Gain-scheduled State-feedback Control Research Articles

Improved Embedding of Nonlinear Systems in Linear Parameter-Varying Models With Polynomial Dependence

In this article, the problem of automated generation of linear parameter-varying (LPV) state-space models to embed the dynamic behavior of nonlinear (NL) systems is addressed. The LPV model depends polynomially on an introduced set of scheduling variables. This set comprises linear combinations of residuals, the differences between NL functions in the original model description with their polynomial approximations, in addition to some of the states. The salient feature of the proposed method (PM) is that the LPV model complexity, LPV model accuracy, and LPV model conservativeness are jointly considered through the embedding procedure. To quantitatively evaluate the LPV model accuracy and conservativeness, two cost functions are introduced based on which the model complexity can be adjusted. Numerical studies reveal that the presented method is capable to deliver more accurate and less conservative LPV models in comparison with the available approaches. The effectiveness of the PM is shown in an empirical case study where the dynamic behavior of a 3DOF gyroscope is embedded into an LPV model. Exploiting the obtained LPV model, a gain-scheduled state feedback controller is designed and validated on the gyroscope, clearly demonstrating the applicability of the presented method.

Feedback Ratiometric Control of Two Microbial Populations in a Single Chemostat

We address the problem of controlling the dilution rate in a chemostat to regulate the ratio between the concentrations of two microbial populations growing in continuous culture. After analyzing the open-loop dynamics of this multicellular system, we present two alternative feedback control strategies, one based on a gain-scheduled state feedback controller, the other on a switching control strategy. We show that both strategies are effective in solving the problem and illustrate the results by a set of representative in-silico experiments.

Robust State-Feedback Controller for Linear Parameter Varying Systems with Time-Invariant Uncertainties

In this paper, the robust gain-scheduled state-feedback controller problem is studied for uncertain linear parameter-varying systems whose state-space representations are the linear combination of the uncertain time-varying parameters including time-invariant parametric uncertainties (TIPU). It is supposed that these uncertainties are bounded by the given intervals and cannot be pulled out as an uncertain block. It is considered a serious challenge because the exact information about the plant dynamics cannot be extracted from the uncertain time-varying parameters. This is while; the gain-scheduled controllers need to have the exact information about the plant dynamics to satisfy the desired control purposes. To handle this challenge, we introduce a state-feedback gain, which is formed by a set of the new scheduling parameters and a secondary time-varying term. The stabilization conditions are obtained in terms of the linear matrix inequalities (LMIs). The effectiveness of the proposed method is shown using an example.

New gain-scheduling control conditions for time-varying delayed LPV systems

This paper investigates the problem of stability and state-feedback control design for linear parameter-varying systems with time-varying delays. The uncertain parameters are assumed to belong to a polytope with bounded known variation rates. The new conditions are based on the Lyapunov theory and are expressed through Linear Matrix Inequalities. An alternative parameter-dependent Lyapunov-Krasovskii functional is employed and its time-derivative is handled using recent integral inequalities for quadratic functions proposed in the literature. As main results, a novel sufficient stability condition for delay-dependent systems as well as a new sufficient condition to design gain-scheduled state-feedback controllers are stated. In the new proposed methodology, the Lyapunov matrices and the system matrices are put separated making it suitable for supporting in a new way the design of the stabilization controller. An example, based on a model of a real-world problem, is provided to illustrate the effectiveness of the proposed method.

