多输入多输出变量带误差模型的最坏情况频域辨识

本文将单输入单输出(SISO)变量带误差(EIV)模型的频域最坏情况辨识方法推广应用于多输入多输出 (MIMO)情况. 类似于SISO情况, 多输入多输出变量带误差(MIMO EIV)模型的辨识模型集合由估计的系统名义模型及 其最坏情况误差界描述. 所估计的系统名义模型表征为正规右图符号, 其最坏情况误差界具有可能的更少保守性, 可利 用EIV 模型的先验信息和后验信息由v-gap度量量化得到. 因此, 这种模型集合非常适合于后期利用Vinnicombe提出 的H1回路成形法设计鲁棒控制器. 最后, 利用一数值仿真实例验证所提出辨识方法的有效性.     

Algorithms for robust identification in ℋ∞ with nonuniformly spaced frequency response data

In this paper, first a two-stage robustly covergent identification algorithm in ℋ_∞ for nonuniformly spaced data is proposed. The worst-case error of the algorithm converges to zero faster than polynomial rates in the noise-free case when the identified system is an exponentially stable discrete-time system. The algorithm is characterized by a rational interpolation step with fixed poles at zero and infinity. Next, a minimax algorithm with better convergence properties is introduced. Sensitivity of the algorithms to small variations in the frequency values is also studied.     

L1鲁棒辨识中的一般中心估计算法及其特例

本文分析具了Ｌ１误差的线性时不变系统的最不利情况的辨识问题，对系统假设的先验信息未知系统的脉冲响应函数控制稳定，并且假定实验数据具有噪声干扰的，本文提出了一般中心估计算法并分析了最不利情况的误差界限，并进一步研究了在某些特殊情况下，中心估计的简单求法及其性质，本文所给算法和辨识结果是面向鲁棒控制的。     

Worst-case control-relevant identification

This paper introduces the reader to several recent developments in worst-case identification motivated by various issues of modelling of systems from data for the purpose of robust control design. Many aspects of identification in H_∞ and ?₁ are covered including algorithms, convergence and divergence results, worst-case estimation of uncertainty models, model validation and control relevancy issues.     

Set Membership identification of nonlinear systems

In the paper the problem of identifying nonlinear dynamic systems, described in nonlinear regression form, is considered, using finite and noise-corrupted measurements. Most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities. A key problem in these methods is the proper choice of the model class, typically realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, based on a Set Membership framework is presented, not requiring assumptions on the functional form of the regression function describing the relations between measured input and output, but assuming only some information on its regularity, given by bounds on its gradient. In this way, the problem of considering approximate functional forms is circumvented. Moreover, noise is assumed to be bounded, in contrast with statistical methods, which rely on assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc., whose validity may be difficult to test reliably and is lost in presence of approximate modeling. In this paper, necessary and sufficient conditions are given for the validation of the considered assumptions. An optimal interval estimate of the regression function is obtained, providing its uncertainty range for any assigned regressor values. The set estimate allows to derive an optimal identification algorithm, giving estimates with minimal guaranteed L_p error on the assigned domain of the regressors. The properties of the optimal estimate are investigated and its worst-case L_p identification error is evaluated. The presented approach is tested and compared with other nonlinear methods on the identification of a water heater, a mechanical system with input saturation and a vehicle with controlled suspensions.     

SM identification of approximating models for H∞ robust control

Set Membership (SM) H_∞ identification of mixed parametric and non-parametric models is investigated, aimed to estimate a low-order approximating model and an identification error, giving a measure of the unmodelled dynamics in a form well suited for H_∞ control methodologies. In particular, the problem of estimating the parameters of the parametric part and the H_∞ bound on the modelling error is solved using frequency domain data, supposing l_∞ bounded measurement errors and that the system to be identified is exponentially stable. The effectiveness of the proposed procedure is tested on some numerical examples, showing the advantages of the proposed methods over the existing non-parametric H_∞ identification approaches, in terms of lower model order and of tightness in the modelling error bounds. Copyright © 1999 John Wiley & Sons, Ltd.     

