Norm-bounded Uncertainties Research Articles

Polynomial chaos-based [formula omitted] output-feedback control of systems with probabilistic parametric uncertainties

H2 static and dynamic output-feedback control problems are investigated for linear time-invariant uncertain systems. The goal is to minimize the averaged H2 performance in the presence of nonlinear dependence on time-invariant probabilistic parametric uncertainties. By applying the polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochastic state uncertainty propagation. Compared to existing polynomial chaos-based control designs, the proposed approach addresses the simultaneous presence of parametric uncertainties and white noises. The effect of truncation errors due to using finite-degree polynomial chaos expansions is captured by time-varying norm-bounded uncertainties, and is explicitly taken into account by adopting a guaranteed cost control approach. This feature avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of expansion truncation errors, thus significantly reducing computational complexity. Moreover, rigorous analysis clarifies the condition under which the stability of the high-dimensional expanded system implies the internal mean square stability of the original system under control. Using iterations between synthesis and post-analysis, a bisection algorithm is proposed to find the smallest bounding parameter on the effect of expansion truncation errors such that robust closed-loop stability is guaranteed. A numerical example is used to illustrate the effectiveness of the proposed approach.

Nonfragile Observer-Based Control for Markovian Jump Systems Subject to Asynchronous Modes

In this paper, the problem of resilient observer-based robust control is considered for discrete-time Markov jump systems subject to asynchronous models and extended dissipativity. The model uncertainty is in the interval type, which has an ability to describe parameter fluctuating phenomenon more accurately than using the norm-bounded uncertainty. A hidden Markov chain is employed to depict the mismatch between the original system and the observer-based controller. Conditions are provided to ensure the stability of the resulting closed-loop system with a desired dissipation performance regardless of the uncertainties. An example is presented to illustrate the effectiveness and potential of the proposed new design techniques.

Non-fragile quasi-synchronization of delayed heterogeneous dynamical networks with memory sampled-data control

This paper investigates the quasi-synchronization problem for a class of heterogeneous dynamical networks based on a non-fragile memory sampled-data controller. Considering the effect of controller gain fluctuation and communication delay, a sampled-data control scheme with norm-bounded uncertainty and a constant signal transmission delay is designed. By introducing a leader, the heterogeneous complex delayed networks can be transformed into the corresponding error systems with bounded disturbances. A sufficient criterion to ensure that the error system can be exponentially stable and converge to a bounded region is established by the Lyapunov-Krasovskii function approach. Based on the criterion, the sampled-data control gain matrix is designed. Finally, the numerical examples are presented to illustrate the effectiveness of the theoretical results in this paper.

Random Sampling Many-Dimensional Sets Arising in Control

Various Monte Carlo techniques for random point generation over sets of interest are widely used in many areas of computational mathematics, optimization, data processing, etc. Whereas for regularly shaped sets such sampling is immediate to arrange, for nontrivial, implicitly specified domains these techniques are not easy to implement. We consider the so-called Hit-and-Run algorithm, a representative of the class of Markov chain Monte Carlo methods, which became popular in recent years. To perform random sampling over a set, this method requires only the knowledge of the intersection of a line through a point inside the set with the boundary of this set. This component of the Hit-and-Run procedure, known as boundary oracle, has to be performed quickly when applied to economy point representation of many-dimensional sets within the randomized approach to data mining, image reconstruction, control, optimization, etc. In this paper, we consider several vector and matrix sets typically encountered in control and specified by linear matrix inequalities. Closed-form solutions are proposed for finding the respective points of intersection, leading to efficient boundary oracles; they are generalized to robust formulations where the system matrices contain norm-bounded uncertainty.

Robust MFC anti-windup scheme for LTI systems with norm-bounded uncertainty

On the basic of the fact that all signals in the practical system are always bounded, this paper proposes a 4-degree-of-freedom (DoF) anti-windup scheme for saturated systems with parametric uncertainty. A fairly straightforward tuning rule is introduced to the robust stability analysis for the proposed anti-windup structure under the framework of IQC (Integral Quadratic Constraint). And the sufficient stability conditions are derived to check the reasonable definiteness of the related transfer function. Moreover, the control design for disturbance response and set-point tracking response are two separate part in this proposed scheme. Numerical example demonstrates the effectiveness and the considerable performance improvement of the anti-windup compensator that is designed by the proposed technique.

Distributed Finite-Horizon Extended Kalman Filtering for Uncertain Nonlinear Systems.

