[3.9 SYSTEM](#page-9-0) STABILITY

← Back to LINEAR SYSTEMS AND SIGNALS Overview

3.9 SYSTEM STABILITY

The concepts and criteria for the BIBO (external) stability and internal (asymptotic) stability for discrete-time systems are identical to those corresponding to continuous-time systems. The comments in Sec. 2.5 for LTIC systems concerning the distinction between external and internal stability are also valid for LTID systems. Let us begin with external (BIBO) stability.

3.9-1 External (BIBO) Stability

Recall that

$$y[n] = h[n] * x[n] = \sum_{m=-\infty}^{\infty} h[m]x[n-m]$$

and

$$|y[n]| = \left|\sum_{m=-\infty}^{\infty} h[m]x[n-m]\right| \le \sum_{m=-\infty}^{\infty} |h[m]| |x[n-m]|$$

If x[n] is bounded, then |x[n−m]| < K¹ < ∞, and

$$|y[n]| \leq K_1 \sum_{m=-\infty}^{\infty} |h[m]|$$

Clearly the output is bounded if the summation on the right-hand side is bounded; that is, if

$$\sum_{n=-\infty}^{\infty} |h[n]| < K_2 < \infty \tag{3.43}$$

This is a sufficient condition for BIBO stability. We can show that this is also a necessary condition (see Prob. 3.9-1). Therefore, if the impulse response h[n] of an LTID system is absolutely summable, the system is (BIBO) stable. Otherwise it is unstable.

All the comments about the nature of external and internal stability in Ch. 2 apply to discrete-time case. We shall not elaborate them further.

3.9-2 Internal (Asymptotic) Stability

For LTID systems, as in the case of LTIC systems, internal stability, called asymptotical stability or stability in the sense of Lyapunov (also the zero-input stability), is defined in terms of the zero-input response of a system.

For an LTID system specified by a difference equation in the form of Eq. (3.15) [or Eq. (3.20)], the zero-input response consists of the characteristic modes of the system. The mode corresponding to a characteristic root γ is γ ⁿ. To be more general, let γ be complex so that

$$\gamma = |\gamma|e^{i\beta}$$

and $\gamma^n = |\gamma|^n e^{i\beta n}$

Since the magnitude of ejβ*ⁿ* is always unity regardless of the value of n, the magnitude of γ ⁿ is |γ | ⁿ. Therefore,

if

$$|\gamma| < 1$$

, then $\gamma^n \to 0$ as $n \to \infty$
if $|\gamma| > 1$ , then $\gamma^n \to \infty$ as $n \to \infty$
and if $|\gamma| = 1$ , then $|\gamma|^n = 1$ for all n

The characteristic modes corresponding to characteristic roots at various locations in the complex plane appear in Fig. 3.27.

These results can be grasped more effectively in terms of the location of characteristic roots in the complex plane. Figure 3.28 shows a circle of unit radius, centered at the origin in a complex plane. Our discussion shows that if all characteristic roots of the system lie inside the unit circle, |γi| < 1 for all i and the system is asymptotically stable. On the other hand, even if one characteristic root lies outside the unit circle, the system is unstable. If none of the characteristic

Figure 3.27 Characteristic roots locations and the corresponding characteristic modes.

(d)

(a)

(b)

roots lie outside the unit circle, but some simple (unrepeated) roots lie on the circle itself, the system is marginally stable. If two or more characteristic roots coincide on the unit circle (repeated roots), the system is unstable. The reason is that for repeated roots, the zero-input response is of the form nr−1γ ⁿ, and if |γ | = 1, then |nr−1γ ⁿ| = nr−¹ → ∞ as n → ∞. † Note, however, that repeated roots inside the unit circle do not cause instability.

Figure 3.28 Characteristic root locations and system stability.

To summarize:

• 1. An LTID system is asymptotically stable if, and only if, all the characteristic roots are inside the unit circle. The roots may be simple or repeated.
• 2. An LTID system is unstable if, and only if, either one or both of the following conditions exist: (i) at least one root is outside the unit circle; (ii) there are repeated roots on the unit circle.
• 3. An LTID system is marginally stable if and only if there are no roots outside the unit circle and there are some unrepeated roots on the unit circle.

3.9-3 Relationship Between BIBO and Asymptotic Stability

For LTID systems, the relation between the two types of stability is similar to those in LTIC systems. For a system specified by Eq. (3.15), we can readily show that if a characteristic root γ*^k*

^† If the development of discrete-time systems is parallel to that of continuous-time systems, we wonder why the parallel breaks down here. Why, for instance, are LHP and RHP not the regions demarcating stability and instability? The reason lies in the form of the characteristic modes. In continuous-time systems, we chose the form of characteristic mode as eλit . In discrete-time systems, for computational convenience, we choose the form to be γ ^{n i} . Had we chosen this form to be ^eλin where ^{γi = e}λⁱ , then the LHP and RHP (for the location of λi) again would demarcate stability and instability. The reason is that if γ = eλ, |γ | = 1 implies |eλ| = 1, and therefore λ = jω. This shows that the unit circle in γ plane maps into the imaginary axis in the λ plane.

302 CHAPTER 3 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS

is inside the unit circle, the corresponding mode γ ^{n k} is absolutely summable. In contrast, if γ*^k* lies outside the unit circle, or on the unit circle, γ ^{n k} is not absolutely summable.†

This means that an asymptotically stable system is BIBO-stable. Moreover, a marginally stable or asymptotically unstable system is BIBO-unstable. The converse is not necessarily true. The stability picture portrayed by the external description is of questionable value. BIBO (external) stability cannot ensure internal (asymptotic) stability, as the following example shows.

