Direct and two-step methods for closed-loop identification: a comparison of asymptotic and finite data set performance

The asymptotic and finite data behavior of some closed-loop identification methods are investigated. It is shown that, when the output power is limited, closed-loop identification can generally identify models with smaller variance than open-loop identification. Several variations on some two-step identification methods are compared with the direct identification method. High order FIR models are used as process models to avoid bias issues arising from inadequate model structures for the processes. Comparisons are, therefore, made based on the variance of the identified process models both for asymptotic situations and for finite data sets. Process model bias resulting from improper selection of the noise and sensitivity function models is also investigated. In this context, the results support the use of direct identification methods on closed-loop data.     

Identifiability of linear stochastic systems operating under linear feedback

The identifiability of multiple input-multiple output stochastic systems operating in closed loop is considered for the case where the plant and the regulator are both linear and time-invariant. Two basic identification methods have been proposed for such systems: the joint input-output method, in which the input and output processes are modelled jointly as the output of a white noise driven system; and the direct method, in which a prediction error method is used on the input-output data as if the system were in open loop. Previously obtained identifiability results for the joint input-output method are extended to a number of new situations, including but extending beyond the identifiability results obtained with the direct method.     

Identification of discrete-time output error model for industrial processes with time delay subject to load disturbance

In this paper, a bias-eliminated output error model identification method is proposed for industrial processes with time delay subject to unknown load disturbance with deterministic dynamics. By viewing the output response arising from such load disturbance as a dynamic parameter for estimation, a recursive least-squares identification algorithm is developed in the discrete-time domain to estimate the linear model parameters together with the load disturbance response, while the integer delay parameter is derived by using a one-dimensional searching approach to minimize the output fitting error. An auxiliary model is constructed to realize consistent estimation of the model parameters against stochastic noise. Moreover, dual adaptive forgetting factors are introduced with tuning guidelines to improve the convergence rates of estimating the model parameters and the load disturbance response, respectively. The convergence of model parameter estimation is analyzed with a rigorous proof. Illustrative examples for open- and closed-loop identification are shown to demonstrate the effectiveness and merit of the proposed identification method.     

Identification of dynamic models in complex networks with prediction error methods—Basic methods for consistent module estimates

The problem of identifying dynamical models on the basis of measurement data is usually considered in a classical open-loop or closed-loop setting. In this paper, this problem is generalized to dynamical systems that operate in a complex interconnection structure and the objective is to consistently identify the dynamics of a particular module in the network. For a known interconnection structure it is shown that the classical prediction error methods for closed-loop identification can be generalized to provide consistent model estimates, under specified experimental circumstances. Two classes of methods considered in this paper are the direct method and the joint-IO method that rely on consistent noise models, and indirect methods that rely on external excitation signals like two-stage and IV methods. Graph theoretical tools are presented to verify the topological conditions under which the several methods lead to consistent module estimates.     

闭环系统的输出过采样辨识方法及其估计精度分析

传统闭环系统辨识方法的可辨识性受到参考设定信号和控制器结构的限制．提出了一种通过对输出过采样实现线性离散时间闭环系统辨识的方法，输出过采样提供了更多的系统结构信息，在传统辨识方法的可辨识条件不满足的情况下，仍能正确辨识系统参数，针对有色噪声干扰，分析其在不同过采样率下的估计精度，得出最优估计的过采样率计算方法．辨识方法实现简单、运算量小、估计精度高．仿真试验验证了其有效性．     

Closed-loop identification of multivariable processes with part of the inputs controlled

In many multivariable industrial processes a subset of the available input signals is being controlled. In this paper it is analysed in which sense the resulting partial closed-loop identification problem is actually a full closed-loop problem, or whether one can benefit from the presence of non-controlled inputs to simplify the identification problem. The analysis focuses on the bias properties of the plant estimate when applying the direct method of prediction error identification, and the possibilities to identify (parts of) the plant model without the need of simultaneously estimating full-order noise models.     

Stochastic parallel model adaptation: theory and applications toactive noise canceling, feedforward control, IIR filtering, andidentification

General stochastic parallel model adaptation problems that consist of an unknown linear time-invariant system and a partially or wholly tunable system connected in parallel, with a common input, are considered. The goal of adaptation is to tune the partially tunable system so that its output matches that of the unknown system, despite the presence of any disturbance which is stochastically uncorrelated with the input. The general formulation allows applications to adaptive feedforward control and adaptive active noise canceling with input contamination, in addition to output error identification and adaptive IIR filtering. It is shown that in all the applications, the goal of adaptation is met whenever a matching condition and a positive real condition are satisfied. A special case of the results therefore resolves the long-standing problem of the convergence and the unbiasedness of the output error identification scheme in the presence of colored noise. A simple general technique for analyzing the strong consistency of parameter estimation with projection is also developed     

