Analytic Feedback System Design:
An Interpolation Approach
Peter Dorato
Preface
Analytic vs Trial & Error
• We can determine
whether or not a
solution to a
problem exists
• If a solution exists
we have an
algorithm to find it
• We can not
determine whether
or not a solution to
a problem exists
• Trial and error
process may not
find the solution
Table of Contents
1.
2.
3.
4.
5.
•
•
Introduction
Design with Stable Compensators
Design with Unstable Compensators
Digital Control Design
Robust Design
A. Interpolation with SPR functions
B Stability Criteria
Point-of-View
• Systems process inputs to produce outputs
• The result of a design is a transfer function
that mathematically describes the
performance of the compensator
• Compensators/feedback add value to plants
• Building compensators is trivial
• Analytic insight is worth the effort
1. Introduction
1. Analytical vs Trial & Error
•
Why analytical techniques? Basic feedback system.
2. Some Analytical Design Methods
•
Why interpolation? Other available techniques.
3. Internal Stability
•
A new important concept.
4. The Interpolation Approach
•
Preview of things to come.
5. Modeling & Numerical Examples
•
What makes control design challenging.
6. Prerequisites
Stability Analysis Techniques
• Routh-Hurwitz
• Root Locus
• Nyquist
• Each stability analysis
technique can be used to
design control systems.
• Choose a controller
• Test for stability
• Repeat
• Trial & Error
Analytic design Techniques
• Conditions for the existence of a solution
• An algorithm that is guaranteed to find the
solution
• Sometimes produce complex compensators
• Need to be matched to the performance
measure
Basic
Feedback
System
R(s) +
-
• +/- signs
• Plant, P(s)
• Compensator, C(s)
E
C(s)
•
•
•
•
D +
U
+
P(s)
Input u; output y;
Input e;
System inputs: r, d
Replace u with m
later
Remember this slide
Y
Trial & Error:
1
P( s) 
2
Fixed Order
s( s  1)
Controller
C (s)  K
1. Choose a particular controller;
2. determine the numerical values;
• Multivariate polynomial inequalities (MPIs)
• Performance measures
• Stability
• Routh-Hurwitz, Nyquist, Root-locus,
Lienard-Chipart, etc.
3. If fail, choose more complicated controller;
1
E ( s) 
R( s )
1  C ( s ) P( s )
2
1
1  C ( s) P( s) s  j
0 3.5
 .01
Algebraic Details: Stability
1
1
s( s  1) 2
s( s  1) 2

 3
 3
2
K
1  C ( s) P( s) 1 
s  2s  s  K s  2s 2  s  K
s ( s  1) 2
s 3  2s 2  s  K  0
s3
s2
s1
s0
1 0
1
20
K
2K
0
2
K 0
Algebraic Details: Performance
1
1
s ( s  1) 2
s ( s  1) 2

 3
 3
2
1  C (s) P(s) 1  K
s  2 s  s  K s  2s 2  s  K
s( s  1) 2
2
1
1  C ( s ) P( s ) s  j
0  3.5
2 2  j (1   2 ) 2 2  j (1   2 )

 .01
2
2
2
2
K  2  j (1   ) K  2  j (1   )
4 4  ( (1   2 )) 2
 .01
2 2
2 2
( K  2 )  ( (1   ))
100  4 4   2 (1  2 2   4 )   ( K  2 2 ) 2   2 (1  2 2   4 )
100  2 4   2   6   K 2   4 K  1  2  2 4   6
K 2   4 K  99   2  198 4  99 6  0
Multivariate Polynomial Inequalities
v0 ( K )  K  0
v1 ( K )  2  K  0
v2 ( K ,  )  K 2   4 K  99   2  198 4  99 6  0
( : 0    3.5)[v0 ( K )  0  v1 ( K )  0  v2 ( K ,  )  0]
  " for all "
  " AND "
  " OR "
  " there exists "
•Solving the MPIs has been
extensively studied.
•If there is no solution, you
must choose a new compensator
structure and start the algebra
over.
Worst Case Analysis?
• Must we consider all possible omegas and
Ks?
• Can we identify the worst case?
– Highest frequency?
– Lowest frequency?
– Intermediate frequency?
• Find K (either plot or use quadratic
formula) for worst case
• Do these K’s satisfy the other two
requirements?
• Do we need a more complex compensator?
Comparison
Design
method
Advantages
Disadvantages
Analytic
Existence theorem
Guaranteed convergence
Limited design objectives
High-order compensator
Trail&Error Design insight
Low-order compensator
No existence theorem
No guarantee of convergence
Numeric
No design insight
Computational complexity
Multiple design objectives
Low-order compensator
Punchline: Analytic design techniques are worthwhile
learning.
2. Some Analytical Design Methods
•
•
•
•
Interpolation
Mean-square (H2), L2
State-Space Design
Hinfinity
3. Internal Stability
• All transfer functions that relate system inputs (R or
D) to possible system outputs (Y or U) with the
basic feedback system must be BIBO stable (proper
and only LHP poles)
Y/D=P/(1+PC); U/R=C/(1+PC); S=E/R=1/(1+PC)
•
•
•
•
External Stability: stability of T=Y/R=PC/(1+PC)
S+T=1
Internal stability implies external stability.
External stability does not imply internal stability.
Example: P(s)=1/s2, C(s)=s+1; Consider T and U/R
• Internal stability implies no “bad” pole/zero
cancellation
4. The Interpolation Approach
5. Modeling and Numerical
Examples
6. Prerequisites
2. Design with Stable Compensators
1.
2.
3.
4.
5.
Introduction
Parameterization with Units
Design with Stable Compensators
Notes & References
Problems
3. Design with Unstable
Compensators
1.
2.
3.
4.
5.
6.
Introduction
Q-Parameter Design
Youla Parameterization
Internal Model Control
Notes & References
Problems
4. Digital Control Design
1.
2.
3.
4.
5.
6.
7.
Introduction
Internal Stability
Deadbeat Response
Digital Unit-Parameter Design
Digital Q-Parameter Design
Notes & References
Problems
5. Robust Design
1.
2.
3.
4.
5.
6.
7.
Introduction
Gain Margin Design
Robust Stabilization
Two-Plant Stabilization
Passive Systems
Notes & References
Problems
A. Interpolation with SPR Functions
1. Preliminaries
2. Interpolation Algorithm
3. Interpolating Units via SPR Functions
B. Stability Criteria
1. Nyquist Stability Criterion
2. Root-Locus
3. Lienard-Chipart Stability Criterion