Signals and Systems - Electrical Engineering

370 C H A P T E R 6: Application to Control and Communications

For some initial conditions and inputx(t), with Laplace transformX(s), we have that the Laplace

transform of the output is

Y(s)=Yzi(s)+Yzs(s)=L[y(t)]=

I(s)

A(s)

+

X(s)B(s)

A(s)

A(s)= 1 +

∑N

k= 1

aksk, B(s)=b 0 +

∑M

m= 1

bmsm

whereI(s)is due to the initial conditions. To find the poles ofH 1 (s)= 1 /A(s), we setA(s)=0,

which corresponds to the characteristic equation of the system and its roots (real, complex conjugate,

simple, and multiple) are the natural modes or eigenvalues of the system.

A causal LTI system with transfer functionH(s)=B(s)/A(s)exhibiting no pole-zero cancellation is said

to be:

n Asymptotically stable if the all-pole transfer functionH 1 (s)= 1 /A(s), used to determine the zero-input

response, has all its poles in the open left-hands-plane (the jaxis excluded), or equivalently

A(s)6= 0 for Re[s]≥ 0 (6.7)

n BIBO stable if all the poles ofH(s)are in the open left-hands-plane (the jaxis excluded), or equivalently

A(s)6= 0 for Re[s]≥ 0 (6.8)

n IfH(s)exhibits pole-zero cancellations, the system can be BIBO stable but not necessarily asymptotically

stable.

Testing the stability of a causal LTI system thus requires finding the location of the roots ofA(s), or the

poles of the system. This can be done for low-order polynomialsA(s)for which there are formulas to

find the roots of a polynomial exactly. But as shown by Abel,^1 there are no equations to find the roots

of higher than fourth-order polynomials. Numerical methods to find roots of these polynomials only

provide approximate results that might not be good enough for cases where the poles are close to

thejaxis. The Routh stability criterion [53] is an algebraic test capable of determining whether the

roots ofA(s)are on the left-hands-plane or not, thus determining the stability of the system.

nExample 6.3: Stabilization of a plant

Consider a plant with a transfer functionG(s)= 1 /(s− 2 ), which has a pole in the right-hand

s-plane and therefore is unstable. Let us consider stabilizing it by cascading it with an all-pass filter

(Figure 6.8(a)) so that the overall system is not only stable but also keeps its magnitude response.

(^1) Niels H. Abel (1802–1829) was a Norwegian mathematician who accomplished brilliant work in his short lifetime. Atage 19, he
showed there is no general algebraic solution for the roots of equations of degree greater than four, in terms of explicit algebraic
operations.