Discrete-time Linear Parameter-varying Systems Research Articles

Set-based guaranteed active fault diagnosis for LPV systems with unknown bounded uncertainties

This paper focuses on designing, via the minimization of a suitable cost function, an input sequence to guarantee active fault diagnosis (AFD) of discrete-time linear parameter-varying systems regardless of how the scheduling variables change in the scheduling set. Convex polyhedrons are used to characterize the sets of system uncertainties such that the designed input sequence leads to outputs which are consistent with a unique fault mode or healthy mode. Considering the smoothness and energy of the input sequence simultaneously, the optimal input sequence is obtained by solving a family of linear constrained quadratic programming (LCQP) problems in a brute-force way, which will lead to inefficiency when the number of facets of the interest region becomes large. In order to handle this problem, by analyzing a geometric property of the objective function, the original optimization problem is shown to be equivalent to the minimization of the radius of a hypersphere containing the origin based on a simple linear mapping, which is much more efficient than the brute force-based resolution. At the end of this paper, a numerical example and a vehicle model are used to illustrate the effectiveness of the proposed method.

A new parametrization for static output feedback control of LPV discrete-time systems

This paper deals with the problem of synthesizing static output feedback controllers for linear parameter-varying discrete-time systems with guaranteed ℓ2 gain performance. The proposed method relies on augmenting the matrices related to the control input with a block of zeros and also augmenting the control gain accordingly with some real matrix. In this way, the closed-loop system remains the same, but extra degrees of freedom can be exploited in the proposed synthesis conditions. These conditions are described in terms of optimization problems having Linear Matrix Inequalities as constraints. Unlike other methods in the literature, all system matrices can depend on the scheduling parameter and, furthermore, the proposed approach is capable of determining a static output feedback control gain by with a one-phase one optimization problem. Numerical examples illustrate that the proposed approach is less conservative when compared to other available methods.

Finite-time estimation algorithms for LPV discrete-time systems with application to output feedback stabilization

This paper deals with new finite-time estimation algorithms for Linear Parameter Varying (LPV) discrete-time systems and their application to output feedback stabilization. Two exact finite-time estimation schemes are proposed. The first scheme provides a direct and explicit estimation algorithm based on the use of delayed outputs, while the second scheme uses two combined asymptotic observers, connected by a condition of invertibility of a certain time-varying matrix, to recover solution of the LPV system in a finite-time. Furthermore, two stabilization strategies are proposed. The first strategy, called Delayed Inputs/Outputs Feedback (DIOF) stabilization method, is based on the use of the explicit estimation algorithm. The second technique, called Two Connected Observers Feedback (2-COF) stabilization method, is based on the use of two combined observers providing exact finite-time estimation. A numerical example is given to show the validity and effectiveness of the proposed algorithms by simulation.

Event-triggered Dynamic Output Feedback Controller for Discrete-time LPV Systems with Constraints

This paper examines an event-triggered control design approach for discrete-time linear parameter-varying (LPV) systems under control constraints. A parameter-dependent dynamic output-feedback controller with an event-triggering condition is designed to ensure the regional asymptotic stability of the closed-loop system while minimizing a quadratic cost function. Sufficient conditions are derived in terms of linear matrix inequalities (LMIs) thanks to the use of a parameter-dependent quadratic Lyapunov function. Also, convex optimization schemes are proposed using these conditions to minimize an upper bound of a given cost function or to maximize the size of the region of closed-loop stability. Examples illustrate the proposal.

On unknown input observers designs for discrete-time LPV systems with bounded rates of parameter variation

This paper concerns the design of a novel unknown input observer (UIO) for discrete-time linear parameter-varying (LPV) systems with bounded rates of parameter variation. The synthesis conditions have been formulated using a less conservative approach to obtain two different structures: a proportional UIO and a proportional-integral UIO. The main highlights of the present design conditions are the ability to deal with discrete-time LPV systems subject to both states and outputs parameter-varying matrices in the state-space representation and also the possibility of bounded rates of parameter variation. Such conditions may be useful to achieve better performance than considering only arbitrarily fast time-varying parameters for the discrete-time LPV representations. In order to obtain the UIO designs, stability and induced L2 norm performance conditions using a poly-quadratic approach in terms of Linear Matrix Inequalities are addressed. Numerical examples are presented to demonstrate the effectiveness of the proposed design methods.

