Cooperative global optimal preview tracking control of linear multi-agent systems: an internal model approach

This paper investigates the cooperative global optimal preview tracking problem of linear multi-agent systems under the assumption that the output of a leader is a previewable periodic signal and the topology graph contains a directed spanning tree. First, a type of distributed internal model is introduced, and the cooperative preview tracking problem is converted to a global optimal regulation problem of an augmented system. Second, an optimal controller, which can guarantee the asymptotic stability of the augmented system, is obtained by means of the standard linear quadratic optimal preview control theory. Third, on the basis of proving the existence conditions of the controller, sufficient conditions are given for the original problem to be solvable, meanwhile a cooperative global optimal controller with error integral and preview compensation is derived. Finally, the validity of theoretical results is demonstrated by a numerical simulation.     

Optimal preview control for a linear continuous-time stochastic control system in finite-time horizon

This paper discusses the design of the optimal preview controller for a linear continuous-time stochastic control system in finite-time horizon, using the method of augmented error system. First, an assistant system is introduced for state shifting. Then, in order to overcome the difficulty of the state equation of the stochastic control system being unable to be differentiated because of Brownian motion, the integrator is introduced. Thus, the augmented error system which contains the integrator vector, control input, reference signal, error vector and state of the system is reconstructed. This leads to the tracking problem of the optimal preview control of the linear stochastic control system being transformed into the optimal output tracking problem of the augmented error system. With the method of dynamic programming in the theory of stochastic control, the optimal controller with previewable signals of the augmented error system being equal to the controller of the original system is obtained. Finally, numerical simulations show the effectiveness of the controller.     

线性连续时滞系统最优预见重复控制

针对一类线性连续时滞系统,提出一种最优预见重复控制设计方法.首先,通过一种等价变换,将被控时滞系统转化为无时滞系统.然后,利用L阶差分算子提升技巧,获得包含状态变量导数和跟踪误差的增广连续系统.在此基础上,通过定义一种新的性能指标,将预见重复控制设计问题转化为连续非自治系统的线性二次调节问题.进一步,基于最优控制理论,得到包含状态反馈、误差积分、重复控制、时滞补偿和预见补偿的最优预见重复控制器.该控制器包含了已有文献的多种控制器形式.最后,通过一个数值仿真实例,说明所提方法的有效性.     

Preview control for a class of linear stochastic systems with multiplicative noise

ABSTRACT

In this paper, the preview control problem for a class of linear continuous time stochastic systems with multiplicative noise is studied based on the augmented error system method. First, a deterministic assistant system is introduced, and the original system is translated to the assistant system. Then, the integrator is employed to ensure the output of the closed-loop system tracking the reference signal accurately. Second, the augmented error system, which includes integrator vector, control vector and reference signal, is constructed based on the system after translation. As a result, the tracking problem is transformed into the optimal control problem of the augmented error system, and the optimal control input is obtained by the dynamic programming method. This control input is regarded as the preview controller of the original system. For a linear stochastic system with multiplicative noise, the difficulty being unable to construct an augmented error system by the derivation method is solved in this paper. And, the existence and uniqueness solution of the Riccati equation corresponding to the stochastic augmented error system is discussed. The numerical simulations show that the preview controller designed in this paper is very effective.     

一类具有输入时滞的时变离散系统的预见控制

针对一类具有输入时滞的时变离散系统,研究其预见控制问题.利用差分算子的性质,对系统的输入时滞项和目标信号进行差分处理,构造包含目标信号但不含时滞的扩大误差系统.基于最优控制和预见控制的相关理论,得到了扩大误差系统带有预见前馈补偿的控制器.进一步,利用矩阵分解方法,将高阶Riccati方程进行降阶处理,从而得到原时滞系统的预见控制器.最后通过仿真实例验证了所提出方法的有效性.     

一类线性离散广义系统最优预见控制器设计

提出一类广义线性离散系统的预见控制问题.首先,利用差分算子构造扩大误差系统,然后通过引入一些恒等式,把可预见的目标信号的差分也包含在扩大误差系统的状态向量中,把所考虑的广义系统的预见控制问题转化为一个形式上的普通广义系统的控制问题.由熟知的广义系统最优控制理论的结论,得到广义系统的带有预见作用的控制器.对扩大误差系统的R-能稳性和能检测性进行了详细讨论.     

