Adaptive robust asymptotic tracking control of nonaffine nonlinear systems

This article investigates the barrier lyapunov function-based adaptive robust control scheme for nonaffine nonlinear systems with unknown system dynamics. First, the nonaffine system is converted into affine system via a combination of first-order filter and coordinate transformation, then a high-gain observer is utilized to reconstruct the immeasurable states of the derived affine system. Second, a robust integral of the sign of the error (RISE) is incorporated into the control design to reject the unknown dynamics. Third, a barrier lyapunov function based design method is used to ensure that the input vector to the parameter estimation remain within a predefined region. Then, an adaptive robust control scheme with only one adaptive parameter by using the upper bound estimation is developed. Finally, numerical simulations validate the efficacy of the developed control scheme.     

Indirect adaptive fuzzy control for a class of nonaffine nonlinear systems with unknown control directions

This article presents an indirect adaptive fuzzy control scheme for a class of nonlinear uncertain nonaffine systems with unknown control directions. The nonlinear nonaffine system is first transformed into an affine form by using a Taylor series expansion, and then fuzzy systems are employed to approximate the equivalent affine system’s unknown nonlinearities. By modifying the estimated input control gain and using a novel smooth robust control term, a stable well-defined adaptive controller is proposed. Simulation results are provided to illustrate the efficiency of the proposed scheme.     

Linear adaptive control of a class of SISO nonaffine nonlinear systems

This paper addresses the problem of linear adaptive control for a class of uncertain continuous-time single-input single-output (SISO) nonaffine nonlinear dynamic systems. Using the implicit function theory, the existence of an ideal controller which can achieve control objectives is firstly demonstrated. However, this ideal controller cannot be known and computed even if the system model is well known. The aim of our work is to construct this unknown ideal controller using a simple linear controller with the free parameters updated online by a stable adaptation mechanism designed to minimise the error between the unknown ideal controller and the used linear controller. Since the mathematical model of the system is assumed unknown in this work, the proposed control scheme can be regarded as a simple model free controller for the studied class of nonaffine systems. We prove that the closed-loop system is stable and all the signals are bounded. An application of the proposed linear adaptive controller for a nonaffine system is illustrated through the simulation results to demonstrate the effectiveness of the proposed control scheme.     

Observer-based direct adaptive fuzzy-neural control for nonaffine nonlinear systems

In this paper, an observer-based direct adaptive fuzzy-neural control scheme is presented for nonaffine nonlinear systems in the presence of unknown structure of nonlinearities. A direct adaptive fuzzy-neural controller and a class of generalized nonlinear systems, which are called nonaffine nonlinear systems, are instead of the indirect one and affine nonlinear systems given by Leu et al. By using implicit function theorem and Taylor series expansion, the observer-based control law and the weight update law of the fuzzy-neural controller are derived for the nonaffine nonlinear systems. Based on strictly-positive-real (SPR) Lyapunov theory, the stability of the closed-loop system can be verified. Moreover, the overall adaptive scheme guarantees that all signals involved are bounded and the output of the closed-loop system will asymptotically track the desired output trajectory. To demonstrate the effectiveness of the proposed method, simulation results are illustrated in this paper.     

控制方向未知的SISO非仿射系统间接自适应模糊输出反馈控制

针对一类单输入单输出(SISO)非仿射非线性系统控制方向未知时出现的控制器奇异问题,提出了一种间接自适应模糊控制方案.利用中值定理将非仿射系统转化为仿射系统,通过模糊逻辑系统逼近该仿射系统中的未知函数,并构造模糊控制器,同时利用Lyapunov稳定性定理设计自适应律,最终克服了控制器的奇异问题;在此基础上,通过构造观测器估计跟踪误差,设计输出反馈自适应模糊控制器,解决了状态不可测时系统控制器设计难题,采用Lyapunov稳定性定理证明控制器能使得跟踪误差收敛同时闭环系统所有信号均有界.仿真结果验证了所设计控制方案的可行性与有效性.     

Globally stabilizing adaptive control design for nonlinearly-parameterized systems

In this note, a new adaptive control design is proposed for nonlinear systems that are possibly nonaffine and contain nonlinearly parameterized unknowns. The proposed control is not based on certainty equivalence principle which forms the foundation of existing and standard adaptive control designs. Instead, a biasing vector function is introduced into parameter estimate; it links the system dynamics to estimation error dynamics, and its choice leads to a new Lyapunov-based design so that affine or nonaffine systems with nonlinearly parameterized unknowns can be controlled by adaptive estimation. Explicit conditions are found for achieving global asymptotic stability of the state, and the convergence condition for parameter estimation is also found. The conditions are illustrated by several examples and classes of systems. Besides global stability and estimation convergence, the proposed adaptive control has the unique feature that it does not contains any robust control part which typically overpowers unknown dynamics, may be conservative, and also interferes with parameter estimation.     

