Advanced - PowerPoint Presentation

Slide1

Advanced Control Systems (ACS)

Dr. Imtiaz Hussainemail: imtiaz.hussain@faculty.muet.edu.pkURL :http://imtiazhussainkalwar.weebly.com/

Lecture-6

State Space Modeling & AnalysisSlide2

Lecture Outline

Basic DefinitionsState EquationsState DiagramState ControllabilityState ObservabilityOutput ControllabilityTransfer Matrix

Solution of State EquationSlide3

Definitions

State of a system: We define the state of a system at time t0 as the amount of information that must be provided at time t0

, which, together with the input signal

u(t)

for

t

 t

0

, uniquely determine the output of the system for all

t  t

0

.

State Variable:

The state variables of a dynamic system are the smallest set of variables that determine the state of the dynamic system.

State Vector:

If

n

variables are needed to completely describe the

behaviour

of the dynamic system then

n

variables can be considered as

n

components of a vector

x

, such a vector is called state vector.

State Space:

The state space is defined as the n-dimensional space in which the components of the state vector represents its coordinate axes. Slide4

Definitions

Let

x

1

and

x

2

are two states variables that define the state of the system completely .

4

Two Dimensional State space

State (t=

t

1

)

State

Vector

State space of a Vehicle

Velocity

Position

State (t=

t

1

)Slide5

State Space Equations

In state-space analysis we are concerned with three types of variables that are involved in the modeling of dynamic systems: input variables, output variables, and state variables.The dynamic system must involve elements that memorize the values of the input for t>

t

1

.

Since integrators in a continuous-time control system serve as memory devices

, the

outputs of such integrators can be considered as the variables that define the

internal state

of the dynamic system.Thus the outputs of integrators serve as state variables.The number of state variables to completely define the dynamics of the system is equal to the number of integrators involved in the system.Slide6

State Space Equations

Assume that a multiple-input, multiple-output system involves integrators. Assume also that there are

inputs

and

outputs

.

Define

outputs of the integrators as state variables:

.

Then the system

may be described by

 Slide7

State Space Equations

The outputs

of the system

may be given as.

If we define

 Slide8

State Space Modeling

State space equations can then be written asIf vector functions f and/or g

involve time

t

explicitly, then the system is called a time varying system.

State Equation

Output EquationSlide9

State Space Modeling

If above equations are linearised about the operating state, then we have the following linearised state equation and output equation:Slide10

State Space Modeling

If vector functions f and g do not involve time t explicitly then the system is called a

time-invariant system.

In this case, state and output equations can be simplified toSlide11

Example-1

Consider the mechanical system shown in figure. We assume that the system is linear. The external force u(t) is the input to the system, and the displacement

y(t)

of the mass is the output

. The

displacement

y(t)

is measured from the equilibrium position in the absence of the

external force. This system is a single-input, single-output system. From

the diagram, the system equation is

This system is of second order. This

means that the system involves two integrators. Let us

define state

variables

and

as

 Slide12

Example-1

Then we obtain

Or

The output equation is

 Slide13

Example-1

In a vector-matrix form,

 Slide14

Example-1

State diagram of the system is

1/s

1/s

-k/m

-b/m

1/m

 Slide15

Example-1

State diagram in signal flow and block diagram format

1/s

1/s

-k/m

-b/m

1/m

 Slide16

Example-2

State space representation of armature Controlled D.C Motor.ea is armature voltage (i.e. input) and 

is output.

e

a

i

a

T

R

a

L

a

J



B

e

b

V

f

=constant

Slide17

Example-2

Choosing

as state variables

Since

is output of the system therefore output equation is given as

 Slide18

State Controllability

A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(to) to any other

desired location

x(t)

in a finite time,

t

o

≤

t ≤ T.controllable

uncontrollableSlide19

State Controllability

Controllability Matrix CM

System is said to be state controllable ifSlide20

State Controllability (Example)

Consider the system given below

State diagram of the system is Slide21

State Controllability (Example)

Controllability matrix CM is obtained as

Thus

Since

therefore system is not completely state controllable.

 Slide22

State Observability

A system is completely observable if and only if there exists a finite time T such that the initial state x(0) can be determined from the observation history y(t)

given the control

u(t)

,

0≤

t ≤

T

.

observable

unobservableSlide23

State Observability

Observable Matrix (OM)

The system is said to be completely state observable ifSlide24

State Observability (Example)

Consider the system given below

OM is obtained as

Where Slide25

State Observability (Example)

Therefore OM is given as

Since

therefore system is not completely state

observable.

 Slide26
Slide27

Output Controllability

Output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval.Output controllability matrix (OCM) is given asSlide28

Home Work

Check the state controllability, state observability and output controllability of the following systemSlide29

Transfer Matrix (State Space to T.F)

Now Let us convert a space model to a transfer function model.Taking Laplace transform of equation (1) and (2) considering initial conditions to zero.From equation (3)

(1)

(2)

(3)

(4)

(5)Slide30

Transfer Matrix (State Space to T.F)

Substituting equation (5) into equation (4) yieldsSlide31

Example#3

Convert the following State Space Model to Transfer Function Model if K=3, B=1 and M=10; Slide32

Example#3

Substitute the given values and obtain A, B, C and D matrices. Slide33

Example#3Slide34

Example#3Slide35

Example#3Slide36

Example#3Slide37

Example#3Slide38

Home WorkObtain the transfer function T(s) from following state space representation.

AnswerSlide39

Forced and Unforced ResponseForced Response, with

u(t) as forcing functionUnforced Response (response due to initial conditions)Slide40

Solution of State Equations

Consider the state equation given belowTaking Laplace transform of the equation (1)

(1)Slide41

Solution of State Equations

Taking inverse Laplace

State Transition MatrixSlide42

Example-4

Consider RLC Circuit obtain the state transition matrix ɸ(t).

V

c

+

-

+

-

V

o

i

LSlide43

Example-4 (cont...)

State transition matrix can be obtained as

Which is further simplified asSlide44

Example-4 (cont...)

Taking the inverse Laplace transform of each elementSlide45

Home Work

Compute the state transition matrix if

SolutionSlide46

State Space TrajectoriesThe unforced response of a system released from any initial point

x(to) traces a curve or trajectory in state space, with time

t

as an implicit function along the trajectory.

Unforced system’s response depend upon initial conditions.

Response due to initial conditions can be obtained asSlide47

State Transition

Any point P in state space represents the state of the system at a specific time t. State transitions provide complete picture of the system

P(

x

1

,

x

2

)

t

0

t

1

t

2

t

3

t

4

t

5

t

6Slide48

Example-5

For the RLC circuit of example-4 draw the state space trajectory with following initial conditions.SolutionSlide49

Example-5 (cont...)

Following trajectory is obtainedSlide50

Example-5 (cont...)Slide51

Equilibrium Point

The equilibrium or stationary state of the system is whenSlide52

Solution of State Equations

Consider the state equation with u(t) as forcing functionTaking Laplace transform of the equation (1)

(1)Slide53

Solution of State Equations

Taking the inverse Laplace transform of above equation.

Natural Response

Forced ResponseSlide54

Example#6

Obtain the time response of the following system:Where u(t) is unit step function occurring at t=0. consider x(0)=0.

Solution

Calculate the state transition matrixSlide55

Example#6

Obtain the state transition equation of the system Slide56

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