Question

Solve in Matlab please.

An ltic system is specified by the eq: D(D+1) y(t) = (D+2)x(t)

a) Find the characteristic polynomial, char. equation, char. roots, and char. modes.

b) find y0(t), zero-input component of the response y(t) for t>=o; if initial conditions is y0(0-) =  y'0(0-) = 1

Answer 1

The system is given by D(D+1)y(t)=(D+2)x(t)

1) The characteristic polynomial is given by €(€+1)

2) The characteristic equation is given by €(€+1)=0.

3) The characteristic roots are given by €=0,€=-1.

4) The corresponding characteristic modes are c₀e^0t , c₁e^-t, i.e., c₀, c₁e^-t

5) The zero input component of the response is given by y₀(t)=c₀+ c₁e^-t by equating t=0 with initial conditions.

Here y'(t) = -c₁e^{-t .}

hence, we have from the initial conditions

c₀+c₁=1

-c₁=1.

Solving them we get c₀=2, c₁=-1.

Hence y₀(t)=2-e^-t.