Class Of Stochastic Nonlinear Systems Research Articles

Adaptive finite‐time control for a class of stochastic nonlinear systems with input saturation constraints: A new approach based on multidimensional Taylor network

AbstractThis article proposes a novel multidimensional Taylor network (MTN)‐based adaptive finite‐time control approach for stochastic nonlinear systems with input saturation constraints. First, by introducing the hyperbolic tangent function, a piecewise smooth function is proposed to approximate the input saturation constraints, which can solve the problem caused by input saturation. Then, with the aid of MTN approximation theorem, a novel adaptive finite‐time control approach is proposed, which can guarantee that all signals of the closed‐loop system are semi‐globally finite‐time stability in probability (SGFTSP), the system output follows the given reference signal, meanwhile, the tracking error converges to a small neighborhood of the origin in finite‐time. Compared with the existing results, the proposed approach has the characteristics of simple structure and low calculation, which can achieve the satisfactory control performance with a reduced computational burden. In the end, the effectiveness of the proposed approach is validated by Lyapunov stability theory and two simulation experiments.

Inverse optimally adaptive neural output‐feedback control of stochastic nonlinear systems

SummaryIn this article, for a class of stochastic nonlinear systems with non‐strict feedback, a neural adaptive inverse optimal output feedback control design scheme is presented. First, according to the existing inverse optimal criterion, an auxiliary system is established. On this basis, a novel observer is built to evaluate the unpredictable states. Second, in the control process, neural networks (NNs) are applied to estimate the unknown functions. Based on NNs and the backstepping technology, an adaptive neural inverse optimal output feedback controller is established. It is indicated that the proposed scheme could ensure the semi‐globally uniformly ultimately bounded of the closed‐loop system and also achieve the objective of inverse optimality. Eventually, an example is applied to testify the feasibility of this scheme.

Fixed-Time Stabilization of a Class of Stochastic Nonlinear Systems

This paper investigates an improved fixed-time stability theory together with a state feedback controller for a class of nonlinear stochastic systems. First, a delicate transformation is performed, and next, a Gamma function is utilized to directly derive the value of the integral function, which ultimately yields a fixed-time stabilization theorem with a higher precision upper bound for the settling time. Unlike the existing estimation process of amplifying twice, we only performed one amplification, which weakens the effect of amplification. Then, a state feedback controller is constructed for stochastic systems by the method of adding a power integrator. Utilizing the proposed stochastic fixed-time stability theory, simulations show that the intended controller ensures that the trivial solution of the suggested system is fixed-time stable in probability. The results of the simulation demonstrate that the suggested control scheme is meaningful.

Output tracking for nonlinear stochastic systems driven by bounded and colored noises

In this paper, we consider practical tracking problem for a class of nonlinear stochastic systems driven by bounded and colored noises by means of active disturbance rejection control approach. The nonlinear extended state observer with homogeneous weighted functions is constructed to estimate in real time the unmeasurable state and the total stochastic disturbance that affects system performance. An observer-based output feedback is designed to guarantee the almost surely practical convergence of the tracking errors. The existence of the global weak solution of the closed-loop system is proved and the tracking convergence accuracy is shown to be higher than that of linear one under the same tuning parameters. Some numerical simulations are carried out to validate the effectiveness of the proposed controller.

Output Feedback Control for Stochastic Nonlinear Systems With Nondifferentiable Measurement Function and Input Saturation.

In this article, the problem of output feedback control for a class of stochastic nonlinear systems in the presence of nondifferentiable measurement function and input saturation is studied. A novel power-auxiliary system is introduced to handle the adverse effects of input saturation. What is more, the common growth assumptions of nonlinear terms can be eliminated by a key lemma. Then, an output feedback controller is constructed to ensure that all the signals in the closed-loop system are globally bounded almost surely. Finally, a simulation shows that the control strategy is effective.

