Least-squares LTI approximation of nonlinear systems and quasistationarity analysis

Least-squares linear time-invariant (LTI) approximation of discrete-time nonlinear systems is studied in a generalized harmonic analysis setting extending an earlier result based on quasistationary signals. The least-squares optimal LTI model is such that the crosscorrelation between the input and the LTI model output equals the crosscorrelation between the input and the output of the nonlinear system. New results for limits of sample averages of signals are derived via Riemann-Stieltjes integration theory. These results are applied to crosscorrelation and quasistationarity analysis of input-output signals for several important classes of nonlinear systems, including stable finite memory, Wiener and Hammerstein systems. This analysis demonstrates that the assumptions used in the least-squares LTI approximation setup are fairly mild. Finally, an illustrative example is provided.     

基于有理平方根逼近的非高斯随机分布系统的故障诊断和容错控制

随机分布系统指的是输入为常规向量而输出为系统输出的概率密度函数所描述的一类随机系统．该类系统控制算法的目标是选择一个控制输入使得系统的实际输出概率密度函数尽可能跟踪一个事先给定的概率密度函数．本文对采用有理平方根B样条逼近其输出概率密度函数的非高斯动态随机分布系统,提出了一种基于非线性自适应观测器的故障诊断方法．该方法可快速有效地诊断出非高斯随机分布系统故障．通过对故障系统的重组,使故障后系统的输出概率密度函数仍能跟踪给定的分布,实现了该随机系统的容错控制,提高了随机系统的可靠性．     

Computational complexity analysis of set membership identification of Hammerstein and Wiener systems

This paper analyzes the computational complexity of set membership identification of Hammerstein and Wiener systems. Its main results show that, even in cases where a portion of the plant is known, the problems are generically NP-hard both in the number of experimental data points and in the number of inputs (Wiener) or outputs (Hammerstein) of the nonlinearity. These results provide new insight into the reasons underlying the high computational complexity of several recently proposed algorithms and point out the need for developing computationally tractable relaxations.     

Threshold superposition in morphological image analysis systems

It is shown that four composite morphological systems, namely morphological edge detection, peak/valley extraction, skeletonization, and shape-size distributions obey a weak linear superposition, called threshold-linear superposition. The output image signal or measurement from each system is shown to be the sum of outputs due to input binary images that result from thresholding the input gray-level image at all levels. These results are generalized to a vector space formulation, e.g. to any finite linear combination of simple morphological systems. Thus many such systems processing gray-level images are reduced to corresponding binary image processing systems, which are easier to analyze and implement     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

Asymptotic behaviours and stability of zero dynamics in nonlinear discrete-time systems using Taylor approach and multirate input

It is well known that the existence of unstable sampled zero dynamics is recognised as a major barrier in many control problems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable sampled zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than or equal to three is of minimum phase. In this paper, we show how an approximate sampled-data model can be obtained for nonlinear systems by the use of multirate input and hold such as a generalised sample hold function (GSHF) in order that discrete zero dynamics of the resulting model can be arbitrarily placed. Furthermore, the properties of sampled zero dynamics are studied and the conditions for ensuring the stability of sampling zero dynamics of the desired model are derived. The results presented here generalise well-known notion of sampling zero dynamics from the linear case to nonlinear systems, and GSHF can provide some advantages over ZOH or FROH in terms of stability of discrete system zero dynamics.     

