Control System Desing

1. Control System Design

2. • Assignment 30% (15+15)
– 4 Exercises from Modern Control Systems,
Richard C. Dorf
– Chapter 1,2,3,4
• Exam 70%
2

3. 3
Definitions
• A system is a group of related parts working
together, or an ordered set of ideas, methods, or
ways of working
• Implementation point-of-view: a system is an
arrangement of physical components connected
or related in such a manner as to form and/or act
as an entire unit
• Mathematical: a system as a mapping of N input
signals onto M output signals; the mapping
carries out a transformation on the input signals
according to a set of rules

4. 4
Basic definitions
• Single-variable system (SISO system) has
only one input and only one output
• Multivariable system (MIMO system) has
more than one input or more than one output
• Input-output relationship (external description) is an
equation that describes the relation between the input
and the output of a system
• Black box concept: the knowledge of the internal
structure of a system is unavailable; the only access to
the system is by means of the input ports and the output
ports

5. 5
Time response
• One-dimensional system: required for
processing a signal that is a function of the single
independent variable
• We assume that the independent variable is time
even in cases where the independent variable is
a physical quantity other than time
• Time response is the output signal as a function
of time, following the application of a set of
prescribed input signals, under specified
operating conditions

6. 6

7. 7
Continuous-time system
Continuous-time system: the input and output signals
are continuous in time domain

8. 8
Discrete-time system
Discrete-time system has discrete-time
input and output signals

9. 9
Digital system
• A discrete-time system is digital if it
operates on discrete-time signals whose
amplitudes are quantized
• Quantization maps each continuous
amplitude level into a number
• The digital system employs digital hardware
1. explicitly in the form of logic circuits
2. implicitly when the operations on the signals are
executed by writing a computer program

10. 10
Analysis and design
• Analysis of a system is investigation of the properties
and the behavior (response) of an existing system
• Design of a system is the choice and arrangement of
systems components to perform a specific task
• Design by analysis is accomplished by modifying the
characteristics of an existing system
• Design by synthesis: we define the form of the system
directly from its specifications

11. 11
Block diagram
• Block diagram is a pictorial representation of a system that
provides a method for characterizing the relationships among the
components
• Single block with one input and one output is the simplest form of
the block diagram
• Interior of the rectangle representing the block contains
(a) component name,
(b) component description, or
(c) the symbol for the mathematical operation to be performed on
input to yield output
• Arrows represent the
direction of signal flow

12. 12
Elements of block diagram
Summing point Takeoff point

13. 13
Interconnections of blocks
Blocks connected
in cascade
Blocks connected
in feedback
Blocks connected
in parallel

14. 14
State
• For some systems, the output at time t0 depends
not only on the input applied at t0, but also on
the input applied before t0
• The state is the information at t0 that, together
with input for t ≥ t0, determines uniquely output
for t ≥ t0
• Dynamical equation is the set of equations that
describes unique relations between the input,
output, and state

15. 15
Causality and stability
• A system is called causal if the output
depends only on the present and past
values of the input
• Intuitively, a stable system is one that will
remain at rest unless excited by an
external source and will return to rest if all
excitations are removed

16. 16
Time-invariant system
• A relaxed system is time-invariant if a time shift in the
input signal causes a time shift in the output signal. In
other words the relationship between the input and
output is independent of time. So if the response to u(t)
is y(t), then the response to u(t − t0) is y(t − t0)
• In the case of discrete-time digital systems, we often use
the term shift-invariant instead of time-invariant
• Characteristics and parameters of a time-invariant
system do not change with time

17. 17
Linear system
• Consider a relaxed system in which there is one
independent variable t
• A linear system is a system which has the
property that if
• input x1(t) produces an output y1(t) and input x2(t)
produces an output y2(t), then input c1 x1(t) + c2
x2(t) produces an output c1 y1(t) + c2 y2(t) for any
x1(t), x2(t) and arbitrary constants c1 and c2

18. 18
Response of an LTI system
• Steady-state response is that part of the
total response which does not approach
zero as time approaches infinity
• Transient response is that part of the total
response which approaches zero as time
goes to infinity (response before steady
state condition)

19. 19
Procedure for analyzing a system
1. Determine the equations
for each system component
2. Choose a model for representing the
system (e.g., block diagram)
3. Formulate the system model by
appropriately connected the components
4. Determine the system characteristics

20. 20
Concept of state space
• State space model: a representation of the dynamics of an
Nth order system as a first order differential equation in an
N-vector, which is called the state.
– Convert the Nth order differential equation that governs the
dynamics into N first-order differential equations
• Characteristics of state: x(t) is called the state of the
system at t because:
– Future output depends only on current state and future input
– Future output depends on past input only through current state
– State summarizes effect of past inputs on future output like the
memory of the system
• For example: Rechargeable flashlight — the state is the
current state of charge of the battery. If you know that
state, then you do not need to know how that level of
charge was achieved (assuming a perfect battery) to
predict the future performance of the flashlight.

