State-constrained Systems Research Articles

Backstepping based neural [formula omitted] optimal tracking control for nonlinear state constrained systems with input delay and disturbances

This paper focuses on the problem of H∞ optimal tracking control for a class of nonlinear state constrained systems with input delay and disturbances. With the aid of Pade approximation, an auxiliary variable is devised to eliminate the effects of input delay. Combining barrier Lyapunov functions (BLFs) with backstepping design technique, a feedforward adaptive controller is designed to transform the tracking control problem of nonlinear state constrained system into an equivalent H∞ control problem of input-affine error system without state constraints, wherein neural networks (NNs) are employed to approximate unknown system dynamics. Then based on single-network adaptive dynamic programming (ADP), an H∞ optimal feedback controller is developed by utilizing a single critic network to learn the Nash equilibrium related to Hamilton–Jacobi–Isaacs (HJI) equation. Therefore, the whole tracking controller can be constructed by integrating the feedforward adaptive controller with the optimal feedback controller. Moreover, it is proven by Lyapunov’s theory that all signals within the closed-loop system are uniformly ultimately bounded (UUB), and the tracking error converges to a small neighborhood of the origin without violating any state constraints. Two simulation examples are also presented to validate the effectiveness of the proposed approach.

Adaptive Fuzzy Output-Feedback Tracking Control for Nonlinear Time-Varying State-Constrained Systems Without the Feasibility Condition

Adaptive Fuzzy Output-Feedback Tracking Control for Nonlinear Time-Varying State-Constrained Systems Without the Feasibility Condition

Adaptive tracking control for stochastic nonlinear state-constrained systems with input delay based on multi-dimensional Taylor network

In this paper, an adaptive tracking control strategy based on multi-dimensional Taylor network (MTN) is proposed for a class of stochastic nonlinear state-constrained systems with input delay. Firstly, to avoid constraint violation, the adaptive backstepping approach and barrier Lyapunov function (BLF) are combined in a unified framework. Then, a suitable auxiliary system with the same order as the considered system is constructed to deal with the effect of input delay. In addition, MTN is used to estimate unknown nonlinear functions during the design of the controller. Furthermore, with the help of the Lyapunov stability theorem, the controller presented in this paper can guarantee that all the closed-loop signals are bounded in probability, the system states remain in the defined compact sets, the output signal can track the reference signal successfully, and the tracking error is bounded by the expected bound. Finally, two examples are given to verify the effectiveness of the designed scheme.

Dynamic event-triggered-based online IRL algorithm for the decentralized control of the input and state constrained large-scale unmatched interconnected system

This article proposed a novel adaptive decentralized control (ADC) method for the continuous-time state-constrained and input-constrained large-scale unmatched interconnection system by the means of the adaptive critic design in the edge dynamic event-triggered (EDET) mechanism. The barrier function is used to transform the state-constrained system to the common system without constrained states. To overcome the influence of the unmatched interconnection terms, the auxiliary systems are devised for each subsystem of the large-scale system. Moreover, the non-quadratic utility functions are introduced to constrain the input of the auxiliary systems. The decentralized control scheme of the large-scale unmatched interconnection system can be realized by solving the optimal control schemes of the auxiliary systems. Then with the help of the single critic neural network (NN), the approximate optimal control policies are acquired by designing the approximate cost functions. After that, an integral reinforcement learning (IRL)-based scheme is proposed to solve the integral Bellman equation rather than the complex coupled Hamilton–Jacobi-Bellman equation (HJBE). Simultaneously, the EDET mechanism is introduced to reduce the computation efforts and communication loads. The asymptotic stability of the constrained large-scale unmatched interconnection system is proved. In addition, the infamous Zeno behavior is effectively avoided. Finally, two simulation cases are given to verify the effectiveness of the proposed algorithm.

Relay-Switching-Based Fixed-Time Tracking Controller for Nonholonomic State-Constrained Systems: Design and Experiment

Dear Editor, This paper investigates the fixed-time trajectory tracking controller design for nonholonomic chained systems with static state constraints. Firstly, a fixed-time tracking control law is given to carry out relay switching, which divides the controller development process into two stages. For the goal of designing fixed-time tracking controller in the second stage and avoiding the explosion of complexity, fixed-time filters are introduced. Under these preparations, a fixed-time tracking controller is elaborated by applying barrier-Lyapunov-function-based backstepping. In addition, a new tracking controller is also proposed to guarantee that the tracking error system is bounded before switching. It has been proved that the resulting closed-loop tracking error system states converge to a small neighborhood around zero within a fixed time and output states are kept in the predefined constrained regions.

