Persistent Excitation Condition Research Articles

A general update rule for Lyapunov‐based adaptive control of mobile robots with wheel slip

SummaryIn this article, we introduce a novel family of Lyapunov‐based adaptive kinematic control laws developed to solve the trajectory tracking problem for a differential‐drive mobile robot under the influence of both longitudinal and lateral wheel slip. Each adaptive controller in this family is constructed by augmenting a nonadaptive nominal controller, originally designed for the slip‐free case, with an update rule capable of estimating the longitudinal slip. In the absence of lateral slip and under constant longitudinal slip, we establish the convergence of the trajectory tracking error to zero and, assuming a persistent excitation condition, we also demonstrate the convergence of the slip estimate error to zero. When lateral slip is present, we analyse a particular control law from our family of adaptive controllers. This law ensures the trajectory tracking error is uniformly ultimately bounded around the origin, demonstrating the robustness of our adaptive control scheme in dealing with time‐varying longitudinal and lateral slip. The validity of our approach is assessed through comprehensive numerical simulations.

Cooperative Finitely Excited Learning for Dynamical Games.

In this article, we propose a way to enhance the learning framework for zero-sum games with dynamics evolving in continuous time. In contrast to the conventional centralized actor-critic learning, a novel cooperative finitely excited learning approach is developed to combine the online recorded data with instantaneous data for efficiency. By using an experience replay technique for each agent and distributed interaction amongst agents, we are able to replace the classical persistent excitation condition with an easy-to-check cooperative excitation condition. This approach also guarantees the consensus of the distributed actor-critic learning on the solution to the Hamilton-Jacobi-Isaacs (HJI) equation. It is shown that both the closed-loop stability of the equilibrium point and convergence to the Nash equilibrium can be guaranteed. Simulation results demonstrate the efficacy of this approach compared to previous methods.

Dynamic event-triggered robust optimal tracking control for multi-player nonzero-sum games with mismatched uncertainties and asymmetric constrained inputs

In this paper, a novel dynamic event-triggered robust optimal tracking control method is proposed for multi-player nonzero-sum game problem with mismatched uncertainties and asymmetric constrained inputs. First, the original asymmetric constrained-input multi-player system is transformed into a new multi-player system with symmetric constrained inputs implemented by the state-space transformation technique, and an augmented system is established based on the new system and the desired trajectory. Next, an auxiliary augmentation system is constructed to handle the robust tracking control problem. Then, a critic neural network with a novel modified weight updating law is established for each player to solve the event-triggered coupled Hamilton–Jacobi equations, which relaxes the restrictions of initial admissible control and persistence of excitation condition. Furthermore, efficient communication and computing are obtained by the newly established dynamic event-triggered mechanism, and the Zeno behavior can be excluded by introducing an exponential term. The uniformly ultimately bounded properties of all signals are proved. Finally, two numerical examples verify the effectiveness of the designed method.

Identification of Parameters of Conservative Multispecies Lotka-Volterra System Based on Sampled Data

In this paper, we propose a method for identifying the parameters of the multispecies Lotka-Volterra model based on sampled data. The conditions (nonhyperplanarity, recurrence) under which the “Stripe” algorithm achieves the identification goals are found. The relation between nonhyperplanar recurrent trajectories and the persistent excitation condition for systems of general form is studied.

Robust optimal control of the multi‐input systems with unknown disturbance based on adaptive integral reinforcement learning Q‐function

