Optimal Tracking Control Law Research Articles

Resilient inverse optimal control for tracking: Overcoming process noise challenges

This paper studies the Inverse Optimal Control (IOC), aiming to identify the underlying cost functions using observed optimal control paths. An innovative IOC algorithm is developed in this paper by leveraging the closed-loop control law of optimal tracking control, without needing to consider any prior knowledge of the process noise. More explicitly, a convex optimization problem is formulated for the IOC problem by encompassing various linear constraints. The contributions of our work include: (i) Robustly handling process noise, ensuring accuracy without excessive data. (ii) Deriving linear conditions for optimal tracking control law, leading to a closed-form IOC solution that can yield the global optimal solution under sufficient conditions. (iii) No extra LMI constraints are needed when dealing with diverse reference signals. The paper concludes by demonstrating our approach’s effectiveness through simulations and comparisons with baseline methods.

Integral sliding mode-based event-triggered optimal fault tolerant tracking control of continuous-time nonlinear systems

In this paper, the integral sliding mode-based event-triggered optimal fault tolerant tracking control of continuous-time nonlinear systems is investigated via adaptive dynamic programming. The developed control scheme consists of two parts, i.e., integral sliding mode control and event-triggered optimal tracking control. For the first part, an integral sliding mode controller is designed to eliminate the affect of actuator fault and the dynamics of nominal nonlinear systems is obtained. For the second part, a novel quadratic cost function with respect to the tracking error and its dynamics is developed such that the feedforward control law or the discount factor is not required, which reduces the complexity of the control method and guarantees the tracking performance. Moreover, a critic-only structure is established to obtain the solution of tracking Hamilton–Jacobi–Bellman equation. It should be noted that the optimal tracking control law is updated only at triggering moments in order to preserve computing and communication resources. Finally, the effectiveness of the present approach is demonstrated through simulation examples of a robotic arm system and a Van der Pol circuit system.

Reinforcement learning‐based optimal trajectory tracking control of surface vessels under input saturations

AbstractThis paper develops a reinforcement learning (RL)‐based optimal trajectory tracking control scheme of surface vessels with unknown dynamics, unknown disturbances, and input saturations of surface vessels. The control scheme is designed by combining the optimal control theory, adaptive neural networks, and the RL method in a unified actor‐critic NN framework. A hyperbolic‐type penalty function of the control input is designed so as to deal with the input saturations of surface vessels. An actor‐critic NN‐based RL mechanism is established to learn the optimal trajectory tracking control law without the knowledge of the surface vessel dynamics and disturbances, where NN weights are tuned online on the basis of devised tuning laws. Theoretical analysis and simulation results prove that the proposed RL‐based optimal trajectory tracking control scheme can ensure surface vessels track the desired trajectory, while guaranteeing the boundedness of all signals in the surface vessel optimal trajectory tracking closed‐loop control system.

Decentralized tracking control design based on intelligent critic for an interconnected spring-mass-damper system

In this paper, the decentralized tracking control (DTC) problem is investigated for a class of continuous-time nonlinear systems with external disturbances. First, the DTC problem is resolved by converting it into the optimal tracking controller design for augmented tracking isolated subsystems (ATISs). %It is investigated in the form of the nominal system. A cost function with a discount is taken into consideration. Then, in the case of external disturbances, the DTC scheme is effectively constructed via adding the appropriate feedback gain to each ATIS. %Herein, we aim to obtain the optimal control strategy for minimizing the cost function with discount. In addition, utilizing the approximation property of the neural network, the critic network is constructed to solve the Hamilton-Jacobi-Isaacs equation, which can derive the optimal tracking control law and the worst disturbance law. Moreover, the updating rule is improved during the process of weight learning, which removes the requirement for initial admission control. Finally, through the interconnected spring-mass-damper system, a simulation example is given to verify the availability of the DTC scheme.

Design of nonlinear optimal tracking control law based on finite-time stability for quadratic performance index

In this paper, a design method of optimal tracking control based on finite-time stability for quadratic performance index is proposed. Finite-time stability of tracking control involves dynamical systems whose actual output can track desired output in finite time while satisfying Lyapunov stability. A nonlinear control law which guaranteed finite-time stability is designed depending on the core idea of dynamic programming. By using Hamilton–Jacobi–Bellman (HJB) equation and finite-time stability theory, sufficient conditions involving V-function are provided, and design steps for nonlinear finite-time tracking control law are derived by constructing augmented systems. In addition, the V-function is constructed to obtain corresponding law for given systems, which verified that the design method is feasible. Simulation examples validate the efficiency of the results.