Adaptive tracking control for a class of stochastic nonlinear systems with input delay using multidimensional Taylor network

AbstractIn this paper, an adaptive tracking control strategy based on multidimensional Taylor network (MTN) is designed for a class of stochastic nonlinear systems with input delay, in which MTN is mainly introduced to approximate uncertain nonlinear functions. Moreover, the effect of input delay is compensated by exploiting an appropriate auxiliary system with the same order as the considered system. By utilizing the backstepping technique and classical adaptive approach, an MTN‐based adaptive tracking control method is proposed, which guarantees that all the signals in the closed‐loop system remain bounded in probability. Furthermore, unlike most existing approaches, the tracking error can be bounded by an explicit expression of design parameters in the mean quartic sense. Finally, two examples are given to verify the validity of the presented scheme.

Optimal regulator for a class of nonlinear stochastic systems with random coefficients

We consider an optimal regulator problem for a class of nonlinear stochastic systems with a square-root nonlinearity and random coefficients, and using the quadratic-linear criterion. This represents a certain nonlinear generalisation of the stochastic linear-quadratic control problem with random coefficients. The solution if found in an explicit closed-form as an affine state-feedback control in terms of a Riccati and linear backward stochastic differential equations. As an application, we give the solution to an optimal investment problem in a market with random coefficients.

Improving the Feasibility of Moment-Based Safety Analysis for Stochastic Dynamics

Given a dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through its exit time moments. Using appropriate semidefinite positive matrix constraints, an SDP moment-based approach can be used to compute moments of the exit time. However, the approach is impeded when analyzing higher dimensional physical systems as the dynamics are limited to polynomials. Computational scalability is also poor as the dimensionality of the state grows, largely due to the combinatorial growth of the optimization program. We propose methods to make feasible the safety analysis of higher dimensional physical systems. The restriction to polynomial dynamics is lifted by using state augmentation which allows one to generate the optimization for a broader class of nonlinear stochastic systems. We then reformulate the constraints to mitigate the computational limitations associated with an increase in state dimensionality. We employ our methods on two example processes to characterize their safety via exit times and show the ability to handle multi-dimensional systems that were previously unsupported by the existing SDP method of moments.

Event-triggered adaptive multi-dimensional Taylor network tracking control for stochastic nonlinear systems

For a class of stochastic nonlinear systems, this paper proposes a novel event-triggered adaptive control scheme by means of multi-dimensional Taylor network (MTN) approach for the first time, which has the advantages of alleviating computational burden and reducing communication frequency. In addition, the event-triggered control (ETC) strategy can effectively save network resource by alleviating the computational burden and reducing the communication frequency. Therefore, the proposed control approach can not only reduce communication frequency but also further alleviate computational burden, thereby saving network resource to a greater extent. The proposed control scheme ensures that all signals of the system are semi-global uniformly ultimately bounded (SGUUB) in probability and the tracking error can be made arbitrarily small by choosing appropriate design parameters. Meanwhile, Zeno behavior can be avoided. Finally, two simulation results are given to illustrate the effectiveness of the proposed scheme.

Adaptive tracking control for a class of stochastic nonlinear systems with full-state constraints and dead-zone

This paper seeks to address the problem of state-feedback tracking control for a class of stochastic systems whose states are constrained and input is perturbed by dead-zone. A tan-type barrier Lyapunov function (BLF) and a finite-time adaptive law are proposed such that all states of the system are constrained in the defined bounded compact sets. Based on stochastic Lyapunov theorem, an adaptive state-feedback tracking controller is designed to ensure that the considered stochastic system is semiglobally finite-time stable in probability (SGFSP) and the tracking error signal is bounded. Different from the existing control strategies for stochastic systems, the form of BLF can be used for asymmetric constraints without changing the controller, so the analysis of symmetric or asymmetric full-state constraints is unified. Finally, illustrative simulations are provided to verify the effectiveness of the proposed control strategy.

