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ControlSystemsDesign Part: Optimisation
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
2007
Control Process Optimisation Optimisation Process
control loop structure design
optimum criteria selection
optimum control parameters computation
control process simulation
control parameters refinement
control process quality evaluation
documentation production
...SAT
Control Process Optimisation
control process qualitycontrol process stability
steady state - process variable deviation
dynamic control process overshooting
time of control process treg
integral criteria f(dev)
non oscillation control processes
Control Process Optimisation control process stability
characteristic polynomial
characteristic polynomial roots negative part of complex roots!
degree of the stability
critical parameters single control loop with P controller
critical GAIN
critical period Tkr
Control Process Optimisation
steady state - process variable deviation should be = 0; Deviation = Set Point - Process
the P controller problem: GAIN has to be as large as possible; (!) stability violation for higher order systems
else process deviation = 0 --- I part of controller; destabilisation of control loop
stability versus quality - solution is compromise
Control Process Optimisation
Dynamic control process optimisation standard forms of a characteristic polynomial
Ziegler Nichols method
method of optimum module
methods of integral criterions
Dynamic control process optimisationstandard forms of a characteristic polynomial
Naslin form of characteristic polynomial
0*0
n
i
ii sa
1 12
. . i i ia a a
is according the
i = 1,2, .... n-1
Dynamic control process optimisationstandard forms of a characteristic polynomial
Naslin form of characteristic polynomial
The parameter depends on the chosen overshooting
xmax according the table:
1.7 1.75 1.8 1.9 2.0
xmax 20 16 12 8 5
Dynamic control process optimisationstandard forms of a characteristic polynomial
Graham - Lathrop form
n Characteristic polynomial q
1 1q
2 1q.4,1q2
3 1q.15,2q.75,1q 23
4 1q.7,2q.4,3q.1,2q 234
5 1q.4,3q.5,5q.0,5q.8,2q 2345
0ω
s
Dynamic control process optimisationZiegler Nichols method
input data:
GAINcr - critical gain
Tcr - critical period
measured or computed at the stability boundary of the single control loop with P - controller
Dynamic control process optimisationZiegler Nichols method
Cont
roller
Para
metersValues
P Kr 0,5 . Kkr
Kr 0,45 . KkrPI
Ti 0,85 . Tkr
Kr 0,6 . Kkr
Ti 0,5 . TkrPID
Td 0,12 . Tkr
Dynamic control process optimisationmethod of optimum module
The transfer function of a controlled system is supposed in the form
The control parameters are for the ideal parallel PID algorithm r0, r-1 and r1:
i
n
ii
s
sa
KsF
.0
0r
KK sr
0
1
r
rTd
1
0
r
rTi
Dynamic control process optimisationmethod of optimum module
PI controller
2021
20
0
1
23
01
..2.5,0.
aaa
a
r
r
aa
aa
Dynamic control process optimisationmethod of optimum module
PID controller
401122
2021
20
1
0
1
345
123
01
..2..2
..2.5,0.
0
aaaaa
aaa
a
r
r
r
aaa
aaa
aa
Dynamic control process optimisationmethods of integral criterions
IAE Integral of Absolute Error
ITAE Integral of Absolute Error multiplied by Time
Dynamic system approximation
by K, T and D:
DsesT
K
.
1.
Dynamic control process optimisationmethods of integral criterions IAE:
A B
P 0,758 -0,861PI
I 1,020 -0,323
P 1,086 -0,869
I 0,740 -1,130
IAE
PID
D 0,348 0,914
Dynamic control process optimisationmethods of integral criterions ITAE:
A B
P 0,586 -0,916PI
I 1,030 -0,165
P 0,965 -0,865
I 0,796 -0,147
ITAE
PID
D 0,308 0,929
Dynamic control process optimisationmethods of integral criterions
B
T
DAY
.
rs KKY .
T
DBA
T
T
i
.
B
d
T
DA
T
T
.
For GAIN
For time constants
For Ti
For Td
Dynamic control process optimisationhalf dumping criterion
Dynamic control process optimisation half dumping criterion
Dynamic system approximation
by K, T and D:
DsesT
K
.
1.
Control Process Optimisation half dumping criterion
PI controller:
946,0
.928,0
T
DY
rs KKY .
583,0
.928,0
T
D
T
Ti
Auxiliary parameter
For GAIN
For integral time constant Ti
Control Process Optimisation half dumping criterion
PID controller:
Auxiliary parameter
95,0
.37,1
T
DY
rs KKY .738,0
.74,0
T
D
T
Ti
95,0
.365,0
T
D
T
Td
For GAIN
For integral time constant Ti
For derivative time constant Td