Event-triggered and self-triggered H ∞ output tracking control for discrete-time linear parameter-varying systems with network-induced delays

In this paper, we address the event-triggered and self-triggered output tracking problem of discrete LPV systems with network-induced delay. Considering the time delay and external disturbance, we formulate the closed-loop event-triggered output tracking problem into a time-delayed discrete polytopic linear parameter-varying system with relative event-triggering mechanism. Then, by constructing the parameter-dependent Lyapunov–Krasovskii functional, we derive a sufficient condition such that the closed-loop system is global asymptotic stable and satisfies the output tracking performance. Further, we develop an approach to design the event-triggering mechanism and output tracking controller. The parameters of event-triggering mechanism and the gains of output tracking controller are obtained by solving linear matrix inequalities. Moreover, we extend the results of the event-triggered control to the self-triggered output tracking control. This not only mitigates the usage of network bandwidth but also avoids the use of additional hardware. Finally, the numerical simulation is included to illustrate the usefulness and effectiveness of the proposed approach.

Stability and stabilization for LPV systems based on Lyapunov functions with non-monotonic terms

This paper presents new conditions for stability analysis, static output-feedback and state-feedback control design for discrete-time linear parameter-varying systems. The proposed methodology is based on the combination of quadratic “Lyapunov-like” terms such that individually each one is not necessarily monotonically decreasing along the state trajectories. Firstly, a new necessary and sufficient stability analysis condition based on the use of non-monotonic terms in the Lyapunov function is proposed. Concerning the stabilization problem, a novel strategy for the static output-feedback control design is proposed and, unlike most approaches in the literature, no structural constraints on the output matrix are imposed, which it is an extra feature advantage of the proposed method as well. Besides that, a new gain-scheduling state-feedback control design solution is derived. All proposed conditions are presented in the form of Linear Matrix Inequalities and their feasibility implies the existence of a Lyapunov function that is monotonically decreasing along trajectories. Numerical experiments illustrate the potential of the proposed techniques.

On Gain-Scheduled State-Feedback Controller Synthesis With Quadratic Stability Condition

This letter shows that, as long as continuous-time linear parameter-varying (LPV) systems are concerned, quadratic-stability-based gain-scheduled state-feedback controller synthesis offers no advantage over quadratic-stability-based fixed (parameter-independent) state-feedback controller synthesis in typical control performance specifications. We derive this counterintuitive result by properly extending the previous results on the robust versions of Finsler's lemma and the elimination lemma. We also show that this counterintuitive result is continuous-time LPV system specific, and in the discrete-time LPV system case quadratic-stability-based gain-scheduled state-feedback controller synthesis does bring improvement. These results give a proper warning about the effectiveness of the quadratic-stability-based gain-scheduled state-feedback controller synthesis.

LMI-Based Filter Design Conditions for Discrete-Time LPV Systems With Bounded Parameter Variation

In this article, we introduce novel filter design conditions for discrete-time linear parameter-varying (LPV) systems. The two main features of the proposed conditions are: the ability of handling LPV systems subject to variation in all matrices of the system and the generalization for systems with bounded rates of parameter variation. Filter stability and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filtering problems are addressed through poly-quadratic Lyapunov functions in terms of linear matrix inequalities. Numerical examples illustrate the proposed designs effectiveness.

Robust Fault Detection and Set-Theoretic UIO for Discrete-Time LPV Systems With State and Output Equations Scheduled by Inexact Scheduling Variables

This paper proposes a novel robust fault detection (FD) approach and designs a set-theoretic unknown input observer (SUIO) for linear parameter-varying (LPV) systems with both state and output equations scheduled by inexact scheduling variables. First, for such LPV systems, we propose a novel robust FD method by combing the set theory with the unknown input observer, which considers the bounds of measurement errors of scheduling variables to generate FD-oriented sets. In general, as long as sensors with sufficiently high precision are equipped to measure the scheduling variables, the bounds of measurement errors of scheduling variables can be less conservative than those direct bounds of scheduling variables, which can reduce robust FD conservatism in this way. Second, we give the unknown input decoupling condition of SUIO for such LPV systems and propose an SUIO design method under this condition for robust state estimation (SE). Besides, stability conditions for the proposed methods are established via matrix inequalities. At the end of this paper, a case study is used to illustrate the effectiveness of the proposed methods.

