Class Of Continuous-time Nonlinear Systems Research Articles

Optimal Adaptive Control of Uncertain Nonlinear Continuous-Time Systems With Input and State Delays.

In this article, an actor-critic neural network (NN)-based online optimal adaptive regulation of a class of nonlinear continuous-time systems with known state and input delays and uncertain system dynamics is introduced. The temporal difference error (TDE), which is dependent upon state and input delays, is derived using actual and estimated value function and via integral reinforcement learning. The NN weights of the critic are tuned at every sampling instant as a function of the instantaneous integral TDE. A novel identifier, which is introduced to estimate the control coefficient matrices, is utilized to obtain the estimated control policy. The boundedness of the state vector, critic NN weights, identification error, and NN identifier weights are shown through the Lyapunov analysis. Simulation results are provided to illustrate the effectiveness of the proposed approach.

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.

Decentralized Neurocontroller Design With Critic Learning for Nonlinear-Interconnected Systems.

We consider the decentralized control problem of a class of continuous-time nonlinear systems with mismatched interconnections. Initially, with the discounted cost functions being introduced to auxiliary subsystems, we have the decentralized control problem converted into a set of optimal control problems. To derive solutions to these optimal control problems, we first present the related Hamilton-Jacobi-Bellman equations (HJBEs). Then, we develop a novel critic learning method to solve these HJBEs. To implement the newly developed critic learning approach, we only use critic neural networks (NNs) and tune their weight vectors via the combination of a modified gradient descent method and concurrent learning. By using the present critic learning method, we not only remove the restriction of initial admissible control but also relax the persistence-of-excitation condition. After that, we employ Lyapunov's direct method to demonstrate that the critic NNs' weight estimation error and the states of closed-loop auxiliary systems are stable in the sense of uniform ultimate boundedness. Finally, we separately provide a nonlinear-interconnected plant and an unstable interconnected power system to validate the present critic learning approach.

Fault Detection Observer Design for Nonlinear Systems via Fuzzy Lyapunov Functions

This article focuses on the design problem of fault detection (FD) observer for a class of continuous-time nonlinear systems. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H_{-}}$ </tex-math></inline-formula> index is employed to measure the worst cast fault sensitivity. First, the nonlinear systems are represented by the Takagi–Sugeno (T–S) fuzzy dynamic models. Then, a novel fuzzy Lyapunov function (FLF) analyzing approach is proposed to reduce the conservatism in stability analysis and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H_{-}}$ </tex-math></inline-formula> index estimation for the T–S fuzzy model. In particular, employing this new type of the FLF, the stability analysis of the T–S fuzzy systems perfectly overcomes the difficulty of evaluating the upper bound of the time derivative of the membership function. Based on the descriptor system approach, some new sufficient conditions on the existence of fuzzy FD observers are obtained in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are given to show the effectiveness of the proposed method.

Self-Triggered Adaptive NN Tracking Control for a Class of Continuous-Time Nonlinear Systems With Input Constraints

This article develops a self-triggered adaptive neural network (NN) tracking controller for a class of continuous-time nonlinear systems, that is, input constrained and with unknown drift and input dynamics. Since the drift and input dynamics are both unknown, an NN is built within a self-triggered update paradigm to approximate the unknown tracking control. The error derivative used in the weight update algorithm is derived using a robust exact differentiator technique. To address input constraints, an auxiliary compensator is designed for the unimplemented control effort. Through rigorous Lyapunov analyses, we can guarantee that all the tracking and weight errors are uniformly ultimately bounded. Finally, to show the effectiveness of the proposed control performance, simulation results of a two-link robot are provided and analyzed.