AbstractConsidering overshoot and chatter caused by the unknown interference, this article studies the adaptive robust optimal controls of continuous‐time (CT) multi‐input systems with an approximate dynamic programming (ADP) based Q‐function scheme. An adaptive integral reinforcement learning (IRL) scheme is proposed to study the optimal solutions of Q‐functions. First, multi‐input value functions are presented, and Nash equilibrium is analyzed. A complex Hamilton–Jacobi–Issacs (HJI) equation is constructed with the multi‐input system and the zero‐sum‐game‐based value function. It is a challenging task to solve the HJI equation for nonlinear system. Thus, A transformation of the HJI equation is constructed as a Q‐function. The neural network (NN) is applied to learn the solution of the transformed Q‐functions based on the adaptive IRL scheme. Moreover, an error information is added to the Q‐function for the issue of insufficient initial incentives to relax the persistent excitation (PE) condition. Simultaneously, an IRL signal of the critic networks is introduced to study the saddle‐point intractable solution, such that the system drift and NN derivatives in the HJI equation are relaxed. The convergence of weight parameters is proved, and the closed‐loop stability of the multi‐system with the proposed IRL Q‐function scheme is analyzed. Finally, a two‐engine driven F‐16 aircraft plant and a nonlinear system are presented to verify the effectiveness of the proposed adaptive IRL Q‐function scheme.

A New Neural Dynamic Learning Framework for Discrete-Time Strict-Feedback Systems: Internal Interaction-Based Weight Adaptive Laws.

This article investigates internal interaction-based dynamic learning control (LC) for uncertain discrete-time strict-feedback systems. On the basis of predict technology, the original system is converted into a common n -step-ahead input-output predict model. The predict model causes every estimated neural weight to converge to n different constants using the existing control framework. To solve such a problem, the predict model is further decomposed into n one-step-ahead subsystems, which can be viewed as n independent agents. Subsequently, the distributed cooperative weight adaptive laws are designed by introducing an undirected and connected interconnection topology among subsystems. By constructing the variable relationship between the subsystems and the n -step-ahead predict model, a new internal weight interaction-based neural dynamic LC framework is proposed for the whole closed-loop system, in which estimated weights at different times share their weight knowledge. The proposed framework ensures the ultimately uniform boundedness of the closed-loop system and achieves the excellent control performance. By combining the consensus theory and a cooperative persistent excitation condition, every estimated weight along the neural input orbit is verified to exponentially converge to a close vicinity of a unique ideal constant, rather than n different constants. Consequently, the developed LC framework facilitates constant weights storage, saves the knowledge storage space, and improves the robustness of knowledge utilization. These characteristics are verified by simulation results.

Finite-dimensional adaptive observer design for linear parabolic systems

A new finite-dimensional adaptive observer is proposed for some linear parabolic systems. The observer is based on the modal decomposition approach and uses a classical persistent excitation condition to ensure practical exponential convergence of both states and parameters estimation.

Double-layer prescribed performance control of free-floating space manipulators after capturing unknown targets

This paper studies the parameter identification and prescribed performance control of the free-floating space manipulator after capturing unknown targets. A novel finite-time concurrent learning (FT-CL) parameter identification algorithm utilizing suitable historical data is proposed. This algorithm guarantees the finite-time convergence of the parameter estimation error without using the persistent excitation condition. Then, we apply the dual modeling approach and view the end-effector states as generalized coordinates of novel virtual dual dynamics. This technique bypasses solving the inverse kinematics and calculating the Jacobi matrix, laying the foundation for Jacobi-free pose control of the end-effector. Besides, stabilizers of prescribed performance reference acceleration are carefully designed in a double-layer form for regulating end-effector pose and base attitude. Finally, the Jacobi-free controllers satisfying the double-layer performance constraints are developed. The efficacy of the proposed parameter identification scheme and control strategy is validated through numerical simulations.

Optimized intelligent tracking control for a quadrotor unmanned aerial vehicle with actuator failures

This work investigates the implementation of tracking control for a quadrotor unmanned aerial vehicle exposed to disturbances and actuator failures. Cost reduction in control processes is not usually considered in conventional control methods. To overcome this limitation, this work proposes an optimized intelligent control scheme that utilizes adaptive dynamic programming. Firstly, a critic-only structure is employed to learn the nominal control strategies, including two critic neural network update laws, lifting the persistent excitation condition. Subsequently, two specifically designed observers are developed on the radial basis function neural network (RBFNN) to reconstruct the disturbances and actuator failures and compensate for the nominal strategies. The observers achieve fixed-time convergence by employing the fixed-time update laws of the RBFNNs. Moreover, two variable vectors using the Gaussian error function are introduced to reduce the impact caused by the observation errors. Additionally, the superior performance of the proposed control scheme is validated through numerical simulations.

