stability of transfer functions in the Laplace domain

The following describes SISO (single input single output) system descriptions.More complex systems require a more sophisticated analysis.

For a general transfer function, $H(s)$ , in Laplace domain, we have

\\displaystyle H(s)

\\displaystyle=\\frac{\\prod_{i=0}^{Z}(s-z_{i})}{\\prod_{i=0}^{P}(s-p_{i})}

(1)

The conclusions below can all be derived and understood by expansion of $H(s)$ interms of partial fractions, and then doing a inverse Laplacetransform fraction by fraction.

$z_{i}$ denotes the zeros and $p_{i}$ denotes the poles of thelinear time invariant system (LTI). Stability of the system $H(s)$ ischaracterized by the location of the poles in the complex s-plane.There are many definitionsof stability in the control system literature, the most common oneused (for transfer functions) is the bounded-input-bounded-outputstability (BIBO), which states that for a BIBO stable system, for anybounded input, or finite amplitude input, the output of the systemwill also be bounded.

For example, a typical second order system such as themass-spring-dashpot system has the following transfer function,

\\displaystyle H(p)

\\displaystyle=\\frac{\\omega^{2}}{p^{2}+2\\zeta\\omega p+\\omega^{2}}

(2)

with a pair of complex poles at $p=-\\zeta\\omega\\pm\\omega\\sqrt{\\zeta^{2}-1}$ . In common control system literature, $\\zeta$ isusually denoted as the damping ratio and $\\omega$ is denoted asthe natural frequency of the system. In the case of themass-spring-dashpot system, we can tell that the oscillation of themass attached to a spring should be a function of the weight of themass and the stiffness of the spring, hence in the literature we have $\\omega=\\sqrt{\\frac{K}{M}}$ , where $K$ is the spring constant and $M$ is the weight of the mass. From the same logic, the amount timefor the mass to stop oscillating should be a function of the springstiffness and the characteristics of the dashpot, so we have $\\zeta=\\frac{D\\omega}{2K}$ , where $D$ is the dashpot constant.

To determine is the system $H(p)$ BIBO stable, the simplest solutionis to scrutinize the time domain solution of 2 viainverse Laplace transform, and such discussion can easily be found incommon control system related text books and online lecture notes.However, such approach tells nothing about the physical natureof the dynamic system, so here I will try to establish relationshipbetween the location of the poles and the stability of the system.Here I first state that the system in 2,specifically the mass-spring-dashpot system, is stable,since it is physically impossible for the system to produce an outputincreasing in amplitude forever as time goes to infinity.

First let’s consider an ideal mass-spring system. If we push the massonce (impulse response), the mass will oscillateforever with the same amplitude and frequency, since there are nodashpot to dampen the motion. In this case, $D=0\\rightarrow\\zeta=0$ , which the pair of complex poles of the system will be located onthe imaginary axis of the complex s-plane, and the stronger the spring( $K$ is large) the further away the poles from the origin. So theimaginary part of the poles $img(p_{i})$ dominates the oscillatory nature of thesystem. Now let’s assume that we have a relatively weak spring compareto the dashpot that we are going to add to the system, so the realpart of the poles $real(p_{i})$ will be dominating. If we push the massthe same way as before, now we can expect the mass to oscillate forsometime then come to a rest, depending on the strength of thedashpot. The stronger the dashpot, the further away the poles fromthe origin along the real axis on the s-plane. We have almost coveredthe whole left-hand side of the s-plane, and so far we have the followingobservations.

•
The imaginary part of complex poles correspond to oscillatory energy storagemechanisms. Note if the original differential equation has real coefficients,the complex poles are always in complex conjugate pairs . As a resultthese can always be represented by sine and cos terms in the time domain.
•
The real parts of poles govern the decay rate of responses tostimulations; with negative values corresponding to decaying terms, and positiveparts to growth terms (which are not stable).
•
The ratio between the real and imaginary parts of the poles govern the shape of theresponse. In particular ratios less than one hide the visible part of the oscillatory terms in thetime domain.
•
Poles that have neither real or imaginary parts, i.e. at the origin,correspond to successive integrations in the time domain. One integration foreach pole. Since these have an unbounded output for a step input, systems like these arenot BIBO; but are common.

For a system consisted of poles with imaginary parts only, $p_{i}=\\pm j\\omega$ , it is usually referred to as a marginally stablesystem. Notice that the real part of the poles for system 2 are always negative, since both $\\zeta$ and $\\omega$ are always positive. If a stable system has negative poles(real or complex), unstable system must has positives poles, in theright-hand side of the s-plane. This can easily be visualize if weplace a amplifier (with $-\\zeta$ ) instead of a dashpot in the system.