Fault tolerant tracking control for nonlinear systems with actuator failures through particle swarm optimization-based adaptive dynamic programming

In this paper, an adaptive dynamic programming based fault tolerant tracking control (FTTC) method is proposed for nonlinear systems with actuator failures. To solve the tracking control problem, the considered system is augmented by combining the tracking error dynamics and the desired trajectory dynamics. By establishing a critic neural network, whose weight vector is updated by particle swarm optimization algorithm, the value function with discount factor is approximated to solve the Hamilton–Jacobi–Bellman equation, and the optimal tracking control law is derived. The neural network-based fault observer is established to compensate the control input online. Then, the augmented system states are guaranteed to be uniformly ultimately bounded under the developed FTTC method according to the Lyapunov stability theorem. Two simulation examples are provided to illustrate the effectiveness of the proposed method.

Application of the Feedback Linearization in Maximum Power Point Tracking Control for Hydraulic Wind Turbine

Taking the hydraulic wind turbine as the research object, the method is studied to improve the utilization ratio of wind energy for hydraulic wind turbine, when the wind speed is lower than the rated wind speed. The hydraulic fixed displacement pump speed and generating power can be used as control output to realize the maximum power point tracking control. The characteristics of the maximum power point tracking control are analyzed for hydraulic wind turbine, and the hydraulic output power is taken as control output based on the comprehensive performance requirements. Because the hydraulic wind turbine is a strong multiplication nonlinear system, the system is globally linearized based the feedback linearization method, and the maximum power point tracking control law is obtained. The simulation and experiment results show that the system has good dynamic performance with the proposed control law. The control provides theoretical guidance for optimal power tracking control law application for hydraulic wind turbine.

Optimal Guaranteed Cost Tracking of Uncertain Nonlinear Systems Using Adaptive Dynamic Programming with Concurrent Learning

In this paper, based on adaptive dynamic programming (ADP) with concurrent learning, the problem of optimal guaranteed cost tracking is studied for a class of uncertain continuous-time nonlinear systems. First, the original uncertain system and a reference system are combined into an augmented uncertain system, and the performance index is transformed into the corresponding augmented form. Afterwards, the solution of the optimal control problem consisting of the nominal augmented system with the modified performance index is proven to be the optimal guaranteed cost tracking control law of the original uncertain system. Moreover, a concurrent learning tuning algorithm based on ADP is presented to approximate the solution of corresponding Hamilton-Jacobi-Bellman (HJB) equation, which relaxes the condition of persistent excitation (PE). A neural network-based approximate optimal guaranteed cost tracking design is developed not only to ensure tracking error convergence to zero for all admissible uncertainties but also to achieve the minimal guaranteed cost. Finally, two simulation examples are considered to verify the effectiveness of the theoretical results.

Data-driven finite-horizon optimal tracking control scheme for completely unknown discrete-time nonlinear systems

This paper proposes finite-horizon optimal tracking control approach based on data for completely unknown discrete-time nonlinear affine systems. First, the identifier is designed by input and output data, which is used to identify system function and system model. And based on tracking error, the system function is transformed to the augmentation system with finite-time optimal performance. In finite time, by minimizing the performance index function, the iterative approximate dynamic programming (ADP) is utilized to solve Hamilton–Jacobi–Bellman (HJB) equation. The idea is carried by the policy iterative (PI) based on the model neural network, which makes the iterative control of the augmentation system available at the each step. At the same time, the action neural network is utilized to acquire the approximate optimal tracking control law and the critic neural network is used for approximating the optimal performance index function for the augmentation system. Afterwards, the paper show the analysis process that the convergence and stability for the iterative ADP algorithm and the weight estimation errors based on the PI, respectively. The end of the paper, a simulation example is applied to show the theoretical results and proposed approach.

Experience replay–based output feedback Q‐learning scheme for optimal output tracking control of discrete‐time linear systems

SummaryThis paper focuses on solving the adaptive optimal tracking control problem for discrete‐time linear systems with unknown system dynamics using output feedback. A Q‐learning‐based optimal adaptive control scheme is presented to learn the feedback and feedforward control parameters of the optimal tracking control law. The optimal feedback parameters are learned using the proposed output feedback Q‐learning Bellman equation, whereas the estimation of the optimal feedforward control parameters is achieved using an adaptive algorithm that guarantees convergence to zero of the tracking error. The proposed method has the advantage that it is not affected by the exploration noise bias problem and does not require a discounting factor, relieving the two bottlenecks in the past works in achieving stability guarantee and optimal asymptotic tracking. Furthermore, the proposed scheme employs the experience replay technique for data‐driven learning, which is data efficient and relaxes the persistence of excitation requirement in learning the feedback control parameters. It is shown that the learned feedback control parameters converge to the optimal solution of the Riccati equation and the feedforward control parameters converge to the solution of the Sylvester equation. Simulation studies on two practical systems have been carried out to show the effectiveness of the proposed scheme.