Driver-Centric Lane-Keeping Assistance System Design: A Noncertainty-Equivalent Neuro-Adaptive Control Approach

Vehicle roadway departure accidents are a major traffic safety concern as they oftentimes result in severe injuries and fatalities. To address such an issue, this article originates a novel driver-centric and neuro-adaptive-control-based lane-keeping assistance system (LKAS). The proposed control strategy synergizes a noncertainty-equivalent adaptive control design scheme, an adaptive radial-basis-function-based neural network (RBFNN) that captures the human driver's lane-keeping steering behavior, and a Gudermannian-function-based smooth parameter projection operator. The benefit and uniqueness of the proposed solution are threefold. First and foremost, the noncertainty-equivalent adaptive control design, which leverages the immersion-and-invariance-like methodology, ensures the asymptotical convergence of the parameter-estimation-error-induced perturbation despite the reference signal's persistency of excitation condition. Second, the LKAS is devised to be driver-centric, i.e., an adaptive RBFNN-based human driver steering model is embedded inside the LKAS's algorithm such that a human driver is assisted in a personalized and adaptive manner. Third, the Gudermannian-function-based smooth parameter projection operator ascertains the prescribed boundedness of the control parameters while maintaining the control action's smoothness. A pilot human-subject study using a high-fidelity moving-base driving simulator is conducted to validate the proposed LKAS. Further, its performance is compared with a baseline certainty-equivalent neuro-adaptive controller.

Convergence analysis of a synchronous gradient estimation scheme for time-varying parameter systems

For the problem of time-varying parameter estimation and its convergence, this paper proposes an invariant matrix to represent the changing laws of time-varying parameters, which suggests a novel expression to describe the parameters by employing finite terms. A joint recursive least squares algorithm is derived by minimizing the criterion function associated with the unknown invariant matrix. For a class of time-varying systems, a joint stochastic gradient algorithm with the computational efficiency is obtained by analogy to achieve the estimation of invariant matrix and system parameters. The convergence analysis is explored to demonstrate that the time-varying parameter estimation error is bounded under the persistent excitation conditions. The simulation results also validate the effectiveness of the proposed synchronous parameter estimation approach.

Adaptive Control With Global Exponential Stability for Parameter-Varying Nonlinear Systems Under Unknown Control Gains.

It is nontrivial to achieve exponential stability even for time-invariant nonlinear systems with matched uncertainties and persistent excitation (PE) condition. In this article, without the need for PE condition, we address the problem of global exponential stabilization of strict-feedback systems with mismatched uncertainties and unknown yet time-varying control gains. The resultant control, embedded with time-varying feedback gains, is capable of ensuring global exponential stability of parametric-strict-feedback systems in the absence of persistence of excitation. By using the enhanced Nussbaum function, the previous results are extended to more general nonlinear systems where the sign and magnitude of the time-varying control gain are unknown. In particular, the argument of the Nussbaum function is guaranteed to be always positive with the aid of nonlinear damping design, which is critical to perform a straightforward technical analysis of the boundedness of the Nussbaum function. Finally, the global exponential stability of parameter-varying strict-feedback systems, the boundedness of the control input and the update rate, and the asymptotic constancy of the parameter estimate are established. Numerical simulations are carried out to verify the effectiveness and benefits of the proposed methods.

