Automatic Control Theory

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Automatic Control Theory

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Title: Automatic Control Theory

1
Automatic Control Theory

Taught by
Prof. ZHANG Duanjin
School of Information Engineering,
Zhengzhou University, P R China
Email djzhang_at_zzu.edu.cn

2
Chapter2 System Modeling

2.1 Mathematical models
2.2 Dynamics equation
2.3 State space and state equation
2.4 Linear differential equation
2.5 State equation
2.6 Transfer function(matrix)
2.7 Transfer function of closed-loop system
2.8 Fundamental parts
2.9 Signal flow graphs
2.10 Impulse response and step response
2.11 Summary Excises

3
2.1 Mathematical Models

The fundamental concept of mathematical
model
The fundamental forms of mathematical
models
The method of modeling
The steps of analyzing and studying a
dynamic system
Instructional objectives
Appendix Property of linear system

4
1.Mathematical model
If the dynamic behavior of a physical system
can be represented by an equation, or a set of
equations, this is referred to as the
mathematical model of the system. Model of
system the relationship between variables in
system. Why do we must study the model of
control system? Quantitative mathematical
models must be obtained to understand and control
complex dynamic systems. (analysis and design)
5
2.The fundamental forms of mathematical model

Mathematical expression
Differential Equation (time domain model) are
used because the systems are dynamic in nature.
Impulse transfer function
state-space equation.
Assumptions are needed because of the
complexity of systems and the ignorance of all
the relevant factors..

Transfer function(s domain model)
Laplace transform is utilized to simplify the
method of solution for linear equations.
Block diagram and Signal flow diagram
Response Curve (non parameter model)
frequency response curve Bode diagram

7
3.The method of modeling

Analytical method It can be constructed from
knowledge of the physical characteristics of the
system
Experimental method
Others. (NN)
Linear Time-invariant (constant) Parameter-lumped

8
4.The steps of analyzing and studying a dynamic
system

Define the system and its components.
Formulate the mathematic model and list the
necessary assumptions.
Write the differential equations describing the
model.
Solve the equations for the desired output
variables.
Examine the solutions and the assumptions.
If necessary, reanalyze or redesign the system.

5.Instructional objectives
Develop dynamic models of physical
components.
Derivation of transfer functions.
Block diagram representation.
Block diagram rules and simplify the block
diagram to determine the closed-loop transfer
function.

10
6.Appendix Property of linear system
The mathematical model of a system is linear, if
it obeys the principle of superposition (or
principle of homogeneity). This principle
implies that if a system model has responses
y1(t) and y2(t) to any two inputs x1(t) and x2(t)
respectively, then the system response to the
linear combination of these inputs ax1(t)
bx2(t) is given by the linear combination of the
individual outputs, i.e. ay1(t) by2(t).
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Figure2.1 Meaning of a linear system
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12
2.2 Dynamics equation
2.2.1 Derivation of the differential
equations 2.2.2 Linear approximations of
nonlinear equation 2.2.3 The differential
equations of complex plants 2.2.4 Single
variable differential equations derivation of
original equations 2.2.5 Dynamics equation of
discrete-time system

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2.2.1 Derivation of the differential equations

In order to analyze the behavior of physical
systems in time domain, we write the differential
equations representing those systems
Electrical systems (KVL and/or KCL, a
electric network can be modeled as a set of nodal
equations using Kirchhoffs current law or
Kirchhoffs voltage law).

Mechanical systems (Newtons laws of motion)
Hydraulic systems (Thermodynamic
Conservation of matter)
Thermal systems (Heat transfer laws,
Conservation of energy)
The standard differential equation is

output
input
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The number of independent energy storage
components in a system determine the order of
that system.
An nth order differential equation implies n
independent energy storing components and you
need n initial conditions.
where nm for a physically realizable system.

