral. Several sufficient conditions are proposed to guarantee the asymptotic stability for dc microgrid systems. Then, a novel sliding mode control (SMC) law is put forward to drive systems to reach and maintain a given sliding mode surface. In addition, the solution of linear matrix inequalities can determine undetermined parameters in the SMC and SMS. Ultimately, a practical example is demonstrated to prove that the proposed adaptive sliding mode controller can ensure the stability of the dc microgrid system under FDI attacks.

LMI-based Control of Singularly Perturbed Switched Affine Systems

In this paper, we tackle the problem of stabilizing singularly perturbed switched affine systems. Even though efficient control strategies based on the solution of Linear Matrix Inequalities (LMIs) have been presented in the switched affine systems literature, the presence of the small singular perturbation parameter may introduce numerical difficulties in the solution of these LMIs due to ill conditioning. Moreover, this parameter may be uncertain or its use in the control law may not be practical. Here, we propose an LMI-based control design strategy that avoids the conditioning issues and that also provides an estimation of the domain of attraction. Furthermore, the resulting controller does not depend on the singular perturbation parameter. The proposed method is illustrated on a numerical example.

State estimation of complex-valued neural networks with leakage delay: A dynamic event-triggered approach

In this paper, the problem of state estimation is investigated for a class of discrete-time complex-valued neural networks (CVNNs) with both leakage delay and discrete time-varying delays. The signal transmission from output sensors to state estimator is implemented via a shared wireless network with limited communication resources. For the aim of reducing the consumption of limited communication resources, the transmission strategy based on dynamic event-triggering is introduced to determine when the updating of the output measurement should be carried out. By taking use of some properties of Hermitian matrix and constructing an appropriate Lyapunov–Krasovskii functional, a sufficient criterion is derived for ensuring the asymptotical stability of the estimation error system without separating the CVNN to its real-part system and imagination one is derived, which is quite different from those approach used in exiting literature. The gain matrix for estimator is designed by resorting to a set of feasible solutions of linear matrix inequalities (LMIs) with complex-valued variables. A numerical example and its simulation results are given to illustrate the validity of the theoretical result.

Overcoming non‐convexity of the robust nonlinear output feedback problem: An adaptive linear matrix inequality solution

AbstractThis article presents a novel solution for adaptive output feedback of uncertain nonlinear systems; it uses nonlinear controller and observer structures to drive a time‐varying function as a Lyapunov one, thus guaranteeing asymptotic stability of the system states and observation error. Exact convex modeling of available terms and polytopic embedding of unavailable ones in the Lyapunov function dynamics, allow linear matrix inequalities to be used, thus overcoming former difficulties over the non‐convex nature of the problem. Moreover, the proposal incorporates a recently appeared factorization to get rid of approximate solutions based on Lipschitz bounds and persistence of excitation. The methodology is successfully put at test against related works in a variety of examples.

Nonfragile Sampled-Data Control of T–S Fuzzy Systems With Time Delay

To reduce the influence of controller uncertainties and lower the computational burden of the system, a novel nonfragile sampled-data control scheme is proposed in this article for the Tagaki–Sugeno fuzzy system with time delay. First, an improved Lyapunov–Krasovskii functional is established with adjusted looped functional, which contains not only the current states, but the delayed states. Then, utilizing the Wirtinger inequality and improved single integral inequality, the new stability criteria are proposed with less conservativeness. By calculating the solution of linear matrix inequalities, the nonfragile controller parameters can be acquired. Furthermore, the sampled-data fuzzy controller leads to a good stable performance for the system, even under the possible gain fluctuations. Finally, three examples are given to verify the effectiveness and superiority of the proposed control scheme.

Control of Sector-Bound Systems with the Guarantee Output Signal in a Given Set

In this paper, we propose a new method for synthesizing the control of plants with sector-bound nonlinearity with a guarantee of finding the controlled signal in given set at any time under conditions of unknown bounded disturbances. The basis of the method consists of two stages. At the first stage, the coordinate transformation is used to reduce the original constrained problem to the problem of studying the input-to-state stability of a new extended system without constraints. Thus, any known control methods can now be applied to stabilize the system in new coordinates. At the same time, to achieve the goal, it is not required to reduce the value of the control error. It is enough to show its boundedness. At the second stage, a control law is synthesized for the extended system, where the adjustable parameter is selected from the solution of linear matrix inequalities. To illustrate the effectiveness of the proposed method, simulation in the MATLAB Simulink is given. The simulation results show the presence of controlled signals in the given set and the boundness of all signals in the control system. It is shown that an increase the value of the gains in the control law improves the quality of disturbance attenuation that is consistent with theoretical results.

