Nonlinear influence in the frequency domain: Alternating series

The nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have Volterra series expansion. It is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities. The magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series. Sufficient (and necessary) conditions in which the output spectrum can be cast into an alternating series are studied. These results reveal a novel frequency-domain insight into the nonlinear influence on a system, and provide a new method for the analysis and design of nonlinear systems in the frequency domain. Examples are given to illustrate the results.     

An algorithm for system immersion into nonlinear observer form: SISO case

The class of nonlinear systems that can be put into nonlinear observer form (linear system with output injection) can be broadened if we employ the system immersion. We provide an immersion algorithm for SISO nonlinear systems with relative degree r. The proposed algorithm is an extension of the previous results and does not require the relative degree 1 assumption. In addition, it is seen that the immersion can always be computed via algebraic computations and simple integrations except in a very special case in which the relative degree equals the system dimension (in this case, only one first order differential equation appears in the algorithm).     

Numerical studies on Boussinesq-type equations via a split-step Fourier method

Boussinesq-type nonlinear wave equations with dispersive terms are solved via split-step Fourier methods. We decompose the equations into linear and nonlinear parts, then solve them orderly. The linear part can be projected into phase space by a Fourier transformation, and resulting in a variable separable ordinary differential system, which can be integrated exactly. Next, by an invert Fourier transformation, the classical explicit fourth-order Runge–Kutta method is adopted to solve the nonlinear subproblem. To examine the numerical accuracy and efficiency of the method, we compare the numerical solutions with exact solitary wave solutions. Additionally, various initial-value problems for all the listed Boussinesq-type system are studied numerically. In the study, we can observe that sech ²-type waves for KdV-BBM system will split into several solitons, which is a very interesting physical phenomenon. The interaction between solitons, including overtaking and head-on collisions, is also simulated.     

Semi‐Global Robust Output Regulation for A Class of Singular Nonlinear Systems with Unknown Algebraic Equations

This paper considers semi‐global robust output regulation problem for a class of singular nonlinear systems whose algebraic equations are not precisely known. Since the algebraic equations are not known, the output regulation problem of singular nonlinear systems cannot be solved by directly reducing the singular nonlinear system into a normal nonlinear system. Based on internal model principle, we convert the robust output regulation problem of singular nonlinear systems into a robust stabilization problem of an augmented singular nonlinear system. The augmented singular nonlinear system is also with unknown algebraic equations. However, without transforming the singular nonlinear system into a normal nonlinear system, it is shown that the augmented singular nonlinear system can be semi‐globally stabilized by a high gain output feedback control law under some reasonable assumptions. Moreover, the semi‐global stabilization control law of the augmented singular nonlinear systems also solves the semi‐global robust output regulation problem of the original singular nonlinear system.     

Efficient computation of higher order frequency response functions for nonlinear systems with,and without,a constant term

The behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation. These changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system's dynamics. In this article, a new and efficient algorithm is presented for computing the higher order Volterra frequency response functions from nonlinear time-domain models that may contain a constant term. A comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method. The algorithm is applicable to systems modelled by nonlinear differential, or difference, equations and is easily automated. Several examples are used to illustrate the method, and to highlight the importance of dc terms in nonlinear system analysis.     

基于饱和系统随机共振的图像去噪算法

利用随机共振（SR）机制，在传输相关信号的非线性系统中加入噪声，在输出端信号可被增强。本文提出一种基于动态饱和非线性系统随机共振的图像去噪算法，首先将图像重采样为一维信号，并调节动态饱和非线性系统的参数，使之达到最优，使动态饱和非线性系统能够产生随机共振。相比一维随机共振，二维随机共振的图像复原效果更接近于原图，输出图像的直方图和峰值信噪比(PSNR)，也明显优于一维随机共振。相比于传统滤波方法，饱和系统的去噪效果更好，同时对于噪声强度的变化具有较好的鲁棒性。     

Immersion and immersion by nonsingular feedback of a discrete-timenonlinear system into a linear system

An analytic discrete-time nonlinear system is considered. Necessary and sufficient conditions for immersion of the system into a linear system are given. Local immersion is defined by nonsingular feedback into a linear system and given necessary and sufficient conditions for this problem. It is shown that a similar approach can also be applied to continuous-time affine nonlinear systems     

基于状态变换的非线性鲁棒控制器设计

本文对具有特定结构的非线性系统,介绍了一种非线性变换使之线性化,并利用线性鲁棒 控制器的设计方法设计非线性鲁棒控制器,从而提高了闭环系统对参数变动的鲁棒性.文中 还通过一个CSTR反应器模型的仿真说明该方法的有效性.     

