[2.5 SYSTEM](#page-8-0) STABILITY

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2.5 SYSTEM STABILITY

Stability is an important system property. Two types of system stability are generally considered: external (BIBO) stability and internal (asymptotic) stability. Let us consider both stability types in turn.

2.5-1 External (BIBO) Stability

To understand the intuitive basis for the BIBO (bounded-input/bounded-output) stability of a system introduced in Sec. 1.7, let us examine the stability concept as applied to a right circular cone. Such a cone can be made to stand forever on its circular base, on its apex, or on its side. For this reason, these three states of the cone are said to be equilibrium states. Qualitatively, however, the three states show very different behavior. If the cone, standing on its circular base, were to be disturbed slightly and then left to itself, it would eventually return to its original equilibrium position. In such a case, the cone is said to be in stable equilibrium. In contrast, if the cone stands on its apex, then the slightest disturbance will cause the cone to move farther and farther away from its equilibrium state. The cone in this case is said to be in an unstable equilibrium. The cone lying on its side, if disturbed, will neither go back to the original state nor continue to move farther away from the original state. Thus it is said to be in a neutral equilibrium. Clearly, when a system is in stable equilibrium, application of a small disturbance (input) produces a small response. In contrast, when the system is in unstable equilibrium, even a minuscule disturbance (input) produces an unbounded response. The BIBO-stability definition can be understood in the light of this concept. If every bounded input produces bounded output, the system is (BIBO) stable.† In contrast, if even one bounded input results in unbounded response, the system is (BIBO) unstable.

For an LTIC system,

y(t) = h(t) * x(t) = \int_{-\infty}^{\infty} h(\tau) x(t - \tau) d\tau

Therefore,

|y(t)| \le \int_{-\infty}^{\infty} |h(\tau)| |x(t - \tau)| d\tau

Moreover, if x(t) is bounded, then |x(t −τ )| < K¹ < ∞, and

|y(t)| \leq K_1 \int_{-\infty}^{\infty} |h(\tau)| d\tau

Hence for BIBO stability,

\int_{-\infty}^{\infty} |h(\tau)| d\tau < \infty \tag{2.45}

This is a sufficient condition for BIBO stability. We can show that this is also a necessary condition (see Prob. 2.5-7). Therefore, for an LTIC system, if its impulse response h(t) is absolutely integrable, the system is (BIBO) stable. Otherwise it is (BIBO) unstable. In addition, we shall show in Ch. 4 that a necessary (but not sufficient) condition for an LTIC system described by Eq. (2.1) to be BIBO-stable is M ≤ N. If M > N, the system is unstable. This is one of the reasons to avoid systems with M > N.

Because the BIBO stability of a system can be ascertained by measurements at the external terminals (input and output), this is an external stability criterion. It is no coincidence that the BIBO criterion in Eq. (2.45) is in terms of the impulse response, which is an external description of the system.

As observed in Sec. 1.9, the internal behavior of a system is not always ascertainable from the external terminals. Therefore, external (BIBO) stability may not be a correct indication of internal stability. Indeed, some systems that appear stable by the BIBO criterion may be internally unstable. This is like a room on fire inside a house: no trace of fire is visible from outside, but the entire house will be burned to ashes.

The BIBO stability is meaningful only for systems in which the internal and the external description are equivalent (controllable and observable systems). Fortunately, most practical systems fall into this category, and whenever we apply this criterion, we implicitly assume that the system, in fact, belongs to this category. Internal stability is all-inclusive, and external stability can always be determined from internal stability. For this reason, we now investigate the internal stability criterion.

^† The system is assumed to be in zero state.

2.5-2 Internal (Asymptotic) Stability

Because of the great variety of possible system behaviors, there are several definitions of internal stability in the literature. Here we shall consider a definition that is suitable for causal, linear, time-invariant (LTI) systems.

If, in the absence of an external input, a system remains in a particular state (or condition) indefinitely, then that state is said to be an equilibrium state of the system. For an LTI system, zero state, in which all initial conditions are zero, is an equilibrium state. Now suppose an LTI system is in zero state and we change this state by creating small nonzero initial conditions (small disturbance). These initial conditions will generate signals consisting of characteristic modes in the system. By analogy with the cone, if the system is stable, it should eventually return to zero state. In other words, when left to itself, every mode in a stable system arising as a result of nonzero initial conditions should approach 0 as t→∞. However, if even one of the modes grows with time, the system will never return to zero state, and the system would be identified as unstable. In the borderline case, some modes neither decay to zero nor grow indefinitely, while all the remaining modes decay to zero. This case is like the neutral equilibrium in the cone. Such a system is said to be marginally stable. Internal stability is also called asymptotic stability or stability in the sense of Lyapunov.

