ENB458 – Exam Resource Sheet
James Mount 2014
General
= ⋯ − = ⋮− [ ]
Controllability and Observability
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
Damping Ratio
Overshoot
Settling Time (2 nd Order)
Peak Time (2 nd Order)
|| = |=±| = = ln%0 1⁄100 ⁄ √ + l n % 100 % = 4=− ⁄ −− × 100 = 1 ∞∞ == ll→iimm → ∞∞ == 111 + ∞− ∞ = 1+ 1+ −−
Final Value Theorem and Steady State Error
FV Theorem (S Transfer Function)
FV Theorem (Z Transfer Function)
SSE
SSE of Open Loop SS (step input)
SSE of Closed Loop SS (step input)
ENB458 – Exam Resource Sheet
James Mount 2014
General
− + − = + −− + ⋯+ + ⋯+ + 00 01 10 ⋯⋯ 00 00 ̇ = 0⋮ 0⋮ 0⋮ ⋯⋱ 1⋮ + 0⋮ [ ] ⋯ 1 − = ⋯ − 1− ⋯⋯ 0 0 0 10 ̇ = [ 0⋮ ⋯⋱ 1⋮ 0⋮ 0⋮ ] + 0⋮ [ ] 0 ⋯ 0 1 0 0 = − ⋯ ⋮− 1⋮ ⋯⋱ 0⋮ 0⋮ −⋮ ̇ = 00 ⋯⋯ 10 01 + 0 ⋯ 0 0] [ = 1 0 ⋯ 0 0 = −+ − + == +
Converting from Transfer Function to State Space
Phase Variable
Controller Canonical
Observer Canonical
Converting from State Space to Transfer Function Time Response of System
To find time response convert SS to TF, but need to consider initial conditions ( the following,
), so use
ENB458 – Exam Resource Sheet
James Mount 2014
Model Development Modelling with Transfer Functions Electrical Systems
1/ = × + = = d et = det =
Component Resistor Inductor Capacitor
Impedance
1. Apply circuit theory such as KVL and KCL around meshes and at nodes a. Look at combining parallel components b. Look for voltage dividers 2. Put into matrix form, 3. Solve using Cramer’s rule
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 Spring Damper Mass
Impedance
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 Cramer’s rule 4. Manipulate current output quantity, from Crame r’s rule result, to desired quantity, if required.
Rotational Systems Component Spring Damper Mass
Impedance
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.
ENB458 – Exam Resource Sheet
James Mount 2014
Model Development Modelling with State Space Electrical Systems Component Resistor Inductor Capacitor
= = = =1 = 1 ∫ . = = ∫ . = = 1 Voltage - Current
Current - Voltage
Voltage – Charge
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 Spring Damper Mass
Translational
Rotational
̇
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)
ENB458 – Exam Resource Sheet
James Mount 2014
Controller Design Method 1 – Using
det( ) = 0
, , , det( ) = 0 , , , det = 0 , , , = − 0 = + − + − + ⋯+ + + + = − + −− + ⋯+ + 0 = = + − − + ⋯+ ,+, , = 0 0 … 1− ̇ ̇ = = 0 00 + 0 +01 0 + ̇ ̇ = = 0 0 1 ℎ, = +
1. Check for controllability 2. Using original state space representation (
) find closed loop characteristic
equation using
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 2. Using open loop characteristic equation find phase variable state space form ( 3. Compute controllability matrices and 4. Calculate the P transform matrix 5. Get desired closed loop characteristic equation from phase variable state space
6. Find desired characteristic equation using pole placement 7. Equate coefficients from equations in steps 5 and 6, and solve for phase variable gains 8. Transform phase variable gains to o riginal state space gains using
Method 3 – Using Ackermann Formula
1. Find desired characteristic equation using pole placement 2. Compute
using original state space representation (
)
3. Calculate the controllability matrix 4. Apply Ackermann Formula
PI Controller
Need to augment the matrix
Open Loop State Space
Closed Loop State Space
If need to transform gains, remember to only apply transform to
not
)
ENB458 – Exam Resource Sheet
James Mount 2014
Observer Design Method 1 – Using
det( ) = 0
, , , det( ) = 0 , , , det = 0 , , , = − 0 = + − + − + ⋯+ + + + = − + −− + ⋯+ + 0 = = + − − + ⋯+ , +, , 0 = − 01⋮
1. Check for observability 2. Using original state space representation (
) find closed loop characteristic
equation using
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 2. Using open loop characteristic equation find observer canonical state space form ( ) 3. Compute observability matrices and 4. Calculate the P transform matrix 5. Get desired closed loop characteristic equation from phase variable state space 6. Find desired characteristic equation using pole placement 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
Method 3 – Using Ackermann Formula
1. Find desired characteristic equation using pole placement 2. Compute
using original state space representation (
3. Calculate the observability matrix 4. Apply Ackermann Formula
)
ENB458 – Exam Resource Sheet
James Mount 2014
Optimal Control Design Methodology
1 max 0 = + + − = −
1. Compute Q and R weighting matrices, by using
, 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 3. Compute the optimal gains
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
ENB458 – Exam Resource Sheet
James Mount 2014
Discrete Systems Z Transformation Methods Backward Difference
Tustin Transform
Pole Zero Mapping
Z Transform Table
= 1 Δ− − 2 1 = Δ 1+− == cosΔ±sinΔ
Courtesy of http://lpsa.swarthmore.edu/LaplaceZTable/LaplaceZFuncTable.html
ENB458 – Exam Resource Sheet
James Mount 2014
Discrete Systems
= / ==− = ⋯
Difference Equations
1. Provided with the transfer function
cross multiply to get
2. Knowing that transform equation from step 2 into a difference equation 3. Manipulate equation from step 3 to get
General Form (Block Diagram)
− − = 20 + 60 1 = 20 + 60 − − 0 = + 20 +60
Determining Stability
1. Get into z form. (i.e. becomes which rearranges to ) 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
1. Need to model the whole plant dynamics including elements such as Z.O.H 2. If Z.O.H need to use
3. Once found write down discretised state space model f rom 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
, , , Γ = ≈ + Δ2! + Δ3! Γ = ∫ . = − ,Γ,,
1. If it is a continuous model: a. Write down the continuous state space representation ( b. Compute and using
2. If is a discrete model: a. Simply write down the discrete state space representation would for a continuous system
)
, like you
ENB458 – Exam Resource Sheet
Topologies and Signal Flow Diagrams Controller Topology
Observer Topology
James Mount 2014
ENB458 – Exam Resource Sheet
Phase Variable Signal Diagram
Controller Canonical Signal Diagram
Observer Canonical Signal Diagram
James Mount 2014
ENB458 – Exam Resource Sheet
Parallel Form Signal Diagram
Cascade Form Signal Diagram
James Mount 2014
