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Approximation of the `True Unfalsified Set'

Citation for published version (APA):

Helvoort, van, J. J. M., Jager, de, A. G., & Steinbuch, M. (2006). Approximation of the `True Unfalsified Set'. In Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands (pp. 100-)

Document status and date: Published: 01/01/2006

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Approximation of the “True Unfalsified Set”

Jeroen van Helvoort, Bram de Jager and Maarten Steinbuch

Technische Universiteit Eindhoven

Mechanical Engineering, Control Systems Technology

PO Box 513, WH -1.125, 5600 MB Eindhoven, The Netherlands

Email: j.j.m.v.helvoort@tue.nl

1 Introduction

Typically, control design focusses on finding one value for the controller parameters, for a given problem. Unfalsified Control (Safonov and Tsao, 1997), however, tries to find the entire set of controller parameters, which are able to satisfy a given performance requirement.

Definition 1 (True Unfalsified Set) The set of controller

parameters, which satisfies the performance requirements (for a given plant)

In Ellipsoidal Unfalsified Control (EUC) (Van Helvoort, De Jager and Steinbuch, 2005), an outer-bounding approxi-mation is made of the True Unfalsified Set, using an easily computable ellipsoidal approximation. In a simulation, a comparison is shown between the True Unfalsified Set and the approximation thereof.

2 Simulation

Consider the plant y(t) = P (s) ∗ u(t), with

P (s) = 0.01s

2+ 0.4s + 100

s2+ 4s + 100 (1)

Also consider the controller

u(t) = 1 θ1  r(t) − θ2 s + 20 s + 2 ∗ y(t)  (2) Here, θ1and θ2are the controller parameters and r(t) is the

reference r(t) = sin(πt). Then the closed loop steady state response Y (s)/R(s) is given by Y (s) R(s) = 1 θ1P (s) 1 +θ2 θ1 s+20 s+2P (s) (3)

The performance requirement is given by 1 − ∆ ≤ Y (s)/R(s) ≤ 1 + ∆, with ∆ ≥ 0 a threshold on the

al-lowable tracking error.

2.1 True Unfalsified Set

The performance requirement can be rewritten as

1 − ∆ ≤ Y (s)/R(s) ≤ 1 + ∆ ⇔ (4)

−∆ ≤ Y (s)/R(s) − 1 ≤ ∆ ⇔ (5)

(Y (s)/R(s) − 1)0(Y (s)/R(s) − 1) ≤ ∆2 (6)

The performance requirement is evaluated at s = jπ (since

r(t) = sin(πt)). From (3) and (6), it can be seen that the

True Unfalsified Set is an ellipsoid in the controller parame-ter space, with size dependent on ∆.

2.2 Ellipsoidal Unfalsified Control

EUC is a data-driven, model-free controller design method, which recursively falsifies controller parameter sets that fail to meet the specified performance requirement. An outer-bounding ellipsoidal approximation is used to describe the set of unfalsified controllers.

2.3 Results i i

“tempimage˙temp” — 2006/1/3 — 10:56 — page 1 — #1

i i i i i i 0.98 0.99 1 1.01 0.026 0.028 0.03 θ2 θ1 ∆ = 0.01 i i

“tempimage˙temp” — 2005/12/22 — 15:08 — page 1 — #1

i i i i i i 0.9954 0.9956 0.02792 0.02794 0.02796 θ2 θ1 ∆ = 0.0001

True Unfalsified Set (grey, solid ellipsoid) and the outer bounding approximation, computed by EUC (black, open ellipsoid), for two values of ∆.

The distinction between the two sets originates from the outer-bounding approach of the EUC algorithm.

3 Remark

In EUC, the approximation of the True Unfalsified Set is used to select suitable controller parameters for implemen-tation.

References

Van Helvoort, J., De Jager, B. and Steinbuch, M. (2005), Unfalsified control using an ellipsoidal unfalsified region applied to a motion system, in ‘Proc. IFAC World Congress’, Prague, Czech Republic, pp. CD–ROM.

Safonov, M. and Tsao, T.-C. (1997), ‘The unfalsified con-trol concept and learning’, IEEE Trans. Automatic Concon-trol

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