A dynamic operability analysis approach for nonlinear processes

Current process operability indicators are mostly restricted to linear approximations of the process dynamics. Other operability analysis approaches that have the capability to include full nonlinear process models rely on mixed integer dynamic optimisation techniques which, in general, require large amount of computations. In this paper we propose a dynamic operability analysis approach for stable nonlinear processes that can be readily applied during process design and can be solved efficiently using a limited amount of computations. The process nonlinear dynamics are approximated by a series interconnection of static nonlinearities and linear dynamics, represented by the so-called Hammerstein–Wiener models. These type of models can often be obtained during process design where detailed steady-state nonlinear models are available, combined with some (usually limited) information on the process dynamics. Using an extended internal model control (IMC) framework, we investigate the interaction between the static nonlinearities and linear dynamics on the operability of the process. The framework extends the well-known equivalence between operability and invertibility of linear processes to nonlinear systems. In particular, by exploiting some results from the theory of passive systems we provide conditions that guarantee the existence of the inverse of the static nonlinearities. We show that the inverse can be attained inside a specific input/output region. This region imposes a constraint on the maximum magnitude of the signals that appear in the closed-loop and represents the effect of the static nonlinearities on the operability of the overall process. Dynamic operability is then quantified using a linear matrix inequality (LMI) optimisation approach that minimises a given performance criterion subject to the constraint imposed by the static nonlinearities.     

Identification of processes having direction-dependent responses,with gas-turbine engine applications

Many processes have dynamic responses which are dependent on the direction in which the process variable is moving. The effects of such nonlinear behaviour on the weighting function model of a process obtained by cross-correlation and on the difference equation model obtained by a generalised least-squares procedure are determined theoretically for a process with first-order dynamics perturbed with pseudo-random binary signals. The theory is confirmed by results from a hybrid computer simulation, and computer-simulated results for processes with second-order dynamics are also presented. The theory is used to explain discontinuities in weighting-function models of a gas-turbine engine in which the input-transducer had direction-dependent dynamic responses. Experimental work on a pilot-scale process is reported, and further examples from the literature are examined.     

Nonlinear model predictive control using parameter varying BP-ARX combination model

A novel back-propagation AutoRegressive with eXternal input (BP-ARX) combination model is constructed for model predictive control (MPC) of MIMO nonlinear systems, whose steady-state relation between inputs and outputs can be obtained. The BP neural network represents the steady-state relation, and the ARX model represents the linear dynamic relation between inputs and outputs of the nonlinear systems. The BP-ARX model is a global model and is identified offline, while the parameters of the ARX model are rescaled online according to BP neural network and operating data. Sequential quadratic programming is employed to solve the quadratic objective function online, and a shift coefficient is defined to constrain the effect time of the recursive least-squares algorithm. Thus, a parameter varying nonlinear MPC (PVNMPC) algorithm that responds quickly to large changes in system set-points and shows good dynamic performance when system outputs approach set-points is proposed. Simulation results in a multivariable stirred tank and a multivariable pH neutralisation process illustrate the applicability of the proposed method and comparisons of the control effect between PVNMPC and multivariable recursive generalised predictive controller are also performed.     

Uniquely connecting frequency domain representations of given order polynomial Wiener–Hammerstein systems

The notion of frequency response functions has been generalized to nonlinear systems in several ways. However, a relation between different approaches has not yet been established. In this paper, frequency domain representations for nonlinear systems are uniquely connected for a class of nonlinear systems. Specifically, by means of novel analytical results, the generalized frequency response function (GFRF) and the higher order sinusoidal input describing function (HOSIDF) for polynomial Wiener–Hammerstein systems are explicitly related, assuming the linear dynamics are known. Necessary and sufficient conditions for this relation to exist and results on the uniqueness and equivalence of the HOSIDF and GFRF are provided. Finally, this yields an efficient computational procedure for computing the GFRF from the HOSIDF and vice versa.     