In this paper, the state estimation problem is investigated for a class of discrete nonlinear systems via sensor networks. A novel robust distributed extended Kalman filter, which can handle norm-bounded uncertainties in both the system model and its Taylor series expansion, is developed with ensured estimation performance. The filter is distributed in the sense that for each sensor, only its own measurements and its neighbors' information are utilized to optimize the upper bound of the estimation error covariance. Besides, a sufficient condition for the proposed algorithm is derived, which is simple and user-friendly since it depends on the property of the original nonlinear system instead of the estimation error covariance calculated at every step. Finally, the simulation results are presented to demonstrate the effectiveness of the filtering algorithm.

Robust Control for Discrete-Time T-S Fuzzy Singular Systems

This paper deals with the robust admissibility and state feedback stabilization problems for discrete-time T-S fuzzy singular systems with norm-bounded uncertainties. By introducing a new approximation technique, the initial membership functions are conveniently expressed in piecewiselinear functions with the consideration of the approximation errors. By utilizing the piecewise-linear membership functions, the fuzzy weighting-based Lyapunov function and the use of auxiliary matrices, the admissibility of the systems is determined by examining the conditions at some sample points. The conditions can be reduced into the normal parallel distributed compensation ones by choosing special values of some slack matrices. Furthermore, the authors design the robust state feedback controller to guarantee the closed-loop system to be admissible. Two examples are provided to illustrate the advantage and effectiveness of the proposed method.

Invariant set design of decentralized control for islanded microgrids under plug-and-play operation

This article presents a new invariant-set design for voltage regulation of islanded microgrids (MGs) composed of several distributed generators (DGs). The advantages of the proposed control are: (1) the control structure is decentralized, and (2) it tracks the reference inputs quickly and maintains robust stability against the MG’s load changes and plug- and-play operation. To achieve these objectives, the global system is decomposed into several subsystems. Each DG subsystem is modeled as a linear time-invariant system with norm-bounded uncertainty representing load changes. For a specific subsystem, the effect of the rest of subsystems is represented as a disturbance. The effect of the disturbance on the subsystem is reduced by minimizing the volume of its invariant ellipsoid (approximating and bounding its invariant set). The changes in the disturbance is modeled by the norm-bounded form. The controller design is based on a derived sufficient condition in the form of nonlinear matrix inequality (solved iteratively by fixing a positive scalar as a convex linear matrix inequality, LMI, optimization problem). To evaluate the effectiveness of the proposed control strategy, different time-domain simulations are conducted using MATLAB/SimPowerSystems Toolbox. Simulations demonstrate that the suggested controller provides excellent performance in terms of voltage tracking under load changes and MG’s plug-and-play operation.

Resilient set-membership state estimation for nonlinear complex networks with time-delay and incomplete measurements

Taking the incomplete measurements and the weighted try-once-discard (WTOD) protocol into account, this paper develops a novel resilient set-membership state estimation (RSMSE) method for time-varying nonlinear complex networks with time-invariant delay. A classic interval matrix technique is utilized to describe incomplete measurements. The Taylor series expansion is applied to dispose the nonlinearities, where the high-order terms of the linearization errors are described by norm-bounded uncertainties. To mitigate the communication burden, the WTOD protocol is introduced, where only one node can send updated data through a shared communication network at each certain transmission step. Using the recursive linear matrix inequalities (RLMIs), a series of ellipsoidal sets including the state vector can be determined. The desirable estimator gain and a smallest possible estimation ellipsoid can be calculated via solving the convex optimization problem. Lastly, we use an illustrative example to show the feasibility of the introduced RSMSE technique.

Robust linear parameter-varying output-feedback control of permanent magnet synchronous motors

This paper investigates the design of a robust linear parameter-varying (LPV) controller for speed regulation of surface permanent magnet synchronous motors (SPMSMs). Motor dynamics is defined in the α–β stationary frame and a parameter-varying model formulation is provided to describe its nonlinear dynamics. A robust gain-scheduled LPV dynamic output-feedback controller is designed to guarantee the asymptotic stability and performance requirements, as well as, guarantee robustness against parameter perturbations and torque load disturbances. As a novel aspect in the SPMSM control, the real-time impact of temperature variation on the winding resistance and magnet flux during motor operation is explicitly considered in the LPV modeling as a scheduling parameter and the subsequent LPV control design compensating for demagnetization effects in the motor response. The controller synthesis conditions guaranteeing stability and robust performance to norm-bounded uncertainty in the system matrices are formulated in a convex linear matrix inequality (LMI) optimization framework. Finally, the validity of the proposed method is assessed in simulation studies. The closed-loop simulation studies demonstrate that the proposed LPV controller provides improved transient response with respect to settling time, overshoot, and disturbance rejection in tracking the velocity profile under the influence of parameter perturbation, temperature variations, and load disturbances.