Distributed output‐feedback finite‐time tracking control of nonaffine nonlinear leader‐follower multiagent systems

The distributed output‐feedback tracking control for a class of networked multiagents in nonaffine pure‐feedback form is investigated in this article. By introducing a low‐pass filter and some auxiliary variables, we first transform the nonaffine system into the affine form. Then, the finite‐time observer is designed to estimate the states of the newly derived affine system. By applying the fraction dynamic surface control approach and the neural network‐based approximation technique, the distributed output‐feedback control laws are proposed and it is proved that the tracking errors converge to an arbitrarily small bound around zero in finite time. Finally, some simulation examples are provided to confirm the effectiveness of the developed method.     

Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems

This paper develops sufficient conditions for a general nonlinear control system to be locally (resp. globally) asymptotically stabilizable via smooth state feedback. In particular, it is shown that as in the case of affine systems, this is possible if the unforced dynamic system of ∑₁ is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems.     

时变时滞非仿射大系统的分散自适应控制

针对一类具有未知时变时滞的非仿射互联大系统基于神经网络的逼近能力, 提出了一种分散自适应神经网络控制方案。该方案利用中值定理对未知非仿射函数进行分离; 利用分离技术和Young's不等式放宽了对未知时滞及时滞互联不确定项的限制, 同时大大减少了在线调节参数的数量。此外, 利用Lyapunov Krasovskii 泛函补偿了未知时滞带来的不确定性。通过理论分析, 证明了闭环系统所有信号是有界的, 输出跟踪误差收敛到原点的一个小邻域内。最后, 仿真结果验证了所提控制方案的有效性。     

不确定非仿射系统的引入动态逆的自抗扰控制器设计

针对一类不确定非仿射严反馈非线性系统, 提出一种引入动态逆的线性自抗扰控制器设计方法. 首先, 利 用微分同胚映射将严反馈非线性系统变换为积分串联型系统, 然后针对积分串联型系统设计线性自抗扰控制器. 提出的线性自抗扰控制器将闭环系统划分为3个时间尺度, 其中线性扩张状态观测器位于最快的时间尺度上, 用来 估计系统的状态和总和扰动, 动态逆位于次快的时间尺度上用以求解非仿射情况下的控制律, 系统动态位于最慢的 时间尺度上. 利用奇异摄动理论分析了闭环系统的稳定性和性能. 提出的自抗扰控制设计方法同样适用于控制增 益不确定的仿射非线性系统. 仿真和实验结果验证了提出的线性自抗扰控制器的可行性.     

Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity

The purpose of this paper is to develop a systematic method for global asymptotic stabilisation in probability of nonlinear control stochastic systems with stable in probability unforced dynamics. The method is based on the theory of passivity for nonaffine stochastic differential systems combined with the technique of Lyapunov asymptotic stability in probability for stochastic differential equations. In particular, we prove that a nonlinear stochastic differential system whose unforced dynamics are Lyapunov stable in probability is globally asymptotically stabilisable in probability provided some rank conditions involving the affine part of the system coefficients are satisfied. In this framework, we show that a stabilising smooth state feedback law can be designed explicitly. A dynamic output feedback compensator for a class of nonaffine stochastic systems is constructed as an application of our analysis.     

Learning from neural control of nonlinear systems in normal form

A deterministic learning theory was recently proposed which states that an appropriately designed adaptive neural controller can learn the system internal dynamics while attempting to control a class of simple nonlinear systems. In this paper, we investigate deterministic learning from adaptive neural control (ANC) of a class of nonlinear systems in normal form with unknown affine terms. The existence of the unknown affine terms makes it difficult to achieve learning by using previous methods. To overcome the difficulties, firstly, an extension of a recent result is presented on stability analysis of linear time-varying (LTV) systems. Then, with a state transformation, the closed-loop control system is transformed into a LTV form for which exponential stability can be guaranteed when a partial persistent excitation (PE) condition is satisfied. Accurate approximation of the closed-loop control system dynamics is achieved in a local region along a recurrent orbit of closed-loop signals. Consequently, learning of control system dynamics (i.e. closed-loop identification) from adaptive neural control of nonlinear systems with unknown affine terms is implemented.     

基于神经网络的非仿射块控非线性系统动态反演控制

即使已知非仿射非线性系统的逆存在,利用隐函数定理求解该显式逆仍然非常困难.为此,针对一类不确定块控非仿射系统,将动态反馈、反演、神经网络和反馈线性化技术相结合,提出一种自适应鲁棒控制器的设计方法.利用神经网络来逼近和消除未知函数,并证明了整个闭环系统在李雅普诺夫意义下是稳定的.仿真结果表明了所提出方法的有效性.     

基于Backstepping 的非仿射非线性系统鲁棒控制

针对一类不确定非仿射非线性系统的跟踪控制问题, 提出一种鲁棒Backstepping 控制策略. 首先, 为利用仿 射非线性方法设计控制器, 给出一种适用于全局的非仿射非线性近似方法; 然后, 设计快速收敛非线性微分器以估计复合干扰和获取虚拟信号的微分, 进而给出不确定非仿射非线性系统的复合控制器, 其中鲁棒项和阻尼项分别用于减少逼近误差和近似方法中动态误差对系统跟踪的影响; 最后, 通过仿真实验验证了所提出方法的有效性.     

