Discrete-Time LPV Controller Synthesis Using Dilated LMIs with Application to an Arm-Driven Inverted Pendulum

In this paper, a linear parameter-varying (LPV) control design method is evaluated experimentally on an active magnetic bearing (AMB) system. Linear matrix inequality (LMI) conditions for control design of affine parameter-dependent systems using parameter-dependent Lyapunov functions are proposed. A speed-dependent LPV model of the AMB system is derived. Speed-dependent model uncertainties are identified using artificial neural networks (ANNs), and a parameter-dependent uncertainty weighting function is approximated for LPV control synthesis. Experiments are conducted to verify the robustness of LPV controllers for a wide range of rotor speeds. This LPV control approach eliminates the need for gain-scheduling, and provides better performance and less conservativeness over a wide range of rotational speeds than controllers designed with constant uncertainty weighting functions.

In this paper, a linear parameter-varying (LPV) control design method is evaluated experimentally on an active magnetic bearing (AMB) system. LMI synthesis conditions for control design of affine parameter-dependent systems using parameter-dependent Lyapunov functions are proposed. A speed-dependent LPV model of the AMB system is derived. Speed-dependent model uncertainties are identified using artificial neural networks (ANNs), and a parameter-dependent uncertainty weighting function is approximated for LPV control synthesis. Experiments are conducted to verify the robustness of LPV controllers for a wide range of rotor speeds. This LPV control approach eliminates the need for gain-scheduling, and provides better performance and less conservativeness over a wide range of rotational speeds than controllers designed with constant uncertainty weighting functions.

This paper investigates the design of a robust linear parameter-varying (LPV) controller for speed regulation of surface permanent magnet synchronous motors (SPMSMs). Motor dynamics is defined in the α–β stationary frame and a parameter-varying model formulation is provided to describe its nonlinear dynamics. A robust gain-scheduled LPV dynamic output-feedback controller is designed to guarantee the asymptotic stability and performance requirements, as well as, guarantee robustness against parameter perturbations and torque load disturbances. As a novel aspect in the SPMSM control, the real-time impact of temperature variation on the winding resistance and magnet flux during motor operation is explicitly considered in the LPV modeling as a scheduling parameter and the subsequent LPV control design compensating for demagnetization effects in the motor response. The controller synthesis conditions guaranteeing stability and robust performance to norm-bounded uncertainty in the system matrices are formulated in a convex linear matrix inequality (LMI) optimization framework. Finally, the validity of the proposed method is assessed in simulation studies. The closed-loop simulation studies demonstrate that the proposed LPV controller provides improved transient response with respect to settling time, overshoot, and disturbance rejection in tracking the velocity profile under the influence of parameter perturbation, temperature variations, and load disturbances.

This paper presents a new dilated linear matrix inequality (LMI) representation to design a state feedback controller and a dynamic feedforward disturbance compensator for linear parameter-varying (LPV) systems. The improved LMIs are convex and finite-dimensional without any iterative approach. The designs are based on a new proposed equivalent bounded real lemma (BRL) by means of matrix dilation for LPV systems and uncertain linear systems under time-varying parametric uncertainties (TVPUs). This dilated BRL provides lower conservative results than existing methods in terms of robust stability. Accordingly, a dynamic disturbance compensator is designed in addition to a state feedback controller. This paper mainly focuses on the design of compensators against disturbances in addition to the design of state feedback controllers. The dynamic matrices of the compensator change with the time-varying parameters of the LPV or uncertain system during operation, assuming that the disturbances and the parameters are measurable or observable. The compensator can be designed to attenuate the disturbances/noises or to improve reference tracking. Finally, numerical and simulation outcomes are presented to prove both the effectiveness and lower conservativeness of the proposed LMIs.

Robust controller synthesis attracts reviving research interest, driven by\nthe rise of learning-based systems where uncertainty and perturbation are\nubiquitous. Facing an uncertain situation, a robustly stabilizing controller\nshould maintain stability while operating under a perturbed system deviating\nfrom its nominal specification. There have been numerous results for robust\ncontroller synthesis in multiple forms and with various goals, including\nmu-synthesis, robust primal-dual Youla, robust input-output, and robust system\nlevel parameterizations. However, their connections with one another are not\nclear, and we lack a general approach to robust controller analysis and\nsynthesis.\n To serve this purpose, we derive robust stability conditions for general\nsystems and formulate the general robust controller synthesis problem. The\nconditions hinge on the realization-stability lemma, a recent analysis tool\nthat unifies existing controller synthesis methods. Not only can the conditions\ninfer a wide range of existing robust results, but they also lead to easier\nderivations of new ones. Together, we demonstrate the effectiveness of the\nconditions and provide a unified approach to robust controller analysis and\nsynthesis.\n

