Discrete-time Linear Parameter Varying Research Articles

Finite-time control for discrete-time nonlinear Markov switching LPV systems with DoS attacks

This work studies the finite-time control problem for discrete-time nonlinear Markov switching linear parameter varying (LPV) systems with denial-of-service (DoS) attacks. The aperiodic DoS attacks are introduced into the systems with partially unknown transition probabilities. First, considering the characteristics of aperiodic DoS attacks, an appropriate feedback controller is designed according to that whether it is attacked or not. Furthermore, the finite-time stability criteria of closed-loop system are derived by means of iterative technique and multiple Lyapunov functions. Finally, a turbofan engine model is taken as an example to verify the effectiveness and feasibility of the proposed finite-time control method.

Transformed Parameter Dependent Sliding Mode Control for Discrete-Time LPV Systems

This paper addresses the problem of sliding mode control for Discrete-time Linear Parameter Varying (DTLPV) systems. In the DTLPV system, a plant includes a time-varying parameter that is measurable and bounded in magnitude. By utilizing the information of the parameter, the transformed parameter-dependent sliding mode controller (TPDSMC) is firstly proposed. The transformed parameter is obtained by scaling and biasing the parameter of the plant. To design the TPDSMC, the sufficient condition is derived by using the transformed parameter-dependent approach. The designed controller not only reduces the effect of disturbance but also provides better control performances when the scale of the transformed parameter decreases. In the numerical example, the mass-spring damper system is considered to provide the robustness and performance of the proposed method.

A New Discrete-Time Interval Estimator for Vehicle Side-Slip Angle Estimation

The information about vehicle side-slip angle is essential given its relation to lateral stability and its importance for active safety control systems. Since direct measurement of the side-slip angle is expensive, developing efficient indirect techniques to extract this information from available sensor measurements is essential. This paper proposes a novel, cost-effective indirect strategy for the robust estimation of the side-slip angle using a state observer that relies on a discrete-time Linear Parameter Varying (LPV) lateral vehicle dynamic model. The proposed algorithm relies on a finite-time interval LMI-based observer, which accounts for the uncertainties on the model parameters (cornering stiffness) with known upper and lower uncertainty bounds. The simulation results show the effective and robust estimation performance of the proposed observer under various changes in system parameters. Moreover, comparison results with ℋ∞ set-membership observer for discrete-time LPV systems show the effectiveness of the developed observer.

Exponential Stabilization of LPV Systems Under Magnitude and Rate Saturating Actuators

We propose a new convex linear parameter varying (LPV) controller design conditions that regionally stabilize LPV discrete-time systems under magnitude and rate saturating actuators. Moreover, the closed-loop system has an ensured exponential stability performance. The design conditions provide regional stabilization in the sense of polyquadratic stability, allowing an estimate of the region of initial conditions yielding trajectories with guaranteed convergence to the origin. A nonlinear first-order model models the magnitude and rate saturation of the actuators. Additionally, we propose a convex optimization procedure to enlarge the estimate of the region of attraction. Our approach is used in real-time nonlinear level control, illustrating the potentialities of the practical application of the method.

Discrete-time LPV Systems with Inexact Scheduling Parameters: an Unknown Input Observer Design approach

In this paper, a novel unknown input observer (UIO) design for discrete-time linear parameter-varying (LPV) with inexact scheduling parameters is addressed. The proposed approach is particularly relevant as most LPV design conditions require that the time-varying parameters of the system are completely known and real-time available. Therefore, such strategies do not present adequate formulations to deal with such types of uncertainty. Moreover, UIOs have significant applications in different areas, such as fault detection and isolation problems. Existence conditions for the UIO are provided in order to properly handle the mismatch along with the presence of unknown inputs in the LPV system. The design conditions are based on an input-to-state stability approach obtained from a parameter-dependent Lyapunov function and are cast in the form of Linear Matrix Inequalities. The effectiveness of the proposed methods is illustrated using a numerical example.

Event-triggered policy for dynamic output stabilization of discrete-time LPV systems under input constraints

This paper concerns the event-triggered dynamic output-feedback control of discrete-time linear parameter-varying (LPV) systems subject to saturating actuators. Two independent event-triggering schemes are introduced to determine whether the current signals should be transmitted (a) from the sensor to the controller and (b) from the controller to the actuator. As a result, the communication resources can be significantly saved. Both the emulation-based problem and the co-design problem are addressed. Sufficient conditions based on linear matrix inequalities (LMIs) are derived to ensure the regional asymptotic stability of the origin for the closed-loop system. A convex optimization procedure is proposed to determine the controller matrices and the event-triggering parameters aiming at reducing the number of updates on the independent channels sensor-to-controller and controller-to-actuator. At last, numerical examples are employed to testify to the validity of the proposed methods. • We propose convex co-design conditions for discrete-time saturated LPV systems. • The proposed controller is a dynamic output feedback controller with anti-windup term. • The output and control updates are independently orchestrated by two ETMs. • The proposed convex co-design condition can be adapted to recover the emulation mode. • Convex optimization procedures are provided to reduce the sampling activity.

