Signals and Systems - Electrical Engineering

216 C H A P T E R 3: The Laplace Transform

whereh(t)=L−^1 [H(s)]andh 1 (t)=L−^1 [H 1 (s)], and

i(t)=L−^1 [I(s)]=

∑N

k= 1

ak





k∑− 1

m= 0

y(m)( 0 )δ(k−m−^1 )(t)





where{δ(m)(t)}aremthderivatives of the impulse signalδ(t)(as indicated before,δ(^0 )(t)=δ(t)).

Zero-State and Zero-Input Responses

Despite the fact that linear differential equations, with constant coefficients, do not represent linear

systems unless the initial conditions are zero and the input is causal, linear system theory is based on

these representations with initial conditions. Typically, the input is causal so it is the initial conditions

not always being zero that causes problems. This can be remedied by a different way of thinking

about the initial conditions. In fact, one can think of the inputx(t)and the initial conditions as two

different inputs to the system, and apply superposition to find the responses to these two different

inputs. This defines two responses. One is due completely to the input, with zero initial conditions,

called thezero-state solution. The other component of the complete response is due exclusively to the

initial conditions, assuming that the input is zero, and is called thezero-input solution.

Remarks

n It is important to recognize that to compute the transfer function of the system

H(s)=

Y(s)

X(s)

according to Equation (3.33) requires that the initial conditions be zero, or I(s)= 0.

n If there is no pole-zero cancellation, both H(s)and H 1 (s)have the same poles, as both have A(s)as

denominator, and as such h(t)and h 1 (t)might be similar.

Transient and Steady-State Responses

Whenever the input of a causal and stable system has poles in the closed left-hand s-plane, poles in

thej-axis being simple, the complete response will be bounded. Moreover, whether the response

exists ast→∞can then be determined without using the inverse Laplace transform.

The complete responsey(t)of an LTI system is made up of transient and steady-state components.

The transient response can be thought of as the system’s reaction to the initial inertia after applying

the input, while the steady-state response is how the system reacts to the input away from the initial

time when the input starts.

If the poles (simple or multiple, real or complex) of the Laplace transform of the output,Y(s), of an LTI system

are in the open left-hands-plane (i.e., no poles on thejaxis), the steady-state response is

yss(t)=lim

t→∞

y(t)= 0