Lecture 35: State-Space Control Design - Controllability, Observability, Pole Placement, Study notes of Mechanical Engineering

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Lecture 35 - Announcements

• HW#8 posted on Web site – Due Nov 29th.
• Final Exam Schedule:
  • Thursday, December 16th
  • 10:00am until 12:00pm – 2 Hours
  • Exam in Weil 270
• Final Exam Format
  • Multiple Choice
• Today’s complete class notes will be posted on on the web site.

Today’s Agenda

• State-Space controller design: Controllability and Observability
• Pole-placement
• Ackerman’s formula

State-Space Control Design

• Frequency domain or Root Locus design is good but: - We place just dominant poles and hope for the best with the remaining poles.
• Wouldn’t it be nice to place all poles where we want them? - Pole placement using state-space design.

State-Space Control Design

• Frequency domain or Root Locus design is good but: - One input & one output
• State-space techniques easily handle multiple input/multiple output systems.

Controllability

• A system is completely controllable if there exists an unconstrained input u(t) that can transfer any initial state x (t 0 ) to any opther desired location x (t) in a finite time t 0 < t < T
• To demonstrate this, the controllability matrix P c must be full rank (det P c can’t be 0):

[ ]

whereBisn and Aisn n

Pc B ABA B An B

× ×

= −

2
1

Observability

• A system is observable if and onl;y if there exists a finite time T such that the initial state x (0) can be determined from the observation history y(t) given the control u(t)
• To be observable, the observability matrix P o must be full rank (det P o can’t be 0):

whereCis nand Aisn n

CA

CA

C

P

n

o × ×

















=

−

1



Full-State Feedback

• u = - Kx
• Assumes we know all the state variables
• No external inputs, response is only a function of initial values, and system is designed to drive those values to zero.
• How is this useful?

Full-State Feedback

• u = - Kx
• Assumes we know all the state variables
• No external inputs, response is only a function of initial values, and system is designed to drive those values to zero.
• How is this useful?

Ackerman’s Formula

Where

• Pc is the controlability matrix
• q(
  ) =
  n^ +  1
  n-1^ + … + n is the desired characteristic equation

K = [ 0 0
1 ] P (^) c^ −^1 q ( A )

State-Space Design in MATLAB

• Define matrices: A=[0 1 0; 0 0 1; -1 -5 -6] B=[0;0;1]
• Controlability: Pc=ctrb(A,B,1) n=det(Pc)
• Ackerman design: K=acker(A,B,P) where P is vector of desired closed-loop poles

A=[0 1 0; 0 0 1; -1 -5 -6]

B=[0;0;1]

P=[-10; -2+4i; -2-4i]

k=acker(A,B,P)

A = 0 1 0 0 0 1 -1 -5 - B = 0 0 1 P = -10. -2.0000 + 4.0000i -2.0000 - 4.0000i k = 199 55 8