Projection Lemma Research Articles

H∞ Filtering of Repeated Scalar Nonlinear Systems: Event-Triggered Communication Case

In this paper, the event-triggered filter design problem is investigated for a class of discrete-time repeated scalar nonlinear systems. An admissible filter is introduced and sufficient conditions to ensure the asymptotical stability of the controlled system with a given performance are established. The solution of the corresponding event-triggered filter problem is settled by the projection lemma, and the cone complementary linearization technique is introduced to derive the filter parameters. An example is provided to demonstrate the effectiveness and potential of the proposed new design method.

Memory fault detection filtering design for uncertain systems with finite frequency specifications

AbstractThis article is concerned with the memory fault detection filtering design in finite frequency domain for discrete‐time systems with polytopic uncertainties. Under the assumption that both disturbances and faults are restricted to finite frequency ranges, that is, the low, middle, or high frequency range, an admissible memory fault detection filter is developed by using historical information of the system. It is shown that the asymptotic stability with prescribed performance of the closed‐loop filtering error system is guaranteed. It is also shown that by using historical system outputs along with parameter‐dependent Lyapunov functions, the better performance of the resulting fault detection filter is obtained. With the aid of Projection lemma, sufficient conditions for the fault detection filtering design are established by solving an optimization problem in the form of a set of linear matrix inequalities (LMIs). Finally, three examples are presented to demonstrate the effectiveness and advantages of the proposed approach.

Fuzzy Static Output Control of T–S Fuzzy Stochastic Systems via Line Integral Lyapunov Function

Considering some unmeasurable states, a fuzzy static output control problem of nonlinear stochastic systems is discussed in this paper. Based on a modelling approach, a Takagi–Sugeno (T–S) fuzzy system, constructed by a family of stochastic differential equations and membership functions, is applied to represent nonlinear stochastic systems. Parallel distributed compensation (PDC) technology is used to construct the static output controller. A line-integral Lyapunov function (LILF) is used to derive some sufficient conditions for guaranteeing the asymptotical stability in the mean square. From the LILF, a potential conservatism produced by the derivative of the membership function is eliminated to increase the relaxation of sufficient conditions. Furthermore, those conditions are transferred into linear matrix inequality (LMI) form via projection lemma. According to the convex optimization algorithm, the feasible solutions are directly obtained to establish the static output fuzzy controller. Finally, a numerical example is applied to demonstrate the effectiveness and usefulness of the proposed design method.

Observer-Based Fuzzy Controller Design for Nonlinear Discrete-Time Singular Systems via Proportional Derivative Feedback Scheme

This paper investigates the observer-based fuzzy controller design method for nonlinear discrete-time singular systems that are represented by Takagi-Sugeno (T-S) fuzzy models. At first, the nonlinearity can be well-approximated with several local linear input-output relationships. The parallel distributed compensation (PDC) technology and the proportional derivative (PD) feedback scheme are then employed to construct the observer-based fuzzy controller. To solve the problem of unmeasured states, the impulsive phenomenon of singular systems, and the PD scheme’s reasonableness, a novel observer-based fuzzy controller is developed. By using the Lyapunov theory and projection lemma, the stability criteria are built in terms of linear matrix inequalities (LMI). Moreover, the gains of fuzzy controller and fuzzy observer can be calculated synchronously by using convex optimization algorithms. Finally, a biological economic system is provided to verify the effectiveness of the proposed fuzzy control method.

Multi-objective Unified qLPV Observer: Application to a Semi-active Suspension System

Abstract This work concerns the design of a multi-objective unified qLPV observer for the state estimation problem in LPV systems with a parameter-dependent control input matrix. The standard forms of the system and the observer are first presented, where the observer matrices are functions of the estimated states (qLPV problem). The effects of bounded unknown input disturbances are decoupled from the estimation error thanks to the parameterization of the observer matrices. To treat the disturbance caused by inexact scheduling parameters, we introduce an upper bound on the parameter estimation error, which is considered uncertainty. Then, the effects of the control input and the random measurement noise on the estimation error are minimized using the H∞ and generalized H2 conditions, respectively, as a multi-objective optimization problem. In the solution of the LMI sets, the projection lemma is applied to reduce the high conservativeness that would otherwise lead to suboptimal results. Then this observer is applied to a semi-active automotive suspension system written in LPV form, using simulations with real data measured from our experiment test platform, and compared with the linear time-invariant H∞ and H2 observers.

