A Numerical Study of Time-Domain Model Validation for Robust Control Sunil L. Kukreja † , Johan L ¨ ofberg † and Benoit Boulet ‡ † Div. of Automatic Control, Dept. of Elect. Eng., Link¨ opings universitet ‡ Dept. of Elect. Eng., CIM, McGill University, E-mail: boulet@cim.mcgill.ca Abstract In this paper, we investigate a discrete-time approach to un- certainty modeling for robust control. We show via simula- tions of one LTI system and experimental data from a one- tank test-bed that a current technique for time-domain uncer- tainty modeling leads to a feasible linear program. Hence, it is useful for developing robust control solutions. 1 Introduction Model validation is an important step in developing strate- gies for robust control. This step is typically preceded by system identification, as well as, system analysis and physi- cal modeling. Model validation is concerned with assessing whether a given nominal model can reproduce data from a plant, collected after some initial experiments to obtain esti- mation data [1]. The model validation problem is really one of model invalidation since a given model can only be said to be not invalidated with the current evidence. Future evi- dence may invalidate the model. The motivation for this study is to investigate whether a time-domain approach to uncertainty modeling for linear time-invariant (LTI) and linear time-varying (LTV) systems developed by Poola et al. [4] can be implemented for robust control applications. While the work provided theoretical development for their approach to uncertainty modeling, it did not provide any simulated example or experimental ap- plication to verify their theory is numerically tractable. To date, the authors are not aware of any simulation study or application to experimental data of Poola et al.’s [4] work. However, a sampled-data approach to model validation de- veloped in [2] was successfully tested by simulation [2] and experimental data [5]. As such, in this paper, we investigate the feasibility of the approach of [4] to experimental data by first studying the behavior of this technique on one simulated causal, LTI system for 1 control and applying this approach to experimental data from a one-tank test-bed. Our results show, this approach to uncertainty modeling pro- vides a numerically tractable solution for noisy simulated data as well as for experimental data. 2 Problem Statement We considered the class of uncertainty models described by an output multiplicative uncertainty, as yk G 0 G 0 W ∆ uk dk ; d is in a convex set (1) where uk and yk are the input-output measurements, G 0 a causal, LTI nominal model, W a causal, LTI normalized uncertainty filter, ∆ 1 a normalized system uncertainty and d D : d ∞ 0 M 1: d ∞ 1 , a noise or disturbance acting on the system. The uncertainty fil- ter W is selected so that Wj ω is an upper bound on G p jω G 0 jω G 0 jω where G p is the perturbed model. Although ∆ is assumed norm-bounded in 1 , this uncertainty struc- ture was selected because it may be capable of describing model mis-specification due to unmodeled dynamics, noise and other disturbances. [6]. Therefore, the model validation problem for this class of uncertainty is stated as [4]: Given input-output sequences u u 0 u 1 u N 1 ℜ and y y 0 y 1 y N 1 ℜ there exists a stable causal, LTI operator ∆ with ∆ i∞ γ where i∞ denotes ∞ induced (2) such that ∆ ˆ u 0 ˆ u 1 ˆ u N 1 ˆ y 0 ˆ y 1 ˆ y N 1 if and only if the following linear programming problem is feasible [4] LP ˆ u ˆ y q γ (3) where ˆ uk π N WG 0 uk ,ˆ yk yk π N G 0 uk and qk the system errors which encompass both the effect of model uncertainty and noise in π N D and π N is the truncation oper- ator keeping N data points [4]. Since the uncertainty filter is normalized the criterion for assessing model invalidation is γ 1. 3 Procedures The efficacy of this model invalidation algorithm was as- sessed using (i) noise corrupted data from a simulated second-order system but nominal model identified as a first- order model and (ii) experimental data from a one-tank test- bed. The simulated system, H 1 z , and nominal model, ˆ H 1 z , used in study are H 1 z 02z 1 0 08z 2 1 0 42z 1 0 32z 2 ˆ H 1 z b 1 z 1 1 a 1 z 1 (4) 0-7803-7896-2/03/$17.00 ©2003 IEEE 3814 Proceedings of the American Control Conference Denver, Colorado June 4-6, 2003