Output regulation boundary control of first-order coupled linear MIMO hyperbolic PIDE systems

ABSTRACT

The work addresses the output regulation problem for coupled linear multiple input multiple output (MIMO) hyperbolic partial integro-differential equation systems with disturbances affecting the systems through the space and boundary input. The exosystems are extended to generate ramp signals and general family of polynomial signals. The system decomposition is applied through the state transformation and yields a decoupled equivalent system. Based on the decoupled form, the backstepping transformation is applied and then in the new coordinate, the full state and output-feedback regulators are designed, respectively. For the state feedback regulator, the corresponding regulator equation is obtained and its solvability conditions are provided to facilitate the regulator design and feasibility. The design of observer-based regulator is based on the decoupling of the observer error system into a PDE subsystem and an ODE subsystem so that the backstepping approach achieves stabilisation by eigenvalue assignment leading to design of observer stabilizing gains.     

Incremental passivity and output regulation for switched nonlinear systems

This paper studies incremental passivity and global output regulation for switched nonlinear systems, whose subsystems are not required to be incrementally passive. A concept of incremental passivity for switched systems is put forward. First, a switched system is rendered incrementally passive by the design of a state-dependent switching law. Second, the feedback incremental passification is achieved by the design of a state-dependent switching law and a set of state feedback controllers. Finally, we show that once the incremental passivity for switched nonlinear systems is assured, the output regulation problem is solved by the design of global nonlinear regulator controllers comprising two components: the steady-state control and the linear output feedback stabilising controllers, even though the problem for none of subsystems is solvable. Two examples are presented to illustrate the effectiveness of the proposed approach.     

Observers and output feedback stabilising controllers for nonlinear strict-feedback systems with sampled observation

The design of observers and output feedback stabilising controllers for continuous-time strict-feedback systems with sampled observation is considered. First two types of observers are designed. One is a discrete-time semiglobal and practical reduced-order observer for the exact model and the other is a continuous-time semiglobal and practical full-order observer for continuous-time strict feedback systems with sampled observation. Then by combining the designed continuous-time observers and continuous-time state feedback laws that are continuous, zero at the origin, and uniformly globally asymptotically stabilise continuous-time systems, output feedback semiglobally practically uniformly asymptotically stabilising controllers are constructed. Numerical examples are given to illustrate the proposed design of observers and output feedback controllers.     

Output regulation of time-varying systems

In this paper the output regulation problem for linear time-varying systems is considered. Replacing the regulator equation by a regulator differential equation we give a necessary and sufficient condition for the problem to be solvable. As in the time-invariant case we first solve the output regulation problem by state feedback and obtain the required condition. Then with the aid of observers we show that the same condition solves the general problem with measurement feedback. We then consider the classes of almost periodic and periodic systems and refine the main results. A simple example of an almost periodic system and simulation results are given to illustrate the theory.     

Quadratic optimal controller to stabilise symmetrical systems

In this paper, we design a controller for stabilising a control system. The technique used for designing the controller includes a linear regulator and an asymptotical observer which form the controller. The linear regulator designed is a feedback of estimated states and also it must minimise a quadratic performance index. The gain matrix of optimal feedback is obtained by solving the Riccati equation, whilst the gain observer matrix is computed by making use of symmetrical systems properties. The properties of symmetrical systems allow us to find the optimal gain matrix of the observer without solving the dual Riccati equation, we only need to compute the matrices of controllability and observability. Having calculated the gain matrices of regulator and of observer, we proceeded to compute the transfer function of the observer-based controller.     

Finite-time output regulation by bounded linear time-varying controls with applications to the satellite formation flying

This article studies the finite-time output regulation problem for linear time-invariant continuous-time systems. By using the solution to a parametric Lyapunov equation (PLE) and regulator equations, three bounded linear time-varying (LTV) state controllers composed of the LTV feedback gain and the LTV feedforward gain are designed, such that (prescribed) finite-time output regulation is solved. As a further result, a linear LTV observer-based controller is also designed. The most significant advantages of this article are that the system under consideration is more general and the output regulation problem is achieved within a user-chosen regulation time. Finally, the developed LTV state controllers are utilized to the design of the satellite formation flying control system and simulation results verify the effectiveness of the proposed approaches.     

