Sumários
Reviews
12 abril 2018, 08:00 • Luis Marcelino Ferreira
Review of last class as per summary
State model from TF -- a derivation of CCF for a 3rd order system
Role of coefficients: numerator vs denominator
The issue of optimal control: giving Q and R
Remember optimal control is K=inv(R)*B’*S
Poles change location
Optimal control on top of traditional PID control
Optimal control all alone -- given the physical system, find the optimal control
How to check other indices, beyond J
Trial-and-error for overshooting
Simple problem: interconnection with an infinite area
Infinite area and zero area (no generation, just a load substation)
A zero area connected to another area
A zero area connected to multiple areas -- eg a triangle connection
Discussion on continuity of quantities such as frequency, phase, power interconnected
Practical class #7 group of Wednesday
11 abril 2018, 09:00 • Célia Maria Santos Cardoso de Jesus
Resolution of problem 45 (conclusion). Quiz #3
Pole placement, CCF, optimal control
10 abril 2018, 11:00 • Luis Marcelino Ferreira
Review of previous class, as per summary
The issue of controllable, then make stable
State feedback, u=-Kx
Controllable, then state feedback can place system poles at desired locations
New characteristic eq, eigenvalues for A-BK
K constant matrix, dimensions
BK is a product -- the importance of B
Controllable canonical form CCF (or reachable canonical form)
Direct derivation from a transfer function
Lewis, optimal control for polynomial systems, page 292
Easy to place poles at desired location: [alpha poly]
K=[alpha poly-characteristic poly]
Handout, cose_part1_3.doc
Pole placement without CCF
How to choose K’s -- especially if there are redundancy
Avoid difficult nonlinearities
Redundancy of K’s -- room for choice
Performance index -- costs of operation
Optimal control -- a special quadratic function (trajectory error and control effort)
Linear quadratic regulator -- a gift from Algebra to Engineering
(A,B) controllable, then define Q>=0 and R>0
Check results for pole location, time response
Practical class #6 group of Thursday
5 abril 2018, 09:30 • Célia Maria Santos Cardoso de Jesus
Resolution of problem 45 (questions 1, 2 and beginning of 3).
State models and linear algebra
5 abril 2018, 08:00 • Luis Marcelino Ferreira
The recurrent theme of linearity and linear operators
dx/dt=Ax+Bu
y=Cx+Du
Symbolic drawing
Integrator, plant matrix, input, output
Where is the control? and perturbations?
State feedback or output feedback?
In general, create an observer, for x^
State control is powerful
What is controllability?
Another drawing, this time with the plant matrix as diagonal
Multiple integrators on a single variable (instead of one matrix on a transformed state vector)
Controllability simply as “iff no row of B is null, then controllable”
The classic controllability matrix (B AB ...)
Iff rank n, then controllable
What controllable means -- can make it go to zero, can make it go to any desired state
Certainly means that the system if unstable can be made stable
Review of formula for transfer matrix form state model -- unique
What is common to all those transfer functions? |sI-A| in the denominator
An example from a two-area control system
How to derive a state model from transfer functions
A simple example in which y=x
Notice that dx/dt=(-1/T) x+...
The importance of being negative -- take look at A for the two-area model
The importance of being diagonal
Remember trace: tr(A) means two things
What is coming next