Robust Fault Detection of Lipschitz Nonlinear Switched Systems Considering Disturbance Attenuation Level and Lipschitz Constant Maximization

This paper deals with robust fault detection (FD) of continuous time Lipschitz nonlinear switched systems based on observer method. A switched observer with average dwell time (ADT) approach is designed using projection lemma wherein the Lipschitz constant is maximized and simultaneously the effect of disturbances is attenuated effectively. Lipschitz constant maximization makes the fault detection system robust against parametric and nonlinear uncertainties. One more degree of freedom is provided by applying projection lemma, which could be used to obtain smaller ADT value and disturbance attenuation level of the observer. Some simulation results are given to demonstrate the efficiency and performance of the proposed FD scheme for nonlinear switched systems.

Finite frequency memory output feedback controller design for discrete‐time systems with state multiplicative noises

AbstractThis article studies the problem of output feedback control for discrete‐time systems with state multiplicative noises. Aiming at reducing the controller design conservatism, a novel finite frequency memory control approach is developed, which makes full use of both current and past output measurements together with the frequency spectrum information of disturbances. By augmenting the control input into the closed‐loop system state, a descriptor system approach is proposed to facilitate the controller synthesis. Then, based on the Fourier transform, Parseval's relation, and Projection lemma, the linear matrix inequalities (LMIs)‐based controller design results are obtained to guarantee the mean‐square asymptotic stability with finite frequency performance for the resulting closed‐loop system. The effectiveness of the developed approach is validated by an antenna motion control system.

Event-triggered dynamic anti-windup augmentation for saturated systems

This paper considers the dynamic output-based event-triggered strategy design for linear systems subject to input saturation. The dynamic event-triggered control reduces the communication burden, and a minimum inter-event time is explicitly provided to exclude Zeno behaviour. A dynamic linear anti-windup augmentation is presented to deal with the input saturation. For the given anti-windup compensator, a criterion is formulated to achieve practical stability with finite gain. Moreover, an algorithm with projection lemma is proposed to design the anti-windup augmentation when it is unknown. For two special cases, i.e. the anti-windup compensator is static or of the plant-order, the feasibility conditions become convex. Simulations are given to illustrate the theoretical results.

Membership-Function-Dependent Fault Detection Filtering Design for Interval Type-2 T–S Fuzzy Systems in Finite Frequency Domain

This article studies the problem of finite frequency fault detection filtering design for uncertain nonlinear systems based on interval type-2 Takagi-Sugeno fuzzy models. It is assumed that the frequencies of disturbances and faults are in finite frequency sets, respectively. The objective is to design an admissible filter such that the fault detection system is asymptotically stable with prescribed finite frequency \mathscr H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> and \mathscr H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> performances. Based on Fourier transform and Projection lemma, finite frequency filtering synthesis results are obtained. Then, a novel membership-function-dependent finite frequency fault detection filtering design approach is proposed by using the information of the lower and upper membership functions together with the footprint of uncertainties. Two algorithms with linear matrix inequality constraints are developed to optimize the finite frequency \mathscr H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance and the finite frequency \mathscr H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> performance, respectively. Finally, simulation studies are provided to show the effectiveness of the proposed method.

Fault estimation for fractional-order linear systems with polytopic uncertainties in the finite frequency domain

This paper investigates the problem of fault estimation for fractional-order systems with polytopic uncertainties. The faults and disturbances are considered to belong to finite frequency domains, and a fractional-order robust fault estimation filter is proposed for fault estimation with the known frequency ranges. As applications of the generalized Kalman–Yakubovich–Popov lemma, the fault estimation filter design problem is transformed into a multi-objective optimisation problem. By using the projection lemma and a linearising change of variables, sufficient conditions for the existence of the robust fault estimator are derived on the basis of a set of linear matrix inequalities. Numerical examples are used to demonstrate the validity of the presented method.

