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Chapter 4

Q. 4.4

OBTAINING A STATE-SPACE MODEL FROM AN IMPULSE-RESPONSE MODEL.

Find an equivalent state-space representation of the following impulse-response model:

y(t) = \frac{K}{\tau} \int_{0}^{t}{e^{-  (t  –  \sigma)/ \tau}  u(\sigma)  d \sigma}         (4.75)

Step-by-Step

Verified Solution

Clearly, in this case, the impulse-response function g(t) is given by:

g(t) = \frac{K}{\tau}  e^{-t/ \tau}        (4.76)

which is differentiable. If we now differentiate Eq. (4.75) with respect to t (with the aid of Eq. (4.74)), we obtain:

\frac{d}{dx} \left\lgroup\int_{A(x)}^{B(x)}{f(x, r)dr} \right\rgroup = \int_{A}^{B}{\frac{\partial f(x,  r)}{\partial r} dr}  +  f(x,  B)\frac{dB}{dx}  –  f(x, A)\frac{dA}{dx}        (4.74)

\frac{dy}{dt} = \frac{K}{\tau}\int_{0}^{t}{- \frac{1}{\tau} e^{-  (t  –  \sigma) / \tau}  u(\sigma)  d \sigma  +  \frac{K}{\tau}  u(t)  –  0}

which simplifies further (using Eq. (4.75) to eliminate the integral) to give:

\frac{dy}{dt} = –  \frac{1}{\tau} y(t)  +  \frac{K}{\tau} u(t)

or

\tau \frac{dy}{dt}  + y(t) = K  u(t)

Again, we may now specify y = x, to obtain:

\tau \frac{dx}{dt}  + x(t) = K  u(t)        (4.62a)

y(t) = x(t)        (4.62b)

as required state-space representation.

It is important to recognize that the impulse-response model is a time-domain input/ output model form that also does not explicitly contain information about the state variable x(t); therefore as with the transform-domain model, conversions to the state space will not be unique.