Fault diagnosis and accommodation for multi-actuator faults of a fixed-wing unmanned aerial vehicle

This study proposes a system design scheme of fault detection, isolation, and accommodation algorithms for the actuator fault of a fixed-wing unmanned aerial vehicle (FW-UAV) by considering system nonlinearity, external disturbance, and multi-actuator faults. The fault diagnosis scheme consists of a comprehensive observer and a bank of fault isolation estimators designed based on the actuator of the FW-UAV. In the diagnosis module, the system model is transformed into two systems by introducing a transformation matrix, so that both the actuator faults and the system disturbance are separated. Then, the concept of equivalent output injection is applied to construct a comprehensive observer to detect the actuator fault and estimate the unknown system disturbance. In the isolation estimation module, a set of sliding mode observers (SMOs) is constructed to isolate the multi-actuator faults, which reveals the fault source precisely. The stability analysis of the proposed SMOs was derived from the solution of linear matrix inequalities. In the event of a fault, the fault estimation provided by the fault diagnosis solution is used to accommodate the fault effects, while maintaining good attitude control and position tracking performance of the FW-UAV. The effectiveness of the proposed scheme is verified by the simulation model of de Havilland DHC-2 ‘Beaver’ aircraft in different fault cases.

H ∞ Consensus Design and Online Scheduling for Multiagent Systems with Switching Topologies via Deep Reinforcement Learning

This paper is devoted to H ∞ consensus design and online scheduling for homogeneous multiagent systems (MASs) with switching topologies via deep reinforcement learning. The model of homogeneous MASs with switching topologies is established based on switched systems theory, in which the switching of topologies is viewed as the switching among subsystems. By employing linear transformation, the closed-loop systems of MASs are converted into reduced-order systems. The problem of H ∞ consensus design can be transformed to the issue of H ∞ control. It is supposed that the consensus protocol is composed of two parts: dynamics-based protocol and learning-based protocol, where dynamics-based protocol is provided to guarantee the convergence and weighted attenuation and learning-based protocol is proposed to improve the transient performance. Then, the multiple Lyapunov function (MLF) method and mode-dependent average dwell time (MDADT) method are combined to ensure the stability and weighted H ∞ disturbance attenuation index of reduced-order systems. The sufficient existing conditions of dynamics-based protocol are given through the feasible solutions of linear matrix inequalities (LMIs). Moreover, the online scheduling is formulated as a Markov decision process, and the deep deterministic policy gradient (DDPG) algorithm in the framework of actor-critic is proposed for the compensation of disturbance to explore optimal control policy. The online scheduling of parameters of MASs is viewed as bounded compensation of dynamics-based protocol, whose stability can be guaranteed by nonfragile control theory. Finally, simulation results are provided to illustrate the effectiveness and superiority of the proposed method.

Event-triggered mechanism based robust fault-tolerant control for networked wind energy conversion system

In this paper, a novel robust fault-tolerant control scheme based on event-triggered communication mechanism for a variable-speed wind energy conversion system (WECS) with sensor and actuator failures is proposed. The nonlinear WECS with event-triggered mechanism is modeled based on the Takagi–Sugeno (T–S) fuzzy model. By Lyapunov stability theory, the parameter expression of the proposed robust fault-tolerant controller with event-triggered mechanisms is proposed based on a feasible solution of linear matrix inequalities. Compared with the existing WECS fault-tolerant control methods, the proposed scheme significantly reduces the pressure of network packet transmission and improves the robustness and reliability of the WECS. Considering a doubly-fed variable speed constant frequency wind turbine, the event- triggered mechanism based fault-tolerant control for WECS is analyzed considering system model uncertainty. Numerical simulation results demonstrate that the proposed scheme is feasible and effective.