Fuzzy Observer-controller Design in Finite Frequency Domain: Application to Wind Turbine

This work presents the generalized Kalman Yakubovich-Popuv (gKYP) combined with the Takagi Sugeno (T-S) fuzzy model to design a fuzzy robust state feedback controller and a fuzzy robust observer-based in finite frequency (FF) domain. T-S fuzzy model is well known for its efficiency to control complex nonlinear systems. However, for wind generator system, the unknown parts are large and produce disturbances parameters. In order to attenuate, the level of the disturbances parameters observer based is utilized to estimate the unknown parts of the wind system. The control design method is based on Lyapunov function, the generalized gKYP with projection lemma, a PDC (Parallel Distributed Compensation) structure and the finite frequency (FF) technique. The proposed approach is formulated linear matrix inequalities (LMIs) to prove the asymptotic stability in (FF) domain. Finally, an example of wind turbine is show the validity of the proposed new approach.

A Relaxed Observer-Based Control for LPV Stochastic Systems Subject to H∞ Performance

A relaxed observer-based control problem of the disturbed Linear Parameter Varying (LPV) stochastic systems is addressed in this paper. For the control problem, some sufficient conditions are derived via a Lyapunov function whose positive definite matrix is not required as the diagonal case. Besides, a general H∞ performance index is proposed to constrain the effect of external disturbance on the state and estimation error. Furthermore, a lemma is proposed to eliminate the oscillation caused by determining performance level of state and estimation error. Another lemma is developed via extending projection lemma to convert the derived conditions into strict Linear Matrix Inequality (LMI) form. By using convex optimization algorithm, all feasible solutions can be obtained by a single-step calculation to establish an observer-based Gain-Scheduled (GS) controller. Based on the designed controller, the asymptotical stability and H∞ performance of the LPV stochastic systems can be guaranteed in the sense of mean square. Finally, two examples are provided to demonstrate the effectiveness and applicability of the proposed method.

Fault detection for fractional‐order linear systems in finite frequency domains

This study is concerned with the fault detection problem for fractional-order linear systems with disturbances. The H − and H ∞ performance indices of fractional-order systems in finite frequency domains are applied to measure the fault sensitivity and the disturbance robustness. A fractional-order fault detection observer is designed to satisfy the two finite frequency performance indices simultaneously. Based on the generalised Kalman–Yakubovich–Popov lemma and the projection lemma, the design conditions are obtained in terms of linear matrix inequalities. Compared with the existing full frequency approaches, the proposed finite frequency one can get better results when the frequency ranges of faults and disturbances are known. The effectiveness of the proposed method is validated by numerical examples.

An improved H∞ filtering for Markovian jump system with time-varying delay based on finite frequency domain

In traditional analysis of system performance, Markovian jump system is often considered in the full frequency domain. However, some unknown factors, such as noise interference, often act on a specific frequency domain, so there is a big defect in putting the system analysis in the full frequency domain. In this paper, the finite frequency domain is introduced into the Markovian jump system. By using generalized KYP lemma, the system is divided into three frequency bands: low, medium and high. The disturbance in different frequency domain is analyzed and the H∞ filter is designed. In addition, in dealing with inequality relations, the improved Wirtinger inequality method is used, and the Projection lemma is used in dealing with filter coupling terms, and time-varying delay problem in finite frequency domain is also considered. Finally, some examples are given to illustrate the effectiveness of our methods and the specific parameters and convergence graph of the finite frequency domain filter are obtained.

A Heuristic algorithm for robustly stable generalized disturbance observer synthesis with closed loop consideration

This paper investigate the problem of disturbance observer (DOB) based disturbance rejection control with closed loop performance guaranteed. A generalized DOB based control framework is presented. Specifically, the novel DOB framework if obtained by taking advantage of Youla factorization of a two degree-of-freedom controller in a nontraditional way. Closed loop analysis clarifies that generalized DOB inherits advantages of the traditional one, while mitigating its restrictions. Through appropriate reconfiguration, the Q-filter synthesizing is transformed into reduced-order controller designing. By taking advantage of the Kalman–Yakubovich–Popov (KYP) lemma and projection lemma, a two-stage heuristic algorithm is proposed: an initial full information controller is firstly established for the reconfigured system, which is used to heuristically obtain the reduced-order controller by alternately solving LMIs in the second stage. Finally, we illustrate the application of these results to minimum phase and non-minimum phase plants. Elaborated comparisons between the two-stage heuristic algorithm based generalized DOB and the state-of-art H∞ based DOB are conducted, the results verify the effectiveness and advantages.

