Eduardo S. Tognetti

Selective

This paper presents new results concerning the stability analysis and design of state-feedback controllers for continuous-time Takagi-Sugeno (T-S) fuzzy systems via fuzzy Lyapunov functions. The membership functions of the T-S fuzzy systems are modeled in a space that is defined by the Cartesian product of simplexes called a multisimplex. If the time derivatives of the membership functions are bounded, the bounds are used to construct a polytope that models the space of the time derivatives of the membership functions. Linear matrix inequality (LMI) relaxations that are based on polynomial matrices are provided for stability analysis and controller design. Extensions for the design of control laws that minimize upper bounds to H2 and H∞ norms are also given. The main novelty of this method is that it allows one to synthesize control gains, which depends only on some premise variables that are selected by the designer. Numerical experiments illustrate the flexibility and advantages of the proposed method.

\hbox{\scr H}_2

This paper presents a general framework to cope with full-order linear parameter-varying (LPV) filter design subject to inexactly measured parameters. The main novelty is the ability of handling additive and multiplicative uncertainties in the measurements, for both continuous and discrete-time LPV systems, in a unified approach. By conveniently modelling scheduling parameters and uncertainties affecting the measurements, the filter design problem can be expressed in terms of robust matrix inequalities that become linear when two scalar parameters are fixed. Therefore, the proposed conditions can be efficiently solved through linear matrix inequality relaxations based on polynomial solutions. Numerical examples are presented to illustrate the improved efficiency of the proposed approach when compared to other methods and, more important, its capability to deal with scenarios where the available strategies in the literature cannot be used.

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This paper presents new results concerning the design of state feedback controllers for continuous-time Takagi Sugeno (T-S) fuzzy systems. The conditions, based on a line integral fuzzy Lyapunov function, are specially suitable for T-S fuzzy systems where no information about the time-derivatives of the membership functions is available. The controller is designed through linear matrix inequalities in a two step procedure: at the first step, a stabilizing fuzzy controller is obtained for a relaxed frozen (i.e. time-invariant) T-S fuzzy system. This control gain is then used as an input data at the second step, that provides a stabilizing control law guaranteed by the line-integral Lyapunov function. An extension to cope with H∞ guaranteed cost control of T-S fuzzy systems is also provided. Numerical examples illustrate the advantages of the proposed method when compared to other techniques available in the literature.

\hbox{\scr H}_\infty

This paper addresses the problem of constant sampling discretization of uncertain time-invariant continuous-time linear systems in polytopic domains. To circumvent the difficulty of dealing with the exponential of uncertain matrices, a new discretization method, based on Taylor series expansion, is proposed. The resulting discrete-time uncertain system is described in terms of homogeneous polynomial matrices with parameters lying in the unit simplex and an additive norm-bounded uncertainty which represents the discretization residual error. As a second contribution, linear matrix inequality (LMI) based conditions for the synthesis of a stabilizing state feedback control for discrete-time linear systems with polynomial dependence on the uncertain parameters and an additive norm-bounded uncertainty are proposed. Numerical experiments illustrate the discretization technique advantages of using higher orders in the Taylor series expansion to obtain more precise approximations. The examples also show that, at the price of simple line searches in a scalar parameter, and using Lyapunov functions of higher degrees, less conservative results for robust state feedback control design of discretized uncertain systems can be obtained.

Stabilization of Takagi–Sugeno Fuzzy Systems

This paper presents new results concerning the stability analysis and the design of state feedback controllers for continuous-time Takagi-Sugeno (T-S) fuzzy systems via fuzzy Lyapunov functions. Using the Cartesian product of simplexes, called multi-simplex, a new modeling is proposed to represent the membership functions of T-S fuzzy systems. In the multisimplex representation, linear matrix inequality relaxations based on homogeneous polynomials matrices are provided for stability analysis and controller design. The time-derivatives of the membership functions are modeled as belonging to polytopic convex sets and may be considered unbounded if nothing is known about their upper limits. As main aspect, the method allows to synthesize control gains depending only on some premise variables selected by the designer. Numerical experiments illustrate the flexibility and advantages of the method.

