Chapter 2: State Space Definitions

Question 1

Advantage of State Space Design (3)

Answer 1

1. Reveals all internal poles/zeros
2. Linear algebra can be used
3. Supports design and analysis of feedback control systems

Question 2

Matrices of State Space Equations (4)

Answer 2

1. A - System Matrix
2. B - Input Matrix
3. C - Output Matrix
4. D - Feedforward Matrix

Question 3

State Space and Transfer Equation

1. C.T.
2. D.T.

Answer 3

1. H(s) = C * inverse(sI-A) * B + D

2. H(z) = C * inverse(zI-A) * B + D

Question 4

Transformation of State Space Matrices where x’ = Tx (4)

Answer 4

1. A’ = T * A * inverse(T)
2. B’ = inverse(T) * B
3. C’ = C * inverse(T)
4. D’ = D

Question 5

Parallel Canonical Form Properties

Answer 5

1. Poles of transfer function are eigenvalues of the system

2. H(s) = bn + c1/(s-p1) + … + cn/(s-pn)

Question 6

Cascade Canonical Properties

Answer 6

1. Diagonal elements are eigenvalues of system matrix and poles of transfer function
2. Series of 1st Order D.E. H(s) = H1(s) * H2(s) * … * Hn(s)

Question 7

Phase Variable Canonical Properties

Answer 7

1. Used when there are no input derivatives.
2. x1 = y, x2 = x1’, … , xn = xn-1’
3. Simplifies control gain selection

Question 8

Controllable Canonical Properties

Answer 8

1. Used to design optimal pole placement
2. Y(s)/U(s) = P(s)/U(s) * Y(s)/P(s)
3. Output of last integrator is x1 = p, xn is output to first integrator

Question 9

Observable Canonical Properties

Answer 9

1. Estimation of state vector by observing input/output of system
2. Observability canonical form is used if observable form doesn’t exist
3. Obtain highest order on left side and integrate

Question 10

Static Equilibrium

1. C.T. State
2. D.T. State
3. Output

Answer 10

1. x’ = f(xe, u0) = 0
2. x(k+1) = f(xe,u0) = xe
3. ye = g(xe,u0)

Question 11

Linearization of Function

Answer 11

1. Taylor Series Approximation

2. f(xe + Dx) = f(xe) + df/dx Dx + 1/2! d2f/dx2 Dx^2 + … where x = xe

Question 12

Small Signal Model

f(xe + D*x)

Answer 12

f(xe) + df/dx D*x where x = xe