For a linear system, x? = Ax + Bu y = Cx + Du A = [0 1; 3...

For a linear system, x? = Ax + Bu y = Cx + Du A = [0 1; 3 -2], B = [1; 0], C = [3 -2], D = 0 and x(0) = 0 Using TWO different methods to find unit-step (u=1) response of this system. Hint: Two approaches to find the solution of the state space equation, (1) State space method x(t) = e^{A(t-t_0)} [x_0 + ?_{t_0}^{t} e^{-A(?-t_0)} Bu(?) d?] and y(t) = C [x_1(t); x_2(t)] (2) Laplace transform method Y(s) = [C(sI - A)^{-1}B + D]U(s) + C(sI - A)^{-1}x_0 and y(t) = L^{-1}[Y(s)]

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00:01 Here we are given x not is equal to a x plus b u and y is equal to cx plus d u and a equal to 2 by 2 matrix with entries 0 1 3 minus 2 b equal to column matrix with entries 1 0 and c of c equal to row matrix with entries 3 minus 2 and d equal to 0 and x of 0 equal 0.

00:26 Now apply the state space method.

00:30 First one we have to apply the state space method so we can have x of t is equal to e power a t minus t minus t not into x of zero plus integral t not to t e power minus a times t minus t b u of tau d tau or we can write x of t s x of t is equal to e power a t into x of t is equal to e power a t into x of 0 plus integral 0 to t e power a times t minus tau b u of tau d tau now we know that e a power t is equal to l inverse of s i minus a the whole inverse which is equal to l inverse of s 0 0 s 0 minus 1 .3 to the whole inverse which is equal to l inverse of s minus 1 minus 3 s minus 2 the whole inverse which is equal to l inverse of s plus 2 1 3 s divided by 3 times s plus 2 minus 3 which is equal to l inverse of s plus 2 divided by 3 times s plus 2 minus 3 which is equal to l inverse of s plus 2 divided by s minus 1 into s plus 3 and 1 divided by s minus 1 s plus 3 3 divided by s minus 1 s plus 3.

02:47 Therefore from this we may have e power 80 is equal to 3 by s minus 1 s minus 1.

02:56 That is we are applying the 4x0 2x4 by s plus 3 and here you will get 1 by 4 divided by s minus 1 minus 1 by 4 divided by s plus 3 and here it is 3 by 4 divided by s minus 1 minus 3 by 4 divided by s plus 3 and it is 1 by 4 divided by s minus 1 plus 3 by 0 0 1 2 by s plus 3 which is equal to 3 by 4 e power t plus 1 by 4e power minus 3 t and 1 by 4 e power t minus 1 by 4 e power minus 3 t.

03:40 3 by 4 e power minus 3 t...

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