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This article proposes an adaptive fuzzy control approach with an event-triggered mechanism and spatial motion constraint for a hybrid spacecraft system. The spacecraft system is composed of a rigid body and a slender flexible panel, with coupled dynamics captured by three ordinary differential equations and two partial differential equations. The overall control objective lies in utilizing an event-triggered control input to regulate the angular velocities of the rigid body and stabilize the vibrations of the flexible panel under unknown input disturbances and prescribed spatial motion performance. We collectively address the posture regulation and disturbance rejection purposes by introducing a barrier Lyapunov function and a fuzzy logic system. The event-triggered solution only updates the control signals at some discrete-time instants, and hence the communication burden is reduced significantly. The potential effectiveness and thrifty efficiency of the developed control strategy are theoretically demonstrated and numerically verified. [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]

In this paper, the problem of cooperative control of robotic sensor networks (RSNs) for monitoring dynamic pollutant plumes in two‐dimensional (2D) space is studied. The pollutant plume propagation dynamics is governed by a 2D advection‐diffusion partial differential equation (PDE), and the plume front is modelled by a level set with a pre‐specified threshold value. Distributed consensus Luenberger‐type PDE observers are first constructed using local concentration measurements from the RSNs for unanimous estimate of the dynamic concentration field over the entire spatial domain. With the aid of the distributed consensus observers, a leaderless cooperative control scheme is then developed for the RSNs to monitor the dynamic plume front expansion. It is proved using the Lyapunov stability method and set stability concept that the proposed cooperative control scheme guarantees tracking of the dynamic plume front expansion and coverage of the plume front. Extensive numerical simulation results demonstrate the effectiveness and merit of the proposed cooperative control scheme. [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 study presents an observer‐based output feedback compensator design for a linear parabolic partial differential equation (PDE), where a finite number of actuators and sensors are active over partial areas and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger‐type PDE observer is first constructed by using the non‐collocated pointwise observation to exponentially track the PDE state. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing a Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for definite integrals, sufficient conditions on the exponential stability of the resulting closed‐loop coupled PDEs are presented in terms of standard linear matrix inequalities. Furthermore, both open‐loop and closed‐loop well‐posedness analysis results are also established by the C0 ‐semigroup method. Numerical simulation results are presented to show the effectiveness of the proposed design method. [ABSTRACT FROM AUTHOR]

This paper employs the observer‐based output feedback control technique to study the problem of local piecewise control for a linear parabolic partial differential equation(PDE) using non‐collocated local piecewise observation. By the non‐collocated local piecewise observation, a Luenberger‐type PDE observer is first constructed to exponentially track the state of the PDE in the sense of both L2 norm and H1 norm. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed for exponential stabilisation of the PDE. Sufficient conditions on exponential stabilisation of the PDE by the suggested feedback controller in the sense of both L2 norm and H1 norm are developed by utilising Lyapunov direct method, integration by parts, Wirtinger's inequality, and first mean value theorem for definite integrals, and presented in terms of standard linear matrix inequalities. The well‐posedness is also analysed for both open‐loop and resulting closed‐loop PDE by using C0 ‐semigroup approach. Finally, numerical simulation results are presented to show the effectiveness of the proposed design method. [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]

This paper discusses the design problem of distributedH∞Luenberger-type partial differential equation (PDE) observer for state estimation of a linear unstable parabolic distributed parameter system (DPS) with external disturbance and measurement disturbance. Both pointwise measurement in space and local piecewise uniform measurement in space are considered; that is, sensors are only active at some specified points or applied at part thereof of the spatial domain. The spatial domain is decomposed into multiple subdomains according to the location of the sensors such that only one sensor is located at each subdomain. By using Lyapunov technique, Wirtinger's inequality at each subdomain, and integration by parts, a Lyapunov-based design of Luenberger-type PDE observer is developed such that the resulting estimation error system is exponentially stable with anH∞performance constraint, and presented in terms of standard linear matrix inequalities (LMIs). For the case of local piecewise uniform measurement in space, the first mean value theorem for integrals is utilised in the observer design development. Moreover, the problem of optimalH∞observer design is also addressed in the sense of minimising the attenuation level. Numerical simulation results are presented to show the satisfactory performance of the proposed design method. [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 deals with the problem of $H_\infty$ fuzzy control design with an input constraint for a class of coupled systems, which consist of an $n$ -dimensional nonlinear subsystem of ordinary differential equations (ODEs) and a scalar linear parabolic subsystem of partial differential equation (PDE) connected in feedback. Initially, the nonlinear coupled system is represented by a Takagi–Sugeno (T–S) fuzzy-coupled ODE-PDE model. Then, based on the fuzzy model and parallel distributed compensation scheme, a fuzzy state feedback control design is developed via Lyapunov's direct method, such that the resulting closed-loop fuzzy-coupled system is exponentially stable, and a prescribed $H_\infty$ performance of disturbance attenuation is satisfied. The existing condition of the proposed $H_\infty$ fuzzy controllers is given in terms of linear matrix inequalities (LMIs). Moreover, in order to make the attenuation level as small as possible while the input constraint is respected to avoid the high magnitude, a suboptimal $H_\infty$-constrained fuzzy control problem is also addressed, which is formulated as an LMI optimization problem. Finally, the proposed method is applied to the control of a hypersonic rocket car to illustrate its effectiveness. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method. [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]

The guaranteed cost distributed fuzzy ( GCDF) observer-based control design is proposed for a class of nonlinear spatially distributed processes described by first-order hyperbolic partial differential equations ( PDEs). Initially, a T-S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer-based control design is developed in terms of a set of space-dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed-loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug-flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366-2378, 2013 [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the problem of designing a distributed fuzzy controller with constraint for a class of nonlinear spatially distributed processes modeled by first-order hyperbolic partial differential equations (PDEs). The purpose of this paper is to design a distributed fuzzy state feedback controller such that the closed-loop PDE system is exponentially stable with a prescribed performance of disturbance attenuation, while the control constraint is respected. Initially, a Takagi–Sugeno (T–S) hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the T–S fuzzy PDE model, a distributed fuzzy controller design with constraint is developed in terms of a set of coupled differential/algebraic linear matrix inequalities (D/ALMIs) in space. Furthermore, a suboptimal distributed fuzzy controller with constraint is proposed to minimize the level of attenuation. The finite difference method in space and the existing linear matrix inequality (LMI) optimization techniques are employed to approximately solve the suboptimal fuzzy control design problem. Finally, the proposed design method is applied to the distributed control of a nonlinear system described by two coupled first-order hyperbolic PDEs to illustrate its effectiveness. [Copyright &y& Elsevier]

In this paper, a distributed fuzzy control design based on Proportional–spatial Derivative (P–sD) is proposed for the exponential stabilization of a class of nonlinear spatially distributed systems described by parabolic partial differential equations (PDEs). Initially, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is proposed to accurately represent the nonlinear parabolic PDE system. Then, based on the T–S fuzzy PDE model, a novel distributed fuzzy P–sD state feedback controller is developed by combining the PDE theory and the Lyapunov technique, such that the closed-loop PDE system is exponentially stable with a given decay rate. The sufficient condition on the existence of an exponentially stabilizing fuzzy controller is given in terms of a set of spatial differential linear matrix inequalities (SDLMIs). A recursive algorithm based on the finite-difference approximation and the linear matrix inequality (LMI) techniques is also provided to solve these SDLMIs. Finally, the developed design methodology is successfully applied to the feedback control of the Fitz–Hugh–Nagumo equation. [ABSTRACT FROM AUTHOR]

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]

Abstract: This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method. [Copyright &y& Elsevier]