Event-triggered-based fuzzy adaptive tracking control for nonstrict-feedback asymmetric state constrained systems

In this article, a fuzzy adaptive command filtered event-triggered control issue is considered for a category of nonstrict-feedback nonlinear systems under the conditions of asymmetric state constraints and time-varying input delay. Firstly, the nonlinear state constrained system is transformed into a new system without state constraints by utilizing nonlinear mappings, which achieves state constraints while removing the feasibility conditions of virtual control signals. Secondly, unknown nonlinear terms in the novel system are approached by using the fuzzy logic systems, a state observer is established to estimate the unmeasurable states in the system, and an auxiliary system is constructed to make up for the influence of time-varying input delay. In addition, the command filtered technology is introduced to overcome the complexity explosion problem in the conventional backstepping design process. It is shown that the fuzzy adaptive event-triggered control scheme can not only ensure the convergence of the tracking error at the origin and the boundedness of all closed-loop signals, but also lighten the communication burden from the controller to the actuator. Finally, a numerical example and an underwater vehicle system are simulated to verify the availability and feasibility of the developed approach.

Neural Adaptive Fixed-Time Control for Nonlinear Systems With Full-State Constraints.

This article aims at this problem of adaptive neural tracking control for state-constrained systems. A general fixed-time stability criterion is first presented, by which an adaptive neural control algorithm is developed. Under the action of the proposed adaptive neural tracking controller, the tracking error converges into a small neighborhood around the origin in fixed time; meanwhile, all system states abide by the corresponding state constraints for all the time. The main difference between the present research and the previous control schemes for state-constrained systems is that this article proposes a novel and feasible approach to ensure that the constructed virtual control signals satisfy the state constraints on the corresponding states viewed as the virtual control inputs. Such an approach guarantees theoretically that all the system states cannot violate their constrained requirements at any time. Finally, two simulation examples provide support to the proposed results.

Diffeomorphism-Based Robust Bounded Control for Permanent Magnet Linear Synchronous Motor With Bounded Input and Position Constraints

This article develops a diffeomorphism-based robust bounded control (DRBC) for the permanent magnet linear synchronous motor system subject to inequality constraints (i.e., bounded input and position constraints) and uncertainties. The uncertainties, including parameter uncertainties and external disturbances, are potentially nonlinear and fast time varying. The bound of the uncertainty is described by a fuzzy set. To overcome the bounded input constraint, a robust bounded control is proposed based on a novel input diffeomorphism scheme, which is in a deterministic form and not if–then rule based. Furthermore, to overcome the bounded position constraint, a transformed system is formulated by a state diffeomorphism scheme, which transforms the bounded state-constrained system to an unconstrained one. Thus, the output of the controlled system can be restricted to a prescribed range. The DRBC guarantees both the uniform boundedness and the uniform ultimate boundedness of the transformed system. A fuzzy performance index, which combines the steady-state performance (the average fuzzy performance) and control effort, is then established based on the fuzzy description of uncertainty. As a result, the design parameter optimization can be solved by minimizing the performance index. Experimental results demonstrate that the DRBC is of superior tracking performance and robustness without violating the prescribed inequality constraints.

Prescribed Finite-Time Adaptive Neural Tracking Control for Nonlinear State-Constrained Systems: Barrier Function Approach.

The purpose of this article is to present a novel backstepping-based adaptive neural tracking control design procedure for nonlinear systems with time-varying state constraints. The designed adaptive neural tracking controller is expected to have the following characters: under its action: 1) the designed virtual control signals meet the constraints on the corresponding virtual control states in order to realize the backstepping design ideal and 2) the output tracking error tends to a sufficiently small neighborhood of the origin with the prescribed finite time and accuracy level. By combining the barrier Lyapunov function approach with the adaptive neural backstepping technique, a novel adaptive neural tracking controller is proposed. It is shown that the constructed controller makes sure that the output tracking error converges to a small neighborhood of the origin with the prespecified tracking accuracy and settling time. Finally, the proposed control scheme is further tested by simulation examples.

Observer-Based Adaptive Fuzzy Control for Nonlinear State-Constrained Systems Without Involving Feasibility Conditions.

