Hybrid Optimal Tracking Control for Linear Continuous-Time Systems by Simultaneous Optimization Approach of System States and Mode Distributions

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.

The optimal output tracking control problem for nonlinear large-scale dynamic systems is considered. A series-based approximate approach for designing the optimal controller is proposed with respect to quadratic performance indexes. By using the approach, the high order, coupling, nonlinear two-point boundary value problem is transformed into a sequence of linear decoupling two-point boundary value problems. By intercepting the frontal finite terms of the solution series, we obtained a suboptimal tracking control law. A simulation example illustrates the efficiency of the optimal output tracking control law designed.

The problem of resilient optimal bipartite tracking control for heterogeneous multi-agent systems with multiple targets under denial-of-service (DoS) attacks is investigated in this paper. A bipartite tracking mechanism is devised in which the agents track the targets under bipartite consensus control, which becomes non-autonomous. This is owing to DoS attacks, as the agents cannot obtain real-time information on the tracked targets and neighboring agents. Consequently, A target observer with a storage module has been developed for the efficient estimation of agent states and the storage of observed information as historical data. By recalling the historical data of the observed state, a new type of distributed resilient optimal controller is formulated, which can achieve the control objective in the case of communication blockage, while minimizing the performance index function of the system. Numerical simulations are performed to verify the proposed secure control design.

ABSTRACTThis paper presents the cooperative preview control method for continuous-time multi-agent systems with a time-invariant directed communication topology. First, the cooperative tracking preview control problem is transformed into the optimal regulation problem of an augmented system. Next, by applying the results of the standard optimal preview control of continuous-time linear systems, a controller of the augmented system is obtained. Furthermore, the stabilisability and detectability of the augmented system are studied under the fixed digraph and the prescribed leader. Then, we derive the controller of multi-agent systems with error integral and preview action that can guarantee the achievement of cooperative optimal preview tracking. Finally, the effectiveness of the controller is shown by numerical simulations.

The optimal output tracking control(OOTC) problem is considered for continuous linear time-invariable systems, which include control delays and external sinusoidal disturbances. We first transform the original system into a new system without time-delay, then construct a new state vector, which containing a external system’s vector. The external system described the aim which we want to track. Following the law of internal model principle, we transform the original and external systems into a new augmented system. Finally we get an optimal control law according to the optimal performance index, and prove the method is effective by emulation.

Existing adaptive optimal tracking controllers for linear continuous-time systems rely on a formulation that hinders learning control policies for general reference trajectories; generalizing approaches are currently limited to discrete-time systems. In addition, existing results usually rely on globally discounted objective functions. We demonstrate that global discounting potentially leads to unstable controllers and propose a partially discounted objective function instead, which we show to have a unique, globally asymptotically stabilizing solution in the linear-quadratic case. Based on this result, we present a model-free adaptive tracking control architecture for linear continuous-time systems. Once trained, the controller can be used to track flexible reference trajectories. We demonstrate the functionality of our approach with an example.

Learning-based optimal cooperative control using edge-based triggering strategy

With the increasing deployment of Unmanned Aerial Vehicle (UAV) swarms in uncertain and dynamically changing environments, optimal cooperative control has become essential for ensuring robust and efficient system coordination. To this end, this paper designs a data-driven optimal bipartite containment tracking control scheme for multi-UAV systems under compound uncertainties. A novel Dynamic Iteration Regulation Strategy (DIRS) is proposed, which enables real-time adjustment of the learning iteration step according to the task-specific demands. Compared with conventional fixed-step data-driven algorithms, the DIRS provides greater flexibility and computational efficiency, allowing for better trade-offs between the performance and cost. First, the optimal bipartite containment tracking control problem is formulated, and the associated coupled Hamilton–Jacobi–Bellman (HJB) equations are established. Then, a data-driven iterative policy learning algorithm equipped with the DIRS is developed to solve the optimal control law online. The stability and convergence of the proposed control scheme are rigorously analyzed. Furthermore, the control law is approximated via the neural network framework without requiring full knowledge of the model. Finally, numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed DIRS-based optimal containment tracking scheme for multi-UAV systems, which can reduce the number of iterations by 88.27% compared to that for the conventional algorithm.

