Multivariable control Flashcards
Draw the SFCIA diagram and state the main equations.
Derive the CL system dynamics state equation
x’ =Ax + Bu
xi’ = r - Ex
u = kixi - kx
z = Ex
[x’] = [A-BK : BKi ] + [0]
[xi`] [ - E : 0 ] [r]
Draw the LMRC diagram
State the error dynamics eqn
——————–{Am,Bm}—-xm—-[E]—–zm
|
r———————-|—-[Kr]—-(X)—u—-{A,B} —–x—–[E]—-z
| |
|______[k]_______|
xe = xm - x
What is the equation for calculating a pseudo inverse?
A^+ = [A^T A]^-1 A^T (on formula sheet)
What is the classicial method for finding x(t)
How would you simplify this to useable form? (A = ???)
x(t) = e^At x(0) + integral(E^A(t-t) B u(t) dt`)
e^At = W E^{Lambda t} W^-1
What is the Laplace method for finding x(t)?
x(t) = L^-1{ (SI - A)^-1 B u(s) }
What is the transfer function matrix for measured output [G(s) = y(s)/u(s)]
C(SI-A)^-1 B
What is the transfer function matrix for controlled output [G(s) = y(s)/u(s)]
E(SI-A)^-1 B
Draw the diagram of Output feedback control
State the main equations
Derive the CL system dynamics state equation
u= -k X̂ +KR r
X̂ ` = (A - G C) X̂ +B u +Gy
xe = x - X̂
leads to
[ x] = [A - B K B K ] + [B KR] r
[xe] [ 0 A - G C ] [ 0 ]
What is young’s inequality?
2ab <= (a^2/gamma) + (gamma*b^2)
for any gamma >0
What are the advantages and disadvantages of SFC?
+ Relatively simple design
- Steady state error is possible if A and B are not perfectly known
- All states need to be measurable
- Plant needs to be controllable
What are the advantages and disadvantages of SFCIA?
+ Steady state error = 0 even if A,B not exactly known
+ K and K_i are determined simultaneously as per SFC methods
- All states need to measurable
- Augmented plant needs to be controllable
- There are extra m evals to locate compared with SFC
What are the advantages and disadvantages of OFC?
+ Does not require all states to be measurable
- Steady state error is possible
- Plant needs to be both controllable and observable
- There are 2 sets of n evals to locate including the Luenberger Observer
What are the advantages and disadvantages of the LO?
+ Simple design
- {A,C} must be observable
- x_e not equal to zero possible if ABC not known exactly
What are the advantages and disadvantages of LMRC?
+ Follows the reference model
+ No eigenvalue problem to solve
+ Simple hand calcs possible
+ Attractive idea
- All states must be measurable
- Plant must be controllable
- Erzberger’s must be satisfied
- residual error possible
What are the advantages and disadvantages of OFCIA?
+ No SSE even if A<b></b>
+ z=r even if A,B are not known exactly
- A,B must be observable
- Augmented plant must be controllable
