EE 362K - Buckman

Fall 2002

**Topical
Overview**

This is an overview of the concepts and topics you need to
know ** and be able to apply** in order to get a good grade. For
brevity, no derivations or examples and minimal explanation and background are present
here. From time to time, I will add
hyperlinks to documents containing further examples and explanations.

**1. The
uncontrolled system, or “plant”:** a
system, described by rational polynomial transfer function, *G*, which is
NOT subject to modification by the designer (shown in blue). Its input and output are **continuous**
functions of time, *i(t) *and *o(t).*

**Diagram:**

** **Flow
of signals or information is left to right unless otherwise specified.

**Plant
Model :** a (linear) differential equation.

_{},

where the *a*’s
and the *b*’s are constants.

**Transfer
function:** Using LaPlace transforms,
you can transform the model to a rational polynomial transfer function,

_{}

Note that whether you work with the model in the time (*t*)
domain, or with the transfer function in the *s*-domain, knowledge of all
of the *a*’s and *b*’s implies a complete description of the
linear, time-invariant system.
Obtaining these models or transfer functions for systems of
technological interest is the subject of entire other courses, such as circuit
theory, electromechanics, dynamics, chemical kinetics, etc.

Text Reference: Ch. 1, Ch. 2.

Handout : Handout
1.1.

Additional help with mechanical
system modeling for ECE’s: mechsys.pdf

**2. Control system
design: adding systems to the “plant”
in order to control it. **Assuming
you can’t directly modify the output of the “plant”, you can modify the input,
and this modification can make use of knowledge of the output.

**Diagram:** A general representation of the use of
additional subsystems, whose design you **can** modify (shown in orange) is
shown below:

The subsystems, *C* and *F* have their own inputs
and outputs, and hence their own transfer functions. The control signal, *r(t)*, represents the output desired by the user. By common convention, the output of *F*
is subtracted from* r(t)* before being fed to *C* as input.

Text Reference: Section 3.1-3.1.8.

Handout 1-2: What can you learn from the transfer function?

**3. Sampled-data version of C or F.**

Nearly all modern control systems use digital computers to
implement the controllers. Computers
can neither accept continuous functions of time as inputs nor provide them as
outputs to the external (analog) world. All they can do is operate on sequences
of numbers (usually in very rapid succession). In order to interface a computer
to the analog world you must implement a system like the one shown below as a
replacement for subsystems *C* or *F* (or sometimes both).

Here the continuous function (solid line in the diagram) r(t), passes through a sample-and-hold subsystem, S&H. The relationship between the input and the output of the sample-and-hold is illustrated below:

Here the white curve is *r(t),* a continuous time function. The output of the S&H block is the blue
line with the stair-step structure. It
is continuous everywhere except when it jumps to new values. Between the jumps,
it is constant. If you listed the
vertical co-ordinates of the blue dots you would have a sequence of
numbers. The digital representations of
these numbers are the output of the analog-to-digital converter, A/D, and
constitute the sequence*, r _{k}*, shown as a dashed line on the
block diagram.

Once this
sequence is formed, the computer has lost all information on the values of *r(t)*
between sampling points. From the
picture of the input and output of the S&H block, you can see that the time
between successive samples, *T*,
is the reciprocal of the sampling frequency, *f _{s}*.

The
discrete subsystem, *D*, transforms the input sequence, *r _{k}*,
into another sequence,

**Discrete system model: **The
model for the discrete system (a digital filter), is the difference equation,

_{}.

**Relationship
of r_{k} to r(t):**
From the subsystem diagram and the plot, each element of the sequence is
related to a sample taken at the corresponding time as

_{}.

**Relationship
of o_{k} to o(t):**
From the subsystem diagram and the plot, each element of the sequence is
related to the constant output of the DAC during the corresponding time
interval as

_{},

where *u(t)* is the unit step function.

**Discrete system transfer
function: **You can use the
z-transform [see Text 8.1-8.2] on the difference
equation to get a discrete transfer function,

_{}.

_{}

_{}

**4. Loss of information
in sampled-data models – modeling a system that includes sampled-data blocks:** Note that knowledge of all of the a’s and
b’s provides a complete description of a linear system. Also, solving the difference equation
generates only the successive outputs, *o _{k}*, without providing
any information about what happens

**4.1. Effect of an upstream sample and hold on a system: **It would be convenient to find the
sampled-data system that is “equivalent” to some continuous system under study:
convenient, but impossible.
Sample&hold-ing will always have some effect on a system. The effect only vanishes if the sampling
frequency becomes infinite, a practical impossibility. Consider the system
below, with only a sample&hold and no other digital subsystems.

A linear
transfer function for this system in the *s*-domain cannot be written,
because of the non-linear nature of the sample&hold . However, you can
write its transfer function in the *z*-domain as

where the operation, ,
is a mathematical transformation from the *s* to the *z*-domain. The
input to this transformation is a rational polynomial in *s*, while the
output is a rational polynomial in *z*.

**The key difference between H(s) and H(z) for
any system is this: The **

More details on sampling: read handout H2.

**4.2. Transfer function, H(z), for an
equivalent digital filter, D(z), replacing subsystem C:** Suppose you replaced subsystem C with a
digital filter having a transfer function,

You can write the transfer function of this system as

,

which says that the digital filter transfer function you need to replicate the continuous subsystem C at a given sampling frequency is

.

