Question: Consider a discrete-time system described by: where x(k), x2(k) and xz(k) are the states and u(k) is the input. Find the necessary and sufficient conditions

Consider a discrete-time system described by:

Consider a discrete-time system described by: where x(k), x2(k) and xz(k) are

where x(k), x2(k) and xz(k) are the states and u(k) is the input. Find the necessary and sufficient conditions on a, b, and c so that the system is controllable. Justify your answer.

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To determine the controllability of the given discretetime system we need to examine the controllability matrix and ensure it has full rank The systems statespace representation is xk1 Axk Buk Where A beginbmatrix 1 0 0 0 2 0 0 0 3 endbmatrix quad B beginbmatrix a b c endbmatrix Step 1 Controllability Matrix The controllability matrix C is given by C beginbmatrix B AB A2B endbmatrix Step 2 Compute AB AB A cdot B beginbmatrix 1 0 0 0 2 0 0 0 3 endbmatrix beginbmatrix a b c endbmatrix beginbmatrix a 2b 3c endbmatrix Step 3 Compute A2B A2 A cdot A beginbmatrix 1 0 0 0 2 0 0 0 3 endbmatrix beginbmatrix 1 0 0 0 2 0 0 0 3 endbmatrix beginbmatrix 1 0 0 0 4 0 0 0 9 endbmatrix A2B A2 cdot B beginbmatrix 1 0 0 0 4 0 0 0 9 endbmatrix beginbmatrix a b c endbmatrix beginbmatrix a 4b 9c endbmatrix Step 4 Construct the Controllability Matrix C beginbmatrix B AB A2B endbmatrix beginbmatrix a a a b 2b 4b c 3c 9c endbmatrix Step 5 Determine the Rank of C The system ... View full answer

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