ControlDesign
Characterize
characterize all PID controllers for pole placement in a desired region
Calling Sequence
Parameters
Options
Description
Examples
Characterize(sys, zeta, omegan, opts)
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
controller : one of P, PI, or PID
Specifies the controller type. The default value is PID.
output : one of relativestability, damping, or all
Specifies the set of inequality conditions that must be returned. The default value is all.
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=jω−ζomegan (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.
withControlDesign:
sys≔DynamicSystems:-NewSystems+2s3+12s2+17s+2
sys≔Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1s
Characterizesys,13,2,controller=P,output=relativestability
0<9kc−29and0<117kc+373
Characterizesys,13,2,controller=P,output=damping
0<kc+12and0<5453−115kcand0<895+31kc−5445kc2−342kc+134375453−115kcand0<5kc2−38kc+1493−5453−115kc9kc2+162kc+153−11664kc+125453−115kc9895+31kc−5445kc2−342kc+134375453−115kcand0<3kc2+54kc+51−3888kc+125453−115kc−8895+31kc−5445kc2−342kc+134375453−115kckc+125kc2−38kc+1493−5453−115kc9kc2+162kc+153−11664kc+125453−115kc9895+31kc−5445kc2−342kc+134375453−115kc
Characterizesys,13,2,controller=P
0<9kc−29and0<117kc+373and0<kc+12and0<5453−115kcand0<895+31kc−5445kc2−342kc+134375453−115kcand0<5kc2−38kc+1493−5453−115kc9kc2+162kc+153−11664kc+125453−115kc9895+31kc−5445kc2−342kc+134375453−115kcand0<3kc2+54kc+51−3888kc+125453−115kc−8895+31kc−5445kc2−342kc+134375453−115kckc+125kc2−38kc+1493−5453−115kc9kc2+162kc+153−11664kc+125453−115kc9895+31kc−5445kc2−342kc+134375453−115kc
Characterizesys,13,2,controller=Π,output=relativestability
0<−18kc+27ki+58and0<2106kc−8406−243kiand0<18kc+27ki−158−252−2016kc+3024ki+64962106kc−8406−243ki
Characterizesys,13,2,controller=Π
0<−18kc+27ki+58and0<2106kc−8406−243kiand0<18kc+27ki−158−252−2016kc+3024ki+64962106kc−8406−243kiand0<ki2and0<−690kc+32718+69kiand0<62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiand0<2kc+ki2+1ki−7776ki2−690kc+32718+69ki−7262kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiki2270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki−366kc2+4374ki+157464kc1458−133ki+102−361296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki−62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki1296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−−690kc+32718+69ki36kc+ki2+1ki−139968ki2−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki3270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiki2216kc2+25832ki+52488kc243+54ki2−2640ki+216−18kc+ki2+1ki−−690kc+32718+69ki36kc+ki2+1ki−139968ki2−690kc+32718+69ki662kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki−270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki12kc+ki2+1ki−46656ki2−690kc+32718+69ki−43262kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiki2270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki66kc2+4374ki+157464kc1458−133ki+102−361296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki−62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki1296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−−690kc+32718+69ki36kc+ki2+1ki−139968ki2−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki3270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiand0<6kc2+4374ki+157464kc1458−133ki+102−361296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki−62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki1296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−−690kc+32718+69ki36kc+ki2+1ki−139968ki2−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki3270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiand0<270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiand0<216kc2+25832ki+52488kc243+54ki2−2640ki+216−18kc+ki2+1ki−−690kc+32718+69ki36kc+ki2+1ki−139968ki2−690kc+32718+69ki662kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki−270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki12kc+ki2+1ki−46656ki2−690kc+32718+69ki−43262kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69kiki2270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki66kc2+4374ki+157464kc1458−133ki+102−361296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki−62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki1296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−−690kc+32718+69ki36kc+ki2+1ki−139968ki2−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki3270kc2−1080kc−4579ki+80622−4374ki+157464kc162−−690kc+32718+69ki18kc2+4374ki+157464kc486−399ki+306−1081296kc2+45832ki+52488kc81+324ki2−15840ki+1296−108kc+ki2+1ki−690kc+32718+69ki62kc−23ki+1790−108270kc2−1080kc−4579ki+80622−4374ki+157464kc162−690kc+32718+69ki
Characterizesys,13,2,controller=PID,output=relativestability
0<27kd+252and0<81kc−351−81−36kd+18kc+27ki−15827kd+252and0<12kd−18kc+27ki+58and0<−36kd+18kc+27ki−158−27kd+25248kd−72kc+108ki+23281kc−351−81−36kd+18kc+27ki−15827kd+252
See Also
ControlDesign[CohenCoon]
ControlDesign[FeasibleGains]
ControlDesign[GainPhaseMargin]
ControlDesign[RegionPoles]
ControlDesign[ZNFreq]
ControlDesign[ZNTimeModified]
Download Help Document