In this paper, a comparison is addressed between two methods, that is, the optimal tracking control methods of unknown nonlinear systems with and without constrained input. Firstly, the optimal tracking problem for a class of affine nonlinear system is formulated. The tracking cost functions are also defined, both for the two methods. The two methods are proved to be equivalent as the actuator bound is large enough. Integral reinforcement learning (IRL) algorithm is employed to solve the optimal tracking problem by using only system data. To facilitate the implementation of the IRL algorithm, the actor-critic neural network technique and the least squares method are employed in approximating the unknown weights iteratively. In the simulation, a detailed comparison is given to demonstrate the relationship between the two methods in the aspects of control input and tracking cost value.

This paper studies the optimal output tracking control and stabilization for networked control systems with packet losses via output feedback control. Both finite-horizon and infinite-horizon cases are considered. For the finite-horizon case, the authors introduce an augmented system according to the state variable and the reference trajectory for the first time. Based on a set of difference Riccati equations, an optimal output feedback tracking controller is proposed by applying the stochastic maximum principle. And an optimal estimator is presented. For the infinite-horizon case, a necessary and sufficient condition for the stabilization of the system is provided. And an optimal output feedback stabilizing tracking controller is obtained by establishing a set of algebraic Riccati equations. Finally, numerical examples are given to verify the proposed results.

For the optimal tracking control problem of affine nonlinear systems, a general value iteration algorithm based on adaptive dynamic programming is proposed in this paper. By system transformation, the optimal tracking problem is converted into the optimal regulating problem for the tracking error dynamics. Then, general value iteration algorithm is developed to obtain the optimal control with convergence analysis. Considering the advantages of echo state network, we use three echo state networks with levenberg-Marquardt (LM) adjusting algorithm to approximate the system, the cost function and the control law. A simulation example is given to demonstrate the effectiveness of the presented scheme.

Optimal tracking control with zero steady-state error for time-delay systems with sinusoidal disturbances

In this paper, online optimal adaptive tracking control of nonlinear discrete-time systems in affine form with uncertain internal dynamics is presented. The augmented system and the cost function over infinite horizon for the augmented state are defined. Two-layer neural network (NN) -based actor-critic framework is introduced to estimate the optimal control input and value function. The temporal difference (TD) error is derived as a function of the difference between actual and estimated value function. The NN weights of critic and actor are tuned at every sampling instant as a function of the instantaneous temporal difference errors and control policy errors, respectively. The proposed scheme ensures the closed-loop stability in the form of boundedness. Simulation results are provided to illustrate the effectiveness of the proposed approach.

A robust optimal output tracking control method for a midwater trawl system is investigated based on T-S fuzzy nonlinear model. A simplified nonlinear mathematical model is first employed to represent a midwater trawl system, and then a T-S fuzzy model is adopted to approximate the nonlinear system. Since the strong nonlinearities and the external disturbance of the trawling system, a mixed H2/H∞ fuzzy output tracking control strategy via T-S fuzzy system is proposed to regulate the trawl depth to follow a desired trajectory. The trawl depth can be regulated by adjusting the winch velocity automatically and the tracking error can be minimized according to the robust optimal criterion. In order to validate the proposed control method, a computer simulation is conducted. The simulation results indicate that the proposed fuzzy robust optimal controller make the trawl net rapidly follow the desired trajectory under the model uncertainties and the external disturbance caused by wave and current.

SUMMARYThe optimal tracking problem for multiple‐input multiple‐output linear‐time‐invariant discrete‐time systems with communication constraints in the feedback path is studied in this paper. The tracking performance is measured by the energy of the error signal between the output of the plant and the reference signal. The objective is to obtain an optimal tracking performance, attainable by all possible stabilizing compensators. It is shown that the optimal tracking performance consists of two parts, one depends on the nonminimum phase zeros and zero direction of the given plant, as well as the reference input signal direction, and the other depends on the nonminimum phase zeros, unstable poles, and pole direction of the given plant, as well as the bandwidth and additive white Gaussian noise of the communication channel. It is also shown that, if the constraint of the communication channel does not exist, the optimal tracking performance reduces to the existing tracking performance of the control system without communication constraints. A typical example is given to illustrate the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.