Control systems resource sheet

ENB458 Exam Resource Sheet James Mount 2014

General

Controllability and Observability = [ 2 1]

=

[

2

1]

Will be controllable/observable if the rank of these matrices is equal to n. Rank will be n if

determinant is non-zero, other ways of determining rank as well.

Transient Characteristics S plane poles =

Sigma (real part of S Pole) || = ||

Damped Frequency = 1 2

Damping Ratio =ln(%0 100 )

2 + ln2(% 100 )

Overshoot % = ( 12 ) 100

Settling Time (2nd Order) =4

Peak Time (2nd Order) =

1 2

Final Value Theorem and Steady State Error FV Theorem (S Transfer Function) () = lim

0()()

FV Theorem (Z Transfer Function) () = lim1

()()

SSE () = 1 ()

SSE of Open Loop SS (step input) () = 1 + 1

SSE of Closed Loop SS (step input) () = 1 + ( )1


General

Converting from Transfer Function to State Space

() =1

1 + + 1 + 0 + 11 + + 1 + 0

Phase Variable

=

[

0 1 0 00 0 1 0 0 0 0 1

0 1 2 1]

+

[

0001]

= [0 1 2 1]

Controller Canonical

=

[ 1 2 1 0

1 0 0 0 0 1 0 00 0 1 0 ]

+

[

1000]

= [1 2 1 0]

Observer Canonical

=

[ 1 1 0 0

2 0 1 01 0 0 10 0 0 0]

+

[

1210 ]

= [1 0 0 0]

Converting from State Space to Transfer Function () = ( )1 +

Time Response of System To find time response convert SS to TF, but need to consider initial conditions (0), so use

the following,

() = ( )1(0 + ()) () = () + ()


Model Development

Modelling with Transfer Functions

Electrical Systems

Component Impedance

Resistor Inductor Capacitor 1/

1. Apply circuit theory such as KVL and KCL around meshes and at nodes

a. Look at combining parallel components =1 2

1 + 2

b. Look for voltage dividers

2. Put into matrix form, [] = [][]

3. Solve using Cramers rule

[12

] = [1 23 4

] [12

]

1 =det ([

1 22 4

])

det ([1 23 4

])

4. If variable 1 is not the quantity you are interested in, sub in equation to get desired

quantity. (i.e. if variable 1, was a current but wanted the voltage through the

component, use = to manipulate it into voltage)

Translational Systems

Component Impedance

Spring Damper

Mass 2

1. Develop system of equations by holding each mass in turn and seeing the forces acting

upon it.

2. Put into matrix form

3. Solve using Cramers rule

4. Manipulate current output quantity, from Cramers rule result, to desired quantity, if

required.

Rotational Systems

Component Impedance

Spring Damper

Mass 2

1. If gearbox present reflect impedances and draw equivalent system

2. Perform same steps as translational systems. Remember to alter the final output quantity if

required, will need to most likely do so if a gearbox was present.


Model Development

Modelling with State Space

Electrical Systems

Component Voltage - Current Current - Voltage Voltage Charge

Resistor = =

=

[]

Inductor =

[] =

1

().

0

= 2

2[]

Capacitor =1

().

0

=

[] =

1

1. Write equations for all energy storing elements. These will be differential equations,

with the differentiated quantities been a possible set of state variables.

2. Apply circuit theory, such as KVL and KCL, to obtain the unknown variables, in the

equations from step 1, in terms of the state variables.

3. Using the information from step 2 write out the state equations, and hence the SS

matrices

Translational and Rotational

Component Translational Rotational

Spring

Damper

[]

[]

Mass 2

2[]

2

2[]

States will generally be displacement ( or ) and velocity ( or )

1. (If there is a gearbox reflect system, and draw equivalent). Write differential motion

equations similar to that when using TF modelling by holding all but one mass still and

seeing the forces acting upon it. (Generally will get two states for every mass element in

the system)

2. Knowing that parts of your states will simply be

[] = , rearrange equations from

step for the remaining differentiated quantity (generally velocity)

3. Write state equations and hence find the state space model. (Be careful with the output

equations for rotational systems, if impedances had to be reflected due to a gearbox)


