Stability

Key Points About Stability

• Stability is an inherent property of a system
• This means its output does not diverge for any reasonable input (impulse, step, etc…)
• Stability of an LTI system is the exponential stability
• Stability of an LTI system is determined by its Poles

How Can You Tell if a System is Stable?

• An LTI system is stable if and only if all poles are in the LHP (for Continuous Time) or inside the unit disk (for Discrete Time) (${\mathbb{C}}^{-}\text{ or }{\mathbb{D}}^{-}$)
• If some poles are on the imaginary axis, and the others are in the LHP, the system is marginally stable (which is technically unstable)
• For a stable system, all coefficients of its characteristic equation defined in 2nd Order Dynamic Systems must be positive
• Computing the roots of a characteristic equation is not very practically useful (used only for analysis but not for design)
• You can also use the Routh Test

Stability in the Time Domain

Let $u\left(t\right)/u(k)$ be a real-valued signal define for $t\ge 0$ or $k\ge 0$.
Then u is bounded if there exists $\overline{u}\in \mathbb{R}$ and $\underline{u}>0$ such that $\left|u\left(t\right)\right|\le \underline{u}$ $\forall t\ge 0$ (with the same definition for $u[k]$.

This basically means, there exists some constant that is real for which all points in a signal is less than.

Info

Here, $\underline{u}$ is some constant

I write $\underline{u}$ here but I mean $\bar{u}$

We use this idea of bounded in BIBO stability.

Stability of Feedback Systems

• External signals: r, d
• Internal signals: u, e, y → Can be affected by our control system

Types of Stability

• Closed Loop and Proving Closed Loop Stability
• Shur Control Systems for testing the stability of higher order discrete time systems