# Control Systems/Stability

## Stability

When a system is unstable, the output of the system may be infinite even
though the input to the system was finite. This causes a number of
practical problems. For instance, a robot arm controller that is
unstable may cause the robot to move dangerously. Also, systems that are
unstable often incur a certain amount of physical damage, which can
become costly. Nonetheless, many systems are inherently unstable - a
fighter jet, for instance, or a rocket at liftoff, are examples of
naturally unstable systems. Although we can design controllers that
stabilize the system, it is first important to understand what stability
is, how it is determined, 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.

For most of this chapter we will be assuming that the system is linear
and can be represented either by a set of transfer functions or in state
space. Linear systems have an associated characteristic polynomial which
tells us a great deal about the stability of the system. If any
coefficient of the characteristic polynomial is zero or negative then
the system is either unstable or at most marginally stable. It is
important to note that even if all of the coefficients of the
characteristic polynomial are positive the system may still be unstable.
We will look into this in more detail below.

## 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 t~o~. 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 t~0~ such that for all t~0~
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**, **Asymptotically 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.

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

## 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) and the system is not improper, 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 and every transfer
function in the transfer function matrix is not improper. 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

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:

$$H_{ol}(s) = KGp(s)Gb(s)$$ \<\-\--Note this definition now contradicts
the updated definition in the Feedback
Loops
section.

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:

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

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 Stability

When the poles of the system in the complex s-domain exist on the
imaginary 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.

