Nonlinear Matrix Inequalities Research Articles

Observer-based control for time-delayed quasi-one-sided Lipschitz nonlinear systems under input saturation

This paper addresses the observer-based controller design problem for nonlinear time-delayed systems under input saturation. The nonlinearities are supposed to satisfy the quasi-one-sided Lipschitz condition, which is less conservative than the one-sided Lipschitz condition. Based on the nonlinear matrix inequalities, control law for nonlinear systems subject to input saturation, time delays, and unavailable states, some sufficient conditions have been developed for an augmented system containing the system state vector and the error vector to ensure the convergence of all states to zero. The paper used a decoupling approach to reduce the complexity of the corresponding observer and controller gain computations. Finally, the effectiveness of the developed results is validated using suitable examples.

A dual scale spatio‐temporal control for complex distributed parameter systems

AbstractIt is very difficult to control the distributed parameter system (DPS) for consistent spatial performance. In this paper, a dual scale spatio‐temporal control is designed for the DPS to achieve a consistent performance across the entire workspace under unknown exogenous disturbances. First, the generalized “spatial observer” should be designed under spectral decomposition/synthesis, to act as the nominal model of the DPS, through which the spatial mismatch between the model and the process could be estimated. Then a distributed disturbance compensator can be constructed to suppress all the undesirable spatial disturbance through the inner loop, and the compensated system will become closer to the nominal model and easier to control. Third, a dual controller will be designed in the outer loop for the final consistent spatial performance. Since the spatial performance is mainly affected by the dominant dynamics on the slow scale, a convex optimization algorithm can be effectively established in terms of nonlinear matrix inequalities for the dual controller. In this way, the nonlinear optimization problem is solved by converting it into the problem of a linear matrix inequalities with constraints. Besides, the spatio‐temporal state variable of the controlled plant is demonstrated to converge in Hilbert space. The feasibility and effectiveness of the proposed control method are verified by temperature control of a catalytic rod, a benchmark in chemical reactors.

Synchronization of Intermittently Coupled Neural Networks With Coupling Delay.

In recent years, the synchronization of coupled neural networks (CNNs) has been extensively studied. However, existing results heavily rely on assuming continuous couplings, overlooking the prevalence of intermittent couplings in reality. In this article, we address for the first time the synchronization challenge posed by intermittently CNNs (ICNNs) with coupling delay. To overcome the difficulties arising from intermittent couplings, we put forward a general piecewise delay differential inequality to characterize the dynamics during both coupled intervals and decoupled intervals. Based on the proposed inequality, we establish delay-independent synchronization criteria (DISCs) for ICNNs, enabling them to tackle general coupling delay. Notably, unlike previous studies, the achievement of synchronization in our approach does not rely on external control. Furthermore, for ICNNs that synchronize only under small delays, we formulate non-linear matrix inequality (LMI)-based delay-dependent synchronization criteria (DDSCs) that are computationally efficient and do not require delay differentiability. Finally, we provide illustrative examples to demonstrate our theoretical results.

Joint non-fragile automatic generation control and multi-event driven mechanism co-design for wind integrated power system under denial of services attacks

Denial of services (DoS) attacks exist in wind integrated power system. DoS attacks can cause network-induced delay and packages loss in information transmission. Meanwhile, considering the parameter perturbation of controller and system model uncertainty in wind integrated power system, these may cause the system dynamic performances degradation or even instability. Based on the above considerations, the joint non-fragile automatic generation robust control of wind integrated power system under DoS attacks is studied in this paper. In order to ensure the expected system performance and more effectively utilize the limited network communication resources under DoS attacks, a novel dynamic multi-event driven mechanism based joint non-fragile H∞ automatic generation control method is proposed. By constructing a suitable Lyapunov-Krasovskii functional and utilizing the Shur complement lemma to handle nonlinear matrix inequality, the sufficient conditions are derived to guarantee the asymptotic stability of wind integrated power system under DoS attacks. Furthermore, the performance of the proposed non-fragile regulator is demonstrated through a four-area wind integrated power system to show the feasibility and applicability. The analysis result indicates that the proposed scheme provides stronger robustness, higher wind energy utilization efficiency and more efficient communication mechanism.

