Continuous Control Systems: a review

Stability

The stability of a dynamic system is related to the asymptotic behavior of the system (when t ^{ } ), starting from an initial condition and for an identically zero input.

For example, consider the first-order system described by the differential Equation 1.1.1 or by the transfer function given in Equation 1.1.9.
Consider the free response of the system given in Equation 1.1.1 for u  0 and from an initial condition y(0) = y₀:

^dy  ¹ y(t)  0 ; y(0)  y

(1.1.10)

dt T ⁰

A solution for y will be of the form

y(t)  Ke ^st
(1.1.11)

in which K and s are to be determined². From Equation 1.1.11 one finds


^dy  s Ke^st
dt
(1.1.12)

and Equation 1.1.10 becomes


Ke ^{st } s  ^{1 }  0

(1.1.13)

 ^T 

from which one obtains


^{s   1 ; K } y
T ⁰

(1.1.14)

and respectively


y(t)  y₀e^{t /T}
(1.1.15)

The response for T > 0 and T < 0 is illustrated in Figure 1.3.

y
o


t
Figure 1.3. Free response of the first-order system

For T > 0, we have s < 0 and, when t   , the output will tend toward zero (asymptotic stability). For T < 0, we have s > 0 and, when t   , the output will diverge (instability). Note that s = -1/T corresponds to the pole of the first-order transfer function of Equation 1.1.9.

We can generalize this result: it is the sign of the real part of the roots of the transfer function denominator that determines the stability or instability of the system.
In order that a system be asymptotically stable, all the roots of the transfer function denominator must be characterized by Re s < 0. If one or several roots of
² The structure of the solution for Equation 1.1.11 results from the theory of linear differential equations.

the transfer function denominator are characterized by Re s > 0, then the system is
unstable.
For Re s = 0 we have a limit case of stability because the amplitude of y(t) remains equal to the initial condition (e.g. pure integrator, dy/dt = u(t); in this case y(t) remains equal to the initial condition).
Figure 1.4 gives the stability and instability domains in the plane of the complex variable s.
Note that stability criteria have been developed, which allow determining the existence of unstable roots of a polynomial without explicitly computing its roots, (e.g. Routh-Hurwitz’ criterion) (Ogata 1990).