Invariant set-based robust fault detection and optimal fault estimation for discrete-time LPV systems with bounded uncertainties

ABSTRACTThis paper proposes an invariant set-based robust fault detection (FD) and optimal fault estimation (FE) method for discrete-time linear parameter varying (LPV) systems with bounded uncertainties. Firstly, a novel invariant-set construction method for discrete-time LPV systems is proposed if and only if the system is poly-quadratically stable, which need not satisfy the condition that there must exist a common quadratic Lyapunov function for all vertex matrices of the system compared to the traditional invariant-set construction methods. Furthermore, by using a shrinking procedure, we provide minimal robust positively invariant (mRPI) set approximations that are always positively invariant at each step of iteration and allow a priori desired precision to obtain a high sensitivity of FD. Owing to the existence of invariant set-based FD phase, the assumption that the initial faults should be bounded by a given set can be avoided for FE. We compute an optimal parametric matrix gain by minimising the Frobenius norm-based size of the corresponding FE set to obtain the optimal FE performance. Theoretically, any trajectory in the FE tube can be chosen as a specific-value estimation for the real fault signals. Finally, a vehicle dynamics system is used to illustrate the effectiveness of the proposed method.

Zonotoptic Fault Estimation for Discrete-Time LPV Systems With Bounded Parametric Uncertainty

This paper presents a novel interval fault estimation approach by using zonotope technique for discrete-time linear parameter-varying systems in the presence of bounded parametric uncertainties, measured perturbation, and system disturbance. First, an augmented descriptor system is generated by using augmentation technique. Thus, the problem of interval fault estimation is transformed into the interval augmented state estimation. Then, an outer approximation of the new augmented state estimation domain is computed by using zonotope method. A zonotope should be consistent with the given outputs, perturbation, disturbance, and parametric uncertainties. Besides, it is minimized at each sampled time via an analytic formulation. Finally, a vehicle lateral dynamic nonlinear model is utilized to show the feasibility and effectiveness of the proposed zonotopic fault estimation technique.

Frequency Response Functions of Linear Parameter-Varying Systems

Frequency-domain system representations offer important system analysis, control design and simulation tools. While this representation concept is well developed for linear time-invariant systems, there has been no general theory established for the linear parameter-varying (LPV) system class. This paper overviews and compares the existing concepts of frozen and instantaneous frequency-domain LPV representations, pointing out the need for a representation form where the effect of the spectrum of a varying scheduling trajectory can be clearly understood and analyzed on the output spectrum. For this purpose, the harmonic frequency response function (hFRF), known from the linear time-varying literature, in introduced for LPV systems. More specifically the paper demonstrates how the hFRF can be computed directly in the frequency domain for discrete-time affine input-output LPV systems.

Invariant Set-Based Analysis of Minimal Detectable Fault for Discrete-Time LPV Systems With Bounded Uncertainties

This paper proposes an invariant-set based minimal detectable fault (MDF) computation method based on the set-separation condition between the healthy and faulty residual sets for discrete-time linear parameter varying (LPV) systems with bounded uncertainties. First, a novel invariant-set computation method for discrete-time LPV systems is developed exclusively based on a sequence of convex-set operations. Notably, this method does not need to satisfy the existence condition of a common quadratic Lyapunov function for all the vertices of the parametric uncertainty compared with the traditional invariant-set computation methods. Based on asymptotic stability assumptions, a family of robust positively invariant (RPI) outer-approximations of minimal robust positively invariant (mRPI) set are obtained by using a shrinking procedure. Based on the mRPI set, the healthy and faulty residual sets can be obtained. Then, by considering the dual case of the set-separation constraint regarding the healthy and faulty residual sets, we transform the guaranteed MDF problem based on the set-separation constraint into a simple linear programming (LP) problem to compute the magnitude of MDF. Since the proposed MDF computation method is robust regardless of the value of scheduling variables in a given convex set, fault detection (FD) can be guaranteed whenever the magnitude of fault is larger than that of the MDF. At the end of the paper, a practical vehicle model is used to illustrate the effectiveness of the proposed method.