Co-Design of Adaptive Event-Triggered Mechanism and Asynchronous H∞ Control for 2-D Markov Jump Systems via Genetic Algorithm.

This article concerns the co-design scheme of the adaptive event-triggered mechanism (AETM) and asynchronous H∞ control for two-dimensional (2-D) Markov jump systems. First, we introduce a hidden Markov model with the observation that the asynchronous phenomenon is inevitable between the plant mode and the controller mode. Besides, for economizing the communication times, an innovative 2-D AETM is constructed, which can dynamically regulate the event-triggered thresholds to strive for better system performance. Then, by utilizing the 2-D Lyapunov stability theory, nonlinear matrix inequalities are built to ensure the asymptotic mean-square stability with an H∞ performance for the closed-loop 2-D system. To avoid introducing any conservatism when handling the above nonlinear matrix inequalities, a binary-based genetic algorithm (BGA) is exploited to treat some variables as known, such that derive some directly solvable linear matrix inequalities. Finally, a simulation example is provided to verify the effectiveness of the proposed 2-D AETM-based asynchronous controller strategy with a BGA.

Dissipativity-based conditions for the feedback stabilization of systems with time-varying input delays

This note is concerned with presenting new dissipativity-based conditions for the stabilizability of discrete-time systems with time-varying delays by linear static output feedback (SOF). We demonstrate that the feasibility of nonlinear matrix inequalities for the design of feedback stabilizing gains derived from the literature is equivalent to the feasibility of a linear matrix inequality establishing dissipativity of the system plus one matrix inequality constraint. An iterative strategy allowing the computation of stabilizing SOF gains is then developed based on a new relaxed sufficient condition. Compared with other popular approaches in the literature, such as the celebrated cone complementarity linearization (CCL) method, the strategy avoids providing initial “guesses” for the matrix variables, which is especially complicated when dealing with a large number of matrices. Due to being a particular case of SOF with an identity output matrix, static state feedback (SSF) gains can also trivially be computed by exploiting the developed conditions.

An improved bounded real lemma for H∞ robust heading control of unmanned surface vehicle

Aiming at a mass variation of unmanned surface vehicle and the poor or even unmeasurable signal of yaw rate signal, the dynamic output feedback [Formula: see text] control method is proposed for heading regulation based on improved bounded real lemma to reduce the conservativeness and guarantee the robustness. First, a linear-varying-parameter heading error model with external disturbances is established by the consideration of mass as a variable parameter. Second, the dynamic output feedback controller with [Formula: see text] performance is presented for the linear-varying-parameter heading error model, and its nonlinear matrix inequality conditions for the stability of the closed-loop system are deduced from improved bounded real lemma by parameter-dependent Lyapunov to maintain the system robustness. Furthermore, the solution of nonlinear matrix inequality conditions relies on the variable substitution method, and improved bounded real lemma reduces the conservativeness of the controller design by introducing auxiliary relaxation variables to derive the equivalent linear matrix inequality condition. Finally, the heading control simulations for unmanned surface vehicle indicate that the dynamic output feedback [Formula: see text] method based on improved bounded real lemma (IDOF) is robust to system parameter variations and external disturbances, and can effectively reduce the conservativeness of the controller design. Moreover, dynamic output feedback H∞ method based on improved bounded real lemma has faster transient processes and smaller steady-state errors.

A vehicle cloud control system considering communication quantization and stochastic delay

AbstractA novel kind of networked feedback controller is designed for the vehicle cloud control system in the presence of both sensor‐controller/controller‐actuator delay and communication quantization. Firstly, the vehicle cloud control system with delayed and quantized communication is modeled using the lateral/longitudinal vehicle dynamic and Markovian jump linear system (MJLS) theory. Then, some efficient stabilization conditions in matrix inequalities forms are derived for the considered cloud‐controlled vehicles. Furthermore, a cone complementary linearization method is utilized to solve the nonlinear matrix inequalities involved in the stabilization conditions. Simulation tests on connected vehicle lateral and longitudinal control are conducted to verify the effectiveness of the analytical results. Compared with the commonly used constant‐parameter controllers, the proposed method is practical to design vehicle cloud controllers under stochastic delay and communication quantization scenarios, with known delay distribution. Lastly, the stability of the vehicle cloud control system is guaranteed under the infection of unreliable communication factors with the proposed control method.

