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7/30/2019 Lecture 09 EAILC Stochastic STR
1/65
Lecture 9: Stochastic / Predictive Self-Tuning Regulators.
Minimum Variance Control.
Moving Average Controller.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 1/14
7/30/2019 Lecture 09 EAILC Stochastic STR
2/65
The stochastic model
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise,
A(q) = qn + a1 qn1 + + an, deg{A} = n
B(q) = b1 qnd0
+ + bn, deg{B} = n d0
C(q) = qn + c1 qn1 + + cn, deg{C} = n
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 2/14
7/30/2019 Lecture 09 EAILC Stochastic STR
3/65
The stochastic model
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise,
A(q) = qn + a1 qn1 + + an, deg{A} = n
B(q) = b1 qnd0
+ + bn, deg{B} = n d0
C(q) = qn + c1 qn1 + + cn, deg{C} = n
Equivalently,
y(t) = a1 y(t 1) an y(t n)
+ bd0 u(t d0) + + bn u(t n)
+ e(t) + c1 e(t 1) + + cn e(t n)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 2/14
7/30/2019 Lecture 09 EAILC Stochastic STR
4/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:
Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
5/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
Apparently, one can also make deg{C} = deg{A} = n byrenaming signals.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
6/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
Apparently, one can also make deg{C} = deg{A} = n byrenaming signals. Let us see it on the example
y(t + 1) + a y(t) = b u(t) + c e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
7/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
Apparently, one can also make deg{C} = deg{A} = n byrenaming signals. Let us see it on the example
y(t + 1) + a y(t) = b u(t) + c e(t)
It can be rewritten as
(q + a) y(t) = b u(t) + c e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
8/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
Apparently, one can also make deg{C} = deg{A} = n byrenaming signals. Let us see it on the example
y(t + 1) + a y(t) = b u(t) + c e(t)
It can be rewritten as
(q + a) y(t) = b u(t) + c q e(t 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
9/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
Apparently, one can also make deg{C} = deg{A} = n byrenaming signals. Let us see it on the example
y(t + 1) + a y(t) = b u(t) + c e(t)
It can be rewritten as
(q + a) y(t) = b u(t) + q enew(t)
introducing enew(t) = c e(t 1).
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
10/65
Remarks on modeling noise.
Assuming that C is stable is not restrictive:Typically, such a model is obtained experimentally fromspectrum characteristics. It does not change if the unstableroots are substituted by stable, which are symmetrical to themwith respect to the unite circle.
Apparently, one can also make deg{C} = deg{A} = n byrenaming signals. Let us see it on the example
y(t + 1) + a y(t) = b u(t) + c e(t)
It can be rewritten as
y(t + 1) + a y(t) = b u(t) + enew(t + 1)
with enew(t) = c e(t 1) being white noise with
var{enew(t)} = c2 var{e(t)}.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 3/14
7/30/2019 Lecture 09 EAILC Stochastic STR
11/65
Minimum-variance control: Example
Consider the model
y(t) = a y(t 1) + b u(t 1) + c e(t 1) + e(t)
where |c| < 1 and {e(t)} is a sequence of random variableswith E{e(t)} = 0 and var{e(t)} = 2.
Our goal is to regulatey(t) to0 as close as possible.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 4/14
7/30/2019 Lecture 09 EAILC Stochastic STR
12/65
Minimum-variance control: Example
Consider the model
y(t) = a y(t 1) + b u(t 1) + c e(t 1) + e(t)
where |c| < 1 and {e(t)} is a sequence of random variableswith E{e(t)} = 0 and var{e(t)} = 2.
Our goal is to regulatey(t) to0 with minimal variance.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 4/14
7/30/2019 Lecture 09 EAILC Stochastic STR
13/65
Minimum-variance control: Example
Consider the model
y(t) = a y(t 1) + b u(t 1) + c e(t 1) + e(t)
where |c| < 1 and {e(t)} is a sequence of random variableswith E{e(t)} = 0 and var{e(t)} = 2.