BLF-based neural dynamic surface control for stochastic nonlinear systems with time delays and full-state constraints

This paper investigates the adaptive neural tracking control problem for a class of stochastic nonlinear systems with time delays and full-state constraints in a unified framework for the first time. The time-delay terms of the controlled systems are compensated by novel Lyapunov–Krasovskii functionals. The asymmetric barrier Lyapunov function (BLF) is adopted to guarantee that the full states are always restricted within prescribed constraints. RBF neural networks are utilised to approximate the lumped unknown functions in the design process. Furthermore, the dynamic surface control (DSC) technique is employed to simplify the process of control design significantly. Stability analysis shows all closed-loop signals are SGUUB, and full-state constraints are not violated. Finally, simulation results confirm the effectiveness of the proposed control scheme.

Sampled-Data Stabilization for Stochastic Systems Subject to Input Saturation and Its Application to Synchronization of Two Chua’s Circuits

In this paper, we focus on the stabilization problem of a class of stochastic nonlinear systems under periodic sampled-data state feedback subject to input saturation. An auxiliary variable is introduced to handle the saturation nonlinearity as well as to characterize the estimate of the region of attraction of the origin(RAO). After discretizing the continuous-time systems and obtaining the iterative relations of sampling instants, sufficient stabilization criteria are established in the form of parametric conditions. Both mean square exponential stability and almost sure stability are considered. Based on our results, the relations among the estimate of RAO and controller gain, sampling period can be analysed clearly, which is demonstrated in the example. An application for synchronization of two Chua’s Circuits is presented to illustrate the effectiveness of our results.

Command-Filtered Neuroadaptive Output-Feedback Control for Stochastic Nonlinear Systems With Input Constraint.

In this article, an adaptive neural-network (NN) command-filtered output-feedback control strategy is proposed for a class of stochastic nonlinear systems (SNSs) with the actuator constraint. The problem of "explosion of complexity" existing in the conventional backstepping design procedure for SNSs is successfully resolved based on the command filter technique, and the error compensation mechanism is introduced to remove effectively the influence of filtered error. By using the NNs to identify the unknown nonlinear functions, a neural-network-based state observer is designed to estimate the unmeasurable states of the SNSs. Based on the quartic Lyapunov function, the stability of stochastic closed-loop systems is analyzed. It is proved that all signals of the closed-loop systems are bounded in probability, and the tracking error approaches a small neighborhood of the origin in probability. Finally, the effectiveness of the developed control algorithm in this article is verified by a comparison example.

Neural dynamic surface control for stochastic nonlinear systems with unknown control directions and unmodelled dynamics

AbstractThe control problem for a class of stochastic nonlinear systems with both unknown control directions and unmodelled dynamics is investigated here for the first time. The technique of dynamics signal is adopted to cope with the unmodelled dynamics in the considered system. The unknown control directions problem are addressed by Nussbaum function. RBF neural networks are employed to approximate the lumped unknown functions, and regardless of the number of neural networks used and the order of the system, only one adaptive parameter requires to be adjusted. Dynamic surface control(DSC) is utilized to cope with the complexity explosion of the backstepping design. Hence, a novel neural control scheme is proposed by means of dynamics signal method, DSC technique and Nussbaum function. Stability analysis proves all closed‐loop signals are SGUUB by choosing the parameters appropriately, and the simulation results demonstrate the correctness and effectiveness of the proposed scheme.

Neural-network-based output feedback adaptive resilient control for stochastic nonlinear systems subject to sensor and actuator attacks

This paper addresses the output feedback adaptive resilient control problem for a class of stochastic nonlinear systems under sensor and actuator attacks. By applying a linear state transformation, the original system is transformed into a new one for which output feedback control becomes possible. The Nussbaum function is exploited to overcome the unknown control direction problem generated by uncertain control coefficients and adversary attacks. Based on the available compromised output signal and the states reconstructed by a new extended state observer, an output feedback adaptive resilient control scheme is proposed for such systems by combining the neural-networks-based approximation approach, backstepping technique and some special methods. The proposed controller can guarantee that all signals in the closed-loop system are semiglobally bounded in probability and the stabilisation error variables remain in an adjustable vicinity of origin. Finally, the effectiveness and applicability of the proposed control methodology is verified by two illustrative examples.