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Example.2.1 Displacement system of
spring-mass-damper
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Example 2.2 Differential equation of a R-L-C
network.
uc(t) is the output voltage and u(t) is the
voltage source.

i(t)
Figure2.2 R-L-C network
25

The equation of the RLC network shown in
Fig.2.2 is obtained by writing the Kirchhoff
voltage equation, yielding

Therefore, solving Eq.(2) for i and
substituting in Eq.(1), we have
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are the time constants of the network.
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Analogous variables and systems
Rewriting the following equations in terms of
displacement and voltage respectively
It is obvious that they are equivalent.
Displacement x(t) and voltage uc(t) are
equivalent variables, usually called analogous
variables, and the systems are analogous systems.
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The concept of analogous systems is a very
useful and powerful technique for system
modeling.
Analogous systems with similar solutions exist
for electrical, mechanical, thermal, and fluid
systems.
The analyst can extend the solution and the
understanding of one system to all analogous
systems with the same describing differential
equations.
Analogous systems have the same form solution.

Conclusion
Different physical system can gain similar
differential equations.
The differential equations reflect the essence
characteristics of system.

30
Example 2.3 Differential equation of dc
motor.
dc motor wiring diagram
31

The armature current is related to the input
voltage applied to the armature as
where Ea is the back electromotive-force
voltage proportional to the motor speed, U is
input voltage, Ia is the armature current,,La is
the motor inductance, Ra is motor resistance.

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M is the motor torque, ML is the load torque,
J is the rotor inertia, O is the (angular)
velocity of the motor bearings.
where kd is defined as the motor constant.
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There are two first-order differential
equations and two algebraic equations, and six
variables U, Ea,, Ia, M, ML, J,O. U and ML are
the input variables, which lead to the motors
movement. If we regard as O be the output
variable, others be the middle variables. The
differential equation of the motor-load
combination is

is the field time constant
of the armature
is the time constant of
the motor armature
Generally,
If Ta is neglected, the equation is

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2.2.2 Linear Approximations of nonlinear
equation
A great majority of physical systems are linear
within some range of the variables. However, all
systems ultimately become nonlinear as the
variables are increased without limit. A system
is defined as linear in terms of the system
excitation and response.
Keywords operating point small-signal
conditions Taylor series continuous
linear approximation
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The relationship of two variables written as
, where f(x) indicates y is a
function of x. The normal operating point is
designated by x0. Because the curve (function) is
continuous over the range of interest, a Taylor
series expansion about the operating point may be
utilized. Then we have

The slope at the operating point,
is a good approximation to the curve over a
small range of (x-x0)(small perturbation), the
deviation from the operating point. Then, as a
reasonable approximation, the equation can be
rewritten as the linear equation
where m is the slope at the operating point.

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Example 2.4 Nonlinear differential equation
Operating point(T0,u0)
Small change

Linear approximation of differential equation

If the dependent variable y depends upon
several excitation variables, then the functional
relationship is written as
The Taylor series expansion about the
operating point x0(x10, x20,, xn0) is useful for
a linear approximation to the nonlinear function.
When the higher-order terms are neglected, the
linear approximation is written as

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2.2.3 The differential equations of complex
plants

Generally steps
(1)Confirm the I/O variables of system and
every components
(2)Write the dynamics equations of every
components using physical laws
(3)Check up the number of equations and decide
if it is equal to the number of unknown
variables or not.
(4)Expurgate the inside variables from the set
of the differential equations, and change to the
standard form.

Example 2.5 Servo-system
Work principle Two same changeable resistor are
supplied by the same dc electrical source The
arm of resistor 1 can rotate by handle 3.
Supposing ??f represent the position of the arm
of the two resistors respectively. If ?is not
equal tof, then the error signal up is formed,
which be amplified by 4, finally the field
current If of the dc dynamotors field winding is
formed, which lead to the voltages change of
dynamotor 5. So the dc motor 8 rotates, which
lead to the load 10 rotates too by the gear 9.
So the arm of resistor 2 starts to move, as far
as?f.

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up
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Write out dynamics equations of every components
(1)the set of changeable resistances
Inputs?,f outputup
(1)
Where kd is the coefficient of the resistance.
(2)amplifier
Inputup , outputIf
or (2)
where ka is the magnified multiple of voltage
Rf is the sum resistance of output circuit Lf is
inductance of field winding 6 TfLf /Rf is the
time constant of field circuit.

