characterize all PID controllers for pole placement in a desired region - MapleSim Help

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ControlDesign[Characterize] - characterize all PID controllers for pole placement in a desired region

Calling Sequence

Characterize(sys, zeta, omegan, opts)

Parameters

sys

-

System; a DynamicSystems system object in continuous-time domain; must be single-input single-output (SISO)

zeta

-

realcons; the specified damping

omegan

-

realcons; the specified natural frequency

opts

-

(optional) equation(s) of the form option = value; specify options for the Characterize command

Description

• 

The Characterize command characterizes all PID controllers for pole placement in a desired region. It returns a Boolean expression of inequalities in terms of the controller parameters that must be satisfied in order to place the closed-loop poles (under unity negative feedback) in the specified desired region. The desired region is specified by zeta and omegan and is defined based on relative stability and damping conditions as follows:

– 

Relative Stability: The desired region is part of the complex left half plane (LHP) with real part less than ζomegan. This is equivalent to the relative stability of the closed-loop system with respect to the line s=jωζomegan (rather than the imaginary axis). Clearly, if zeta or omegan are set to zero, the relative stability reduces to the absolute stability with respect to the imaginary axis.

– 

Damping: The desired region is part of the complex left half plane (LHP) inside the angle +/-arccosζ measured from the negative real axis.

• 

The controller parameters are kc for a P controller, kc,ki for a PI controller, and kc,ki,kd for a PID controller, where kc is the proportional gain, ki is the integral gain, and kd is the derivative gain. The controller transfer function is then obtained as: Cs=kc, Cs=kc+kis, or Cs=kc+kis+kds for the P, PI, and PID controllers, respectively.

Examples

withControlDesign:

sysDynamicSystems:-NewSystems+2s3+12s2+17s+2

sys:=Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1s

(1)

Characterizesys,13,2,controller=P,output=relativestability

0<9kc29and0<117kc&plus;373

(2)

Characterizesys&comma;13&comma;2&comma;controller&equals;P&comma;output&equals;damping

0<115kc&plus;5453and0<31kc&plus;8955445kc2342kc&plus;13437115kc&plus;5453and0<5kc238kc&plus;149319115kc&plus;54539kc2&plus;162kc&plus;15354216kc2&plus;432kc&plus;216115kc&plus;545331kc&plus;8955445kc2342kc&plus;13437115kc&plus;5453and0<3kc2&plus;54kc&plus;5118216kc2&plus;432kc&plus;216115kc&plus;54531331kc&plus;8955445kc2342kc&plus;13437115kc&plus;545324kc2&plus;48kc&plus;245kc238kc&plus;149319115kc&plus;54539kc2&plus;162kc&plus;15354216kc2&plus;432kc&plus;216115kc&plus;545331kc&plus;8955445kc2342kc&plus;13437115kc&plus;5453and0<kc2&plus;2kc&plus;1

(3)

Characterizesys&comma;13&comma;2&comma;controller&equals;P

0<9kc29and0<117kc&plus;373and0<115kc&plus;5453and0<31kc&plus;8955445kc2342kc&plus;13437115kc&plus;5453and0<5kc238kc&plus;149319115kc&plus;54539kc2&plus;162kc&plus;15354216kc2&plus;432kc&plus;216115kc&plus;545331kc&plus;8955445kc2342kc&plus;13437115kc&plus;5453and0<3kc2&plus;54kc&plus;5118216kc2&plus;432kc&plus;216115kc&plus;54531331kc&plus;8955445kc2342kc&plus;13437115kc&plus;545324kc2&plus;48kc&plus;245kc238kc&plus;149319115kc&plus;54539kc2&plus;162kc&plus;15354216kc2&plus;432kc&plus;216115kc&plus;545331kc&plus;8955445kc2342kc&plus;13437115kc&plus;5453and0<kc2&plus;2kc&plus;1

(4)

Characterizesys&comma;13&comma;2&comma;controller&equals;&Pi;&comma;output&equals;relativestability

0<18kc&plus;27ki&plus;58and0<2106kc8406243kiand0<18kc&plus;27ki1582522016kc&plus;3024ki&plus;64962106kc8406243ki

(5)

Characterizesys&comma;13&comma;2&comma;controller&equals;&Pi;

0<18kc&plus;27ki&plus;58and0<2106kc8406243kiand0<18kc&plus;27ki1582522016kc&plus;3024ki&plus;64962106kc8406243kiand0<ki2and0<690kc&plus;32718&plus;69kiand0<62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69kiand0<270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki18kc2&plus;9kcki&plus;324kc399ki&plus;3061081296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69kiand0<2kcki&plus;ki2&plus;2ki7776ki2690kc&plus;32718&plus;69ki7262kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69kiki2270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki18kc2&plus;9kcki&plus;324kc399ki&plus;3061081296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki366kc2&plus;3kcki&plus;108kc133ki&plus;102361296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki1362kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki1296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki36kcki&plus;18ki2&plus;36ki139968ki2690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki18kc2&plus;9kcki&plus;324kc399ki&plus;3061081296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69kiki2216kc2&plus;30kcki&plus;45ki2&plus;432kc2658ki&plus;21616690kc&plus;32718&plus;69ki36kcki&plus;18ki2&plus;36ki139968ki2690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki16270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki18kc2&plus;9kcki&plus;324kc399ki&plus;3061081296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki12kcki&plus;6ki2&plus;12ki46656ki2690kc&plus;32718&plus;69ki43262kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69kiki2270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki18kc2&plus;9kcki&plus;324kc399ki&plus;3061081296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki6kc2&plus;3kcki&plus;108kc133ki&plus;102361296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki1362kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki1296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki36kcki&plus;18ki2&plus;36ki139968ki2690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki18kc2&plus;9kcki&plus;324kc399ki&plus;3061081296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69kiand0<6kc2&plus;3kcki&plus;108kc133ki&plus;102361296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki1362kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki1296kc2&plus;180kcki&plus;270ki2&plus;2592kc15948ki&plus;1296690kc&plus;32718&plus;69ki36kcki&plus;18ki2&plus;36ki139968ki2690kc&plus;32718&plus;69ki62kc23ki&plus;1790108270kc227kcki2052kc4579ki&plus;80622690kc&plus;32718&plus;69ki270kc227kcki2052kc