On the worst-case divergence of the least-squares algorithm

In this paper, we provide a _∞-norm lower bound on the worst-case identification error of least-squares estimation when using FIR model structures. This bound increases as a logarithmic function of model complexity and is valid for a wide class of inputs characterized as being quasi-stationary with covariance function falling off sufficiently quickly.     

Estimation of model quality

This paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems. This is a very large field. We therefore concentrate on approaches that have been motivated by the need for reliable models for control system design. This will involve a discussion of efforts that go under the titles of ‘estimation in ’, ‘worst-case estimation’, ‘estimation in ℓ₁’ and ‘stochastic embedding of undermodelling’. A central theme of this survey is to examine these new methods with reference to the classic bias/variance tradeoff in model structure selection.     

l_1系统辨识中的代数算法及其Worst－case误差

讨论单输入单输出、离散时不变因果系统的ｌ１系统辨识问题。首先提出基于代数方法的代数算法，并分析了该算法的特点；然后估计其Ｗｏｒｓｔ－ｃａｓｅ误差，并证明了该算法的收敛性；最后讨论了在某些特殊情况下该算法的相应形式。所给结果是面向鲁棒控制的。     

The least squares algorithm, parametric system identification and bounded noise

The least squares parametric system identification algorithm is analyzed assuming that the noise is a bounded signal. A bound on the worst-case parameter estimation error is derived. This bound shows that the worst-case parameter estimation error decreases to zero as the bound on the noise is decreased to zero.     

Risk sensitive identification of linear stochastic systems

Risk-sensitive identification of AR-processes was first considered in [12]. The purpose of this paper is to extend this original approach to ARMA-processes and even to multi-variable linear stochastic systems. We provide a new definition of a risk-sensitive identification criterion. For this we first consider a recursive identification procedure which is parameterized by a weight-matrix K acting on the stochastic gradient. Using the asymptotic theory of recursive estimation a suitably scaled version of the error process will be approximated by a stationary Gaussian process, see Chapter 4.5, Part II of [1]. The new risk sensitive criterion will be defined in terms of this associated stationary Gaussian process in a familiar manner via an exponential-quadratic cost. The main result of the paper is the minimization of the proposed new criterion with respect to the weight-matrix K over a feasible set E_K, see (22), where the cost function is known to be finite, Theorem 6.1. This results will then be extended to the case when minimization over a feasible set E°_K is considered, see (26), on the complement of which the cost function is known to be infinite, Theorem 6.1. The starting point of our analysis is an expression of the cost function given in LEQG-theory, in particular a result of [10]. A new expression for the cost function will be also given, using stochastic realization theory, as the mutual information rate between two stochastic processes.This research was supported in part by grants from the Swedish Research Council for Engineering Sciences (TFR), the Göran Gustafsson Foundation, the National Research Foundation of Hungary (OTKA) under Grants T015668, T16665, T020984 and T032932.     

Optimal input design for worst-case system identification in l1/l2/l∞

In this paper, we examine optimal sequences that generate worst-case parameters estimation errors in the l₁, l₂ and l_∞ norm context for algorithms identifying linear, time-invariant discrete-time, finite impulse response systems excited by bounded sequences and with l_∞ norm measurement error.     

Worst-case analysis of the least-squares method and related identification methods

We consider worst-case analysis of system identification under less restrictive assumptions on the noise than the l_∞ bounded error condition. It is shown that the least-squares method has a robust convergence property in l₂ identification, but lacks a corresponding property in l₁ identification (as well as in all other non-Hilbert space settings). The latter result is in stark contrast with typical results in asymptotic stochastic analysis of the least-squares method. Furthermore, it is shown that the Khintchine inequality is useful in the analysis of least l^p identification methods.     

Control of Selective Catalytic Reduction Systems Using Feedback Linearisation

This paper discusses the identification and control of a selective catalytic reduction (SCR) system. SCR after‐treatment systems form an important technology for reducing the nitrogen oxides, NO_x, produced by diesel engines. To be able to control the system, i.e. reducing the output NO_x, good models of the after‐treatment system are essential. In this paper a nonlinear black‐box model is identified using a recursive prediction error method. The nonlinear model is applied for design of a controller using feedback linearization techniques including an adaptive strategy. A linear quadratic Gaussian controller is used for the control of the linearized system. A total of 17 parameters were estimated for the nonlinear model. The results indicate that output NO_x control using feedback linearization based on a second order black‐box nonlinear model is feasible, provided that identification or adaptivity is used for model tuning. The latter requirement is a result of a study of the robustness. In summary, the paper indicates that significant improvements as compared to linear control can be obtained with the proposed strategy.     

