SAT midterm 2

X(s)

Inv(sI-A) B U(s)

Y(s)

(C inv(sI-A) B + D) U(s)

G(s)

C inv(sI-A) B + D

Natural response

Inv(sI-A) x0

Forced response

Inv(sI-A) B U(s)

Find eigenvalues

Det(lamb I - A) = 0

Find eigenvectors

(Lamb I - A) x = 0

Find characteristic polynomial

Det(lamb I - A)

Big X invertible

Unique eigenvalues, or repeated eigenvalues have enough linearly independent eigenvectors

If eigenvectors are linearly independent

Big lamb = big X A inv big X

Generalized eigenvector

(Lamb I - A) xk = -x k-1

A [eigenvectors | generalized eigenvectors]

[eigenvectors | generalized eigenvectors] Jordan blocks

eAt: A diagonal

e aii t on diagonal

eAt: A diagonalizable

Big X e big lamb t inv big X

eAt: not enough LI eigenvectors

[eigenvectors with generalized] eJt inv [eigenvectors with generalized]

e Jit

e lambi t t e lambi t t²/2 e lambi t etc (diagonal down)

Cayley Hamilton definition

Matrix A satisfies its own characteristic polynomial subbing in for lambda

Cayley Hamilton implication

Any power of A can be written as sum of previous powers multiplied by constants

eAt taylor

I + At + A²t²/2! + etc…

Zero order hold

Maintains value through sampler time

y(k-1)

Delta y(k)

A^k: A diagonalizable

Big X (big lamb)^k inv big X

A^k: Cayley Hamilton

an-1(k) A^n-1 + … + a1(k) A + a0(k) I

z^-1

x(k) → x(k-1)

G dt (z): Euler difference

G ct ((1-z^-1)/Ts)

G dt (z): tustin

(1-z^-1)/(1+z^-1) 2/Ts

Continuous time stability

Real part of poles negative

Discrete time stability

Real and imaginary parts of pole inside unit circle

Controllability check

[B AB A²B …] rank = n

Controllable subspace

Image of [B AB A²B …]

Observability check

[C ; CA; CA²; …] rank = n

Observable subspace

Image of [C ; CA; CA²; …]

Controllability meaning

Can reach any other state in n steps

Observability meaning

Can distinguish and initial state with observed inputs and outputs for n steps

Canonical decomposition for Observability

T2 spans unobservable space, T full rank

Canonical decomposition for controllability

T1 Spence unobservable space, T full rank

Control law

u = -Kx + v

A CL

A - BK

K

[kn kn-1 … k1]

Desired characteristic polynomial

Get from desired eigenvalues

Controller canonical form

Used to find K for pole placement

PAP^-1

Ones above diagonal, -ai in reverse order on bottom

P^-1 B

Zeros, 1 on bottom

K hat

ai hat - ai in reverse order

K hat for z to K for x

K = K hat P^-1

Requirement for pole placement

System is controllable

Minimal representation

State representation that is both observable and controllable