Our goal is to regulatey(t) to0 with minimal variance.
In the absence of noise:
y(t) = a y(t 1) + b u(t 1)
the best strategy (dead beat design) is
u(t) =a
by(t) = y(t + 1) = 0
What should we do when noise is present?
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 4/14
7/30/2019 Lecture 09 EAILC Stochastic STR
14/65
Minimum-variance control: Example
Consider the model
y(t) = a y(t 1) + b u(t 1) + c e(t 1) + e(t)
where |c| < 1 and {e(t)} is a sequence of random variableswith E{e(t)} = 0 and var{e(t)} = 2.
Our goal is to regulatey(t) to0 with minimal variance.
In the absence of noise or when c = 0:
y(t) = a y(t 1) + b u(t 1) + e(t)
the best strategy (dead beat design) is
u(t) =a
by(t) = y(t + 1) = e(t + 1)
What should we do when noise dynamics are present?
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 4/14
7/30/2019 Lecture 09 EAILC Stochastic STR
15/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
16/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
Let us try to compute e(t) symbolically
(q + a) y(t) = b u(t) + (q + c) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
17/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
Let us try to compute e(t) symbolically
e(t) =q + a
q + cy(t)
b
q + cu(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
18/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
Let us try to compute e(t) symbolically
e(t) =q + a
q + cy(t)
b
q + cu(t)
Substituting back into the model
y(t+1) = a y(t)+b u(t)+e(t+1)+c q + a
q + cy(t)
b
q + cu(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
19/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
Let us try to compute e(t) symbolically
e(t) =q + a
q + cy(t)
b
q + cu(t)
Substituting back into the model
y(t + 1) =(c a) q
q + cy(t) +
b q
q + cu(t)
known and independent from e(t + 1)!
+e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
20/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
Let us try to compute e(t) symbolically
e(t) =q + a
q + cy(t)
b
q + cu(t)
Substituting back into the model
y(t + 1) =(c a) q
q + cy(t) +
b q
q + cu(t)
known and independent from e(t + 1)!
+e(t + 1)
Clearly, var{y(t + 1)} var{e(t + 1)} =
2
.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
21/65
Minimum-variance control: Example (contd)
We have the process
y(t + 1) + a y(t) = b u(t) + e(t + 1) + c e(t)
Let us try to compute e(t) symbolically
e(t) =q + a
q + cy(t)
b
q + cu(t)
Substituting back into the model
y(t + 1) =(c a) q
q + cy(t) +
b q
q + cu(t)
known and independent from e(t + 1)!
+e(t + 1)
The minimal variance is achieved with: u(t) = c a
b y(t).
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 5/14
7/30/2019 Lecture 09 EAILC Stochastic STR
22/65
Minimum-variance control: General case
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise with var{e(t)} = 2.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 6/14
7/30/2019 Lecture 09 EAILC Stochastic STR
23/65
Minimum-variance control: General case
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise with var{e(t)} = 2.
Additional assumptions:
The polynomials A and B are monic (right rescaling).
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 6/14
7/30/2019 Lecture 09 EAILC Stochastic STR
24/65
Minimum-variance control: General case
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise with var{e(t)} = 2
.
Additional assumptions:
The polynomials A and B are monic (right rescaling). deg{A} = deg{C} = n 1 (always can be done).
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 6/14
7/30/2019 Lecture 09 EAILC Stochastic STR
25/65
Minimum-variance control: General case
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise with var{e(t)} = 2
.
Additional assumptions:
The polynomials A and B are monic (right rescaling). deg{A} = deg{C} = n 1 (always can be done).
deg{A} deg{B} = d0 1 known relative degree.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 6/14
7/30/2019 Lecture 09 EAILC Stochastic STR
26/65
Minimum-variance control: General case
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise with var{e(t)} = 2
.
