Control For Stochastic Nonlinear Systems Research Articles

Temporal Logic Control of Nonlinear Stochastic Systems Using a Piecewise-Affine Abstraction

Automatically synthesizing controllers for continuous-state nonlinear stochastic systems, while giving guarantees on the probability of satisfying (infinite-horizon) temporal logic specifications crucially depends on abstractions with a quantified accuracy. For this similarity quantification, approximate stochastic simulation relations are often used. To handle the nonlinearity of the system effectively, we use finite-state abstractions based on piecewise-affine approximations together with tailored simulation relations that leverage the local affine structure. In the end, we synthesize a robust controller for a nonlinear stochastic Van der Pol oscillator.

Adaptive Event-Triggered Control of Stochastic Nonlinear Systems With Unknown Dead Zone

We consider the tracking control problem for a class of stochastic nonlinear systems in strict-feedback structure via output feedback signal in this article. The controlled plant is assumed subject to unknown dead-zone input. By utilizing the fact that the unknown dead-zone input function can be modeled as a time-varying nonlinear function and a bounded disturbance, and selecting appropriate design parameters, we show that the effect of unknown dead zone can be compensated. We design a fuzzy state observer via fuzzy logic modeling technique to estimate the unknown system states. Then, we prove, via Lyapunov stability analysis, that the controlled plant is bounded in probability. In addition, all the signals in the closed-loop system are guaranteed to be globally bounded in probability. The tracking errors are also ensured to converge to a small neighborhood of the origin. Finally, to show the effectiveness of the proposed control strategy, a simulation example of one-link manipulator is presented in the simulation section.

Finite-time mean-nonovershooting control for stochastic nonlinear systems

The problem of finite-time mean-nonovershooting control is investigated for a class of stochastic nonlinear systems in this paper. For general stochastic strict-feedback nonlinear systems, a new controller is designed, which uses the framework of the backstepping technique, to assure that the mean of overshoot can be adjusted arbitrarily small (by changing controller gains) while all the states of the closed-loop system in finite-time to get bounded in probability. In contrast with corresponding deterministic systems, it has better performance that control gains did not rely on the initial conditions of system states and derivatives of the reference trajectory. Finally, a simulation example is showing the appealing performance of the proposed scheme.

Fixed-time adaptive neural self-triggered decentralized control for stochastic nonlinear systems with strong interconnections

This article develops a fixed-time adaptive self-triggered decentralized control strategy based on the neural network for a class of stochastic nonlinear systems with strong interconnections. With the help of the special property of the Gaussian function, the strong interconnected functions in stochastic systems can be handled successfully without matching conditions assumptions, and the processing conditions of strong interconnected functions are relaxed. Moreover, a self-triggered mechanism is designed to decrease the waste of the system communication resources. On this basis, a fixed-time adaptive self-triggered decentralized control scheme is constructed by utilizing the fixed-time stability theory such that the stochastic nonlinear system with strong interconnections is fixed-time stable in probability. Finally, the feasibility of the developed control strategy can be ensured by the simulation result.

Mixed H2/H∞ Control for Nonlinear Stochastic Systems With Cooperative and Non-Cooperative Differential Game

This technical note discusses the design process of the cooperative differential game (CDG) with the mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{2}/H_{\infty }$ </tex-math></inline-formula> control to obtain Pareto solutions for nonlinear stochastic systems by adaptive dynamic programming (ADP). In order to obtain Pareto solutions, a non-cooperative differential game (NCDG) for nonlinear stochastic systems is also studied. It is worth mentioning that the mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{2}/H_{\infty }$ </tex-math></inline-formula> control can achieve the expected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{2}$ </tex-math></inline-formula> performance under the condition of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> continuous attenuation. Firstly, the two-player nonzero-sum differential game for stochastic systems is given to obtain Nash equilibrium. Then, on this basis, a joint cost function is established, which is derived from Pareto strategies and the corresponding Hamiltonian function. Furthermore, two critic neural networks (NNs) are used to realize Nash equilibrium and Pareto equilibrium, respectively. Finally, a detailed algorithm and a numerical example are given.

Observer-Based Adaptive Optimized Control for Stochastic Nonlinear Systems With Input and State Constraints.

In this work, an adaptive neural network (NN) optimized output-feedback control problem is studied for a class of stochastic nonlinear systems with unknown nonlinear dynamics, input saturation, and state constraints. A nonlinear state observer is designed to estimate the unmeasured states, and the NNs are used to approximate the unknown nonlinear functions. Under the framework of the backstepping technique, the virtual and actual optimal controllers are developed by employing the actor-critic architecture. Meanwhile, the tan-type Barrier optimal performance index functions are developed to prevent the nonlinear systems from the state constraints, and all the states are confined within the preselected compact sets all the time. It is worth mentioning that the proposed optimized control is clearly simple since the reinforcement learning (RL) algorithm is derived based on the negative gradient of a simple positive function. Furthermore, the proposed optimal control strategy ensures that all the signals in the closed-loop system are bounded. Finally, a practical simulation example is carried out to further illustrate the effectiveness of the proposed optimal control method.