Direct adaptive controller for nonaffine nonlinear systems using self-structuring neural networks

A direct adaptive state-feedback controller is proposed for highly nonlinear systems. We consider uncertain or ill-defined nonaffine nonlinear systems and employ a neural network (NN) with flexible structure, i.e., an online variation of the number of neurons. The NN approximates and adaptively cancels an unknown plant nonlinearity. A control law and adaptive laws for the weights in the hidden layer and output layer of the NN are established so that the whole closed-loop system is stable in the sense of Lyapunov. Moreover, the tracking error is guaranteed to be uniformly asymptotically stable (UAS) rather than uniformly ultimately bounded (UUB) with the aid of an additional robustifying control term. The proposed control algorithm is relatively simple and requires no restrictive conditions on the design constants for the stability. The efficiency of the proposed scheme is shown through the simulation of a simple nonaffine nonlinear system.     

Robust adaptive control of nonaffine nonlinear plants with small input signal changes

Assuming small input signal magnitudes, ARMA models can approximate the NARMA model of nonaffine plants. Recently, NARMA-L1 and NARMA-L2 approximate models were introduced to relax such input magnitude restrictions. However, some applications require larger input signals than allowed by ARMA, NARMA-L1 and NARMA-L2 models. Under certain assumptions, we recently developed an affine approximate model that eliminates the small input magnitude restriction and replaces it with a requirement of small input changes. Such a model complements existing models. Using this model, we present an adaptive controller for discrete nonaffine plants with unknown system equations, accessible input-output signals, but inaccessible states. Our approximate model is realized by a neural network that learns the unknown input-output map online. A deadzone is used to make the weight update algorithm robust against modeling errors. A control law is developed for asymptotic tracking of slowly varying reference trajectories.     

Generalized PI control design for a class of unknown nonaffine systems with sensor and actuator faults

This work deals with the tracking control problem of a class of unknown nonaffine dynamic systems that involve unpredictable sensor and actuation failures. As the control inputs enter into and influence the dynamic behavior of the nonaffine system through a nonlinear and implicit way, control design for such system becomes quite challenging. The underlying problem becomes even more complex if the system dynamics are unavailable for control design yet involving unanticipated sensor and/or actuator faults. In this work, a structurally simple and computationally inexpensive control scheme is proposed to achieve uniformly ultimately bounded (UUB) stable tracking control of a class of nonaffine systems. The proposed control is of a generalized PI form and is able to accommodate both sensor and actuator faults. The effectiveness of the proposed control strategy is confirmed by theoretical analysis and numerical simulations.     

Adaptive dynamic surface control for uncertain nonaffine nonlinear systems

This paper focuses on the problem of adaptive control for uncertain nonaffine nonlinear systems. The original nonaffine systems are transformed into the augmented affine systems via adding an auxiliary integrator, which makes the explicit control design possible. By introducing a modified sliding mode filter in each step, a novel adaptive dynamic surface controller is proposed, where the ‘explosion of complexity’ problem inherent in the backstepping design is avoided. It is proven rigorously that for any initial control condition, the proposed adaptive scheme is able to ensure the semiglobal uniformly ultimately boundedness of all signals in the closed loop. An illustrative example is carried out to verify the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems

An adaptive fuzzy control approach is proposed for a class of multiple-input-multiple-output (MIMO) nonlinear systems with completely unknown nonaffine functions. The MIMO systems are composed of n subsystems and each of subsystems is in the nested lower triangular form. It is difficult and complicated to control this class of systems due to the existence of unknown nonaffine functions and the couplings among the nested subsystems. This difficulty is overcome by introducing some special type Lyapunov functions and taking advantage of the mean-value theorem, the backstepping design method and the approximation property of the fuzzy systems. The proposed control approach can guarantee that all the signals in the closed-loop system are bounded. A simulation experiment is utilized to verify the feasibility of the proposed approach.     

Robust adaptive prescribed performance control for a class of nonlinear pure‐feedback systems

An adaptive prescribed performance control design procedure for a class of nonlinear pure‐feedback systems with both unknown vector parameters and unmodeled dynamics is presented. The unmodeled dynamics lie within some bounded functions, which are assumed to be partially known. A state transformation and an auxiliary system are proposed to avoid using the cumbersome formula to handle the nonaffine structure. Simultaneously, a parameter‐type Lyapunov function and L function are designed to ensure the prescribed performance of the pure‐feedback system. As illustrated by examples, the proposed adaptive prescribed performance control scheme is shown to guarantee global uniform ultimate boundedness. At the same time, this method not only guarantees the prescribed performance of the system but also makes the tracking error asymptotically close to a certain value or stable.