Additional assumptions:
The polynomials A and B are monic (right rescaling). deg{A} = deg{C} = n 1 (always can be done).
deg{A} deg{B} = d0 1 known relative degree.
The polynomial C is stable (always can be done).
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 6/14
7/30/2019 Lecture 09 EAILC Stochastic STR
27/65
Minimum-variance control: General case
Assume the plant is represented by the special ARMAX model
A(q) y(t) = B(q) u(t) + C(q) e(t)
where {e(t)} is white noise with var{e(t)} = 2
.
Additional assumptions:
The polynomials A and B are monic (right rescaling). deg{A} = deg{C} = n 1 (always can be done).
deg{A} deg{B} = d0 1 known relative degree.
The polynomial C is stable (always can be done).
The polynomial B is stable a serious restriction.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 6/14
7/30/2019 Lecture 09 EAILC Stochastic STR
28/65
Minimum-variance control: General case (contd)
We have the process
A(q) y(t) = B(q) u(t) + C(q) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 7/14
7/30/2019 Lecture 09 EAILC Stochastic STR
29/65
Minimum-variance control: General case (contd)
We have the process
A(q) y(t) = B(q) u(t) + C(q) e(t)
Let us solve it for y(t)
y(t) =B(q)
A(q)u(t) +
C(q)
A(q)e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 7/14
7/30/2019 Lecture 09 EAILC Stochastic STR
30/65
Minimum-variance control: General case (contd)
We have the process
A(q) y(t) = B(q) u(t) + C(q) e(t)
Shifting by the relative degree
y(t + d0) =B(q)
A(q)u(t + d0) +
C(q)
A(q)e(t + d0)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 7/14
7/30/2019 Lecture 09 EAILC Stochastic STR
31/65
Minimum-variance control: General case (contd)
We have the process
A(q) y(t) = B(q) u(t) + C(q) e(t)
Shifting by the relative degree and rewriting
y(t + d0) =B(q)
A(q)u(t + d0) +
C(q) qd01
A(q)e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 7/14
7/30/2019 Lecture 09 EAILC Stochastic STR
32/65
Minimum-variance control: General case (contd)
We have the process
A(q) y(t) = B(q) u(t) + C(q) e(t)
Shifting by the relative degree and rewriting
y(t + d0) =B(q)
A(q)u(t + d0) +
C(q) qd01
A(q)e(t + 1)
Using polynomial long division
C(q) qd01
A(q) = F(q) +
G(q)
A(q)
whereG(q)A(q)
is strictly proper and
deg{F} = deg{C(q) qd01} deg{A} = d0 1.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 7/14
7/30/2019 Lecture 09 EAILC Stochastic STR
33/65
Minimum-variance control: General case (contd)
We have rewritten the process as
y(t + d0) =B(q)
A(q)u(t + d0) +
F(q) +
G(q)
A(q)
e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
34/65
Minimum-variance control: General case (contd)
Finally the process can be represented by
y(t + d0) =qd0 B(q)
A(q) proper fraction
u(t) +q G(q)
A(q) proper fraction
e(t) + F(q) e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
35/65
Minimum-variance control: General case (contd)
Finally the process can be represented by
y(t + d0) =qd0 B(q)
A(q) proper fraction
u(t) +q G(q)
A(q) proper fraction
e(t) + F(q) e(t + 1)
Solving the model for e(t), we have
e(t) =A(q)
C(q)y(t)
B(q)
C(q)u(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
36/65
Minimum-variance control: General case (contd)
Finally the process can be represented by
y(t + d0) =qd0 B(q)
A(q) proper fraction
u(t) +q G(q)
A(q) proper fraction
e(t) + F(q) e(t + 1)
Solving the model for e(t), we have
e(t) =A(q)
C(q)y(t)
B(q)
C(q)u(t)
After substituting it back
y(t+d0) =qd0 B
Au(t)+
q G
A
A
Cy(t)
B
Cu(t)
+F e(t+1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
37/65
Minimum-variance control: General case (contd)
