Control Systems/Stability

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| ← Nichols Charts                                         | State-Space Stability → | Glossary     |             |

Stability

When a system becomes unstable, the output of the system approaches infinity (or negative infinity), which often poses a security problem for people in the immediate vicinity. Also, systems which become unstable often incur a certain amount of physical damage, which can become costly. This chapter will talk about system stability, what it is, and why it matters.

The chapters in this section are heavily mathematical, and many require a background in linear differential equations. Readers without a strong mathematical background might want to review the necessary chapters in the Calculus and Ordinary Differential Equations books (or equivalent) before reading this material. Negativeness of any coefficient of a characteristic polynomial indicates that the system is either unstable or at most marginally stable. If any coefficient is zero/negative then we can say that the system is unstable.

BIBO Stability

A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output over the time interval $[t_0, \infty)$. This must hold for all initial times to. So long as we don't input infinity to our system, we won't get infinity output.

A system is defined to be uniformly BIBO Stable if there exists a positive constant k that is independent of t0 such that for all t0 the following conditions:

$\|u(t)\| \le 1$
$t \ge t_0$

implies that

$\|y(t)\| \le k$

There are a number of different types of stability, and keywords that are used with the topic of stability. Some of the important words that we are going to be discussing in this chapter, and the next few chapters are: BIBO Stable, Marginally Stable, Conditionally Stable, Uniformly Stable, Asymptoticly Stable, and Unstable. All of these words mean slightly different things.

Determining BIBO Stability

We can prove mathematically that a system f is BIBO stable if an arbitrary input x is bounded by two finite but large arbitrary constants M and -M:

$-M < x \le M$

We apply the input x, and the arbitrary boundaries M and -M to the system to produce three outputs:

$y_x = f(x)$
$y_M = f(M)$
$y_{-M} = f(-M)$

Now, all three outputs should be finite for all possible values of M and x, and they should satisfy the following relationship:

$y_{-M} \le y_x \le y_M$

If this condition is satisfied, then the system is BIBO stable.

For a SISO linear time-invariant (LTI) system is BIBO stable if and only if $g(t)$ is absolutely integrable from [0,∞] or from:

$\int_{0}^{\infty} |g(t)| \,dt \leq M < {\infty}$

Example

Consider the system:

$h(t) = \frac{2}{t}$

We can apply our test, selecting an arbitrarily large finite constant M, and an arbitrary input x such that -M < x < M.

As M approaches infinity (but does not reach infinity), we can show that:

$y_{-M} = \lim_{M \to \infty} \frac{2}{-M} = 0^-$

And:

$y_M = \lim_{M \to \infty} \frac{2}{M} = 0^+$

So now, we can write out our inequality:

$y_{-M} \le y_x \le y_M$
$0^- \le x < 0^+$

And this inequality should be satisfied for all possible values of x. However, we can see that when x is zero, we have the following:

$y_x = \lim_{x \to 0} \frac{2}{x} = \infty$

Which means that x is between -M and M, but the value yx is not between y-M and yM. Therefore, this system is not stable.

Poles and Stability

When the poles of the closed-loop transfer function of a given system are located in the right-half of the S-plane (RHP), the system becomes unstable. When the poles of the system are located in the left-half plane (LHP), the system is shown to be stable. A number of tests deal with this particular facet of stability: The Routh-Hurwitz Criteria, the Root-Locus, and the Nyquist Stability Criteria all test whether there are poles of the transfer function in the RHP. We will learn about all these tests in the upcoming chapters.

If the system is a multivariable, or a MIMO system, then the system is stable if and only if every pole of every transfer function in the transfer function matrix has a negative real part. For these systems, it is possible to use the Routh-Hurwitz, Root Locus, and Nyquist methods described later, but these methods must be performed once for each individual transfer function in the transfer function matrix.

Poles and Eigenvalues

Note:
Every pole of G(s) is an eigenvalue of the system matrix A. However, not every eigenvalue of A is a pole of G(s).

The poles of the transfer function, and the eigenvalues of the system matrix A are related. In fact, we can say that the eigenvalues of the system matrix A are the poles of the transfer function of the system. In this way, if we have the eigenvalues of a system in the state-space domain, we can use the Routh-Hurwitz, and Root Locus methods as if we had our system represented by a transfer function instead.

On a related note, eigenvalues and all methods and mathematical techniques that use eigenvalues to determine system stability only work with time-invariant systems. In systems which are time-variant, the methods using eigenvalues to determine system stability fail.

Transfer Functions Revisited

We are going to have a brief refesher here about transfer functions, because several of the later chapters will use transfer functions for analyzing system stability.

Let us remember our generalized feedback-loop transfer function, with a gain element of K, a forward path Gp(s), and a feedback of Gb(s). We write the transfer function for this system as:

$H_{cl}(s) = \frac{KGp(s)}{1 + H_{ol}(s)}$

Where $H_{cl}$ is the closed-loop transfer function, and $H_{ol}$ is the open-loop transfer function. Again, we define the open-loop transfer function as the product of the forward path and the feedback elements, as such:

$H_{ol}(s) = KGp(s)Gb(s)$

Now, we can define F(s) to be the characteristic equation. F(s) is simply the denominator of the closed-loop transfer function, and can be defined as such:

[Characteristic Equation]

$F(s) = 1 + H_{ol} = D(s)$

We can say conclusively that the roots of the characteristic equation are the poles of the transfer function. Now, we know a few simple facts:

1. The locations of the poles of the closed-loop transfer function determine if the system is stable or not
2. The zeros of the characteristic equation are the poles of the closed-loop transfer function.
3. The characteristic equation is always a simpler equation than the closed-loop transfer function.

These functions combined show us that we can focus our attention on the characteristic equation, and find the roots of that equation.

State-Space and Stability

As we have discussed earlier, the system is stable if the eigenvalues of the system matrix A have negative real parts. However, there are other stability issues that we can analyze, such as whether a system is uniformly stable, asymptotically stable, or otherwise. We will discuss all these topics in a later chapter.

Marginal Stablity

When the poles of the system in the complex S-Domain exist on the complex frequency axis (the vertical axis), or when the eigenvalues of the system matrix are imaginary (no real part), the system exhibits oscillatory characteristics, and is said to be marginally stable. A marginally stable system may become unstable under certain circumstances, and may be perfectly stable under other circumstances. It is impossible to tell by inspection whether a marginally stable system will become unstable or not.

We will discuss marginal stability more in the following chapters.
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