modeling in the time domain - state-space. mathematical models classical or frequency-domain...

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Modeling in the Time Domain - State-Space

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Modeling in the Time Domain - State-Space

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.

Example 1:Epidemic Disease

Discrete State Space Model(Linear System)

Discrete State Space(Nonlinear System)

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

Example 3.4

equivalentblock diagram showingphase-variables.Note: y(t) = c(t)

Transfer Function with Numerator Polynomial

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

Example 3.5

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:

Example 3.6 p.p.152-153

• Solve by hand

• Solve using MATLAB’s ‘ss’ and ‘tf’ commands

Linearization

• Demonstration of how to linearize a nonlinear state space model about a nominal solution.

• Application of linearization to guidance and control of a lunar mission.