Finally the process can be represented by
y(t + d0) =qd0 B(q)
A(q) proper fraction
u(t) +q G(q)
A(q) proper fraction
e(t) + F(q) e(t + 1)
Solving the model for e(t), we have
e(t) =A(q)
C(q)y(t)
B(q)
C(q)u(t)
After substituting it back and collecting terms
y(t + d0) =q G
Cy(t) +
q B
A C
qd0 C G
u(t) + F e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
38/65
Minimum-variance control: General case (contd)
Finally the process can be represented by
y(t + d0) =qd0 B(q)
A(q) proper fraction
u(t) +q G(q)
A(q) proper fraction
e(t) + F(q) e(t + 1)
Solving the model for e(t), we have
e(t) =A(q)
C(q)y(t)
B(q)
C(q)u(t)
After substituting it back and collecting terms
y(t + d0) =q G
Cy(t) +
q B
C
(qd0 C G)
Au(t) + F e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
39/65
Minimum-variance control: General case (contd)
Finally the process can be represented by
y(t + d0) =qd0 B(q)
A(q) proper fraction
u(t) +q G(q)
A(q) proper fraction
e(t) + F(q) e(t + 1)
Solving the model for e(t), we have
e(t) =A(q)
C(q)y(t)
B(q)
C(q)u(t)
Substituting F(q) fromC(q) qd01
A(q) = F(q) +G(q)A(q)
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t) + F(q) e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 8/14
7/30/2019 Lecture 09 EAILC Stochastic STR
40/65
Minimum-variance control: General case (contd)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Mi i i l G l ( d)
7/30/2019 Lecture 09 EAILC Stochastic STR
41/65
Minimum-variance control: General case (contd)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
Clearly, var{y(t + d0)} var{F(q) e(t + 1)}
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Mi i i t l G l ( td)
7/30/2019 Lecture 09 EAILC Stochastic STR
42/65
Minimum-variance control: General case (contd)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
var{y(t+d0)} var{F(q) e(t+1)} = 1 + f21 + + f
2d01
2
where F(q) = qd01 + f1 qd02 + + fd01.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Mi i i t l G l ( td)
7/30/2019 Lecture 09 EAILC Stochastic STR
43/65
Minimum-variance control: General case (contd)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
var{y(t+d0)} var{F(q) e(t+1)} = 1 + f21 + + f
2d01
2
where F(q) = qd01 + f1 qd02 + + fd01.
The minimal variance is achieved with: u(t) =
G(q)
B(q) F(q) y(t) .
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Minimum variance control: General case (contd)
7/30/2019 Lecture 09 EAILC Stochastic STR
44/65
Minimum-variance control: General case (contd)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
var{y(t+d0)} var{F(q) e(t+1)} = 1 + f21 + + f
2d01
2
where F(q) = qd01 + f1 qd02 + + fd01.
The minimal variance is achieved with: u(t) =
G(q)
B(q) F(q) y(t) .The closed-loop system is
A(q) y(t) = B(q) u(t) + C(q) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Minimum variance control: General case (contd)
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Minimum-variance control: General case (cont d)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
var{y(t+d0)} var{F(q) e(t+1)} = 1 + f21 + + f
2d01
2
where F(q) = qd01 + f1 qd02 + + fd01.
The minimal variance is achieved with: u(t) =
G(q)
B(q) F(q) y(t) .The closed-loop system is
A(q) y(t) = B(q) G(q)
B(q) F(q) y(t) + C(q) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Minimum variance control: General case (contd)
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Minimum-variance control: General case (cont d)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
var{y(t+d0)} var{F(q) e(t+1)} = 1 + f21 + + f
2d01
2
where F(q) = qd01 + f1 qd02 + + fd01.