21. 21
Parts of a state space representation
• State Variables: a subset of system variables which if
known at an initial time to along with subsequent inputs
are determined for all time t>+t0
• State Equations: n linearly independent first order
differential equations relating the first derivatives of the
state variables to functions of the state variables and the
inputs.
• Output equations: algebraic equations relating the state
variables to the system outputs

22. 22
Mathematical form of state space

23. 23
• Matrix A
Matrix A is the system matrix, and relates how the current state affects
the state change x' . If the state change is not dependent on the current
state, A will be the zero matrix. The exponential of the state matrix, eAt
is called the state transition matrix, and is an important function that
we will describe below.
• Matrix B
Matrix B is the control matrix, and determines how the system input
affects the state change. If the state change is not dependent on the
system input, then B will be the zero matrix.
• Matrix C
Matrix C is the output matrix, and determines the relationship between
the system state and the system output.
• Matrix D
Matrix D is the feed-forward matrix, and allows for the system input to
affect the system output directly. A basic feedback system like those we
have previously considered do not have a feed-forward element, and
therefore for most of the systems we have already considered, the D
matrix is the zero matrix.

24. 24
Typical state space model

25. 25
Properties-state variables
• The internal state variables are the smallest possible subset of
system variables that can represent the entire state of the system at
any given time.
• State variables must be linearly independent; a state variable cannot
be a linear combination of other state variables.
• The minimum number of state variables required to represent a
given system, n, is usually equal to the order of the system's
defining differential equation.
• If the system is represented in transfer function form, the minimum
number of state variables is equal to the transfer function's
denominator after it has been reduced to a proper fraction.
• It is important to understand that converting a state space realization
to a transfer function form may lose some internal information about
the system, and may provide a description of a system which is
stable, when the state-space realization is unstable at certain points.

26. 26
State space model for LTI system

27. 27
Various representations in SS

28. 28
Second order mass-spring system

29. 29
State space commands in Matlab
In order to enter a state space model into
MATLAB, the variables much be given numerical
value, because MATLAB cannot manipulate
symbolic variables without the symbolic toolbox.
• Define state space model:
Enter the coefficient matrices A, B, C, and D into
MATLAB. The syntax for defining a state space
model in MATLAB is:
statespace = ss(A, B, C, D)
where A, B, C, and D are from the standard vector-
matrix form of a state space model.

30. 30
• Extracting A, B, C, D matrices from a
state space model
In order to extract the A, B, C, and D
matrices from a previously defined state
space model, use MATLAB's ssdata
command.
[A, B, C, D] = ssdata(statespace)
where statespace is the name of the state
space system

31. 31
• Step response using the state space model
Once the state space model is entered into
MATLAB it is easy to calculate the response to a
step input. To calculate the response to a unit
step input, use:
step(statespace)
where statespace is the name of the state space
system.
For steps with magnitude other than one, calculate
the step response using:
step(u * statespace)
where u is the magnitude of the step and statespace is
the name of the state space system.

32. 32
• Impulse response using the state space
model
MATLAB can also plot the impulse response of
a state space model. The syntax for the
impulse response function in MATLAB is:
impulse(u * statespace)
where u is the magnitude of the impulse
and statespace is the name of the state space
system.

33. 33
• Bode plot using the state space model
MATLAB’s bode command plots the
frequency response of a system as a bode
plot. The syntax for the bode plot function
in MATLAB is:
bode(statespace)
where statespace is the name of the state
space system.

34. 34
• Transfer function from state space
MATLAB can determine the transfer function from the
state space representation in two ways. To find the
numerator matrix and denominator vector of the transfer
function of the system from the matrices of the state space
model use MATLAB's ss2tf command:
 [num, den] = ss2tf(A, B, C, D)
where num is defined as the numerator matrix, and den is defined
as the denominator matrix, and A, B, C, and D are the coefficients
of the state space model.
• In order to find the entire transfer function system from the
state space model, use the following command:
 transferfunction = tf(statespace)
where statespace is the name of the state space system, and
transferfunction is the name of the generated transfer function
system.