(3)dynamotor-motor
InputIf output O.
From example 2.2
(3)
(4)drive institution
InputOoutputf.
(4)
where kt is the drive ratio of the drive
institution.

Equation(1)(4)are the mathematics models of
the servo-system. It consists of one second-order
differential equation, two first-order
differential equations and one algebraic
equation. There are six variables such as??f? up
? If ? ML ?O.To the whole system, the inputs are
? and ML, others are the inside dependent
variables, and the number is equal to the number
of the equations.

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2.2.4 Scalar differential equations derivation
of original equations
(1)Confirm the I/O variables of system and every
components (2)Write the dynamics equations of
every components using physical laws (3)Check up
the number of equations and decide if it is
equal to the number of unknown variables or
not. (4)Expurgate the inside variables from the
set of the differential equations, and change to
the standard form.

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2.2.5 Dynamics equation of discrete-time system

The discrete-time approximation is based on the
division of the time axis into sufficiently small
time increments. The values of the input/output
are evaluated at the successive time intervals,
that is, t0,T,2T,, where T is the increment of
time T. If the time increment T is
sufficiently small compared with the time
constants of the system, the discrete-time series
will be reasonably accurate.

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Difference equation reflect the relationship of
the input and output series at the every sample
points of the discrete time system. Using T
denotes the space between two sampling time.
Difference equation describes the movement of
these plants. Details in chapter nine.
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General backward difference equation of linear
time-invariant system
or the forward difference equation

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2.3 State space and state equation

2.3.1 State vector and state space
2.3.2 State equation and output equation
2.3.3 State equations derivation of original
equations

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2.3.1 State vector and state space

Dynamic system a system that can store
input information.
Example

m the mass a(t) the acceleration of m at time
t F(t) the force applied to m for time
t0,t v(t) the velocity of m at time
t x(t)the displacement of m.
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According to Newtons law ,we have

Where x(t0) and v(t0) are the initial states
value.
If we choose x(t) and v(t) as two state
variables,then
x(t0-), v(t0-) F(t)(tgtt0)_
x(t) and v(t) can describe the behavior of
system completely.
determine
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Statethe minimum set of variables (called the
state variables)which at some initial time t0 ,
together with the inputs signal u(t) for time
,suffices to determine the future behavior of
the system for time . For a dynamic
system, the state of a system is described in
terms of a set of state variables x1(t), x2(t),
,xn(t). Example
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State variables are those variables that
determine the future behavior of a system when
the present state of the system and the
excitation signals are known.
Example displacement x(t) and velocity v(t)
are two state variables, any one of them can not
describe the mass system completely.

State vectorVector which consists of n state
variables that described entirely the dynamics
action of a known system. that is described as

or
Example
are state vector of mass system
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State space is n dimensions space which takes
x1,x2xn as coordinate.
The manner in which the state variables change
as a function of time may be thought of as a
trajectory in n dimensional space ,called State
space.

v
Example
x
0
two dimension state space
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Example
Laplace
then1)if t0,u(t) is known,and the initial
condition is known too, then t0 the
solution of equation can be determined
only.,namely the behavior of the system to be
determined for x(t0-). So, the minimum set of
variables can be defined to be
the state variables.

2) If t0,u(t) is known,and the beginning
condition is known too, then t0 the
solution of equation can be determined
only,namely the behavior of the system to be
determined for tto. So, the minimum set of
variables can also be
defined to be the state variables.

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The properties of state variable

Linear independent each other
Not unique and may be selected to suit the
problem being studied.
The number of state variables is unique
The method , how to select the set of state
variables ,is not unique.

Conclusion
For a passive network, the number of state
variables required is equal to the number of
independent energy storage elements.
The state variables that describe a system are
not a unique set.(linear transformation)
A widely used choice is a set of state variables
that can be readily measured.

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2.3.2 State Equation and output equation

State Equation A set of first-order
differential equations.
Output equation A set of algebraic
equation which describe the relationship of
output vector and input vector, state vector.