Linear Proportional-Integral-Differential-Robustified Continuous-Time Optimal Predictive Control for a Class of Nonlinear Systems

This paper presents a novel robust optimal predictive control approach for a class of nonlinear continuous-time systems perturbed by unknown disturbances. First, a new error state with a linear proportional-integral-differential (PID) structure considering current, accumulative, and derivative tracking errors is defined. Second, prediction of the error state within the predictive periods is expressed by the error state and its high-order derivatives according to the Taylor series expansion. Last, the proposed control law as well as the main result of this paper are derived by minimizing the prediction of the error state. Numerical validation for designing a missile autopilot shows that, due to minimizing the accumulative tracking error included in the PID-structuralized new error state, the proposed approach can generate smaller steady-state tracking errors than two commonly applied continuous-time optimal predictive control approaches whether the disturbances encountered by the missile are constant or time-varying.

Sampled-data estimator for nonlinear systems with uncertainties and arbitrarily fast rate of convergence

We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation.

Optimal tracking control based on reinforcement learning value iteration algorithm for time-delayed nonlinear systems with external disturbances and input constraints

This article investigates the design of an optimal tracking controller for a class of nonlinear continuous-time systems with time-delay, mismatched external disturbances and input constraints. The technique of integral reinforcement learning (IRL) is utilized for determining the control input that optimizes an objective function. To enable the usage of an estimation of the external disturbances in the recursive objective function, a disturbance observer is designed. For the derivation of the optimal control input, a Hamilton-Jacobi-Bellman (HJB) equation is employed and solved using the iterative IRL algorithm. The proposed approach guarantees that in the presence of mismatched disturbances, the output of the time-delayed nonlinear system tracks the desired trajectory with bounded error. A critic neural network is designed for the implementation of the proposed approach. The efficiency of the method is illustrated by a simulation example.

Regional stabilization of nonlinear sampled-data control systems: A quasi-LPV approach

This paper addresses the asymptotic stabilization of a class of continuous-time nonlinear systems under a sampled-data control. The proposed approach is based on a quasi linear parameter varying (quasi-LPV) model for the nonlinear system and the use of a parameter dependent looped-functional to deal with the aperiodic sampling effects. Explicitly taking into account that the model parameters are functions of the state and therefore are bounded only in a given region of the state space, quasi-LMI conditions are proposed to compute a regional stabilizing nonlinear state feedback control law under aperiodic sampling. These conditions are then incorporated in convex optimization problems to compute the control law aiming at the maximization of an estimate of the region of attraction of the origin or the maximization of an upper bound on the intersampling time, with a guaranteed region of stability.

Approximate Optimal Tracking Control of Nondifferentiable Signals for a Class of Continuous-Time Nonlinear Systems.

In this article, for a class of continuous-time nonlinear nonaffine systems with unknown dynamics, a robust approximate optimal tracking controller (RAOTC) is proposed in the framework of adaptive dynamic programming (ADP). The distinguishing contribution of this article is that a new Lyapunov function is constructed, by using which the derivative information of tracking errors is not required in computing its time derivative along with the solution of the closed-loop system. Thus, the proposed method can make the system states follow nondifferentiable reference signals, which removes the common assumption that the reference signals have to be continuous for tracking control of continuous-time nonlinear systems in the literature. The theoretical analysis, simulation, and application results well illustrate the effectiveness and superiority of the proposed method.

Model Identification and Adaptive State Observation for a Class of Nonlinear Systems

In this article, we consider the joint problems of state estimation and model identification for a class of continuous-time nonlinear systems in the output-feedback canonical form. An adaptive observer is proposed that combines an extended high-gain observer and a discrete-time identifier. The extended observer provides the identifier with a dataset permitting the identification of the system model and the identifier adapts the extended observer according to the new estimated model. The design of the identifier is approached as a system identification problem and sufficient conditions are presented that, if satisfied, allow different identification algorithms to be used for the adaptation phase. The cases of recursive least-squares and multiresolution black-box identification via wavelet-based identifiers are specifically addressed. Stability results are provided relating the asymptotic estimation error to the prediction capabilities of the identifier. Robustness with respect to additive disturbances affecting the system equations and measurements is also established in terms of an input-to-state stability property relative to the noiseless estimates.