Output feedback preview tracking control for discrete-time polytopic time-varying systems

This paper focuses on the problem of static output feedback preview tracking control for discrete-time systems with time-varying parameters subject to a previewable reference signal. We develop a design method of a robust controller with integral and preview actions achieving robust tracking performance. First, an augmented error system including future information on reference signal is constructed by introducing two new related auxiliary variables to the original system state and input. This leads to a tracking problem being transformed into a regulator problem. Then, a previewable reference signal is fully utilised through reformulation of the output equation for the augmented error system while considering a static output feedback. Meanwhile, the static output preview control gains are solved explicitly by the proposed conditions. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.     

Design of Preview Controller for Linear Continuous-time Systems with Input Delay

This paper makes it possible to design a preview controller for linear continuous-time systems with input delay, which expands the preview control theory. An augmented error system is established. And a variable substitution is used to transform the delay preview control system into a delay-free one whose preview controller uses the future value of the augmented state vector. The future value of the augmented state vector is estimated by the information of the current state vector, the control input on the past time window, and the reference signal on the future time window. Under the assumption of zero initial conditions, a simplified controller with both delay compensation and preview compensation is obtained for the original system. In addition, a preview controller with full-order observer is offered. Finally, simulation results are presented to illustrate the effectiveness and robustness of the controller.     

时变参数不确定离散时间系统的预见控制

基于具有参数依赖的Lyapunov函数方法及LMI技巧,研究一类时变参数不确定离散时间系统的预见跟踪控制问题.首先,采用预见控制理论中误差系统的方法,引入两个与状态变量和输入变量有关的辅助信号,构造出包含未来目标值信号和干扰信号的信息的扩大误差系统,将原系统的预见跟踪问题转化为扩大误差系统的镇定问题;然后,针对扩大误差系统,考虑输出反馈时,通过改造输出方程融合可预见信号的未来信息.研究表明,通过求解LMI,即可确定静态输出反馈预见控制器增益矩阵的参数矩阵.数值仿真表明了结果的有效性.     

Parallel Control for Optimal Tracking via Adaptive Dynamic Programming

This paper studies the problem of optimal parallel tracking control for continuous-time general nonlinear systems. Unlike existing optimal state feedback control, the control input of the optimal parallel control is introduced into the feedback system. However, due to the introduction of control input into the feedback system, the optimal state feedback control methods can not be applied directly. To address this problem, an augmented system and an augmented performance index function are proposed firstly. Thus, the general nonlinear system is transformed into an affine nonlinear system. The difference between the optimal parallel control and the optimal state feedback control is analyzed theoretically. It is proven that the optimal parallel control with the augmented performance index function can be seen as the suboptimal state feedback control with the traditional performance index function. Moreover, an adaptive dynamic programming (ADP) technique is utilized to implement the optimal parallel tracking control using a critic neural network (NN) to approximate the value function online. The stability analysis of the closed-loop system is performed using the Lyapunov theory, and the tracking error and NN weights errors are uniformly ultimately bounded (UUB). Also, the optimal parallel controller guarantees the continuity of the control input under the circumstance that there are finite jump discontinuities in the reference signals. Finally, the effectiveness of the developed optimal parallel control method is verified in two cases.      

Distributed cooperative stabilisation of continuous-time uncertain nonlinear multi-agent systems

This paper addresses the distributed cooperative stabilisation problem of continuous-time uncertain nonlinear multi-agent systems. By approximating the uncertain dynamics using neural networks, a distributed adaptive cooperative controller, based on the state information of the neighbouring agents, is proposed. The control design is developed for any undirected connected communication topologies without requiring the accurate model of each agent. This result is further extended to the output feedback case. An observer-based distributed cooperative controller is devised and a parameter dependent Riccati inequality is employed to prove stability of the overall multi-agent systems. This design is less complex than the other design methods and has a favourable decouple property between the observer design and the controller design for uncertain nonlinear multi-agent systems. For both cases, the developed controllers guarantee that all signals in the closed-loop network are uniformly ultimately bounded, and the states of all agents cooperatively converge to a small neighbourhood of origin. A comparative study is given to show the efficacy of the proposed method.     

The preview control of a class of linear systems and its application in the fault-tolerant control theory

In the fault-tolerant control theory based on model following control, the desired signal of the control system is the output of a reference system. This paper is concerned with the design of the preview controller for a class of fault systems. A composite vector is introduced by including error vector, fault system state vector and reference system state vector. Then, we derived an augmented system from the known system equation, in which the reference input has equal status with the desired signal in the traditional preview control theory. Therefore, we can use the known theory to design the preview controller for the augmented system, then the preview controller of the original fault system can be obtained by the integration method. This paper strictly discusses the connection between stabilisation and detectability of the augmented system and the corresponding characteristics of the original system. Finally, by applying this theory to a real steam generator water level control system, it is found that the actions of the reference input preview and the fault signal preview can effectively eliminate the effect of the fault signal on the water level of the steam generator. The simulation shows the effectiveness of the controller designed.     

