Farokhi KTH Short Course Report

1 Designing Robust Controller for Magnetic Suspension System as a Project for Robust Control & Convex Optimization Course Farhad Farokhi I. I NTRODUCTION Thi s rep ort deals with the pro ble m of des ign ing robust controller for a magnetic suspension system. Feedback control is esse ntia l for magnetic suspens ion systems beca use they are unsta ble systems natur ally . The math emat ical model of the se sys tems ha ve va rio us uncert ain tie s suc h as par ame- ters identification errors, unmodeled dynamics, and neglected nonlinearities. Because of these uncertainties in modeling of the system, the controller is required to have robustness for stability and performance. The rest of this report is organized as follows. In section II, the math emat ical modelin g of the syste m with uncertai ntie s are discussed. In section III, H ∞ and μ-synthesis controller designing appr oache s are presented and simu lati on resu lts are given. Finally, some concluding remarks are presented in section IV. TABLE I PARAMETERS OF ELECTROMAGNETIC SUSPENSION SYSTEM Pa rame ter Minimum V al ue Nomi na l Value Maxi mum Val ue R (Ω) 25.6 26.6 27.6 L (H) 0.518 0.558 0.608 M (Kg) 1.70 1.75 1.80 I (A) 1.08 1.15 1.23 x 0 (m) 4.70 × 10 −3 5.00 × 10 −3 5.30 × 10 −3 k (Nm 2 /A 2 ) 2.64 × 10 −4 2.84 × 10 −4 2.08 × 10 −4 E (V) 27.6 30.6 33.9 II. MODEL STRUCTURE A. Electr omagnetic Suspen sion System The structure of the electromagnetic suspension system is shown schematically in figure 1. The objective of our control experiments is to suspend the ball made of iron with attractive electromagnetic forces. It should be noted that this system is essentially unstable. The nominal value, the minimum value and the maximum value of different parameters of the system are mentioned in table I. B. Model of Electro magnetic Suspensio n System My purpose in this section of the report is to introduce the model of the system in different working situations. We will Farhad Farokhi is a PhD student in Automatic Control Laboratory, School of Electrical Engineering, The Royal Institute of Technology (KTH), Stockholm, Sweden. (e-mails: [email protected]) Swedish Personal Number:870515-6795 Fig. 1. Schematic di agram of the sys tem [1 ]. 10 −2 10 0 10 2 10 4 10 −4 10 −3 10 −2 10 −1 10 0 ω ( P ( j ω ) P ( j ω ) ) / P ( j ω ) Multiplicative Uncertainty Model Fig. 2. (P ( jω ) − P nom ( jω ))/P nom ( jω ) and W m ( jω ) versus ω for one hundred random parameters. use two different model structures for the system shown in figure 1. These two models are finite dimensional, linear, and time-invariant of the following state-space representation ˙ z = Az + Bu, y = Cz , (1) where z = [ x ˙ x i ] T , u = e, y = z. (2) For the defi nition of x, i, e chec k the Fi g. 1. Si nc e the beha vior of elec tromagne tic force is nonli near , we then use the linearization method around the operating point. 1) Idea l Math emati cal Model - Const ant Inducta nce: The following assumptions are used in the procedure of modeling the electromagnetic suspension system.

Farokhi KTH Short Course Report

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