Standard form of linear time-invariant system
state equation and output equation

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System(state) matrix
Input(control) matrix
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State-output matrix
Input-output matrix
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State- space model
comment
n-order system n state variables
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Diagram of state variables

Standard form of linear time-varying system
state equation
A time-varying control system is a system for
which one or more of the parameters of the system
may vary as a function of time.

nonlinear time-invariant system
nonlinear time-varying system

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Classical control theory
Which can handle SISO system
Which can handle linear and time-invariant
system.
Which describe the relationship between input and
output.
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Modern Control Theory and State Space Model
An approach for handling MIMO system is Modern
Control Theory
Which can handle nonlinear system and
time-varying system.
Which provides additional insights into system
behavior that TF analysis does not.
Which relies heavily on computers except for
simple systems.
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The time-domain representation of control
systems is an essential basis for modern control
theory and system optimization.

Example

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2.3.3 State equations derivation of original
equations
Differential equation
Transfer function
State-space model
Block diagram
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State Space Model from Differential Equation 1.
No differential of input
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matrix
D0

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2. differential of input
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State equation

Example

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2.4 Linear differential equation

2.4.1 Regular solution of linear
differential equation
2.4.2 Laplace transform solution of linear
differential equation
2.4.3 Kinetic mode

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2.4.1 Regular solution of linear differential
equation

In order to compute the time response of a
dynamic system, it is necessary to solve the
differential equations for given inputs. There
are a number of analytical and numerical
techniques available to do this.

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Regular solution
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2.4.2 Laplace transform solution of linear
differential equation
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Example RC network, determine output
1.Laplace transform

2.solution of algebra equation
3.solution of original differential equation

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2.4.3 Kinetic mode

Modethe exponential form of eigenvalues,
which are the free solution of a control system.
Mode is related to the structure and
parameters of system, it is independent of the
select of input or output.

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2.5 State equation

2.5.1 Matrix exponential
2.5.2 Time-domain solution of state equations
2.5.3 State Transition Matrix
2.5.4 Invariability of systems eigenvalues
and modes

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2.5.1 Matrix exponential

1. Solution of force-free equation
Laplace transform method

Progression method
Scalar quantity equation
Vector equation
Solution form

Compare the coefficients

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Matrix exponential is defined as
(sIA)-1 is called resolvent matrix of A.

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Characteristic
1.differential
2.
3.

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4. is nonsingular for all A.
5.

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6. Matrix exponential of diagonal matrix
7.Laplace transform

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compute
1.definition
2.Laplace transform
3.Matrix exponential via diagonalizable linear
transformation (canonical method)

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Suppose A is diagonalizable, i.e., A has a set
of n linearly independent eigenvectors, say,
Put the equations together, (where T is
invertible, why?)

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The characteristic polynomial of A is defined
as
F(s) is a polynomial of degree n.
Roots of F(s) are the eigenvalues of A.
F(s) has real coefficients, so the eigenvalues
of A are either real or occur in
complex-conjugate pairs.
There are n eigenvalues.

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2.5.2 Time-domain solution of state equations

In this section, we consider the question of
determining the solution to the state equations
for the case when the input is zero or not zero.
These case is respectively referred to as the
force-free problem response and forced response.
1. Free motion of dynamic system

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2. Force motion of dynamic system Method 1
using Laplace transform The nature of x(t)
depends on the matrix
Forced respond
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Which can be written as
is State Transition Matrix of
the system.
To linear time-varying system,

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The solution can be written as
free-response component
forced-response component
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2.5.3 State Transition Matrix

Definition
As to linear time-invariant system, the
solution which satisfied the
following matrix equation and initial condition
is called the state transition matrix.

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Character
1.
2.
3.
4.compute equation

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5.State-transition matrix is nn function
matrix with variable of t.
6.As to a linear constant system,
It is only determined by A matrix, it is have
nothing to do with the input and output.

