10. StateSpace Feedback Control

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10. State-Space Feedback Control

• Radu Stoleru

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Outline

• State Space Analysis (Recap of 7)
• State Feedback Control
• Static State
• Precompensated Static State
• Dynamic State
• Design Techniques
• Pole Placement
• LQR Optimal Control
• Problems

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State Space Analysis (Recap of 7)
Can you define a different state?
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State Space Analysis (Recap of 7)

• State Space Model

C?
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State Space Analysis (Recap of 7)
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Static State Feedback Control

• Analogous to P Control

• Control Law

gains
What signal is missing?
Characteristic polynomial - p.217
Closed-Loop Model
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Static State Example Tandem Queue

• Control objective

How did we get this?
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Static State Example Tandem Queue
disturbance
Initial cond. R13.5 R27.5
input
Why R1 has bigger variation?
state
output
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Static State - Other Examples
http//www.urbanrail.net
http//www.ccsn.nevada.edu
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Precompensated Static State Feedback Control

• Variation of Static State, that can track a
  reference
• Let r reference input
• xss steady-state value of state vector
• uss steady-state control input

How do we get these?

• Control Law

How does N affect the poles?
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Precompensated Static State Feedback Control

• Remember?

• At steady-state

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Precompensated Static State Feedback Control
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Precompensated Static Example Tandem Queue
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Precompensated Static Example Tandem Queue
disturbance
R 9-gt19 No disturbance
input
state
output
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Precompensated Static Example Tandem Queue
disturbance
R 9-gt9 Disturbance
input
state
Whats the problem?
output
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Dynamic State Feedback Control

• Analogous to PI Control

• Include integrated (accumulated) control error as
  part of state

Where is the disturbance?
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Dynamic State Feedback Control

• Augmented State Space model

Remember this ?
Closed-Loop Model
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Dynamic State Example Tandem Queue

• The closed loop model becomes

Somehow (we will see next time), we get KP115.3
KP215.5 KI-9.43
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Dynamic State Example Tandem Queue
Disturbance
Input
State

• Step increase in reference input

Output
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Dynamic State Example Tandem Queue

• Step increase in disturbance

Disturbance
Input
State
Output
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Dynamic State - Other Examples
http//cot.marine.usf.edu
http//www.fd.net.au/
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Comparison of State Space Techniques

• Simulations of 3 architectures
• Questions
• Reference input?
• Disturbance rejection?
• Settling time?

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Comparison of State Space Techniques
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Summary

• Static State Feedback Control
• Similar to P Control
• No reference input
• Precompensated Static Feedback Control
• Poor disturbance rejection
• Tracks reference input
• Dynamic Feedback Control
• Similar to PI Control
• Can track references and reject disturbances

Remember this ?
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Outline

• State Space Analysis (Recap of 7)
• State Feedback Control
• Static State
• Precompensated Static State
• Dynamic State
• Design Techniques
• Summary of State-Space FC Architectures
• Pole Placement
• LQR Optimal Control
• Problems

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Static State Feedback Control

• Control Law

gains
Characteristic polynomial - p.217
Closed-Loop Model
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Precompensated Static State Feedback Control

• Variation of Static State, that can track a
  reference

gains

• Control Law

Characteristic polynomial - p.217
Closed-Loop Model
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Dynamic State Feedback Control

• Include accumulated control error as part of
  state

• Control Law

Characteristic poly ?
Remember this ?
Closed-Loop Model
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Outline

• State Space Analysis (Recap of 7)
• State Feedback Control
• Static State
• Precompensated Static State
• Dynamic State
• Design Techniques
• Summary of State-Space FC Architecture
• Pole Placement
• LQR Optimal Control
• Problems

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Pole Placement

• Want to answer How to get Feedback Gains K?
• Not to exceed ks settling time, with MP maximum
  overshoot
• Extension of the PI Pole Placement design
• Compute the closed loop poles (assume complex)
• Construct desired characteristic polynomial p.
  304
• Construct modeled characteristic polynomial
• Solve for Ki (equate the coefficient of Step 2
  and Step 3)
• Verify the result (poles are within unit circle)

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Pole Placement Static State Example 1

• Step 1
• Step 2
• Step 3
• Step 4

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Pole Placement Static State Example 2

• Step 1
• Step 2
• Step 3
• Step 4

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Pole Placement Static State Example 2
Disturbance
Input
Initial cond. R13.5 R27.5
State
Output
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Pole Placement Dynamic State Example

• Step 1
• Step 2
• Step 3
• Step 4

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LQR Optimal Control Design

• Another approach
• Consider Control Errors (?x(k)) and Control
  Effort (?u(k)), where x is the state vector and u
  is control input
• Reduce Control Errors reduce settling time and
  overshoot
• Reduce Control Effort reduce noise sensitivity
• Linear Quadratic Regulation (LQR) objective
  minimize the impulse response of the states and
  the control expenditure, in a quadratic sense.
• LQR
• Two input parameters weighting matrices Q and R
• Q quantifies the cost of divergence of system
  state Control Errors
• R specifies the cost the Control Effort

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LQR Optimal Control Design

• LQR - Minimize
• Design steps
• Define Q and R
• Compute Ks
• Predict performance from model, or run simulation
• If unsatisfactory, choose new Q and R and repeat
• Compute Ks by hand
• Initialize S and K (temporary matrices) to 0
• Repeat until K converges

How?
No easy answer!
How?
Easy answer Matlab
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LQR Dynamic State Example

• Notes
• q11 q22 so the two state are weighted equally
• Only qii ! 0, so interactions among states is
  ignored
• R1, control effort and control error are equally
  important

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LQR Dynamic State Example
Disturbance
Input
State
good
Step increase in reference input
Output
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LQR Dynamic State Example
Disturbance

• Step increase in disturbance

Input
State
Even better
Output
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Outline

• State Space Analysis (Recap of 7)
• State Feedback Control
• Static State
• Precompensated Static State
• Dynamic State
• Design Techniques
• Pole Placement
• LQR Optimal Control
• Problems

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Apache HTTP Server

• Use Dynamic State Feedback Control
• Regulate CPU and MEM
• Control Input MC (max clients) and KA (keep
  alive)
• Operating Points?

• What are we looking for?

Closed-Loop Model and Ks

• State vector?

• Control Input?

• State Space model?

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Apache HTTP Server

• Do you remember this?

Dynamic state feedback

• We instead use

Why?
Closed-Loop Model
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Pole Placement Apache HTTP Server

• Step 1
• Step 2
• Step 3
• Step 4

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Pole Placement Apache HTTP Server
bad
good
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LQR Apache HTTP Server

• LQR Design steps
• Define Q and R
• Compute Ks (by hand, or Matlab - dlqr)
• Predict performance from model, or run simulation
• If unsatisfactory, choose new Q and R and repeat

• From Matlab (dlqr)

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LQR Apache HTTP Server
better
good
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Summary

• Static State Feedback Control
• Similar to P Control
• No reference input
• Precompensated Static Control
• Poor disturbance rejection
• Tracks reference input
• Dynamic Feedback Control
• Similar to PI Control
• Can track references and reject disturbances
• Pole Placement Control Design
• Obtain dominant poles from ks and Mp
• Obtain Ks from coefficients of desired and
  modeled characteristic equations
• LQR Control Design
• Use Matlab