Continuous-time Linear Parameter-varying Systems Research Articles

Sampled-Data State Estimation for LSTM.

This article first introduces a sampled-data state estimator design method for continuous-time long short-term memory (LSTM) neural networks with irregularly sampled output. To this end, the structure of the LSTM is addressed to obtain its dynamic equation. As a result, the LSTM neural network is modeled as a continuous-time linear parameter-varying system that is dependent on the gate units. For this system, the sampled-data Luenberger-and Arcak-type state estimator design methods are presented in terms of linear matrix inequalities (LMIs) by using the properties of the gate units. Lastly, the proposed method not only provides a numerical example for analyzing absolute stability but also demonstrates it in practice by applying a pre-trained behavior generation model of a robot manipulator.

Direct identification of continuous-time LPV state-space models via an integral architecture

In this paper, we present a block-structured architecture for direct identification of continuous-time Linear Parameter-Varying (LPV) state-space models. The proposed architecture consists of an LPV model followed by an integral block. This structure is used to approximate the continuous-time LPV system dynamics. The unknown LPV model matrices are estimated along with the state sequence by minimizing a properly constructed dual-objective criterion. A coordinate-descent algorithm is employed to optimize the desired objective, which alternates between computing the unknown LPV matrices and estimating the state sequence. Thanks to the linear parametric structure induced by the LPV model, the optimization variables within each coordinate-descent step can be updated analytically via ordinary least squares. Furthermore, in order to handle large-size datasets, we discuss how to perform optimization based on short-size subsequences. The effectiveness of the proposed methodology is demonstrated via an academic example and two case studies. The first case study consists of identifying an LPV model describing the behaviour of an electronic bandpass filter from benchmark experimental data. The second case study involves identification of the plasma safety factor from a tokamak plasma simulator.

Bounding the Trajectories of Continuous-Time LPV Systems with Parameters known in Real Time

Predicting the exact future of a Linear Parameter-Varying (LPV) system with its parameter exclusively known in real time is by definition an impossible task. In particular, it is difficult to quantify the error induced by a Zero-Order Hold (ZOH) discretization of the parameter which is not verified in practice (e.g. when the parameter depends on the states, inputs or outputs of the continuous-time system). Under a Lipschitz assumption, this paper upper bounds — in terms of uncertain matrices — the greatest possible discrepancy between the real future of the system and its estimate based on the last known values of the input and parameter. This not only upper bounds the error due to the ZOH discretization, but also provides sufficient conditions for controllability and observability of the system in the near future by bounding its Gramians.

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.

Adaptive Threshold Generation for Vehicle Fault Detection Using Switched T–S Interval Observers

This paper is concerned with the robust passive fault detection problem for switched continuous-time linear parameter-varying systems with mensurable and unmeasurable scheduling parameters. A switched Takagi–Sugeno (TS) interval observer is designed to estimate the set of admissible state values. Using multiple fuzzy input to state stability-Lyapunov function and average dwell time (ADT) concept, sufficient conditions to guarantee the convergence and the robustness of the proposed observer are obtained. These conditions are formulated as a linear matrix inequality (LMI) problem. In contrast to the existing results based on multiple Lyapunov function and ADT switching, the derived conditions lead to less conservative LMIs characterization. Subsequently, the residual intervals are generated using the designed interval observer and used directly for fault detection (FD) decision making. Finally, the proposed methodology is tested using two examples. First, an academic example is used to illustrate the obtained relaxation. Second, a nonlinear vehicle model corrupted by faults is considered. Longitudinal velocity and cornering stiffness coefficients are treated, respectively, as the measurable and unmeasurable scheduling parameters. Simulation results based on experimental data show the effectiveness of the proposed FD schema.

State-feedback control for continuous-time LPV systems with polynomial vector fields

This paper is concerned with the design of state-feedback controllers for Linear Parameter Varying (LPV) polynomial continuous-time systems. The vector field presents polynomial dependence on the states. Two synthesis conditions are proposed, the first one considers arbitrary rates of variation in the time-varying parameters, while the second one provides LPV controllers that are constructed based on a smoothed approximation of the time-varying parameter. The sum of squares matrix decomposition is employed to solve the proposed conditions. The L2 gain is also considered to give a robustness measure of the proposed controllers. The results are illustrated with examples from the literature.