Leader-follower Consensus of Upper-triangular Nonlinear Multi-agent Systems

This paper is concerned with the leader-follower consensus problem by using both state and output feedback for a class of nonlinear multi-agent systems. The agents considered here are all identical upper-triangular nonlinear systems which satisfy the Lipschitz growth condition. First, it is shown that the leader-follower consensus problem is equivalent to the control design problem of a high-dimensional multi-variable system. Second, by introducing an appropriate state transformation, the control design problem can be converted into the problem of finding a constant parameter, which can be obtained by solving the Lyapunov equation and estimating the nonlinear terms of the given system. At last, an example is given to verify effectiveness of the proposed consensus algorithms.      

Optimal tracking control for nonlinear large-scale systems with persistent disturbances

This paper studies the optimal control with zero steady-state error problem for nonlinear large-scale systems affected by external persistent disturbances. The nonlinear large-scale system is transformed into N nonlinear subsystems with interconnect terms. Based on the internal model principle, a disturbance compensator is constructed such that the ith subsystem with external persistent disturbances is transformed into an augmented subsystem without disturbances. According to the sensitivity approach, the optimal tracking control law for the ith nonlinear subsystem can be obtained. The optimal tracking control law for the nonlinear large-scale systems can be obtained. A numerical simulation shows that the method is effective.     

Linear pole-placement anti-windup control for input saturation nonlinear system based on Takagi Sugeno fuzzy model

An anti-windup controller for multi-input nonlinear system is proposed in this paper. The proposed method does not need to calculate every time for every fuzzy rules comparing with traditional linear pole placement of T-S(Takagi-Sugeno) fuzzy. This means fewer LMIs(linear matrix inequality) will be needed and its solution will be guaranteed as much as possible. For different saturation limit, different D-stable disk center and radius of pole placement can be selected to eliminate input saturation effect directly. Nonlinear system will be transferred to T-S fuzzy model first. Then, by employing a series of transition matrix, nonlinear system will be transferred into a nearly linear format accompanied by a nonlinear part. Finally, by designing a proper controller, linear pole placement method can be used and the controller gains can be calculated out with LMIs.

     

Quantized stabilization of strict-feedback nonlinear systems based on ISS cyclic-small-gain theorem

This paper proposes a new tool for quantized nonlinear control design of dynamic systems transformable into the dynamically perturbed strict-feedback form. To address the technical challenges arising from measurement and actuator quantization, a new approach based on set-valued maps is developed to transform the closed-loop quantized system into a large-scale system composed of input-to-state stable (ISS) subsystems. For each ISS subsystem, the inputs consist of quantization errors and interacting states, and moreover, the ISS gains can be assigned arbitrarily. Then, the recently developed cyclic-small-gain theorem is employed to guarantee input-to-state stability with respect to quantization errors and to construct an ISS-Lyapunov function for the closed-loop quantized system. Interestingly, it is shown that, under some realistic assumptions, any n-dimensional dynamically perturbed strict-feedback nonlinear system can be globally practically stabilized by a quantized control law using 2n three-level dynamic quantizers.     

The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems

The method of multiple scales is implemented in Maple V Release 2 to generate a uniform asymptotic solutionO(ϵ^r) for a weakly nonlinear oscillator.In recent work, it has been shown that the method of multiple scales also transforms the differential equations into normal form, so the given algorithm can be used to simplify the equations describing the dynamics of a system near a fixed point.These results are equivalent to those obtained with the traditional method of normal forms which uses a near-identity coordinate transformation to get the system into the “simplest” form.A few Duffing type oscillators are analysed to illustrate the power of the procedure. The algorithm can be modified to take care of systems of ODEs, PDEs and other nonlinear cases.     

Input linearization of nonlinear systems via pulse-width control

In this note, it is shown that a general nonlinear system can be transformed to an affine nonlinear system by virtue of the use of pulsewidth control. Therefore, the combined system from the pulse width control input to the nonlinear system output behaves as an affine (linear-in-control) system. By this way, a cumbersome nonlinear system model can be transformed to a simpler linear-in-control form without increasing the system dimension; this in turn enables simpler control design. Furthermore, using this methodology, some control design methods developed only for affine systems can be adapted to general nonlinear systems.     