Output-Synchronization of Discrete-Time Multiagent Systems: A Cooperative Event-Triggered Dissipative Approach

This paper investigates output synchronization of heterogenous time-delay discrete-time multiagent systems via output couplings. Firstly, sufficient conditions are provided to guarantee dissipativity for an individual agent modeled by the descriptor dynamics. Fully distributed consensus protocol based on the outputs of the neighboring agents is proposed to show that the individual dissipativity leads to output synchronization of the overall multiagent system. The proposed synchronization scheme exhibits a strong robustness facing norm-bounded uncertainties. Moreover, the counterparts of the result of the cooperation based on dissipativity for the multiagent system with event triggering mechanism is also investigated. The proposed triggering mechanism and the consensus protocol solve the synchronization for all admissible uncertainties and time varying delay. Finally, a numerical heterogeneous model is carried out to illustrate the effectiveness of the proposed strategies.

Novel Robust Stability Criteria of Uncertain Systems with Interval Time-Varying Delay Based on Time-Delay Segmentation Method and Multiple Integrals Functional

Interval time-varying delay is common in control process, e.g., automatic robot control system, and its stability analysis is of great significance to ensure the reliable control of industrial processes. In order to improve the conservation of the existing robust stability analysis method, this paper considers a class of linear systems with norm-bounded uncertainty and interval time-varying delay as the research object. Less conservative robust stability criterion is put forward based on augmented Lyapunov-Krasovskii (L-K) functional method and reciprocally convex combination. Firstly, the delay interval is partitioned into multiple equidistant subintervals, and a new Lyapunov-Krasovskii functional comprising quadruple-integral term is introduced for each subinterval. Secondly, a novel delay-dependent stability criterion in terms of linear matrix inequalities (LMIs) is given by less conservative Wirtinger-based integral inequality approach. Three numerical comparative examples are given to verify the superiority of the proposed approach in reducing the conservation of conclusion. For the first example about closed-loop control systems with interval time-varying delays, the proposed robust stability criterion could get MADB (Maximum Allowable Delay Bound) about 0.3 more than the best results in the previous literature; and, for two other uncertain systems with interval time-varying delays, the MADB results obtained by the proposed method are better than those in the previous literature by about 0.045 and 0.054, respectively. All the example results obtained in this paper clearly show that our approach is better than other existing methods.

Fuzzy Resilient $\mathcal {H}_\infty$ Filter Design for Continuous-Time Nonlinear Systems

In this article, the resilient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal H_\infty$</tex-math></inline-formula> filter design problem for continuous-time nonlinear systems is addressed. A Takagi–Sugeno fuzzy model with norm-bounded uncertainties is used to represent the nonlinear plant. Meanwhile, the fuzzy filter to be designed is assumed to have gain variations. A useful matrix inequality decoupling approach is proposed to separate the product terms with different types of uncertainties. Then, the resulting design condition of the resilient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal H_\infty$</tex-math></inline-formula> filter is described by strict linear matrix inequalities. Compared with the existing fuzzy resilient filtering results, the proposed design method shows a strong advantage of reducing design conservatism. Finally, a simulation example is provided to demonstrate the feasibility and the advantage of the proposed design method.

Sub-predictors for network-based control under uncertain large delays

This paper, for the first time, provides sub-predictors for networked control systems (NCSs) under uncertain large communication delays. We use a time-delay approach to NCS and employ sub-predictors to partially compensate large uncertain transmission delays in the sensor-to-controller and controller-to-actuator channels by dividing the long delay into small pieces. We consider systems with norm-bounded uncertainties, and take into account Round-Robin scheduling protocol in sensor-to-controller channel. In comparison with the traditional reduction-based classical predictor involving distributed input, the sub-predictor-based feedback is more friendly in the presence of norm-bounded uncertainties and is simpler for implementation. The sub-predictor-based feedback is further extended to decentralized control of interconnected systems provided that the couplings are not strong. The stability analysis of the closed-loop system is based on the Lyapunov–Krasovskii method and the stability conditions are given in terms of linear matrix inequalities.

Resilient State Estimation for Complex Dynamic Networks With System Model Perturbation

This article investigates the resilient estimation problem for a class of complex networks with disturbances. Specifically, there exist uncertainties in system matrices and inner coupling simultaneously, described by Gaussian noise or deterministic norm-bounded matrices. For each kind of uncertainty, a novel class of resilient estimation algorithms is designed from “centralized” and “distributed” perspectives, respectively. Here, “centralized” means that global information is utilized by each node while “distributed” indicates that only the local information of each node's own and its neighbors is used in the estimation process. Particularly, for norm-bounded uncertainties, an extended-state method is proposed, where adaptive system matrices are introduced to improve the estimation performance. A recursive upper bound of the estimation error covariance for each node is derived to obtain the estimator gains. Such designed estimation algorithms avoid solving linear matrix inequalities. Furthermore, an easy-to-check condition only about the system matrices is provided to guarantee the feasibility of the estimation algorithms on the infinite horizon. Finally, the effectiveness of the theoretical results is demonstrated by several numerical examples.