Distributed Adaptive Inverse Differential Game Approach to Leader's Behavior Learning for Multiple Autonomous Followers

In this study, the leader's behavior learning problem is investigated for a class of leader-follower systems, where the leader's behavior is assumed not a priori known to all followers that are autonomous. A multi-player nonzero-sum differential game is introduced to indicate the control interaction between agents, where the leader's behavior is modelled by an unknown cost function. The autonomous followers aim to retrieve the weighting matrix in the leader's cost function collaboratively. A distributed online adaptive inverse differential game (IDG) approach to the leader's behavior learning is proposed for the autonomous followers. Specifically, a concurrent learning (CL) based adaptive law and an interactive game controller are first developed for each autonomous follower to learn the leader's feedback gain matrix online, while the feedback Nash equilibrium of the game can be achieved. Then, a linear matrix inequality (LMI) optimization problem is formulated to determine the weighting matrix of the leader's cost function for each autonomous follower. The proposed method simply requires that all followers share their interactive feedback gain matrices, not their private intents, and use only the system state data, not both the system state data and the leader's control input data. The main advantages of the proposed method are that it can be implemented online without requiring the persistent excitation condition and needs less computational power for all followers. Finally, numerical simulations are provided to demonstrate the effectiveness and feasibility of the developed method.

Dynamic learning from adaptive neural control for full-state constrained strict-feedback nonlinear systems

This study focuses on the learning and control issues of strict-feedback systems with full-state constraints. To achieve learning capability under constraints, transformation mapping is utilized to convert the original system with full-state constraints into a quasi-pure-feedback unconstrained system. Utilizing the system transformation technique, only a single neural network (NN) is required to identify the unknown dynamics within the transformed system. Combining the dynamic surface control design, a novel adaptive neural control scheme is developed to ensure that all closed-loop signals are uniformly bounded, and every system state remains within the predefined constraint range. In addition, the precise convergence of NN weights is further transformed into an exponential stability problem for a category of linear time-varying systems under persistent excitation conditions. Subsequently, the converged NN weights are efficiently stored and utilized to create a learning controller to achieve better control performance while abiding by the full-state constraints. The viability of this control strategy is demonstrated via simulations.

Two‐layer leader‐follower optimal affine formation maneuver control for networked unmanned surface vessels with input saturations

AbstractThis article studies the two‐layer leader‐follower optimal affine formation maneuver (AFM) control problem for multiple unmanned surface vessels (USV). The basic idea of the developed two‐layer control framework is to realize the time‐varying formation for the leader group in advance, and then let the follower USVs agilely respond to the target formation affinely localized by leaders. To realize this goal, a time‐varying formation reference system is first synthesized to describe leaders' target formation in the affine map of the nominal configuration. The reference system is set up by virtual vessel velocity command and unrotated geometric command, which make it possesses an intuitive application structure for operators to plan AFM commands in real‐time sensed surrounding environments. Then, by means of optimized backstepping design and the adaptive dynamic programming technique, actor‐only reinforcement learning controllers are established for USVs in both two layers to realize the optimal target formation tracking objective. For the proposed optimal controllers, the persistent excitation condition that is required by common optimal control schemes is also removed, which endows the presented scheme with lower computational complexity. Finally, theoretical analysis and simulation results illustrate the closed‐loop stability and the effectiveness of the presented scheme.

Adaptive-Dynamic-Programming-Based Robust Control for a Quadrotor UAV with External Disturbances and Parameter Uncertainties

Thiswork addresses the trajectory-tracking-control problem for a quadrotor unmanned aerial vehicle with external disturbances and parameter uncertainties. A novel adaptive-dynamic-programming-based robust control method is proposed to eliminate the effects of lumped uncertainties (including external disturbances and parameter uncertainties) and to ensure the approximate optimal control performance. Its novelty lies in that two radial basis function neural network observers with fixed-time convergence properties were first established to reconstruct the lumped uncertainties. Notably, they tune only the scalar parameters online and have low computational complexities. Subsequently, two actor–critic neural networks were designed to approximate the optimal cost functions and control policies for the nominal system. In this design, two new actor–critic neural network weight update laws are proposed to eliminate the persistent excitation condition. Then, two adaptive-dynamic-programming-based robust control laws were obtained by integrating the observer reconstruction information and the nominal control policies. The uniformly ultimately bounded stability of the closed-loop tracking control systems was ensured using the Lyapunov methodology. Finally, numerical results are shown to verify the effectiveness and superiority of the proposed control scheme.

