**Subject :**Advance Control System

## ZERO-INPUT AND ASYMPTOTIC STABILITY OF CONTINUOUS-DATA SYSTEMS

**ZERO-INPUT AND ASYMPTOTIC STABILITY OF CONTINUOUS-DATA SYSTEMS**

Zero-input stability refers to the stability condition when the input is zero, and the system is driven only by its initial conditions. We shall show that the zero-input stability also depends on the roots of the characteristic equation. Let the input of an nth-order system be zero and the output due to the initial conditions be y(t). Then, y(t) can be expressed as

Where

And g_{k}(t) denotes the zero-input response due to y ^{(k)}( t_{0}). The zero-input stability is defined as follows: If the zero-input response y(t), subject to the finite initial conditions, y ^{(k)} (t_{0}), reaches zero as t approaches infinity, the system is said to be zero-input stable, or stable; otherwise, the system is unstable. Mathematically, the foregoing definition can be stated: A linear time-invariant system is zero-input stable if, for any set of finite y^{(k)}(t_{0}), there exists a positive number M, which depends on y^{(k)}(t_{0}), such that

1.

And

2.

Zero as time approaches infinity, the zero-input stability is also known at the asymptotic stability. Taking the absolute value on both sides of Eq. (2-240), we get

Because all the initial conditions are assumed to be finite, the condition in Eq. (2-242) requires that the following condition be true:

Let the n characteristic equation roots be expressed as S_{i} = σ_{i} jw_{i}, i = 1,2. . . . . n. Then, if m of the n roots are simple, and the rest are of multiple order, y(t) will be of the form:

Where K_{i}, and L_{i},- are constant coefficients. Because the exponential terms eSi' in the last equation control the response y(t) as t→∞, to satisfy the two conditions in Eqs. (2-242) and (2-243), the real parts of S_{i} must be negative. In other words, the roots of the characteristic equation must all be in the left-half .y-plane. From the preceding discussions, we see that,/or linear time-invariant systems, BIBO, zero-input, and asymptotic stability all have the same requirement that the roots of the characteristic equation must all be located in the left-half s-plane. Thus, if a system is BIBO stable, it must also be zero-input or asymptotically stable. For this reason, we shall simply refer to the stability condition of a linear system as stable or unstable. The latter condition refers to the condition that at least one of the characteristic equation roots is not in the left-half s-plane.

For practical reasons, we often refer to the situation in which the characteristic equation has simple roots on they jω-axis and none in the right-half plane as marginally stable or marginally unstable. An exception to this is if the system were intended to be an integrator (or, in the case of control systems, a velocity control system); then the system would have root(s) at s = 0 and would be considered stable. Similarly, if the system were designed to be an oscillator, the characteristic equation would have simple roots on the jω-axis, and the system would be regarded as stable.