Non-fragile H ∞ control of periodic piecewise time-varying systems based on matrix polynomial approach

This paper investigates the problem of non-fragile control for periodic piecewise time-varying systems. Based on a Lyapunov function with continuous time-varying Lyapunov matrix polynomial, and combining with the positiveness and negativeness properties of matrix polynomials, the performance analysis is first accomplished. Then consider two types of controller gain perturbations that are formulated by time-varying matrix parameters and norm-bounded uncertainties. The additive and multiplicative non-fragile controllers to guarantee the performance of the system are formed, of which the controller gain could be solved with linear matrix inequalities directly. The designed non-fragile controller is desirable in applications. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

H 2/H ∞ formulation of LQR controls based on LMI for continuous-time uncertain systems

The classical theory of linear matrix inequality (LMI)-based linear quadratic regulator (LQR) control that is well established in literature does not provide, in the controller project phase, the occurrence of exogenous inputs acting on the plants. Therefore, this study proposes sufficient conditions for the robust synthesis of a mixed / control of the LQR problem. A new mixed polytopic and norm-bounded uncertainties representation for linear time-invariant uncertain systems allows to express different uncertainty models in one single LMI by using the S-procedure. As an additional objective, we propose the study of the robustness of controllers concerned with the sensibility of their coefficients, in such a way that the variations in controller gain matrix are described by norm-bounded uncertainties. A practical implementation in Quanser active suspension evaluates the control projects through the rejection level of vibrations caused by track irregularities.

A Phase-Shaping Approach to the Control of Parametric Systems and Its Application

The robust performance design of parametric systems is a long-standing unsolved problem, even though its analysis has seen significant success. Most existing design methods usually treat the parameter uncertainty as other types, such as norm-bounded uncertainty, and apply the corresponding approach. However, such treatment inevitably broadens the range of uncertainty and brings about design conservatism consequentially. To overcome such difficulty and establish a less conservative performance design method for parametric systems, this article looks back at the passivity theory and put forward a phase-shaping approach. This approach is composed of a generalized Popov transformation and a phase-shaping method for the nominal system. The key idea is to transform an uncertain but positive parameter into a positive real function while shaping the phase of the nominal system via a meta-heuristic method. This design freedom of phase shaping makes it possible to achieve a higher performance. Furthermore, this method is applied to the control design of drivetrain system: a test bed for automobile drivetrains. Its superiority is validated experimentally on an industrial setup.

Design of robust fuzzy Sliding-Mode control for a class of the Takagi-Sugeno uncertain fuzzy systems using scalar Sign function

This article presents a fuzzy sliding-mode control scheme for a class of Takagi-Sugeno (T-S) fuzzy which are subject to norm-bounded uncertainties in each subsystem. The proposed stabilization method can be adopted to explore T-S uncertain fuzzy systems (TSUFS) with various local control inputs. Firstly, a new design is proposed to transform TSUFS into sliding-mode dynamic systems.In addition, the proposed stabilization condition for the sliding-mode dynamic system can be formulated in terms of linear matrix inequalities (LMIs) using the Lyapunov functions. Finally, the validity of the proposed design strategy is demonstrated through the simulation of an inverted pendulum on a cart. According to simulation results, the performance indexes of the proposed method can achieve satisfactory results with minimum energy control effort, faster response, and zero steady-state error.

Strong Stability Analysis of Linear Delay-Difference Equations With Multiple Time Delays

This article is concerned with the strong stability problem of linear continuous-time delay-difference equations with multiple time delays. A family of linear matrix inequalities (LMIs), indexed by a positive integer k, is derived to assess strong stability. A time-domain interpretation of the proposed LMI-based condition is given in terms of a quadratic integral Lyapunov functional, which allows us to reveal relations with an existing result. The LMI condition can easily be reformulated in a form where the dependence on the coefficients of the delay-difference equation is affine, which is instrumental to establishing a sufficient LMI condition for robust strong stability of delay-difference equations with norm-bounded uncertainty. A necessary and sufficient condition for robust strong stability is also given, in the form of a structured singular value characterization. Examples are given to illustrate the effectiveness of the presented results.