Research and analysis of manipulator control method based on deep learning

Aiming at the stability and accuracy of grasping objects, this paper studies the adaptive neural network control and learning of manipulator control system with unknown system dynamics. First, a stable adaptive neural network (NN) controller is designed, and the unknown closed-loop system dynamics of the manipulator is approximated by using radial basis function (RBF) neural network. The LSTM long and short memory algorithm is introduced, and the RBF based hybrid neural network model is constructed. The LSTM algorithm is used to design the input gate, forgetting gate and output gate structures to suppress the gradient expansion problem during coordinate data training, and accurate trajectory correction instructions are given. Lyapunov stability theorem is used to analyze stability. Partial persistent excitation (PE) conditions of some internal signals in the closed-loop system are satisfied in the control process of tracking the circular reference trajectory. Under the PE condition, the proposed adaptive NN controller can accurately identify the dynamic uncertainties of the manipulator in the stability control process. Then, a new NN learning control method is proposed, which effectively uses the knowledge learned without re adapting to the unknown manipulator control system dynamics to achieve closed-loop stability and improve control performance. The effectiveness of the proposed method is studied through simulation.Keyboard

Trajectory tracking control of nonholonomic wheeled mobile robots using only measurements for position and velocity

This paper studies the trajectory tracking control of nonholonomic wheeled mobile robots, and only measurements for Cartesian position and velocity are used. A full-order observer is proposed to estimate the unmeasurable attitude and filter the position signal. Its effectiveness relies on the persistent excitation (PE) condition of non-zero linear velocity. Using the observer states, we design a local tracking controller that guarantees: (a) the linear velocity input is positive, and (b) the position tracking error is convergent to zero locally asymptotically. Experiments performed on a robot built in-house illustrate the control algorithm.

Hierarchical Sliding-Mode Surface-Based Adaptive Critic Tracking Control for Nonlinear Multiplayer Zero-Sum Games via Generalized Fuzzy Hyperbolic Models

This paper investigates the hierarchical sliding-mode surface (HSMS)-based adaptive critic tracking control problem for nonlinear multiplayer zero-sum games (ZSGs). First, a generalized fuzzy hyperbolic models-based identifier is employed to approximate the unknown nonlinear functions. According to the derived data-driven model, a static control policy is proposed to transform the considered optimal tracking control problem into an optimal regulation control problem of an induced error system. Subsequently, based on sliding mode control technology and the concept of hierarchical design, a novel optimal feedback control policy is developed to regulate the tracking error by minimizing a performance index function related to the HSMS. The solution to the Hamilton-Jacobi-Isaacs equation of nonlinear multiplayer ZSGs is obtained via the HSMS-based critic neural network. Furthermore, the historical stored data is utilized to conquer the difficulty associated with the persistence of excitation conditions. Based on Lyapunov stability theory, it is strictly proven that the tracking error and the critic neural network weight are uniformly ultimately bounded. Finally, two examples are presented to demonstrate the effectiveness of the proposed control scheme.

Identification of polynomial nonlinear systems based on center manifold

A center-manifold (CM) based system identification (SID) method is developed to identify multi-input-multi-output continuous-time polynomial nonlinear systems in the state–space form. On one hand, the CM method is a natural extension of the frequency-domain subspace method to nonlinear SID problems. It decomposes the nonlinear SID problem into a series of linear least-square problems (in a precise sense), which can be solved analytically. This is in contrast to many existing methods that rely on iterative search of the global minimum. On the other hand, it is also a generalization of the invariant-subspace (ISP) based SID method. It inherits the nice property of the ISP method that the advantages of time-domain and frequency-domain SID methods are combined. In the paper, a persistence of excitation (PE) condition for the CM method is derived, which is described by the rank of certain Khatri–Rao product. Under the PE condition, the identification error is shown to be bounded within an arbitrarily small neighborhood of the origin when measurement noise is presented.