Artificial Neural Network-Based Gain-Scheduled State Feedback Speed Controller for Synchronous Reluctance Motor

Abstract This paper focuses on designing a gain-scheduled (G-S) state feedback controller (SFC) for synchronous reluctance motor (SynRM) speed control with non-linear inductance characteristics. The augmented model of the drive with additional state variables is introduced to assure precise control of selected state variables (i.e. angular speed and d-axis current). Optimal, non-constant coefficients of the controller are calculated using a linear-quadratic optimisation method. Non-constant coefficients are approximated using an artificial neural network (ANN) to assure superior accuracy and relatively low usage of resources during implementation. To the best of our knowledge, this is the first time when ANN-based gain-scheduled state feedback controller (G-S SFC) is applied for speed control of SynRM. Based on numerous simulation tests, including a comparison with a signum-based SFC, it is shown that the proposed solution assures good dynamical behaviour of SynRM drive and robustness against q-axis inductance, the moment of inertia and viscous friction fluctuations.

Dwell-time based stability analysis and L2 control of LPV systems with piecewise constant parameters and delay

Abstract Dwell-time based stability conditions for a class of LPV systems with piecewise constant parameters under time-varying delay are derived using clock-dependent Lyapunov–Krasovskii functional. Sufficient synthesis conditions for clock-dependent gain-scheduled state-feedback controllers ensuring L 2 -performance are also provided. Several numerical and practical examples, to illustrate the efficacy of the results, are given.

State-Feedback Control for Cyber-Physical LPV Systems Under DoS Attacks

This letter addresses the state-feedback control problem for Cyber-physical Linear Parameter Varying systems. Both the feedback state and the time-varying parameter are affected by denial of service (DoS) attacks. The main goal is to design gain-scheduled state-feedback controllers that can mitigate the interference of this type of attack in the system. The system under attack is modeled as a switched linear parameter-varying system, and new conditions in the form of parameter-dependent Linear Matrix Inequalities are presented to provide stabilizing controllers with exponential decay rate. Moreover, lifted conditions based on the use of post-attack dynamics, that may improve the performance of the obtained controllers are presented. Numerical experiments illustrate the effectiveness of the proposed method.

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.

Online Estimation and Control for Feed Drive Systems With Unmeasurable Parameter Variations

This paper develops an online parameter estimation and control method for both rigid and flexible feed drive systems with unmeasurable parameter variations. The perturbations of the state-space model caused by the parameter variations are formulated, thereby making it possible to obtain the parameter variations in real-time by estimating the perturbations. To estimate the perturbations, they are regarded as the extended states and estimated through the extended-state-observer. With the estimation method, a novel state feedback control structure with double integrators is further proposed, which can eliminate the steady-state tracking error at constant velocities. The H$_\infty $ optimization technique is used to design the proposed state feedback controller. A simulation is conducted that integrates the estimator and the controller for a ball screw setup with mass-dependent resonant modes, where several proposed state feedback controllers are linearly interpolated into a gain-scheduling controller and scheduled by the estimated mass. The results demonstrate that the designed gain-scheduling state feedback controller outperforms a linear-time-invariant state feedback controller and an adaptive backstepping sliding mode controller. The proposed method is experimentally validated on a rigid linear-motor-driven motion stage, of which the results indicate the proposed estimation method can accurately estimate the parameter variations.

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.

Gain‐scheduled constrained state feedback control of DC–DC buck power converter

A gain-scheduled constrained state feedback controller (SFC) for a SiC MOSFET direct current (DC)-DC buck power converter with a non-synchronous control strategy is presented. First, the synthesis process of a gain-scheduled SFC for a control system with a state-dependent parameter is proposed. Next, constraints of important states and control variables are a posteriori introduced into control algorithm to provide satisfactory dynamic behaviour and safe operation of the plant. The boundary values of control signals which provide permissible values of the future state-space variables are derived from the models of power converters that take into account two modes of operation (i.e. charging and discharging mode). The novelty of the proposed approach lies in efficient control algorithm that utilises the gain-scheduled SFC with the parameter-dependent constraint handling method. The designed control algorithm is extensively tested through simulation and experiments at a switching frequency of 36 kHz. The proposed solution is further compared with a conventional cascade control structure based on two proportional integral controllers for voltage and current loops as well as with a non-constrained state feedback control structure to illustrate its performance and low computational complexity.