Rapid stabilization of stochastic quantum systems in a unified framework

Rapid stabilization of general stochastic quantum systems is investigated based on the rapid stability of stochastic differential equations. We introduce a Lyapunov–LaSalle-like theorem for a class of nonlinear stochastic systems first, based on which a unified framework of rapidly stabilizing stochastic quantum systems is proposed. According to the proposed unified framework, we design the switching state feedback controls to achieve the rapid stabilization of single-qubit systems, two-qubit systems, and N-qubit systems. From the unified framework, the state space is divided into two state subspaces, and the target state is located in one state subspace, while the other system equilibria are located in the other state subspace. Under the designed state feedback controls, the system state can only transit through the boundary between the two state subspaces no more than two times, and the target state is globally asymptotically stable in probability. In particular, the system state can converge exponentially in (all or part of) the state subspace where the target state is located. Moreover, the effectiveness and rapidity of the designed state feedback controls are shown in numerical simulations by stabilizing GHZ states for a three-qubit system.

Modified Kalman Filtering for Stochastic Nonlinear Systems Under Non-Gaussian–Lévy Noise and Cyber Attacks

This article studies the filtering problem for a class of stochastic nonlinear system under non-Gaussian–Lévy noise and cyber attacks, where the denial-of-service (DoS) attacks and the false data-injection (FDI) attacks are both considered. Since the covariance of the Lévy noise is unknown and infinite, the standard Kalman filter fails to estimate system states. By exploiting saturation functions, a modified Kalman filter is proposed, where the extremely large values of the measurement outputs caused by the Lévy noises can be clipped. In the presence of Lévy noise and cyber attacks, an upper bound for the error covariance is guaranteed and can be minimized via designing the filter parameter. Besides, a sufficient condition is provided to guarantee the boundedness of the upper bound, and the convergence analysis of the filtering error is presented. Finally, the simulation results are given to verify the algorithm.

Self‐triggered adaptive prescribed‐time tracking control for stochastic nonlinear systems

SummaryThis paper proposes a self‐triggered (ST) adaptive prescribed‐time tracking control method for a class of stochastic nonlinear systems. Different from the existing results, an improved ST mechanism is proposed by adding a judgment condition to reduce the negative effect of excessive design interval on system performance. Based on the one‐to‐one mapping and backstepping technique, an adaptive prescribed‐time tracking control method is proposed, which can make the error converge to the predefined precision set within the predetermined time. Simultaneously, applying the Lyapunov stability method, the boundedness of all signals in the closed‐loop system can be ensured. Finally, a detailed simulation example is provided to show the effectiveness of the proposed control strategy.

A novel network-based controller design for a class of stochastic nonlinear systems with multiple faults and full state constraints

In this paper, the control issue of adaptive fault-tolerant is studied for a class of stochastic nonlinear systems with multiple faults and full state constraints, with multiple faults including the actuator faults and the external system fault. The problem with full state constraints are solved by constructing a logarithmic barrier Lyapunov functions (BLFs). By integrating multi-dimensional Taylor network (MTN) technology into the backstepping process, a new adaptive MTN-based fault-tolerant controller is designed. On the basis of considering multiple faults, the proposed control strategy can ensure that all signals in the closed-loop system are semi-global ultimately uniformly bounded (SGUUB) in probability, and all states of the system are constrained within the given boundary. Finally, three simulation examples are given to illustrate the effectiveness and practicability of the proposed control strategy.

Memoryless output feedback control for a class of stochastic nonlinear systems with large delays in the state and input

Memoryless output feedback control for a class of stochastic nonlinear systems with large delays in the state and input