Nonfragile H∞ Control of Delayed Active Suspension Systems in Finite Frequency Under Nonstationary Running

The vehicle active suspension has drawn considerable attention due to its superiority in improving the vehicle dynamic performance. This paper investigates the nonfragile H∞ control of delayed vehicle active suspension in a finite frequency range under nonstationary running. The control objective is to improve ride quality in a finite frequency band and ensure suspension constraints, and a quarter car model of the active suspension is established for a controller design. Then, the input delay, actuator uncertainty, and external disturbances are considered in the controller design. Moreover, a further generalization of the strict S-procedure is utilized to derive a sufficient condition in terms of linear matrix inequality (LMI) to capture performance in the concerned frequency range. Furthermore, a multi-objective controller is designed based on projection lemma in the framework of the solution of LMIs. A nonstationary road profile is established, and numerical simulations are also conducted to show the effectiveness and robustness of the proposed controller. Finally, experimental tests on a quarter-car test rig are implemented to examine the performance of the proposed controller for real applications.

A Novel Piecewise Affine Filtering Design for T-S Fuzzy Affine Systems Using Past Output Measurements.

This paper tackles the problem of piecewise affine memory filtering design for the discrete-time norm-bounded uncertain Takagi-Sugeno fuzzy affine systems. The objective is to design an admissible filter using past output measurements of the system, guaranteeing the asymptotic stability of the filtering error system with a given [Formula: see text] performance index. Based on the piecewise fuzzy Lyapunov functions and the projection lemma, a new sufficient condition for [Formula: see text] filtering performance analysis is first derived, and then the filter synthesis is carried out. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities. In addition, it is also shown that the filtering performance can be improved with the increasing number of past output measurements used in the filtering design. Finally, two examples are presented to show the advantages and effectiveness of the proposed approach.

A Novel Memory Filtering Design for Semi-Markovian Jump Time-Delay Systems

This paper investigates the problem of delay-dependent $\boldsymbol {\mathscr {H}_{\infty }}$ memory filtering for continuous-time semi-Markovian jump linear systems (MJLSs) with time-varying delay in an input-output framework. Differing from the constant transition rates (TRs) in the conventional MJLSs, the TRs of the semi-MJLSs depend on the random sojourn-time and are thus with time-varying characteristics. By utilizing a two-term approximation for the terms with time-varying delay, it is first shown that the filtering error system (FES) can be reformulated into a feedback interconnection form and the stability and performance analysis problem of the FES can be recast as the scaled small gain (SSG) problem of an interconnected system. Then, based on a semi-Markovian Lyapunov–Krasovskii formulation of SSG condition combined with projection lemma, the $\boldsymbol {\mathscr {H}_{\infty }}$ filter synthesis for the underlying semi-MJLSs is formulated in terms of linear matrix inequalities. Finally, simulation studies are provided to evaluate the effectiveness and superiority of the proposed design method.

H∞ Synchronization of Semi-Markovian Jump Neural Networks with Randomly Occurring Time-Varying Delays

Based on the Lyapunov stability theory, this paper mainly investigates the H∞ synchronization problem for semi-Markovian jump neural networks (semi-MJNNs) with randomly occurring time-varying delays (TVDs). The continuous-time semi-MJNNs, where the transition rates are dependent on sojourn time, are introduced to make the issue under our consideration more general. One of the main characteristics of our work is the handling of TVDs. In addition to using the improved Jensen inequality and the reciprocal convexity lemma to deal with the integral inequality, we also employ Schur complement and the projection lemma to achieve the decoupling between the square term of TVDs. Finally, we verify the validity and feasibility of our method by a couple of simulation examples.

Finite Frequency Filtering Design for Uncertain Discrete-Time Systems Using Past Output Measurements

This paper tackles the problem of finite frequency $\mathcal H_{\infty }$ filtering design for polytopic uncertain discrete-time systems using past output measurements. The noise is assumed to be restricted in a finite frequency range, i.e., the low, middle, or high frequency range. The objective is to design an admissible filter with past output measurements of the system, guaranteeing the asymptotic stability of the filtering error system with a prescribed finite frequency $\mathcal H_{\infty }$ disturbance attenuation level. Based on past output measurements together with the parameter-dependent Lyapunov function and projection lemma, a new sufficient condition for robust finite frequency $\mathcal H_{\infty }$ performance is first derived, and then, the filter synthesis is developed. It is shown that the filter gains can be obtained by solving a set of linear-matrix-inequalities. Finally, two examples are given to demonstrate the advantages and effectiveness of the proposed approach.