Stability and stabilization of piecewise‐affine slab systems subject to Wiener process noise

The main contribution of this paper is to propose a convex formulation of sufficient conditions for both stability analysis and synthesis of stabilizing controllers for stochastic piecewise affine (PWA) systems with multiplicative noise. One of the main difficulties in PWA systems is the fact that the affine terms in the dynamics make it extremely difficult to formulate the synthesis problem as a convex optimization or even convex feasibility program. The presence of multiplicative noise modeled as a Wiener process adds an additional level of difficulty to the analysis and synthesis procedures. Sufficient conditions for stability of stochastic PWA slab systems in the mean square sense are developed first using a stochastic globally quadratic Lyapunov function. Second, PWA state feedback controllers are designed such that the closed‐loop system is stochastically exponentially mean square stable. The conditions for both stability and stabilization are formulated as LMIs, which can then be solved efficiently using currently available software packages. A numerical example shows the effectiveness of the approach. Copyright © 2013 John Wiley & Sons, Ltd.     

CLUSTERED BASED TAKAGI‐SUGENO NEURO‐FUZZY MODELING OF A MULTIVARIABLE NONLINEAR DYNAMIC SYSTEM

This research frame work investigates the application of a clustered based Neuro‐fuzzy system to nonlinear dynamic system modeling from a set of input‐output training patterns. It is concentrated on the modeling via Takagi‐Sugeno (T‐S) modeling technique and the employment of fuzzy clustering to generate suitable initial membership functions. Hence, such created initial memberships are then employed to construct suitable T‐S sub‐models. Furthermore, the T‐S fuzzy models have been validated and checked through the use of some standard model validation techniques (like the correlation functions). Compared to other well‐known approximation techniques such as artificial neural networks, fuzzy systems provide a more transparent representation of the system under study, which is mainly due to the possible linguistic interpretation in the form of rules. Such intelligent modeling scheme is very useful once making complicated systems linguistically transparent in terms of fuzzy if‐then rules. The developed T‐S Fuzzy modeling system has been then applied to model a nonlinear antenna dynamic system with two coupled inputs and outputs. Validation results have resulted in a very close antenna sub‐models of the original nonlinear antenna system. The suggested technique is very useful for development transparent linear control systems even for highly nonlinear dynamic systems.     

Distribution function tracking filter design using hybrid characteristic functions

A new tracking filtering algorithm for a class of multivariate dynamic stochastic systems is presented. The system is expressed by a set of time-varying discrete systems with non-Gaussian stochastic input and nonlinear output. A new concept, such as hybrid characteristic function, is introduced to describe the stochastic nature of the dynamic conditional estimation errors, where the key idea is to ensure the distribution of the conditional estimation error to follow a target distribution. For this purpose, the relationships between the hybrid characteristic functions of the multivariate stochastic input and the outputs, and the properties of the hybrid characteristic function, are established. A new performance index of the tracking filter is then constructed based on the form of the hybrid characteristic function of the conditional estimation error. An analytical solution, which guarantees the filter gain matrix to be an optimal one, is then obtained. A simulation case study is included to show the effectiveness of the proposed algorithm, and encouraging results have been obtained.     

A robust adaptive control method for Wiener nonlinear systems

This paper proposes a new robust adaptive control method for Wiener nonlinear systems with uncertain parameters. The considered Wiener systems are different from the previous ones in the sense that we consider nonlinear block approximation error, process noise, and measurement noise. The parameterization model is obtained based on the inverse of the nonlinear function block. The adaptive control method is derived from a modified criterion function that can overcome non‐minimum phase property of the linear subsystem. The parameter adaptation is performed by using a robust recursive least squares algorithm with a deadzone weighted factor. The control law compensates the model error by incorporating the unmodeled dynamics estimation. Theoretical analysis indicates that the closed‐loop system stability can be guaranteed under mild conditions. Numerical examples including an industrial problem are studied to validate the results. Copyright © 2016 John Wiley & Sons, Ltd.     

一般齐次T-S模糊系统的逼近性能

为剖析一般齐次T-S模糊系统的逼近性能,通过广泛总结常用模糊集的特点,明确定义了一种具有普遍意义的输入空间的一般模糊划分(GFP).基于输入采用GFP的一般齐次T-S模糊系统的解析结构,证明了该类一般齐次T-S模糊系统能够以任意精度逼近任意非线性函数,并得到了一个其作为通用逼近器的充分条件.作为GFP的一种退化,进一步研究了输入采用线性模糊划分(LFP)的一般齐次T-S模糊系统的一阶逼近性能.仿真实例验证了所得理论结果的有效性,并考察了充分条件的保守性.这为基于齐次T S模糊模型的复杂系统建模与控制提供了理论指导.     

