State derivative feedback for adaptive cancellation of unmatched disturbances in unknown strict-feedback LTI systems

Solutions exist for the problem of canceling sinusoidal disturbances by the measurement of the state or by the measurement of an output for linear and nonlinear systems. In this paper, an adaptive backstepping controller is designed to cancel sinusoidal disturbances forcing an unknown linear time-invariant system in controllable canonical form which is augmented by a linear input subsystem with unknown system parameters. The state-derivatives of the original subsystem and the state of the input subsystem are the only measurements that are used in the design of the controller. The design is based on four steps, (1) parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters, (2) design of an adaptive disturbance observer for both disturbance and its derivative, (3) design of an adaptive controller for the virtual control input, and (4) design of the final adaptive controller by using the backstepping procedure. It is proven that the equilibrium of the closed-loop adaptive system is stable and the state of the considered original subsystem converges to zero as t→∞$t\to \infty$ with perfect disturbance estimation. The effectiveness of the controller is illustrated with a simulation example of a third order system.     

含未知输入的广义系统的状态与输入估计

研究了含未知输入的广义系统的输入解耦观测器设计问题. 在系统脉冲能控的条件下通过系统输入-状态对的非奇异变换, 把此问题等价地转化为正常状态空间系统的相应问题. 用大家熟知的方法设计正常状态空间系统的观测器, 从而得到广义系统的输入解耦观测器. 然后用广义系统的观测器状态和系统输出的线性组合渐近估计系统的状态与未知输入.     

Time‐varying input delay compensation for nonlinear systems with additive disturbance: An output feedback approach

This paper addresses the output feedback tracking control of a class of multiple‐input and multiple‐output nonlinear systems subject to time‐varying input delay and additive bounded disturbances. Based on the backstepping design approach, an output feedback robust controller is proposed by integrating an extended state observer and a novel robust controller, which uses a desired trajectory‐based feedforward term to achieve an improved model compensation and a robust delay compensation feedback term based on the finite integral of the past control values to compensate for the time‐varying input delay. The extended state observer can simultaneously estimate the unmeasurable system states and the additive disturbances only with the output measurement and delayed control input. The proposed controller theoretically guarantees prescribed transient performance and steady‐state tracking accuracy in spite of the presence of time‐varying input delay and additive bounded disturbances based on Lyapunov stability analysis by using a Lyapunov‐Krasovskii functional. A specific study on a 2‐link robot manipulator is performed; based on the system model and the proposed design procedure, a suitable controller is developed, and comparative simulation results are obtained to demonstrate the effectiveness of the developed control scheme.     

非线性微分——代数系统的输出反馈镇定控制

对满足线性增长条件的非线性微分--代数系统, 研究了其输出反馈镇定控制问题. 通过将状态观测器与控制器耦合在一起设计, 构造出一种非初始化的线性高增益状态观测器, 具有良好的鲁棒性. 基于反推设计方法构造出一个线性的动态输出补偿器, 使得整个闭环系统是渐近稳定的. 仿真结果验证了本文所提控制方法的有效性.     

多智能体系统的鲁棒故障估计观测器的设计

针对线性变参数多智能体系统设计了有限频域鲁棒故障估计观测器。首先,根据每个智能体的绝对可测输出和相对可测输出建立了每个节点的动力学方程,结合无向通讯拓扑图及拉普拉斯矩阵得到了多智能体系统的动力学方程,通过合适的变换对多智能体系统模型进行了解耦;然后,根据解耦后的系统动力学方程设计了故障估计观测器,并通过优化技术得到了故障估计观测器增益矩阵和优良的鲁棒性能指标;最后,通过微型飞行器纵向飞行运动的例子验证了所设计的故障估计观测器的有效性,及系统参数在一定的范围内发生变动的时候,故障估计观测器依然可以准确的估计系统所发生的故障。     

Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large‐scale systems using DSC technique

In this paper, an adaptive fuzzy decentralized backstepping output feedback control approach is proposed for a class of uncertain large‐scale stochastic nonlinear systems without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. Using the designed fuzzy state observer, and by combining the adaptive backstepping technique with dynamic surface control technique, an adaptive fuzzy decentralized output feedback control approach is developed. It is shown that the proposed control approach can guarantee that all the signals of the resulting closed‐loop system are semi‐globally uniformly ultimately bounded in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing appropriate design parameters. A simulation example is provided to show the effectiveness of the proposed approaches. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive output-feedback stabilization of non-local hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system’s functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.     