Stability Analysis and State-Feedback Control of LPV Systems With Piecewise Constant Parameters Subject to Spontaneous Poissonian Jumps

Linear parameter-varying systems with piecewise constant parameters subject to spontaneous Poissonian jumps are a class of systems that does not seem to have been thoroughly considered in the literature. We partially fill this gap here by providing sufficient stability and performance analysis conditions stated in terms of infinite-dimensional LMI problems that can be solved using sum of squares programming. A particularity of the obtained conditions lies in the presence of an integral term leading to some technical difficulties when attempting to obtain convex conditions for the design of a gain-scheduled state-feedback controller. This difficulty is circumvented by relying on a recent result for time-delay systems analysis and an equivalent integral-free LMI condition is obtained. The approach is illustrated through several examples.

Robust Disturbance Rejection by the Attractive Ellipsoid Method – Part I: Continuous-time Systems

Abstract This paper develops sufficient conditions for the constrained robust stabilization of continuous-time polytopic linear systems with unknown but bounded perturbations. The attractive ellipsoid method (AEM) is employed to determine a robustly controllable invariant set, known as attractive ellipsoid, such that the state trajectories of the system asymptotically converge to a small neighborhood of the origin despite the presence of non-vanishing perturbations. To solve the stabilization problem, we employ the Finsler’s lemma and derive new linear matrix inequality (LMI) conditions for robust state-feedback control design, ensuring convergence of state trajectories of the system to a minimal size ellipsoidal set. We also consider the state and control constrained problem and derive extended LMI conditions. Under certain conditions, the obtained LMIs guarantee that the attractive ellipsoid is nested inside the bigger ellipsoids imposed by the control and state constraints. Finally, we extend our AEM approach to the gain-scheduled state-feedback control problem, where the scheduling parameters governing the time-variant system are unknown in advance but can be measured in real-time. Two examples demonstrate the feasibility of the proposed AEM and its improvements over previous works.

Robust Disturbance Rejection by the Attractive Ellipsoid Method – Part II: Discrete-time Systems

Abstract This paper presents sufficient conditions for the robust stabilization of discrete-time polytopic systems subject to control constraints and unknown but bounded perturbations. The attractive ellipsoid method (AEM) is extended and applied to cope with this problem. To tackle the stabilization problem, new linear matrix inequality (LMI) conditions for robust state-feedback control are developed. These conditions ensure the convergence of state trajectories of the system to a minimal size ellipsoidal set despite the presence of non-vanishing disturbances. The developed LMI conditions for the AEM are extended to deal with the problem of gain-scheduled state-feedback control, where the scheduling parameters governing the time-variant dynamical system are unknown in advance but can be measured in real-time. A feature of the obtained conditions is that the state-space matrices and Lyapunov matrix are separated. The desired robust control laws are obtained by convex optimization. Numerical simulations are given to illustrate the feasibility of the proposed AEM for robust disturbance rejection.

Gain-scheduled H∞ admissibilisation of LPV discrete-time systems with LPV singular descriptor

ABSTRACTThis paper deals with the admissibilisation with guaranteed H∞ performance of linear parameter-varying (LPV) discrete-time singular systems under gain-scheduled (GS) state-feedback control. In particular, we focus on systems with descriptor matrix of LPV form and unknown/unmeasurable disturbance input. To our best knowledge and despite its practical interest, this problem has not been fully investigated and still poses relevant issues, which motivates our study. By means of a proposed descriptor redundancy approach, in combination with the bounded real lemma and Finsler's Lemma, a method based on linear matrix inequalities is proposed to design a GS state-feedback control with maximum H∞ performance. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

Stabilization of linear time-varying systems with state and input constraints using convex optimization

The stabilization problem of linear time-varying systems with both state and input constraints is considered. Sufficient conditions for the existence of the solution to this problem are derived and a gain-switched(gain-scheduled) state feedback control scheme is built to stabilize the constrained timevarying system. The design problem is transformed to a series of convex feasibility problems which can be solved efficiently. A design example is given to illustrate the effect of the proposed algorithm.

