In a continuous time linear system whenever any of the characteristic roots, say λ, has a positive realpart, ie R_{e}(λ) = λ_{R} > 0, then the system is unstable. Stability is ensured i 3 each root λ has a negative real part, ie λ_{R} > 0. Typical time responses for stable systems are shown in figure 4.1 and unstable systems in figure 4.2. Note the unstable response is the reverse time plot of the stable plot with the initial and final values interchanged ; clearly this is because reversing the sign of the roots is equivalent to reversing the time variable ie (-λ)(+t) = (+λ)(-t).
In a discrete time, sampled data, systems the stability criterion is that the characteristic roots have magnitude less than 1, ie lie in the unit circle in the complex plane. However this course only considers continuous systems.
Real Roots
Modes of purely real roots are non-oscillatory. When a C.E. such as p(s) = 0 (where p(s) is a polynomial of order n) has a real root it can be factored p(s) = (s-λ_{R})q(s) = 0 where q(s) is a polynomial of orderone less than p(s). Then s = λ_{R} clearly satisfies the equation. The variable s is a complex frequency expressible in units of [s^{-1}]. The inverse of -λ_{R} is termed the time-constant of the root τ that is τ = -1/λ_{R }expressible in units of [s].