A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Control and filtering of time-varying linear systems via parameter dependent Lyapunov functions

The main contribution of this dissertation is to propose conditions for linear filter and controller design, considering both robust and parameter dependent structures, for discrete time-varying systems. The controllers, or filters, are obtained through the solution of optimization problems, formulated in terms of bilinear matrix inequalities, using a method that alternates convex optimization problems described in terms of linear matrix inequalities. Both affine and multi-affine in different instants of time (path dependent) Lyapunov functions were used to obtain the design conditions, as well as extra variables introduced by the Finsler\u27s lemma. Design problems that take into account an H-infinity guaranteed cost were investigated, providing robustness with respect to unstructured uncertainties. Numerical simulations show the efficiency of the proposed methods in terms of H-infinity performance when compared with other strategies from the literature

Delay-Scheduled State-Feedback Design for Time-Delay Systems with Time-Varying Delays - A LPV Approach

International audienceThis paper is concerned with the synthesis of delay-scheduled state-feedback controllers which stabilize linear systems with time-varying delays. In this framework, it is assumed that the delay is approximately known in real-time and used in the controller in a scheduling fashion. First, a new model transformation turning a time-delay system into an uncertain LPV system is introduced. Using this transformation, a new delay-dependent stability test based on the so-called full block S-procedure is developed and from this result, a new delay-dependent stabilization result is derived. Since the resulting LMI conditions depend polynomially on the parameters, a relaxation result is then applied in order to obtain a tractable finite set of finite-dimensional LMIs. The interests of the approach resides in 1) the synthesis of a new type of controllers scheduled by the delay value which has a lower memory consumption than controllers with memory (since it is not necessary to store past values of the state), and 2) an easy consideration of uncertainties on the delay knowledge

Switching State-Feedback LPV Control with Uncertain Scheduling Parameters

This paper presents a new method to design Robust Switching State-Feedback Gain-Scheduling (RSSFGS) controllers for Linear Parameter Varying (LPV) systems with uncertain scheduling parameters. The domain of scheduling parameters are divided into several overlapped subregions to undergo hysteresis switching among a family of simultaneously designed LPV controllers over the corresponding subregion with the guaranteed H-infinity performance. The synthesis conditions are given in terms of Parameterized Linear Matrix Inequalities that guarantee both stability and performance at each subregion and associated switching surfaces. The switching stability is ensured by descent parameter-dependent Lyapunov function on switching surfaces. By solving the optimization problem, RSSFGS controller can be obtained for each subregion. A numerical example is given to illustrate the effectiveness of the proposed approach over the non-switching controllers

H_∞ Static Output-Feedback Gain-Scheduled Control for Discrete LPV Time-Delay Systems^⁎

This paper proposes new synthesis conditions to design H∞ static output-feedback controllers for discrete-time linear systems affected by time-varying parameters and time-varying delays. The design conditions are provided in terms of sufficient parameter-dependent linear matrix inequalities with a scalar parameter, being capable of synthesizing either robust or gain-scheduled controllers. The main motivations to deal with such problem are that many real-world plants can be modeled in terms of discrete-time linear parameter-varying (LPV) time-delay models and the lack of methods to deal with such systems considering an output-feedback based approach. The technique presented in this paper is quite generalist, allowing an arbitrary structure for the measured output matrix. Numerical examples are provided to illustrate the effectiveness of the synthesis conditions, tractable in terms of LMI relaxations, for robust or gain-scheduled H∞ output-feedback for LPV time-delayed systems

From Fixed-Order Gain-Scheduling to Fixed-Structure LPV Controller Design

This thesis focuses on the development of some fixed-order controller design methods in the gain-scheduling/Linear Parameter Varying (LPV) framework. Gain-scheduled controllers designed using frequency-domain Single Input Single Output (SISO) models are considered first, followed by LPV controller design in the SISO transfer function setting and, finally, by Multiple Input Multiple Output (MIMO) LPV controller design in the state-space setting. In addition to the guarantee of closed-loop stability, each of the methods optimizes some classical performance measure, such as the $\mathscr{H}_\infty$ or $\mathscr{H}_2$ performance metrics. In the LPV state-space setting, the practical assumption of bounded scheduling parameter variations is taken into account in order to allow a higher performance level to be achieved. The fixed-order gain-scheduled controller design method is based on frequency-domain models dependent on the scheduling parameters. Based on the linearly parameterized gain-scheduled controllers and desired open-loop transfer functions, the $\mathscr{H}_\infty$ performance of the weighted closed-loop transfer functions is presented in the Nyquist diagram as a set of convex constraints. No a posteriori interpolation is needed, so the stability and performance level are guaranteed for all values of scheduling parameters considered in the design. Controllers designed with this method are successfully applied to the international benchmark in adaptive regulation. These low-order controllers ensure good rejection of the multisinusoidal disturbance with time-varying frequencies on the active suspension testbed. One issue related to the gain-scheduled controller design using the frequency response model is the computational burden due to the constraint sampling in the frequency domain. The other is a guarantee of stability and performance for all the values of scheduling parameters, not just those treated in design. To overcome these issues, a method for the design of fixed-order LPV controllers with the transfer function representation is proposed. The LPV controller parameterization considered in this approach leads to design variables in both the numerator and denominator of the controller. Stability and $\mathscr{H}_\infty$ performance conditions for all fixed values of scheduling parameters are presented in terms of Linear Matrix Inequalities (LMIs). With a problem of rejection of a multisinusoidal disturbance with time-varying frequencies in mind, LPV controller is designed for an LTI plant with a transfer function model. The extension of these methods from SISO to MIMO systems is far from trivial. The state-space setting is used for this reason, as there the transition from SISO to MIMO systems is natural. A method for fixed-order output-feedback LPV controller design for continuous-time state-space LPV plants with affine dependence on scheduling parameters is proposed. Bounds on the scheduling parameters and their variation rates are exploited in design through the use of affine Parameter Dependent Lyapunov Functions (PDLFs). The exponential decay rate, induced $\mathscr{L}_2$-norm and $\mathscr{H}_2$ performance constraints are expressed through a set of LMIs. The proposed method is applied to the 2DOF gyroscope experimental setup. In practice control is performed using digital computers, so some effort needs to be put into the LPV controller discretization. If the discrete-time LPV model of the system is available [...

LPV Analysis and Control Using Fast Iterative Solutions to Rationally Parametric Lyapunov and Riccati Equations

Abstract-We consider the problem of analysis and control of Linear Parameter Varying(LPV) systems. With regard to such problems, solving rationally parametric Lyapunov and Riccati equations for parametric matrices often arises. In this paper, we develop computationally efficient iterative methods for finding rational approximations to solutions to such problems to arbitrary accuracy

Fuzzy control turns 50: 10 years later

In 2015, we celebrate the 50th anniversary of Fuzzy Sets, ten years after the main milestones regarding its applications in fuzzy control in their 40th birthday were reviewed in FSS, see [1]. Ten years is at the same time a long period and short time thinking to the inner dynamics of research. This paper, presented for these 50 years of Fuzzy Sets is taking into account both thoughts. A first part presents a quick recap of the history of fuzzy control: from model-free design, based on human reasoning to quasi-LPV (Linear Parameter Varying) model-based control design via some milestones, and key applications. The second part shows where we arrived and what the improvements are since the milestone of the first 40 years. A last part is devoted to discussion and possible future research topics.Guerra, T.; Sala, A.; Tanaka, K. (2015). Fuzzy control turns 50: 10 years later. Fuzzy Sets and Systems. 281:162-182. doi:10.1016/j.fss.2015.05.005S16218228