Mechatronic systems identification using an impulse response recursive algorithm

A recursive identification algorithm is used to identify mechatronic systems using impulse response data. The algorithm is based on an auto regressive moving average (ARMA) model with a steepest descent method to minimize the least square error between the original and predicted outputs. Two mechatronic systems are tested: DC motor and gyroscope. Impulse voltage input is used to excite the system and the angular speed output is measured. In both systems, the torque and angular velocity outputs are dependent on the voltage and current inputs. This relationship is governed by characteristics such as inductance, resistance, moment of inertia, friction, load, and system constants. Once the ARMA model is constructed, the transfer function is realized. Then the input voltage is varied and the identified model results are compared with the original system. Simulation results using Simulink and experimental results using Labview with data acquisition card (DAQ) are presented. Results show that the recursive identification algorithm is able to identify the two systems with minimal error.     

Sliding mode dynamics in continuous feedback control for distributed discrete-event scheduling

A continuous feedback control approach for real-time scheduling of discrete events is presented in this paper motivated by the need for control theoretic techniques to analyze and design such systems in distributed manufacturing applications. These continuous feedback control systems exhibit highly nonlinear and discontinuous dynamics. Specifically, when the production demand in the manufacturing system exceeds the available resource capacity then the control system “chatters” and exhibits sliding modes. This sliding mode behavior is advantageously used in the scheduling application by allowing the system to visit different schedules within an infinitesimal region near the sliding surface. In this paper, an analytical model is developed to characterize the sliding mode dynamics. This model is then used to design controllers in the sliding mode domain to improve the effectiveness of the control system to “search” for schedules with good performance. Computational results indicate that the continuous feedback control approach can provide near-optimal schedules and that it is computationally efficient compared to existing scheduling techniques.     

Identification of Wiener systems with binary-valued output observations

This work is concerned with identification of Wiener systems whose outputs are measured by binary-valued sensors. The system consists of a linear FIR (finite impulse response) subsystem of known order, followed by a nonlinear function with a known parametrization structure. The parameters of both linear and nonlinear parts are unknown. Input design, identification algorithms, and their essential properties are presented under the assumptions that the distribution function of the noise is known and the nonlinearity is continuous and invertible. It is shown that under scaled periodic inputs, identification of Wiener systems can be decomposed into a finite number of core identification problems. The concept of joint identifiability of the core problem is introduced to capture the essential conditions under which the Wiener system can be identified with binary-valued observations. Under scaled full-rank conditions and joint identifiability, a strongly convergent algorithm is constructed. The algorithm is shown to be asymptotically efficient for the core identification problem, hence achieving asymptotic optimality in its convergence rate. For computational simplicity, recursive algorithms are also developed.     

Unknown input observer design and analysis for networked control systems

The insertion of communication networks in the feedback loops of control systems is a defining feature of modern control systems. These systems are often subject to unknown inputs in a form of disturbances, perturbations, or attacks. The objective of this paper is to design and analyse an observer for networked dynamical systems with unknown inputs. The network effect can be viewed as either a perturbation or time-delay to the exchanged signals. In this paper, we (1) review an unknown input observer (UIO) design for a non-networked system, (2) derive the networked unknown input observer (N_etUIO) dynamics, (3) design a N_etUIO such that the effect of higher delay order terms are nullified and (4) establish stability-guaranteeing bounds on the networked-induced time-delay and perturbation. The formulation and results derived in this paper can be generalised to scenarios and applications where the signals are perturbed due to a different source of perturbation or delay.