A new approach to handle additive and multiplicative uncertainties in the measurement for LPV filtering

This paper investigates the problems of uncertain sampling rate discretisation and the networked control of uncertain time-invariant continuous-time linear systems in polytopic domains. The sampling period is assumed to be unknown but belonging to a given interval. To avoid the difficulty of dealing with the exponential of uncertain matrices, a discrete-time model is obtained by applying a Taylor series expansion of degree ℓ to the original system. The resulting discrete-time model is composed of homogeneous polynomial matrices with parameters lying in the Cartesian product of simplexes, called a multi-simplex, plus an additive norm-bounded term representing the discretisation residual error. The original continuous-time system is controlled through a communication network that introduces a time delay in the process. Linear matrix inequality relaxations that include a scalar parameter search are proposed for the design of a digital robust state feedback controller that guarantees the closed-loop stability of the networked control system. Numerical experiments are presented to illustrate the versatility of the proposed method, which can be applied to a more general class of networked control problems than the existing approaches in the literature.

Improved stabilization conditions for Takagi-Sugeno fuzzy systems via fuzzy integral lyapunov functions

Abstract 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.

A new procedure for discretization and state feedback control of uncertain linear systems

The problem of state feedback control design for discrete-time Takagi–Sugeno (TS) (T–S) fuzzy systems is investigated in this paper. A Lyapunov function, which is quadratic in the state and presents a multi-polynomial dependence on the fuzzy weighting functions at the current and past instants of time, is proposed.This function contains, as particular cases, other previous Lyapunov functions already used in the literature, being able to provide less conservative conditions of control design for TS fuzzy systems. The structure of the proposed Lyapunov function also motivates the design of a new stabilising compensator for Takagi–Sugeno fuzzy systems. The main novelty of the proposed state feedback control law is that the gain is composed of matrices with multi-polynomial dependence on the fuzzy weighting functions at a set of past instants of time, including the current one. The conditions for the existence of a stabilising state feedback control law that minimises an upper bound to the or norms are given in terms of linear matrix inequalities. Numerical examples show that the approach can be less conservative and more efficient than other methods available in the literature.

Selective stabilization of Takagi-Sugeno fuzzy systems

This paper proposes a design procedure for reduced-order dynamic output feedback (DOF) gain-scheduling controllers with H∞ guaranteed cost for linear parameter-varying (LPV) continuous-time systems, where the measurement of the scheduling parameters may be affected by uncertainties. Thanks to the flexibility of the proposed modeling, the LPV-DOF controllers can be implemented in terms of a selected set of parameters, which are supposed to be available for measurement in real time. The design conditions can cope with both additive and multiplicative noises, considered as time-varying uncertainties, affecting the measures. All parameters and uncertainties are modeled through the multi-simplex framework, i.e., the Cartesian product of simplexes. The problem is solved through a two-stage procedure based on linear matrix inequalities.

Discretisation and control of polytopic systems with uncertain sampling rates and network-induced delays

Abstract The problem of static output feedback control design for continuous-time Takagi-Sugeno (T–S) fuzzy systems is addressed in this paper. The membership functions are modeled in a space defined by the Cartesian product of simplexes, called multi-simplex, and are allowed to vary arbitrarily (i.e. no bounds on the time-derivative of the membership functions are assumed). The static output feedback fuzzy controller is obtained through a two-step procedure: first, a stabilizing fuzzy state feedback control gain is determined by means of linear matrix inequalities (LMIs). Then, the state feedback gain matrices are used in LMI conditions that, if satisfied, provide the fuzzy static output feedback control law. A fuzzy line integral Lyapunov function with arbitrary polynomial dependence on the premise variables is used to assess closed-loop stability. The main appeal of the approach is that the output feedback gains can have independent and arbitrary polynomial dependence on some specific premise variables, selected by the designer, with great advantages for practical applications. An example illustrates that the proposed strategy can provide less conservative results when compared to other methods from the literature for output feedback stabilization of continuous-time T–S fuzzy systems.