Direct data-driven filter design for uncertain LTI systems with bounded noise

This paper investigates the filter design problem for linear time-invariant dynamic systems when no mathematical model is available, but a set of initial experiments can be performed where also the variable to be estimated is measured. Instead of using the initial experimental data to identify a model on the basis of which a filter is designed, these data are used to directly design a filter. Assuming norm-bounded disturbances and noises, a Set Membership formulation is followed. For classes of filters with exponentially decaying impulse response, approximating sets are determined that guarantee to contain all the solutions to the optimal filtering problem, where the aim is the minimization of the induced norm from disturbances to the estimation error. A method is proposed for designing almost-optimal linear filters with finite impulse response, whose worst-case filtering error is at most twice the lowest achievable one. In the H_∞ SISO case, an efficient technique is presented, that allows the evaluation of bounds on the guaranteed worst-case filtering error of the designed filter. Numerical examples illustrate the effectiveness of the proposed solution.     

The least-squares identification of FIR systems subject to worst-case noise

The least-squares identification of FIR systems is analyzed assuming that the noise is a bounded signal and the input signal is a pseudo-random binary sequence. A lower bound on the worst-case transfer function error shows that the least-square estimate of the transfer function diverges as the order of the FIR system is increased. This implies that, in the presence of the worst-case noise, the trade-off between the estimation error due to the disturbance and the bias error (due to unmodeled dynamics) is significantly different from the corresponding trade-off in the random error case: with a worst-case formulation, the model complexity should not increase indefinitely as the size of the data set increases.     

MODEL SET IDENTIFICATION IN FREQUENCY-DOMAIN AND ITS APPLICATION TO JOINT DESIGN WITH ROBUST CONTROL

In this paper, we propose a new model set identification method for robust control, which determines both nominal models and uncertainty bounds in frequency-domain using periodgrams obtained from experimental data. This method also gives less conservative model sets when we have more experimental data, which is one of the distinguished features compared with the existing model set identification methods. To this end, first, we construct a new noise model set in terms of periodgrams, which consists of hard-bounded (or deterministic) noises but takes account of a low correlation property of noise signals, simultaneously. Then, based on the noise model, we show how to compute the nominal models and the upper bounds of modeling error via convex optimization, which minimize given cost functions. Furthermore, by introducing a weighting function compatible with control performance criterion into the identification cost function, we consider a joint design method of the proposed model set identification and H_∞ control. Numerical examples show the effectiveness of the proposed method.     

Identification of switched linear systems via sparse optimization

The work presented in this paper is concerned with the identification of switched linear systems from input-output data. The main challenge with this problem is that the data are available only as a mixture of observations generated by a finite set of different interacting linear subsystems so that one does not know a priori which subsystem has generated which data. To overcome this difficulty, we present here a sparse optimization approach inspired by very recent developments from the community of compressed sensing. We formally pose the problem of identifying each submodel as a combinatorial ?₀ optimization problem. This is indeed an NP-hard problem which can interestingly, as shown by the recent literature, be relaxed into a (convex) ?₁-norm minimization problem. We present sufficient conditions for this relaxation to be exact. The whole identification procedure allows us to extract the parameter vectors (associated with the different subsystems) one after another without any prior clustering of the data according to their respective generating-submodels. Some simulation results are included to support the potentialities of the proposed method.     

On the time complexity of worst-case system identification

In this paper we treat a general worst-case system identification problem. This problem is worst-case with respect to both noise and system modeling uncertainty. We consider this problem under various a priori information structures. We determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst-case error bound. Our results are algorithm independent. We show that this minimum duration is prohibitively long. Based on our results, we suggest that worst-case (with respect to noise) system identification requires unrealistic amounts of experimental data