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This article addresses the issue of fuzzy control design subject to a mixed $H_2/H_\infty$ performance constraint and guidance law design for semilinear parabolic distributed parameter systems (DPSs) with mobile collocated actuator/sensor pairs. Initially, via the local sector nonlinearity method, a Takagi–Sugeno (T–S) fuzzy model is constructed to accurately describe the spatiotemporal dynamics of the DPSs. Then, based on the obtained T–S fuzzy model and Lyapunov technique, a membership-function-dependent mixed $H_2/H_\infty$ fuzzy control design is developed and the mobile actuator/sensor guidance laws are also determined simultaneously, such that the resulting closed-loop system is exponentially stable while providing an $H_2$ performance bound under the given $H_\infty$ performance of disturbance attenuation, and the transient response of closed-loop state is improved. Moreover, a suboptimal mixed $H_2/H_\infty$ fuzzy control design is derived in the sense of minimizing the upper bound of the given $H_2$ performance function by applying the existing linear matrix inequality optimization techniques. At last, some simulation results for a numerical example are presented to verify the proposed method. [ABSTRACT FROM AUTHOR]

This study addresses the problem of dynamic compensator design for exponential stabilisation of linear space‐varying parabolic multiple‐input–multiple‐output (MIMO) partial differential equations (PDEs) subject to periodic boundary conditions. With the aid of the observer‐based feedback control technique, an observer‐based dynamic feedback compensator, whose implementation requires only a few actuators and sensors active over partial areas of the spatial domain, is constructed such that the resulting closed‐loop coupled PDEs is exponentially stable. The spatial domain is divided into multiple subdomains according to the minimum of the actuators' number and the sensors' one. By Lyapunov direct method and two general variants of Poincaré‐Wirtinger inequality at each subdomain, sufficient conditions for the existence of such feedback compensator are developed and presented in terms of algebraic linear matrix inequalities (LMIs) in space. Based on the extreme value theorem, LMI‐based sufficient and necessary conditions are presented for the feasibility of algebraic LMIs in space. Finally, numerical simulation results are presented to support the proposed design method. [ABSTRACT FROM AUTHOR]

This paper presents a Lyapunov and partial differential equation (PDE)-based methodology to solve static collocated piecewise fuzzy control design of quasi-linear parabolic PDE systems subject to periodic boundary conditions. Two types of piecewise control, i.e., globally piecewise control and locally piecewise control are considered, respectively. A Takagi–Sugeno (T–S) fuzzy PDE model that is constructed via local sector nonlinearity method is first employed to accurately describe spatiotemporal dynamics of quasi-linear PDEs. Based on the T–S fuzzy PDE model, a static collocated piecewise fuzzy feedback controller is constructed to guarantee the locally exponential stability of the resulting closed-loop system. Sufficient conditions for the existence of such fuzzy controller are developed by applying vector-valued Poincaré–Wirtinger inequality and its variants and a linear matrix inequality (LMI) relaxation technique. These sufficient conditions are presented in terms of standard LMIs. Finally, the performance of the suggested fuzzy controller is illustrated by numerical simulation results of a nonlinear PDE system described by quasi-linear FitzHugh–Nagumo equation with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

This paper employs a Takagi–Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\Vert \cdot \Vert _\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincaré–Wirtinger inequality in one-dimensional spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\Vert \cdot \Vert _\infty$ , if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of exponential stabilization for a linear distributed parameter system (DPS) using pointwise control and non-collocated pointwise observation, where the system is modeled by a parabolic partial differential equation (PDE). The main objective of this paper is to construct an output feedback controller for pointwise exponential stabilization of the linear parabolic PDE system using the non-collocated pointwise observation. The observer-based output feedback control technique is utilized to overcome the design difficulty caused by the non-collocation pointwise observation. We construct a Luenberger-type PDE state observer to exponentially track the state of the PDE system. A collocated pointwise feedback controller is proposed based on the estimated state. A variation of Poincaré – Wirtinger inequality is derived and the spatial domain is decomposed into multiple sub-domains according to the number of the actuators and sensors. By using Lyapunov’s direct method, integration by parts, and the variation of Poincaré – Wirtinger inequality at each sub-domain, sufficient conditions for the existence of the observer-based out feedback controller are developed such that the resulting closed-loop system is exponentially stable, and presented in terms of standard linear matrix inequalities (LMIs). Furthermore, the closed-loop well-posedness analysis is also provided by the C 0 -semigroup approach. Extensive numerical simulation results are presented to show the performance of the proposed output feedback controller. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of exponential stabilization for nonlinear parabolic distributed parameter systems using the Takagi–Sugeno (T–S) fuzzy partial differential equation (PDE) model, where a finite number of actuators are active only at some specified points of the spatial domain (these actuators are referred to as pointwise actuators). Three cases of state feedback are respectively considered in this study as follows: full state feedback, piecewise state feedback, and collocated pointwise state feedback. It is initially assumed that a T–S fuzzy PDE model obtained via the sector nonlinearity approach is employed to accurately represent the semilinear parabolic PDE system. Based on the obtained T–S fuzzy PDE model, Lyapunov-based design methodologies of fuzzy feedback control laws are subsequently derived for the above three state feedback cases by using the vector-valued Wirtinger's inequality to guarantee locally exponential pointwise stabilization of the semilinear PDE system, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, the favorable property offered by sharing all the same premises in the T–S fuzzy PDE models and fuzzy controllers is not applicable for the case of collocated pointwise state feedback. A parameterized LMI is introduced for this case to enhance the stabilization ability of the fuzzy controller. Finally, the merit and effectiveness of the proposed design methods are demonstrated by numerical simulation results of two examples. [ABSTRACT FROM PUBLISHER]