EXAMPLE 3.26 A BIBO-Stable but Asymptotically Unstable System

An LTID systems consists of two subsystems S¹ and S² in cascade (Fig. 3.29). The impulse response of these systems are h1[n] and h2[n], respectively, given by

h1[n] = 4δ[n] −3(0.5) n ^u[n] and ^h2[n] = ²ⁿ u[n]

Investigate the BIBO and asymptotic stability of the composite system.

The composite system impulse response h[n] is given by

$$h[n] = h_1[n] * h_2[n] = h_2[n] * h_1[n] = 2nu[n] * (4\delta[n] - 3(0.5)nu[n])$$

= 4(2)ⁿu[n] - 3\left[\frac{2^{n+1} - (0.5)ⁿ⁺¹}{2 - 0.5}\right]u[n]
= (0.5)ⁿu[n]

If the composite cascade system were to be enclosed in a black box with only the input and the output terminals accessible, any measurement from these external terminals would show that the impulse response of the system is (0.5)nu[n], without any hint of the unstable system sheltered inside the composite system.

† This conclusion follows from the fact that (see Sec. B.8-3)

$$\sum_{n=-\infty}^{\infty} |\gamma_k^n| u[n] = \sum_{n=0}^{\infty} |\gamma_k|^n = \frac{1}{1 - |\gamma_k|} \qquad |\gamma_k| < 1$$

Moreover, if |γ | ≥ 1, the sum diverges and goes to ∞. These conclusions are valid also for the modes of the form nr γ n k .

The composite system is BIBO-stable because its impulse response (0.5)nu[n] is absolutely summable. However, the system S² is asymptotically unstable because its characteristic root, 2, lies outside the unit circle. This system will eventually burn out (or saturate) because of the unbounded characteristic response generated by intended or unintended initial conditions, no matter how small.

The system is asymptotically unstable, though BIBO-stable. This example shows that BIBO stability does not necessarily ensure asymptotic stability when a system is uncontrollable, unobservable, or both. The internal and the external descriptions of a system are equivalent only when the system is controllable and observable. In such a case, BIBO stability means the system is asymptotically stable, and vice versa.

Fortunately, uncontrollable or unobservable systems are not common in practice. Henceforth, in determining system stability, we shall assume that unless otherwise mentioned, the internal and the external descriptions of the system are equivalent, implying that the system is controllable and observable.

EXAMPLE 3.27 Investigating Asymptotic and BIBO Stability

Determine the internal and external stability of systems specified by the following equations. In each case plot the characteristic roots in the complex plane.

• (a) y[n+2] +2.5y[n+1] +y[n] = x[n+1] −2x[n]
• (b) y[n] −y[n−1] +0.21y[n−2] = 2x[n−1] +3x[n−2]
• (c) ^y[n+3] +2y[n+2] + ^{3 2 y}[n+1] + ^{1 2} y[n] = x[n+1]
• (d) (E² −E +1)²y[n] = (3E +1)x[n]

(a) The characteristic polynomial is

$$\gamma^2 + 2.5\gamma + 1 = (\gamma + 0.5)(\gamma + 2)$$

The characteristic roots are −0.5 and −2. Because | − 2| > 1 (−2 lies outside the unit circle), the system is BIBO-unstable and also asymptotically unstable (Fig. 3.30a).

(b) The characteristic polynomial is

$$\gamma^2 - \gamma + 0.21 = (\gamma - 0.3)(\gamma - 0.7)$$

The characteristic roots are 0.3 and 0.7, both of which lie inside the unit circle. The system is BIBO-stable and asymptotically stable (Fig. 3.30b).

(c) The characteristic polynomial is

$$\gamma^{3} + 2\gamma^{2} + \frac{3}{2}\gamma + \frac{1}{2} = (\gamma + 1)(\gamma^{2} + \gamma + \frac{1}{2}) = (\gamma + 1)(\gamma + 0.5 - j0.5)(\gamma + 0.5 + j0.5)$$

304 CHAPTER 3 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS

The characteristic roots are −1, −0.5 ± j0.5 (Fig. 3.30c). One of the characteristic roots is on the unit circle and the remaining two roots are inside the unit circle. The system is BIBO-unstable but marginally stable.

(d) The characteristic polynomial is

$$(\gamma^2 - \gamma + 1)^2 = \left(\gamma - \frac{1}{2} - j\frac{\sqrt{3}}{2}\right)^2 \left(\gamma - \frac{1}{2} + j\frac{\sqrt{3}}{2}\right)^2$$

The characteristic roots are (1/2)±j( ^√3/2) ^{= 1}e±j(π/3) repeated twice, and they lie on the unit circle (Fig. 3.30d). The system is BIBO-unstable and asymptotically unstable.

DR ILL 3.21 Assessing Stability by Characteristic Roots

Using the complex plane, locate the characteristic roots of the following systems, and use the characteristic root locations to determine external and internal stability of each system.

(a) (E +1)(E² +6E +25)y[n] = 3Ex[n]

(b) (E −1)2(E +0.5)y[n] = (E² +2E +3)x[n]

ANSWERS

Both systems are BIBO-and asymptotically unstable.