Controller Design

Method 1 Using det( ( )) = 0 1. Check for controllability

2. Using original state space representation (, , , ) find closed loop characteristic

equation using det( ( )) = 0

3. Find desired characteristic equation using pole placement

4. Equate coefficients from the two equations in steps 2 and 3, and solve for gains

Method 2 Using P Transformation 1. Using original state space representation (, , , ) find open loop characteristic

equation using det( ) = 0

2. Using open loop characteristic equation find phase variable state space form ( , , , )

3. Compute controllability matrices and

4. Calculate the P transform matrix = 1

5. Get desired closed loop characteristic equation from phase variable state space

0 = + (1 + )1 + + (1 + 2) + (0 + 1)


7. Equate coefficients from equations in steps 5 and 6, and solve for phase variable gains

8. Transform phase variable gains to original state space gains using = 1

Method 3 Using Ackermann Formula 1. Find desired characteristic equation using pole placement

0 = + 1

1 + + 1 + 0 2. Compute () using original state space representation (, , , )

() = + 1

1 + + 1 + 0 3. Calculate the controllability matrix

4. Apply Ackermann Formula = [0 0 1]1()

PI Controller Need to augment the matrix

Open Loop State Space

[

] = [ 0

0] + [

0] + [

01]

= [ 0]

Closed Loop State Space

[

] = [

0] + [

01]

= [ 0] , = +

If need to transform gains, remember to only apply transform to not


Observer Design

Method 1 Using det( ( )) = 0 1. Check for observability

2. Using original state space representation (, , , ) find closed loop characteristic

equation using det( ( )) = 0


4. Equate coefficients from the two equations in steps 2 and 3, and solve for gains

Method 2 Using P Transformation 1. Using original state space representation (, , , ) find open loop characteristic

equation using det( ) = 0

2. Using open loop characteristic equation find observer canonical state space form

( , , , )

3. Compute observability matrices and

4. Calculate the P transform matrix = 1

5. Get desired closed loop characteristic equation from phase variable state space

0 = + (1 + )1 + + (1 + 2) + (0 + 1)


7. Equate coefficients from equations in steps 5 and 6, and solve for observer canonical gains

8. Transform observer canonical gains to original state space gains using = 1

Method 3 Using Ackermann Formula 1. Find desired characteristic equation using pole placement

0 = + 1

1 + + 1 + 0 2. Compute () using original state space representation (, , , )

() = + 1

1 + + 1 + 0 3. Calculate the observability matrix

4. Apply Ackermann Formula = ()1 [

001

]


Optimal Control

Design Methodology

1. Compute Q and R weighting matrices, by using 1 (max)2 , will get diagonal matrices

2. Solve Riccatti equation with infinite horizon, and take the non-negative solution, remember

S will be symmetric about the diagonal

0 = + + 1 3. Compute the optimal gains

= 1

If want integral control augment to open loop state space and use new A and B matrices in

the Riccatti equation. Will also need to compute new Q matrix as there is an added element


Discrete Systems

Z Transformation Methods

Backward Difference

=1 1

Tustin Transform

=2

1 1

1 + 1

Pole Zero Mapping

=

= (cos() sin())

Z Transform Table

Courtesy of http://lpsa.swarthmore.edu/LaplaceZTable/LaplaceZFuncTable.html


Discrete Systems

Difference Equations 1. Provided with the transfer function () = ()/() cross multiply to get

()() = ()() 2. Knowing that () = 1 transform equation from step 2 into a difference equation

3. Manipulate equation from step 3 to get =

General Form (Block Diagram)

Determining Stability 1. Get into z form. (i.e. = 201 + 602 becomes 1 = 20

1 + 602 which rearranges

to 0 = 2 + 20 + 60)

2. Find roots of z form equation. If magnitude of the roots are outside unit circle then the

system is unstable.

Discretising a Model

If Given Plant Model/Diagram That Does Not Include All Dynamic Aspects 1. Need to model the whole plant dynamics including elements such as Z.O.H

2. If Z.O.H need to use () = (1 1) {()

}

3. Once found () write down discretised state space model from the discrete transfer

function, same as if it was a continuous model with a continuous transfer function

If Given Transfer Function/State Space That Does Include All Dynamic Aspects 1. If it is a continuous model:

a. Write down the continuous state space representation (, , , )

b. Compute and using

= + 22

2! +33

3!

= ( .

0) = 1( )

2. If is a discrete model:

a. Simply write down the discrete state space representation (, , , ), like you

would for a continuous system


Topologies and Signal Flow Diagrams

Controller Topology

Observer Topology


Phase Variable Signal Diagram

Controller Canonical Signal Diagram

Observer Canonical Signal Diagram


Parallel Form Signal Diagram

Cascade Form Signal Diagram

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