Control of an unstable reaction–diffusion PDE with long input delay

A Smith Predictor-like design for compensation of arbitrarily long input delays is available for general, controllable, possibly unstable LTI finite-dimensional systems. Such a design has not been proposed previously for problems where the plant is a PDE. We present a design and stability analysis for a prototype problem, where the plant is a reaction–diffusion (parabolic) PDE, with boundary control. The plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay, with an unbounded input operator. The predictor-based feedback design extends fairly routinely, within the framework of infinite-dimensional backstepping. However, the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ODE. The unbounded character of the input operator requires that the stability be characterized in terms of the H₁ (rather than the usual L₂) norm of the actuator state. The analysis involves an interesting structure of interconnected PDEs, of parabolic and first-order hyperbolic types, where the feedback gain kernel for the undelayed problem becomes an initial condition in a PDE arising in the compensator design for the problem with input delay. Space and time variables swap their roles in an interesting manner throughout the analysis.     

Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback

This paper considers the problem of finite dimensional disturbance observer based control (DOBC) via output feedback for a class of nonlinear parabolic partial differential equation (PDE) systems. The external disturbance is generated by an exosystem modeled by ordinary differential equations (ODEs), which enters into the PDE system through the control channel. Motivated by the fact that the dominant dynamic behavior of parabolic PDE systems can be characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of finite dimensional ODEs. Subsequently, based on the slow subsystem and the exosystem, a disturbance observer (DO) and a slow mode observer (SMO) are constructed to estimate the disturbance and the slow modes. Moreover, an observation spillover observer (OSO) is also constructed to cancel approximately the effect of the observation spillover. Then, a finite dimensional DOBC design via output feedback is developed to estimate and compensate the disturbance, such that the closed-loop PDE system is exponentially stable in the presence of the disturbance. The condition for the existence of the proposed controller is given in terms of bilinear matrix inequality. Two algorithms based on the linear matrix inequality (LMI) technique are provided for solving control and observer gain matrices of the proposed controller. Finally, the developed design method is applied to the control of a one-dimensional diffusion-reaction process to illustrate its effectiveness.     

Fixed-time output regulation of linear delay systems by smooth time-varying control

This article investigates the fixed-time output regulation problem (FxTORP) for linear systems in the presence of input delay. A linear controller consisting of the linear periodic delayed feedback (PDF) gain and the feedforward gain obtained by solving regulator equations is designed, such that FxTORP is addressed. If only the measurable output can be used for feedback, a linear observer with periodic coefficient and artificial delay is designed so that its state converges to the state of the augmented system at a prescribed finite time. Based on the estimated state, the output regulation problem can also be solved by using observer-based output feedback. The most significant advantages of this article are that the PDF gain can be taken as smooth and the output regulation problem is achieved within a prespecified regulation time. Finally, a simulation example is given to substantiate the validity of the proposed approaches.     

数据驱动的保证收敛速率最优输出调节

针对具有外部系统扰动的线性离散时间系统的输出调节问题,提出了可保证收敛速率的数据驱动最优输出调节方法,包括状态可在线测量系统的基于状态反馈的算法,与状态不可在线测量系统的基于输出反馈的算法.首先,该问题被分解为输出调节方程求解问题与反馈控制律设计问题,基于输出调节方程的解,通过引入收敛速率参数,建立了可保证收敛速率的最...     

Adaptive Output Regulation of a Class of Nonlinear Output Feedback Systems With Unknown High Frequency Gain

This paper presents an output feedback design approach based on the adaptive control scheme developed for nonlinearly parameterized systems, to achieve global output regulation for a class of nonlinear systems in output feedback form. We solve the output regulation problem without the knowledge of the sign and the value of the high frequency gain a priori. It is not necessary to have both the limiting assumptions that the exogenous signal ω and the unknown parameter μ belong to a prior known compact set and the high frequency gain has a determinate lower and upper bounds. The effectiveness of the proposed algorithm is shown with the help of an example.      

A new static output feedback approach to the suboptimal mixed H2/H∞ problem

In this paper we propose a new approach to solve the static output feedback suboptimal mixed H₂/H_∞ control problem using a state fixed‐structure feedback design. We formulate the static output feedback problem as a constrained static state feedback problem and obtain three coupled design equations: one Riccati equation, one Lyapunov equation, and a gain equation. We will prove the equivalence of the proposed solution to the existing solution. A very simple iterative algorithm is then presented to solve the design equations for the stabilizing output feedback gain that minimizes an upper bound of H₂ norm while satisfying the H_∞ disturbance attenuation requirement. A unique feature of the new approach is that it admits the Kalman gain as an initial stabilizing gain to start the above iterative solution procedure, which is computationally attractive and advantageous compared to the direct approach, as the latter has to deal with the difficult algorithm initialization problem. Some illustrative numerical examples are given to demonstrate the effectiveness of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

