Robust optimal experiment design for system identification

This paper develops the idea of min-max robust experiment design for dynamic system identification. The idea of min-max experiment design has been explored in the statistics literature. However, the technique is virtually unknown by the engineering community and, accordingly, there has been little prior work on examining its properties when applied to dynamic system identification. This paper initiates an exploration of these ideas. The paper considers linear systems with energy (or power) bounded inputs. We assume that the parameters lie in a given compact set and optimise the worst case over this set. We also provide a detailed analysis of the solution for an illustrative one parameter example and propose a convex optimisation algorithm that can be applied more generally to a discretised approximation to the design problem. We also examine the role played by different design criteria and present a simulation example illustrating the merits of the proposed approach.     

Dynamic input signal design for the identification of constrained systems

In current model predictive control (MPC) practise, the accuracy of the model from system identification is often the crucial factor for the final success. This makes the input signal design a very important step in MPC applications. Because the identification task should move the outputs within some constraints, a constrained design method is needed. Previous constrained signal design methods are usually based on the steady-state gain matrix of a process. Ignoring the system dynamics makes these designs either too conservative when the dynamics are overdamped or allows them to violate the output constraints in the case of underdamped dynamics. In order to address these problems, a new design method making use of the prior approximate estimate of the system dynamics is proposed in this paper. Furthermore, an iterative method of signal design for identification experiments is proposed, and a criterion is defined to compare the accuracy of two successive dynamic models. An example on a subsystem of the challenging Tennessee Eastman process is used to prove the effectiveness of the proposed method.     

On the estimation of transfer functions

This paper treats the close conceptual relationships between basic approaches to the estimation of transfer functions of linear systems. The classical methods of frequency and spectral analysis are shown to be related to the well-known time domain methods of prediction error type via a common “empirical transfer function estimate”. Asymptotic properties of the estimates obtained by the respective methods are also described and discussed. An important feature that is displayed by this treatment is a frequency domain weighting function that determines the distribution of bias in case the true system cannot be exactly described within the chosen model set. The choice of this weighting function is made in terms of noise models for time-domain methods. The noise model thus has a dual character from the system approximation point of view.     

For model-based control design, closed-loop identification gives better performance

We compare open loop versus closed loop identification when the identified model is used for control design, and when the system itself belongs to the model class, so that only variance errors are relevant. Our measure of controller performance (which is used as our design criterion for identification) is the variance of the error between the output of the ideal closed loop system (with the ideal controller) and that of the actual closed loop system (with the controller computed from the identified model). Under those conditions, we show that, when the controller is a smooth function of the input-output dynamics and the disturbance spectrum, the best controller performance is achieved by performing the identification in closed loop with an operating controller that we characterize. For minimum variance and model reference control design criteria, we show that this ‘optimal operating controller for identification’ is the ideal controller. This then leads to a suboptimal but feasible iterative scheme.     

Identification of multi-input systems: variance analysis and input design issues

This paper examines the identification of multi-input systems. Motivated by an experiment design problem (should one excite the various inputs simultaneously or separately), we examine the effect of an additional input on the variance of the estimated coefficients of parametrized rational transfer function models, with special emphasis on the commonly used FIR, ARX, ARMAX, OE and BJ model structures. We first show that, for model structures that have common parameters in the input-output and noise models (e.g. ARMAX), any additional input contributes to a reduction of the covariance of all parameter estimates. We then show that the accuracy improvement extends beyond the case of common parameters in all transfer functions, and we show exactly which parameter estimates are improved when a new input is added. We also conclude that it is always better to excite all inputs simultaneously.     

Frequency domain versus time domain methods in system identification

The basic techniques of time domain and frequency domain identification, including the maximum entropy methods, are outlined. Then connections and distinctions between the methods are explored. This includes the derivation of some analytic relationships together with a discussion of the restrictions inherent in choosing certain methods, and their ease of use in different experimental conditions. It is concluded that these are complementary rather than competing techniques.     

A moving horizon approach to input design for closed loop identification

The identification of high fidelity models is a critical element in the implementation of high performance model predictive control (MPC) applications in the industry. These controllers can vary in size with input–ouput dimensions ranging from 5 × 10 to 50 × 100. Identifying models of this scale accurately is a time consuming and demanding exercise. We present a novel approach wherein an information rich test signal is generated in closed loop by maximizing the MPC objective, as opposed to minimization that is done in the standard controller. We show that the proposed input design approach is similar to T-optimal (trace optimal) experiment design method. Our approach automatically accounts for the input and output constraints and is implemented in a moving horizon manner. It is demonstrated through simulation examples on both well and ill-conditioned processes.     

Closed loop experiment design for linear time invariant dynamical systems via LMIs

All stationary experimental conditions corresponding to a discrete-time linear time-invariant causal internally stable closed loop with real rational system and feedback controller are characterized using the Youla-Kucera parametrization. Finite dimensional parametrizations of the input spectrum and the Youla-Kucera parameter allow a wide range of closed loop experiment design problems, based on the asymptotic (in the sample size) covariance matrix for the estimated parameters, to be recast as computationally tractable convex optimization problems such as semi-definite programs. In particular, for Box-Jenkins models, a finite dimensional parametrization is provided which is able to generate all possible asymptotic covariance matrices. As a special case, the very common situation of a fixed controller during the identification experiment can be handled and optimal reference signal spectra can be computed subject to closed loop signal constraints. Finally, a brief numerical comparison with closed loop experiment design based on a high model order variance expression is presented.     