More about Digital Controllers Inside Analog Feedback Loops

**5. Stability from the transfer function: **a stable
system produces bounded (finite) outputs from any bounded input.

**Continuous
system:** a continuous system is
stable if all the poles of *H(s)* lie in the left half of the complex
plane.

**Discrete
system:** a discrete system is stable of all the poles of *H(z) *lie
within a unit circle whose center is at the origin in the complex plane.

Alternative criteria for stability exist, but are not needed as long as the poles of the transfer function can be found. Pole locations determine stability.

**6. Relative settling time from the transfer function:** In general,

**Continuous
systems** respond faster to step-wise changes in the control signal, *r(t),*
if all poles of *H(s)* lie further to the left in the complex plane.

**Sampled-data
systems** respond faster to step-wise changes in the control signal, *r(t*),
if all poles of *H(z)* lie closer to center of a unit circle at the origin
in the complex plane.

This property lets you use pole locations as a rough design tool to determine stability and speed of response.

**7. Frequency domain analysis and design**

For systems
whose transfer functions have many poles and zeros, frequency-domain analysis
provides better insight into system performance than does observing pole and
zero locations. You calculate the magnitude, |*H*|, and phase, *f*, of system transfer functions vs. angular
frequency, *w = 2pf*, as follows:

**Continuous
systems:**

_{}

**Sampled-data
systems:**

_{}

where *T* is the sampling period. In the above equations, we just made the
substitution, ,
for continuous systems and for sampled-data systems.

The
objective of frequency-domain design is to achieve a magnitude, |*H(**w)|, *that satisfies the conditions illustrated
below, and is stable.

Good designs have |*H(**w)|* =1 within a small tolerance over a wide bandwidth.

** **

** **

**8. State-space analysis and design: **The usefulness of state-space analysis in
control system design arises primarily from three of its properties:

1) It lets you deal with systems having more than one feedback loop,

2) It lets you deal with systems having multiple inputs and outputs,

3) It lets you find a set of feedback parameters that place the system poles at any locations you want, if all the state variables are available for measurement (observable).

**8.1. Continuous systems:**

**Time-domain:** You can model
the dynamics of any linear system having:

P states (state vector **x** is
a column vector, length P),

Q inputs(input vector** i** is
a column vector, length Q), and

R outputs, (output vector **o**
is a column vector, length R).

with a set of simultaneous, first-order linear differential
equations in the set of state variables (termed the state vector),** x**, in
the form,

, the state equation,

along with a linear output equation,

.

The components of these equations have the following forms,

·
**A** is a P x P square matrix, called the *state*
matrix,

·
**B** is a P x Q rectangular matrix, called the*
input* matrix

·
**C** is an R x P rectangular matrix, called the *output*
matrix

·
**D** is an R x Q rectangular matrix, called *the
feedforward* matrix.

You design the control of a state-variable system by feeding back a weighted sum of the states to each system input can control (there may be inputs you cannot control: these are disturbances).

** s-Domain:** You can take the LaPlace transform of the
state equation to get

,

and then solve for the output in terms of the input as

,

where I is the square identity matrix. This equation relates all of the inputs to
all of the outputs in the *s*-domain, and thus plays the role of a
transfer function, even though it can’t be written as a simple ratio of
polynomials.
[For SISO systems (only one input and one output) you get a rational
polynomial transfer function. To see
how state variable models apply to SISO systems, including some simple
examples, read the handout on that subject.]

The poles of this system are the roots of the equation,

**Manipulating
pole locations with state feedback: **Start with the state-variable
representation of a system as shown below, with blue representing the
uncontrolled system or “plant” as before:

Assuming that all states are observable, you can feed back a weighted sum of these states to each input, allowing for the fact that in general, this weighting can be different for each accessible input. Now you have the system shown below,

which can be modeled by the state equation

where **K** is in general a Q x P rectangular
matrix. Obviously you can rearrange
this equation so it has the form of a state equation with a new state matrix, **A _{f}**,
as

,

whose poles are the roots of

The state-variable
approach to control system design consists of adjusting the elements of the
state feedback matrix, **K**, to produce desired pole locations consistent
with other design specifications.

**8.2. Discrete and sampled-data
systems**

** Time-domain: **You could also write state equations for
any system in the form of a set of difference equations, giving

for the state equation and

for the output equation.
The dimensions of the matrices and vectors are the same as for the
continuous case. The vectors are now sequences, and **the
matrices now depend on the sampling frequency as well as on the parameters of
the system**.

** z-Domain**: Taking the z-transform of the state equation
yields

,

from which you can get the transfer function-like matrix relationship,

,

The poles of the uncontrolled system are the roots of

**Manipulating pole locations with
state feedback: **Similarly to state
feedback for a continuous system, you can choose a feedback matrix, **K**,
to adjust pole locations. **K** has
the same dimensions as for the continuous case, and the new pole locations with
feedback are the roots of

**8.3. Frequency-domain design for state-variable systems**

There is a transfer-function relationship between each output and each input which you can evaluate in the frequency domain. Write the matrix input-output relations for both cases as

, and substitute , OR

and substitute .

You can use transfer functions you get from these operations to design a control system with multiple inputs and outputs.