# Control Systems/System Identification

## Systems

Systems, in one sense, are devices that take input and produce an
output. A system can be thought to **operate** on the input to produce
the output. The output is related to the input by a certain relationship
known as the **system response**. The system response usually can be
modeled with a mathematical relationship between the system input and
the system output.

## System Properties

Physical systems can be divided up into a number of different
categories, depending on particular properties that the system exhibits.
Some of these system classifications are very easy to work with and have
a large theory base for analysis. Some system classifications are very
complex and have still not been investigated with any degree of success.
By properly identifying the properties of a system, certain analysis and
design tools can be selected for use with the system.

The early sections of this book will focus primarily on **linear
time-invariant** (LTI) systems. LTI systems are the easiest class of
system to work with, and have a number of properties that make them
ideal to study. This chapter discusses some properties of systems.

Later chapters in this book will look at time variant systems and
nonlinear systems. Both time variant and nonlinear systems are very
complex areas of current research, and both can be difficult to analyze
properly. Unfortunately, most physical real-world systems are
time-variant, nonlinear, or both.

An introduction to system identification and least squares techniques
can be found
here.
An introduction to parameter identification techniques can be found
here.

## Initial Time

The **initial time** of a system is the time before which there is no
input. Typically, the initial time of a system is defined to be zero,
which will simplify the analysis significantly. Some techniques, such as
the **Laplace Transform** require that the initial time of the system be
zero. The initial time of a system is typically denoted by t~0~.

The value of any variable at the initial time t~0~ will be denoted with
a 0 subscript. For instance, the value of variable x at time t~0~ is
given by:

$$x(t_0) = x_0$$

Likewise, any time t with a positive subscript are points in time *after
t~0~*, in ascending order:

$$t_0 \le t_1 \le t_2 \le \cdots \le t_n$$

So t~1~ occurs after t~0~, and t~2~ occurs after both points. In a
similar fashion above, a variable with a positive subscript (unless
specifying an index into a vector) also occurs at that point in time:

$$x(t_1) = x_1$$

$$x(t_2) = x_2$$

This is valid for all points in time t.

## Additivity

A system satisfies the property of **additivity** if a sum of inputs
results in a sum of outputs. By definition: an input of
$x_3(t) = x_1(t) + x_2(t)$ results in an output of
$y_3(t) = y_1(t) + y_2(t)$. To determine whether a system is additive,
use the following test:

Given a system f that takes an input x and outputs a value y, assume two
inputs (x~1~ and x~2~) produce two outputs:

$$y_1 = f(x_1)$$

$$y_2 = f(x_2)$$

Now, create a composite input that is the sum of the previous inputs:

$$x_3 = x_1 + x_2$$ Then the system is additive if the following
equation is true:

$$y_3 = f(x_3) = f(x_1 + x_2) = f(x_1) + f(x_2) = y_1 + y_2$$

Systems that satisfy this property are called **additive**. Additive
systems are useful because a sum of simple inputs can be used to analyze
the system response to a more complex input.

### Example: Sinusoids

## Homogeneity

A system satisfies the condition of **homogeneity** if an input scaled
by a certain factor produces an output scaled by that same factor. By
definition: an input of $ax_1$ results in an output of $ay_1$. In other
words, to see if function *f()* is **homogeneous**, perform the
following test:

Stimulate the system *f* with an arbitrary input *x* to produce an
output *y*:

$$y = f(x)$$

Now, create a second input *x~1~*, scale it by a multiplicative factor
*C* (*C* is an arbitrary constant value), and produce a corresponding
output *y~1~*:

$$y_1 = f(Cx_1)$$

Now, assign x to be equal to *x~1~*:

$$x_1 = x$$

Then, for the system to be homogeneous, the following equation must be
true:

$$y_1 = f(Cx) = Cf(x) = Cy$$

Systems that are homogeneous are useful in many applications, especially
applications with gain or amplification.

### Example: Straight-Line

## Linearity

A system is considered **linear** if it satisfies the conditions of
Additivity and Homogeneity. In short, a system is linear if the
following is true:

Take two arbitrary inputs, and produce two arbitrary outputs:

$$y_1 = f(x_1)$$

$$y_2 = f(x_2)$$

Now, a linear combination of the inputs should produce a linear
combination of the outputs:

$$f(Ax_1 + Bx_2) = f(Ax_1) + f(Bx_2) = Af(x_1) + Bf(x_2) = Ay_1 + By_2$$

This condition of additivity and homogeneity is called
**superposition**. A system is linear if it satisfies the condition of
superposition.

