Question

(3) Using the complete controllability theorem show that the model of the two masses attached by springs discussed in the review problems is completely controllable re- gardless of the parameters k1, k2, m1, m2 (assuming they are all non-zero). Write the system in the form * = Ax + Bu for appropriate A and B.

Answer 1

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Please see the answer in the attachment.

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Answer - . It consists of two masses m, & m2 & two springs with stiffness coefficients kifka. Displacement of the mass es from their equillibrium positions is determined by the coordinates di and x2. we find now the eans of motion using Lagrangian formalism, 2 2 2 V. Ka+ K, (X2-21) V 121 + M2 X2 te mi? 4 mg xit , v= kimi kz (x2+xv)? LET-V Le maint me teper - Kyx?- ką (23–24, 2 As we know the differential Langrange equating to e con & ( ) - Poze 2 2 = mise ol e-K,X, + ką (X2-31) ob maitz - K2(*2 = x, ) 1.25 = ( kita yax, 14 min) az 4 Q2 = (he) *, + (-k) *2 form ë = AX + BU . This is the required