Optimal control of continuous stochastic systems with functional inequality constraints

We investigate the problem of optimal control of nonlinear stochastic systems whose mathematical model is described by an ordinary Ito stochastic differential equation with functional inequality constraints on the trajectory. Analogues of the stochastic Pontryagin maximum principle, the linearized maximum condition are established.

Dynamic Event-Based Adaptive Finite-Time Tracking Control for Nonlinear Stochastic Systems Under State Constraints

This article focuses on the problem of adaptive finite-time tracking control for nonlinear stochastic systems under asymmetric constraints based on dynamic event-triggering control. Different from the existing works, a novel adaptive tracking control algorithm is proposed with asymmetric time-varying constraints and dynamic event-triggering mechanism. First, to constrain the state variable within given time-varying boundaries, a novel predefined-time performance function is constructed. Second, a novel barrier function related to state variable is constructed, by means of which the state variable is directly constrained within the asymmetric time-varying boundaries without the virtual controller. In addition, by establishing a novel dynamic function, we propose a dynamic event-triggering mechanism, and then design controller accordingly, which can reduce computation burdens and save the network resources. By the aid of the Lyapunov stability theory, it is proved that the system tracking error converges to an adjustable bounded set in probability in a finite time and all state variables are successfully constrained into the asymmetric time-varying boundaries. Finally, the effectiveness of the proposed control algorithm is verified by a simulation example.

Adaptive Asymptotic Tracking Control for Stochastic Nonlinear Systems with Unknown Backlash-Like Hysteresis

Adaptive Asymptotic Tracking Control for Stochastic Nonlinear Systems with Unknown Backlash-Like Hysteresis

Event-Triggered Impulsive Control for Nonlinear Stochastic Systems.

We study the stabilization problem for nonlinear stochastic systems via an event-triggered impulsive control (ETIC) scheme, where the impulsive control time sequence is generated by the event-triggered mechanism (ETM). Both continuous ETM and periodic ETM are developed by continuous measuring and periodic sampling, respectively. The continuous ETM with time regularization is proposed to exclude the Zeno behavior. The upper bound of the sampling period is given for the periodic ETM. By means of the continuous ETM and periodic ETM, sufficient conditions are given to guarantee the p th moment uniform stability and the p th moment exponential stability of related systems. Moreover, LMI-based conditions of exponential stability in the mean square are established for linear stochastic systems under ETIC. Finally, two examples are presented to illustrate the proposed ETIC schemes, in which an example of the consensus of linear stochastic multiagent systems is considered.

Event-Based Adaptive Neural Asymptotic Tracking Control for Networked Nonlinear Stochastic Systems

This paper investigates the adaptive asymptotic tracking control for networked nonlinear stochastic systems. Different from having the necessity of prior knowledge of the unknown control coefficients in the conventional adaptive control of nonlinear stochastic systems, in this study, the limitation of control coefficients in the stability analysis is relaxed by constructing a new Lyapunov function that contains the lower bounds of the control gain function. By constructing a smooth function with a positive time-varying integral function and utilizing the boundary estimation method, asymptotic tracking control can be guaranteed. At the same time, for nonlinear stochastic systems with unknown control coefficients, a neural adaptive event-triggered strategy that greatly saves communication resources while ensuring system performance is proposed. Finally, simulation results show that the proposed control scheme can guarantee the realization of the control objectives.

Stability Verification in Stochastic Control Systems via Neural Network Supermartingales

We consider the problem of formally verifying almost-sure (a.s.) asymptotic stability in discrete-time nonlinear stochastic control systems. While verifying stability in deterministic control systems is extensively studied in the literature, verifying stability in stochastic control systems is an open problem. The few existing works on this topic either consider only specialized forms of stochasticity or make restrictive assumptions on the system, rendering them inapplicable to learning algorithms with neural network policies. In this work, we present an approach for general nonlinear stochastic control problems with two novel aspects: (a) instead of classical stochastic extensions of Lyapunov functions, we use ranking supermartingales (RSMs) to certify a.s. asymptotic stability, and (b) we present a method for learning neural network RSMs. We prove that our approach guarantees a.s. asymptotic stability of the system and provides the first method to obtain bounds on the stabilization time, which stochastic Lyapunov functions do not. Finally, we validate our approach experimentally on a set of nonlinear stochastic reinforcement learning environments with neural network policies.