Preface Notation Part I. Introduction. Robust Decision Problems in Engineering: A linear matrix inequality approach L. El Ghaoui and S.-I. Niculescu Part II. Algorithms and Software: Mixed Semidefinite-Quadratic-Linear Programs J.-P. A. Haeberly, M. V. Nayakkankuppam and M. L. Overton Nonsmooth algorithms to solve semidefinite programs C. Lemarechal and F. Oustry sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure S.-P. Wu and S. Boyd Part III. Analysis: Parametric Lyapunov Functions for Uncertain Systems: The Multiplier Approach M. Fu and S. Dasgupta Optimization of Integral Quadratic Constraints U. Jonsson and A. Rantzer Linear Matrix Inequality Methods for Robust H2 Analysis: A Survey with Comparisons F. Paganini and E. Feron Part IV. Synthesis. Robust H2 Control K. Y. Yang, S. R. Hall and E. Feron Linear Matrix Inequality Approach to the Design of Robust H2 Filters C. E. de Souza and A. Trofino Robust Mixed Control and Linear Parameter-Varying Control with Full Block Scalings C. W. Scherer Advanced Gain-Scheduling Techniques for Uncertain Systems P. Apkarian and R. J. Adams Control Synthesis for Well-Posedness of Feedback Systems T. Iwasaki Part V. Nonconvex Problems. Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints K. M. Grigoriadis and E. B. Beran Bilinearity and Complementarity in Robust Control M. Mesbahi, M. G. Safonov and G. P. Papavassilopoulos Part VI. Applications:Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization J. Oishi and V. Balakrishnan Multiobjective Robust Control Toolbox for LMI-Based Control S. Dussy Multiobjective Control for Robot Telemanipulators J. P. Folcher and C. Andriot Bibliography Index.

This paper deals with the stabilization and the H 2 control synthesis of Linear Parameter Varying (LPV) descriptor systems. The compensator design problem is examined through a parameter-dependent observer-based control structure. The proposed approach uses parameter-dependent Lyapunov functions and involves slack variables in order to achieve less restrictive synthesis conditions. These synthesis conditions are expressed as a Linear Matrix Inequality (LMI) solvability problem. The admissibility and the H 2 performance are ensured by computing appropriate gains of the state-feedback controller and the observer. Simulation results obtained with a numerical example, show the effectiveness of the proposed approach.

This paper considers the optimal control of polytopic, discrete-time linear parameter varying (LPV) systems with a guaranteed l2 to l∞ gain. Additionally, to guarantee robust stability of the closed-loop system under parameter variations, H∞ performance criterion is also considered as well. Controllers with a guaranteed l2 to l∞ gain and a guaranteed H∞ performance (l2 to l2 gain) are a special family of mixed H2=H∞ controllers. Normally, H2 controllers are obtained by considering a quadratic cost function that balances the output performance with the control input needed to achieve that performance. However, to obtain an optimal controller with a guaranteed l2 to l∞ gain (closely related to the physical performance constraint), the cost function used in the H2 control synthesis minimizes the control input subject to maximal singular-value performance constraints on the output. This problem can be efficiently solved by a convex optimization with linear matrix inequality (LMI) constraints. The main contribution of this paper is the characterization of the control synthesis LMIs used to obtain an LPV controller with a guaranteed l2 to l∞ gain and >H∞ performance. A numerical example is presented to demonstrate the effectiveness of the convex optimization.

This paper presents an active flutter suppression design using linear parameter-varying (LPV) control method for a nonlinear aeroservoelastic model that captures wing-section stall flutter. The loss-of-effectiveness fault is considered for the control surface. The resulting model is a function of speed as well as effectiveness factor of the control surface. These two parameters are treated as the varying parameters to formulate an LPV representation of the wing-section model using Jacobian linearization. An H2 gain-scheduled controller based on a parameter-dependent Lyapunov function is designed for the LPV system. The simulation shows that it can suppress limit cycle oscillations over a range of speed and work under a certain range of control surface effectiveness loss.

The paper presents stabilization of a Networked Control System (NCS) with uncertain, time-varying delays using two stabilization approaches namely, Parameter Dependant Lyapunov Function (PDLF) and Quadratic Lyapunov Function (QLF). The PDLF approach depends upon the time varying delay parameters. Thus, for larger delays the system's stability is ensured. In NCS, the uncertainties arising out from the variations in time-delay is modeled using polytopic approach and the stability region of the system is presented in terms of controller gain and maximum tolerable delay. The overall NCS is represented in discrete domain. The proposed PDLF stabilization results have been compared with that of the QLF approach. Lyapunov-based stability criterion is presented in Linear Matrix inequalities (LMIs). The stabilized state feedback controller is designed using LMIs and considering the bounded time-varying delays. The proposed approach is illustrated by two numerical examples. From the results obtained, it is observed that the PDLF approach improves the stability region of the system.