On closed-loop system identification using polyspectral analysis given noisy input–output time-domain data

The problem of closed-loop system identification given noisy input–output measurements is considered. It is assumed that the closed-loop system operates under an external non-Gaussian input which is not measured. If the external input has non-vanishing integrated bispectrum (IB) and data IB is used for identification, then the various disturbances/noise processes affecting the system are assumed to be zero-mean stationary with vanishing IB. If the external input has non-vanishing integrated trispectrum (IT) and data IT is used for identification, then the various disturbances/noise processes affecting the system are assumed to be zero-mean stationary Gaussian. Noisy measurements of the (direct) input and output of the plant are assumed to be available. The closed-loop system must be stable but it is allowed to be unstable in open loop. Parametric modeling of the various noise sequences affecting the system is not needed. First the open-loop transfer function is estimated using the integrated polyspectrum and cross-polyspectrum of the time-domain input–output measurements. Then two existing techniques for parametric system identification given consistent estimates of the underlying transfer function, are exploited. The parameter estimators are strongly consistent. Asymptotic performance analysis is also carried out. A computer simulation example using an unstable open-loop system is presented to illustrate the proposed approach.     

Controller performance assessment by frequency domain techniques

A system identification based method for assessing the performance of closed-loop systems is proposed, utilizing measures which coincide naturally with classical and modern frequency domain design specifications. Standard robust control system design methodologies seek to maximize closed-loop performance, subject to strict robustness requirements and include specifications for bandwidth and peak magnitude of the sensitivity and complementary sensitivity functions. Estimates of these transfer functions can be obtained by exciting the reference input with a zero mean, pseudo random binary sequence, observing the process output and error response, and developing a closed-loop model. Performance assessment is based on the comparison between the observed frequency response characteristics and the design specifications. Selection of appropriate model structures, experiment design, and model validation which will ensure reasonable estimates of the closed-loop transfer functions are considered in this paper. A case study involving the performance assessment of a packed bed tubular reactor control system is presented.     

Identification of closed-loop linear systems via cyclic spectralanalysis given noisy input-output time-domain data

The problem of closed-loop system identification given noisy time-domain input-output measurements is considered. It is assumed that the various disturbances affecting the system are zero-mean stationary whereas the closed-loop system operates under an external cyclostationary input which is not measured. Noisy measurements of the (direct) input and output of the plant are assumed to be available. The closed-loop system must be stable, but it is allowed to be unstable in open loop. No knowledge about the linear-feedback mechanism is assumed. Two identification algorithms are investigated using cyclic spectral analysis of noisy input-output data. For both approaches, the open-loop transfer function is first estimated using the cyclic spectrum and cyclic cross-spectrum of the input-output data. These transfer function estimates are then used as “data” for the proposed algorithms. Both classes of parameter estimators are shown to be weakly consistent in any stationary noise (both at input as well as output). Asymptotic performance analysis of the proposed parameter estimators is also provided. Computer simulation examples are presented in support of the proposed approaches     

Predicting the output error of the suboptimal state estimator to improve the performance of the MPC-based artificial pancreas

The error of single step-ahead output prediction is the information traditionally used to correct the state estimate while exploiting the new measurement of the system output. However, its dynamics and statistical properties can be further studied and exploited in other ways. It is known that in the case of suboptimal state estimation, this output prediction error forms a correlated sequence, hence it can be effectively predicted in real time. Such a suboptimal scenario is typical in applications where the process noise model is not known or it is uncertain. Therefore, the paper deals with the problems of analytical and empirical modeling, identification, and prediction of the output error of the suboptimal state estimator for the sake of improving the output prediction accuracy and ultimately the performance of the model predictive control. The improvements are validated on an empirical model of type 1 diabetes within an in-silico experiment focused on glycemia prediction and implementation of the MPC-based artificial pancreas.     

A cumulant based algorithm for the identification of input-output quadratic systems

The identification of a special class of polynomial models is pursued in this paper. In particular a parameter estimation algorithm is developed for the identification of an input-output quadratic model excited by a zero mean white Gaussian input and with the output corrupted by additive measurement noise. Input-output crosscumulants up to the fifth order are employed and the identification problem of the unknown model parameters is reduced to the solution of successive triangular linear systems of equations that are solved at each step of the algorithm. Simulation studies are carried out and the proposed methodology is compared with two least squares type identification algorithms, the output error method and a combination of the instrumental variables and the output error approach. The proposed cumulant based algorithm and the output error method are tested with real data produced by a robotic manipulator.     