Co-design of an event-triggered dynamic output feedback controller for discrete-time LPV systems with constraints

This paper investigates an event-triggered control design approach for discrete-time linear parameter-varying (LPV) systems under control constraints. The proposed conditions can simultaneously design a parameter-dependent dynamic output feedback controller and an event generator, ensuring the closed-loop system’s regional asymptotic stability. Based on the Lyapunov stability theory, these conditions are given in terms of linear matrix inequalities (LMIs). Moreover, using some proposed optimization procedures, it is possible to minimize the number of sensor transmissions, maximize the estimation of the region of attraction of the origin, and incorporate optimal control criteria into the formulation. Through numerical examples, some comparisons with other approaches in the literature evidence the proposed technique’s efficacy.

Causal Gain-scheduled output feedback controllers using parameter-dependent Lyapunov Functions

This paper addresses the design problem of Gain-Scheduled Output Feedback (GSOF) controllers, in which causality with respect to (w.r.t) scheduling parameters in the GSOF controllers is kept, for continuous-/discrete-time Linear Parameter-Varying (LPV) systems using Parameter-Dependent Lyapunov Functions (PDLFs). In general, continuous-time GSOF controllers designed by conventional methods, i.e. so-called change-of-variables using PDLFs, depend on the derivatives of scheduling parameters, and discrete-time GSOF controllers designed by conventional methods or extended Linear Matrix Inequality (LMI) technique using parameter-dependent auxiliary matrices both depend on the one-step-ahead scheduling parameters. These mean that the designed GSOF controllers are not implementable to practical systems due to the non-causality w.r.t. scheduling parameters. On this issue, we propose a formulation which circumvents the causality problem by replacing the term that introduces the causality issue with another term via the reverse use of the Elimination lemma which is also known as “S-variable” approach. A toy example and a practical example (lateral-directional motion control around wing level flight of JAXA’s research airplane MuPAL-α) are included to demonstrate the effectiveness compared to the existing methods.

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.

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.

Stability analysis of discrete-time LPV switched systems

This paper addresses the stability problem for discrete-time switched systems under autonomous switching. Each mode of the switched system is modeled as a Linear Parameter Varying (LPV) system, the time-varying parameters can vary arbitrarily fast and are represented in a polytopic form. The Lyapunov theory is employed to get new conditions in the form of parameter-dependent LMIs. The constructed Lyapunov function takes advantage of using an augmented state vector with shifted states in its construction. In this sense, the Lyapunov function employed in this paper can be viewed as a discrete-time LPV switched Lyapunov function. Numerical experiments illustrate the efficacy of the technique in providing stability certificates.

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.

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.

Aircraft Gust Alleviation Preview Control with a Discrete-Time LPV Model in Consideration of the Elastic Mode

This paper deals with a gust alleviation (GA) control system using gain-scheduled (GS) discrete-time preview control. A linear parameter-varying (LPV) model of an aircraft including a single elastic mode is developed. In order to effectively attenuate the vertical acceleration of a flexible aircraft in turbulence, we propose a two-input control system, in which the aileron in addition to the elevator is used as a control surface. It is shown that when turbulence is acting on the aircraft, the vertical acceleration of the aircraft is effectively reduced with the proposed control system through some numerical simulations.

Discrete-Time LPV Observer With Nonlinear Bounded Varying Parameter and Its Application to the Vehicle State Observer

In this paper, we designed a unique structure and developed a novel linear parameter varying (LPV) formulation design for a vehicle-model-based state observer. The design problem for a vehicle state observer was totally reformulated into an LPV system that depended on online accessible time-varying parameters. The proposed LPV formulation, which is presented for the first time in this paper, did not require high computational costs compared to a conventional approach using an extended Kalman filter (EKF) including a nonlinear term. With regard to vehicle state estimation using a global positioning system, we considered the measured course angle as a varying parameter. Observer gain scheduling was determined using an $H_2$ filter based on the linear matrix inequality approach. The state observer gain using a discrete-time $H_2$ filter could minimize the error from a disturbance to a state. The stability of the proposed algorithm is proven, and the performance of the proposed method was demonstrated with an experimental test through a performance comparison with the EKF.