Interval observer design for continuous-time linear parameter-varying systems

This paper studies interval observer design for linear parameter-varying systems subject to uncertainties in parameters, input and output. Based on a general assumption that these uncertainties are unknown but bounded by known intervals, a novel interval observer is presented to estimate the bounds of state. To attenuate the effect of uncertainties and obtain tight state intervals, a novel L∞ norm based design method is proposed. Numerical simulations are conducted to demonstrate the performance of the proposed method.

Event-Triggered Fault Detection for Networked LPV Systems

This paper is concerned with the fault detection problem for a class of continuous-time linear parameter-varying systems with signal transmission delays. An effective event-triggered communication scheme is introduced to reduce the burden of the shared network where the current sampled data will be sent only when the certain condition is satisfied. By properly designing a novel fault detection filter and augmenting the states of the original system, the addressed fault detection problem can be transformed into a $$H_\infty $$ filtering problem for the filtering error system with uncertain parameters. According to the parameter-dependent Lyapunov–Krasovskii functional method and free-weighting matrix technique, the sufficient conditions, which guarantee the filtering error system satisfying the prescribed $$H_\infty $$ performance constraint, are derived in the form of parameterized linear matrix inequalities. The basic functions and gridding technique are used to deal with the corresponding parameterized convex problem. Here, the LPVTools is used to convert the infinite-dimensional feasibility conditions into a finite-dimensional set of LMIs. Then, the MATLAB LMI toolbox is applied for solving the LMI problem of finite dimensions. Moreover, the explicit expressions of the target filter parameters are also obtained. Finally, two simulation examples are provided to illustrate the validity of the proposed fault detection method.

Pre-filtering in continuous-time quadratic gain-scheduled and robust state-feedback control

ABSTRACTWe revisit the issue of quadratic gain-scheduled and robust state-feedback control with a focus on linear matrix inequalities widely used in the literature. It has been established that for continuous-time linear parameter-varying systems which depend affinely on the scheduling parameter, gain-scheduled stabilisability implies robust stabilisability. In other words, as far as stabilisation is concerned, using a more complex gain-scheduled controller is not advantageous. The proof of such equivalence is non constructive and, although very general, limited to stabilisability. One contribution of the present paper is to establish constructive proofs in the quadratic case that can be extended to performance optimisation in terms of quadratic and norms. One issue investigated in detail is the strategy of using pre-filters to handle parameter-variation in the input matrix. Our results show that said technique, which has been popular since the 1980s, is rarely productive in the sense that solvability of quadratic gain-scheduled control design problems for the original system augmented with an arbitrary pre-filter implies existence of a lower-dimensional quadratic robust controller for the original system, which we construct explicitly via projections.

State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers

ABSTRACTThis paper proposes a linear parameter varying (LPV) interval observer for state estimation and unknown inputs decoupling in uncertain continuous-time LPV systems. Two different problems are considered and solved: (1) the evaluation of the set of admissible values for the state at each instant of time; and (2) the unknown input observation, i.e. the design of the observer in such a way that some information about the nature of the unknown inputs affecting the system can be obtained. In both cases, analysis and design conditions, which rely on solving linear matrix inequalities (LMIs), are provided. The effectiveness and appeal of the proposed method is demonstrated using an illustrative application to a two-joint planar robotic manipulator.

Data-driven predictive control for continuous-time linear parameter varying systems with application to wind turbine

A new data-driven predictive control method based on subspace identification for continuous-time linear parameter varying (LPV) systems is presented in this paper. It is developed by reformulating the continuous-time LPV system which utilizes Laguerre filters to obtain the subspace prediction of output. The subspace predictors are derived by QR decomposition of input-output and Laguerre matrices obtained by input-output data. The predictors are then applied to design the model predictive controller. It is shown that the integrated action is incorporated in the control effect to eliminate the steady-state offset. We control the continuous-time LPV systems to obtain the attractive performance with the proposed data-driven predictive control method. The proposed controller is applied to a wind turbine to verify its effectiveness and feasibility.

Discretization and event triggered digital output feedback control of LPV systems

This paper investigates the problem of discretization and digital output feedback control design for continuous-time linear parameter-varying (LPV) systems subject to a time-varying networked-induced delay. The proposed discretization procedure converts a continuous-time LPV system into an equivalent discrete-time LPV system based on an extension of the Taylor series expansion and using an event-based sampling. The scheduling parameters are continuously measured and modeled as piecewise constant. A new transmission of the measured output to the controller is triggered by significant changes in the parameters, yielding time-varying transmission intervals. The obtained discretized model has matrices with polynomial dependence on the time-varying parameters and an additive norm-bounded term representing the discretization residual error. A two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the LPV model. The conditions can also be used to provide robust (i.e., independent of the scheduling parameter) static output feedback controllers. The viability of the proposed design method is illustrated through numerical examples.