The minimal variance is achieved with: u(t) =
G(q)
B(q) F(q) y(t) .The closed-loop system is
B(q) (A(q) F(q) + G(q)) y(t) = B(q) F(q) C(q) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Minimum variance control: General case (contd)
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Minimum-variance control: General case (cont d)
We have rewritten the process as
y(t + d0) =q G(q)
C(q)y(t) +
q B(q) F(q)
C(q)u(t)
known and independent from F(q) e(t + 1)
+F(q) e(t + 1)
var{y(t+d0)} var{F(q) e(t+1)} = 1 + f21 + + f
2d01
2
where F(q) = qd01 + f1 qd02 + + fd01.
The minimal variance is achieved with: u(t) =
G(q)
B(q) F(q) y(t) .
The closed-loop system (using: qd01 C = A F + G) is
qd0
1 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 9/14
Minimum-variance control: Remarks
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Minimum-variance control: Remarks
(1) The closed-loop system
qd01 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
defines the noise-to-output relation
y(t) =B(q) F(q) C(q)
qd01 B(q) C(q)e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 10/14
Minimum-variance control: Remarks
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Minimum variance control: Remarks
(1) The closed-loop system
qd01 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
defines the noise-to-output relation
y(t) =F(q)
qd01e(t) =
qd01 + f1 qd02 + + fd01
qd01e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 10/14
Minimum-variance control: Remarks
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Minimum variance control: Remarks
(1) The closed-loop system
qd01 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
defines the noise-to-output relation
y(t) = (1 + f1 q1 + + f
d0+1) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 10/14
Minimum-variance control: Remarks
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Minimum variance control: Remarks
(1) The closed-loop system
qd01 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
defines the noise-to-output relation
y(t) = e(t) + f1 e(t 1) + + fd0+1 e(t d0 + 1)
which is a moving average of order d0
1.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 10/14
Minimum-variance control: Remarks
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u a a ce co t o e a s
(1) The closed-loop system
qd01 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
defines the noise-to-output relation
y(t) = e(t) + f1 e(t 1) + + fd0+1 e(t d0 + 1)
which is a moving average of order d0
1.
(2) The closed-loop poles consist of (a) zeros of B(q), (b) zerosof C(q), and (c) d0 1 zeros.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 10/14
Minimum-variance control: Remarks
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(1) The closed-loop system
qd01 B(q) C(q) y(t) = B(q) F(q) C(q) e(t)
defines the noise-to-output relation
y(t) = e(t) + f1 e(t 1) + + fd0+1 e(t d0 + 1)
which is a moving average of order d0
1.
(2) The closed-loop poles consist of (a) zeros of B(q), (b) zerosof C(q), and (c) d0 1 zeros.
(3) The minimum variance controller
u(t) = G(q)
B(q) F(q)y(t)
can be interpreted as a pole-placement controller.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 10/14
Minimum-variance control as pole-placement
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p p
Sinceqd01 C(q) = A(q) F(q) + G(q)
the closed-loop characteristic polynomial
Ac(q) = qd01 B(q) C(q) = A(q) B(q) F(q) R(q)
+B(q) G(q) S(q)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 11/14
Minimum-variance control as pole-placement
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p p
Sinceqd01 C(q) = A(q) F(q) + G(q)
the closed-loop characteristic polynomial
Ac(q) = qd01 B(q) C(q) = A(q) B(q) F(q) R(q)
+B(q) G(q) S(q)
Note that for the Diophantine equation Ac = A R+ B S
deg{S(q)} = deg{G(q)} = n 1
andS/R
is proper since
deg{R} = deg{B} +deg{F} = (n d0) + (d0 1) = n 1.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 11/14
Minimum-variance control as pole-placement
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Sinceqd01 C(q) = A(q) F(q) + G(q)
the closed-loop characteristic polynomial
Ac(q) = qd01 B(q) C(q) = A(q) B(q) F(q) R(q)
+B(q) G(q) S(q)
Note that for the Diophantine equation Ac = A R+ B S
deg{S(q)} = deg{G(q)} = n 1
andS/R
is proper since
deg{R} = deg{B} +deg{F} = (n d0) + (d0 1) = n 1.