A new robust LMI-based model predictive control for continuous-time uncertain nonlinear systems

ABSTRACT This paper presents a new robust predictive controller for a special class of continuous-time non-linear systems with uncertainty. These systems have bounded disturbances with unknown upper bound as well as constraints on input states. The controller is designed in the form of an optimization problem of the ‘worst-case’ objective function over an infinite moving horizon. Through this objective function, constraints and uncertainties can be applied explicitly on the controller design, which guarantees the system stability. Next, LMI tool is used to improve the calculation time and complexity. To do this, in order to find the optimum gain for state-feedback, the optimization problem is solved using LMI method in each time step. Finally, to show the efficiency and effectiveness of the proposed algorithm, a surge phenomenon avoidance problem in centrifugal compressors is solved.

Contract-Based Design of Symbolic Controllers for Safety in Distributed Multiperiodic Sampled-Data Systems

This article presents a symbolic control approach to the design of distributed safety controllers for a class of continuous-time nonlinear systems. More precisely, we consider systems made of components where each component is equipped with a sampled-data controller with its own sampling period, resulting globally in a distributed multiperiodic sampled-data system. Moreover, controllers receive partial information on the state of the other components. We propose a component-based approach to controller synthesis, which relies on the use of abstractions and continuous-time assume-guarantee contracts. The abstractions describe the dynamics of the system from the point of view of each component based on the information structure, whereas assume-guarantee contracts specify guarantees that a component must satisfy if assumptions on the other components are met. We show that our approach makes it possible to decompose a global safety control problem into local ones that can be solved independently. We then show how symbolic control techniques can be used to synthesize controllers that enforce the local control objectives. Illustrative applications in building automation and vehicle platooning are shown.

LMI-Based Robust Stabilization of a Class of Input-Constrained Uncertain Nonlinear Systems with Application to a Helicopter Model

This paper is concerned with the robust stabilization of a class of continuous-time nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, via an affine state-feedback control law using the linear matrix inequality (LMI) approach. The nonlinear dynamics is subject to norm-bounded parametric uncertainties and disturbances. In addition, the problem of actuator nonlinearity is addressed by considering the saturation effect of the control law. We demonstrate first that the synthesis problem of the saturated controller is expressed in terms of bilinear matrix inequalities (BMIs). Thanks to the Schur complement lemma and the matrix inversion lemma, we convert these BMIs into LMIs allowing the simultaneous computation of the two gains of the affine controller. Furthermore, we address in this work the estimation problem of the domain of attraction using the invariant set concept. This is solved by computing the largest attractive invariant ellipsoid. Compared with previous works, the research procedure of such ellipsoidal set is achieved in a single step with a reduced number of LMI constraints and then with less conservative conditions. A portfolio of numerical results is presented. The effectiveness and robustness of the proposed saturated controller in the stabilization of the adopted helicopter pitch model toward parametric uncertainties and disturbances are illustrated through simulation results.

Decentralized Event-Triggered Control for a Class of Nonlinear-Interconnected Systems Using Reinforcement Learning.

In this article, we propose a novel decentralized event-triggered control (ETC) scheme for a class of continuous-time nonlinear systems with matched interconnections. The present interconnected systems differ from most of the existing interconnected plants in that their equilibrium points are no longer assumed to be zero. Initially, we establish a theorem to indicate that the decentralized ETC law for the overall system can be represented by an array of optimal ETC laws for nominal subsystems. Then, to obtain these optimal ETC laws, we develop a reinforcement learning (RL)-based method to solve the Hamilton-Jacobi-Bellman equations arising in the discounted-cost optimal ETC problems of the nominal subsystems. Meanwhile, we only use critic networks to implement the RL-based approach and tune the critic network weight vectors by using the gradient descent method and the concurrent learning technique together. With the proposed weight vectors tuning rule, we are able to not only relax the persistence of the excitation condition but also ensure the critic network weight vectors to be uniformly ultimately bounded. Moreover, by utilizing the Lyapunov method, we prove that the obtained decentralized ETC law can force the entire system to be stable in the sense of uniform ultimate boundedness. Finally, we validate the proposed decentralized ETC strategy through simulations of the nonlinear-interconnected systems derived from two inverted pendulums connected via a spring.

