On the equivalence of least costly and traditional experiment design for control

49th IEEE Conference on Decision and Control (CDC)

Hjalmarsson

Stoica

2010

Abstract-This paper studies optimal input excitation design for parametric frequency response estimation. The objective is to minimize the uncertainty of functions of the frequency response estimate at a specified frequency ω while limiting the power of the input signal. We focus on least-squares estimation of Finite Impulse Response (FIR) models and minimum variance input design. The optimal input problem is formulated as a convex optimization problem (semi-definite program) in the second order statistics of the input signal. We analytically characterize the optimal solution for first order FIR systems with two parameters, and perform a numerical study to obtain insights in the optimal solution for higher order models. The optimal solution is compared to the case when a sinusoidal input signal, with frequency ω and amplitude that gives the same accuracy as the optimal input, is used as excitation signal. For first order FIR models with two parameters the input signal power can be reduced at best by a factor of two by using the optimal input signal compared with such a sinusoidal input signal. Numerical studies show that less is in general gained for higher order systems, for which a sinusoidal input signal with frequency ω often is optimal. We consider estimation of the H ∞ -norm of a stable linear system, that is the maximum of the absolute value of the corresponding frequency response. An asymptotic error variance expression for H ∞ -norm estimates is derived.

“…The optimal solution will satisfy r 0 = γ. As shown in [10] this is closely related to the least costly identification optimization problem, [1], P2 : min…”

Section: Propositionmentioning

confidence: 99%

“…It is possible to re-scale P2 to allow for other variance values. Also scaling the solution of P2 to obtain r 0 = γ gives the solution to problem P1, see [10].…”

Section: Propositionmentioning

confidence: 99%

See 1 more Smart Citation

On optimal input signal design for frequency response estimation

49th IEEE Conference on Decision and Control (CDC)

Hjalmarsson

Stoica

2010

“…As shown in [20] many classical methods for input design can be reformulated as convex optimization problems. The unifying framework for optimal input design for system identification presented in [11] provides a transparent way to connect the performance of the estimated model in the intended application to the system identification experiment conditions.…”

Section: Introductionmentioning

confidence: 99%

On optimal input signal design for identification of output error models

IEEE Conference on Decision and Control and European Control Conference

Annergren

Rojas

2011

Self Cite

Abstract-This paper extends recent results on minimum variance input signal design for identification of Finite Impulse Response (FIR) models to the Output Error (OE) system identification case. The idea is to use "the useful input parametrization" for OE models proposed by Stoica and Söderstrom (1982). The advantage of this parametrization is that the Toeplitz covariance matrix structure instrumental in the FIR analysis also holds for this OE model input representation after a transformation. However, an issue is that the corresponding minimum variance cost function for the OE case will be more complicated than for FIR models, and that the dimension of the optimization problem will be of one degree higher than for the corresponding FIR case. The proposed OE framework is applied to minimum variance input signal design in system identification frequency response estimation and model predictive control. The results are illustrated by numerical examples. The unifying framework for optimal input design for system identification presented in [11] provides a transparent way to connect the performance of the estimated model in the intended application to the system identification experiment conditions. The identification objective is to guarantee, with a given probability, that the estimated model will be in the set of models that satisfies given specifications. The papers [23], [24] have analyzed and evaluated this framework, in detail, by using the fact that the corresponding input minimum variance optimization problem has a very simple structure for Finite Impulse Response (FIR) models. The problem is more involved for Output Error (OE), even if it is possible to find the optimal input for such models using numerical convex optimization. However, by using the input parametrization proposed in [22], several structural FIR results can be extended to identification of OE, and also BoxJenkins, models. This parametrization also allows for a direct implementation of the corresponding SDP/LMI optimization problem. I. INTRODUCTIONThe outline of the paper is as follows: In Section II application motivated system identification and optimal input design are introduced. Section III considers the output error model case in detail. Frequency response estimation is studied in Section IV, while Section V and VI deals with control application examples. Section VII concludes the paper.

“…System identification for control has been an active area of research for many years, see [15] for an overview. Recently, there has been extensive progress in optimal experiment design, [22], [5], [18], [3], [17], [26], [24], [6], [23], [1]. The key idea is to convexify this type of optimization problems, see [12], [16], [2], [4], [14], [8].…”

Section: Introductionmentioning

confidence: 99%

On optimal input design in system identification for control

49th IEEE Conference on Decision and Control (CDC)

Hjalmarsson

Annergren

2010

Abstract-This paper considers a recently proposed framework for experiment design in system identification for control. We study model based control design methods, such as Model Predictive Control, where the model is obtained by means of a prediction error system identification method. The degradation in control performance due to uncertainty in the model estimate is specified by an application cost function. The objective is to find a minimum variance input signal, to be used in system identification experiment, such that the control application specification is guaranteed with a given probability when using the estimated model in the control design. We provide insight in the potentials of this approach by finite impulse response model examples, for which it is possible to analytically solve the optimal input problem. The examples show how the control specifications directly affect the excitation conditions in the system identification experiment.