### Example: Linear Differential Equations

## Memory

A system is said to have **memory** if the output from the system is
dependent on past inputs (or future inputs!) to the system. A system is
called **memoryless** if the output is only dependent on the current
input. Memoryless systems are easier to work with, but systems with
memory are more common in digital signal processing applications.

Systems that have memory are called **dynamic** systems, and systems
that do not have memory are **static** systems.

## Causality

Causality is a property that is very similar to memory. A system is
called **causal** if it is only dependent on past and/or current inputs.
A system is called **anti-causal** if the output of the system is
dependent only on future inputs. A system is called **non-causal** if
the output depends on past and/or current and future inputs.

## Time-Invariance

A system is called **time-invariant** if the system relationship between
the input and output signals is not dependent on the passage of time. If
the input signal $x(t)$ produces an output $y(t)$ then any time shifted
input, $x(t + \delta)$, results in a time-shifted output $y(t + \delta)$
This property can be satisfied if the transfer function of the system is
not a function of time except expressed by the input and output. If a
system is time-invariant then the system block is commutative with an
arbitrary delay. This facet of time-invariant systems will be discussed
later.

To determine if a system f is time-invariant, perform the following
test:

Apply an arbitrary input x to a system and produce an arbitrary output
y:

$$y(t) = f(x(t))$$

Apply a second input x~1~ to the system, and produce a second output:

$$y_1(t) = f(x_1(t))$$

Now, assign x~1~ to be equal to the first input x, time-shifted by a
given constant value δ:

$$x_1(t) = x(t - \delta)$$

Finally, a system is time-invariant if y~1~ is equal to y shifted by the
same value δ:

$$y_1(t) = y(t - \delta)$$

## LTI Systems

A system is considered to be a **Linear Time-Invariant** (LTI) system if
it satisfies the requirements of time-invariance and linearity. LTI
systems are one of the most important types of systems, and they will be
considered almost exclusively in the beginning chapters of this book.

Systems which are not LTI are more common in practice, but are much more
difficult to analyze.

## Lumpedness

A system is said to be **lumped** if one of the two following conditions
are satisfied:

1.  There are a finite number of states that the system can be in.
2.  There are a finite number of state variables.

The concept of \"states\" and \"state variables\" are relatively
advanced, and they will be discussed in more detail in the discussion
about **modern controls**.

Systems which are not lumped are called **distributed**. A simple
example of a distributed system is a system with delay, that is,
$A(s)y(t)=B(s)u(t-\tau)$, which has an infinite number of state
variables (Here we use $s$ to denote the Laplace variable). However,
although distributed systems are quite common, they are very difficult
to analyze in practice, and there are few tools available to work with
such systems. Fortunately, in most cases, a delay can be sufficiently
modeled with the Pade approximation. This book will not discuss
distributed systems much.

## Relaxed

A system is said to be **relaxed** if the system is causal and at the
initial time t~0~ the output of the system is zero, i.e., there is no
stored energy in the system. The output is excited solely and uniquely
by input applied thereafter.

$$y(t_0) = f(x(t_0)) = 0$$

In terms of differential equations, a relaxed system is said to have
\"zero initial states\". Systems without an initial state are easier to
work with, but systems that are not relaxed can frequently be modified
to approximate relaxed systems.

## Stability

**Stability** is a very important concept in systems, but it is also one
of the hardest function properties to prove. There are several different
criteria for system stability, but the most common requirement is that
the system must produce a finite output when subjected to a finite
input. For instance, if 5 volts is applied to the input terminals of a
given circuit, it would be best if the circuit output didn\'t approach
infinity, and the circuit itself didn\'t melt or explode. This type of
stability is often known as \"**Bounded Input, Bounded Output**\"
stability, or **BIBO**.

There are a number of other types of stability, most of which are based
on the concept of BIBO stability. Because stability is such an important
and complicated topic, an entire section of this text is devoted to its
study.

## Inputs and Outputs

Systems can also be categorized by the number of inputs and the number
of outputs the system has. Consider a television as a system, for
instance. The system has two inputs: the power wire and the signal
cable. It has one output: the video display. A system with one input and
one output is called **single-input, single output**, or SISO. a system
with multiple inputs and multiple outputs is called **multi-input,
multi-output**, or MIMO.

These systems will be discussed in more detail later.