Harmonic Distortion Reduction of Transformer-Less Grid-Connected Converters by Ellipsoidal-Based Robust Control

A photovoltaic generator connected to a large network and supplying a nonlinear load (source of harmonics) injects distorted current into the grid. This manuscript presents an invariant-ellipsoid set design of a robust controlled active power filter to inject current into the large grid with minimum total harmonic distortion (THD). The nonlinear load current is considered an external disturbance to minimize its effect on the injected grid current. Moreover, the large grid is modeled as a fixed voltage source in a series with a Thevenin impedance whose value changes within an interval. Using the invariant-ellipsoid technique, the problem is cast as a robust disturbance-rejection tracking control. The volume of the ellipsoid is minimized, which results in minimizing the effect of disturbance on system performance and keeping the trajectories as close as possible to the origin. The design is cast into a set of nonlinear matrix inequalities that are linearized by fixing a scalar. The resulting convex optimization is solved iteratively by linear matrix inequalities (LMIs). The simulation and experimental findings show that the proposed design is successful in reducing THD injected into the grid when grid impedance is uncertain and variable loads are applied (balanced and unbalanced cases).

Actuator fault tolerance based on continuous‐time fuzzy singularly perturbed model

AbstractActuator fault tolerance based on fuzzy singularly perturbed model (FSPM) for nonlinear singularly perturbed systems (NSPSs) is investigated. A continuous‐time FSPM with subtractive faults on control input is presented to describe NSPSs with actuator faults, which is less conservative than slow–fast submodel methods. A hybrid fuzzy controller (HFC) consisting of a fuzzy detector and a fault‐tolerant controller can realize real‐time fault tolerance, such that the target system with actuator faults is stabilized. The theorem to solve controller gains is derived by using Lyapunov, singular perturbation, , and Schur complement theories, which can avoid the problem that the feasible solutions cannot be obtained by solving linear matrix inequalities (LMIs). A nonlinear item was replaced by a linear item, which provides a good idea for solving nonlinear matrix inequalities. The sudden and remittent actuator faults of a motor system are suppressed by using the above approaches.

Sliding mode control of interval type-2 T-S fuzzy systems with redundant channels

This paper investigates the sliding mode control (SMC) of interval type-2 (IT2) T-S fuzzy systems. The measurement outputs are propagated via redundant channels for reducing the probability of packet loss and improving the reliability of data transmission. A key feature for the above problem is that the premise variables and the measurement signals may not be available by the controller, which brings difficulty to stabilize the nonlinear systems. Accordingly, a crucial issue is how to synthesize an implementable SMC law under the redundant channels. To this end, the characteristic of the redundant channels is firstly analyzed and the model of available measurement output signals is established. By employing these available measurements as the premise variables and utilizing the upper and lower bounds of the system membership functions (MFs), new MFs are constructed and the sliding mode controller is synthesized. By introducing some null terms carrying the information of MFs, sufficient conditions are derived in terms of nonlinear matrix inequalities to ensure the stochastically ultimate boundedness of the closed-loop system and the reachability of the specified sliding surface. Besides, a binary genetic algorithm (GA) is introduced to solve the nonlinear criteria via the objective function reflecting the control performance. Finally, a numerical example illustrates the effectiveness of the proposed methods.