Input design for structured nonlinear system identification

This paper is concerned with the input design problem for a class of structured nonlinear models. This class contains models described by an interconnection of known linear dynamic systems and unknown static nonlinearities. Many widely used model structures are included in this class. The model class considered naturally accommodates a priori knowledge in terms of signal interconnections. Under certain structural conditions, the identification problem for this model class reduces to standard least squares. We treat the input design problem in this situation.An expression for the expected estimate variance is derived. A method for synthesizing an informative input sequence that minimizes an upper bound on this variance is developed. This reduces to a convex optimization problem. Features of the solution include parameterization of the expected estimate variance by the input distribution, and a graph-based method for input generation.     

Control relevant identification for robust optimal control

This paper presents an approach to designing the input signal for an identification experiment, in which the process model estimate is to be used to formulate and solve for a robust (in a worst case sense) optimal controller. The input signal is designed to contain the information that is relevant for the end use of the model, that is for control purposes. The proposed approach uses sensitivity analysis to determine the input signal frequencies that are important with respect to a certain measure of achievable controller performance in conjunction with a frequency sampling filter model of the process. Based on the sensitivity analysis, an iterative experimental design methodology is suggested.     

Non-parametric methods of system identification

The non-parametric identification of systems in terms of unparametrized representations such as the impulse response and frequency response is considered. Basic approaches are outlined in a retrospective setting as are the relationships between non-parametric and parametric identification models. The article concludes with an assessment of non-parametric methods which is conducted in terms of typical industrial applications.     

Optimal input design for an AR model with output constraints

The output power constraint problem of optimal input design for parameter estimation for an autoregressive model is considered. A simple method to obtain an optimal design by solving two sets of $\text{p}\text{2-}\text{linear}$ simultaneous equations and a polynomial equation of $\text{p}\text{2}\text{th}$ order is proposed and two nontrivial examples are given to illustrate this methodology.     

Input signal design for identification of max-plus-linear systems

Max-plus-linear systems efficiently describe the dynamics of event time sequences of a class of discrete event systems. The present contribution addresses the problem of designing adequate input signals for state space identification of max-plus-linear systems. It is shown that the input signal design problem can be rewritten as a set of upper bound constraints and therefore solved using an existing algorithm. This input signal design method allows to incorporate additional objectives and constraints, e.g. minimum or maximum input event separation, time order constraints, etc., which are desirable or even required for the input signals and the resulting process behavior.     

最优实验设计的改进随机进化算法

详细阐述构造最优实验设计的原始随机进化算法,并在原始算法的基础上,拓展广度搜索,改进深度搜索,以提高最优实验设计的计算速度。通过不同规模和不同优化准则的拉丁超立方体最优实验设计,验证改进算法的应用效果。算例分析表明,改进算法能够比原始算法节省约30%～60%的机时完成最优实验设计,而且改进算法对应于优化准则的最优值与原始算法最优值的差别仅为1%～3%。可见,改进算法能够兼顾最优实验设计的计算时间和优化质量,明显提高最优实验设计的构造效率。     

Objective functions for optimal building design

In this paper, the authors describe 13 investor motivations for initiating a project. Quantitative criteria to measure success in achieving these motivations are listed. One of the these criteria, net present value, is selected as most capable of reflecting key investor motivations. This measure is manipulated to yield a life-cycle costing relationship which can be used as an objective function in the design decision-making process. It is shown that the life-cycle cost equation in its most general form embodies within it a family of commonly used objective functions which describe financial, technical performance and user criteria. Results from a case study in which different objective functions were used are presented.     

Least costly identification experiment for control

All approaches to optimal experiment design for control have so far focused on deriving an input signal (or input signal spectrum) that minimizes some control-oriented measure of plant/model mismatch between the nominal closed-loop system and the actual closed-loop system, typically under a constraint on the total input power. In practical terms, this amounts to finding the (constrained) input signal that minimizes a measure of a control-oriented model uncertainty set. Here we address the experiment design problem from a “dual” point of view and in a closed-loop setting: given a maximum allowable control-oriented model uncertainty measure compatible with our robust control specifications, what is the cheapest identification experiment that will give us an uncertainty set that is within the required bounds? The identification cost can be measured by either the experiment time, the performance degradation during experimentation due to the added excitation signal, or a combination of both. Our results are presented for the situation where the control objective is disturbance rejection only.     

Multiple control input design for identification of light aircraft

This paper establishes a design technique for control input histories resulting in higher confidence levels in the estimate of lateral stability and control derivatives of light aircraft, and which cause dynamic motion favorable to reductions in numerical divergence tendencies in the identification algorithms. A combined aileron and rudder input design is demonstrated for the Cessna 172 aircraft.     

Optimal input design for set-membership identification of Hammerstein models

In this paper the problem of optimal input design for the identification of Hammerstein models is considered under the assumption that the linear dynamic part of the model is a FIR and that lower and upper bounds are available for the additive measurement errors. The parameters of the Hammerstein model can then be estimated via the identification of a linearized augmented Hammerstein model . External approximations of the feasible intervals for the parameters of the original Hammerstein models are then derived (which may correspond to the actual feasible intervals). This paper deals with the design of input sequences minimizing parameter uncertainty for the linearized augmented Hammerstein model . Some new results are also reported about optimal input design for polynomial non-linear blocks, that may be part of Hammerstein models.