For a system characterized by Eq. (2.1), we can restate the internal stability criterion in terms of the location of the N characteristic roots λ1, λ2, …, λ*^N* of the system in a complex plane. The characteristic modes are of the form e^λkt or t r e^λkt . The locations of various roots in the complex plane and the corresponding modes are shown in Fig. 2.17. These modes → 0 as t → ∞ if Re λ*^k* < 0. In contrast, the modes → ∞ as t → ∞ if Reλ*^k* > 0.†

From Fig. 2.17, we see that a system is (asymptotically) stable if all its characteristic roots lie in the LHP, that is, if Reλ*^k* < 0 for all k. If even a single characteristic root lies in the RHP, the system is (asymptotically) unstable. Modes due to roots on the imaginary axis (λ = ±jω0) are of the form e±jω0*^t* . Hence, if some roots are on the imaginary axis, and all the remaining roots are in the LHP, the system is marginally stable (assuming that the roots on the imaginary axis are not repeated). If the imaginary axis roots are repeated, the characteristic modes are of the form t r e±jωkt , which do grow with time indefinitely. Hence, the system is unstable. Figure 2.18 shows stability regions in the complex plane.

To summarize:

1. An LTIC system is asymptotically stable if, and only if, all the characteristic roots are in the LHP. The roots may be simple (unrepeated) or repeated.
1. An LTIC system is unstable if, and only if, one or both of the following conditions exist: (i) at least one root is in the RHP; (ii) there are repeated roots on the imaginary axis.
1. An LTIC system is marginally stable if, and only if, there are no roots in the RHP, and there are some unrepeated roots on the imaginary axis.

\lim_{t \to \infty} e^{\lambda t} = \lim_{t \to \infty} e^{(\alpha + j\beta)t} = \lim_{t \to \infty} e^{\alpha t} e^{j\beta t} = \begin{cases} 0 & \alpha < 0 \\ \infty & \alpha > 0 \end{cases}

This conclusion is also valid for the terms of the form t r eλt .

^† This may be seen from the fact that if α and β are the real and the imaginary parts of a root λ, then

Figure 2.17 Location of characteristic roots and the corresponding characteristic modes.

2.5-3 Relationship Between BIBO and Asymptotic Stability

External stability is determined by applying an external input with zero initial conditions, while internal stability is determined by applying the nonzero initial conditions and no external input. This is why these stabilities are also called the zero-state stability and the zero-input stability, respectively.

Recall that h(t), the impulse response of an LTIC system, is a linear combination of the system characteristic modes. For an LTIC system, specified by Eq. (2.1), we can readily show that when a characteristic root λ*^k* is in the LHP, the corresponding mode e^λkt is absolutely integrable. In

contrast, if λ*^k* is in the RHP or on the imaginary axis, e^λkt is not absolutely integrable.† 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; that is, BIBO stability does not necessarily inform us about the internal stability of the system. For instance, if a system is uncontrollable and/or unobservable, some modes of the system are invisible and/or uncontrollable from the external terminals [3]. Hence, the stability picture portrayed by the external description is of questionable value. BIBO (external) stability cannot assure internal (asymptotic) stability, as the following example shows.

EXAMPLE 2.13 A BIBO-Stable but Asymptotically Unstable System

An LTID system consists of two subsystems S¹ and S² in cascade (Fig. 2.19). The impulse response of these systems are h1(t) and h2(t), respectively, given by

h_1(t) = \delta(t) - 2e^{-t}u(t)

and $h_2(t) = e^t u(t)$

Comment on the BIBO and asymptotic stability of the composite system.

\int_{-\infty}^{\infty} |e^{\lambda \tau} u(\tau)| d\tau = \int_{0}^{\infty} e^{\alpha \tau} d\tau = \begin{cases} -1/\alpha & \alpha < 0\\ \infty & \alpha \ge 0 \end{cases}

This conclusion is also valid when the integrand is of the form |t keλ*^t u*(t)|.

^† Consider a mode of the form eλ*^t* , where ^λ ⁼ ^α ⁺jβ. Hence, ^eλ^t ⁼ ^eα^t ejβ*^t* and ^|eλ*^t* | = ^eα^t . Therefore,

The composite system impulse response h(t) is given by

h(t) = h_1(t) * h_2(t) = h_2(t) * h_1(t) = e^t u(t) * [\delta(t) - 2e^{-t} u(t)]

= $e^t u(t) - 2 \left[ \frac{e^t - e^{-t}}{2} \right] u(t)$
= $e^{-t} u(t)$

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 e−^t u(t), without any hint of the dangerously unstable system the system is harboring within.