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2.5.4 Invariability of systems eigenvalues and
modes

Assume x and x are two different state
variables of the same system. P is a nonsingular
matrix.
The same system can be described by two sate
equation

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Substitute xPx into the first equation, then
we can obtain

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Conclusion no matter how to select the state
variables, the following value keep the same
1. The characteristic polynomial of matrix A.
2. Eigenvalues of matrix A. 3. The modes of
systems free motion.
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2.6 Transfer Function(matrix)

2.6.1 Transfer function
2.6.2 Block Diagram
2.6.3 Poles and Zeros of TF
2.6.4 Poles and zeros cancellation
2.6.5 Transfer Function Matrix

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Laplace Transform (Review)

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2.6.1 Transfer function

One of the most powerful tools for control
system analysis and design is the transfer
function.
For a SISO system with input u(t) and output
y(t), the transfer function is defined as(Zero
initial conditions) Laplace transform of the
input-output relation of a system.

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S-plane
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Step
Define the system and its components and
identify the input and output of the system.
Formulate the mathematical model and list the
necessary assumptions.
Write the differential equations describing the
system.
Linearize the differential equation.()
Take Laplace transform with Zero initial
conditions.
Determine the ratio output/input which is called
the Transfer Function.

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w Consider a linear time invariant system defined
by
w where n m for a physically realizable
system.Taking Laplace transform of this equation
with zero initial conditions, we get
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w The above transfer function can be written as
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n lk Þ Order of the system zi
Þ Zeros of the system K bm / an Þ Gain of the
system pi Þ Poles of the system l Þ
Type of the system Poles and zeros are either
real, or form complex conjugate pairs.
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Character
1. The transfer function of a system is a
mathematic model, it is an operational method of
expressing the differential equation that relates
the output variable to the input variable.
2. Transfer function cannot provide any
information concerning the physical structure of
the system. Transfer function of different
systems can be identical.

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3. Transfer function is a property of system
itself, independent of the magnitude and nature
of the input or driving function.
4. If the transfer function of a system is
known, the output or response can be studied for
various forms of inputs with a vies toward
understanding the nature of the system.

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5. Transfer function may be established
experimentally by introducing known input and
studying the output of the system.
6. Transfer function gives a full description
of the dynamic characteristics of the system.
7. Transfer function is defined only for a
linear system, and, strictly, only for
time-invariant systems.

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Transfer function is the Laplace transform of the
impulse response and vice versa.
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Exp. A system is defined by the following
differential equation. Considering x to be the
input and y the output, determine the transfer
function of the system, and calculate the output
if the input is a unit ramp function. Assume all
initial conditions are zero for both input and
output.
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Solution Taking the Laplace transform of both
sides, s2Y3sY8Y2XsX
If the input x(t) t ( unit ramp) then
X(s)1/s2 Hence
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Solving the above produces the partial-fraction
coefficients
The output is given by the inverse transform
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Example

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2.6.2 Block Diagram

w Introduction
The Block Diagram Model
which consists of block, arrow, summing
junction and branch point.
A block diagram represents the flow of
information and the function performed by each
component in the system.
Arrows are used to show the direction of the
flow of information.

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wBlock diagram representation
The block represents the the function or
dynamic characteristics of the component and is
represented by a transfer function.
The complete block diagram shows how the
functional components are connected and the
mathematic equations that determine the response
of each component.

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summing junction
branch point
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Block Diagram Algebra
Rule 1 ( Combining serial blocks )
Rule 2 ( Combining parallel blocks )
Rule 3 ( Closing a feedback loop )
Rule 4 ( Moving a summing junction ahead of a
block )
Rule 5 ( Moving a summing junction past a block
)
Rule 6 ( Moving a branch(pickoff) point ahead of
a block )
Rule 7 ( Moving a branch point past a block )

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Block diagram manipulations
Rule 1 ( Combining serial blocks ) Associative
and Commutative Properties
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Rule 2 ( Combining parallel blocks )
principle of superposition

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Rule 3 ( Closing a feedback loop )

Closed-loop Transfer function
Open-loop transfer function
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Example

Solution
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Block Diagram of RC network

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Example
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take Laplace transform and obtaining
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2.6.3 Poles and Zeros of TF

Effect of zeros and poles on the transient
response
1. Poles produce some motion modalities, which
are generic to the plant itself and irrespective
to the input signal, they are main factor which
affect the systems performance.

