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By Joao P. Hespanha

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1. 3. This is done using the ARX model that will be constructed below. 1 are related by αm z m + αm−1 z m−1 + · · · + α1 z + α0 Y (z) = H(z) = , U (z) z n + βn−1 z n−1 + · · · + β1 z + β0 25 26 Jo˜ao P. Hespanha y(k) u(k) H(z) =? 1. System identification from input/output experimental data where the αi and the βi denote the coefficients of the numerator and denominator polynomials of H(z). Multiplying the numerator and denominator of H(z) by z −n , we obtain the transfer function expressed in negative powers of z: αm z −n+m + αm−1 z −n+m−1 + · · · + α1 z −n+1 + α0 z −n Y (z) = U (z) 1 + βn−1 z −1 + · · · + β1 z −n+1 + β0 z −n and therefore Y (z) + βn−1 z −1 Y (z) + · · · + β1 z −n+1 Y (z) + β0 z −n Y (z) = αm z −n+m U (z) + αm−1 z −n+m−1 U (z) + · · · + α1 z −n+1 U (z) + α0 z −n U (z).

Important: write MATLAB scripts to automate these procedures. These scripts should take as inputs the simulation data u(k), y(k), and the integers m, n. 3. 01. For the best values of m and n determined above, plot the SSE vs. the step size. Which step-size leads to the best model? Exercise 7 (Model order). Use the data provided to identify the transfer function of the system. Use the procedure outlined above to determine the order of the numerator and denominator polynomials. Plot the largest and smallest singular value of R and the SSE as a function of n.

1. Spring-mass-damper system. 49 50 Jo˜ao P. Hespanha following transfer function from the applied force u to the spring position y P (s) = 1 . 1) Typically, the mass m, the friction coefficient b, and the spring constant k would be measured experimentally (or taken from some specifications sheet) leading to confidence intervals for these parameters and not just a single value: m ∈ [m0 − δ1 , m0 + δ1 ], b ∈ [b0 − δ2 , b0 + δ2 ], k ∈ [k0 − δ3 , k0 + δ3 ].

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Topics in Undergraduate Control Systems Design (Draft) by Joao P. Hespanha

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