ADP-Based Robust Tracking Control for a Class of Nonlinear Systems With Unmatched Uncertainties

In this paper, an approximately optimal control strategy is developed for the tracking control of a class of continuous-time nonlinear systems with unmatched uncertainties. By transforming the unmatched uncertain term, the auxiliary system associated with the uncertain nonlinear system is established. The auxiliary system is divided into steady and transient parts, and the related controllers are separately solved, meanwhile the transient tracking error system is also obtained by introducing the reference system. A neural network-based adaptive dynamic programming method is used to get the approximately optimal tracking control law of uncertain nonlinear systems with a predefined cost function. Furthermore, the ultimately uniform boundedness of neural network weights and the stability of tracking error systems are both proved through Lyapunov theory. Two cases of nonlinear systems with unmatched uncertainties are investigated to illustrate the effectiveness of the proposed robust tracking control strategy.

Trajectory Tracking and Re-planning with Model Predictive Control of Autonomous Underwater Vehicles

The trajectory tracking of Autonomous Underwater Vehicles (AUV) is an important research topic. However, in the traditional research into AUV trajectory tracking control, the AUV often follows human-set trajectories without obstacles, and trajectory planning and tracking are separated. Focusing on this separation, a trajectory re-planning controller based on Model Predictive Control (MPC) is designed and added into the trajectory tracking controller to form a new control system. Firstly, an obstacle avoidance function is set up for the design of an MPC trajectory re-planning controller, so that the re-planned trajectory produced by the re-planning controller can avoid obstacles. Then, the tracking controller in the MPC receives the re-planned trajectory and obtains the optimal tracking control law after calculating the object function with a Sequential Quadratic Programming (SQP) optimisation algorithm. Lastly, in a backstepping algorithm, the speed jump can be sharp while the MPC tracking controller can solve the speed jump problem. Simulation results of different obstacles and trajectories demonstrate the efficiency of the proposed MPC trajectory re-planning tracking control algorithm for AUVs.

Tracking control using hierarchical fuzzy-regulated optimal approach for robotic manipulators

Robotic manipulators have captured the attention of many specialists owing to its importance in academic research and industrial automation. This paper proposes a novel hierarchical self-organizing fuzzy optimal controller for the trajectory tracking control of robotic manipulator systems. The hierarchical self-organizing fuzzy optimal controller employs a self-organizing fuzzy logic system as a superior control strategy regulator for a subordinate optimal tracking controller. Using the self-organizing learning and fuzzy inference operation, the weighting matrix in the optimal controller is configured adaptively according to the robotic dynamical behavior. The optimal tracking control law under this hierarchical architecture is derived using the maximum principle. Stability and robustness of the hierarchical self-organizing fuzzy optimal controller are then analyzed and proved through the Lyapunov stability approach and Barbalat's Lemma. A simulation study demonstrates the effectiveness and feasibility of this hierarchical system, and compares it with a self-organizing fuzzy controller.

Optimal placement of an actuator in a two degrees of freedom system to generate desired vibrations

In this work, a new optimization criterion is proposed for optimal placement of an actuator over a structure to generate desired vibrations. Linear quadratic tracking (LQT) control is used to generate desired vibrations. To solve optimization problem, genetic algorithm (GA) has been employed as an optimization technique. Proposed optimization criterion is applied on a simple two degrees of freedom system having a hypothetical actuator. Second order model of two degrees of freedom system is converted into a state space model. Thereafter, optimal tracking control law is applied on the state space model. Effect of location of an actuator on the performance of optimal control system is observed. It is found that by placing the hypothetical actuator at an optimal location suggested by GA, optimal control strategy is very effective in tracking the desired vibrations.

