Simulation diagram, signalflow graph

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Simulation diagram, signal-flow graph
state-variable model

• Digital Control Theory
• Lecture 3

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Outline

• Overview graphical representation
• Basic element in graphical representation
• Simulation diagram and signal-flow graph
• State-variable method
• Discrete-time state space equation
• From z-transform TF to state-variable model via
  signal-flow graph
• Canonical forms
• Similarity transformation

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Overview graphical representation

• In the study of continuous-time systems, we often
  resort to graphical means for showing the
  input-output relationship between controlling
  variables and controlled variables. We have
  studied block diagrams and signal-flow graphs in
  the subject CON. The graphical representation
  greatly facilitates system analysis and design.
  Same idea is applicable to discrete-time systems.
  
• The graphical representation used for
  discrete-time LTI systems includes the following
  two equivalent diagrams
• Simulation diagram (block diagram)
• Signal-flow graph

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Basic element in graphical representation

• Recall that a discrete-time LTI system can be
  described by a difference equation of the form

The basic element used to construct simulation
diagrams for systems described by difference
equations is a shift register, also known as time
delay. Every T seconds, a number is shifted into
the register, and at that instant, the number
that was stored in the register is shifted out.
A symbolic representation of this memory device
is shown below.
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Basic element in graphical representation (contd)

• Consider the time-delay device shown in the
  figure. Using the real translation property, the
  z-transform of e(k-1) is

Hence the transfer function of the time-delay
element is z-1.
What is the basic element of block diagrams or
signal-flow graphs for continuous-time systems?
What is the transfer function for that element?
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z-transform transfer function
Both block diagrams and signal-flow graphs are
closely associated with transfer functions (TF).
For continuous-time LTI systems, the Laplace
transform is taken to obtain the TF from the
differential equation that represents the system
under consideration. Accordingly, for
discrete-time LTI systems, taking the z-transform
will yield the TF of the difference equation that
represents the system under consideration.
Consider a general nth-order difference equation
where m(k) denotes the output sequence and
e(k) input sequence. Taking the z-transform of
this equation yields
This difference equation may then be represented
by the TF
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Simulation diagram

• The simulation diagram of the difference equation
  

is given in the figure.
This is only one of many simulation diagrams that
can represent the above difference equation.
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We have derived the z-transform TF of the
difference equation
which is
The signal-flow graph of this TF can be drawn
using Masons gain formula. Given their
importance, let us review main points about the
signal-flow graph and Masons gain formula.
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Review signal-flow graph
A signal-flow graph consists of a network in
which nodes are connected by directed branches.
It depicts the flow of signals from one point of
a system to another and gives the relationships
among the signals.

• Node - a point representing a signal or variable.
• Branch unidirectional line segment joining two
  nodes.
• Path a branch or a continuous sequence of
  branches that can be traversed from one node to
  another node.
• Loop a closed path that originates and
  terminates on the same node and along the path no
  node is met twice.
• Nontouching loops two loops are said to be
  nontouching if they do not have a common node.

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Review Masons gain formula.
The linear dependence Tij between input signal xi
and output signal xj is
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Signal-flow graph of the z-transform TF
A special case of the Masons gain formula is
when all feedback loops are touching and also
touch all forward paths. In this case, Masons
gain formula is simplified to
Coming back to the z-transform transfer function
we compare it with the simplified Masons gain
formula. To draw the signal-flow graph of this
TF, we need to consider

• forward paths
• feedback loops
• time-delay elements

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Signal-flow graph of the z-transform TF (contd)
One form of the signal-flow graph of
is shown in the figure. Note that this
signal-flow graph representation is non-minimal
in that the flow graph contains 2n delays, but an
nth-order system can be represented with only n
delays.
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State-variable method
One important application of simulation diagrams
or signal-flow graphs is to develop
state-variable models from TFs. Before studying
the connection between the state-variable model
and graphical representation, we shall review
some basics about the state-variable method.