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Poles and Zeros in the Complex Plane
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2. Zeros determine the scale of each modality
which is shown in the response. Sometimes they
can interdict some modality being components in
input to be transfer to the output.
ConclusionPoles and zeros determine the
systems dynamic performances.

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2.6.4 Poles and zeros cancellation

1. The cancellation occur on input path
(input decoupling zero)
2. The cancellation occur on output path
(output decoupling zero)

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3. The cancellation occur on both input and
output path (input and output decoupling zero)
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2.6.5 Transfer Function Matrix

The definition of transfer function is easily
extended to multivariable system.
1. Transfer Function Matrix
u(t) l,y(t) m,G(s) ml

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2. Transfer Function Matrix from State Space
Model
State equation

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3. Characteristic polynomial of transfer
function matrix is equal to the characteristic
polynomial of A.
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2.7 Transfer function of closed-loop system

2.7.1 Block Diagram Rules
2.7.2 Transfer function of closed-loop system
2.7.3 Transfer function matrix of closed-loop
system

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2.7.1 Block Diagram rules
How to Simplify the block diagram of a complex
feedback control systems? -------to
rearrange the block diagram based on some usual
rules. Precondition invariability or
equivalent (Ensure to maintain outputs
of rearranged blocks )
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Rule 4 ( Moving a summing junction ahead of
a block )

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Rule 5 ( Moving a summing junction past a
block ) Distributive property

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Moving summing junctions between each
other.(Interchange)

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Rule 6 ( Moving a pickoff point ahead of a
block )

Y GX
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Rule 7 ( Moving a pickoff point past a block )

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Moving a pickoff points between each other.
(Interchange)

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Exp12 Simplify the following block diagram.

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Closing a feedback loop
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Example
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2.7.2 Transfer function of closed-loop system

Multiple inputs for output

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Multiple inputs for error

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Open-loop transfer function the ratio of the
feedback signal B(s) to the actuating error
signal E(s) is called the open-loop transfer
function.

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Characteristic equation
Denominator polynomial of the close-loop
transfer function. The poles are the roots of the
characteristic equation
For a system to be stable, the poles (roots of
the characteristic equation) need to have
negative real parts, i.e. be on the left half of
the complex plane.

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2.7.3 Transfer function matrix of closed-loop
system
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2.8 Fundamental Parts

1. Proportional part
2. Inertial part(first-order systems standard
TF)
T is the time-constant.

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3. Oscillating part(second-order systems
standard TF)

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4. Integral part

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5. Unstable part
the poles have positive real parts, i.e. be on
the right half of the complex plane.

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6. Differential part
The output contain the derivation of input.
Cannot exist singly.

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7. Delaying part(lag)

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2.9 Signal flow graphs (SFG)

The composing of SFG
Node (summing junction and pickoff point)
a. Addition of the signals on all incoming
branches
b. Transmission of the total node signal (the
sum of all incoming signals) to all outgoing
branches.
Arrow (block and arrow which expresses the
direction of signal flowing)

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Draw the SFG according to the block diagram.
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SFG Algebra
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Terms
1. Source is a node with only outgoing branches.
2. Sink is a node with only incoming branches.
3. Path is a group of connected branches having
the same sense of direction.
4. Forward path is a path originating from a
source and no node is encountered more than once.
5.Path gain is the product of the coefficient
associated with the branches along the path.

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6. Feedback loop is a path originating from a
node and terminating at the same node. In
addition, a node cannot be encountered more than
once.
7. Loop gain is the product of coefficients
associated with the branching forming a feedback
loop.
8. Nontouching loops loops are nontouching
if they do not possess any common nodes or
branches.

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be, cf is loop, becf is not loop.
abcd is path,aecd, abebcd is not path.

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The Mason Gain Formula
The overall transmittance (gain) can be
obtained from the Mason gain formula
Y(s) is output, and R(s) is input.