具有状态和控制滞后的 线性时变时滞系统的输出反馈镇定

研究具有状态和控制滞后的线性时变时滞系统的静态输出反馈镇定，其设计过程只需解一个特殊的代数Ｒｉｃｃａｔｉ方程。并且指出，系统的动态输出反馈镇定问题可等价为广义系统的静态输出反馈镇定问题。最后通过实例论证了本方法的有效性。     

Cooperative output regulation of heterogeneous multi-agent systems based on passivity

This paper investigates the problem of cooperative output regulation of heterogeneous linear multi-agent systems. A passive framework is presented for the stabilisation analysis of cooperative output regulation, which can overcome the difficulty caused by the fact that the global dynamics of heterogeneous multi-agent systems depends on the global communication structure. An adaptive distributed observer is proposed to estimate the state of the exosystem, and the proposed distributed observer is independent of any global information of the communication graph. Based on passivity design and adaptive distributed observer, both a distributed state feedback and a distributed output feedback protocol are designed for output synchronisation of heterogeneous multi-agent systems. The gain matrices of the distributed protocols and observers are obtained by a Riccati equation design approach. Furthermore, sufficient local conditions for solving the problem of cooperative output regulation of heterogeneous multi-agent systems are presented. Finally, numerical simulation results are given to illustrate the effectiveness of the proposed distributed control schemes.     

A global state feedback output regulating control for uncertain systems in strict feedback form

A family of single-input, single-output, nonlinear systems in strict feedback form with uncertain constant parameters which appear linearly and belong to a known compact set Π is considered. A global robust output regulation problem via state feedback is addressed and solved under the assumptions that the regulator equations are solvable in Π, the output equation does not depend on uncertain parameters, no modelled disturbances affect the system and the output reference signal is generated by a known exosystem whose state is bounded and available for feedback. Robust adaptive techniques are used to guarantee closed loop boundedness and to achieve, without requiring persistency of excitation, global asymptotic output regulation for a class of feedback linearizable systems which may have unbounded uncertain tracking dynamics or may not have a well-defined global relative degree.     

一类带不确定输入动态非线性系统的输出反馈鲁棒镇定

研究一类带不确定输入动态非线性系统的输出反馈鲁棒镇定问题.通过在高增益观测器引入新的设计参数,改进了通常的高增益反馈控制的设计方法.在输入动态满足零相对阶最小相位的假设下,基于非分离设计原则给出了动态输出反馈控制器的设计方法,所设计的控制器实现了对任意可允许不确定输入动态的全局鲁棒镇定.     

A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems

Low gain feedback, a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero, has found several applications in constrained control systems, robust control and nonlinear control. In the continuous-time setting, there are currently three ways of constructing low gain feedback laws: the eigenstructure assignment approach, the parametric ARE based approach and the parametric Lyapunov equation based approach. The eigenstructure assignment approach leads to feedback gains explicitly parameterized in the low gain parameter. The parametric ARE based approach results in a Lyapunov function along with the feedback gain, but requires the solution of an ARE for each value of the parameter. The parametric Lyapunov equation based approach possesses the advantages of the first two approaches and results in both an explicitly parameterized feedback gains and a Lyapunov function. The first two approaches have been extended to discrete-time setting. This paper develops the parametric Lyapunov equation based approach to low gain feedback design for discrete-time systems.     

Adaptive error feedback regulation problem for 1D wave equation

By using the adaptive control approach, we solve the error feedback regulator problem for the one‐dimensional wave equation with a general harmonic disturbance anticollocated with control and with two types of disturbed measurements, ie, one collocated with control and the other anti‐collocated with control. Different from the classical error feedback regulator design, which is based on the internal mode principle, we give the adaptive servomechanism design for the system by making use of the measured tracking error (and its time derivative) and the estimation mechanism for the parameters of the disturbance and of the unknown reference. Constructing auxiliary systems and observer and applying the backstepping method for infinite‐dimensional system play important roles in the design. The control objective, which is to regulate the tracking error to zero and to keep the states bounded, is achieved.     

Semi‐global leader‐following consensus of discrete‐time linear multi‐agent systems subject to actuator position and rate saturation

This paper deals with the leader‐following consensus of discrete‐time multi‐agent systems subject to both position and rate saturation. Each agent is described by a discrete‐time general linear dynamic with actuator subject to position and rate saturation. A modified algebraic Riccati equation and low‐gain feedback design technique are used to construct both state feedback and output feedback control protocols. It is established that a semi‐global leader‐following consensus can be achieved when the system is asymptotically null controllable with bounded controls and a leader agent has a directed path to every follower agent. Finally, several simulations are carried out to illustrate the results. Copyright © 2016 John Wiley & Sons, Ltd.