Gain-Scheduled Vehicle Handling Stability Control Via Integration of Active Front Steering and Suspension Systems

This paper proposes an integrated vehicle dynamics control system that aims to enhance vehicles handling stability and safety performance by coordinating active front steering (AFS) and active suspension systems (ASS). The integrated controller design is based on the lateral stability region described by phase plane approach that is employed to bound the vehicle stability and coordinate AFS and ASS. During normal steering conditions, the vehicle states lie inside the lateral stability region, only the AFS is involved for vehicle steerability enhancement. Whereas, when the vehicle reaches the handling limits and the vehicle states go outside the lateral stability region under extreme steering maneuvers, both AFS and ASS collaborate together to improve vehicle handling stability. The linear parameter-varying (LPV) polytopic vehicle model is built, which depends affinely on the time-varying longitudinal speed that is described by a polytope with finite vertices. The resulting gain-scheduling state-feedback controller is designed and solved utilizing a set of linear matrix inequalities derived from quadratic H∞ performance. Simulation using matlab/simulink-carsim® is carried out to evaluate the performance of the integrated controller. The simulation results show the effectiveness of the proposed controller.

Stability analysis and control of a class of LPV systems with piecewise constant parameters

Stability criteria characterizing the asymptotic stability of a class of LPV systems with piecewise constant parameters under constant and minimum dwell-time are derived. It is shown that, for such systems, the conditions for the stability under minimum dwell-time can be seen as a unifying stability concept lying in between quadratic and robust stability, thereby including them as extremal cases. The results are then extended to address the stabilization problem using a particular class of time-dependent gain-scheduled state-feedback controllers. Several examples are given for illustration.

Gain-scheduled robust control for lateral stability of four-wheel-independent-drive electric vehicles via linear parameter-varying technique

This paper proposes a robust gain-scheduled H∞ controller for lateral stability control of four-wheel-independent-drive electric vehicles via linear parameter-varying technique. The controller aims at tracking the desired yaw rate and vehicle sideslip angle by controlling the external yaw moment. In the design of controller, uncertain factors such as vehicle mass and tire cornering stiffness in vehicle lateral dynamics are represented via the norm-bounded uncertainty. To address the importance of time-varying longitudinal velocity for vehicle lateral stability control, a linear parameter-varying polytopic vehicle model is built, and the built vehicle model depends affinely on the time-varying longitudinal speed that is described by a polytope with finite vertices. In order to reduce conservative, the hyper-rectangular polytope is replaced by a hyper-trapezoidal polytope. Simultaneously, the quadratic D-stability is also applied to improve the transient response of the closed-loop system. The resulting gain-scheduling state-feedback controller is finally designed, and solved utilizing a set of linear matrix inequalities derived from quadratic H∞ performance and D-stability. Simulations using Matlab/Simulink-Carsim® are carried out to verify the effectiveness of the proposed controller with a high-fidelity, CarSim®, full-vehicle model. It is found from the results that the robust gain-scheduled H∞ controller suggested in this paper provides improved vehicle lateral stability, safety and handling performance.

Gain-Scheduled State Feedback Controllers for Discrete-Time LPV systems Using Scheduling Parameters Affected by Absolute and Proportional Uncertainties

This paper is concerned with the design problem of Gain-Scheduled State Feedback (GSSF) controllers for discrete-time Linear Parameter-Varying (LPV) systems. By considering that exact scheduling parameters are not available in many practical systems, the scheduling parameters provided to GSSF controllers are supposed to be affected by absolute as well as proportional uncertainties. We introduce a set covering the admissible set for the pair of scheduling parameters and the inexact ones, and propose a design method of GSSF controllers using the set. Numerical examples illustrate the effectiveness and possible conservatism of our method.