A New Power Flow Model With a Single Nonconvex Quadratic Constraint: The LMI Approach

In this paper, we propose a new mathematical model for power flow problem based on the linear and nonlinear matrix inequality theory. We start with rectangular model of power flow (PF) problem and then reformulate it as a Bilinear Matrix Inequality (BMI) model. A <b>Theorem</b> is proved which is able to convert this BMI model to a Linear Matrix Inequality (LMI) model along with One Nonconvex Quadratic Constraint (ONQC). Our proposed LMI-ONQC model for PF problem has only one single nonconvex quadratic constraint irrespective of the network size, while in the rectangular and BMI models the number of nonconvex constraints grows as the network size grows. This interesting property leads to reduced complexity level in our LMI-ONQC model which in turn makes it easier to solve for finding a PF solution. The non-conservativeness, iterative LMI solvability, well-defined and easy-to-understand geometry and pathwise connectivity of feasibility region are other important properties of proposed LMI-ONQC model which are discussed in this paper. An illustrative two-bus example is carefully studied to show different properties of our LMI-ONQC model. We have also tested our LMI-ONQC model on 30 different power-system cases including four ill-conditioned systems and compared it with a group of existing approaches. The numerical results show the promising performance of our LMI-ONQC model and its solution algorithm to find a PF solution.

Robust Feedback Control for Discrete-Time Systems Based on Iterative LMIs with Polytopic Uncertainty Representations Subject to Stochastic Noise

This paper deals with the design of linear observer-based state feedback controllers with constant gains for a class of nonlinear discrete-time systems in the form of a quasi-linear representation in presence of stochastic noise. For taking into account nonlinearities in the design of linear observer-based state feedback controllers, a polytopic modeling approach is investigated. An optimization problem is formulated to reduce the sensitivity of the controlled system towards stochastic input, state, and output noise with a predefined covariance. Due to the nonlinearities, the separation principle does not hold, thus, the controller and the observer have to be designed simultaneously. For this purpose, a Lyapunov-based method is used, which provides, in addition to the controller and observer gains, a stability proof for the nonlinear closed loop in a predefined polytopic domain. In general, this leads to nonlinear matrix inequalities. To solve these nonlinear matrix inequalities efficiently, we propose an approach based on linear matrix inequalities (LMIs) with a superposed iteration rule. When using this iterative LMI approach, a minimization task can be solved additionally, which desensitizes the closed loop to stochastic noise. The proposed method additionally enables the consideration of different linear closed loop structures by a unified Lyapunov-based framework. The efficiency of the proposed approach is demonstrated and compared with a classical LQG approach for a nonlinear overhead traveling crane.

Switching cluster synchronization control of networked harmonic oscillators subject to denial-of-service attacks

This paper is concerned with the average cluster synchronization control problem of networked harmonic oscillators under denial-of-service (DoS) attacks. Different from some existing DoS attack models that often necessitate specific statistical characteristics, only the worse-case duration bound of the DoS attacks is required during the control design procedure, which represents the least a priori knowledge of realistic DoS attacks. Then, a novel switching cluster synchronization control scheme, which leverages a position-based feedback control protocol under a non-small delay and a position velocity-based feedback control protocol with a small delay, is developed such that the above two control protocols are selected based on the occurrence of the DoS attacks. Via formulating the resultant synchronization system as a switched time-delay system, a complete-type Lyapunov–Krasovskii functional (LKF) method is further proposed to establish sufficient controller design criteria associated with the duration and frequency of attacks for both synchronous and asynchronous DoS attacks among clusters. Furthermore, an iterative algorithm is designed to calculate the control gain matrices by solving a set of nonlinear matrix inequalities (NLMIs). Finally, a multi-vehicle cooperative control system is presented to demonstrate the validity of the proposed control scheme.

Research on time-delay-dependent H∞/H2 optimal control of magnetorheological semi-active suspension with response delay

Considering the influence of the magnetorheological damper response time delay on the low-frequency vibration (body vibration) and multi-objective control requirements of the semi-active suspension system, a novel H∞/H2 controller (time-delay-dependent H∞/H2 optimal controller) is proposed in this paper. Based on the Lyapunov–Krasovskii function and robust control theory, a time-delay-dependent H∞/H2 controller in consideration of the MR damper response time delay is designed to reduce and remove the effects of time delay on the semi-active suspension system. To obtain better suspension control performances, the optimal selection strategy of anti-interference coefficients in the time-delay-dependent H∞/H2 controller is proposed. Then, the time-delay-dependent H∞/H2 optimal controller is structured built on the optimal selection strategy. With the application of matrix properties and the cone-compensation linearization algorithm, the nonlinear matrix inequality for controller design is linearized, and the feedback gain matrix of the controller is solved iteratively. Finally, comparative simulation and experimental research are conducted to verify the effectiveness and superiority of the designed optimal controller.