For nonlinear full-state-constrained systems with unmeasured states, an adaptive output feedback control strategy is developed. The main challenge of this article is how to avoid that the unmeasured states exceed the constrained spaces. To achieve a good tracking performance for the considered systems, a stable state observer is structured to estimate unmeasured states which are not available in the control design. In addition, the constraints existing in most practical engineering are the source of reducing control performance and causing the system instability. The main limitation of current barrier Lyapunov functions is the feasibility conditions for intermediate controllers. The nonlinear mappings are used to achieve the satisfaction of full-state constraints directly and avoid feasibility conditions for intermediate controllers. By the Lyapunov theorem, the closed-loop system stability is proven. Simulation results are given to confirm the validity of the developed strategy.

Reinforcement learning based adaptive optimal control for constrained nonlinear system via a novel state-dependent transformation

Existing schemes for state-constrained systems either impose feasibility conditions or ignore the optimality. In this article, an adaptive optimal control scheme for the strict-feedback nonlinear system is proposed, which benefits from two design steps. Firstly, a novel nonlinear state-dependent function (NSDF) is formulated to equivalently transform the system into a non-constrained one to deal with state constraints without the requirements on feasibility conditions. Secondly, an adaptive optimal control scheme is designed for the non-constrained system, in which reinforcement learning (RL) is utilized to yield the optimal controller in each designing procedure. Updating rules of the actor and critic neural network are driven by the modified adaptive laws, used to approximate the optimal virtual and actual controllers. It is proved that all the signals in the closed-loop system are bounded and the output tracking error converges to an adjustable neighborhood of the origin not affected by the proposed NSDF. Two simulation examples are presented illustrating the effectiveness of the proposed scheme.

Adaptive fixed-time consensus control for a class of non-strict feedback multi-agent systems subject to input nonlinearities, state constraints, unknown control directions, and actuator faults

In this work, an adaptive fixed-time controller has been designed for a class of uncertain non-strict feedback multi-agent systems (MASs) subject to different types of input nonlinearities, time-varying asymmetric state constraints, unknown control directions, infinite number of actuator faults and external disturbances. Compared to the existing consensus control schemes, the proposed controller can handle input nonlinearities, actuator faults and unknown control directions simultaneously, while guaranteeing fixed-time convergence of the consensus tracking error and satisfying state constraints for non-strict feedback MASs. By employing a nonlinear mapping, the constrained MAS has been transformed into an unconstrained one, hence the feasibility analysis required for barrier Lyapunov function (BLF)-based controllers for state-constrained systems has been avoided. Different input nonlinearities including backlash, hysteresis, dead-zone and saturation have been tackled by employing a unified framework, hence each agent can have a different type of input nonlinearity. By using fuzzy logic systems (FLSs), system unknown dynamics have been approximated. Utilizing Lyapunov stability theorem, it has been proved that all closed-loop signals are semi-globally practically fixed-time stable (SPFTS), state constraints are satisfied and the consensus tracking is achieved. Finally, the effectiveness of the designed control scheme has been shown via two simulation studies.

Adaptive Neural Partial State Tracking Control for Full-State-Constrained Uncertain Singularly Perturbed Nonlinear Systems and Its Applications to Electric Circuit

This paper is concerned with the adaptive neural network (NN) partial tracking control problem for a class of completely unknown multi-input multi-output (MIMO) singularly perturbed nonlinear systems possessing time-varying asymmetric state constraints. To satisfy the constraints, we utilize the state-depended transformation technique to convert the original state-constrained system to an equivalent unconstrained one, then the state constraint problem can be solved by ensuring its stability. Partial state tracking can be achieved without the violation of state constraints. The adaptive tracking controllers are designed by using singular perturbation theory and the adaptive control method, in which NNs are used to approximate unknown nonlinear functions. The ill-conditioned numerical problems lurking in the controller design process are averted and the closed-loop system stability can be guaranteed by introducing an appropriate Lyapunov function with singular perturbation parameter. Finally, a practical example is given to demonstrate the effectiveness of our proposed adaptive NN tracking control scheme.

Disturbance-observer-based tracking controller for a flexible-joint robotic manipulator with full-state constraints

In this study, a nonlinear tracking controller is designed for a flexible-joint (FJ) robotic manipulator system subject to full-state constraints and external disturbances. First, a nonlinear disturbance observer (NDO) is introduced to guarantee that the respective disturbance can be estimated asymptotically. Then, state-dependent function (SDF) transformations are used to convert the original state-constrained system into a new unconstrained system. Subsequently, a NDO-based tracking controller is proposed to enable the output tracking error to converge to a predefined compact set that can be adjusted to be arbitrarily small. The boundedness of all the closed-loop system signals is rigorously proved. Finally, simulation results are presented that show the effectiveness of the proposed control algorithm.

