The utmost important requirement in control system design is the stability. We would revisit some of the definitions related to stability of a system.

1. Revisiting the basics

Let us consider the following system.

(1)

where $A \in R^{n\times n}$ , $B \in R^{n\times 1}$ , $C \in R^{1\times n}$ .

Zero State Response: The output response of system (1) that is due to the input only (initial states are zero) is called zero state response.

Zero Input Response: The output response of system (1) that is driven by the initial states only (in absence of any external input) is called zero input response.

BIBO Stability: If for any bounded input u(k) , the output y(k) is also bounded, then the system is said to be BIBO stable.

Bounded Input Bounded State Stability: If for any bounded input u(k), the states are also bounded, then the system is said to be Bounded Input Bounded State stable.

L² Norm: L² norm of a state vector x(k) is defined as $\displaystyle \vert\vert \boldsymbol{x}(k) \vert\vert _2 = \left [ \sum_{i=1}^n x_i^2(k) \right ]^{\frac{1}{2}}$

x(k) is said to be bounded if $\vert\vert \boldsymbol{x}(k) \vert\vert<M$ for all k where M is finite.

Zero Input or Internal Stability: If the zero input response of a system subject to a finite initial condition is bounded and reaches zero as k → ∞, then the system is said to be internally stable.

The above condition can be formulated as

$\displaystyle 1. \;\;\;\;\;\; \vert y(k)\vert \le M < \infty$
$\displaystyle 2. \;\;\;\;\;\; \lim_{k\rightarrow \infty} \vert y(k)\vert = 0\;\;\;$

The above conditions are also requirements for asymptotic stability .

To ensure all possible stability for an LTI system, the only requirement is that the roots of the characteristic equations are inside the unit circle.