This paper presents a controller design method for dealing with the induced ℒ 2 -norm problem for switched linear parameter-varying (LPV) systems. Considering the arbitrary switchings caused by the parameters varying, a common parameter-dependent Lyapunov function is employed to derive sufficient linear matrix inequality (LMI) conditions for the switched LPV systems. A family of LPV controllers are designed according to the LMI conditions, and each of them is suitable for the corresponding parameter region. The proposed switching LPV control method is applied to an F-16 aircraft longitudinal model and simulation results demonstrate the effectiveness of the approach.

We propose a less conservative linear matrix inequalities (LMIs) condition for ℋ∞ performance analysis and synthesis of discrete-time linear parameter varying (LPV) systems. The time varying parameter is represented by a linear fractional form. Sufficient conditions for ℋ∞ performance and controller synthesis are given in terms of a finite number of LMIs using parameter dependent Lyapunov functions.

Applications of a quadrotor with payload, particularly for chemical spraying, have increased in recent times. The variation in payload mass over time causes a change in the moments of inertia (MOI). Moreover, large tilt angles are required for fast reference tracking and external disturbance rejection. These variations in plant parameters (i.e., mass and inertia) and large tilt angles can degrade the control scheme’s performance and stability. This article proposes a linear matrix inequalities (LMIs)-based linear parameter varying (LPV) control scheme for a quadrotor subject to time-varying mass, time-varying inertia, mass flow rate, and large tilt angles. The control strategy is designed by solving LMIs derived from quadratic H∞ performance and D-stability. The robust stability and quadratic H∞ performance are assessed by LMIs. The efficacy of the proposed methodology is established using numerical simulations, and its performance is compared to the linear time-invariant (LTI) H∞ design with pole placement constraints. The results obtained show that the LPV control scheme gives better tracking performance in the presence of time-varying parameters, noise, and external disturbances without actuator saturation. In comparison to the LTI design technique, the proposed LPV scheme improves the rise time (tr), settling time (ts), and mean squared error (MSE) by up to 14%, 15%, and 30%, respectively. Moreover, smooth transitions are observed in the tilt angles and control signals with the LPV scheme, contrary to the LTI controller, which exhibits significant oscillations.

A linear parameter-varying (LPV) control approach is examined to determine if it is practical to use for largeenvelope e ight control designs. The basic approach is combined with π analysis to reduce conservatism, and the resulting mixed LPV/ π approach is used to design a longitudinal LPV controller for the F-16 Variable Stability In-Flight Simulator Test Aircraft. Implementation issues are discussed, and the controller is modie ed accordingly. The e nal closed-loop system is tested for robust stability and handling qualities throughout the subsonic e ight envelope. High-e delity nonlinear simulations are also used to verify performance. I. Introduction T HE challenge of most control problems is to obtain as much performance as possible in the presence ofplant variations and uncertainty. Unfortunately, robustness often must be sacrie ced to gain performance and vice versa. In some problems adequate levels of performance and robustness cannot be achieved by a e xed controller. This problem occursin e ight control because aircraft dynamicsvary signie cantlyduring operation.To combat this problem, the e ight control designer is forced to design a controller that also varies during operation. The classical approach to developing a varying controller is to design several point controllers throughout the operating region and connect them with some type of blending or interpolation. This process is referred to as gain scheduling (Ref. 1, p. 1) in the literature, and it has several drawbacks. First, designing several point controllersisatime-consumingandtediousprocess.Second,interpolatingorblendingbetweenthepointcontrollersisoftenatrialanderror procedure with historically very little theoretical guidance. Third, anyperformance androbustness guaranteesinthe individualoperatingregionsarelostinthetransitionregionbetweenpointcontrollers. Some modern e ight control methods have eliminated some of the problems with gain scheduling, but not all of them. Dynamic inversion methods (Ref.1,p.25), for example, avoid the scheduling problem by using nonlinear feedback to cancel the dynamics of the aircraft. This enables the controlled dynamics to be specie ed by the designer. Unfortunately, the cancellation is not perfect, and so like classical gain scheduling, dynamic inversion techniques lack solid performance and robustness guarantees. Several new linear parameter-varying (LPV) control approaches 2i 10 have emerged in the last few years, however, that may improve the e ight control designer’ s task. These approaches explicitly take into account the relationship between real-time parameter variations and performance. This enables controllers to be designed for whole ranges of operating conditions with theoretical guarantees of performance and robustness throughout the region. All of the approaches are based (at least in part ) on linear matrix inequalities (LMI), 11 and so they all can be solved numerically with some efeciency.

The purpose of this study is to introduce a novel ”completed” LPV controller and observer design approach for control of particular class of nonlinear systems. The developed tools allow to use classical linear controller design theorems via LPV framework without the advanced Linear Matrix Inequality (LMI) theorems. The developed method combines the classical state feedback theorems and a supplementary controller and observer structure which basis is the special properties of the parameter space of the linear parameter varying (LPV) system. The main benefit of the proposed method is that the controller design does not require highly advanced mathematical tools and high computational capacity. We have proven the usability of the method in case of a highly nonlinear compartmental model. The results have shown that the completed LPV controller is able to handle the system with good performance.