On the Necessity of LMI-Based Design Conditions for Discrete Time LPV Filters

In this note, we provide a proof of necessity for one of the results in [W. M. H. Heemels, J. Daafouz, and G. Millerioux, “Observer-based control of discrete-time LPV systems with uncertain parameters,” Automatic Control, IEEE Transactions on, vol. 55, no. 9, pp. 2130–2135, 2010]. This result is essential to the design of discrete-time linear parameter varying (LPV) filters using parameter-dependent poly-quadratic Lyapunov functions. We also provide a brief discussion ofits broader implications on the design of discrete-time LPV filters.

Relaxed observer-based controller design method of discrete-time multiplicative noised LPV systems via an extended projective lemma

ABSTRACTThe aim of this paper is to discuss a relaxed observer-based control problem of multiplicative noised Linear Parameter Varying (LPV) systems with unmeasurable states. For discussing the problem, positive definite matrices without neglecting any element are employed to construct a novel parameter-dependent Lyapunov function. Based on the Lyapunov function, some relaxed sufficient conditions are derived to hold the freedom in searching feasible solutions. Furthermore, an extended projective lemma is developed to convert those conditions into Linear Matrix Inequality (LMI) form. Solving these LMI conditions, controller gain and observer gain can be simultaneously obtained in one-step procedure via using convex optimization algorithm. Therefore, an observer-based controller can be established to guarantee the asymptotical stability of the multiplicative noised LPV systems in the sense of mean square. At last, two numerical examples are used to show effectiveness and applicability of the proposed design method.

On exploiting inexact scheduling parameters for gain-scheduled control of linear parameter-varying discrete-time systems

This paper is concerned with the problem of gain-scheduled controller design for parameter-varying discrete-time systems. In practice, the scheduling parameters are only available with limited accuracy due to the measurement errors. In the present paper, these uncertainties are systematically taken into account for the gain-scheduled controller synthesis to achieve improved performance in practical situations. It is assumed that all the matrices of the state-space model of the plant have a homogeneous polynomial dependency of arbitrary degree on the scheduling parameters. The merit of the proposed method is its capability in dealing with the measurement error less conservatively than available approaches while it can also encompass the traditional methods employing the exact scheduling parameters in a less conservative manner. A comprehensive numerical example demonstrates the effectiveness of the proposed method.

Actuator and sensor faults estimation for discrete-time descriptor linear parameter-varying systems in finite frequency domain

ABSTRACTThis paper investigates the problem of fault estimation for discrete-time descriptor linear parameter-varying systems with actuator faults, sensor faults and external disturbances. First, in order to estimate the actuator and sensor faults simultaneously, an augmented system is constructed, where the auxiliary state vector consists of the original state vector and the sensor fault vector. Then, considering the fact that many faults usually occur in finite frequency ranges, a novel fault estimation observer with finite frequency specifications is designed, where the conservatism can be reduced compared with the traditional fault estimation design methods in full frequency domain. By employing the generalised Kalman–Yakubovich–Popov lemma and two finite frequency H∞ performance indices, the sufficient conditions of the proposed observer are given in terms of linear matrix inequalities. Furthermore, by using slack variables, improved results on the fault estimation observer design are obtained, which are convenient to calculate fault estimation observer parameters for different frequency ranges. An example is presented to illustrate the effectiveness of the proposed method.

Discrete-time H∞ control of linear parameter-varying systems

ABSTRACTWe introduce new conditions for the H∞ synthesis of discrete-time linear parameter-varying systems in the form of linear matrix inequalities. A distinctive feature of the proposed conditions is the ability to handle variation in both the dynamics and the input matrices without resorting to dynamic augmentation or iterative procedures. We show that this new condition contains the existing poly-quadratic H∞ synthesis result as a particular case. We also derive a corollary which shows improvement even in the stronger case of quadratic H∞ synthesis. Additionally, we show that, surprisingly, a dynamic gain-scheduled quadratic H∞ controller can result in inferior performance compared to a static robust controller. Numerical examples illustrate our results.

Interval Observer Design for Actuator Fault Estimation of Linear Parameter-Varying Systems

This work is devoted to fault estimation of discrete-time Linear Parameter-Varying (LPV) systems subject to actuator additive faults and external disturbances. Under the assumption that the measurement noises and the disturbances are unknown but bounded, an interval observer is designed, based on decoupling the fault effect, to compute a lower and upper bounds for the unmeasured state and the faults. Stability conditions are expressed in terms of matrices inequalities. A case study is used to illustrate the effectiveness of the proposed approach.