Bumpless [formula omitted] control for periodic piecewise linear systems

This paper investigates the bumpless H∞ control problem based on exponential stability and L2-gain analyses for a class of periodic piecewise linear systems. A novel characterization of bumpless transfer among a variety of subsystem controllers satisfying some interpolation constraints is introduced. By utilizing adjustable interpolation functions and multiple Lyapunov functions, more flexibility is provided so as to guarantee a desirable closed-loop performance and trajectory smoothness. Since the problem formulated boils down to a feasibility problem involving nonlinear matrix inequalities, an iterative algorithm which solves a series of convex optimization problems is developed. Numerical examples are presented to show the validity of the proposed methods.

Annulus-event-based fault detection for state-saturated nonlinear systems with time-varying delays

In this paper, the annulus-event-based fault detection method is proposed for a class of nonlinear systems subject to state saturation and time-varying delays. In particular, in order to save the network resources, a novel communication strategy called annulus-event-based triggering mechanism is considered. That is to say, the measurement output is sent to the filter only when the error between the measured value at the current moment and the data at the last transmission moment is within the prescribed triggering-annulus. Moreover, the system states at each instant are bounded by the convex hull via introducing a free matrix whose infinite norm is less than or equal to 1. Subsequently, sufficient conditions are obtained to ensure the global asymptotic stability and the H∞ performance of the filtering error dynamics system, which are expressed in line with some nonlinear matrix inequalities that can be handled via the given iterative linear matrix inequality method. Finally, two numerical simulations are employed to illustrate the validity of the developed fault detection filtering algorithm.

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.

Robust static output feedback based iterative learning control design with a finite‐frequency‐range two‐dimensional specification for batch processes subject to nonrepetitive disturbances

AbstractFor industrial batch processes subject to time‐varying uncertainties and nonrepetitive disturbances, this article proposes a robust static output feedback (SOF) based iterative learning control (ILC) design with a finite‐frequency‐range two‐dimensional (2D) performance specification. An important advantage of the proposed design lies in its simple structure and easy implementation, with no need to store the state information of the learning controller as used in the existing 2D dynamic output feedback based ILC methods. Based on a 2D Roesser system setting, sufficient conditions are established in terms of nonlinear matrix inequalities to guarantee robust stability of the resulting closed‐loop ILC systems along both the time and batch directions, by using the matrix dilatation technique. Moreover, a finite‐frequency‐range 2D performance specification for attenuating nonrepetitive disturbances is introduced for control synthesis. To derive the feasible SOF based ILC controller gains, a two‐stage heuristic approach is developed for iterative computation based on predesigning the corresponding state feedback based ILC gains. An illustrative example is used to demonstrate the effectiveness and merits of the proposed method.

Linear Matrix Inequalities for an Iterative Solution of Robust Output Feedback Control of Systems with Bounded and Stochastic Uncertainty

Linear matrix inequalities (LMIs) have gained much importance in recent years for the design of robust controllers for linear dynamic systems, for the design of state observers, as well as for the optimization of both. Typical performance criteria that are considered in these cases are either or measures. In addition to bounded parameter uncertainty, included in the LMI-based design by means of polytopic uncertainty representations, the recent work of the authors showed that state observers can be optimized with the help of LMIs so that their error dynamics become insensitive against stochastic noise. However, the joint optimization of the parameters of the output feedback controllers of a proportional-differentiating type with a simultaneous optimization of linear output filters for smoothening measurements and for their numeric differentiation has not yet been considered. This is challenging due to the fact that the joint consideration of both types of uncertainties, as well as the combined control and filter optimization lead to a problem that is constrained by nonlinear matrix inequalities. In the current paper, a novel iterative LMI-based procedure is presented for the solution of this optimization task. Finally, an illustrating example is presented to compare the new parameterization scheme for the output feedback controller—which was jointly optimized with a linear derivative estimator—with a heuristically tuned D-type control law of previous work that was implemented with the help of an optimized full-order state observer.