ControlDesign
FeasibleGains
find feasible proportional(-integral) controller gains for pole placement in a desired region
Calling Sequence
Parameters
Options
Description
Examples
FeasibleGains(sys, zeta, omegan, opts)
sys
-
System; system object
zeta
realcons; desired damping ratio
omegan
realcons; desired natural frequency
opts
(optional) equation(s) of the form option = value; specify options for the FeasibleGains command
controller = P or PI
Specifies the controller type. The default value is PI.
condition = relativestability, damping, or all
Specifies the conditions defining the desired region. The default value is all.
posgains = true or false
Specifies whether only positive controller gains must be considered. The default value is true.
output = samplepoints or region
Specifies the type of output that the command returns. If samplepoints is specified, a sequence of sample feasible values for the controller gains is returned. If region is specified, for the PI controller the admissible region of the controller gains is plotted in the 2-D K−Ki space and for the P controller the admissible intervals for K are given. The plot may contain solid and dashed lines. The feasibility of the points on the solid lines is not guaranteed (although they might be still feasible). The points on the dashed lines are feasible. The admissible intervals are given as strict inequalities meaning that the feasibility of the boundary points is not guaranteed (although they might be still feasible). The default value is samplepoints.
color = color specification (see plot/color)
The admissible region for K−Ki is filled with the specified color. If the samplepoints output or the P controller are selected, this option is disregarded. The default color is Blue.
The FeasibleGains command finds feasible proportional (P) or proportional-integral (PI) controller gains for pole placement in a 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 system sys is a continuous-time linear system object created using the DynamicSystems package. The system object must single-input single-output (SISO) and one of the following types: TransferFunction (tf), ZeroPoleGain(zpk), Coefficients(coeff), StateSpace(ss), DiffEquations (de).
The FeasibleGains command returns admissible values of the controller gains for which the closed-loop poles (under unity negative feedback) are in the specified desired region.
The controller parameters are K for a P controller and K,Ki for a PI controller, where K is the proportional gain and Ki is the integral gain.
The controller transfer function is then obtained as:
P: Cs=K
PI: Cs=K+Kis
withControlDesign:
Example 1
sys1≔DynamicSystems:-NewSystems+3s3+12s2+12s+7:
ζ1≔25;omegan1≔32
ζ1≔25
omegan1≔32
The following gives sample solutions for the PI controller:
solution≔FeasibleGainssys1,ζ1,omegan1,output=samplepoints,controller=Π
solution≔K=48,Ki=36,K=76,Ki=62,K=85,Ki=88
The admissible region for K−Ki can also be plotted:
FeasibleGainssys1,ζ1,omegan1,output=region,posgains=true,controller=Π
The following gives sample solutions for the P controller:
solution≔FeasibleGainssys1,ζ1,omegan1,output=samplepoints,controller=P
solution≔K=66
And the admissible interval for K is obtained by:
solution≔FeasibleGainssys1,ζ1,omegan1,output=region,controller=P
solution≔2.226666667<K<129.6601807
Example 2
The following system is open loop unstable:
sys2≔DynamicSystems:-NewSystems+3s2−3s+5:
ζ2≔35;omegan2≔2
ζ2≔35
omegan2≔2
solution≔FeasibleGainssys2,ζ2,omegan2,output=samplepoints,controller=Π
solution≔K=16,Ki=28,K=18,Ki=34,K=997496413294674398046511104,Ki=34
FeasibleGainssys2,ζ2,omegan2,output=region,controller=Π
solution≔FeasibleGainssys2,ζ2,omegan2,output=samplepoints,controller=P
solution≔K=14,K=19
FeasibleGainssys2,ζ2,omegan2,output=region,controller=P
10.14252948<K<18.59166305or18.59166305<K
Example 3
sys3≔DynamicSystems:-NewSystems+12s3+123s2+12s+20:
ζ3≔0.7;omegan3≔2
ζ3≔0.7
omegan3≔2
solution≔FeasibleGainssys3,ζ3,omegan3,output=samplepoints,controller=P
solution≔K=3780,K=4709,K=6170
FeasibleGainssys3,ζ3,omegan3,output=region,controller=P
2929.002531<K<4630.529993or4630.529993<K<4787.368518or4787.368518<K<7553.341051
solution≔FeasibleGainssys3,ζ3,omegan3,output=samplepoints,controller=Π
solution≔K=3694,Ki=5711,K=4130,Ki=6880,K=4874,Ki=6880,K=4216,Ki=8759,K=4941,Ki=8759,K=5601,Ki=8759,K=4434,Ki=13910,K=5073,Ki=13910,K=6306,Ki=13910,K=6349,Ki=49511
FeasibleGainssys3,ζ3,omegan3,output=region,condition=damping,controller=Π
FeasibleGainssys3,ζ3,omegan3,output=region,condition=all,controller=Π
See Also
ControlDesign[Characterize]
ControlDesign[CohenCoon]
ControlDesign[GainPhaseMargin]
ControlDesign[RegionPoles]
ControlDesign[ZNFreq]
ControlDesign[ZNTimeModified]
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