Adaptive NN output‐feedback decentralized stabilization for a class of large‐scale stochastic nonlinear strict‐feedback systems

In this paper, the decentralized adaptive neural network (NN) output‐feedback stabilization problem is investigated for a class of large‐scale stochastic nonlinear strict‐feedback systems, which interact through their outputs. The nonlinear interconnections are assumed to be bounded by some unknown nonlinear functions of the system outputs. In each subsystem, only a NN is employed to compensate for all unknown upper bounding functions, which depend on its own output. Therefore, the controller design for each subsystem only need its own information and is more simplified than the existing results. It is shown that, based on the backstepping method and the technique of nonlinear observer design, the whole closed‐loop system can be proved to be stable in probability by constructing an overall state‐quartic and parameter‐quadratic Lyapunov function. The simulation results demonstrate the effectiveness of the proposed control scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Output‐based disturbance rejection control for 1‐D anti‐stable Schrödinger equation with boundary input matched unknown disturbance

In this paper, we are concerned with the output feedback control design for a system (plant) described by a boundary controlled anti‐stable one‐dimensional Schrödinger equation. Our output measure signals are the displacements at both side. An untraditional infinite‐dimensional disturbance estimator is developed to estimate the disturbance. Based on the estimator, we propose a state observer that is exponentially convergent to the original system and then design a stabilizing control law consisting of two parts: The first part is to compensate the disturbance by using its approximated value and the second part is to stabilize the observer system by applying the classical backstepping approach. The resulting closed‐loop system is shown to be exponentially stable with guaranteeing that all internal systems are uniformly bounded. An effective output‐based disturbance rejection control algorithm is concluded. An application, namely, a cascade of ODE–wave systems, is investigated by the developed control algorithm. Numerical experiments are carried out to illustrate the effectiveness of the proposed control law. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimation of states,faults and unknown disturbances in non-linear systems

In this paper, we extend existing theory on non-linear unknown input observer design to a wider class of non-linear systems where the systems are subject to unknown input and output disturbances and experience faults. The approach used is to decouple the faults and unknown disturbances from the rest of the system through a series of transformations on state and output equations. Once total disturbance decoupling is achieved, an appropriate observer for the disturbance free part of the non-linear system is proposed and designed. The designed observer is, subsequently, used for estimation of unknown inputs, outputs, and fault signals. Two examples are given for illustration.     

Global output feedback control of nonlinear time-delay systems with input matching uncertainty and unknown output function

This paper studies the problem of global output feedback control for nonlinear time-delay systems with input matching uncertainty and the unknown output function, whose nonlinearities are bounded by lower triangular linear unmeasured states multiplying the unknown constant, polynomial-of-output and polynomial-of-input growth rates. By constructing a new extended state observer and skillfully combining the dynamic gain method, backstepping method and Lyapunov–Krasovskii theorem, a delay-independent output feedback controller can be developed with only one dynamic gain. It is proved that all the signals of the closed-loop system are bounded, the states of the original system and the corresponding observer converge to zero, and the estimation of input matching uncertainty converges to its actual value. Two examples demonstrate the effectiveness of the control scheme.     

Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions

So far there is still no result available for backstepping based decentralized adaptive stabilization of unknown systems with interactions directly depending on subsystem inputs, even though such interactions commonly exist in practice. In this paper, we provide a solution to this problem by considering both input and output dynamic interactions. To clearly illustrate our approaches, we will start with linear systems and then extend the results to nonlinear systems. Each local controller, designed simply based on the model of each subsystem by using the standard adaptive backstepping technique without any modification, only employs local information to generate control signals. It is shown that the designed decentralized adaptive backstepping controllers can globally stabilize the overall interconnected system asymptotically. The L₂ and L_∞ norms of the system outputs are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters.     

Observer‐based adaptive neural control for a class of nonlinear singular systems

The problem of observer‐based adaptive neural control via output feedback for a class of uncertain nonlinear singular systems is studied in this article. The nonlinear singular systems can be regarded as two subsystems that are coupled with each other: differential subsystem and algebraic subsystem. The differential systems can be nonstrict feedback structures. To guarantee that the singular system is regular and impulse‐free, two new conditions are proposed. By the conditions, the linear controller and observer, which are used to estimate the immeasurable state variables, are obtained. Then, an output feedback scheme through adaptive neural backstepping is proposed to ensure that all states of the closed‐loop system are semiglobally uniformly ultimately bounded and converge to a small neighborhood of the origin. Simulation examples illustrate the effectiveness of the presented method.     

Adaptive output control of a class of uncertain chaotic systems

In this paper, a new observer-based backstepping output control scheme is proposed for stabilizing and controlling a class of uncertain chaotic systems. The controller is designed through the use of a robust observer and backstepping technique. We firstly show that many chaotic systems as paradigms in the research of chaos can be transformed into a class of nonlinear systems in the feedback form. Secondly, the synchronization problem is converted to the tracking problem from control theory, thereby leading to the use of state observer design techniques. A new observer is utilized to estimate the unmeasured states. Unlike some existing methods for chaos control, no priori knowledge on the system parameters is required and only the output signal is available for control purpose. The Lyapunov functions are quadratic in the state estimates, the observer errors and the parameter estimation error based on the backstepping technique. It is shown that not only global stability is guaranteed by the proposed controller, but also both transient and asymptotic tracking performances are quantified as explicit functions of the design parameters so that designers can tune the design parameters in an explicit way to obtain the desired closed-loop behavior.     