The composite system is BIBO-stable because its impulse response, e−^t u(t), is absolutely integrable. Observe, however, the subsystem S² has a characteristic root 1, which lies in the RHP. Hence, S² is asymptotically unstable. Eventually, S² will burn out (or saturate) because of the unbounded characteristic response generated by intended or unintended initial conditions, no matter how small. We shall show in Ex. 10.12 that this composite system is observable, but not controllable. If the positions of S¹ and S² were interchanged (S² followed by S1), the system is still BIBO-stable, but asymptotically unstable. In this case, the analysis in Ex. 10.12 shows that the composite system is controllable, but not observable.

This example shows that BIBO stability does not always imply asymptotic stability. However, asymptotic stability always implies BIBO stability.

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

EXAMPLE 2.14 Investigating Asymptotic and BIBO Stability

Investigate the asymptotic and the BIBO stability of LTIC system described by the following equations, assuming that the equations are internal system descriptions:

(a) (D+1)(D² +4D+8)y(t) = (D−3)x(t)
(b) (D−1)(D² +4D+8)y(t) = (D+2)x(t)
(c) (D+2)(D² +4)y(t) = (D² +D+1)x(t)
(d) (D+1)(D² +4)²y(t) = (D² +2D+8)x(t)

The characteristic polynomials of these systems are

(a) (λ+1)(λ² +4λ+8) = (λ+1)(λ+2−j2)(λ+2+j2)
(b) (λ−1)(λ² +4λ+8) = (λ−1)(λ+2−j2)(λ+2+j2)
(c) (λ+2)(λ² +4) = (λ+2)(λ−j2)(λ+j2)
(d) (λ+1)(λ² +4)² = (λ+2)(λ−j2)2(λ+j2)²

Consequently, the characteristic roots of the systems are (see Fig. 2.20):

(a) −1, −2±j2
(b) 1, −2±j2
(c) −2, ±j2
(d) −1, ±j2, ±j2

System (a) is asymptotically stable (all roots in LHP), system (b) is unstable (one root in RHP), system (c) is marginally stable (unrepeated roots on imaginary axis) and no roots in RHP, and system (d) is unstable (repeated roots on the imaginary axis). BIBO stability is readily determined from the asymptotic stability. System (a) is BIBO-stable, system (b) is BIBO-unstable, system (c) is BIBO-unstable, and system (d) is BIBO-unstable. We have assumed that these systems are controllable and observable.

DR ILL 2.15 Assessing Stability by Characteristic Roots

For each case, plot the characteristic roots and determine asymptotic and BIBO stabilities. Assume the equations reflect internal descriptions.

(a) D(D+2)y(t) = 3x(t)
(b) D²(D+3)y(t) = (D+5)x(t)
(c) (D+1)(D+2)y(t) = (2D+3)x(t)
(d) (D² +1)(D² +9)y(t) = (D² +2D+4)x(t)
(e) (D+1)(D² −4D+9)y(t) = (D+7)x(t)

ANSWERS

(a) Marginally stable, but BIBO-unstable
(b) Unstable in both senses
(c) Stable in both senses
(d) Marginally stable, but BIBO-unstable
(e) Unstable in both senses.

IMPLICATIONS OF STABILITY

All practical signal-processing systems must be asymptotically stable. Unstable systems are useless from the viewpoint of signal processing because any set of intended or unintended initial conditions leads to an unbounded response that either destroys the system or (more likely) leads it to some saturation conditions that change the nature of the system. Even if the discernible initial conditions are zero, stray voltages or thermal noise signals generated within the system will act as initial conditions. Because of exponential growth of a mode or modes in unstable systems, a stray signal, no matter how small, will eventually cause an unbounded output.

Marginally stable systems, though BIBO unstable, do have one important application in the oscillator, which is a system that generates a signal on its own without the application of an external input. Consequently, the oscillator output is a zero-input response. If such a response is to be a sinusoid of frequency ω0, the system should be marginally stable with characteristic roots at ±jω0. Thus, to design an oscillator of frequency ω0, we should pick a system with the characteristic polynomial (λ−jω0)(λ+jω0) = λ² +ω⁰ 2. A system described by the differential equation

(D2 + \omega_02)y(t) = x(t)

will do the job. However, practical oscillators are invariably realized using nonlinear systems.