Question: Consider the dynamical system, * = Ax + Bu; y = Cx + Du, given by -1 0 0 1 0 0 0 0

Consider the dynamical system, * = Ax + Bu; y = Cx + Du, given by -1 0 0 1 0 0 0 0 -1 -1 0 -1 1 [****** B= D

Consider the dynamical system, * = Ax + Bu; y = Cx + Du, given by -1 0 0 1 0 0 0 0 -1 -1 0 -1 1 [****** B= D a. Find the transfer function of the underlying dynamical system b. Determine the controllability for the following inputs: (i) u, (ii) U, (iii) all u a. Compute the infinity controllability Gramian (P) and hence deduce the energy required to reach the ith unit vector e,, i.c, vector of zeros with 1 on the ith. position using (1) u, (ii) u, (iii) all u respectively. Comment your results. c. Determine the observability from each of the outputs: (i) y. (ii) y2. (iii) all y a. Compute the infinity observability gramian (0) and hence deduce the energy obtained by observing the ith unit vector e,, i.e, vector of zeros with 1 on the ith. position using (1) y, (ii) y, (iii) all y respectively. Comment your results. d. What can you say with regard to minimality of the system?

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