Preview control of polytopic time-varying systems subject to actuator saturation

In this study, a novel method is proposed to track a previewable reference signal in the polytopic time-varying system with input saturation. Firstly, an augmented model containing future information is constructed using a new formal variable. This leads to the tracking control problem of polytopic time-varying system with input saturation is transformed into a stability problem of augmented error system. Next, the state and static output feedback preview controls are introduced, and the corresponding controller gains are produced by the proposed conditions. Two examples are presented to validate the effectiveness of the proposed preview controller.     

多采样率不确定离散时滞系统的鲁棒预见控制

研究一类输入多采样率型不确定离散时滞系统的鲁棒预见控制问题.首先,利用离散提升技术从形式上消除输入时滞和多采样率特点,将多采样率不确定系统的鲁棒预见控制问题转化为一个普通的单采样率不确定系统的鲁棒预见控制问题;然后根据预见控制的基本方法,构造出包含未来目标信息的扩大误差系统,并对其相应的标称系统设计预见控制器;最后,根据所设计的控制器和Lyapunov稳定性理论,给出不确定闭环系统的鲁棒稳定性判据.数值仿真结果验证了所提出设计方法的有效性.     

Output feedback preview control for polytopic uncertain discrete‐time systems with time‐varying delay

This paper considers the preview tracking control problem of polytopic uncertain discrete‐time systems with a time‐varying delay subject to a previewable reference signal. First, a model transformation is employed and a discrete‐time system with a time‐invariant delay and an external disturbance is obtained. The difference operator method can be extended to derive an augmented error system that includes future information on the reference signal. Then, a previewable reference signal is fully utilized through reformulation of the output equation while considering the output feedback. Based on the small gain theorem, a static output feedback controller with preview actions is designed such that the output can asymptotically track the reference signal. Finally, numerical simulation examples also illustrate the superiority of the desired preview controller for the uncertain system in the paper.     

Cooperative semi-global robust output regulation for a class of nonlinear uncertain multi-agent systems

In this paper, we study the cooperative semi-global robust output regulation problem for a class of minimum phase nonlinear uncertain multi-agent systems. This problem is a generalization of the leader-following tracking problem in the sense that it further addresses such issues as disturbance rejection, robustness with respect to parameter uncertainties. To solve this problem, we first introduce a type of distributed internal model that converts the cooperative semi-global robust output regulation problem into a cooperative semi-global robust stabilization problem of the so-called augmented system. We then solve the semi-global stabilization problem via distributed dynamic output control law by utilizing and combining a block semi-global backstepping technique, a simultaneous high gain feedback control technique, and a distributed high gain observer technique.     

Cooperative global output regulation of heterogeneous second-order nonlinear uncertain multi-agent systems

In this paper, we study the cooperative global output regulation problem for a class of heterogeneous second order nonlinear uncertain multi-agent systems. We first introduce a type of distributed internal model that converts the cooperative global output regulation problem into the global robust stabilization problem of the so-called augmented multi-agent system. Then we further globally stabilize this augmented multi-agent system via a distributed state feedback control law, thus leading to the solution of the original problem. A special case of our result leads to the solution of the global leader-following consensus problem for the second order nonlinear multi-agent systems without satisfying the global Lipschitz condition.     

Online Adaptive Approximate Optimal Tracking Control with Simplified Dual Approximation Structure for Continuous-time Unknown Nonlinear Systems

This paper proposes an online adaptive approximate solution for the infinite-horizon optimal tracking control problem of continuous-time nonlinear systems with unknown dynamics. The requirement of the complete knowledge of system dynamics is avoided by employing an adaptive identifier in conjunction with a novel adaptive law, such that the estimated identifier weights converge to a small neighborhood of their ideal values. An adaptive steady-state controller is developed to maintain the desired tracking performance at the steady-state, and an adaptive optimal controller is designed to stabilize the tracking error dynamics in an optimal manner. For this purpose, a critic neural network (NN) is utilized to approximate the optimal value function of the Hamilton-Jacobi-Bellman (HJB) equation, which is used in the construction of the optimal controller. The learning of two NNs, i.e., the identifier NN and the critic NN, is continuous and simultaneous by means of a novel adaptive law design methodology based on the parameter estimation error. Stability of the whole system consisting of the identifier NN, the critic NN and the optimal tracking control is guaranteed using Lyapunov theory; convergence to a near-optimal control law is proved. Simulation results exemplify the effectiveness of the proposed method.