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Example
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Example

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Solution1.

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Return
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2.10 Response of impulseand step signal

Test input signals
The response of to a specific input signal
will provide several measures of the performance
of control system.
The actual input signal of system is usually
unknown. A standard test input signal is normally
chosen. Many control systems experience input
signals very similar to the standard test
signals.
Unit impulse input, step input, ramp input,
acceleration input, Sinusoid input etc.

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1.Impulse signal (Dirac pulse)

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2.(Unit) Step
3.Ramp
4.Acceleration

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5. function table

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2.10.1 Impulse function
2.10.2 Impulse response
2.10.3 Step response

Return
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2.10.1 Impulse function

1. Impulse (Dirac) function

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2. Laplace transform of impulse function is
equal to the area of the function. (The impulse
function whose area is unity is called a unit
impulse .
3. Relation of unit impulse and unit step

Return
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2.10.2 Impulse response

Concept The response of a linear constant
system to unit impulse function with zero initial
condition, marked as g(t).
1. The relation between g(t) and G(s)

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Conclusion Transfer function of a plant is the
Laplace transform of its impulse response.

2. First-order system

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3. The response of a system to any input function

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4. State equation

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2.10.3 Step response

Concept The response to unity step function
with zero initial condition, marked as h(t).

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The relationship between unit impulse
(response) and unit step (response)

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The Relationship between input signals, or output
signals of a linear control system
And we can find
It suits every linear time-invariant system.
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Time response of first order systems
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2.11 Conclusion

Quantitative mathematical models of control
components and systems using differential
equations.
Small-signal linear approximation using a Taylor
series expansion about the operating point for
nonlinear components.
Laplace transformation and its related
input-output relationship, the transfer function,
for linear systems.

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Block diagram models using transfer function
notations.
Masons flow graph gain formula investigated,
useful for obtaining the relationship between
system variables in a complex feedback system
without requiring any reduction or manipulation
of the flow graph.
The utility of the computer simulation of linear
and nonlinear systems.

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Exercises

1.Block diagram and transfer function.

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2. Transfer function.
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3.?????,???????????,???Id???u ????????,
, ???? ,?????
,??????
???????? ?

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4. Transfer function.
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How to analysis and design using a system model?
Return
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END OF
CHAPTER TWO

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num1 2den1 3 3 4 1 gtf(num,den) printsys(
num,den)
Transfer function s 2 ---------------------
---------------- s4 3 s3 3 s2 4 s
1 num/den s 2 ---------------------------
---------- s4 3 s3 3 s2 4 s 1
258
A1 21 1B11C1 0D0 gss(A,B,C,D)
a x1 x2 x1 1 2 x2 1 1 b
u1 x1 1 x2 1 c x1
x2 y1 1 0 d u1 y1
0 Continuous-time model.
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A1 21 1B11C1 0D0 gss(A,B,C,D) g1
tf(g)
Transfer function s 1 --------------- s2
- 2 s - 1
260
kgain5z-1p-2 -5 -10 gzpk(z,p,kgain)
Zero/pole/gain 5 (s1) ------------------
----- (s2) (s5) (s10)
261
kgain5z-1p-2 -5 -10 gzpk(z,p,kgain) g1
tf(g)
Transfer function 5 s
5 --------------------------------- s3 17 s2
80 s 100
262
A1 21 1B11C1 0D0 gss(A,B,C,D) g1
tf(g) g2zpk(g)
Zero/pole/gain (s1) -----------------
-------- (s-2.414) (s0.4142)
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A1 21 1B11C1 0D0 gss(A,B,C,D) T
0 11 0 g3ss2ss(g,T)
a x1 x2 x1 1 1 x2 2 1
b u1 x1 1 x2 1 c
x1 x2 y1 0 1 d u1 y1 0
Continuous-time model.
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num1 7 24 24den1 10 35 50
24 r,p,kresidue(num,den,0)
r -1.0000 2.0000 -1.0000 -1.0000
1.0000 p -4.0000 -3.0000 -2.0000
-1.0000 0 k
Step response

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