self-tuning regulators (str)
TRANSCRIPT
Contents
1.1 Indirect Self-Tuning Regulators (STR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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1.1. INDIRECT SELF-TUNING REGULATORS (STR) Adaptive Control
1.1 Indirect Self-Tuning Regulators (STR)
STR is one of the adaptive control methods that automatically finds its parameters in the control law by solving the Diophantine
equation.
Diophantine Equation
G(z−1) =z−dB(z−1)
A(z−1)
Gc(z−1) =
S(z−1)
R(z−1)
Gcl(z−1) =
y
uc=
z−dBS
AR + z−dBS
A(z−1) = 1 + a1z−1 + a2z
−2 + ...+ anaz−na
B(z−1) = b0 + b1z−1 + b2z
−2 + ...+ bnbz−nb
R(z−1) = 1 + r1z−1 + r2z
−2 + ...+ rnrz−nr
S(z−1) = s0 + s1z−1 + s2z
−2 + ...+ snsz−ns
α(z−1) = 1 + α1z−1 + α2z
−2 + ...+ αnαz−nα
nr = nb+ d− 1 , ns = na− 1 , nα = na+ nb+ d− 1 = nr + ns+ 1
C/C Eqn. : AR + z−dBS = AmA0 = αc
1 0 0 0a1 1 0 0a2 a1 1 0a3 a2 a1 1a4 a3 a2 a1
. a4 a3 a2
ana . a4 a3
0 ana . a4
0 0 ana .0 0 0 ana
nα x nr
[0 0 0 00 0 0 0
](d−1) x (ns+1)
b0 0 0 0b1 b0 0 0b2 b1 b0 0. b2 b1 b0bnb . b2 b10 bnb . b20 0 bnb .0 0 0 bnb
nα x nα
r1
r2
.
.
.rnr
s0
s1
s2
.
.
.sns
=
α1
α2
α3
α4
α5
α6
.
.
.αnα
−
a1
a2
a3
.
.
.ana0...0
STR Configurations
Feed forward Controller Feed forward and Feedback Controller
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1.2. EXAMPLES Adaptive Control
R(z) Configurations
B = B−B+
B+: monic polynomial whose zeros are stable and well damped that they can be canceled
B− : monic polynomial whose zeros are unstable poorly damped that they cannot be canceled or a simple constant b0
Indirect STR without zero cancellation
Use Basic Diophantine Equation
Indirect STR with zero cancellation
B+ = B/b0 , B− = b0
C/C Eqn. : AR′ + z−dB−S = αcR = B+R′
T (z) Configurations
1. T (z) = Simple constant, to improve DCgain
2. T (z) = A0
3. Choose T (z) to cancel error dynamics
C/C Eqn. : ML+ z−dBT = AmA0 = αcL(z−1) = 1 + l1z
−1 + l2z−2 + ...+ lnlz
−nl
T (z−1) = t0 + t1z−1 + t2z
−2 + ...+ tntz−nt
Where uc =N(z)M(z)
, nl = nb+ d− 1 , nt = nm− 1 , nα = nm+ nb+ d− 1 = nl + nt+ 1
1 0 0m1 1 0m2 m1 1m3 m2 m1
m4 m3 m2
. m4 m3
mnm . m4
0 mnm .0 0 mnm
nα x nl
[0 0 0 00 0 0 0
](d−1) x (nt+1)
b0 0 0 0b1 b0 0 0b2 b1 b0 0. b2 b1 b0bnb . b2 b10 bnb . b20 0 bnb .0 0 0 bnb
nα x nα
l1l2...lnl
t0t1t2...tnt
=
α1
α2
α3
α4
α5
α6
.
.
.αnα
−
m1
m2
m3
.
.
.mnm
0...0
4. choose T (z) as T (z)B(z) = z−n
1.2 Examples
The first step of STR is to estimate the system parameters by one of the linear estimation methods [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], then
design STR controller, the results of some STR techniques is presented,
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1.2. EXAMPLES Adaptive Control
1. RLS estimation and model following without zero cancellation controller
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1.2. EXAMPLES Adaptive Control
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1.2. EXAMPLES Adaptive Control
2. RLS estimation and model following with zero cancellation controller
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1.2. EXAMPLES Adaptive Control
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1.2. EXAMPLES Adaptive Control
3. RLS estimation and model following without zero cancellation and controller and T = A0
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1.2. EXAMPLES Adaptive Control
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1.2. EXAMPLES Adaptive Control
4. RLS estimation and model following with zero cancellation controller and T = A0
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1.2. EXAMPLES Adaptive Control
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1.2. EXAMPLES Adaptive Control
5. RLS estimation and model following without zero cancellation controller and T is selectedto cancel error dynamics
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1.2. EXAMPLES Adaptive Control
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1.2. EXAMPLES Adaptive Control
6. RLS estimation and model following with zero cancellation controller and T is selected tocancel error dynamics
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1.2. EXAMPLES Adaptive Control
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1.2. EXAMPLES Adaptive Control
7. RLS estimation and model following controller and T (z)B(z) = z−n
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1.2. EXAMPLES Adaptive Control
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1.3. MATLAB CODES Adaptive Control
1.3 MATLAB Codes
1.1 http://goo.gl/9qweI1
1.4 References
1. http://goo.gl/e7J2kq
2. http://goo.gl/3q6Yc6
3. http://goo.gl/SCPvEW
4. http://goo.gl/JnrdNh
5. http://goo.gl/xjpHha
6. http://goo.gl/6wVeuW
7. http://goo.gl/vuKeaL
8. http://goo.gl/mL0RCz
9. http://goo.gl/vzViYE
10. Karl Johan Astrom, Adaptive Control, 2nd Edition.
11. Leonid B. Freidovich, lecture 12.
1.5 Contacts
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