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sn#187702 filedate 1975-11-18 generic text, type T, neo UTF8
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␈↓␈↓↓␈↓α␈↓β␈↓∧␈↓␈↓ ↓H
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␈↓ ∧}PROGRESS REPORT␈↓ ↓H
␈↓ ∧<JULY TO OCTOBER 1975␈↓ ↓H
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␈↓ ¬-M.D. RODERICK␈↓ ↓H
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␈↓↓␈↓ ¬hCONTENTS␈↓ ↓H
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PART I: Classical Analysis of the Stanford Arm Control System␈↓ ↓H␈↓!⊃ ∧␈↓ ↓H␈↓!∩ ∧␈↓ ↓H␈↓!∪ ∧␈↓ ↓H␈↓!∀ ∧␈↓
L␈↓ ↓H
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I. Introduction ␈↓ S2␈↓ ↓H
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II. Present Arm Configuration ␈↓ S3␈↓ ↓H
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III. Experimental Determination of Arm Parameters ␈↓ S9␈↓ ↓H
␈↓ ↓H
␈↓ αHA. Closed Loop Transient Response ␈↓ S9␈↓ ↓H
␈↓ ↓H
␈↓ αHB. Open Loop Transient Response ␈↓ @13␈↓ ↓H
␈↓ ↓H
␈↓ αHC. Closed Loop Stability Analysis ␈↓ @17␈↓ ↓H
␈↓ ↓H
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IV. Recommendations ␈↓ @19␈↓ ↓H
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␈↓ αHA. Sampling Time Independence ␈↓ @19␈↓ ↓H
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␈↓ αHB. Inertia Independence ␈↓ @20␈↓ ↓H
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PART II: The Automatic Tuning of Controllers Using Search Techniques␈↓ ↓H␈↓!⊃
8␈↓ ↓H␈↓!∩
8␈↓ ↓H␈↓!∪
8␈↓ ↓H␈↓!∀
8␈↓ ␈↓ ↓H
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I. Introduction ␈↓ S2␈↓ ↓H
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II. Tuning Package ␈↓ S5␈↓ ↓H
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␈↓ αHA. Search Routines ␈↓ S5␈↓ ↓H
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␈↓ αHB. Stability Criteria ␈↓ @15␈↓ ↓H
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␈↓ αH␈↓ βH1. Stability Ratio ␈↓ @15␈↓ ↓H
␈↓ ↓H
␈↓ αH␈↓ βH2. Maximum Output ␈↓ @17␈↓ ↓H
␈↓ ↓H
␈↓ αH␈↓ βH3. Negative Feedback Gains ␈↓ @17␈↓ ↓H
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␈↓ αH␈↓ βH4. Steady State Error ␈↓ @18␈↓ ↓H
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␈↓ αHC. Error Criteria ␈↓ @18␈↓ ↓H
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␈↓ αHD. Global Search Capability ␈↓ @19␈↓ ↓H
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III. Experimental Results ␈↓ @24␈↓ ↓H
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␈↓ αHA. Description of Experimental System ␈↓ @24␈↓ ↓H
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␈↓ αHB. Integral Squared Error Criterion ␈↓ @27␈↓ ↓H
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␈↓ αHC. Absolute Error Criterion ␈↓ @45␈↓ ↓H
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IV. Theoretical Analysis ␈↓ @52␈↓ ↓H
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V. Conclusions ␈↓ @55␈↓ ↓H
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VI. References ␈↓ @59␈↓ ↓H
␈↓ε ␈↓ ↓H␈↓"
␈↓ ε%page 2␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
I. INTRODUCTION␈↓ ↓H␈↓!⊃↓y␈↓ ↓H␈↓!∩↓y␈↓ ↓H␈↓!∪↓y␈↓ ↓H␈↓!∀↓y␈↓ βA␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α⊃following␈α⊃is␈α⊃a␈α⊃two␈α⊃part␈α⊃report␈α⊂covering␈α⊂progress␈α⊂made␈α⊂at␈α⊂the␈α⊂Artificial␈α⊂Intelligence␈↓ ↓H␈↓"
␈↓ ↓HLaboratory␈α∞from␈α∞July␈α∞through␈α∞October␈α∞of␈α∞1975.␈α∞The␈α∞first␈α∞part␈α∞of␈α∞this␈α
report␈α
covers␈α
work␈↓ ↓H␈↓"
␈↓ ↓Haimed␈α⊂at␈α⊂characterizing␈α⊂the␈α⊂control␈α⊂system␈α⊂of␈α⊂the␈α⊂Stanford␈α⊂arm.␈α∂Previously,␈α∂the␈α∂arm␈α∂has␈↓ ↓H␈↓"
␈↓ ↓Hbeen␈α⊂modeled␈α⊂in␈α⊂the␈α⊂continuous␈α∂Laplace␈α∂transform␈α∂domain.␈α∂Since␈α∂the␈α∂arm␈α∂control␈α∂system␈↓ ↓H␈↓"
␈↓ ↓Hincludes␈α
a␈α
discrete␈α
digital␈α
computer,␈α
however,␈α
Laplace␈α
transforms␈α cannot␈α accurately␈α model␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α⊂system␈α⊂unless␈α⊂the␈α⊂sampling␈α⊂rate␈α⊂is␈α⊂kept␈α⊂considerably␈α⊂higher␈α⊂than␈α⊂the␈α⊂system's␈α⊂time␈↓ ↓H␈↓"
␈↓ ↓Hconstants.␈α
Since␈α
one␈α
of␈α
the␈α
goals␈α
of␈α
my␈α
work␈α
will␈α
be␈α to␈α investigate␈α the␈α feasibility␈α of␈α using␈↓ ↓H␈↓"
␈↓ ↓Hlower␈α⊃sampling␈α⊃rates,␈α⊃an␈α⊃analysis␈α⊃was␈α⊃needed␈α⊂that␈α⊂could␈α⊂accurately␈α⊂represent␈α⊂both␈α⊂the␈↓ ↓H␈↓"
␈↓ ↓Hdiscrete␈α and␈α continuous␈α portions␈α of␈α the␈α control␈α system.␈α In␈α the␈α sections␈α that␈α follow,␈α the␈α arm␈↓ ↓H␈↓"
␈↓ ↓Hhas␈α∞been␈α∞modeled␈α∞in␈α∞the␈α∞Z␈α∞transform␈α∞domain.␈α∞Unlike␈α∞Laplace␈α∞transforms,␈α∞Z␈α
transforms␈α
can␈↓ ↓H␈↓"
␈↓ ↓Hmodel␈α a␈α sampled-data␈α system␈α exactly␈α at␈α each␈α sampling␈α instant.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HModeling␈α∀the␈α∀arm␈α∀in␈α∀the␈α∀Z␈α∪transform␈α∪domain␈α∪is␈α∪the␈α∪first␈α∪step␈α∪in␈α∪a␈α∪program␈α∪aimed␈α∪at␈↓ ↓H␈↓"
␈↓ ↓Hdetermining␈α⊂the␈α⊂effects␈α⊂of␈α⊂sampling␈α⊂time␈α⊂on␈α⊂the␈α∂arm's␈α∂performance.␈α∂A␈α∂state␈α∂space␈α∂state␈↓ ↓H␈↓"
␈↓ ↓Hestimate␈α↔feedback␈α↔(SEF)␈α↔controller␈α↔will␈α↔subsequently␈α↔be␈α⊗designed␈α⊗for␈α⊗the␈α⊗arm.␈α⊗The␈↓ ↓H␈↓"
␈↓ ↓Hresponse␈α
of␈α
the␈α
SEF␈α
controller␈α will␈α then␈α be␈α compared␈α with␈α the␈α existing␈α classical␈α controller␈↓ ↓H␈↓"
␈↓ ↓Hto␈α determine␈α which␈α controller␈α performs␈α best␈α at␈α low␈α sampling␈α rates.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"∧ ␈↓ ε%page 3␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧II. PRESENT ARM CONFIGURATION␈↓ ↓H␈↓!⊃βq␈↓ ↓H␈↓!∩βq␈↓ ↓H␈↓!∪βq␈↓ ↓H␈↓!∀βq␈↓ ¬9␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α∃arm's␈α∃trajectory␈α∃is␈α∃determined␈α∃by␈α∀evaluating␈α∀the␈α∀fifth␈α∀order␈α∀polynomial␈α∀given␈α∀in␈↓ ↓H␈↓"∧␈↓ ↓Hequation␈α (1)␈α below.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧F␈↓ε(k) = a␈↓#qo␈↓# + a␈↓#q1␈↓#[kT] + a␈↓#q2␈↓#[kT]␈↓#∂2␈↓# + a␈↓#q3␈↓#[kT]␈↓#∂3␈↓#␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ + a␈↓#q4␈↓#[kT]␈↓#∂4␈↓# + a␈↓#q5␈↓#[kT]␈↓#∂5␈↓# ␈↓ =(1)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ where ␈↓∧F␈↓ε(k) = predicted joint position ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ T = sampling period␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ a␈↓#qo␈↓# to a␈↓#q5␈↓# = polynomial coefficients␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α∞predicted␈α∞arm␈α∞velocity,␈α∞␈↓∧F␈↓ε'(k),␈α∞is␈α∞determined␈α∞by␈α∞differencing␈α∞the␈α∞current␈α
and␈α
previous␈↓ ↓H␈↓"∧␈↓ ↓Hpositions␈α and␈α then␈α dividing␈α by␈α the␈α sampling␈α period.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧F␈↓ε'(k) = ␈↓∧F␈↓ε(k) - ␈↓∧F␈↓ε(k-1)␈↓ ∧c␈↓!β↓C␈↓ ∧c␈↓!∧↓C␈↓ ∧c␈↓!¬↓C␈↓ ∧c␈↓!ε↓C␈↓ ε& ␈↓ %(2) ␈↓ ↓H␈↓"∧ T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α∪arm's␈α∪predicted␈α∩acceleration,␈α∩␈↓∧F␈↓ε''(k),␈α∩is␈α∩determined␈α∩by␈α∩differencing␈α∩the␈α∩current␈α∩and␈↓ ↓H␈↓"∧␈↓ ↓Hprevious␈α velocities␈α and␈α then␈α dividing␈α by␈α the␈α sampling␈α period.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧F␈↓ε''(k) = ␈↓∧F␈↓ε'(k) - ␈↓∧F␈↓ε'(k-1)␈↓ ∧k␈↓!β↓S␈↓ ∧k␈↓!∧↓S␈↓ ∧k␈↓!¬↓S␈↓ ∧k␈↓!ε↓S␈↓ ε>␈↓ ↓H␈↓"∧ T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ = ␈↓∧F␈↓ε(k) - 2␈↓∧F␈↓ε(k-1) + ␈↓∧F␈↓ε(k-2)␈↓ ∧s␈↓!βα←␈↓ ∧s␈↓!∧α←␈↓ ∧s␈↓!¬α←␈↓ ∧s␈↓!εα←␈↓ πR ␈↓ %(3) ␈↓ ↓H␈↓"∧ ␈↓#qT␈↓#2␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α z␈α transforms␈α of␈α equations␈α (1),␈α (2),␈α and␈α (3)␈α are␈α given␈α in␈α equations␈α (4),␈α (5),␈α and␈α (6).␈↓ ↓H␈↓"∧ ␈↓ ε%page 4␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓#q␈↓∧␈↓#qF␈↓ε␈↓#q(z) =␈↓# a␈↓#qo␈↓#(z)␈↓ ∧≡␈↓!⊃F␈↓ ∧≡␈↓!∩F␈↓ ∧≡␈↓!∪F␈↓ ∧≡␈↓!∀F␈↓ ∧d ␈↓#q+␈↓# a␈↓#q1␈↓#(z)Tz␈↓ ¬∞␈↓!⊃j␈↓ ¬∞␈↓!∩j␈↓ ¬∞␈↓!∪j␈↓ ¬∞␈↓!∀j␈↓ ¬x ␈↓#q+␈↓# a␈↓#q2␈↓#(z)T␈↓#∂2␈↓#z(z+1)␈↓ ε"␈↓!⊃↓G␈↓ ε"␈↓!∩↓G␈↓ ε"␈↓!∪↓G␈↓ ε"␈↓!∀↓G␈↓ πi ␈↓#q+␈↓# a␈↓#q3␈↓#(z)T␈↓#∂3␈↓#z(z␈↓#∂2␈↓# + 4z + 1)␈↓ λ∪␈↓!⊃α>␈↓ λ∪␈↓!∩α>␈↓ λ∪␈↓!∪α>␈↓ λ∪␈↓!∀α>␈↓
Q␈↓ ↓H␈↓"∧ ␈↓#q(z-1)␈↓# ␈↓#q(z-1)␈↓#2 ␈↓#q(z-1)␈↓#3 ␈↓#q(z-1)␈↓#4␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓#q+␈↓# a␈↓#q4␈↓#(z)T␈↓#∂4␈↓#z(z␈↓#∂3␈↓# + 11z␈↓#∂2␈↓# + 11z + 1)␈↓ ∧*␈↓!⊃βC␈↓ ∧*␈↓!∩βC␈↓ ∧*␈↓!∪βC␈↓ ∧*␈↓!∀βC␈↓ πm␈↓ ↓H␈↓"∧ ␈↓#q(z-1)␈↓#5␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓#q+␈↓# a␈↓#q5␈↓#(z)T␈↓#∂5␈↓#z(z␈↓#∂4␈↓# + 26z␈↓#∂3␈↓# + 66z␈↓#∂2␈↓# + 26z + 1)␈↓ ∧6␈↓!⊃∧5␈↓ ∧6␈↓!∩∧5␈↓ ∧6␈↓!∪∧5␈↓ ∧6␈↓!∀∧5␈↓ λk ␈↓ =(4)␈↓ ↓H␈↓"∧ ␈↓#q(z-1)␈↓#6␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧F␈↓ε'(z) = ␈↓∧F␈↓ε(z) - z␈↓#∂-1␈↓#␈↓∧F␈↓ε(z)␈↓ ∧b␈↓!β↓P␈↓ ∧b␈↓!∧↓P␈↓ ∧b␈↓!¬↓P␈↓ ∧b␈↓!ε↓P␈↓ ε2␈↓ ↓H␈↓"∧ T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ = ␈↓∧F␈↓ε(z)[z-1]␈↓ ∧g␈↓!β↓¬␈↓ ∧g␈↓!∧↓¬␈↓ ∧g␈↓!¬↓¬␈↓ ∧g␈↓!ε↓¬␈↓ ¬l ␈↓ =(5)␈↓ ↓H␈↓"∧ Tz␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧F␈↓ε''(z) = ␈↓∧F␈↓ε'(z) - z␈↓#∂-1␈↓#␈↓∧F␈↓ε'(z)␈↓ ∧j␈↓!β↓`␈↓ ∧j␈↓!∧↓`␈↓ ∧j␈↓!¬↓`␈↓ ∧j␈↓!ε↓`␈↓ εJ␈↓ ↓H␈↓"∧ T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ = ␈↓∧F␈↓ε(z)[z-1]␈↓#∂2␈↓#␈↓ ∧g␈↓!β↓_␈↓ ∧g␈↓!∧↓_␈↓ ∧g␈↓!¬↓_␈↓ ∧g␈↓!ε↓_␈↓ ¬␈ ␈↓ =(6)␈↓ ↓H␈↓"∧ ␈↓#qT␈↓#2␈↓#qz␈↓#2␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α
actual␈α
joint␈α
position,␈α
␈↓∧Q␈↓ε(k),␈α
is␈α
monitored␈α
with␈α
potentiometers.␈α
The␈α
actual␈α
joint␈α velocity,␈↓ ↓H␈↓"∧␈↓ ↓H␈↓∧Q␈↓ε'(k),␈α_is␈α_either␈α_measured␈α_with␈α↔a␈α↔tachometer␈α↔(joints␈α↔1,␈α↔2,␈α↔and␈α↔3)␈α↔or␈α↔calculated␈α↔by␈↓ ↓H␈↓"∧␈↓ ↓Hdifferencing␈α∪the␈α∪current␈α∪and␈α∪previous␈α∪arm␈α∪positions␈α∩and␈α∩then␈α∩dividing␈α∩by␈α∩the␈α∩sampling␈↓ ↓H␈↓"∧␈↓ ↓Hperiod.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧Q␈↓ε'(k) = ␈↓∧Q␈↓ε(k) - ␈↓∧Q␈↓ε(k-1)␈↓ ∧b␈↓!β↓A␈↓ ∧b␈↓!∧↓A␈↓ ∧b␈↓!¬↓A␈↓ ∧b␈↓!ε↓A␈↓ ε# ␈↓ =(7)␈↓ ↓H␈↓"∧ T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α z␈α transform␈α of␈α equation␈α (7)␈α is␈α given␈α by␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧Q␈↓ε'(z) = ␈↓∧Q␈↓ε(z)[z-1]␈↓ ∧a␈↓!β↓∧␈↓ ∧a␈↓!∧↓∧␈↓ ∧a␈↓!¬↓∧␈↓ ∧a␈↓!ε↓∧␈↓ ¬e ␈↓ =(8)␈↓ ↓H␈↓"∧ T␈↓ ↓H␈↓"∧ ␈↓ ε%page 5␈↓ ↓H␈↓"∧ ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α motor␈α torque␈α difference␈α equation␈α is␈α given␈α by␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧T␈↓ε(k) = J(k)␈↓∧F␈↓ε''(k) + G(k) + F - k␈↓#qc␈↓#k␈↓#qe␈↓#[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)]␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ - k␈↓#qc␈↓#k␈↓#qv␈↓#J(k)[␈↓∧Q␈↓ε'(k) - ␈↓∧F␈↓ε'(k)] - k␈↓#qc␈↓#k␈↓#qi␈↓#/J(k) ␈↓∧S␈↓ε[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] ␈↓ =(9)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ where ␈↓∧T␈↓ε(k) = motor torque (oz-in)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧F␈↓ε(k) = predicted joint position (degrees)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧Q␈↓ε(k) = actual joint position (degrees)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ J(k) = joint inertia (oz-in-sec squared)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ G(k) = joint gravity loading (oz-in)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ F = joint friction with same sign as velocity (oz-in)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ k␈↓#qe␈↓# = proportional feedback constant ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ k␈↓#qv␈↓# = derivative feedback constant ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ k␈↓#qi␈↓# = integral feedback constant␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ k␈↓#qc␈↓# = constant to convert degrees to radians = .01745␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α gravity␈α loading,␈α G(k),␈α is␈α calculated␈α using␈α a␈α first␈α order␈α polynomial.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ G(k) = g␈↓#qo␈↓# + ␈↓∧d␈↓εg kT/S ␈↓ *(10)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ where g␈↓#qo␈↓# = initial gravity loading␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧d␈↓εg = change in gravity loading␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ T = sampling period␈↓ ↓H␈↓"∧ ␈↓ ε%page 6␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ kT = elapsed time in segment␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ S = total time required to pass through segment␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ kT/S = fraction of time through segment (0→1.0)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α inertia,␈α J(k),␈α is␈α also␈α calculated␈α using␈α a␈α first␈α order␈α polynomial.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ J(k) = j␈↓#qo␈↓# + ␈↓∧d␈↓εj kT/S ␈↓ *(11)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ where j␈↓#qo␈↓# = initial inertia␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧d␈↓εj = change in inertia ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ kT/S = fraction of time through segment (0→1.0)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HCombining␈α⊃equations␈α⊃(2),(3),(7),(9),(10),␈α⊂and␈α⊂(11)␈α⊂gives␈α⊂the␈α⊂complete␈α⊂difference␈α⊂equation␈↓ ↓H␈↓"∧␈↓ ↓Hfor␈α the␈α motor␈α torque,␈α ␈↓∧T␈↓ε(k).␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓∧T␈↓ε(k) = (j␈↓#qo␈↓# + ␈↓∧d␈↓εj kT/S)[␈↓∧F␈↓ε(k) - 2␈↓∧F␈↓ε(k-1) + ␈↓∧F␈↓ε(k-2)]/T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ + g␈↓#qo␈↓# + ␈↓∧d␈↓εg kT/S + F␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ - k␈↓#qe␈↓#[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)]␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ - k␈↓#qv␈↓#(j␈↓#qo␈↓# + ␈↓∧d␈↓εj kT/S)[␈↓∧Q␈↓ε(k) - ␈↓∧Q␈↓ε(k-1) - ␈↓∧F␈↓ε(k) + ␈↓∧F␈↓ε(k-1)]/T␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ - k␈↓#qi␈↓# ␈↓∧S␈↓ε[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)]/(j␈↓#qo␈↓# + ␈↓∧d␈↓εj kT /S) ␈↓ *(12)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HA␈α block␈α diagram␈α of␈α the␈α system␈α described␈α in␈α equation␈α (12)␈α is␈α given␈α in␈α Figure␈α 1.␈↓ ↓H␈↓"∧ ␈↓ ε%page 7␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓π␈↓ ↓H
⊂αααα⊃␈↓ ↓H
⊂αααααααααα→~ FF ~ααααααααααααααααααααααα⊃␈↓ ↓H
~ %αααα$ ~␈↓ ↓H
~ ↓␈↓ ↓H
R(z) ~ ⊂ααα⊃ ⊂αααα⊃ ⊂αααα⊃ ⊂ααα⊃ ⊂ααααα⊃ ⊂αααα⊃ ␈↓∧Q␈↓π(z)␈↓ ↓H
ααααα∀αα→~ + ~ααααα→~ G␈↓#oc␈↓# ~ααααα→~ k␈↓#oc␈↓# ~ααααα→~ + ~ααααα→~ D/A ~ααααα→~ G␈↓#op␈↓# ~αααααπαααα→␈↓ ↓H
%ααα$ %αααα$ %αααα$ %ααα$ %ααααα$ %αααα$ ~␈↓ ↓H
↑ ↑ ~␈↓ ↓H
~ ~ ~␈↓ ↓H
~ ~ ~ ␈↓ ↓H
~ F + G(z) ~␈↓ ↓H
~ ~␈↓ ↓H
~ ⊂αααααα⊃ ⊂ααααα⊃ ~␈↓ ↓H
%αααααααααααααααααα~ 1/k␈↓#oc␈↓# ~←αααααααα~ A/D ~←αααααααααααααααααααααααα$␈↓ ↓H
%αααααα$ %ααααα$␈↓ ↓H
␈↓ ↓H␈↓"
␈↓ε Figure 1 - Present Arm Control System Configuration␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where G␈↓#qc␈↓# = transfer function of controller (feedback terms only)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qp␈↓# = transfer function of motor and arm␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
FF = feedforward compensation for inertia = J(k)␈↓∧F␈↓ε''␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
D/A = digital to analog converter␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
A/D = analog to digital converter␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧F␈↓ε(z) = system input (predicted arm trajectory in degrees)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧Q␈↓ε(z) = system output (actual arm position in radians)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α transfer␈α function␈α of␈α the␈α controller␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ε%page 8␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qc␈↓# = k␈↓#qe␈↓# + k␈↓#qv␈↓#J(k)(z-1)␈↓ ∧S␈↓!⊃↓!␈↓ ∧S␈↓!∩↓!␈↓ ∧S␈↓!∪↓!␈↓ ∧S␈↓!∀↓!␈↓ ¬t + k␈↓#qi␈↓#z ␈↓ ε≡␈↓!⊃↓π␈↓ ε≡␈↓!∩↓π␈↓ ε≡␈↓!∪↓π␈↓ ε≡␈↓!∀↓π␈↓ π% . ␈↓ *(13)␈↓ ↓H␈↓"
Tz J(k)(z-1)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α transfer␈α function␈α of␈α the␈α motor␈α and␈α arm␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qp␈↓# = (z-1)␈↓ ∧
␈↓!⊃H␈↓ ∧
␈↓!∩H␈↓ ∧
␈↓!∪H␈↓ ∧
␈↓!∀H␈↓ ∧R ␈↓∧Z␈↓ε(G␈↓#qp␈↓#(s)␈↓ ∧⎇␈↓!⊃L␈↓ ∧⎇␈↓!∩L␈↓ ∧⎇␈↓!∪L␈↓ ∧⎇␈↓!∀L␈↓ ¬I) = (z-1)␈↓ ¬␈␈↓!⊃H␈↓ ¬␈␈↓!∩H␈↓ ¬␈␈↓!∪H␈↓ ¬␈␈↓!∀H␈↓ εG ␈↓∧Z␈↓ε( 1 ␈↓ εr␈↓!⊃↓S␈↓ εr␈↓!∩↓S␈↓ εr␈↓!∪↓S␈↓ εr␈↓!∀↓S␈↓ λE)␈↓ ↓H␈↓"
z s z s(J(k)s␈↓#∂2␈↓# + Bs)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
= (TB + J(k)(␈↓∧b␈↓ε-1))z - TB␈↓∧b␈↓ε - J(k)(␈↓∧b␈↓ε-1)␈↓ ∧∨␈↓!⊃∧β␈↓ ∧∨␈↓!∩∧β␈↓ ∧∨␈↓!∪∧β␈↓ ∧∨␈↓!∀∧β␈↓ λ" . ␈↓ *(14)␈↓ ↓H␈↓"
B␈↓#∂2␈↓#(z-1)(z-␈↓∧b␈↓ε)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HFinally,␈α
the␈α
open␈α
loop␈α
transfer␈α function␈α of␈α the␈α combined␈α controller,␈α motor,␈α and␈α arm␈α is␈α given␈↓ ↓H␈↓"
␈↓ ↓Hby␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qtot␈↓#(z) = K N/D ␈↓ *(15)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where K = transfer function gain = (TB + J(k)(␈↓∧b␈↓ε-1))(k␈↓#qe␈↓#TJ(k) + k␈↓#qv␈↓#J(k)␈↓#∂2␈↓# + k␈↓#qi␈↓#T␈↓ ε¬␈↓!⊃∧K␈↓ ε¬␈↓!∩∧K␈↓ ε¬␈↓!∪∧K␈↓ ε¬␈↓!∀∧K␈↓
P)␈↓ ↓H␈↓"
TJ(k)B␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
N = transfer function numerator ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓#q=␈↓# ␈↓#q[␈↓#z - (TB␈↓∧b␈↓ε + J(k)(␈↓∧b␈↓ε-1))␈↓ βj␈↓!⊃↓{␈↓ βj␈↓!∩↓{␈↓ βj␈↓!∪↓{␈↓ βj␈↓!∀↓{␈↓ ¬e␈↓#q]␈↓#␈↓#q[␈↓#z␈↓#∂2␈↓# + (-k␈↓#qe␈↓#TJ(k) - 2k␈↓#qv␈↓#J(k)␈↓#∂2␈↓#)z + k␈↓#qv␈↓#J(k)␈↓#∂2␈↓#␈↓ εI␈↓!⊃β↑␈↓ εI␈↓!∩β↑␈↓ εI␈↓!∪β↑␈↓ εI␈↓!∀β↑␈↓
'␈↓#q]␈↓#␈↓ ↓H␈↓"
TB + J(k)(␈↓∧b␈↓ε-1) k␈↓#qe␈↓#TJ(k) + k␈↓#qv␈↓#J(k)␈↓#∂2␈↓# + k␈↓#qi␈↓#T␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
D = transfer function denominator = z(z-␈↓∧b␈↓ε)(z-1)␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ε%page 9␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
III. EXPERIMENTAL DETERMINATION OF ARM PARAMETERS␈↓ ↓H␈↓!⊃εT␈↓ ↓H␈↓!∩εT␈↓ ↓H␈↓!∪εT␈↓ ↓H␈↓!∀εT␈↓ λ≤ ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α⊃the␈α⊃experiments␈α⊃that␈α⊃follow,␈α⊃the␈α⊃arm␈α⊃was␈α⊃excited␈α⊃by␈α⊃step␈α⊂inputs␈α⊂to␈α⊂determine␈α⊂each␈↓ ↓H␈↓"
␈↓ ↓Hjoint's␈α↔inertia␈α↔and␈α⊗damping.␈α⊗For␈α⊗simplicity,␈α⊗motions␈α⊗were␈α⊗selected␈α⊗which␈α⊗produced␈α⊗a␈↓ ↓H␈↓"
␈↓ ↓Hconstant␈α∞inertia␈α∞throughout␈α∞the␈α∞motion.␈α∞Thus,␈α∞in␈α∞the␈α∞equations␈α∞given␈α∞below,␈α
J(k)␈α
has␈α
been␈↓ ↓H␈↓"
␈↓ ↓Hreplaced␈α by␈α the␈α constant␈α J.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
A. CLOSED LOOP TRANSIENT RESPONSE␈↓ α∧␈↓!⊃∧J␈↓ α∧␈↓!∩∧J␈↓ α∧␈↓!∪∧J␈↓ α∧␈↓!∀∧J␈↓ εN␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
In this test, all of the terms in the output torque equation were deleted␈↓ ↓H␈↓"
except friction, F, and the proportional feedback constant, k␈↓#qe␈↓#,␈↓ ↓H␈↓"
which was set to 2000.0. A block diagram of the resulting system is given ␈↓ ↓H␈↓"
in figure 2 below. ␈↓ ↓H␈↓"
␈↓π␈↓ ↓H
R(s) ⊂ααα⊃ ⊂αααα⊃ ⊂αααα⊃ ⊂ααα⊃ ⊂ααααα⊃ ⊂αααααααααα⊃ ␈↓∧Q␈↓π(s)␈↓ ↓H
αααα→~ + ~ααα→~ k␈↓#oe␈↓# ~ααα→~ k␈↓#oc␈↓# ~ααα→~ + ~ααα→~ D/A ~ααα→~ 1 ␈↓ λH␈↓!⊃↓␈↓ λH␈↓!∩↓␈↓ λH␈↓!∪↓␈↓ λH␈↓!∀↓␈↓ H ~ααααααπαα→␈↓ ↓H
%ααα$ %αααα$ %αααα$ %ααα$ %ααααα$ ~ Js␈↓#⊃2␈↓# + Bs ~ ~␈↓ ↓H
~ ↑ %αααααααααα$ ~␈↓ ↓H
~ ~ ~␈↓ ↓H
~ F ~␈↓ ↓H
~ ~␈↓ ↓H
~ ⊂αααααα⊃ ⊂ααααα⊃ ~␈↓ ↓H
%ααααααααααααααααααα~ 1/k␈↓#oc␈↓# ~←αααααα~ A/D ~←ααααααααααααααααααααααα$␈↓ ↓H
%αααααα$ %ααααα$␈↓ ↓H
␈↓ε␈↓ ↓H␈↓"
Figure 2 - Block Diagram of a Single Joint with Proportional Feedback only␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α transfer␈α function␈α for␈α the␈α system␈α shown␈α in␈α Figure␈α 2␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓#qH(s) =␈↓# k␈↓#qc␈↓#k␈↓#qe␈↓# ␈↓ ∧≠␈↓!⊃↓E␈↓ ∧≠␈↓!∩↓E␈↓ ∧≠␈↓!∪↓E␈↓ ∧≠␈↓!∀↓E␈↓ ¬` . ␈↓ *(16)␈↓ ↓H␈↓"
Js␈↓#∂2␈↓# + Bs + k␈↓#qe␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ε≤page 10␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HFor␈α a␈α step␈α input␈α of␈α magnitude␈α 1.0,␈α the␈α response␈α of␈α the␈α arm␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓#q␈↓∧␈↓#qQ␈↓ε␈↓#q(s) =␈↓# k␈↓#qc␈↓#k␈↓#qe␈↓# ␈↓ ∧)␈↓!⊃↓i␈↓ ∧)␈↓!∩↓i␈↓ ∧)␈↓!∪↓i␈↓ ∧)␈↓!∀↓i␈↓ ε∩ ␈↓ ↓H␈↓"
s(Js␈↓#∂2␈↓# + Bs + k␈↓#qe␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓#q=␈↓# k␈↓#qc␈↓#k␈↓#qe␈↓#/J ␈↓ ∧+␈↓!⊃α:␈↓ ∧+␈↓!∩α:␈↓ ∧+␈↓!∪α:␈↓ ∧+␈↓!∀α:␈↓ εe ␈↓ ↓H␈↓"
s(s␈↓#∂2␈↓# + B/J s + k␈↓#qe␈↓#/J)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓#q=␈↓# k␈↓#qc␈↓#␈↓∧w␈↓ε␈↓#qn␈↓#␈↓#∂2␈↓# ␈↓ ∧+␈↓!⊃α#␈↓ ∧+␈↓!∩α#␈↓ ∧+␈↓!∪α#␈↓ ∧+␈↓!∀α#␈↓ εN ␈↓ *(17)␈↓ ↓H␈↓"
s(s␈↓#∂2␈↓# + 2␈↓∧z␈↓ε␈↓∧w␈↓ε␈↓#qn␈↓# + ␈↓∧w␈↓ε␈↓#qn␈↓#␈↓#∂2␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where k␈↓#qe␈↓#/J = ␈↓∧w␈↓ε␈↓#qn␈↓#␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
B/J = 2␈↓∧z␈↓ε␈↓∧w␈↓ε␈↓#qn␈↓#␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧z␈↓ε = damping ratio␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧w␈↓ε␈↓#qn␈↓# = undamped natural frequency (rad/sec) .␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α the␈α time␈α domain,␈α the␈α step␈α response␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧Q␈↓ε(t) = k␈↓#qc␈↓#[1 - e␈↓#∂-␈↓∧␈↓#∂z␈↓ε␈↓#∂␈↓∧␈↓#∂w␈↓ε␈↓#∂␈↓#n␈↓#∂t␈↓#(cos ␈↓∧w␈↓ε␈↓#qd␈↓#t + ␈↓∧z␈↓ε ␈↓ π∧␈↓!⊃↓∨␈↓ π∧␈↓!∩↓∨␈↓ π∧␈↓!∪↓∨␈↓ π∧␈↓!∀↓∨␈↓ λ# sin ␈↓∧w␈↓ε␈↓#qd␈↓#t)] . ␈↓ *(18)␈↓ ↓H␈↓"
(1 - ␈↓∧z␈↓ε␈↓#∂2␈↓#)␈↓#∂1/2␈↓#␈↓ ↓H␈↓"
␈↓ ε≤page 11␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α∂typical␈α∂step␈α∂response␈α∞is␈α∞shown␈α∞in␈α∞figure␈α∞3.␈α∞The␈α∞peak␈α∞time␈α∞t␈↓#qp␈↓#,␈α∞at␈α∞which␈α∞the␈α∞arm␈α∞begins␈↓ ↓H␈↓"
␈↓ ↓Hto␈α⊂reverse␈α⊂its␈α⊂motion␈α⊂can␈α⊂be␈α⊂obtained␈α⊂by␈α⊂differentiating␈α⊂␈↓∧Q␈↓ε(t)␈α⊂and␈α⊂setting␈α⊂the␈α∂derivative␈↓ ↓H␈↓"
␈↓ ↓Hequal␈α to␈α zero.␈α The␈α peak␈α time␈α obtained␈α from␈α equation␈α (18)␈α is␈α given␈α in␈α equation␈α (19).␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
t␈↓#qp␈↓# = π/␈↓∧w␈↓ε␈↓#qd␈↓# ␈↓ *(19)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α amount␈α of␈α overshoot,␈α M␈↓#qp␈↓#,␈α can␈α be␈α obtained␈α by␈α evaluating␈α ␈↓∧Q␈↓ε(t)␈α at␈α t␈↓#qp␈↓#.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
M␈↓#qp␈↓# = exp[-␈↓∧p␈↓ε␈↓∧z␈↓ε/(1-␈↓∧z␈↓ε␈↓#∂2␈↓#)␈↓#∂1/2␈↓#] ␈↓ *(20)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α⊃our␈α⊃case,␈α⊃t␈↓#qp␈↓#␈α⊃and␈α⊃M␈↓#qp␈↓#␈α⊃can␈α⊃be␈α⊃measured␈α⊃experimentally␈α⊃from␈α⊃the␈α⊃arm's␈α⊃step␈α⊃response␈↓ ↓H␈↓"
␈↓ ↓Hdisplayed␈α∩in␈α∩figure␈α∩2.␈α∩When␈α∩these␈α∩parameters␈α⊃are␈α⊃known,␈α⊃the␈α⊃arm's␈α⊃damping␈α⊃ratio␈α⊃and␈↓ ↓H␈↓"
␈↓ ↓Hundamped␈α∂natural␈α∂frequency␈α∂and␈α∂finally␈α∂its␈α∂inertia␈α∂and␈α∂damping␈α∞can␈α∞be␈α∞obtained␈α∞from␈α∞the␈↓ ↓H␈↓"
␈↓ ↓Hfollowing␈α equations.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓#q␈↓∧␈↓#qz␈↓ε␈↓#q = [␈↓# (ln(1/M␈↓#qp␈↓#)␈↓#∂2␈↓# ␈↓ ∧:␈↓!⊃↓z␈↓ ∧:␈↓!∩↓z␈↓ ∧:␈↓!∪↓z␈↓ ∧:␈↓!∀↓z␈↓ ε4 ␈↓#q]␈↓#1/2 ␈↓ *(21)␈↓ ↓H␈↓"
␈↓∧p␈↓ε␈↓#∂2␈↓# + (ln(1/M␈↓#qp␈↓#)␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧w␈↓ε␈↓#qn␈↓# = ␈↓∧p␈↓ε ␈↓ ∧8␈↓!⊃↓_␈↓ ∧8␈↓!∩↓_␈↓ ∧8␈↓!∪↓_␈↓ ∧8␈↓!∀↓_␈↓ ¬P ␈↓ *(22)␈↓ ↓H␈↓"
t␈↓#qp␈↓#(1-␈↓∧z␈↓ε␈↓#∂2␈↓#)␈↓#∂1/2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
J = k␈↓#qe␈↓#/␈↓∧w␈↓ε␈↓#qn␈↓#␈↓#∂2␈↓# ␈↓ *(23)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
B = 2␈↓∧z␈↓ε␈↓∧w␈↓ε␈↓#qn␈↓#J ␈↓ *(24)␈↓ ↓H␈↓"
␈↓ ε≤page 12␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ∧∂Figure 3 Closed Loop Step Response of Arm␈↓ ↓H␈↓"
␈↓ ∧jWith Proportional Feedback Only␈↓ ↓H␈↓"
␈↓ ε≤page 13␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HFrom␈α⊂figure␈α⊂3,␈α∂t␈↓#qp␈↓#␈α∂and␈α∂M␈↓#qp␈↓#␈α∂were␈α∂determined␈α∂to␈α∂be␈α∂1.045␈α∂sec␈α∂and␈α∂1.077␈α∂respectively␈α∂for␈↓ ↓H␈↓"
␈↓ ↓Hjoint␈α⊃1␈α⊃with␈α⊃the␈α⊃arm␈α⊃starting␈α⊃in␈α⊃park␈α⊂position␈α⊂and␈α⊂moving␈α⊂+30␈α⊂degrees.␈α⊂These␈α⊂numbers␈↓ ↓H␈↓"
␈↓ ↓Hyield␈α the␈α following␈α results␈α for␈α joint␈α 1␈α (of␈α the␈α yellow␈α arm):␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧z␈↓ε = .0235; ␈↓∧w␈↓ε␈↓#qn␈↓# = 3.007 rad/sec; J = 221.169 oz-in-sec squared; B = 31.268 oz-in-sec .␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
B. OPEN LOOP TRANSIENT RESPONSE␈↓ α∧␈↓!⊃∧&␈↓ α∧␈↓!∩∧&␈↓ α∧␈↓!∪∧&␈↓ α∧␈↓!∀∧&␈↓ ε*␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α general␈α equation␈α for␈α the␈α torque␈α developed␈α by␈α a␈α motor␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧T␈↓ε(t) = J␈↓∧Q␈↓ε''(t) + J'␈↓∧Q␈↓ε'(t) + B␈↓∧Q␈↓ε'(t) + F ␈↓ *(25)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HSince␈α∩motions␈α∩have␈α∩been␈α∩selected␈α∩which␈α∩keep␈α∩J␈α∩constant,␈α∩J'␈α∩=␈α∩0.␈α∩At␈α∩low␈α⊃speeds,␈α⊃the␈↓ ↓H␈↓"
␈↓ ↓Hdamping␈α
term,␈α
B␈↓∧Q␈↓ε'(t),␈α
can␈α
be␈α
ignored.␈α
Taking␈α the␈α Laplace␈α transform␈α of␈α the␈α remaining␈α terms,␈↓ ↓H␈↓"
␈↓ ↓Happlying␈α⊂a␈α⊂step␈α⊂input␈α⊂in␈α⊂torque,␈α⊂and␈α⊂assuming␈α⊂that␈α⊂the␈α⊂friction␈α⊂is␈α⊂a␈α∂step␈α∂starting␈α∂at␈α∂t=0␈↓ ↓H␈↓"
␈↓ ↓Hgives␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧T␈↓ε = Js␈↓#∂3␈↓#␈↓∧Q␈↓ε(s) + F .␈↓ *(26)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where ␈↓∧T␈↓ε = magnitude of the step input in torque␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HEquation␈α (26)␈α can␈α be␈α solved␈α for␈α J␈α giving␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
J = ␈↓∧T␈↓ε - F␈↓ βt␈↓!⊃T␈↓ βt␈↓!∩T␈↓ βt␈↓!∪T␈↓ βt␈↓!∀T␈↓ ∧H = [␈↓∧T␈↓ε - F]t␈↓#∂2␈↓#␈↓ ¬ ␈↓!⊃↓ ␈↓ ¬ ␈↓!∩↓ ␈↓ ¬ ␈↓!∪↓ ␈↓ ¬ ␈↓!∀↓ ␈↓ ε⊗ .␈↓ *(27)␈↓ ↓H␈↓"
s␈↓#∂3␈↓#␈↓∧Q␈↓ε(s) 2␈↓∧Q␈↓ε(t)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ε≤page 14␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α
open␈α
loop␈α
step␈α
response␈α
obtained␈α
by␈α
setting␈α
␈↓∧T␈↓ε(t)␈α
=␈α
1604␈α
oz-in␈α
and␈α F␈α =␈α 497␈α oz-in␈α is␈↓ ↓H␈↓"
␈↓ ↓Hdisplayed␈α∞in␈α∞Figure␈α∞4.␈α∞At␈α∞␈↓∧Q␈↓ε(t)␈α∞=␈α∞20␈↓#∂o␈↓#,␈α∞J␈α∞=␈α∞213␈α∞oz-in␈α∞-sec␈↓#∂2␈↓#.␈α∞To␈α
check␈α
the␈α
accuracy␈α
of␈α
this␈↓ ↓H␈↓"
␈↓ ↓Hmethod,␈α∞the␈α∞inertia␈α∞was␈α∞recomputed␈α∞with␈α∞the␈α∞wrist␈α∞extended␈α∞to␈α∞a␈α∞horizontal␈α∞position␈α∞and␈↓ ↓H␈↓"
␈↓ ↓Hfound␈α∞to␈α∞be␈α∞215␈α∞oz-in-sec␈↓#∂2␈↓#.␈α∞A␈α∞clamp␈α∞weighing␈α∞31␈α∞oz␈α∞was␈α∞then␈α∞mounted␈α∞in␈α∞the␈α∞hand␈α∞at␈α
a␈↓ ↓H␈↓"
␈↓ ↓Hdistance␈α of␈α 23␈α inches␈α from␈α joint␈α 1.␈α The␈α inertia␈α added␈α by␈α the␈α clamp␈α was␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
J␈↓#qa␈↓# = (31 oz-in)(23 in)␈↓#∂2␈↓# ␈↓ ∧∧␈↓!⊃αK␈↓ ∧∧␈↓!∩αK␈↓ ∧∧␈↓!∪αK␈↓ ∧∧␈↓!∀αK␈↓ εO = 42.4 oz-in-sec␈↓#∂2␈↓# .␈↓ ↓H␈↓"
(32.2 ft/sec␈↓#∂2␈↓#)(12 in/ft)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α⊂arm␈α⊂was␈α⊂then␈α⊂given␈α⊂a␈α⊂second␈α⊂step␈α∂input␈α∂identical␈α∂to␈α∂the␈α∂first,␈α∂except␈α∂this␈α∂time␈α∂the␈↓ ↓H␈↓"
␈↓ ↓Harm␈α⊂was␈α⊂holding␈α⊂the␈α⊂clamp.␈α∂The␈α∂response␈α∂obtained␈α∂from␈α∂the␈α∂second␈α∂step␈α∂is␈α∂displayed␈α∂in␈↓ ↓H␈↓"
␈↓ ↓HFigure␈α⊃5.␈α⊃At␈α⊃␈↓∧Q␈↓ε(t)␈α⊃=␈α⊃20␈↓#∂o␈↓#,␈α⊃the␈α⊃inertia␈α⊃of␈α⊃the␈α⊃arm␈α⊃was␈α⊃258␈α⊃oz-in-sec␈↓#∂2␈↓#␈α⊃or␈α⊃43␈α⊂oz-in-sec␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓Hgreater␈α than␈α it␈α was␈α without␈α the␈α clamp.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α
more␈α
exact␈α
analysis␈α
can␈α
be␈α
obtained␈α
by␈α
solving␈α equation␈α (26)␈α for␈α ␈↓∧Q␈↓ε(s)␈α with␈α the␈α damping␈↓ ↓H␈↓"
␈↓ ↓Hterm␈α included.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧Q␈↓ε(s) = ␈↓∧T␈↓ε - F ␈↓ ∧≥␈↓!⊃↓≤␈↓ ∧≥␈↓!∩↓≤␈↓ ∧≥␈↓!∪↓≤␈↓ ∧≥␈↓!∀↓≤␈↓ ¬9 ␈↓ *(28)␈↓ ↓H␈↓"
s(Js␈↓#∂2␈↓# + Bs)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
In the time domain,␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧Q␈↓ε(t) = (␈↓∧T␈↓ε - F␈↓ ∧%␈↓!⊃T␈↓ ∧%␈↓!∩T␈↓ ∧%␈↓!∪T␈↓ ∧%␈↓!∀T␈↓ ∧y)[t - J/B (1 - exp(-Bt/J)] . ␈↓ *(29)␈↓ ↓H␈↓"
B␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HUsing␈α∞the␈α∞inertia␈α∞of␈α∞213␈α
oz-in-sec␈↓#∂2␈↓#␈α
calculated␈α
above␈α
and␈α
the␈α
damping␈α
of␈α
31.27␈α
oz-in-sec␈↓ ↓H␈↓"
␈↓ ↓Hdetermined␈α
in␈α
the␈α
closed␈α
loop␈α
transient␈α
response␈α
test␈α
on␈α
page␈α
8,␈α
the␈α predicted␈α ␈↓∧Q␈↓ε(t)␈α were␈↓ ↓H␈↓"
␈↓ ↓Hcalculated␈α and␈α compared␈α with␈α the␈α actual␈α ␈↓∧Q␈↓ε(t).␈α The␈α results␈α are␈α displayed␈α in␈α Table␈α 1.␈↓ ↓H␈↓"
␈↓ ε≤page 15␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ∧≠Figure 4 Open Loop Step Response of Arm␈↓ ↓H␈↓"
␈↓ ε≤page 16␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ αoFigure 5 Open Loop Step Response of Arm With Weight in Fingers␈↓ ↓H␈↓"
␈↓ ε≤page 17␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
TABLE 1 - COMPARISON OF PREDICTED AND ACTUAL POSITIONS␈↓ α≤␈↓!⊃π*␈↓ α≤␈↓!∩π*␈↓ α≤␈↓!∪π*␈↓ α≤␈↓!∀π*␈↓ F␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧Q␈↓ε(t)␈↓#qpred.␈↓#␈↓ αp␈↓!⊃↓␈↓ αp␈↓!∩↓␈↓ αp␈↓!∪↓␈↓ αp␈↓!∀↓␈↓ βp ␈↓∧Q␈↓ε(t)␈↓#qact.␈↓#␈↓ ∧h␈↓!⊃i␈↓ ∧h␈↓!∩i␈↓ ∧h␈↓!∪i␈↓ ∧h␈↓!∀i␈↓ ¬Q % ERROR␈↓ εI␈↓!⊃⎇␈↓ εI␈↓!∩⎇␈↓ εI␈↓!∪⎇␈↓ εI␈↓!∀⎇␈↓ πF␈↓ ↓H␈↓"
5.147␈↓#∂o␈↓# 5␈↓#∂o␈↓# 2.94%␈↓ ↓H␈↓"
10.073␈↓#∂o␈↓# 10␈↓#∂o␈↓# 0.73%␈↓ ↓H␈↓"
19.674␈↓#∂o␈↓# 20␈↓#∂o␈↓# 1.63%␈↓ ↓H␈↓"
29.296␈↓#∂o␈↓# 30␈↓#∂o␈↓# 2.34%␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
C. CLOSED LOOP STABILITY ANALYSIS␈↓ α∧␈↓!⊃∧0␈↓ α∧␈↓!∩∧0␈↓ α∧␈↓!∪∧0␈↓ α∧␈↓!∀∧0␈↓ ε4␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HAnother␈α
method␈α
for␈α
verifying␈α
the␈α
accuracy␈α
of␈α
the␈α
model␈α
developed␈α
thus␈α far␈α for␈α the␈α arm␈α is␈↓ ↓H␈↓"
␈↓ ↓Hto␈α∩increase␈α∩the␈α∩feedback␈α∩gains␈α⊃until␈α⊃the␈α⊃arm␈α⊃becomes␈α⊃unstable.␈α⊃The␈α⊃gain␈α⊃values␈α⊃which␈↓ ↓H␈↓"
␈↓ ↓Hcause␈α∞borderline␈α∞stability␈α∞can␈α∞then␈α∞be␈α∞compared␈α∞with␈α∞the␈α∞predicted␈α∞values␈α
obtained␈α
from␈↓ ↓H␈↓"
␈↓ ↓Ha␈α∞root␈α∞locus␈α∞analysis.␈α∞In␈α∞the␈α∞first␈α
test␈α
of␈α
this␈α
type,␈α
all␈α
terms␈α
in␈α
the␈α
motor␈α
torque␈α
equation␈↓ ↓H␈↓"
␈↓ ↓H(equation␈α
9)␈α
except␈α
proportional␈α
feedback,␈α
k␈↓#qe␈↓#,␈α
and␈α
friction␈α
were␈α
deleted.␈α Also,␈α the␈α motion␈↓ ↓H␈↓"
␈↓ ↓Hselected␈α∀had␈α∀a␈α∀constant␈α∀inertia␈α∀throughout␈α∪the␈α∪motion.␈α∪The␈α∪resulting␈α∪block␈α∪diagram␈α∪is␈↓ ↓H␈↓"
␈↓ ↓Hidentical␈α∃to␈α∃the␈α∃one␈α∃given␈α∃earlier␈α∃in␈α∃Figure␈α∃2.␈α∃The␈α∃value␈α∀of␈α∀k␈↓#qe␈↓#␈α∀was␈α∀then␈α∀increased␈↓ ↓H␈↓"
␈↓ ↓Hgradually␈α to␈α a␈α value␈α of␈α 3,150␈α at␈α which␈α joint␈α 1␈α became␈α unstable.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HTo␈α⊂determine␈α⊂analytically␈α⊂the␈α⊂predicted␈α⊂value␈α∂of␈α∂k␈↓#qe␈↓#␈α∂which␈α∂should␈α∂make␈α∂joint␈α∂1␈α∂unstable,␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α∞transfer␈α∞function␈α∞given␈α∞in␈α∞equation␈α∞(15)␈α
for␈α
the␈α
controller␈α
and␈α
arm␈α
was␈α
used␈α
with␈α
the␈↓ ↓H␈↓"
␈↓ ↓Hintegral␈α∪and␈α∪derivative␈α∪feedback␈α∪gains␈α∪set␈α∪to␈α∪zero.␈α∪The␈α∪simplified␈α∪transfer␈α∪function␈α∪is␈↓ ↓H␈↓"
␈↓ ↓Hgiven␈α below.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qtot␈↓#(z) = (TB + J(␈↓∧b␈↓ε-1))k␈↓#qe␈↓#␈↓ ∧F␈↓!⊃↓e␈↓ ∧F␈↓!∩↓e␈↓ ∧F␈↓!∪↓e␈↓ ∧F␈↓!∀↓e␈↓ ε+ z-␈↓∧a␈↓ε ␈↓ ε7␈↓!⊃↓∩␈↓ ε7␈↓!∩↓∩␈↓ ε7␈↓!∪↓∩␈↓ ε7␈↓!∀↓∩␈↓ πI ␈↓ *(30)␈↓ ↓H␈↓"
B␈↓#∂2␈↓# (z-␈↓∧b␈↓ε)(z-1)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where ␈↓∧a␈↓ε = TB␈↓∧b␈↓ε + J(␈↓∧b␈↓ε-1)␈↓ β(␈↓!⊃↓?␈↓ β(␈↓!∩↓?␈↓ β(␈↓!∪↓?␈↓ β(␈↓!∀↓?␈↓ ∧g␈↓ ↓H␈↓"
TB + J(␈↓∧b␈↓ε-1)␈↓ ↓H␈↓"
␈↓ ε≤page 18␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α closed␈α loop␈α transfer␈α function␈α is␈α then␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qcl␈↓#(z) = k␈↓#qc␈↓# G␈↓#qtot␈↓#(z) ␈↓ ∧4␈↓!⊃↓&␈↓ ∧4␈↓!∩↓&␈↓ ∧4␈↓!∪↓&␈↓ ∧4␈↓!∀↓&␈↓ ¬Z . ␈↓ *(31)␈↓ ↓H␈↓"
1 + G␈↓#qtot␈↓#(z)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α characteristic␈α equation␈α obtained␈α from␈α the␈α closed␈α loop␈α transfer␈α function␈α is␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
z␈↓#∂2␈↓# + [(TB + J(␈↓∧b␈↓ε-1))k␈↓#qe␈↓#␈↓ ∧⊂␈↓!⊃↓e␈↓ ∧⊂␈↓!∩↓e␈↓ ∧⊂␈↓!∪↓e␈↓ ∧⊂␈↓!∀↓e␈↓ ¬u - ␈↓∧b␈↓ε - 1]z + ␈↓∧b␈↓ε - (TB␈↓∧b␈↓ε + J(␈↓∧b␈↓ε-1))k␈↓#qe␈↓#␈↓ πi␈↓!⊃↓w␈↓ πi␈↓!∩↓w␈↓ πi␈↓!∪↓w␈↓ πi␈↓!∀↓w␈↓ ` = 0 . ␈↓ *(32)␈↓ ↓H␈↓"
B␈↓#∂2␈↓# B␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HUsing␈α⊂k␈↓#qe␈↓#␈α⊂=␈α⊂3,150,␈α⊂J␈α⊂=␈α⊂213␈α⊂oz-in-sec␈α⊂squared,␈α⊂B␈α∂=␈α∂31.27␈α∂oz-in-sec,␈α∂and␈α∂T␈α∂=␈α∂1/60,␈α∂the␈↓ ↓H␈↓"
␈↓ ↓Hroots␈α∞of␈α∞equation␈α∞(21)␈α∞were␈α∞found␈α∞to␈α∞be␈α∞z␈α∞=␈α
.99775␈α
+␈α
or␈α
-␈α
j.06402.␈α
The␈α
radius␈α
of␈α
these␈↓ ↓H␈↓"
␈↓ ↓Hroots␈α∞is␈α∞0.99981␈α∞which␈α∞indicates␈α∞that␈α
the␈α
system␈α
is␈α
stable␈α
with␈α
a␈α
k␈↓#qe␈↓#␈α
of␈α
3,150.␈α
If␈α
the␈α
gain␈↓ ↓H␈↓"
␈↓ ↓Hk␈↓#qe␈↓#␈α∂is␈α∂increased␈α∂until␈α∂the␈α∂roots␈α∞become␈α∞unstable,␈α∞it␈α∞is␈α∞found␈α∞the␈α∞the␈α∞predicted␈α∞gain␈α∞which␈↓ ↓H␈↓"
␈↓ ↓Hproduces␈α marginal␈α stability␈α is␈α 3,750.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∂error␈α∞between␈α∞the␈α∞actual␈α∞gain␈α∞of␈α∞3150␈α∞and␈α∞the␈α∞predicted␈α∞gain␈α∞of␈α∞3750␈α∞is␈α∞the␈α∞result␈↓ ↓H␈↓"
␈↓ ↓Hof␈α⊃an␈α⊃inaccuracy␈α⊃in␈α⊃the␈α⊃compensation␈α⊃for␈α⊃friction.␈α⊃Since␈α⊃the␈α⊃model␈α⊃used␈α⊃in␈α⊂this␈α⊂analysis␈↓ ↓H␈↓"
␈↓ ↓Hdoes␈α∂not␈α∂include␈α∂friction,␈α∂a␈α∂constant␈α∂term␈α∂has␈α∂been␈α∞added␈α∞to␈α∞the␈α∞output␈α∞torque␈α∞equation␈↓ ↓H␈↓"
␈↓ ↓Hto␈α↔compensate␈α↔for␈α↔friction.␈α↔Small␈α↔changes␈α⊗in␈α⊗the␈α⊗friction␈α⊗compensation␈α⊗term␈α⊗have␈α⊗a␈↓ ↓H␈↓"
␈↓ ↓Hdramatic␈α
effect␈α
of␈α
the␈α
system's␈α
dynamics.␈α
If␈α
the␈α
friction␈α compensation␈α term␈α was␈α fine␈α tuned␈↓ ↓H␈↓"
␈↓ ↓Hto␈α match␈α the␈α arm's␈α friction␈α more␈α closely,␈α the␈α error␈α in␈α gain␈α would␈α be␈α significantly␈α reduced.␈↓ ↓H␈↓"
␈↓ ε≤page 19␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
IV. RECOMMENDATIONS␈↓ ↓H␈↓!⊃α\␈↓ ↓H␈↓!∩α\␈↓ ↓H␈↓!∪α\␈↓ ↓H␈↓!∀α\␈↓ ∧$␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
A. SAMPLING TIME INDEPENDENCE␈↓ α∧␈↓!⊃β}␈↓ α∧␈↓!∩β}␈↓ α∧␈↓!∪β}␈↓ α∧␈↓!∀β}␈↓ εα␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∩feedback␈α∩portion␈α∩of␈α∩the␈α∩output␈α∩torque␈α∩equation␈α⊃now␈α⊃used␈α⊃in␈α⊃the␈α⊃blue␈α⊃and␈α⊃yellow␈↓ ↓H␈↓"
␈↓ ↓Harms␈α is␈α repeated␈α from␈α equation␈α (9)␈α below.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧T␈↓ε␈↓#qfb␈↓#(k) = -k␈↓#qc␈↓#{ k␈↓#qe␈↓#[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] + k␈↓#qv␈↓#J(K)[␈↓∧Q␈↓ε'(k) - ␈↓∧F␈↓ε'(k)]␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
+ k␈↓#qi␈↓#/J(K) ␈↓∧S␈↓ε[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] } ␈↓ *(33)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HTo␈α⊗be␈α⊗independent␈α⊗of␈α⊗sampling␈α⊗time,␈α⊗the␈α⊗integral␈α⊗term␈α⊗should␈α⊗be␈α⊗multiplied␈α⊗by␈α∃the␈↓ ↓H␈↓"
␈↓ ↓Hsampling␈α⊂time␈α∂T.␈α∂The␈α∂derivative␈α∂term␈α∂is␈α∂already␈α∂independent␈α∂of␈α∂time␈α∂since␈α∂on␈α∂joints␈α∂1,2,␈↓ ↓H␈↓"
␈↓ ↓Hand␈α∞3,␈α∞the␈α∞velocity␈α∞is␈α
measured␈α
with␈α
tachometers.␈α
On␈α
joints␈α
4,5,␈α
and␈α
6␈α
of␈α
the␈α
yellow␈α
arm,␈↓ ↓H␈↓"
␈↓ ↓Hwhere␈α position␈α differencing␈α is␈α used,␈α the␈α full␈α equation␈α for␈α the␈α derivative␈α term␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧T␈↓ε␈↓#qderiv.␈↓#(k) = k␈↓#qc␈↓#k␈↓#qv␈↓#J(k)[␈↓∧Q␈↓ε(k) - ␈↓∧Q␈↓ε(k-1) - ␈↓∧F␈↓ε(k) + ␈↓∧F␈↓ε(k-1)]/T .␈↓ *(34)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIt␈α can␈α be␈α seen␈α that␈α sampling␈α time␈α is␈α already␈α taken␈α into␈α account␈α in␈α the␈α derivative␈α portion␈α of␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α∪output␈α∪torque␈α∪equation.␈α∪In␈α∩the␈α∩proportional␈α∩portion␈α∩of␈α∩the␈α∩equation,␈α∩sampling␈α∩time␈↓ ↓H␈↓"
␈↓ ↓Hneed␈α⊃not␈α⊃be␈α⊃taken␈α⊂into␈α⊂account.␈α⊂The␈α⊂sampling␈α⊂time␈α⊂independent␈α⊂feedback␈α⊂equation␈α⊂thus␈↓ ↓H␈↓"
␈↓ ↓Hbecomes␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧T␈↓ε␈↓#qfb␈↓#(k) = -k␈↓#qc␈↓#{ k␈↓#qe␈↓#[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] + k␈↓#qv␈↓#J(k)[␈↓∧Q␈↓ε'(k) - ␈↓∧F␈↓ε'(k)]␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
+ k␈↓#qi␈↓#T/J(k) ␈↓∧S␈↓ε[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] } .␈↓ *(35)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ε≤page 20␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
B. INERTIA INDEPENDENCE␈↓ α∧␈↓!⊃βλ␈↓ α∧␈↓!∩βλ␈↓ α∧␈↓!∪βλ␈↓ α∧␈↓!∀βλ␈↓ ¬ ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∩feedback␈α∩portion␈α∩of␈α∩the␈α∩output␈α∩torque␈α∩equation␈α∩which␈α∩is␈α∩presently␈α∩being␈α∩used␈α⊃to␈↓ ↓H␈↓"
␈↓ ↓Hcontrol␈α the␈α blue␈α arm␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ ␈↓∧T␈↓ε␈↓#qfb␈↓#(k) = Prop + J(k)(Der) + (Int)/J(k) ␈↓ *(36)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where Prop = proportional feedback term␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Der = derivative feedback term␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Int = integral feedback term␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( J(k) = inertia .␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α
feedback␈α
equation␈α
for␈α the␈α yellow␈α arm␈α is␈α identical␈α except␈α that␈α the␈α inertia␈α term,␈α J(k),␈α is␈↓ ↓H␈↓"
␈↓ ↓Hreplaced␈α by␈α J(k)␈↓#∂1/2␈↓#.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HWhen␈α∩the␈α⊃output␈α⊃torque␈α⊃equation␈α⊃has␈α⊃the␈α⊃above␈α⊃form,␈α⊃the␈α⊃roots␈α⊃of␈α⊃the␈α⊃arm's␈α⊃transfer␈↓ ↓H␈↓"
␈↓ ↓Hfunction␈α∞will␈α∞be␈α∞shifting␈α∞around␈α∞during␈α∞any␈α∞motion␈α∞which␈α∞has␈α∞a␈α∞varying␈α∞inertia.␈α∞This␈α
means␈↓ ↓H␈↓"
␈↓ ↓Hthat␈α∪the␈α∪arm's␈α∪rise␈α∪time,␈α∪settling␈α∪time,␈α∪overshoot,␈α∪and␈α∩the␈α∩rest␈α∩of␈α∩the␈α∩arm's␈α∩transient␈↓ ↓H␈↓"
␈↓ ↓Hresponse␈α∞parameters␈α∞will␈α∞be␈α∞varying␈α
throughout␈α
a␈α
motion.␈α
To␈α
minimize␈α
the␈α
dependence␈α
of␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α∪feedback␈α∪equation␈α∪on␈α∪inertia,␈α∪the␈α∪inertia␈α∪terms␈α∪should␈α∪be␈α∪rearranged␈α∪to␈α∪give␈α∪the␈↓ ↓H␈↓"
␈↓ ↓Hfollowing␈α output␈α torque␈α equation.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ ␈↓∧T␈↓ε␈↓#qfb␈↓#(k) = J(k)[Prop + Der + Int] ␈↓ *(37)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α⊂detailed␈α⊂equation␈α⊂for␈α⊂the␈α⊂feedback␈α⊂portion␈α⊂of␈α⊂the␈α⊂output␈α⊂torque␈α⊂equation,␈α∂with␈α∂the␈↓ ↓H␈↓"
␈↓ ↓Hcorrections␈α given␈α above␈α for␈α inertia␈α and␈α sampling␈α time,␈α will␈α then␈α appear␈α as␈α follows.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓∧T␈↓ε␈↓#qfb␈↓#(k) = -k␈↓#qc␈↓#J(k){ k␈↓#qe␈↓#[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] + k␈↓#qv␈↓#[␈↓∧Q␈↓ε'(k) - ␈↓∧F␈↓ε'(k)]␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
+ k␈↓#qi␈↓#T ␈↓∧S␈↓ε[␈↓∧Q␈↓ε(k) - ␈↓∧F␈↓ε(k)] } ␈↓ *(38)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓β␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ¬}PART II␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α]THE AUTOMATED TUNING OF CONTROLLERS␈↓ ↓H␈↓"
␈↓ β,USING RANDOM SEARCH TECHNIQUES␈↓ ↓H␈↓"
␈↓ε ␈↓ ↓H␈↓"
␈↓ ε%page 2␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
I. INTRODUCTION␈↓ ↓H␈↓!⊃↓y␈↓ ↓H␈↓!∩↓y␈↓ ↓H␈↓!∪↓y␈↓ ↓H␈↓!∀↓y␈↓ βA␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∩tuning␈α∩of␈α∩controllers␈α∩has␈α∩traditionally␈α∩been␈α∩a␈α∩black␈α∩art.␈α∩Feedback␈α∩gains␈α⊃for␈α⊃simple␈↓ ↓H␈↓"
␈↓ ↓Hcontrol␈α∂systems␈α∂can␈α∂be␈α∂calculated␈α∂analytically,␈α∂but␈α∂as␈α∂the␈α∂size␈α∂and␈α∂complexity␈α∂of␈α∂today's␈↓ ↓H␈↓"
␈↓ ↓Hcontrol␈α∪systems␈α∪continue␈α∪to␈α∪increase,␈α∪the␈α∪feasibility␈α∪of␈α∪calculating␈α∩a␈α∩system's␈α∩optimum␈↓ ↓H␈↓"
␈↓ ↓Hfeedback␈α⊂gains␈α⊂rapidly␈α⊂decreases.␈α∂In␈α∂industry,␈α∂it␈α∂is␈α∂often␈α∂impractical␈α∂and␈α∂uneconomical␈α∂to␈↓ ↓H␈↓"
␈↓ ↓Hcharacterize␈α_a␈α_control␈α↔system␈α↔analytically␈α↔to␈α↔determine␈α↔its␈α↔optimum␈α↔feedback␈α↔gains.␈↓ ↓H␈↓"
␈↓ ↓HInstead,␈α⊃the␈α⊃gains␈α⊃are␈α⊃often␈α⊃varied␈α⊃manually␈α⊃to␈α⊂determine␈α⊂compromise␈α⊂values␈α⊂which␈α⊂fall␈↓ ↓H␈↓"
␈↓ ↓Hbetween␈α optimum␈α tuning␈α and␈α economical␈α tuning.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HControl␈α_systems␈α_frequently␈α_have␈α_varying␈α_parameters␈α↔which␈α↔continuously␈α↔change␈α↔the␈↓ ↓H␈↓"
␈↓ ↓Hoptimum␈α∞values␈α
of␈α
the␈α
system's␈α
feedback␈α
gains.␈α
An␈α
example␈α
of␈α
such␈α
a␈α
system␈α
is␈α
a␈α
control␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α
for␈α controlling␈α the␈α temperature␈α of␈α a␈α tank␈α which␈α is␈α being␈α filled␈α with␈α chemicals.␈α Since␈↓ ↓H␈↓"
␈↓ ↓Hit␈α∞is␈α∞usually␈α∞impractical␈α∞to␈α∞vary␈α∞a␈α∞system's␈α∞feedback␈α∞gains␈α∞continuously␈α∞as␈α∞its␈α∞parameters␈↓ ↓H␈↓"
␈↓ ↓Hvary,␈α a␈α set␈α of␈α compromise␈α values␈α must␈α be␈α chosen.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThis␈α∞paper␈α∞describes␈α∞a␈α∞tuning␈α∞package␈α∞which␈α∞can␈α∞be␈α∞used␈α∞to␈α∞automatically␈α∞tune␈α∞a␈α
control␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α↔without␈α↔a␈α↔detailed␈α↔knowledge␈α⊗of␈α⊗the␈α⊗system's␈α⊗dynamics␈α⊗or␈α⊗parameters.␈α⊗The␈↓ ↓H␈↓"
␈↓ ↓Hpackage␈α can␈α be␈α used␈α to␈α tune␈α a␈α system␈α at␈α startup,␈α or␈α it␈α can␈α be␈α used␈α to␈α continuously␈α tune␈α a␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α∂as␈α∂its␈α∂parameters␈α∞vary.␈α∞In␈α∞normal␈α∞use,␈α∞the␈α∞package␈α∞is␈α∞connnected␈α∞to␈α∞an␈α∞operating␈↓ ↓H␈↓"
␈↓ ↓Hcontroller␈α∀which␈α∀has␈α∀a␈α∀stable␈α∀set␈α∀of␈α∀feedback␈α∪gains.␈α∪The␈α∪package␈α∪monitors␈α∪the␈α∪error␈↓ ↓H␈↓"
␈↓ ↓Hbetween␈α∩the␈α∩system's␈α∩normal␈α∩input␈α∩and␈α∩output␈α∩signals␈α∩and␈α⊃then␈α⊃uses␈α⊃a␈α⊃random␈α⊃search␈↓ ↓H␈↓"
␈↓ ↓Htechnique␈α∀to␈α∀adjust␈α∪the␈α∪system's␈α∪feedback␈α∪gains␈α∪to␈α∪minimize␈α∪the␈α∪system's␈α∪error.␈α∪This␈↓ ↓H␈↓"
␈↓ ↓Happroach␈α∀avoids␈α∀the␈α∀need␈α∪to␈α∪excite␈α∪the␈α∪system␈α∪with␈α∪external␈α∪test␈α∪signals␈α∪that␈α∪could␈↓ ↓H␈↓"
␈↓ ↓Hdisturb␈α the␈α operation␈α of␈α the␈α system.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α∞more␈α∞detailed␈α∞description␈α∞of␈α∞the␈α∞procedure␈α∞used␈α∞by␈α∞this␈α∞tuning␈α
package␈α
is␈α
given␈α
in␈α
the␈↓ ↓H␈↓"
␈↓ ↓Hfive␈α steps␈α below.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(1. Starting with an initial set of stable feedback gains supplied by the user, the␈↓ ↓H␈↓"
␈↓ α( package runs the system for a predetermined interval of time and measures the␈↓ ↓H␈↓"
␈↓ α( resulting system error.␈↓ ↓H␈↓"
␈↓ ε%page 3␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(2. On the second time interval, the package perturbs the initial feedback gains,␈↓ ↓H␈↓"
␈↓ α( runs the system with the new gains, and again measures the resulting error.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(3. If the error from the second time interval is less than that obtained on the first,␈↓ ↓H␈↓"
␈↓ α( the package replaces the initial gains with the gains from the second interval.␈↓ ↓H␈↓"
␈↓ α( If the error from the second interval is larger, however, the initial gains are␈↓ ↓H␈↓"
␈↓ α( left unchanged.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(4. On the third time interval, the package perturbs the previous set of gains␈↓ ↓H␈↓"
␈↓ α( which produced the least error, runs the system with the new gains, and measures␈↓ ↓H␈↓"
␈↓ α( the error.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(5. The package repeats the above procedure until the improvement in system error␈↓ ↓H␈↓"
␈↓ α( is negligible.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIf,␈α⊂at␈α⊂any␈α⊂time␈α⊂during␈α⊂the␈α⊂above␈α⊂procedure,␈α⊂the␈α⊂tuning␈α⊂package␈α⊂detects␈α⊂signs␈α⊂of␈α∂system␈↓ ↓H␈↓"
␈↓ ↓Hinstability,␈α∞it␈α∞immediately␈α∞aborts␈α∞the␈α∞present␈α∞time␈α∞interval,␈α∞switches␈α∞back␈α
to␈α
the␈α
previous␈↓ ↓H␈↓"
␈↓ ↓Hbest␈α∞set␈α∞of␈α∞feedback␈α∞gains,␈α∞and␈α∞then␈α∞runs␈α∞the␈α∞system␈α∞for␈α∞the␈α∞next␈α∞time␈α
interval␈α
with␈α
the␈↓ ↓H␈↓"
␈↓ ↓Hinput␈α signal␈α set␈α to␈α zero␈α to␈α allow␈α the␈α system␈α to␈α stabilize.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HAlthough␈α→a␈α→random␈α_search␈α_has␈α_been␈α_used␈α_exclusively␈α_in␈α_this␈α_report,␈α_other␈α_search␈↓ ↓H␈↓"
␈↓ ↓Htechniques␈α⊃can␈α⊃also␈α⊃be␈α⊃used␈α⊃with␈α⊃equivalent␈α⊃or␈α⊃better␈α⊃results.␈α⊃The␈α⊃main␈α⊂thrust␈α⊂of␈α⊂this␈↓ ↓H␈↓"
␈↓ ↓Hreport␈α∩is␈α∩not␈α∩the␈α∩search␈α∩technique␈α∩itself,␈α∩but␈α∩how␈α∩a␈α⊃search␈α⊃technique␈α⊃can␈α⊃be␈α⊃used␈α⊃to␈↓ ↓H␈↓"
␈↓ ↓Hminimize␈α⊂a␈α⊂control␈α⊂system's␈α∂error␈α∂without␈α∂losing␈α∂control␈α∂of␈α∂the␈α∂system.␈α∂It␈α∂should␈α∂also␈α∂be␈↓ ↓H␈↓"
␈↓ ↓Hnoted␈α∪that␈α∪a␈α∪considerable␈α∪amount␈α∪of␈α∪effort␈α∪has␈α∪been␈α∪expended␈α∪to␈α∪keep␈α∪this␈α∩package␈↓ ↓H␈↓"
␈↓ ↓Hgeneral.␈α∀The␈α∀only␈α∀parameters␈α∀that␈α∀need␈α∀to␈α∀be␈α∪specified␈α∪concerning␈α∪the␈α∪system␈α∪being␈↓ ↓H␈↓"
␈↓ ↓Htuned␈α∞are␈α∞its␈α∞largest␈α
time␈α
constant␈α
and␈α
a␈α
stable␈α
set␈α
of␈α
starting␈α
feedback␈α
gains.␈α
Obviously,␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α∂convergence␈α∞of␈α∞a␈α∞search␈α∞can␈α∞be␈α∞improved␈α∞dramatically␈α∞if␈α∞the␈α∞dynamics␈α∞of␈α∞the␈α∞system␈↓ ↓H␈↓"
␈↓ ↓Hbeing␈α∪tuned␈α∪are␈α∪known.␈α∪No␈α∪attempt␈α∪has␈α∪been␈α∪made␈α∪to␈α∩utilize␈α∩such␈α∩information␈α∩in␈α∩this␈↓ ↓H␈↓"
␈↓ ↓Htuning␈α⊃package,␈α⊃however,␈α⊂since␈α⊂it␈α⊂has␈α⊂been␈α⊂assumed␈α⊂that␈α⊂this␈α⊂package␈α⊂will␈α⊂be␈α⊂used␈α⊂for␈↓ ↓H␈↓"
␈↓ ↓Hsystems␈α whose␈α dynamics␈α are␈α either␈α unknown␈α or␈α rapidly␈α changing.␈↓ ↓H␈↓"
␈↓ ε%page 4␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
In Section II of this report, the tuning procedure is explained␈↓ ↓H␈↓"
in detail. Section III uses a simulated third order control system to␈↓ ↓H␈↓"
illustrate the tuning package's ability to tune control systems under many␈↓ ↓H␈↓"
different circumstances. In Section IV, the feedback gains for one of the␈↓ ↓H␈↓"
systems tuned in Section III are calculated analytically and compared with␈↓ ↓H␈↓"
the gains obtained by the tuning package. Finally, conclusions are drawn ␈↓ ↓H␈↓"
in Section V concerning the effective use of the tuning package presented␈↓ ↓H␈↓"
in this report.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"∧ ␈↓ ε∨page 5 ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧II. TUNING PACKAGE␈↓ ↓H␈↓!⊃α/␈↓ ↓H␈↓!∩α/␈↓ ↓H␈↓!∪α/␈↓ ↓H␈↓!∀α/␈↓ βw␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ A. SEARCH TECHNIQUE␈↓ α(␈↓!⊃αT␈↓ α(␈↓!∩αT␈↓ α(␈↓!∪αT␈↓ α(␈↓!∀αT␈↓ ∧|␈↓ ↓H␈↓"∧␈↓ α(␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HTwo␈α⊃search␈α⊃methods␈α⊃were␈α⊃investigated␈α⊃in␈α⊃this␈α⊂study:␈α⊂the␈α⊂random␈α⊂search␈α⊂and␈α⊂the␈α⊂linear␈↓ ↓H␈↓"∧␈↓ ↓Hrandom␈α search.␈α These␈α two␈α methods␈α are␈α described␈α in␈α equations␈α 2-1␈α and␈α 2-2␈α below.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ W(k+1) = W(k) + V*N(k) ␈↓ ~(2-1)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ W(k+1) = W(k) - V*N(k)*[E(k+1) - E(k)] ␈↓ ~(2-2)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ where W(k+1) = latest weight vector␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ W(k) = previous weight vector␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ N(k) = random variable with a gaussian distribution ␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ V = variance of the random number generator␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ E(k+1) = latest error vector␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ E(k) = previous error vector␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HLearning␈α⊂curves␈α⊂for␈α∂a␈α∂single␈α∂random␈α∂search␈α∂and␈α∂for␈α∂an␈α∂average␈α∂of␈α∂100␈α∂random␈α∂searches␈↓ ↓H␈↓"∧␈↓ ↓Hare␈α⊂displayed␈α⊂in␈α⊂Figures␈α⊂2-1a␈α⊂and␈α⊂2-1b.␈α⊂The␈α⊂data␈α⊂in␈α⊂Figure␈α⊂2-1b␈α⊂is␈α⊂replotted␈α⊂on␈α⊂a␈α⊂log␈↓ ↓H␈↓"∧␈↓ ↓Hscale␈α⊃in␈α⊃Figure␈α⊃2-1c␈α⊃to␈α⊃expand␈α⊃the␈α⊃lower␈α⊃portion␈α⊃of␈α⊂the␈α⊂learning␈α⊂curve.␈α⊂The␈α⊂curves␈α⊂in␈↓ ↓H␈↓"∧␈↓ ↓HFigures␈α∞2-1␈α∞were␈α∞obtained␈α∞while␈α∞the␈α∞random␈α∞search␈α∞routine␈α∞was␈α
minimizing␈α
equation␈α
2-3␈↓ ↓H␈↓"∧␈↓ ↓Hwith␈α⊃an␈α⊃initial␈α⊃value␈α⊃of␈α⊂X␈α⊂=␈α⊂500.␈α⊂A␈α⊂value␈α⊂of␈α⊂5␈α⊂was␈α⊂chosen␈α⊂for␈α⊂the␈α⊂variance␈α⊂because␈α⊂it␈↓ ↓H␈↓"∧␈↓ ↓Hminimized␈α⊂the␈α⊂final␈α⊂value␈α⊂for␈α⊂E(k).␈α⊂The␈α⊂final␈α⊂value␈α⊂obtained␈α⊂for␈α⊂E(k)␈α⊂in␈α⊂Figure␈α∂2-1b␈α∂was␈↓ ↓H␈↓"∧␈↓ ↓H1645.␈α It␈α can␈α be␈α seen␈α from␈α equation␈α 2-3␈α that␈α the␈α minimum␈α value␈α for␈α E(k)␈α is␈α 100.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ E(k) = 100 + X(k)␈↓#∂2␈↓# ␈↓ ~(2-3)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HWhen␈α∞the␈α∞"linear"␈α∞random␈α∞search␈α
was␈α
used␈α
to␈α
minimize␈α
equation␈α
2-3,␈α
it␈α
was␈α
found␈α
that␈α
it␈↓ ↓H␈↓"∧␈↓ ↓Htended␈α∞to␈α∞decelerate␈α∞too␈α∞quickly,␈α∞even␈α∞when␈α
the␈α
variance␈α
was␈α
increased.␈α
The␈α
best␈α
value␈↓ ↓H␈↓"∧␈↓ ↓Hobtained␈α∞for␈α∞E(k),␈α∞after␈α∞several␈α∞runs␈α∞with␈α
different␈α
variances,␈α
was␈α
10,460.␈α
To␈α
reduce␈α
the␈↓ ↓H␈↓"∧␈↓ ↓Hrate␈α⊂of␈α∂deceleration,␈α∂the␈α∂gradient␈α∂term␈α∂in␈α∂the␈α∂linear␈α∂random␈α∂search␈α∂algorithm␈α∂was␈α∂raised␈↓ ↓H␈↓"∧␈↓ ↓Hto␈α the␈α .25␈α power.␈α The␈α modified␈α linear␈α random␈α search␈α algorithm␈α is␈α given␈α in␈α equation␈α 2-4.␈↓ ↓H␈↓"∧ ␈↓ ε%page 9␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ W(k+1) = W(k) - V*N(k)*[E(k+1) - E(k)]␈↓#∂.25␈↓# ␈↓ ~(2-4)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HLearning␈α
curves␈α
for␈α
the␈α
modified␈α linear␈α random␈α search␈α are␈α given␈α in␈α Figures␈α 2-2a␈α and␈α 2-2b␈↓ ↓H␈↓"∧␈↓ ↓Hfor␈α
a␈α
single␈α
search␈α
and␈α
for␈α
an␈α
average␈α
of␈α
100␈α
searches.␈α
Figure␈α
2-2b␈α is␈α replotted␈α on␈α a␈α log␈↓ ↓H␈↓"∧␈↓ ↓Hscale␈α∪in␈α∪Figure␈α∪2-2c.␈α∩The␈α∩linear␈α∩random␈α∩search␈α∩tends␈α∩to␈α∩be␈α∩more␈α∩oscillatory␈α∩than␈α∩the␈↓ ↓H␈↓"∧␈↓ ↓Hrandom␈α∞search,␈α∞but␈α∞it␈α∞converges␈α∞to␈α∞a␈α∞value␈α
which␈α
is␈α
closer␈α
to␈α
the␈α
optimum␈α
value,␈α
since␈α
it␈↓ ↓H␈↓"∧␈↓ ↓Hdecelerates␈α⊃as␈α⊃the␈α⊃error␈α⊃gradient,␈α⊃[E(k+1)␈α⊃-␈α⊃E(k)],␈α⊃decreases.␈α⊃For␈α⊃a␈α⊃variance␈α⊃of␈α⊂5,␈α⊂the␈↓ ↓H␈↓"∧␈↓ ↓Hfinal␈α∪value␈α∪of␈α∪E␈α∪in␈α∪Figure␈α∪2-2b␈α∪was␈α∪284.␈α∪Note␈α∪that␈α∪Figures␈α∩2-1␈α∩and␈α∩2-2␈α∩have␈α∩been␈↓ ↓H␈↓"∧␈↓ ↓Hnormalized,␈α⊂so␈α⊂they␈α⊂appear␈α⊂similar,␈α⊂even␈α⊂though␈α⊂the␈α⊂values␈α⊂on␈α⊂their␈α∂vertical␈α∂axes␈α∂differ␈↓ ↓H␈↓"∧␈↓ ↓Hconsiderably.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HThe␈α∞learning␈α∞curves␈α∞in␈α∞Figure␈α∞2-1␈α∞and␈α∞2-2␈α∞were␈α∞obtained␈α∞with␈α∞no␈α∞noise.␈α∞In␈α∞equation␈α
2-5␈↓ ↓H␈↓"∧␈↓ ↓Hbelow,␈α∀the␈α∀function␈α∪minimized␈α∪earlier␈α∪has␈α∪been␈α∪modified␈α∪to␈α∪include␈α∪a␈α∪large␈α∪amount␈α∪of␈↓ ↓H␈↓"∧␈↓ ↓Hsimulated␈α∀process␈α∀noise.␈α∀The␈α∀logs␈α∀of␈α∀the␈α∀new␈α∪learning␈α∪curves,␈α∪with␈α∪noise␈α∪added,␈α∪are␈↓ ↓H␈↓"∧␈↓ ↓Hdisplayed␈α⊃in␈α⊃Figures␈α⊃2-3␈α⊃and␈α⊃2-4␈α⊃for␈α⊃the␈α⊃random␈α⊂search␈α⊂and␈α⊂the␈α⊂linear␈α⊂random␈α⊂search.␈↓ ↓H␈↓"∧␈↓ ↓HBoth␈α⊂figures␈α⊂represent␈α∂an␈α∂average␈α∂of␈α∂100␈α∂iterations.␈α∂The␈α∂final␈α∂values␈α∂obtained␈α∂for␈α∂E␈α∂by␈↓ ↓H␈↓"∧␈↓ ↓Hthe␈α∂random␈α∂search␈α∂with␈α∂noise␈α∂was␈α∂15,482,␈α∂while␈α∂the␈α∞final␈α∞value␈α∞obtained␈α∞with␈α∞the␈α∞linear␈↓ ↓H␈↓"∧␈↓ ↓Hrandom␈α search␈α with␈α noise␈α was␈α 6,632.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ E(K+1) = 100 + X(k)␈↓#∂2␈↓# + .25*E(K)*N(k) + 100*N(k) ␈↓ ~(2-5)␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HIt␈α⊂can␈α⊂be␈α⊂seen␈α⊂that␈α⊂noise␈α⊂affects␈α⊂the␈α⊂rate␈α∂of␈α∂convergence␈α∂and␈α∂the␈α∂accuracy␈α∂of␈α∂the␈α∂final␈↓ ↓H␈↓"∧␈↓ ↓Hvalue␈α∂in␈α∂both␈α∂search␈α∂methods.␈α∂The␈α∂linear␈α∂random␈α∂search,␈α∂however,␈α∂still␈α∂converges␈α∂faster␈↓ ↓H␈↓"∧␈↓ ↓Hand␈α arrives␈α at␈α a␈α more␈α accurate␈α estimate␈α of␈α the␈α optimum␈α value.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧ ␈↓ ε≤page 15␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ α(B. STABILITY CRITERIA␈↓ α(␈↓!⊃αS␈↓ α(␈↓!∩αS␈↓ α(␈↓!∪αS␈↓ α(␈↓!∀αS␈↓ ∧{␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HSince␈α∃the␈α∃search␈α∃package␈α∃has␈α∃very␈α∃little␈α∃information␈α∃concerning␈α∀the␈α∀dynamics␈α∀of␈α∀the␈↓ ↓H␈↓"∧␈↓ ↓Hsystem␈α
it␈α
is␈α
tuning,␈α it␈α will␈α occasionally␈α choose␈α a␈α set␈α of␈α feedback␈α gains␈α which␈α will␈α cause␈α the␈↓ ↓H␈↓"∧␈↓ ↓Hsystem␈α∂to␈α∂become␈α∂unstable.␈α∂When␈α∂this␈α∂happens␈α∂the␈α∂tuning␈α∂package␈α∂must␈α∂immediately␈α∞be␈↓ ↓H␈↓"∧␈↓ ↓Hable␈α∂to␈α∂recognize␈α∂the␈α∂onset␈α∂of␈α∂instability␈α∂so␈α∞that␈α∞it␈α∞can␈α∞retreat␈α∞to␈α∞the␈α∞best␈α∞previous␈α∞set␈↓ ↓H␈↓"∧␈↓ ↓Hof␈α∪stable␈α∪feedback␈α∪gains.␈α∪Four␈α∩criteria␈α∩have␈α∩been␈α∩used␈α∩is␈α∩this␈α∩report␈α∩as␈α∩measures␈α∩of␈↓ ↓H␈↓"∧␈↓ ↓Hsystem␈α stability.␈α They␈α are␈α listed␈α and␈α then␈α explained␈α below.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ α(␈↓ βλ1) Stability Ratio␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ α(␈↓ βλ2) Maximum Output␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ α(␈↓ βλ3) Negative Feedback Gains␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ α(␈↓ βλ4) Steady State Error␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ α(␈↓ βλ1. STABILITY RATIO␈↓ βλ␈↓!⊃α&␈↓ βλ␈↓!∩α&␈↓ βλ␈↓!∪α&␈↓ βλ␈↓!∀α&␈↓ ¬.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓ ↓HFor␈α∂the␈α∂typical␈α∂control␈α∂system␈α∞illustrated␈α∞in␈α∞Figure␈α∞2-5,␈α∞the␈α∞ratio␈α∞of␈α∞E/R␈α∞will␈α∞tend␈α∞to␈α∞fall␈↓ ↓H␈↓"∧␈↓ ↓Hbetween␈α .5␈α and␈α 10␈α if␈α the␈α system␈α has␈α been␈α well␈α tuned.␈↓ ↓H␈↓"∧␈↓ ↓H␈↓"∧␈↓π␈↓ ↓H
␈↓ αH␈↓ βH␈↓ ∧H␈↓ ¬H ⊂αααααααααα⊃␈↓ ↓H
␈↓ αH R(t) + ⊂ααα⊃ E(t) ~ ~ Y(t)␈↓ ↓H
αααααααααα→~ + ~αααααααααα→~ ~αααααααααααααα→␈↓ ↓H
␈↓ αH␈↓ βH %ααα$ ~ ~ ~␈↓ ↓H
␈↓ αH␈↓ βH - ↑ %αααααααααα$ ~␈↓ ↓H
␈↓ αH␈↓ βH ~␈↓ ¬H␈↓ εH␈↓ πH␈↓ λH~␈↓ ↓H
␈↓ αH␈↓ βH ~␈↓ ¬H␈↓ εH␈↓ πH␈↓ λH~␈↓ ↓H
␈↓ αH␈↓ βH %αααααααααααααααααααααααααααααααα$␈↓ ↓H
␈↓ ↓H␈↓"
␈↓ε Figure 2-5 Block Diagram of Basic Control System␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where R(t) = system input␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Y(t) = system output␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( E(t) = system error␈↓ ↓H␈↓"
␈↓ ε≤page 16␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HSince␈α⊃the␈α⊃system␈α⊃being␈α⊃tuned␈α⊃has␈α⊃a␈α⊃time␈α⊃constant,␈α⊃it␈α⊃is␈α⊃possible␈α⊃to␈α⊂quickly␈α⊂change␈α⊂the␈↓ ↓H␈↓"
␈↓ ↓Hinput␈α∪R␈α∪from␈α∪1␈α∪to␈α∪0␈α∪and␈α∪create␈α∪a␈α∪ratio␈α∪of␈α∪E/R␈α∪=␈α∪infinity.␈α∪To␈α∩prevent␈α∩the␈α∩ratio␈α∩from␈↓ ↓H␈↓"
␈↓ ↓Hbecoming␈α⊃unreasonably␈α⊃high,␈α⊃the␈α⊃value␈α⊃of␈α⊃R␈α⊃can␈α⊃be␈α⊃replaced␈α⊃by␈α⊂the␈α⊂output␈α⊂of␈α⊂a␈α⊂digital␈↓ ↓H␈↓"
␈↓ ↓Hfilter␈α
whose␈α
input␈α
is␈α
R␈α
and␈α
which␈α
has␈α
a␈α
time␈α
constant␈α
of␈α
1/2␈α
to␈α
1/10␈α of␈α the␈α system␈α being␈↓ ↓H␈↓"
␈↓ ↓Htuned.␈α⊂To␈α⊂prevent␈α⊂the␈α⊂ratio␈α⊂of␈α⊂E/R␈α⊂from␈α⊂approaching␈α⊂infinity␈α⊂when␈α⊂the␈α⊂input␈α⊂remains␈α∂at␈↓ ↓H␈↓"
␈↓ ↓Hzero␈α∪for␈α∪prolonged␈α∪periods␈α∪of␈α∩time,␈α∩the␈α∩maximum␈α∩desired␈α∩output␈α∩value,␈α∩MO,␈α∩has␈α∩been␈↓ ↓H␈↓"
␈↓ ↓Hadded␈α⊃to␈α⊃the␈α⊃denominator.␈α⊃(MO␈α⊃will␈α⊃be␈α⊃defined␈α⊃on␈α⊃page␈α⊃17.)␈α⊃The␈α⊃new␈α⊃ratio,␈α⊂which␈α⊂will␈↓ ↓H␈↓"
␈↓ ↓Hhereafter␈α be␈α refered␈α to␈α as␈α the␈α stability␈α ratio,␈α is␈α given␈α by␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
SR(k) = E(k)/[F(k) + MO] ␈↓ ~(2-5)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where SR(k) = stability ratio␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( E(k) = system error␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( F(k) = output of digital filter ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
= (1-c)F(k-1) + c[R(k) + R(k-1)]␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
R(k) = system input signal␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( MO = maximum desired system output␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( c = constant = tan [(w␈↓#qo␈↓#t)/2] .␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HWhen␈α∂the␈α∂predefined␈α∂maximum␈α∂stability␈α∂ratio,␈α∂MSR,␈α∂is␈α∞exceeded,␈α∞the␈α∞search␈α∞package␈α∞has␈↓ ↓H␈↓"
␈↓ ↓Hbeen␈α
structured␈α
to␈α
abort␈α
the␈α
present␈α
search␈α
interval,␈α
set␈α
the␈α
input␈α
R(k)␈α
to␈α zero,␈α and␈α then␈↓ ↓H␈↓"
␈↓ ↓Hrun␈α
the␈α
next␈α
interval␈α of␈α the␈α search␈α with␈α the␈α previous␈α best␈α set␈α of␈α feedback␈α gains␈α to␈α insure␈↓ ↓H␈↓"
␈↓ ↓Hthat␈α the␈α system␈α has␈α stabilized␈α before␈α a␈α new␈α set␈α of␈α feedback␈α gains␈α is␈α tested.␈α Thus␈α the␈α rule␈↓ ↓H␈↓"
␈↓ ↓Hfor␈α condition␈α 1␈α is:␈↓ ↓H␈↓"
␈↓↓␈↓ ↓H␈↓"
CONDITION 1: IF SR(k) > MSR THEN RETREAT␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ε where MSR = prespecified maximum stability ratio␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( RETREAT = set the input to zero and go back to previous best set of stable␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ feedback gains for the next search interval.␈↓ ↓H␈↓"
␈↓ ε≤page 17␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ2. MAXIMUM OUTPUT␈↓ βλ␈↓!⊃αA␈↓ βλ␈↓!∩αA␈↓ βλ␈↓!∪αA␈↓ βλ␈↓!∀αA␈↓ ¬I␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HMost␈α⊃systems␈α⊃have␈α⊃a␈α⊃maximum␈α⊃allowable␈α⊃output,␈α⊂MAO,␈α⊂beyond␈α⊂which␈α⊂actuators␈α⊂become␈↓ ↓H␈↓"
␈↓ ↓Hsaturated␈α↔and␈α↔manipulators␈α↔collide␈α↔with␈α↔surrounding␈α↔objects.␈α↔For␈α↔safety,␈α⊗a␈α⊗maximum␈↓ ↓H␈↓"
␈↓ ↓Hdesired␈α∞output,␈α∞MO,␈α∞is␈α∞defined␈α∞for␈α∞use␈α∞with␈α∞this␈α∞package␈α∞which␈α∞ranges␈α
between␈α
.1␈α
and␈α
.9␈↓ ↓H␈↓"
␈↓ ↓Htimes␈α∀MAO.␈α∀An␈α∀output␈α∀larger␈α∀than␈α∪MO␈α∪is␈α∪an␈α∪indication␈α∪that␈α∪the␈α∪output␈α∪is␈α∪closer␈α∪than␈↓ ↓H␈↓"
␈↓ ↓Hdesired␈α to␈α the␈α MAO.␈α Thus␈α the␈α rule␈α for␈α condition␈α 2␈α is:␈↓ ↓H␈↓"
␈↓↓␈↓ ↓H␈↓"
CONDITION 2: IF Y(k) > MO THEN RETREAT␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ε where MO = prespecified maximum output value.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HWhen␈α⊃the␈α⊃tuning␈α⊃package␈α⊃is␈α⊃used␈α⊃for␈α⊃the␈α⊃first␈α⊃time,␈α⊃the␈α⊃value␈α⊂of␈α⊂MO␈α⊂should␈α⊂be␈α⊂set␈α⊂to␈↓ ↓H␈↓"
␈↓ ↓H.1*MAO.␈α∂As␈α∂the␈α∞tuning␈α∞package␈α∞improves␈α∞the␈α∞system's␈α∞response,␈α∞the␈α∞value␈α∞of␈α∞MO␈α∞can␈α∞be␈↓ ↓H␈↓"
␈↓ ↓Hincreased.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ3. NEGATIVE FEEDBACK GAINS␈↓ βλ␈↓!⊃βK␈↓ βλ␈↓!∩βK␈↓ βλ␈↓!∪βK␈↓ βλ␈↓!∀βK␈↓ εS␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIf␈α∂the␈α∂tuning␈α∂package␈α∂chooses␈α∞a␈α∞negative␈α∞value␈α∞for␈α∞a␈α∞feedback␈α∞gain,␈α∞the␈α∞resulting␈α∞system␈↓ ↓H␈↓"
␈↓ ↓Hresponse␈α⊃will␈α⊂be␈α⊂poor␈α⊂at␈α⊂best.␈α⊂To␈α⊂avoid␈α⊂a␈α⊂possible␈α⊂unstable␈α⊂set␈α⊂of␈α⊂feedback␈α⊂gains,␈α⊂the␈↓ ↓H␈↓"
␈↓ ↓Htuning␈α∪package␈α∪has␈α∪been␈α∩structured␈α∩to␈α∩immediately␈α∩recompute␈α∩a␈α∩new␈α∩set␈α∩of␈α∩feedback␈↓ ↓H␈↓"
␈↓ ↓Hgains␈α∂wherever␈α∂a␈α∂negative␈α∂value␈α∂is␈α∂computed␈α∞for␈α∞one␈α∞of␈α∞the␈α∞feedback␈α∞gains.␈α∞Condition␈α∞3␈↓ ↓H␈↓"
␈↓ ↓Hcan␈α be␈α summarized␈α by:␈↓ ↓H␈↓"
␈↓↓␈↓ ↓H␈↓"
CONDITION 3: IF K(I) < 0 THEN RECOMPUTE␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ε where K(I) = value chosen for i'th feedback gain␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( RECOMPUTE = compute new set of feedback gains.␈↓ ↓H␈↓"
␈↓ ε≤page 18␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ4. STEADY STATE ERROR␈↓ βλ␈↓!⊃α{␈↓ βλ␈↓!∩α{␈↓ βλ␈↓!∪α{␈↓ βλ␈↓!∀α{␈↓ εβ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α∀some␈α∪systems,␈α∪small␈α∪steady␈α∪state␈α∪errors␈α∪can␈α∪be␈α∪disastrous.␈α∪An␈α∪example␈α∪of␈α∪such␈α∪a␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α∞is␈α∞a␈α∞speed␈α∞control␈α∞system␈α∞for␈α∞controlling␈α∞the␈α∞speed␈α∞of␈α∞a␈α∞continuous␈α∞strip␈α∞of␈α
steel␈↓ ↓H␈↓"
␈↓ ↓Hin␈α∞a␈α∞steel␈α∞mill.␈α∞A␈α∞small␈α∞steady␈α∞state␈α∞error␈α
in␈α
speed␈α
will␈α
lead␈α
to␈α
a␈α
build␈α
up␈α
of␈α
steel␈α
at␈α
one␈↓ ↓H␈↓"
␈↓ ↓Hpoint␈α∀in␈α∀the␈α∀process␈α∀resulting␈α∪in␈α∪what␈α∪is␈α∪called␈α∪a␈α∪"cobble"␈α∪where␈α∪the␈α∪steel␈α∪becomes␈↓ ↓H␈↓"
␈↓ ↓Hentangled␈α in␈α the␈α rolling␈α equipment.␈α To␈α avoid␈α this␈α situation,␈α condition␈α 4␈α was␈α formulated.␈↓ ↓H␈↓"
␈↓↓␈↓ ↓H␈↓"
CONDITION 4: IF IE > MIE THEN RETREAT␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ε where IE = absolute value of the integral of the system error␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( MIE = prespecified maximum integral of system error␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(C. ERROR CRITERIA␈↓ α(␈↓!⊃α,␈↓ α(␈↓!∩α,␈↓ α(␈↓!∪α,␈↓ α(␈↓!∀α,␈↓ ∧T␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∪response␈α∪of␈α∪a␈α∪system␈α∪tuned␈α∩with␈α∩the␈α∩tuning␈α∩package␈α∩is␈α∩largely␈α∩a␈α∩function␈α∩of␈α∩the␈↓ ↓H␈↓"
␈↓ ↓Hspecification␈α⊂of␈α⊂system␈α⊂error.␈α⊂If␈α⊂system␈α⊂error␈α⊂is␈α⊂defined␈α⊂to␈α⊂be␈α⊂the␈α⊂integral␈α⊂of␈α∂the␈α∂error␈↓ ↓H␈↓"
␈↓ ↓Hsquared,␈α≥ISE,␈α≥the␈α≥system␈α≥response␈α≥to␈α≥a␈α≥step␈α≥input␈α≥will␈α≥be␈α≥underdamped␈α≥with␈↓ ↓H␈↓"
␈↓ ↓Happroximately␈α∪0.5␈α∪damping.␈α∪When␈α∪integral␈α∪of␈α∪absolute␈α∪error,␈α∪IAE,␈α∪is␈α∪used␈α∩as␈α∩an␈α∩error␈↓ ↓H␈↓"
␈↓ ↓Hcriterion,␈α the␈α damping␈α ratio␈α will␈α be␈α 0.7.␈α (see␈α reference␈α [1])␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIf␈α⊃the␈α⊃system␈α⊃being␈α⊃controlled␈α⊃is␈α⊃an␈α⊃actuator␈α⊃or␈α⊃manipulator,␈α⊃an␈α⊃underdamped␈α⊂response␈↓ ↓H␈↓"
␈↓ ↓Hmay␈α~be␈α~detrimental,␈α~since␈α~the␈α~manipulator␈α~may␈α~overshoot␈α~and␈α~bump␈α~into␈α→objects␈↓ ↓H␈↓"
␈↓ ↓Hsurrounding␈α≠the␈α~manipulator's␈α~path.␈α~An␈α~overdamped␈α~response␈α~can␈α~be␈α~obtained␈α~by␈↓ ↓H␈↓"
␈↓ ↓Hestablishing␈α a␈α high␈α penalty␈α in␈α the␈α error␈α criterion␈α for␈α overshoot.␈α For␈α example:␈↓ ↓H␈↓"
␈↓↓␈↓ ↓H␈↓"
IF ABS[Y(k)] > ABS[R(k)] THEN E = E + ∞␈↓ ↓H␈↓"
␈↓ε␈↓ ↓H␈↓"
where ␈↓↓∞␈↓ε = infinity (10↑10 is a good practical approximation for ␈↓↓∞␈↓ε).␈↓ ↓H␈↓"
␈↓ ε≤page 19␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ε where Y(k) = system output␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( R(k) = system input␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( E = system error.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HAlmost␈α⊃any␈α⊃type␈α⊃of␈α⊃desired␈α⊃response␈α⊂can␈α⊂be␈α⊂obtained␈α⊂by␈α⊂defining␈α⊂an␈α⊂appropriate␈α⊂error␈↓ ↓H␈↓"
␈↓ ↓Hcriterion.␈α∂Detailed␈α∂information␈α∂on␈α∂this␈α∂subject␈α∂can␈α∂be␈α∂obtained␈α∂from␈α∂reference␈α∂[1]␈α∞at␈α∞the␈↓ ↓H␈↓"
␈↓ ↓Hend␈α of␈α this␈α report.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(D. GLOBAL SEARCH CAPABILITY␈↓ α(␈↓!⊃βe␈↓ α(␈↓!∩βe␈↓ α(␈↓!∪βe␈↓ α(␈↓!∀βe␈↓ ε
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HSearch␈α∩routines␈α∩can␈α∩usually␈α∩find␈α∩a␈α⊃local␈α⊃minimum␈α⊃fairly␈α⊃easily.␈α⊃It␈α⊃is␈α⊃much␈α⊃more␈α⊃difficult,␈↓ ↓H␈↓"
␈↓ ↓Hhowever,␈α∩to␈α∩find␈α∩the␈α∩smallest␈α∩or␈α∩the␈α∩global␈α∩minimum.␈α∩Several␈α∩extensive␈α∩methods␈α∩have␈↓ ↓H␈↓"
␈↓ ↓Hbeen␈α∪developed␈α∪for␈α∪finding␈α∪the␈α∩global␈α∩minimum.␈α∩Reference␈α∩[2]␈α∩discusses␈α∩a␈α∩method␈α∩for␈↓ ↓H␈↓"
␈↓ ↓Hremembering␈α∩the␈α⊃parts␈α⊃of␈α⊃an␈α⊃error␈α⊃surface␈α⊃where␈α⊃one␈α⊃has␈α⊃been␈α⊃so␈α⊃that␈α⊃one␈α⊃can␈α⊃later␈↓ ↓H␈↓"
␈↓ ↓Hmake␈α∞more␈α∞intelligent␈α
decisions␈α
when␈α
looking␈α
for␈α
the␈α
global␈α
minimum.␈α
This␈α
technique␈α
might␈↓ ↓H␈↓"
␈↓ ↓Hwork␈α⊃well␈α⊃for␈α⊃systems␈α⊃whose␈α⊃error␈α⊃surfaces␈α⊃change␈α⊃very␈α⊃slowly␈α⊃with␈α⊃time,␈α⊃but␈α⊂this␈α⊂is␈↓ ↓H␈↓"
␈↓ ↓Hoften␈α
not␈α
the␈α
case.␈α
For␈α
example,␈α
a␈α
manipulator␈α
can␈α
pick␈α
up␈α
an␈α
object␈α and␈α instantly␈α change␈↓ ↓H␈↓"
␈↓ ↓Hits␈α parameters␈α and␈α error␈α surfaces.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α∞simplified␈α∞approach␈α∞has␈α∞been␈α∞taken␈α∞in␈α∞this␈α∞project.␈α∞With␈α
the␈α
stability␈α
criteria␈α
discussed␈↓ ↓H␈↓"
␈↓ ↓Hearlier,␈α∞it␈α∞is␈α∞possible␈α∞to␈α∞increase␈α∞the␈α∞variance␈α∞of␈α∞the␈α∞random␈α
number␈α
generator,␈α
try␈α
a␈α
set␈↓ ↓H␈↓"
␈↓ ↓Hof␈α⊗feedback␈α⊗gains,␈α⊗and␈α⊗then␈α⊗quickly␈α⊗retreat␈α⊗if␈α⊗the␈α∃system␈α∃shows␈α∃signs␈α∃of␈α∃becoming␈↓ ↓H␈↓"
␈↓ ↓Hunstable.␈α∞If␈α∞the␈α∞control␈α∞system␈α∞being␈α
tuned␈α
is␈α
controlling␈α
a␈α
chemical␈α
process,␈α
the␈α
resulting␈↓ ↓H␈↓"
␈↓ ↓Hproduct␈α⊃may␈α⊃be␈α⊃poor␈α⊃while␈α⊃the␈α⊃tuning␈α⊃package␈α⊃makes␈α⊃large␈α⊃excursions␈α⊃in␈α⊂the␈α⊂system's␈↓ ↓H␈↓"
␈↓ ↓Hfeedback␈α
gains,␈α
but␈α
the␈α
system␈α
should␈α
remain␈α
stable␈α
as␈α
the␈α
tuning␈α
package␈α searches␈α for␈α a␈↓ ↓H␈↓"
␈↓ ↓Hglobal␈α∂minimum.␈α∂Once␈α∂the␈α∂global␈α∂minimum␈α∂has␈α∂been␈α∂found␈α∞and␈α∞the␈α∞global␈α∞search␈α∞is␈α∞turned␈↓ ↓H␈↓"
␈↓ ↓Hoff,␈α the␈α product␈α should␈α be␈α equal␈α to␈α or␈α superior␈α to␈α that␈α produced␈α before␈α the␈α global␈α search.␈↓ ↓H␈↓"
␈↓ ε≤page 20␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∀basic␈α∀global␈α∀search␈α∀feature␈α∀developed␈α∀for␈α∀this␈α∪project␈α∪iterates␈α∪until␈α∪the␈α∪filtered␈↓ ↓H␈↓"
␈↓ ↓Hchange␈α⊃in␈α⊃the␈α⊃error␈α⊃is␈α⊃less␈α⊃than␈α⊂20%␈α⊂of␈α⊂the␈α⊂error,␈α⊂indicating␈α⊂that␈α⊂the␈α⊂tuning␈α⊂package␈α⊂is␈↓ ↓H␈↓"
␈↓ ↓Hclose␈α
to␈α
a␈α
local␈α minimum.␈α At␈α this␈α point,␈α the␈α tuning␈α package␈α switches␈α from␈α a␈α local␈α mode␈α to␈α a␈↓ ↓H␈↓"
␈↓ ↓Hglobal␈α
mode␈α
and␈α increases␈α the␈α variance␈α of␈α the␈α random␈α number␈α generator␈α by␈α a␈α factor␈α of␈α 50␈↓ ↓H␈↓"
␈↓ ↓Hto␈α⊃enable␈α⊃the␈α⊃package␈α⊃to␈α⊃locate␈α⊃other␈α⊂valleys.␈α⊂When␈α⊂a␈α⊂new␈α⊂error␈α⊂is␈α⊂found␈α⊂that␈α⊂is␈α⊂less␈↓ ↓H␈↓"
␈↓ ↓Hthan␈α
1.25␈α
times␈α
the␈α
previous␈α
error,␈α
the␈α
package␈α switches␈α back␈α to␈α the␈α local␈α mode,␈α reduces␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α⊂variance␈α⊂of␈α⊂the␈α⊂random␈α⊂number␈α⊂generator␈α∂to␈α∂its␈α∂original␈α∂value,␈α∂and␈α∂begins␈α∂searching␈↓ ↓H␈↓"
␈↓ ↓Hfor␈α the␈α minimum␈α point␈α in␈α the␈α new␈α valley.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α∪simulated␈α∪one␈α∪dimensional␈α∪error␈α∪response␈α∪surface␈α∩is␈α∩described␈α∩in␈α∩equation␈α∩2-6␈α∩and␈↓ ↓H␈↓"
␈↓ ↓Hgraphed␈α∂in␈α∂Figure␈α∞2-5.␈α∞The␈α∞global␈α∞minimum␈α∞for␈α∞this␈α∞surface␈α∞is␈α∞approximately␈α∞23.6.␈α∞Figure␈↓ ↓H␈↓"
␈↓ ↓H2-6␈α∂shows␈α∂the␈α∂values␈α∂chosen␈α∂for␈α∂X␈α∂by␈α∂the␈α∂tuning␈α∂package␈α∂as␈α∂it␈α∂searched␈α∂for␈α∞this␈α∞global␈↓ ↓H␈↓"
␈↓ ↓Hminimum.␈α The␈α log␈α of␈α the␈α learning␈α curve␈α for␈α this␈α search␈α is␈α given␈α in␈α Figure␈α 2-7.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
E = 100 + (X-25)␈↓#∂2␈↓# + 100*sin(.4πx) ␈↓ ~(2-6)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∞probably␈α∞of␈α∞finding␈α∞the␈α∞global␈α∞minimum␈α∞can␈α∞be␈α∞increased␈α
by␈α
increasing␈α
the␈α
number␈α
of␈↓ ↓H␈↓"
␈↓ ↓Hiterations␈α⊂over␈α⊂which␈α⊂the␈α⊂tuning␈α∂package␈α∂searches␈α∂with␈α∂the␈α∂larger␈α∂variance.␈α∂There␈α∂is␈α∂no␈↓ ↓H␈↓"
␈↓ ↓Hguarantee,␈α⊃however,␈α⊂that␈α⊂the␈α⊂above␈α⊂procedure␈α⊂will␈α⊂find␈α⊂the␈α⊂global␈α⊂minimum.␈α⊂In␈α⊂fact,␈α⊂the␈↓ ↓H␈↓"
␈↓ ↓Htuning␈α
package␈α
could␈α
find␈α
the␈α
same␈α
minimum␈α
several␈α
times.␈α
If␈α
this␈α
cannot␈α
be␈α
tolerated␈α
and␈↓ ↓H␈↓"
␈↓ ↓Ha␈α∂substantial␈α∂amount␈α∂of␈α∂memory␈α∂is␈α∂available␈α∂for␈α∂use␈α∞by␈α∞a␈α∞global␈α∞search␈α∞routine,␈α∞then␈α∞the␈↓ ↓H␈↓"
␈↓ ↓Htechniques␈α for␈α finding␈α a␈α global␈α minimum␈α described␈α in␈α reference␈α [2]␈α should␈α be␈α considered.␈↓ ↓H␈↓"
␈↓ ε≤page 24␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
III. EXPERIMENTAL RESULTS␈↓ ↓H␈↓!⊃β≠␈↓ ↓H␈↓!∩β≠␈↓ ↓H␈↓!∪β≠␈↓ ↓H␈↓!∀β≠␈↓ ∧c␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
A. DESCRIPTION OF EXPERIMENTAL SYSTEM␈↓ α∧␈↓!⊃∧⎇␈↓ α∧␈↓!∩∧⎇␈↓ α∧␈↓!∪∧⎇␈↓ α∧␈↓!∀∧⎇␈↓ π↓␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HTo␈α
test␈α
the␈α
tuning␈α
package␈α
described␈α
in␈α
Section␈α
III,␈α
a␈α second␈α order␈α analog␈α plant,␈α controlled␈↓ ↓H␈↓"
␈↓ ↓Hby␈α∃a␈α∃digital␈α∀controller␈α∀with␈α∀proportional␈α∀and␈α∀integral␈α∀feedback,␈α∀was␈α∀simulated␈α∀on␈α∀the␈↓ ↓H␈↓"
␈↓ ↓HPDP-10␈α≡computer␈α≡at␈α≡the␈α≡Stanford␈α≡Artificial␈α≡Intelligence␈α≡Laboratory.␈α≡The␈α≥Laplace␈↓ ↓H␈↓"
␈↓ ↓Htransforms␈α⊂of␈α⊂the␈α⊂second␈α⊂order␈α⊂plant␈α⊂and␈α⊂the␈α⊂controller␈α⊂are␈α⊂given␈α∂in␈α∂equations␈α∂3-1␈α∂and␈↓ ↓H␈↓"
␈↓ ↓H3-2.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qplant␈↓#(s) = ␈↓∧b␈↓ε ␈↓ ∧c␈↓!⊃↓R␈↓ ∧c␈↓!∩↓R␈↓ ∧c␈↓!∪↓R␈↓ ∧c␈↓!∀↓R␈↓ ε5 ␈↓ ~(3-1)␈↓ ↓H␈↓"
s␈↓#∂2␈↓# + ␈↓∧a␈↓εs + ␈↓∧b␈↓ε␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qcont␈↓#(s) = k␈↓#qp␈↓#s + k␈↓#qi␈↓#␈↓ ∧Y␈↓!⊃s␈↓ ∧Y␈↓!∩s␈↓ ∧Y␈↓!∪s␈↓ ∧Y␈↓!∀s␈↓ ¬L ␈↓ ~(3-2)␈↓ ↓H␈↓"
s␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where G␈↓#qplant␈↓#(s) = Laplace transform transfer function of plant␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( G␈↓#qcont␈↓#(s) = Laplace transform transfer function of controller␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧b␈↓ε = T␈↓#q1␈↓#*T␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧a␈↓ε = T␈↓#q1␈↓# + T␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( T␈↓#q1␈↓# = first time constant of plant␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( T␈↓#q2␈↓# = second time constant of plant␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( k␈↓#qp␈↓# = proportional feedback gain␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( k␈↓#qi␈↓# = integral feedback gain␈↓ ↓H␈↓"
␈↓ ε≤page 25␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HFor␈α∩the␈α∩simulation,␈α∩the␈α∩combined␈α∩plant-controller␈α⊃was␈α⊃modeled␈α⊃using␈α⊃a␈α⊃zero-order-hold␈↓ ↓H␈↓"
␈↓ ↓Happroximation.␈α∪The␈α∪resulting␈α∪Z␈α∪transform␈α∩transfer␈α∩function␈α∩and␈α∩difference␈α∩equation␈α∩are␈↓ ↓H␈↓"
␈↓ ↓Hgiven␈α in␈α equations␈α 3-3␈α and␈α 3-4.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G␈↓#qdiscrete␈↓#(z) = D␈↓#q1␈↓#z␈↓#∂2␈↓# + D␈↓#q2␈↓#z + D␈↓#q3␈↓# ␈↓ ¬¬␈↓!⊃¬6␈↓ ¬¬␈↓!∩¬6␈↓ ¬¬␈↓!∪¬6␈↓ ¬¬␈↓!∀¬6␈↓
; ␈↓ ~(3-3)␈↓ ↓H␈↓"
z␈↓#∂3␈↓# + (D␈↓#q1␈↓# - ␈↓∧Q␈↓ε - 1)z␈↓#∂2␈↓# + (D␈↓#q2␈↓# + F + ␈↓∧Q␈↓ε)z + (D␈↓#q3␈↓# - F)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
Y(k+3) = -(D␈↓#q1␈↓# - ␈↓∧Q␈↓ε - 1)Y(k+2) - (D␈↓#q2␈↓# + F + ␈↓∧Q␈↓ε)Y(k+1) - (D␈↓#q3␈↓# - F)Y(k)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
+ D␈↓#q1␈↓#R(k+2) + D␈↓#q2␈↓#R(k+1) + D␈↓#q3␈↓#R(k) ␈↓ ~(3-4)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where G␈↓#qdiscrete␈↓#(z) = Z transform transform of combined plant-controller␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Y(k) = plant output␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( R(k) = plant input␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧Q␈↓ε = G + H␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( G = exp(-T␈↓#q1␈↓#T␈↓#qs␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( H = exp(-T␈↓#q2␈↓#T␈↓#qs␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( T␈↓#qs␈↓# = sampling period␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( F = exp(-␈↓∧a␈↓εT␈↓#qs␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧a␈↓ε = T␈↓#q1␈↓# + T␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( D␈↓#q1␈↓# = ␈↓∧g␈↓ε(␈↓∧Q␈↓ε - 1) + k␈↓#qi␈↓#T␈↓#qs␈↓# + J/␈↓∧u␈↓ε␈↓ ↓H␈↓"
␈↓ ε≤page 26␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( D␈↓#q2␈↓# = ␈↓∧g␈↓ε(1 - ␈↓∧Q␈↓ε - F) - k␈↓#qi␈↓#T␈↓#qs␈↓#␈↓∧Q␈↓ε - 2J/␈↓∧u␈↓ε␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( D␈↓#q3␈↓# = F(␈↓∧g␈↓ε + k␈↓#qi␈↓#T␈↓#qs␈↓#) + J/␈↓∧u␈↓ε␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧u␈↓ε = T␈↓#q1␈↓# - T␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( J = QH - PG␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( P = T␈↓#q1␈↓#␈↓∧g␈↓ε - k␈↓#qi␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Q = T␈↓#q2␈↓#␈↓∧g␈↓ε - k␈↓#qi␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧g␈↓ε = (k␈↓#qi␈↓#␈↓∧a␈↓ε)/␈↓∧b␈↓ε - k␈↓#qp␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ␈↓∧b␈↓ε = T␈↓#q1␈↓#*T␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HFor␈α⊂the␈α⊂tests␈α⊂in␈α⊂this␈α⊂Section␈α⊂of␈α⊂the␈α⊂report,␈α⊂the␈α⊂time␈α⊂constants␈α⊂of␈α⊂the␈α⊂plant,␈α∂T␈↓#q1␈↓#␈α∂and␈α∂T␈↓#q2␈↓#,␈↓ ↓H␈↓"
␈↓ ↓Hwere␈α∞set␈α∞to␈α
1.0␈α
and␈α
0.5␈α
seconds␈α
respectively.␈α
Only␈α
the␈α
linear␈α
random␈α
search␈α
was␈α
used␈α
in␈↓ ↓H␈↓"
␈↓ ↓Hthis␈α∂Section␈α∂since␈α∂it␈α∂was␈α∞shown␈α∞in␈α∞Section␈α∞III-A␈α∞that␈α∞the␈α∞linear␈α∞random␈α∞search␈α∞is␈α∞superior␈↓ ↓H␈↓"
␈↓ ↓Hto␈α the␈α random␈α search.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α∂the␈α∂following␈α∂tests,␈α∂it␈α∂was␈α∂assumed␈α∂that␈α∂the␈α∂maximum␈α∂allowable␈α∂output,␈α∂MAO,␈α∞was␈α∞10␈↓ ↓H␈↓"
␈↓ ↓Hvolts.␈α∞For␈α∞a␈α∞manipulator␈α∞system,␈α∞10␈α∞volts␈α∞might␈α∞correspond␈α∞to␈α∞a␈α∞rotation␈α
of␈α
100␈α
degrees,␈↓ ↓H␈↓"
␈↓ ↓Hat␈α
which␈α
point␈α
the␈α
manipulator␈α
would␈α
encounter␈α
a␈α
physical␈α
stop.␈α
To␈α
avoid␈α
exceeding␈α MAO,␈↓ ↓H␈↓"
␈↓ ↓Ha␈α⊃maximum␈α⊃desired␈α⊃output␈α⊃value,␈α⊃MO,␈α⊃was␈α⊃defined␈α⊃and␈α⊃set␈α⊃equal␈α⊃to␈α⊃.5*MAO␈α⊃or␈α⊂5.␈α⊂The␈↓ ↓H␈↓"
␈↓ ↓Hmaximum␈α∞stability␈α∞ratio,␈α
MSR,␈α
was␈α
set␈α
equal␈α
to␈α
3.␈α
Recommendations␈α
concerning␈α
the␈α
choice␈↓ ↓H␈↓"
␈↓ ↓Hof␈α values␈α for␈α these␈α and␈α other␈α stability␈α criterion␈α are␈α given␈α in␈α Section␈α V,␈α Conclusions.␈↓ ↓H␈↓"
␈↓ ε≤page 27␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
B. INTEGRAL SQUARED ERROR CRITERION␈↓ α∧␈↓!⊃∧T␈↓ α∧␈↓!∩∧T␈↓ α∧␈↓!∪∧T␈↓ α∧␈↓!∀∧T␈↓ εX␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∪response␈α∪of␈α∪the␈α∪second␈α∪order␈α∪system␈α∪described␈α∪above␈α∪to␈α∪a␈α∪one␈α∪volt␈α∪step␈α∪input␈↓ ↓H␈↓"
␈↓ ↓Hstarting␈α∞at␈α∞t=0␈α∞is␈α∞shown␈α∞in␈α∞Figure␈α∞3-1a␈α∞for␈α∞initial␈α∞feedback␈α
gains␈α
of␈α
k␈↓#qp␈↓#=5.0␈α
and␈α
k␈↓#qi␈↓#=5.0.␈α
In␈↓ ↓H␈↓"
␈↓ ↓HFigure␈α∩3-1b,␈α∩the␈α∩response␈α∩of␈α∩the␈α∩system␈α⊃is␈α⊃shown␈α⊃after␈α⊃it␈α⊃was␈α⊃tuned␈α⊃with␈α⊃the␈α⊃tuning␈↓ ↓H␈↓"
␈↓ ↓Hpackage␈α
for␈α
150␈α
iterations.␈α
The␈α
final␈α
gains␈α
obtained␈α
for␈α
k␈↓#qp␈↓#␈α
and␈α k␈↓#qi␈↓#␈α were␈α 8.12␈α and␈α 0.82␈α and␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α⊂final␈α⊂integral␈α⊂squared␈α⊂error␈α⊂(ISE)␈α⊂was␈α⊂3.4944.␈α⊂The␈α∂log␈α∂of␈α∂the␈α∂learning␈α∂curve␈α∂for␈α∂the␈↓ ↓H␈↓"
␈↓ ↓Htuning␈α∃of␈α∃this␈α∃system␈α∃is␈α∃shown␈α∃in␈α∃Figure␈α∀3-2.␈α∀It␈α∀will␈α∀subsequently␈α∀be␈α∀shown␈α∀in␈α∀the␈↓ ↓H␈↓"
␈↓ ↓Htheoretical␈α∂analysis␈α∂in␈α∂Section␈α∂IV␈α∞that␈α∞the␈α∞optimum␈α∞values␈α∞for␈α∞the␈α∞feedback␈α∞gains␈α∞for␈α∞this␈↓ ↓H␈↓"
␈↓ ↓Hcase␈α⊃are␈α⊃k␈↓#qp␈↓#=8.13␈α⊃and␈α⊃k␈↓#qi␈↓#=0.87␈α⊃and␈α⊂that␈α⊂the␈α⊂minimum␈α⊂ISE␈α⊂is␈α⊂3.4977.␈α⊂The␈α⊂optimum␈α⊂ISE␈α⊂is␈↓ ↓H␈↓"
␈↓ ↓Hslightly␈α→higher␈α→than␈α→the␈α→experimental␈α_ISE␈α_because␈α_it␈α_is␈α_an␈α_infinite␈α_sum,␈α_while␈α_the␈↓ ↓H␈↓"
␈↓ ↓Hexperimental␈α ISE␈α was␈α summed␈α for␈α only␈α 20␈α seconds.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HWhen␈α
the␈α
tuning␈α
package␈α
is␈α
used␈α
to␈α
tune␈α
mechanical␈α
control␈α
systems,␈α
an␈α
operation␈α can␈α be␈↓ ↓H␈↓"
␈↓ ↓Hperformed,␈α∃the␈α∃system␈α∃can␈α∀be␈α∀stopped␈α∀and␈α∀reset␈α∀to␈α∀its␈α∀original␈α∀state,␈α∀and␈α∀then␈α∀the␈↓ ↓H␈↓"
␈↓ ↓Hoperation␈α_can␈α_be␈α_repeated.␈α_The␈α_tuning␈α_of␈α↔a␈α↔chemical␈α↔process,␈α↔however,␈α↔is␈α↔not␈α↔as␈↓ ↓H␈↓"
␈↓ ↓Hstraightforward␈α→since␈α→the␈α→process␈α→cannot␈α→be␈α→simply␈α_reset,␈α_but␈α_must␈α_be␈α_controlled␈↓ ↓H␈↓"
␈↓ ↓Hcontinuously.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α↔second␈α↔test␈α↔was␈α↔a␈α↔simulation␈α↔of␈α⊗a␈α⊗chemical␈α⊗process␈α⊗which␈α⊗must␈α⊗be␈α⊗controlled␈↓ ↓H␈↓"
␈↓ ↓Hcontinuously.␈α⊂The␈α⊂responses␈α∂of␈α∂the␈α∂system␈α∂as␈α∂it␈α∂was␈α∂being␈α∂tuned␈α∂by␈α∂the␈α∂tuning␈α∂package␈↓ ↓H␈↓"
␈↓ ↓Hare␈α∞given␈α∞in␈α∞Figures␈α∞3-3a␈α
through␈α
3-3e␈α
for␈α
25␈α
iterations␈α
of␈α
the␈α
tuning␈α
package.␈α
It␈α
can␈α
be␈↓ ↓H␈↓"
␈↓ ↓Hseen␈α⊃that,␈α⊃in␈α⊃this␈α⊂case,␈α⊂the␈α⊂control␈α⊂is␈α⊂continuous.␈α⊂That␈α⊂is,␈α⊂if␈α⊂the␈α⊂graphs␈α⊂in␈α⊂Figures␈α⊂3-3a␈↓ ↓H␈↓"
␈↓ ↓Hthrough␈α∂3-3e␈α∂were␈α∂laid␈α∂end␈α∂to␈α∂end,␈α∂they␈α∂would␈α∂form␈α∂a␈α∂continuous␈α∂graph.␈α∂The␈α∂log␈α∂of␈α∞the␈↓ ↓H␈↓"
␈↓ ↓Hlearning␈α curve␈α for␈α this␈α test␈α is␈α displayed␈α in␈α Figure␈α 3-4.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α
input␈α
signal␈α
for␈α
test␈α 2␈α described␈α above␈α was␈α a␈α one␈α volt␈α square␈α wave␈α turned␈α on␈α at␈α t=0␈↓ ↓H␈↓"
␈↓ ↓Hand␈α⊃off␈α⊃at␈α⊃t=10␈α⊃seconds.␈α⊂For␈α⊂this␈α⊂test␈α⊂only,␈α⊂condition␈α⊂3␈α⊂of␈α⊂the␈α⊂stability␈α⊂criterion,␈α⊂which␈↓ ↓H␈↓"
␈↓ ↓Havoids␈α∞the␈α∞use␈α∞of␈α∞negative␈α∞feedback␈α∞gains,␈α
was␈α
purposely␈α
omitted␈α
to␈α
illustrate␈α
the␈α
tuning␈↓ ↓H␈↓"
␈↓ ↓Hpackage's␈α∂ability␈α∂to␈α∂recognize␈α∞unstable␈α∞conditions␈α∞and␈α∞take␈α∞appropriate␈α∞corrective␈α∞action.␈↓ ↓H␈↓"
␈↓ ↓HBecause␈α of␈α this␈α omission,␈α the␈α tuning␈α package␈α chose␈α negative␈α values␈α for␈α the␈α feedback␈↓ ↓H␈↓"
␈↓ ε≤page 37␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓Hgains␈α∞on␈α∞interval␈α∞numbers␈α∞3,␈α∞8,␈α∞and␈α∞12␈α∞of␈α∞Figures␈α∞3-1a,␈α∞b,␈α∞and␈α∞c.␈α∞In␈α∞each␈α∞of␈α∞these␈α
cases,␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α
tuning␈α
package␈α
recognized␈α
the␈α
onset␈α of␈α unstability,␈α retreated␈α to␈α the␈α previous␈α best␈α set␈↓ ↓H␈↓"
␈↓ ↓Hof␈α∩stable␈α∩feedback␈α∩gains,␈α⊃set␈α⊃the␈α⊃input␈α⊃R␈α⊃to␈α⊃zero,␈α⊃and␈α⊃brought␈α⊃the␈α⊃system␈α⊃back␈α⊃under␈↓ ↓H␈↓"
␈↓ ↓Hcontrol␈α before␈α the␈α maximum␈α allowable␈α output␈α of␈α 10␈α volts␈α was␈α exceeded.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α∀iteration␈α∀number␈α∀8,␈α∀for␈α∀example,␈α∀a␈α∀value␈α∀of␈α∪-1.3␈α∪was␈α∪chosen␈α∪for␈α∪the␈α∪proportional␈↓ ↓H␈↓"
␈↓ ↓Hfeedback␈α∞gain␈α∞k␈↓#qp␈↓#.␈α∞After␈α∞3␈α∞seconds,␈α∞the␈α∞search␈α∞package␈α∞recognized␈α∞that␈α∞the␈α∞output␈α
of␈α
the␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α≥had␈α≥exceeded␈α≥the␈α≥prespecified␈α≥maximum␈α≥output,␈α≥MO.␈α≥The␈α≥package␈α≤then␈↓ ↓H␈↓"
␈↓ ↓Himmediately␈α⊃aborted␈α⊃iteration␈α⊃#8,␈α⊃switched␈α⊃to␈α⊃the␈α⊂previous␈α⊂best␈α⊂set␈α⊂of␈α⊂feedback␈α⊂gains,␈↓ ↓H␈↓"
␈↓ ↓Hand␈α∂set␈α∂the␈α∂input␈α∂R␈α∞equal␈α∞to␈α∞zero.␈α∞It␈α∞can␈α∞be␈α∞seen␈α∞that␈α∞on␈α∞the␈α∞following␈α∞iteration,␈α∞#9,␈α∞the␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α was␈α brought␈α back␈α under␈α control␈α without␈α its␈α output␈α ever␈α exceeding␈α 6.5␈α volts.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α
the␈α
third␈α
test,␈α
a␈α
one␈α
volt␈α
square␈α
wave␈α
input␈α
was␈α
also␈α
used,␈α
but␈α in␈α this␈α case␈α the␈α square␈↓ ↓H␈↓"
␈↓ ↓Hwave␈α
was␈α
not␈α
synchronized␈α with␈α the␈α search␈α intervals.␈α Each␈α iteration␈α of␈α the␈α tuning␈α package␈↓ ↓H␈↓"
␈↓ ↓Hlasted␈α
43␈α
seconds,␈α
but␈α
the␈α
period␈α
of␈α
the␈α
square␈α
wave␈α was␈α 40␈α seconds.␈α The␈α values␈α of␈α the␈↓ ↓H␈↓"
␈↓ ↓Hfeedback␈α∂gains␈α∂obtained␈α∞after␈α∞approximately␈α∞150␈α∞iterations␈α∞of␈α∞the␈α∞search␈α∞were␈α∞6.27␈α∞and␈↓ ↓H␈↓"
␈↓ ↓H0.81␈α∞for␈α∞k␈↓#qp␈↓#␈α∞and␈α∞k␈↓#qi␈↓#␈α∞respectively.␈α∞The␈α∞responses␈α∞obtained␈α∞before␈α∞and␈α∞after␈α
the␈α
search␈α
are␈↓ ↓H␈↓"
␈↓ ↓Hgiven␈α in␈α Figures␈α 3-5a␈α and␈α b.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HSince␈α
the␈α
input␈α
signal␈α
for␈α
test␈α
3␈α
is␈α
identical␈α
to␈α
the␈α
one␈α
used␈α
for␈α
test␈α
1,␈α
except␈α that␈α it␈α not␈↓ ↓H␈↓"
␈↓ ↓Hsynchronized␈α
with␈α
the␈α
tuning␈α
package␈α intervals,␈α the␈α tuning␈α package␈α should␈α obtain␈α the␈α same␈↓ ↓H␈↓"
␈↓ ↓Hvalues␈α∞for␈α∞the␈α∞optimum␈α∞feedback␈α∞gains.␈α∞To␈α∞see␈α∞if␈α∞this␈α∞was␈α∞indeed␈α
the␈α
case,␈α
the␈α
search␈α
in␈↓ ↓H␈↓"
␈↓ ↓Htest␈α∃3␈α∃was␈α∃allowed␈α∃to␈α∃run␈α∃for␈α∃an␈α∃additional␈α∃50␈α∃iterations.␈α∃At␈α∀that␈α∀time,␈α∀the␈α∀values␈↓ ↓H␈↓"
␈↓ ↓Hrecommended␈α∞for␈α∞k␈↓#qp␈↓#␈α∞and␈α∞k␈↓#qi␈↓#␈α∞were␈α∞7.94␈α∞and␈α∞0.32,␈α∞so␈α∞it␈α∞can␈α∞be␈α∞seen␈α
that␈α
the␈α
package␈α
was␈↓ ↓H␈↓"
␈↓ ↓Happroaching␈α
the␈α
correct␈α
values.␈α It␈α should␈α be␈α noted,␈α however,␈α that␈α the␈α number␈α of␈α iterations␈↓ ↓H␈↓"
␈↓ ↓Hneeded␈α→to␈α→approach␈α→the␈α→optimum␈α→values␈α→is␈α→greater␈α→when␈α→the␈α→input␈α→signal␈α→is␈α_not␈↓ ↓H␈↓"
␈↓ ↓Hsynchronized␈α with␈α the␈α search␈α intervals.␈↓ ↓H␈↓"
␈↓ ε≤page 41␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α∩the␈α∩fourth␈α∩and␈α∩final␈α∩test␈α∩with␈α⊃this␈α⊃error␈α⊃criteria,␈α⊃the␈α⊃system␈α⊃was␈α⊃excited␈α⊃by␈α⊃a␈α⊃sine␈↓ ↓H␈↓"
␈↓ ↓Hwave␈α∂that␈α∂was␈α∂not␈α∂synchronized␈α∂with␈α∂the␈α∂period␈α∂of␈α∂the␈α∂search␈α∂package.␈α∂Also,␈α∞noise␈α∞was␈↓ ↓H␈↓"
␈↓ ↓Hadded␈α⊃to␈α⊃simulate␈α⊃process␈α⊃noise␈α⊃that␈α⊃would␈α⊃be␈α⊃encountered␈α⊃in␈α⊃a␈α⊃real␈α⊃life␈α⊂situation.␈α⊂An␈↓ ↓H␈↓"
␈↓ ↓Hunexpected␈α∩result␈α∩was␈α∩obtained␈α∩when␈α∩the␈α∩sine␈α∩wave␈α∩input␈α⊃was␈α⊃first␈α⊃used.␈α⊃The␈α⊃tuning␈↓ ↓H␈↓"
␈↓ ↓Hpackage␈α⊂began␈α⊂to␈α⊂minimize␈α⊂the␈α⊂error,␈α⊂as␈α⊂usual,␈α⊂and␈α⊂by␈α⊂the␈α⊂50th␈α⊂interval,␈α∂the␈α∂error␈α∂was␈↓ ↓H␈↓"
␈↓ ↓Hsignificantly␈α∞reduced.␈α∞On␈α∞the␈α∞55th␈α∞interval,␈α∞however,␈α∞the␈α∞error␈α∞suddenly␈α∞blew␈α∞up␈α∞and␈α∞the␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α⊃went␈α⊃out␈α⊃of␈α⊃control.␈α⊃The␈α⊃log␈α⊃of␈α⊃the␈α⊃learning␈α⊃curve␈α⊃for␈α⊃this␈α⊃test␈α⊃is␈α⊂displayed␈α⊂in␈↓ ↓H␈↓"
␈↓ ↓HFigure␈α 3-6.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∀cause␈α∀of␈α∀the␈α∀sudden␈α∀upset␈α∀was␈α∀traced␈α∀to␈α∀the␈α∀fact␈α∪that␈α∪the␈α∪tuning␈α∪package␈α∪had␈↓ ↓H␈↓"
␈↓ ↓Hgradually␈α⊃increased␈α⊃the␈α⊃gain␈α⊃of␈α⊃the␈α⊃feedback␈α⊃gains␈α⊃to␈α⊃the␈α⊃point␈α⊂where␈α⊂they␈α⊂were␈α⊂near␈↓ ↓H␈↓"
␈↓ ↓Hoptimal␈α
for␈α
small␈α
errors␈α
between␈α
the␈α
sine␈α
wave␈α
input␈α
and␈α
the␈α
system's␈α
output.␈α
For␈α
larger␈↓ ↓H␈↓"
␈↓ ↓Herrors,␈α∪however,␈α∪the␈α∪feedback␈α∪gains␈α∪were␈α∪too␈α∪high.␈α∪On␈α∪the␈α∪55th␈α∪iteration,␈α∪the␈α∪tuning␈↓ ↓H␈↓"
␈↓ ↓Hpackage␈α⊗chose␈α⊗a␈α⊗poor␈α⊗set␈α∃of␈α∃feedback␈α∃gains␈α∃which␈α∃caused␈α∃the␈α∃system␈α∃response␈α∃to␈↓ ↓H␈↓"
␈↓ ↓Hdegrade␈α⊂and␈α⊂the␈α⊂error␈α⊂to␈α⊂increase.␈α⊂The␈α⊂tuning␈α∂package␈α∂detected␈α∂the␈α∂sudden␈α∂increase␈α∂in␈↓ ↓H␈↓"
␈↓ ↓Herror␈α∂and␈α∂switched␈α∂back␈α∂to␈α∂what␈α∞it␈α∞thought␈α∞was␈α∞a␈α∞good␈α∞set␈α∞of␈α∞feedback␈α∞gains.␈α∞The␈α∞gains␈↓ ↓H␈↓"
␈↓ ↓Hto␈α which␈α it␈α switched␈α were␈α too␈α high,␈α however,␈α so␈α the␈α system␈α began␈α to␈α oscillate.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α→problem␈α→was␈α→solved␈α_by␈α_adding␈α_derivative␈α_feedback.␈α_Derivative␈α_feedback␈α_adds␈↓ ↓H␈↓"
␈↓ ↓Hdamping␈α⊂to␈α⊂the␈α⊂system␈α⊂and␈α⊂enables␈α⊂it␈α⊂to␈α⊂handle␈α⊂larger␈α⊂errors.␈α⊂To␈α⊂prevent␈α∂this␈α∂situation␈↓ ↓H␈↓"
␈↓ ↓Hfrom␈α∩occuring␈α∩with␈α∩a␈α∩real␈α∩system,␈α∩derivative␈α∩feedback␈α∩should␈α∩always␈α∩be␈α∩included.␈α∩The␈↓ ↓H␈↓"
␈↓ ↓Hresponses␈α∩obtained␈α∩with␈α⊃derivative␈α⊃feedback,␈α⊃before␈α⊃and␈α⊃after␈α⊃tuning,␈α⊃are␈α⊃displayed␈α⊃in␈↓ ↓H␈↓"
␈↓ ↓HFigures␈α 3-7a␈α and␈α b.␈α Learning␈α curves␈α are␈α displayed␈α in␈α Figure␈α 3-8.␈↓ ↓H␈↓"
␈↓ ε≤page 45␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
C. ABSOLUTE ERROR CRITERION␈↓ α∧␈↓!⊃βK␈↓ α∧␈↓!∩βK␈↓ α∧␈↓!∪βK␈↓ α∧␈↓!∀βK␈↓ ¬O␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α⊂this␈α∂Section,␈α∂the␈α∂error␈α∂criterion␈α∂was␈α∂chosen␈α∂so␈α∂that␈α∂the␈α∂resulting␈α∂tuned␈α∂system␈α∂would␈↓ ↓H␈↓"
␈↓ ↓Hhave␈α
an␈α
overdamped␈α
response.␈α
This␈α
type␈α
of␈α
response␈α
would␈α
be␈α
needed␈α
for␈α
the␈α control␈α of␈↓ ↓H␈↓"
␈↓ ↓Ha␈α∂system␈α∂such␈α∂as␈α∂a␈α∂manipulator␈α∂where␈α∂an␈α∂overshoot␈α∂could␈α∂cause␈α∂the␈α∞arm␈α∞to␈α∞collide␈α∞with␈↓ ↓H␈↓"
␈↓ ↓Han␈α object␈α it␈α was␈α approaching.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
The error criteria used for this case was:␈↓ ↓H␈↓"
␈↓↓␈↓ ↓H␈↓"
IF C≤1 THEN ESUM = ESUM + ERROR↑2␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ αH ␈↓ βH IF C>1 THEN ESUM = ESUM + ∞␈↓ ↓H␈↓"
␈↓ε␈↓ ↓H␈↓"
where C = ABS[Y(k)]/ABS[R(k)]␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Y(k) = system output␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( R(k) = system input␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( ESUM = sum of errors␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∂error␈α∂criterion␈α∂defined␈α∂above␈α∂includes␈α∂a␈α∂high␈α∂penalty␈α∞for␈α∞overshoot.␈α∞In␈α∞the␈α∞first␈α∞test␈↓ ↓H␈↓"
␈↓ ↓Hwith␈α∪this␈α∪error␈α∪criterion,␈α∪test␈α∪5,␈α∪the␈α∪input␈α∪was␈α∪a␈α∩square␈α∩wave␈α∩synchronized␈α∩with␈α∩the␈↓ ↓H␈↓"
␈↓ ↓Hperiod␈α
of␈α
the␈α
tuning␈α
package.␈α
The␈α
responses␈α
obtained␈α
before␈α
and␈α after␈α 100␈α iterations␈α are␈↓ ↓H␈↓"
␈↓ ↓Hdisplayed␈α in␈α Figures␈α 3-9a␈α and␈α b.␈α The␈α log␈α of␈α the␈α learning␈α curve␈α is␈α shown␈α in␈α Figure␈α 3-10.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIn␈α→the␈α→second␈α_case,␈α_test␈α_6,␈α_the␈α_system␈α_input␈α_was␈α_a␈α_two␈α_volt␈α_triangle␈α_wave␈α_not␈↓ ↓H␈↓"
␈↓ ↓Hsynchronized␈α∩with␈α∩the␈α∩period␈α∩of␈α∩the␈α∩tuning␈α⊃package.␈α⊃Noise␈α⊃was␈α⊃also␈α⊃added␈α⊃to␈α⊃simulate␈↓ ↓H␈↓"
␈↓ ↓Hprocess␈α∂noise.␈α∂For␈α∂this␈α∂test,␈α∂the␈α∂maximum␈α∂allowable␈α∂output␈α∂was␈α∂changed␈α∂from␈α∂+10␈α∞volts␈↓ ↓H␈↓"
␈↓ ↓Hto␈α∞+␈α∞or␈α∞-␈α∞10␈α∞volts.␈α∞The␈α∞responses␈α∞obtained␈α∞before␈α
and␈α
after␈α
100␈α
iterations␈α
of␈α
the␈α
tuning␈↓ ↓H␈↓"
␈↓ ↓Hpackage␈α are␈α given␈α in␈α Figures␈α 3-11a␈α and␈α b␈α and␈α the␈α log␈α of␈α the␈α learning␈α curve␈α is␈α displayed␈α in␈↓ ↓H␈↓"
␈↓ ↓HFigure␈α
3-12.␈α
Notice␈α
that␈α
the␈α
search␈α
interval␈α
was␈α
lengthened␈α
in␈α
this␈α
case.␈α Best␈α results␈α are␈↓ ↓H␈↓"
␈↓ ↓Hobtained␈α⊃when␈α⊃the␈α⊃search␈α⊃interval␈α⊃is␈α⊃several␈α⊂times␈α⊂longer␈α⊂than␈α⊂the␈α⊂period␈α⊂of␈α⊂the␈α⊂input␈↓ ↓H␈↓"
␈↓ ↓Hsignal.␈↓ ↓H␈↓"
␈↓ ε≤page 52␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
IV. THEORETICAL ANALYSIS␈↓ ↓H␈↓!⊃β⊗␈↓ ↓H␈↓!∩β⊗␈↓ ↓H␈↓!∪β⊗␈↓ ↓H␈↓!∀β⊗␈↓ ∧↑␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HA␈α∀discussion␈α∀of␈α∀this␈α∀tuning␈α∪package␈α∪would␈α∪not␈α∪be␈α∪complete␈α∪without␈α∪showing␈α∪that␈α∪the␈↓ ↓H␈↓"
␈↓ ↓Hvalues␈α⊂obtained␈α⊂by␈α⊂the␈α⊂tuning␈α⊂package␈α⊂closely␈α⊂approximate␈α⊂the␈α⊂optimum␈α⊂feedback␈α⊂gains␈↓ ↓H␈↓"
␈↓ ↓Hfor␈α∩the␈α∩system␈α∩in␈α∩question.␈α∩In␈α∩this␈α∩Section,␈α∩test␈α∩1␈α∩in␈α∩Section␈α⊃III-B␈α⊃will␈α⊃be␈α⊃investigated␈↓ ↓H␈↓"
␈↓ ↓Hanalytically␈α⊃to␈α⊃determine␈α⊃if␈α⊃the␈α⊃tuning␈α⊂package␈α⊂converged␈α⊂on␈α⊂the␈α⊂proper␈α⊂values␈α⊂for␈α⊂the␈↓ ↓H␈↓"
␈↓ ↓Hfeedback␈α↔gains.␈α↔In␈α↔this␈α↔case␈α↔the␈α↔system␈α↔was␈α↔excited␈α↔by␈α↔a␈α↔square␈α↔wave␈α⊗that␈α⊗was␈↓ ↓H␈↓"
␈↓ ↓Hsynchronized␈α∞with␈α∞the␈α∞period␈α∞of␈α∞the␈α∞tuning␈α∞package.␈α∞The␈α∞derivation␈α∞of␈α∞the␈α∞Z␈α∞transform␈α∞of␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α error␈α for␈α this␈α case␈α is␈α given␈α below.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
R(z) = z ␈↓ ∧≠␈↓!⊃?␈↓ ∧≠␈↓!∩?␈↓ ∧≠␈↓!∪?␈↓ ∧≠␈↓!∀?␈↓ ∧Z ␈↓ ~(4-1)␈↓ ↓H␈↓"
z-1␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
G(z) = D␈↓#q1␈↓#z␈↓#∂2␈↓# + D␈↓#q2␈↓#z + D␈↓#q3␈↓# ␈↓ ∧≥␈↓!⊃β"␈↓ ∧≥␈↓!∩β"␈↓ ∧≥␈↓!∪β"␈↓ ∧≥␈↓!∀β"␈↓ π? ␈↓ ~(4-2)␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ␈↓ βh z␈↓#∂3␈↓# - (␈↓∧Q␈↓ε+1)z␈↓#∂2␈↓# + (F + ␈↓∧Q␈↓ε)z - F␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
E(z) = R(z) ␈↓ ∧~␈↓!⊃t␈↓ ∧~␈↓!∩t␈↓ ∧~␈↓!∪t␈↓ ∧~␈↓!∀t␈↓ ¬∞ ␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ␈↓ βh 1 + G(z)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
= z␈↓#∂3␈↓# - ␈↓∧Q␈↓εz␈↓#∂2␈↓# + Fz ␈↓ ∧∪␈↓!⊃¬,␈↓ ∧∪␈↓!∩¬,␈↓ ∧∪␈↓!∪¬,␈↓ ∧∪␈↓!∀¬,␈↓ ? ␈↓ ~(4-3)␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ␈↓ βh z␈↓#∂3␈↓# + (D␈↓#q1␈↓# - ␈↓∧Q␈↓ε - 1)z␈↓#∂2␈↓# + (D␈↓#q2␈↓# + F + ␈↓∧Q␈↓ε)z + (D␈↓#q3␈↓# - F)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where R(z) = system input␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( G(z) = transfer function of combined plant and controller␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( E(z) = system error␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
and D␈↓#q1␈↓#, D␈↓#q2␈↓#, D␈↓#q3␈↓#, ␈↓∧Q␈↓ε, and F are defined on pages 25 and 26␈↓ ↓H␈↓"
␈↓ ε≤page 53␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIf␈α∞it␈α∞is␈α∞assumed␈α∞that␈α∞the␈α∞error␈α∞function␈α∞approaches␈α∞zero␈α
before␈α
the␈α
end␈α
of␈α
search␈α
period,␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α∃sum␈α∃of␈α∃the␈α∃errors␈α∃can␈α∃be␈α∀approximated␈α∀by␈α∀using␈α∀the␈α∀Z␈α∀transform␈α∀operation␈α∀for␈↓ ↓H␈↓"
␈↓ ↓Hsummation␈α∂and␈α∂then␈α∂applying␈α∂the␈α∞final␈α∞value␈α∞theorem.␈α∞The␈α∞resulting␈α∞sum␈α∞of␈α∞errors,␈α∞ESUM,␈↓ ↓H␈↓"
␈↓ ↓His␈α given␈α in␈α equation␈α 4-4.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
ESUM = ␈↓∧S␈↓ε error(k)␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ␈↓ βh = a␈↓#qo␈↓#B␈↓#qo␈↓#Q␈↓#qo␈↓# - a␈↓#qo␈↓#B␈↓#q1␈↓#Q␈↓#q1␈↓# + a␈↓#qo␈↓#B␈↓#q2␈↓#Q␈↓#q2␈↓# - B␈↓#q3␈↓#Q␈↓#q3␈↓# ␈↓ ∧+␈↓!⊃ε%␈↓ ∧+␈↓!∩ε%␈↓ ∧+␈↓!∪ε%␈↓ ∧+␈↓!∀ε%␈↓
P ␈↓ ~(4-4)␈↓ ↓H␈↓"
␈↓ βλ␈↓ βh [(a␈↓#qo␈↓#␈↓#∂2␈↓# - a␈↓#q3␈↓#␈↓#∂2␈↓#)Q␈↓#qo␈↓# - (a␈↓#qo␈↓#a␈↓#q1␈↓# - a␈↓#q2␈↓#a␈↓#q3␈↓#)Q␈↓#q1␈↓# + (a␈↓#qo␈↓#a␈↓#q2␈↓# - a␈↓#q1␈↓#a␈↓#q3␈↓#)Q␈↓#q2␈↓#]a␈↓#qo␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
where B␈↓#qo␈↓# = b␈↓#qo␈↓#␈↓#∂2␈↓# + b␈↓#q1␈↓#␈↓#∂2␈↓# + b␈↓#q2␈↓#␈↓#∂2␈↓# + b␈↓#q3␈↓#␈↓#∂2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( B␈↓#q1␈↓# = 2(b␈↓#qo␈↓#b␈↓#q1␈↓# + b␈↓#q1␈↓#b␈↓#q2␈↓# + b␈↓#q2␈↓#b␈↓#q3␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( B␈↓#q2␈↓# = 2(b␈↓#qo␈↓#b␈↓#q2␈↓# + b␈↓#q1␈↓#b␈↓#q3␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( B␈↓#q3␈↓# = 2b␈↓#qo␈↓#b␈↓#q3␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Q␈↓#qo␈↓# = a␈↓#qo␈↓#e␈↓#q1␈↓# - a␈↓#q3␈↓#e␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Q␈↓#q1␈↓# = a␈↓#qo␈↓#a␈↓#q1␈↓# - a␈↓#q2␈↓#a␈↓#q3␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Q␈↓#q2␈↓# = a␈↓#q1␈↓#e␈↓#q2␈↓# - a␈↓#q2␈↓#e␈↓#q1␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( Q␈↓#q3␈↓# = (a␈↓#q1␈↓# - a␈↓#q3␈↓#)(e␈↓#q2␈↓#␈↓#∂2␈↓# - e␈↓#q1␈↓#␈↓#∂2␈↓#) + a␈↓#qo␈↓#(a␈↓#qo␈↓#e␈↓#q2␈↓# - a␈↓#q3␈↓#e␈↓#q1␈↓#)␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( e␈↓#q1␈↓# = a␈↓#qo␈↓# + a␈↓#q2␈↓#␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( e␈↓#q2␈↓# = a␈↓#q1␈↓# + a␈↓#q3␈↓#␈↓ ↓H␈↓"
␈↓ ε≤page 54␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( a␈↓#qo␈↓# = 1␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( a␈↓#q1␈↓# = D␈↓#q1␈↓# - ␈↓∧Q␈↓ε - 1␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( a␈↓#q2␈↓# = D␈↓#q2␈↓# + F + ␈↓∧Q␈↓ε␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( a␈↓#q3␈↓# = D␈↓#q3␈↓# - F␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( b␈↓#qo␈↓# = 1␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( b␈↓#q1␈↓# = -␈↓∧Q␈↓ε␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( b␈↓#q2␈↓# = F␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( b␈↓#q3␈↓# = 0␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α( and D␈↓#q1␈↓#, D␈↓#q2␈↓#, D␈↓#q3␈↓#, ␈↓∧Q␈↓ε, and F are defined on pages 25 and 26␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α∞optimum␈α∞values␈α∞of␈α∞the␈α∞feedback␈α∞gains␈α∞k␈↓#qp␈↓#␈α∞and␈α∞k␈↓#qi␈↓#␈α∞were␈α∞determined␈α∞by␈α
using␈α
a␈α
random␈↓ ↓H␈↓"
␈↓ ↓Hsearch␈α∩routine␈α∩to␈α∩minimize␈α∩equation␈α∩4-4.␈α∩The␈α∩values␈α∩obtained␈α∩for␈α⊃k␈↓#qp␈↓#␈α⊃and␈α⊃k␈↓#qi␈↓#␈α⊃are␈α⊃given␈↓ ↓H␈↓"
␈↓ ↓Hbelow:␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ␈↓ βhk␈↓#qp␈↓# = 8.13␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ␈↓ βhk␈↓#qi␈↓# = 0.87␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ ESUM = 3.4977␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThese␈α∂values␈α∂are␈α∂very␈α∂close␈α∂to␈α∂those␈α∞obtained␈α∞experimentally␈α∞by␈α∞the␈α∞tuning␈α∞in␈α∞test␈α∞1␈α∞in␈↓ ↓H␈↓"
␈↓ ↓HSection␈α∞III.␈α∞The␈α∞values␈α∞obtained␈α∞experimentally␈α∞for␈α∞k␈↓#qp␈↓#␈α∞and␈α∞k␈↓#qi␈↓#␈α∞were␈α∞8.12␈α
and␈α
0.82␈α
and␈α
the␈↓ ↓H␈↓"
␈↓ ↓Hexperimental␈α ISE␈α was␈α 3.4944.␈↓ ↓H␈↓"
␈↓ ε≤page 55␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
V. CONCLUSIONS␈↓ ↓H␈↓!⊃↓z␈↓ ↓H␈↓!∩↓z␈↓ ↓H␈↓!∪↓z␈↓ ↓H␈↓!∀↓z␈↓ βB␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HThe␈α
effectiveness␈α
of␈α
the␈α
tuning␈α
package␈α has␈α been␈α illustrated␈α in␈α a␈α variety␈α of␈α circumstances␈↓ ↓H␈↓"
␈↓ ↓Hin␈α
Section␈α
III.␈α
I␈α
believe␈α
that␈α
this␈α
package␈α
has␈α
been␈α
developed␈α to␈α the␈α point␈α where␈α it␈α can␈α be␈↓ ↓H␈↓"
␈↓ ↓Hused␈α successfully␈α on␈α actual␈α systems.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HIt␈α∩is␈α∩possible␈α∩to␈α∩optimize␈α∩a␈α∩system␈α∩for␈α∩a␈α∩given␈α∩input,␈α⊃but␈α⊃it␈α⊃is␈α⊃not␈α⊃possible␈α⊃to␈α⊃tune␈α⊃a␈↓ ↓H␈↓"
␈↓ ↓Hsystem␈α⊃that␈α⊃will␈α⊃have␈α⊃a␈α⊃minimum␈α⊃error␈α⊃for␈α⊃all␈α⊃types␈α⊃of␈α⊃inputs.␈α⊃Thus␈α⊃a␈α⊂system␈α⊂that␈α⊂has␈↓ ↓H␈↓"
␈↓ ↓Hbeen␈α∃tuned␈α∃for␈α∃a␈α∃ramp␈α∀input␈α∀will␈α∀have␈α∀a␈α∀large␈α∀error␈α∀(because␈α∀of␈α∀overshoots)␈α∀when␈↓ ↓H␈↓"
␈↓ ↓Hsubjected␈α∂to␈α∂a␈α∂step␈α∂input.␈α∂In␈α∂view␈α∂of␈α∂this␈α∂fact,␈α∂there␈α∂are␈α∞several␈α∞ways␈α∞a␈α∞system␈α∞can␈α∞be␈↓ ↓H␈↓"
␈↓ ↓Htuned:␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(o If the event being tuned is short, the entire event should␈↓ ↓H␈↓"
␈↓ α( be included as one tuning interval.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(o If the event is long, the system should be tuned for a representative␈↓ ↓H␈↓"
␈↓ α( subevent. This subevent should include the more strenuous portions␈↓ ↓H␈↓"
␈↓ α( of the event.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(o If the capability to change feedback gains during an event␈↓ ↓H␈↓"
␈↓ α( is available, the event can be broken into subevents which can␈↓ ↓H␈↓"
␈↓ α( each be tuned individually. ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓HLike␈α∂any␈α∂other␈α∂type␈α∂of␈α∂control,␈α∂engineering␈α∂judgment␈α∂must␈α∂be␈α∂exercised␈α∂when␈α∂using␈α∂this␈↓ ↓H␈↓"
␈↓ ↓Hpackage.␈α∂It␈α∂is␈α∂possible␈α∂for␈α∞a␈α∞system␈α∞to␈α∞become␈α∞unstable␈α∞and␈α∞go␈α∞out␈α∞of␈α∞control␈α∞while␈α∞it␈α∞is␈↓ ↓H␈↓"
␈↓ ↓Hbeing␈α∂tuned␈α∞with␈α∞this␈α∞package␈α∞if␈α∞the␈α∞constants␈α∞for␈α∞this␈α∞package␈α∞are␈α∞not␈α∞chosen␈α∞correctly.␈↓ ↓H␈↓"
␈↓ ↓HThis␈α
should␈α
not␈α
happen,␈α
however,␈α
if␈α the␈α guidelines␈α given␈α on␈α the␈α next␈α page␈α are␈α followed.␈α In␈↓ ↓H␈↓"
␈↓ ↓Hgeneral,␈α∩conservative␈α∩values␈α⊃should␈α⊃be␈α⊃chosen␈α⊃for␈α⊃the␈α⊃constants␈α⊃specified␈α⊃below␈α⊃when␈↓ ↓H␈↓"
␈↓ ↓Hthe␈α∞tuning␈α∞package␈α∞is␈α∞first␈α∞used␈α∞with␈α∞a␈α∞system.␈α∞Conservative␈α∞values␈α∞increase␈α
the␈α
number␈↓ ↓H␈↓"
␈↓ ↓Hof␈α∞iterations␈α∞required␈α∞by␈α∞the␈α∞tuning␈α∞package␈α∞to␈α∞converge␈α∞on␈α∞the␈α∞optimum␈α
feedback␈α
gains,␈↓ ↓H␈↓"
␈↓ ↓Hbut␈α∩they␈α∩also␈α∩minimize␈α∩the␈α∩chances␈α∩of␈α∩a␈α∩system␈α∩going␈α∩out␈α∩of␈α⊃control.␈α⊃After␈α⊃the␈α⊃tuning␈↓ ↓H␈↓"
␈↓ ↓Hpackage␈α has␈α improved␈α a␈α system's␈α response,␈α the␈α values␈α of␈α the␈α constants␈α can␈α be␈α increased.␈↓ ↓H␈↓"
␈↓ ε≤page 56␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
Guidelines for Using This Tuning Package␈↓ α@␈↓!⊃∧L␈↓ α@␈↓!∩∧L␈↓ α@␈↓!∪∧L␈↓ α@␈↓!∀∧L␈↓ π ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ1. Determine the maximum allowable output value, MAO. This␈↓ ↓H␈↓"
is the output value at which actuators saturate or manipulators␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ bump into their stops.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ2. Choose a value for the maximum desired output value, MO. ␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ The first time the tuning package is used, a conservative␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ value of .1*MAO should be used. After the system has␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ been tuned and its response has improved, MO can be␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ increased to between .5 or .7*MAO.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ3. Initially choose a value of 2 for the maximum stability␈↓ ↓H␈↓"
␈↓ α( ratio, MSR. This value can later be decreased to .5 for ␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ additional safety, or if the system is well behaved, it␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ can be increased to a maximum of 10 to speed up the rate␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ of convergence of the tuning package.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ4. If a small steady state error is acceptable, the␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ maximum integral error should be set to a high value, such ␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ as 10↑10, to deactivate this feature. Otherwise,␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ a suitable value for MIE should be chosen by starting␈↓ ↓H␈↓"
␈↓ α( ␈↓ βλ with a value that will guarantee safety, such as 1. This␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ value can then be increased gradually as it becomes␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ apparent what level of steady state error is acceptable.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ5. Choose a value for the variance of the search package.␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ A good starting value is between 1 and 5% of the smallest␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ initial feedback gain. If this value proves to be␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ too small, the variance can be increased. One word of␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ caution, however: if the constants MO, MSR, or MIE have␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ been increased while using a particular variance, they ␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ should be reset to their former values before the variance␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ is increased.␈↓ ↓H␈↓"
␈↓ ε≤page 57␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ6. Choose a value for the time constant, T, of the digital␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ filter used in the computation of the stability ratio.␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ T should fall between 1/10 and 1/2 of the largest plant␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ time constant.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ7. All three of the common modes of feedback should be␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ employed: proportional, derivative, and integral.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ8. Unless the event being tuned is extremely short, the␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ tuning package will work best when the tuning intervals␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ are long, especially when the input in not synchronized␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ with the intervals of the tuning package. Long tuning␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ intervals, however, lengthen the time required by the␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ tuning package to approach optimum feedback gains, so a␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ compromise must be made.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλAdvantages of This Tuning Package␈↓ βλ␈↓!⊃β{␈↓ βλ␈↓!∩β{␈↓ βλ␈↓!∪β{␈↓ βλ␈↓!∀β{␈↓ πβ␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ1. This package can be used without a detailed knowledge␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ of control theory or of the dynamics of the process␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ being tuned. The only information that is required␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ is a stable set of initial feedback gains and an␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ estimate of the largest system time constant.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ2. This package can be used to determine which types of ␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ feedback are best suited for a particular system. For␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ example, in addition to the proportional, derivative, and␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ integral feedback, feedback terms can be added which␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ are proportional to the error cubed, to the second␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ derivative of the error, etc. After the system has␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ been tuned, the relative importance of each of the terms␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ can be determined by looking at the gains obtained for␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ each term.␈↓ ↓H␈↓"
␈↓ ε≤page 58␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ3. If feedback gains can be changed during a motion or an␈↓ ↓H␈↓"
␈↓ α( event, the package can be set up to break up a motion into␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ segments and then tune each segment individually. The resulting␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ system response should be far superior to that which would␈↓ ↓H␈↓"
␈↓ α(␈↓ βλ be obtained from a single set of feedback gains.␈↓ ↓H␈↓"
␈↓ ε≤page 59␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
VI. REFERENCES␈↓ ↓H␈↓!⊃↓t␈↓ ↓H␈↓!∩↓t␈↓ ↓H␈↓!∪↓t␈↓ ↓H␈↓!∀↓t␈↓ β<␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[1] D. Graham and R.C. Lathrop, "The Synthesis of Optimum Transient Response:␈↓ ↓H␈↓"
Criteria and Standard Forms," Trans AIEE, Vol 72, pt. II, 1953, pp. 273-288.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[2] R.A. Jarvis, "Adaptive Global Search by the Process of Competitive Evaluation,"␈↓ ↓H␈↓"
IEEE Trans on Systems, Man, and Cybernetics, Vol SMC-5, No. 3, May 1975.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[3] M.D. Roderick, "The Automatic Tuning of Controllers Using Search Techniques,"␈↓ ↓H␈↓"
Masters Thesis, University of Pennsylvania, August 1974.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[4] K. Ogata, "Modern Control Engineering,", Prentice-Hall, Englewood Cliffs,␈↓ ↓H␈↓"
N.J., 1970.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[5] J.A. Cadzow, "Discrete Time Systems," Prentice-Hall, Englewood Cliffs, N.J.,␈↓ ↓H␈↓"
1973.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[6] J.T. Tou, "Digital and Sampled-data Control Systems," McGraw-Hill, New York,␈↓ ↓H␈↓"
1959, pp. 518-525.␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
␈↓ ↓H␈↓"
[7] J.A. Cadzow and H.R. Martins, "Discrete-Time and Computer Control Systems,"␈↓ ↓H␈↓"
Prentice-Hall, Englewood Cliffs, N.J., 1970.␈↓ ↓H␈↓"