An H_\infty fuzzy control design is presented for the disturbance attenuation of a class of coupled systems described by a set of nonlinear ordinary differential equations (ODEs) and a semi-linear parabolic partial differential equation (PDE). The fuzzy control scheme consists of an ODE state feedback fuzzy subcontroller for the ODE subsystem and a PDE static output feedback fuzzy subcontroller for the PDE subsystem by using piecewise uniform actuators and pointwise sensors. Initially, the original nonlinear system is accurately represented by employing a Takagi–Sugeno fuzzy coupled parabolic PDE–ODE model. Then, an H_\infty fuzzy controller is developed to exponentially stabilize the fuzzy coupled system while satisfying a prescribed H_\infty performance of disturbance attenuation, whose existence condition is given by linear matrix inequalities. Finally, simulation results on a hypersonic rocket car are given to show the effectiveness of the proposed design method. [ABSTRACT FROM PUBLISHER]

This paper gives a membership-function-dependent approach to solve the design problem of fuzzy pointwise state feedback controller for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs), where only a few actuators are discretely distributed in space. In the proposed design method, a Takagi–Sugeno (T–S) fuzzy PDE model obtained by using the sector nonlinearity method is first utilized to accurately describe the nonlinear spatiotemporal dynamics of the PDE system. As only the state information at some known specified points in the spatial domain (i.e., the pointwise state information) is available for the controller design, the favorable property offered by sharing all the same premises in the fuzzy PDE plant model and fuzzy controller cannot be employed to develop the fuzzy control design method. To overcome this drawback, a linear matrix inequality (LMI) relaxation technique is developed to enhance the stabilization ability of the fuzzy controller. Based on the T–S fuzzy PDE model, a membership-function-dependent fuzzy pointwise state feedback control design is then proposed by employing the Lyapunov technique, integration by parts, the vector-valued Wirtinger’s inequality and the LMI relaxation technique, and presented in term of standard LMIs. Finally, the satisfactory and better performance of the proposed design method are demonstrated by the extensive numerical simulation results of two numerical examples. [ABSTRACT FROM AUTHOR]

This paper considers the design problem of a fuzzy output tracking controller via a finite number of actuators for a class of nonlinear distributed parameter systems with unknown matched perturbations described by semi-linear first-order hyperbolic partial differential equations (PDEs). Firstly, a T–S fuzzy hyperbolic PDE model is assumed to be used to accurately represent the semi-linear first-order hyperbolic PDE system. Subsequently, based on the T–S fuzzy PDE model, the classic variable structure control technique is adopted to develop a fuzzy output tracking controller such that the output of the semi-linear PDE system exponentially tracks a desired reference and the state of the system is uniformly bounded in the Filippov sense. By constructing a strict Lyapunov function, the sufficient condition on the existence of the fuzzy output tracking controller is derived and presented in terms of algebraic linear matrix inequalities (LMIs) in space with an equality constraint. It has been shown that when these algebraic LMIs with an equality constraint are feasible, fuzzy tracking control design can be obtained via the existing convex optimization techniques. Finally, the developed design methodology is successfully applied to the control of a semi-linear Lotka–Volterra PDE system and the achieved results illustrate its effectiveness. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of static output feedback (SOF) control design for a class of diffusion partial differential equation (PDE) and ordinary differential equation (ODE) cascades, where the ODE model is used to describe the dynamics of the multi-input and multi-output (MIMO) plant and the diffusion PDE model is employed to represent the dynamics of actuators. The objective of this paper is to develop a simple as well as effective SOF controller via the Lyapunov's direct method such that the resulting closed-loop system is globally exponentially stable. By constructing a quadratic Lyapunov function, the sufficient condition on the globally exponential stability of the closed-loop cascaded system is presented in terms of linear matrix inequality (LMI). Then, an LMI-based design method of the SOF controller is developed on the basis of the obtained stability analysis result. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed design method. [ABSTRACT FROM PUBLISHER]

This paper deals with the exponential stabilization problem for a class of nonlinear spatially distributed processes that are modeled by semilinear parabolic partial differential equations (PDEs), for which a finite number of actuators are used. A fuzzy control design methodology is developed for these systems by combining the PDE theory and the Takagi–Sugeno (T–S) fuzzy-model-based control technique. Initially, a T–S fuzzy parabolic PDE model is proposed to accurately represent a semilinear parabolic PDE system. Then, based on the T–S fuzzy model, a Lyapunov technique is used to design a continuous fuzzy state feedback controller such that the closed-loop PDE system is exponentially stable with a given decay rate. The stabilization condition is presented in terms of a set of spatial differential linear matrix inequalities (SDLMIs). Furthermore, a recursive algorithm is presented to solve the SDLMIs via the existing linear matrix inequality optimization techniques. Finally, numerical simulations on the temperature profile control of a catalytic rod are given to verify the effectiveness of the proposed design method. [ABSTRACT FROM PUBLISHER]