modeling in the time domain - state-space. mathematical models classical or frequency-domain...
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Mathematical Models
• Classical or frequency-domain technique
• State-Space or Modern or Time-Domain technique
Classical or Frequency-Domain Technique
• Advantages– Converts differential
equation into algebraic equation via transfer functions.
– Rapidly provides stability & transient response info.
• Disadvantages– Applicable only to
Linear, Time-Invariant (LTI) systems or their close approximations.
LTI limitation became a problem circa 1960 when space applications became important.
LTI limitation became a problem circa 1960 when space applications became important.
State-Space or Modern or Time-Domain Technique
• Advantages– Provides a unified
method for modeling, analyzing, and designing a wide range of systems using matrix algebra.
– Nonlinear, Time-Varying, Multivariable systems
• Disadvantages– Not as intuitive as
classical method.
– Calculations required before physical interpretation is apparent
State-Space RepresentationAn LTI system is represented in state-space format by thevector-matrix differential equation (DE) as:
( ) ( ) ( )
( ) ( ) ( )
( ).
x t Ax t Bu t
y t Cx t Du t
with t t and initial conditions x t
0 0
The vectors x, y, and u are the state, output and input vectors.The matrices A, B, C, and D are the system, input, output, andfeedforward matrices.
Dynamic equation(s)
Measurement equations
Definitions
• System variables: Any variable that responds to an input or initial conditions.
• State variables: The smallest set of linearly independent system variables such that the initial condition set and applied inputs completely determine the future behavior of the set.
Linear Independence: A set of variables is linearly independent if none of the variables can be written as a linear combination of the others.
Linear Independence: A set of variables is linearly independent if none of the variables can be written as a linear combination of the others.
Definitions (continued)
• State vector: An (n x 1) column vector whose elements are the state variables.
• State space: The n-dimensional space whose axes are the state variables.
Graphic representationof state spaceand a state vector
The minimum number of state variables is equal to:
• the order of the DE’s describing the system.
• the order of the denominator polynomial of its transfer function model.
• the number of independent energy storage elements in the system.
Remember the state variables must be linearly independent! If not, you may not be able to solve for all the other system variables, or even write the state equations.
Remember the state variables must be linearly independent! If not, you may not be able to solve for all the other system variables, or even write the state equations.
Case Study: Pharmaceutical Drug Absorption
Advantages of the state-space approach are the ability to focus on
component parts of the system and to represent multiple-input, multiple-
output systems.
Pharmaceutical Drug Absorption Problem
We wish to describe the distribution of a drug in the body bydividing the process into compartments: dosage, absorption site,blood, peripheral compartment, and urine.
Each isthe amount of drugin that particularcompartment.
xi
Pharmaceutical Drug Absorption Solution
1. Assume dosage is released at a rate proportional to concentration.
d
dtx K x1 1 1
2. Assume rate of accumulation at a site is a linear function of thedosage of the donor compartment and the resident dosage. Then,
d
dtx K x K x
d
dtx K x K x K x K x
d
dtx K x K x
d
dtx K x
2 1 1 2 2
3 2 2 3 3 4 4 5 5
4 5 3 4 4
5 3 3
Pharmaceutical Drug Absorption Solution
3. Define the state vector as the dosage amount in eachcompartment and assume we measure the dosage in eachcompartment. Then,
d
dt
x
x
x
x
x
K
K K
K K K K
K K
K
x
x
x
x
x
1
2
3
4
5
1
1 2
2 3 5 4
5 4
3
1
2
3
4
5
0 0 0 0
0 0 0
0 0
0 0 0
0 0 0 0
( )
y I x
Converting a Transfer Function to State Space
• State variables are not unique. A system can be accurately modeled by several different sets of state variables.
• Sometimes the state variables are selected because they are physically meaningful.
• Sometimes because they yield mathematically tractable state equations.
• Sometimes by convention.
Phase-variable Format1. Consider the DE
d y
dta
d y
dt
d y
dta y b u
n
n n
n
n
1
1
1 0 0
where y is the measure variable and u is the input.
2. The minimum number of state variables is n since the DE is of nth order.3. Choose the output and its derivatives as state variables.
x y
x y x x
xd y
dtx x
xd y
dta x a x a x b u
n
n
n n n
n
n
n n n
1
2 1 2
1
1 1
0 1 1 2 1 0
First row of state equations
Last row of state equations
Phase-variable Format (continued)
4. Arrange in vector-matrix format
d
dt
x
x
x
x a a a a
x
x
x
x b
y
x
x
x
x
n
n
n
n
n
n
1
2
1
0 0 0 0
1
2
1
0
1
2
1
0 1 0 0
0 0 1 0
0 0 0 1
0
0
0
1 0 0 0
Note the transfer function formatNote the transfer function format
Y s
U s
b
s a s a s ann
n
( )
( )
0
11
11
0
Transfer Function with Numerator Polynomial
(continued)1. From the first block: X s
R s
a
sa
as
a
as
a
a
1 3
3 2
3
2 1
3
0
3
1( )
( )
/
2. Therefore,
d
dt
x
x
xa
a
a
a
a
a
x
x
xa
u1
2
30
3
1
3
2
3
1
2
3
3
0 1 0
0 0 1
0
01
Transfer Function with Numerator Polynomial
(continued)3. The measurement (observation) equation is obtained from the second transfer function.
C s Y s b s b s b X s b s X b sX b X
sX X X
Y s b X b X b X
y b b b
x
x
x
( ) ( ) ( )
( )
22
1 0 1 22
1 1 1 0 1
1 2 3
2 3 1 2 0 1
0 1 2
1
2
3
But, and sX
So,
2
Converting from State Space to a Transfer Function
( ) ( ) ( )
( ) ( ) ( )
x t Ax t Bu t
y t Cx t Du t
with t t and zero initial conditions
.0
Taking the Laplace transform,
s X s AX s BU s X s sI A BU s
Y s C X s DU s C sI A BU s DU s
C sI A B D U s
Y s
U sC sI A B D
Cadj sI A B sI A D
sI A
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( )
( )
( )
det
det
1
1
1
1
In the case of SISO (Single - Input, Single - Output) systems: