Stability and Stabilization
6.3.1 Stability and Stabilization
A very important question related to the performance of systems is: How do we know that a given causal system has finite zero-input, zero-state, or steady-state responses? This is the stability prob- lem of great interest in control. Thus, if the system is represented by a linear differential equation with constant coefficients the stability of the system determines that the zero-input, the zero-state, as well as the steady-state responses may exist. The stability of the system is also required when con- sidering the frequency response in the Fourier analysis. It is important to understand that only the Laplace transform allows us to characterize stable as well as unstable systems; the Fourier transform does not.
Two possible ways to look at the stability of a causal LTI system are:
When there is no input so that the response of the system depends on initial energy in the system. This is related to the zero-input response of the system.
When there is a bounded input and no initial condition. This is related to the zero-state response of the system.
Relating the zero-input response of a causal LTI system to stability leads to asymptotic stability. An LTI system is said to be asymptotically stable if the zero-input response (due only to initial conditions in the system) goes to zero as t increases—that is,
(6.5) for all possible initial conditions.
y zi ( t) →0
t →∞
The second interpretation leads to the bounded-input bounded-output (BIBO) stability, which we defined in Chapter 2. A causal LTI system is BIBO stable if its response to a bounded input is also bounded. The condition we found in Chapter 2 for a causal LTI system to be BIBO stable was that the impulse response of the system be absolutely integrable—that is
|h(t)|dt < ∞
Such a condition is difficult to test, and we will see in this section that it is equivalent to the poles of the transfer function being in the open left-hand s-plane, a condition that can be more easily visualized and for which algebraic tests exist.
Consider a system being represented by the differential equation
y(t) M X d ℓ x(t)
y(t) +
=b 0 x(t) +
M<N
dt
dt
C H A P T E R 6: Application to Control and Communications
For some initial conditions and input x(t), with Laplace transform X(s), we have that the Laplace transform of the output is
I(s)
X(s)B(s)
Y(s) =Y zi ( s) +Y zs ( s) = L[y(t)] =
A(s)
A(s)
X A(s) X =1+ a
k s k , B(s) =b 0 +
k =1
m =1
where I(s) is due to the initial conditions. To find the poles of H 1 ( s) = 1/A(s), we set A(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 function H(s) = B(s)/A(s) exhibiting no pole-zero cancellation is said to be:
Asymptotically stable if the all-pole transfer function H 1 ( s) = 1/A(s), used to determine the zero-input
response, has all its poles in the open left-hand s-plane (the j axis excluded), or equivalently
A(s) 6= 0 for Re[s] ≥ 0
BIBO stable if all the poles of H(s) are in the open left-hand s-plane (the j axis excluded), or equivalently
A(s) 6= 0 for Re[s] ≥ 0
If H(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 of A(s), or the poles of the system. This can be done for low-order polynomials A(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 the j axis. The Routh stability criterion [53] is an algebraic test capable of determining whether the roots of A(s) are on the left-hand s-plane or not, thus determining the stability of the system.