Basal power reconstruction during cycling using a robust discrete-time PI observer

The basal metabolic rate characterizes the energy consumption of the human body at rest. It can be estimated by using respiratory gas exchange analyzers and indirect calorimetry. During cycling this value could vary due to several adaptation processes such as an increase in breathing and blood circulation, controlling body temperature, among others. In this paper we propose to reconstruct the instantaneous value of this basal metabolic rate, referred here as basal power (measured in Watts), by using a robust discrete-time proportional integral (PI) observer. The observer design is based on a solution of linear matrix inequalities and uses an uncertain linear parameter varying model of the gas exchange dynamics. The proposed methodology allows the reconstruction of the basal power while making the respiratory gas exchange estimation robust to bounded uncertainties and disturbances. The PI observer has been validated in simulation.

Sampled-data in space nonlinear control of scalar semilinear parabolic and hyperbolic systems

The paper describes a novel method of sampled-data in space (spatial variable) nonlinear control of scalar semilinear parabolic and hyperbolic systems with unknown parameters, distributed disturbances and finite number of measurements along the spatial variable. Differently from recent results based on piecewise constant control laws, the proposed one is used piecewise nonlinear functions choosing by designer for providing some properties in the closed-loop system. In particular, we propose several types of functions providing reduced control. The gain design in the control law is found as a solution of linear matrix inequalities with minimum ultimate bound guarantee. The simulations confirm theoretical results and show the efficiency of the proposed control scheme compared with some existing ones.

Distributed event‐triggered cooperative H∞ load frequency control for interconnected networked power systems

Interconnected communication time-varying delays between subsystems for large scale distributed networked power systems (DNPSs) are studied in this work. Distributed event-triggered mechanisms-based interconnected DNPSs and their cooperative H ∞ stability control strategy are proposed for load frequency control in multi-area interconnected power systems. Considering the distributed interconnected DNPSs, the subsystems have their own state time-delay, and there are time-varying delays and package dropout in subsystems interconnection; in order to reduce the amount of transmitted signals and improve the efficiency of the network communication resource, a distributed event-triggered mechanisms-based system model and a novel design method for networked cooperative H ∞ control strategy are proposed. Based on the Lyapunov stability analysis method, a suitable Lyapunov Krasovskii function is constructed, and the distributed controller design scheme is derived based on the feasible solution of linear matrix inequality (LMI). The sufficient conditions for the system asymptotical stability is given. Finally, a four-area networked interconnected hybrid power system is considered and the system performance under networked load fluctuations is analysed; the simulation results show that the proposed scheme is feasible and effective.

Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.

Backstepping Based Nonlinear Sensorless Control of Induction Motor System

This work proposes a sensorless control strategy for the induction motor (IM) using a Backstepping control and a nonlinear observer based on the circle-criterion approach. The Backstepping is a powerful control strategy that deals with nonlinear higher-order systems and includes non-measurable parameters related to the (IM). The nonlinear observer approach is intended to determine these important parameters. The circle-criterion approach is employed to determine the observer gain matrices as a solution of LMI (linear matrix inequalities) that guarantee the stability conditions of the designed observer. The main objective of this method is to solve the problem of the nonlinearities of the system which ensure the global asymptotic convergence of the observed dynamics and to improve the performance of the induction motors. The efficiency and correctness of the proposed scheme are proven by several numerical simulations.

Reference tracking and transient response shaping of uncertain quasi one-sided Lipschitz nonlinear systems

This work addresses the reference tracking problem for uncertain systems with quasi one-sided Lipschitz nonlinearity. The uncertainty is assumed to be of a norm bound parametric type. Moreover, transient response shaping using the concept of ‘return time’ is also proposed. The controller design relies on the solution of Linear Matrix Inequalities (LMIs) and hence is computationally efficient. The proposed control law is linear in states, and thus the implementation is often straightforward. To illustrate the capability and simplicity of the proposed theory, three design examples are included.

Input‐to‐state learning of recurrent neural networks with delay and disturbance

SummaryThis article deals with the issue of input‐to‐state stabilization for recurrent neural networks with delay and external disturbance. The goal is to design a suitable weight‐learning law to make the considered network input‐to‐state stable with a predefined ‐gain. Based on the solution of linear matrix inequalities, two schemes for the desired learning law are presented via using decay‐rate‐dependent and decay‐rate‐independent Lyapunov functionals, respectively. It is shown that, in the absence of external disturbance, the proposed learning law also guarantees the exponential stability of the network. To illustrate the applicability of the present weight‐learning law, two numerical examples with simulations are given.