Observer-Based Adaptive Fuzzy Finite-Time Fault-Tolerant Control for Stochastic Nonlinear Systems with State Constraint

Observer-Based Adaptive Fuzzy Finite-Time Fault-Tolerant Control for Stochastic Nonlinear Systems with State Constraint

Adaptive tracking control for stochastic nonlinear systems with time-varying delays using multi-dimensional Taylor network

For stochastic nonlinear systems (SNSs) perturbed by compound uncertainties, the conventional model-based control approaches assume that the evolution behavior of uncertain variables is known. Unfortunately, such approaches are often conservative for most practical scenarios with the slow convergence speed and unsatisfactory anti-interference performance. For this sake, an adaptive control scheme based on deep deterministic policy gradient (DDPG) and multi-dimensional Taylor network (MTN) is proposed here to address the tracking problem for a category of SNSs subject to fast time-varying uncertainties, stochastic disturbance and unknown time-varying delays. The effect of time delay is embedded in the reproducing kernel Hilbert space through the error coordinate transformation. In the framework of DDPG, the MTN-based surrogate is utilized to construct the online network and target network via the temporal-difference method, which promises more desirable real-time performance due to its concise structure than conventional NN-based surrogates. In order to enhance the robustness of the system under fast time-varying uncertainties, a novel persistent excitation (PE) mechanism is designed to ensure that the control policy is appropriately rewarded or punished. Based on the PE condition, weights of MTNs converge exponentially and animate the system to evolve towards the target persistently. The tracking error and closed-loop state signals are proved theoretically to be uniformly ultimately bounded (UUB) via Lyapunov–Krasovskii functional. A numerical simulation from the process industry verifies the effectiveness of the proposed method.

Prescribed tracking error fixed-time control of stochastic nonlinear systems

The issue of prescribed tracking error fixed-time control of stochastic nonlinear systems is investigated in this article. Different from the conventional quartic Lyapunov function (LF) on tracking error, a novel LF based on two important tuning functions is constructed. By means of the fixed-time command filtered dynamic surface control (DSC) technique with the newly error compensating signals (ECSs), the designed controller can ensure that the tracking error and state tracking errors can be predicted in advance without any state transformation. Meanwhile, the problem of “curse of dimensionality” is avoided, and the filtering errors are effectively compensated for. Furthermore, an improved event-triggering mechanism (ETM) is designed to save network resources. A simulation result verifies the scheme developed.

Neurodynamic Programming and Tracking Control for Nonlinear Stochastic Systems by PI Algorithm

In this paper, for nonlinear stochastic systems, the optimal tracking control problem (OTCP) is solved by adaptive dynamic programming (ADP) techniques. Firstly, the complex OTCP is transformed into a stable control optimization problem by reconstructing a new stochastic augmented system. Then, simplifying the actor-critic architecture and reducing the computational load, critic neural networks (NNs) is used in iterative learning. And by using Lyapunov method, the ultimate uniform boundedness (UUB) of the tracking system is proved. To be precise, no literature has been published on the OTCP for nonlinear Itô type stochastic systems via the ADP method. This work is the first attempt in this field. Finally, in simulation, the method is applied to sinusoidal waveform and periodic rectangular step signal, and even unbounded exponential waveform.

Chaos synchronization control for stochastic nonlinear systems of interior PMSMs based on fixed-time stability theorem

This paper proposes an enhanced chaos synchronization control scheme of stochastic nonlinear systems for the permanent magnet synchronous motors(PMSMs) applications for dynamic operating mechanism under different working environment. Firstly, a novel fixed-time stochastic stability theorem and convergence for the stochastic nonlinear system is valid through formulating a novel Lyapunov function. Different from what has been achieved on the fixed-time stability, it’s not the usual range but it’s more precise for settling time which improving control performance effectually. Secondly, the chaos synchronization issue of driven-response PMSMs with stochastic noise is firstly investigated based on the proposed Lyapunov theorem synchronizing the complex dynamic systems considering practical significance. Finally, the fixed-time synchronization control scheme is developed via applying the adaptive algorithms, furthermore, through constructing a reasonable synchronization dynamic error system achieving stability in probability of fixed time. A simulation analysis is demonstrated to verify the availability of the proposed control strategy.

Dynamic Event-Triggered Impulsive Control for Stochastic Nonlinear Systems With Extension in Complex Networks

This study proposes a novel dynamic event-triggered impulsive control (ETIC) scheme to study the exponential stabilization of general stochastic nonlinear systems, where the impulsive sequence is determined by a dynamic event-triggered mechanism. The dynamic ETIC can effectively reduce controller updates and significantly save energy under the same decay rate compared to traditional static event-triggered impulsive generators. Additionally, there is a guaranteed positive minimum inter-event time for each sample path solution of systems. Furthermore, the proposed dynamic ETIC scheme is employed to stabilize stochastic complex networks based on the graph theory and the Lyapunov method. Finally, we provide two illustrative examples to verify the effectiveness and correctness of the proposed dynamic ETIC scheme.

Event-triggered predictive control of nonlinear stochastic systems with output delay

In this paper, we focus on the stabilization of nonlinear stochastic systems with output delay. Based on the predictive scheme and event-triggering mechanism (ETM), a novel event-triggered predictive control is proposed to exponentially stabilize such stochastic systems. Moreover, this novel control law allows non-uniform sampled measurements and large delays involved in the output. Simultaneously, the control input error induced by ETM is considered and a positive lower bound on the inter-event times is obtained. Finally, an illustrative example is given to demonstrate the obtained results.

Event-triggered-based fixed-time adaptive neural fault-tolerant control for stochastic nonlinear systems under actuator and sensor faults

Event-triggered-based fixed-time adaptive neural fault-tolerant control for stochastic nonlinear systems under actuator and sensor faults