• Modern control theory is based on state space
  description and is applicable to systems, which
  may be linear or nonlinear, time invariant or
  time varying.
• A modern complex system may have many inputs and
  outputs, which are interrelated in a complicated
  manner. The analysis of such a MIMO
  (multiple-input multiple-output) system is
  facilitated by digital computers. The state
  space approach to system analysis is best suited
  from this viewpoint.

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State-variable method (contd)

• State. The state of a dynamic system is the
  smallest set of variables (called state
  variables) such that the knowledge of these
  variables and the input functions completely
  determines the system behaviour.
• State vector. The vector whose components are
  state variables is called a state vector.
• State space. The n-dimensional space consisting
  of the x1 axis, the x2 axis, the xn axis, where
  x1, x2, xn are state variables, is called a
  state space. Any state can be represented by a
  point in the state space.
• In state space representation, there are state
  vector, input vector and output vector.

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Discrete-time state space equation
A symbolic state-variable representation for a
discrete-time system is shown in the figure. The
state space equation of the system is
where
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Discrete-time state space equation (contd)
If the discrete-time system is linear, the state
space equation
reduces to
If the system is LTI, then matrices in the above
equation are all constant, and hence the state
space equation further reduces to
state equation
output equation
which is the discrete-time LTI state space
representation that we use throughout the course.
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State-variable model
We now come back to our original task how to
obtain a state-variable model from a given
z-transform TF written in the general form
Here we assume that the order of the numerator is
less than that of the denominator.
Introduce the auxiliary variable E(z) and rewrite
G(z) as
From this expression we obtain two equations
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A signal-flow graph of the system can be easily
derived from the above equations. Or
alternatively, draw the signal-flow graph
directly without using the auxiliary variable
E(z) if you are familiar with the correspondence
between the TF and Masons gain formula. Notice
the similarity of the two entities below.
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From z-transform TF to state-variable model via
signal-flow graph
The signal-flow graph sets up the link from
z-transform TFs to state-variable models. This
is one way of obtaining state-variable models
from TFs. We present the procedure below.

• Draw a signal-flow graph from the z-transform TF
  using the given method.
• Assign a state variable to each delay output.
• Write the state space equation for each delay
  input and each system output in terms of only the
  delay outputs and the system input.

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From z-transform TF to state-variable model via
signal-flow graph (contd)
We have derived the signal-flow graph of the
z-transform TF
Following the procedure presented in the previous
slide, the state space equation is
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Control canonical form or phase variable
canonical form
The state space equation is written in matrix
form as
The control canonical form corresponds to
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Example - Control canonical form
Given the following TF, derive a signal-flow
graph and corresponding state space equation in
control canonical form.
Rewriting the given TF and using Masons gain
formula, we can draw the signal-flow graph in
control canonical form. Following the procedure,
we obtain the state space equation.
control canonical form
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Observer canonical form
The z-transform TF
may have state space representation in other
canonical forms, e.g., the simulation diagram (or
block diagram) of the observer canonical form is
shown below.
Using Masons gain formula, can you verify the TF
of the system?
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Similarity transformation
Given a transfer function, there exist infinitely
many input-output equivalent state-variable
models. All these state-variable models are
linked through similarity transformation.
Consider the state space equation
we can apply the following linear transformation,
where P is a constant n by n matrix and w(k) is
the new state vector. It is necessary that P-1
exists. Then the similarity transformation leads
to
which is a new state space equation.
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Similarity transformation (contd)
The characteristic equation of a matrix A is
defined by the determinant

• and the eigenvalues or characteristic values of
  the matrix are the roots of the characteristic
  equation.
• A similarity transformation P-1AP has the
  following properties
• The eigenvalues of the matrix are unchanged under
  the similarity transformation.
• The determinant of (P-1AP) (which is the product
  of eigenvalues) is equal to the determinant of A.
• The trace of (P-1AP) is equal to the trace of A

Trace of a matrix is the sum of the diagonal
elements and is equal to the sum of its
eigenvalues.