Can we use an analogous pole-placement technique for thecase whenB(q) has unstable zeros?
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 11/14
Moving Average Controller
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Design the controller
u(t) = S(q)
R(q)y(t) for A(q) y(t) = B(q) u(t)+C(q) e(t)
as follows.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 12/14
Moving Average Controller
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Design the controller
u(t) = S(q)
R(q)y(t) for A(q) y(t) = B(q) u(t)+C(q) e(t)
as follows.
Factor B as B = B+ B, where B+ is monic and stable.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 12/14
Moving Average Controller
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Design the controller
u(t) = S(q)
R(q)y(t) for A(q) y(t) = B(q) u(t)+C(q) e(t)
as follows.
Factor B as B = B+ B, where B+ is monic and stable.
Set d = deg{A} deg{B+
}.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 12/14
Moving Average Controller
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Design the controller
u(t) = S(q)
R(q)y(t) for A(q) y(t) = B(q) u(t)+C(q) e(t)
as follows.
Factor B as B = B+ B, where B+ is monic and stable.
Set d = deg{A} deg{B+
}. Find Rp and S solving the Diophantine equation
qd1 C = A Rp + B S, d d0
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 12/14
Moving Average Controller
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Design the controller
u(t) = S(q)
R(q)y(t) for A(q) y(t) = B(q) u(t)+C(q) e(t)
as follows.
Factor B as B = B+ B, where B+ is monic and stable.
Set d = deg{A} deg{B+
}. Find Rp and S solving the Diophantine equation
qd1 C = A Rp + B S, d d0
Let R = Rp B+, where deg{Rp} = d 1.
The obtained transfer function for the closed-loop system is
y(t) = q1d Rp(q) e(t) = e(t)+r1 e(t1)+ +rd1 e(td+1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 12/14
Example 4.3
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Consider the plant described by
(q2 + a1 q + a2) y(t) = (b0 q + b1) u(t) + (q2 + c1 q + c2) e(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 13/14
Example 4.3
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Consider the plant described by
(q2 + a1 q + a2) y(t) = (b0 q + b1) u(t) + (q2 + c1 q + c2) e(t)
1. If |b1/b0| < 1 minimum variance (MV) controller can be
designed noticing that
q11 (q2 + c1 q + c2)
q2 + a1 q + a2= 1 +
(c1 a1) q + (c2 a2)
q2 + a1 q + a2
as follows
u(t) = G(q)
B(q) F(q)y(t) =
(c1 a1) q + (c2 a2)
(b0 q + b1) 1y(t)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 13/14
Example 4.3
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Consider the plant described by
(q2 + a1 q + a2) y(t) = (b0 q + b1) u(t) + (q2 + c1 q + c2) e(t)
1. If |b1/b0| < 1 minimum variance (MV) controller can be
designed noticing that
q11 (q2 + c1 q + c2)
q2 + a1 q + a2= 1 +
(c1 a1) q + (c2 a2)
q2 + a1 q + a2
as follows
u(t) = G(q)
B(q) F(q)y(t) =
(c1 a1) q + (c2 a2)
(b0 q + b1) 1y(t)
2. If |b1/b0| > 1 minimum variance (MV) controller cannot bedesigned but we can apply moving average controller (MA) withd = 2 using the solution of the Diophantine equation
q (q2+c1 q+c2) = (q2+a1 q+a2)(q+r1)+(b0 q+b1)(s0 q+s1)
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 13/14
Next Lecture / Assignments:
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Next meeting (May 10, 13:00-15:00, in A208Tekn): Recitations.
Homework problems: Consider the process in Example 4.3
with a1 = 1.5, a2 = 0.7, b0 = 1, c1 = 1, and c2 = 0.2.
Determine the variance of the output in the closed-loop systemas a function of b1 when the Moving Average controller is used.
Compare with the lowest achievable variance.
c Leonid Freidovich. A ril 28, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 9 . 14/14