Sampled‐data MPC for LPV systems with input saturation

In this work, a sampled-data model predictive control (MPC) design method is proposed for continuous-time linear parameter varying (LPV) systems. The input saturation and parameter uncertainties are both considered. Using a method to deal with actuator saturation, the MPC controller is permitted to saturate. Using the measurable parameter vector, a scheduled state-feedback MPC controller is computed at each time instant which fully exploits the real-time information on the variations of the plant characteristics. By modelling the closed-loop systems of the continuous-time LPV systems with a piecewise constant sampled-data control input as linear impulsive systems, the stability properties of the proposed MPC are studied. The sampling interval in this work is not required to be periodic. The proposed MPC design method is expected to further reduce the conservativeness. The improvements of the proposed sampled-data MPC method w.r.t. other existing MPC techniques are demonstrated by an example.

Output-feedback sampled-data control design for linear parameter-varying systems with delay

In this paper, we address the sampled-data output-feedback control design problem for continuous-time linear parameter-varying systems with time-varying delay in the system states. Due to the combination of the plant's continuous-time dynamics and the controller's discrete-time dynamics connected through A/D and D/A converter devices, the closed-loop system is a hybrid system. In order to analyse this hybrid system from stability and performance perspectives we use the input-delay approach to map the closed-loop system into the continuous-time domain with delay in the states. This results in a closed-loop system containing two types of delays, the system internal delay and the one imposed by the mapping. Next, we use delay-dependent conditions for analysis of stability and -norm performance which result in a sampled-data control synthesis procedure. The proposed output-feedback sampled-data controller is obtained based on the solution to a linear matrix inequality optimisation problem using a set of appropriately defined slack variables. A numerical example of a milling machine is presented to demonstrate the viability of the proposed sampled-data control design method to satisfy the stability and performance objectives even with a varying sampling rate.

Gain-Scheduledℋ2Controller Synthesis for Continuous-Time Polytopic LPV Systems

This paper is concerned with the problem of gain-scheduledℋ2controller synthesis for continuous-time linear parameter-varying systems. In this problem, the system matrices in the state-space form are polytopic and patameterized and the admissible values of the parameters are assumed to be measurable on-line in a polytope space. By employing a basis-parameter-dependent Lyapunov function and introducing some slack variables to the well-established performance conditions, sufficient conditions for the existence of the desired gain-scheduledℋ2state feedback and dynamic output feedback controllers are established in terms of parameterized linear matrix inequalities. Based on the polytopic characteristic of the dependent parameters and a convexification method, the corresponding controller synthesis problem is then cast into finite-dimensional convex optimization problem which can be efficiently solved by using standard numerical softwares. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.

Sampled-data filtering for linear parameter varying systems

In this paper, we address the sampled-data filter design problem for continuous-time linear parameter-varying (LPV) systems. The filtering error system obtained from augmenting a continuous-time LPV system and the sampled-data filter is a hybrid system. The sampled-data filter design objective is to ensure the error system stability and a prescribed level of the induced energy-to-energy gain (or norm) from the disturbance input to the estimation error. To this purpose, we employ a lifting method to derive an equivalent discrete-time LPV representation for the continuous-time LPV system. In the present study, the sampled-data filter synthesis conditions are formulated in terms of linear matrix inequality optimisation problems. The viability of the proposed design method to cope with variable sampling rates is illustrated through numerical examples, where reliable estimation of the LPV system outputs is achieved.

On Interval Observer Design for a Class of Continuous-Time LPV Systems

This work is devoted to interval observer design for Linear Parameter-Varying (LPV) systems under assumption that the vector of scheduling parameters is not available for measurements. Stability conditions are expressed in terms of matrix inequalities, which can be solved using standard numerical solvers. Robustness and estimation accuracy with respect to model uncertainty is analyzed using L ∞ /L 1 framework. Two solutions are proposed for nonnegative systems and for a generic case. The efficiency of the proposed approach is demonstrated through computer simulations.