State space control design Flashcards
What is state space form?
x_dot = Ax + Bu y = Cx + Du
Why do we normally neglect the D term of SS form?
Not sure, just normally a null matrix for most physical systems.
What is x_dot = Ax + Bu?
A differential equation describing the dynamics of the internal state of the system
What is y = Cx + Du?
Describes how the output relies on the system state.
How do you check for stability when something is in SS form?
Stable if all eigenvalues of A have negative real parts. Find eigenvalues by solving: det(A-lambdaI) = 0
What is reachability?
Aka controllability. Any final state can be reached from any initial state in finite time by choice of control input.
How can you tell if a system is reachable?
W = [B AB] has full rank
What is an observer?
A system that measures y and produces an estimate xhat of x.
What is a common observer structure?
xhat_dot = Axhat + Bu + L(y-Cxhat)
What is choosing L a balance of?
Predictions based on the model and the current estimate xhat AND new information arriving in measurement y. (Large L = trust measurements)
How can you tell if a system is observable?
W = [C CA] downwards has full rank
What is observability?
The state x at any finite time can be computed from measurements of y and u.
How do you convert from SS to TF?
G(s) = C(sI-A)^-1 B
How do you decide pole placement when deciding a closed loop system?
- Choose 2nd order dominant poles based on desired transient response
- Place all other poles (including observer poles) so they are less dominant (5-10x further left on imaginary axis so low zeta)
How are poles related to damping?
Distinct real poles = overdamped
Equal poles = critically damped
Complex poles = underdamped
How do you design an observer?
xsquig_dot = (A-LC)xhat
Make char poly by det(sI-(A-LC)) = 0
Compare coefficents with desired char poly
How do you find the TF from y to yhat?
xhat_dot = (A- LC)xhat + Ly, yhat = [1 0]
TF = newC(sI - newA)^-1 * newB
What is xhat vs xsquig?
xhat is the estimate
xsquig = x - xhat
How do you design a state feedback controller?
Find desired char poly.
System char poly: det(sI-(A-BK)) = 0
Equate coefficients
What is the characteristic second order polynomial?
s^2 + 2zetawn*s + wn^2