Adaptive neural smooth finite-time controller for large-scale stochastic nonlinear state-constrained systems with feasibility removal and time delays

This paper studies the adaptive smooth finite-time neural controller design for large-scale stochastic nonlinear systems subject to command filter and time-delay interactions. The main features are (1) the communication interconnection has the output delay between subsystems; (2) the smooth finite time strategy is given for stochastic disturbances; (3) all states are confined in the constrained space without feasibility conditions. First, the state time-delay interactions are tackled by the appropriate establishment of Lyapunov–Krasovskii functionals. Second, for full-state constraints, state-related mappings are employed to remove the feasibility. Then, by constructing the switching virtual control signals, the smooth fast finite-time feature is attached to the controller. Finally, based on finite-time stability theory in probability, command filter, and backstepping technique, the smooth fast finite-time neural controller is constructed to make that all system states converge in finite time, all closed-loop variables are bounded in probability, and full-state constraints are satisfied. Simulation results are provided to test the correctness of the method.

Command-filter-based adaptive neural tracking control for a class of nonlinear MIMO state-constrained systems with input delay and saturation

This paper investigates the problem of adaptive tracking control for a class of nonlinear multi-input and multi-output (MIMO) state-constrained systems with input delay and saturation. During the process of the control scheme, neural network is employed to approximate the unknown nonlinear uncertainties and the appropriate barrier Lyapunov function is introduced to prevent violation of the constraint. In addition, for the issue of input saturation with time delay, a smooth non-affine approximate function and a novel auxiliary system are utilized, respectively. Moreover, adaptive neural tracking control is developed by combining the command filtering backstepping approach, which effectively avoids the explosion of differentiation and reduces the computation burden. The introduced filtering error compensating system brings a significant improvement for the system tracking performance. Finally, the simulation result is presented to verify the feasibility of the proposed strategy.

Cone-Copositive Lyapunov Functions for Complementarity Systems: Converse Result and Polynomial Approximation

This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation which introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing stability of such systems is addressed by constructing cone-copositive Lyapunov functions. As a first analytical result, we show that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials. This later class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms for precisely that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. Some examples are given to illustrate our approach.

Prescribed performance tracking control for a class of nonlinear system considering input and state constraints

This article develops a new anti-saturation tracking approach for effectively controlling a type of state-constrained systems under actuator failure. To construct a feedback control loop possessed the predetermined indexes, an auxiliary variable incorporated with a performance guider is first introduced into the design process. Then, a robust fault tolerant control law with the variable-gains is devised to guarantee that the tracking errors can be suppressed in the specified range after a predetermined time. In order to dispose of the input constraint problem, an anti-saturation algorithm is designed without compromising the prescribed capability indexes in control process, it is shown that the proposed feedback control loop can efficiently fulfill the fast and accurate requirement of the constraint tasks. Finally, computer simulation related with robot manipulator is taken to evaluate the validity of designed method.

Fuzzy Observer Constraint Based on Adaptive Control for Uncertain Nonlinear MIMO Systems With Time-Varying State Constraints.

This article presents an adaptive output feedback approach of nonlinear multi-input-multi-output (MIMO) systems with time-varying state constraints and unmeasured states. An adaptive approximator is designed to approximate the unknown nonlinear functions existing in the state-constrained systems with immeasurable states. To deal with the tracking problem of such systems, a state observer with time-varying barrier Lyapunov functions (BLFs) is introduced in the controller design procedure. The backstepping design with time-varying BLFs is utilized to guarantee that all system states remain within the time-varying-constrained interval. The constant constraint is only the special case of the time-varying constraint which is more general in the real systems. The proposed control approach guarantees that all signals in the closed-loop systems are bounded and the tracking errors converge to a bounded compact set, and time-varying full-state constraints are never violated. A simulation example is given to confirm the feasibility of the presented control approach in this article.

Adaptive Practical Fast Finite-Time Consensus Protocols for High-Order Nonlinear Multi-Agent Systems With Full State Constraints

This paper mainly addresses the issue of fast finite-time consensus (FTC) algorithms for a group of high-order uncertain nonlinear non-strict-feedback Multi-agent Systems (MASs) with time-varying asymmetric full state constraints. In order to guarantee the state constraints are satisfied, an appropriate asymmetric nonlinear mapping (NM) is employed to transform the original full state constrained system into a corresponding unconstrained one. Combining neural networks technology, graph theory, fast finite-time control theory and backstepping recursive design scheme, the uncertainties and non-strict-feedback form are considered to propose a distributed adaptive FTC protocol which can guarantee tracking error reaching a region in finite time. Finally, an example is given to show that the designed control law can perform effective control.