Quantized Fuzzy Output Feedback ${\mathcal{H}}_{\infty}$ Control for Nonlinear Systems With Adjustment of Dynamic Parameters

This paper investigates the output feedback $ {\mathcal {H}_{\infty }}$ control problem for a class of continuous-time nonlinear systems with both output and input quantization. The nonlinear plant is represented by a Takagi–Sugeno fuzzy model. The measurement output and the control input signals will be quantized by dynamic quantizers. The attention of this paper is focused on the design of the output feedback controller and quantizers such that the fuzzy closed-loop system is asymptotically stable and also achieves a prescribed $ {\mathcal {H}_{\infty }}$ noise attenuation level with respect to the effect of quantization. In the presence of quantization, two novel $ {\mathcal {H}_{\infty }}$ performance analysis criteria are presented for the considered fuzzy systems based on the descriptor representation approach. Then, sufficient conditions for the existence of the feedback $ {\mathcal {H}_{\infty }}$ controller gain and the quantizers’ dynamic parameters are expressed in terms of linear matrix inequalities. An adjusting scheme on quantizers’ parameters is proposed for the fuzzy system. Simulation examples are provided to show the effectiveness of the proposed design method.

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.

Dynamic output-feedback control for nonlinear continuous-time systems based on parametric uncertain subsystem and interval type-2 fuzzy model

This paper addresses the interval type-2 fuzzy robust dynamic output-feedback control problem for a class of nonlinear continuous-time systems with parametric uncertainties and immeasurable premise variables. First, the parametric uncertainties are assumed to be a subsystem based on the control input matrix and output matrix, and described as a linear fractional. Secondly, the nonlinear continuous-time systems are described by the interval type-2 fuzzy model. Thirdly, the new dynamic output feedback controller is designed based on the interval type-2 fuzzy model and the linear fractional (parametric uncertainties), the sufficient conditions for robust stabilization are given in the form of linear matrix inequalities (LMIs). Compared with previous work, the developed methods not only have abilities to handle the fuzzy system with immeasurable premise variables but also can deal with the parametric uncertainties effectively. The results are further extended to a mobile robot case and a chemical process case. Finally, two simulation examples are performed to show the effectiveness of the propose methods.

Output feedback tracking control of a class of continuous-time nonlinear systems via adaptive dynamic programming approach

In this paper, we propose an output-based tracking control scheme for a class of continuous-time nonlinear systems via the adaptive dynamic programming (ADP) technique. A neural networks (NNs) observer is constructed to reconstruct immeasurable information of the nonlinear systems, and, by introducing a new state vector and appropriate coordinate transformation, tracking control issues are converted into optimal regulation problems where critic-actor neural networks structures are developed for the solution of Hamilton–Jacobi–Bellman (HJB) equation corresponding to tracking errors. In addition, a robust term is introduced to eliminate effects from approximation errors. It is proven that all signals in the closed-loop system are uniformly ultimately bounded (UUB) by the Lyapunov approach. Finally, simulation examples are provided for illustration of the theoretical claims.

Near-optimal output tracking controller design for nonlinear systems using an event-driven ADP approach

Abstract In this paper, a novel output tracking control scheme for a class of continuous-time nonlinear system is proposed by using an event-driven adaptive dynamic programming (ADP) approach. Based on the controlled system and the desired reference trajectory, an augmented system is constructed for the optimal tracking control problem (OTCP), where the control law is composed of the steady-state part and the feedback part. According to the standard solution of the OTCP, the feedback part associated to the cost function is to be solved and the steady-state part of the control can be obtained directly. Thus the event-driven near-optimal control method is proposed to solve feedback control of the OTCP, where a single network ADP approach is developed to approximate the optimal cost function. The output tracking error dynamic get converged to zero and the parameters of critic neural network (NN) get converged to the optimal ones by using the designed weight tuning law and event-triggering condition. The stability of the augmented system under the novel control scheme is guaranteed based on Lyapunov theory and two simulation examples are presented to demonstrate the effectiveness of the proposed method.