An integrated data-driven Markov parameters sequence identification and adaptive dynamic programming method to design fault-tolerant optimal tracking control for completely unknown model systems

In this paper, an integrated design of data-driven fault-tolerant tracking control is addressed relying on the Markov parameters sequence identification and adaptive dynamic programming techniques. For the unknown model systems, the sequence of Markov parameters together with the covariance of innovation signal is firstly estimated by least square method. After a transformation of value function from stochastic to deterministic, a policy iteration adaptive dynamic programming algorithm is then formulated to find the optimal tracking control law. In order to eliminate the influence of unpredicted faults, an active fault-tolerant supervisory control strategy is further constructed by synthesizing fault detection, isolation, estimation and compensation. All these involved designs are performed in the data-driven manner, and thus avoid the information requirement about system drift dynamics. From the perspective of system operation management, the above integrated control scheme provides a framework to achieve the tracking performance optimization, monitoring and maintaining simultaneously. The effectiveness of these conclusions is finally verified via two case studies.

Observer-based approximate optimal tracking control for time-delay systems with external disturbances

This paper proposes a successive approximation design approach of observer-based optimal tracking controllers for time-delay systems with external disturbances. To solve a two-point boundary value problem with time-delay and time-advance terms and obtain the optimal tracking control law, two sequences of vector differential equations are constructed first. Second, the convergence of the sequences of the vector differential equations is proved to guarantee the existence and uniqueness of the control law. Third, a design algorithm of the optimal tracking control law is presented and the physically realisable problem is addressed by designing a disturbance state observer and a reference input state observer. An example of an industrial electric heater is given to demonstrate the efficiency of the proposed approach.

Internal Model-Based Optimal Tracking Control for Multi-Mission Networks

This paper addresses the minimizing power control problem over networks with multi-mission application. The hierarchical decentralized control system is modeled. According to Pontryagin maximum principle, the optimal tracking control law is obtained by deriving difference Riccati equation and Stein equation. Based on constructing internal model, the targets can be tracked without steady-state errors. The dynamical optimal controller is given considering unmeasurable disturbance. The simulation of a simple two-level hierarchical system consisted by electronic inverted pendulums is carried out. The simulation results validate the effectiveness of the designed controller. The proposed approach is illustrated feasible and efficient.

Optimal tracking control for discrete-time systems with multiple input delays under sinusoidal disturbances

This study researches the tracking control problem for discrete-time systems with multiple input delays affected by sinusoidal disturbances. This study is organized around the expression of sinusoidal and disturbances and the delay-free transformation. First, based on the periodic characteristic of the sinusoidal disturbance, the sinusoidal disturbances are considered as the output of an exosystem. By proposing a discrete variable transformation, the discrete-time system with multiple input delays and the quadratic performance index are transformed into equivalent delay-free ones. Then, by constructing an augmented system that comprises the states of the exosystems of sinusoidal disturbances, the reference input, and the delay-free transformation systems, the original tracking problem is transformed into the optimal tracking problem for a delay-free system with respect to the simplified performance index. The optimal tracking control (OTC) law is obtained from Riccati and Stein equations. The existent and uniqueness of the optimal control law is proved. A reduced-order observer is constructed to solve the problem of physically realizable for the items of the reference input and sinusoidal disturbances. Finally, the feasibility and effectiveness of the proposed approaches are validated by numerical examples.

Optimal tracking control for networked control systems with random time delays and packet dropouts

This paper studies the problem of optimal output tracking control for networked control system with uncertain time delays and packet dropouts. Active time-varying sampling period strategy is proposed to ensure the random variable time delays always shorter than one sampling period. Hybrid driven modes are adopted by sensor to solve the issues of long time delay and packet dropout. By using augmentation approach, the tracking problem of this formulated within-one-step delayed discrete-time system is transformed into a general problem of non-delayed state linear quadratic regulator. A “gridding” approach is introduced to guarantee the realization of optimal output feedback control law by the solution of a series of Riccati matrix equations from an offline database that is constructed by different combination of time delays and packet dropouts. Simulation results demonstrate the effectiveness of the optimal tracking control law.

Finite‐Time Optimal Tracking Control for Dynamic Systems on Lie Groups

AbstractThis paper investigates the problem of finite‐time optimal tracking control for dynamic systems on Lie groups for the situation when the tracking time and/or the cost functions need to be considered. The specific results are illustrated on SE(3) (the specific Euclidean groups of rigid body motions). The tracking time is given according to task requirements in advance. By using Pontryagin's maximum principle (PMP) on Lie groups and the backstepping method, a finite‐time optimal tracking control law is designed to track a desired reference trajectory at the given time. Simultaneously, the corresponding cost functions are guaranteed to be optimal. Compared with existing results of optimal control on Lie groups, it is noteworthy that we consider the finite‐time tracking control for dynamic systems rather than kinematic systems. Furthermore, the obtained optimal control law is described by explicit formulations, which is significant for practical applications.