Journal of Process Control 21 (2011) 111–118 Contents lists available at ScienceDirect Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont Optimal input design for parameter estimation in a single and double tank system through direct control of parametric output sensitivities Hassan Akbari Chianeh a , J.D. Stigter b , Karel J. Keesman a,∗ a Systems & Control Group, Wageningen University, PO Box 17, 6700 AA Wageningen, The Netherlands b Biometris, Wageningen University, PO Box 100, 6700 AC Wageningen, The Netherlands article info Article history: Received 22 December 2009 Received in revised form 21 September 2010 Accepted 20 October 2010 Keywords: Optimal input design Parameter estimation Bernoulli’s law Sensitivity abstract In this paper the traditional and well-known problem of optimal input design for parameter estimation is considered. In particular, the focus is on input design for the estimation of the flow exponent present in Bernoulli’s law. The theory will be applied to a water tank system with a controlled inflow and free outflow. The problem is formulated as follows: Given the model structure (f, g), which is assumed to be affine in the input, and the specific parameter of interest (), find a feedback law that maximizes the sensitivity of the model output to the parameter under different flow conditions in the water tank. The input design problem is solved analytically. The solution to this problem is used to estimate the parameter of interest with a minimal variance. Real-world experimental results are presented and compared with theoretical solutions. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction The problem of optimal input design (OID) has received ample attention in the identification literature. It is one of the classical identification problems [1,2] that seeks to address an essential question for the model developer, namely whether it is possible to design an experiment in such a way that the parameters in the model structure can be estimated with minimum variance. More specifically, the question that needs to be addressed is how to design an input signal that minimizes (c.q. maximizes) an a pri- ori chosen norm of the Fisher information matrix associated with the specific experimental setup. In their early work, Kalaba and Spingarn focused on the dynamic behaviour of parametric output sensitivities as a means to arrive at better experiments (read input signals) that yield more informa- tion on the values of the parameters in the model [3,4]. Because the analysis is rather complex, only low dimensional models were analyzed. Approximately one decade later, Munack discussed the OID problem for more complex models and approached the prob- lem numerically by solving a complex optimization problem that includes as a goal function the D-criterion of the Fisher Informa- tion Matrix [5]. In their work, optimal input designs for estimation of the maximum specific growth rate and Monod-constant in a ∗ Corresponding author. Tel.: +31 317 483780; fax: +31 317 484957. E-mail address: karel.keesman@wur.nl (K.J. Keesman). Michaelis–Menten type of model for microbial growth in a biore- actor were calculated. Versyck et al. [6] studied a similar case and presented an optimal solution for a problem that includes the mod- ified E criterion in the goal function. The solution encompasses an input signal for a maximal de-correlation of the parametric uncer- tainty in the estimate. Extension of this problem in a numerical setting can be found in [7]. Numerical solutions were also presented in [8] when solving an adaptive optimal input design problem. The quoted literature in the above, and related to the field of process control, is only a small number of references on the subject of combined optimal input design–parameter estimation. This list can be extended easily, especially when including other disciplines, see e.g. [9–15]. In addition, recent progress in the literature demon- strates that the OID problem and the related problem of structural identifiability (see [16] for an application) still is a very topical issue. Current literature shows a continuous interest in issues related to especially practical identification that always arise when estimat- ing parameters from real-life data, see e.g. [17a] for an overview. This demonstrates that the issues presented in the following are indeed relevant. In the recent past, several analytical solutions have been derived to a question that is very much related to the familiar OID problem, i.e. can we find an optimal control strategy u * (t) that, given the cur- rent state of the system, optimally controls the input-state-output dynamics in such a way that the parametric output sensitivities with respect to one (or several) specific parameters contain as much information as possible on the value(s) of these parameter(s) in the model structure?’ In previous work, it has been demonstrated 0959-1524/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2010.10.012