Realizations of BELS as WIV method in both direct and indirect closed-loop system identification

The bias eliminated least squares (BELS) method, which is known as efficient for unknown parameter estimation of transfer function in the correlated noise case, has been developed and applied effectively to the closed-loop system identification. In this paper, under the general settings, the realizations of the BELS method as a weighted instrumental variables (WIV) method in both direct and indirect closed-loop system identification are established through constructing an appropriate weighting matrix in the WIV method. The constructed structures are similar in both cases, which reveals that all the proof procedures of the two realizations are the same. Thus, the unified realizations of the BELS as the WIV method for the closed-loop system identification can be built. A simulation example is given to validate our theoretical analysis. Supported by the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 60625104), the Ministerial Foundation of China (Grant No. A2220060039), and the Fundamental Research Foundation of BIT (Grant No. 1010050320810)     

Consistency of noise covariance estimation in joint input–output closed-loop subspace identification with application in LQG benchmarking

The LQG trade-off curve has been used as a benchmark for control loop performance assessment. The subspace approach to estimating the LQG benchmark has been proposed in the literature which requires certain intermediate matrices in subspace identification as well as the covariance matrix of the noise. It is shown in this paper that many existing closed-loop identification methods do not give a consistent estimate of the noise covariance matrix. As a result, we propose an alternative subspace formulation for the joint input–output closed-loop identification for which the consistency of the required subspace matrices and noise covariance is guaranteed. Simulation studies and experimental results are provided to demonstrate the utility of the proposed method.     

Identification of probabilistic system uncertainty regions byexplicit evaluation of bias and variance errors

A procedure is developed for identification of probabilistic system uncertainty regions for a linear time-invariant system with unknown dynamics, on the basis of time sequences of input and output data. The classical framework is handled in which the system output is contaminated by a realization of a stationary stochastic process. Given minor and verifiable prior information on the system and the noise process, frequency response, pulse response, and step response confidence regions are constructed by explicitly evaluating the bias and variance errors of a linear regression estimate. In the model parametrizations, use is made of general forms of basis functions. Conservatism of the uncertainty regions is limited by focusing on direct computational solutions rather than on closed-form expressions. Using an instrumental variable method for identification, the procedure is suitable also for input-output data obtained from closed-loop experiments     

复杂多变量过程模型的闭环频域辨识

针对模型预测控制中模型辨识存在的问题,提出一种多变量过程闭环辨识方法.首先通过对多变量闭环系统正常运行产生的输入输出信号进行信号分解和频谱分析,得出多变量过程对象在重要频率段上的频率响应特性矩阵;然后采用最小二乘法,在幅值和相位两方面拟合一个二阶加纯滞后模型结构;最终获得一个多变量传递函数模型矩阵.仿真实验表明,该闭环辨识方法适用于广泛的多变量过程对象,具有很好的鲁棒性和精确性.     

Subspace model identification under load disturbance with unknown transient and periodic dynamics

To overcome the influence from load disturbance with unknown transient and periodic dynamics, as often encountered when performing identification tests in engineering applications, a bias-eliminated subspace model identification method is proposed to realize consistent estimation, which can be used for both open- and closed-loop systems. By decomposing the output response into disturbed and undisturbed components, an oblique projection is subtly introduced to eliminate the disturbance and noise impact so as to obtain unbiased estimation on the deterministic system state matrices, while the disturbance response dynamics could be estimated. In particular, a specific algorithm based on minimizing the output prediction error is given to find out the disturbance period if exists, such that the disturbance effect can be eliminated by the above projection regardless of the disturbance waveform and magnitude. A shift-invariant approach is then given to retrieve the deterministic state matrices. Consistent estimation on the deterministic system matrices is analyzed with a proof. A benmark example from the literature and an industrial injection molding process are used to demonstrate the effectiveness and merit of the proposed method.     

Virtual reference direct design method: an off-line approach todata-based control system design

This paper presents a direct model-reference approach to the off-line design of linear controllers, suited to deal with plants described by a single set of open-loop I/O measurements only. The method is direct inasmuch as the controller parameters are directly estimated with no preliminary identification of any model to describe the plant. The design can be carried out off-line and, in the present formulation, leads to a nonadaptive controller. The basic idea is that of interpreting the open-loop I/O measurements of the plant as closed-loop data produced by a “virtual” reference signal that can be computed by backpropagating the measured output of the plant through the reference model; thus, the controller design reduces to a standard identification problem, in which the “output” signal to be matched is the measured input of the plant. Both a deterministic (noise-free) and a stochastic setting, are considered     

A bias-eliminated least-squares method for continuous-time model identification of closed-loop systems

Schemes for system identification based on closed-loop experiments have attracted considerable interest lately. However, most of the existing approaches have been developed for discrete-time models. In this paper, the problem of continuoustime model identification is considered. A bias correction method without noise modelling associated with the Poisson moment functionals approach is presented for indirect identification of closed-loop systems. To illustrate the performances of the proposed method, the bias-eliminated least-squares algorithm is applied to the parameter estimation of a simulated system via Monte Carlo simulations.