Robust control of nonlinear semi‐strict feedback systems using finite‐time disturbance observers

The output tracking controller design problem is dealt with for a class of nonlinear semi‐strict feedback systems in the presence of mismatched nonlinear uncertainties, external disturbances, and uncertain nonlinear virtual control coefficients of the subsystems. The controller is designed in a backstepping manner, and to avoid the shortcoming of ‘explosion of terms’, the dynamic surface control technique that employs a group of first‐order low‐pass filters is adopted. At each step of the virtual controller design, a robust feedback controller employing some effective nonlinear damping terms is designed to guarantee input‐to‐state practical stable property of the corresponding subsystem, so that the system states remain in the feasible domain. The virtual controller is enhanced by a finite‐time disturbance observer that estimates the disturbance term in a finite‐time. The properties of the composite control system are analyzed theoretically. Furthermore, by exploiting the cascaded structure of the control system, a simplified robust controller is proposed where only the first subsystem employs a disturbance observer. The performance of the proposed methods is confirmed by numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Observer-based adaptive output-constrained control design of switched stochastic nonlinear systems with input saturation

In this article, the problem of adaptive fuzzy control for output-constrained switched stochastic nonlinear systems subject to input saturation is addressed. By employing the trigonometric function mapping method, the constrained systems are transformed into unconstrained ones, and the control goals of the original constrained systems are not affected. Meanwhile, an auxiliary system is established to deal with the issue of input saturation, and an observer is constructed to estimate the unmeasured states. Then, the unknown nonlinear functions in the system are approximated by the fuzzy logic systems (FLSs). Based on the backstepping technique and Lyapunov function method, an output feedback control strategy is designed, where the dynamic surface control technique is applied in the backstepping design process to overcome the issue of a large number of online calculations. The designed controller can guarantee that all the signals of the system satisfy bounded conditions, and the output can track given reference signals within a small error range. Finally, a simulation example is given to verify the effectiveness of the proposed control scheme.     

Adaptive output feedback control for nonholonomic systems with uncertain chained form

The output feedback adaptive control problem is investigated for nonholonomic systems with strongly nonlinear uncertainties and unknown virtual control directions. A nonlinear output feedback switching controller based on the output measurement of the first subsystem is employed in order to make the state scaling effective and ensure the convergence of the system states. The novel observer/estimator is introduced for state and unknown parameter estimates. The integrator backstepping technique by the use of a constructive recursive is applied to the design of the adaptive controller and to overcome the unknown virtual control directions. The simulation result validates the effectiveness of the proposed scheme.     

Output feedback control of a class of discrete MIMO nonlinear systems with triangular form inputs

In this paper, adaptive neural network (NN) control is investigated for a class of discrete-time multi-input-multi-output (MIMO) nonlinear systems with triangular form inputs. Each subsystem of the MIMO system is in strict feedback form. First, through two phases of coordinate transformation, the MIMO system is transformed into input-output representation with the triangular form input structure unchanged. By using high-order neural networks (HONNs) as the emulators of the desired controls, effective output feedback adaptive control is developed using backstepping. The closed-loop system is proved to be semiglobally uniformly ultimate bounded (SGUUB) by using Lyapunov method. The output tracking errors are guaranteed to converge into a compact set whose size is adjustable, and all the other signals in the closed-loop system are proved to be bounded. Simulation results show the effectiveness of the proposed control scheme.     

Robust Output Regulation of T--S Fuzzy Systems With Multiple Time-Varying State and Input Delays

This paper proposes a robust output regulator design for Takagi–Sugeno (T–S) fuzzy systems with uncertain time delays. For generalization of the control problem, we assume that there exist affine dynamics, unknown multiple time-varying delays in state and input, uncertain parameters, disturbance, and partial state feedback. Both continuous-time and discrete-time cases are considered in a unified theoretical derivation. Here, we introduce a memoryless fuzzy observer and a fuzzy integral compensator. In comparison with previous studies, the proposed scheme drops the need for coordinate transformation, calculation of the desired operational point, which is known state and input delays, and exact fuzzy modeling. The control result is asymptotic output regulation for systems without disturbance and guaranteed $H^{infty}$ performance for systems with disturbance. Moreover, a piecewise constant output regulation is achieved without controller redesign. In addition, the developed fuzzy regulator can be applied to both affine and nonaffine uncertain time-delay systems. Finally, numerical simulations are performed by using two typical time-delay systems to validate the expected performance.      

Boundary control and estimation of reaction–diffusion equations on the sphere under revolution symmetry conditions

ABSTRACT

Recently, the problem of designing boundary controllers and observers for unstable linear constant-coefficient reaction–diffusion equation on N-balls has been solved by means of the backstepping method. However, the extension of these results to spatially varying coefficients is far from trivial. This work deals with radially varying reaction coefficients under revolution symmetry conditions on a sphere (the three-dimensional case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, a similar type of singularity appears in the backstepping control and observer kernel equations. However, with a simple scaling transformation, we are able to reduce the singular equation to a regular equation, which turns out to be the same kernel equations appearing when using the one-dimensional backstepping method. In addition, the scaling transformation allows us to prove stability in the H ¹ space.