Deterministic learning and nonlinear observer design

A “deterministic learning” (DL) theory was recently proposed for identification of nonlinear system dynamics under full‐state measurements. In this paper, for a class of nonlinear systems undergoing periodic or recurrent motions with only output measurements, firstly, it is shown that locally‐accurate identification of nonlinear system dynamics can still be achieved. Specifically, by using a high gain observer and a dynamical radial basis function network (RBFN), when state estimation is achieved by the high gain observer, along the estimated state trajectory, a partial persistence of excitation (PE) condition is satisfied, and locally‐accurate identification of system dynamics is achieved in a local region along the estimated state trajectory. Secondly, by embedding the learned knowledge of system dynamics into a RBFN‐based nonlinear observer, it is shown that correct state estimation can be achieved according to the internal matching of the underlying system dynamics, rather than by using high gain domination. The significance of this paper is that it reveals that the difficult problems in nonlinear observer design can be successfully resolved by incorporating the deterministic learning mechanisms. Simulation studies are included to demonstrate the effectiveness of the approach. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Feedback control of multiinput nonlinear systems by extendedlinearization

Control of a nonlinear system is addressed by considering its family of linearizations. Given that a family of linear state-feedback controllers has been designed to meet some (linear) control objective, it is shown how these controllers can be pieced together into an overall, nonlinear, state-feedback controller. The resulting closed-loop system has the property that its linearizations are equal to the desired linearly controlled systems. A similar approach can be used to design nonlinear state estimators. The approach is illustrated by applying it to the control of a planar two-link manipulator     

Direct adaptive interval type-2 fuzzy control of multivariable nonlinear systems

A fuzzy logic controller equipped with a training algorithm is developed such that the H^∞ tracking performance should be satisfied for a model-free nonlinear multiple-input multiple-output (MIMO) system, with external disturbances. Due to universal approximation theorem, fuzzy control provides nonlinear controller, i.e., fuzzy logic controllers, to perform the unknown nonlinear control actions and the tracking error, because of the matching error and external disturbance is attenuated to arbitrary desired level by using H^∞ tracking design technique. In this paper, a new direct adaptive interval type-2 fuzzy controller is developed to handle the training data corrupted by noise or rule uncertainties for nonlinear MIMO systems involving external disturbances. Therefore, linguistic fuzzy control rules can be directly incorporated into the controller and combine the H^∞ attenuation technique. Simulation results show that the interval type-2 fuzzy logic system can handle unpredicted internal disturbance, data uncertainties, very well, but the adaptive type-1 fuzzy controller must spend more control effort in order to deal with noisy training data. Furthermore, the adaptive interval type-2 fuzzy controller can perform successful control and guarantee the global stability of the resulting closed-loop system and the tracking performance can be achieved.     

A functional equations approach to nonlinear discrete-time feedback stabilization through pole-placement

The present work proposes a new approach to the nonlinear discrete-time feedback stabilization problem with pole-placement. The problem's formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived. Using tools from functional equations theory, one can prove that the solution to the above system of nonlinear functional equations is locally analytic, and an easily programmable series solution method can be developed. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete-time state feedback control law that are both computed through the solution of the system of nonlinear functional equations, the feedback stabilization with pole-placement design objective can be attained under rather general conditions. The key idea of the proposed single-step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach. However, since the proposed method does not involve an external reference input, it cannot meet other control objectives such as trajectory tracking and model matching.     

Controller design for nonlinear systems with time delay using model algorithm control (MAC)

This paper presents a digital model algorithm control (MAC) for nonlinear time-delayed systems. The MAC controller is designed based on the sampled-data representation of the dynamic system. A new discretization method using the Taylor series and zero order hold assumption is proposed to discretize the continuous nonlinear time-delayed system. The proposed discretization method can provide accurate time discretization results and convert the discrete-time system into a general nonlinear system, so that the MAC method can be employed. Several simulations are conducted to verify the performance of the proposed method.     

Existence of Carathe/spl acute/odory solutions in nonlinear systems with discontinuous switching feedback controllers

In this note, we consider the existence of a Carathe/spl acute/odory solution in a single-input-single-output nonlinear system with a discontinuous switching feedback controller. The main contribution is to show that if the nonlinear system can be transformed into a global normal form, then we can specify the value of the discontinuous switching feedback controller on the switching hypersurface so that the closed-loop system has a Carathe/spl acute/odory solution.