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ControlDesign[FeasibleGains] - find feasible proportional(-integral) controller gains for pole placement in a desired region

Calling Sequence

FeasibleGains(sys, zeta, omegan, opts)

Parameters

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

Description

• 

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=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 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

Examples

withControlDesign:

Example 1

sys1:=DynamicSystems:-NewSystems+3s3+12s2+12s+7:

ζ1:=25;omegan1:=32

ζ1:=25

omegan1:=32

(1)

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

(2)

The admissible region for KKi 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

(3)

And the admissible interval for K is obtained by:

solution:=FeasibleGainssys1,ζ1,omegan1,'output'=region,'controller'=P

solution:=2.226666667<KandK<129.6601807

(4)

Example 2

The following system is open loop unstable:

sys2:=DynamicSystems:-NewSystems&plus;3s23s&plus;5&colon;

&zeta;2:=35&semi;omegan2:=2

&zeta;2:=35

omegan2:=2

(5)

solution:=FeasibleGainssys2&comma;&zeta;2&comma;omegan2&comma;&apos;output&apos;&equals;samplepoints&comma;&apos;controller&apos;&equals;&Pi;

solution:=K&equals;16&comma;Ki&equals;28&comma;K&equals;18&comma;Ki&equals;34&comma;K&equals;6234352583093274877906944&comma;Ki&equals;34

(6)

FeasibleGainssys2&comma;&zeta;2&comma;omegan2&comma;&apos;output&apos;&equals;region&comma;&apos;controller&apos;&equals;&Pi;

solution:=FeasibleGainssys2&comma;&zeta;2&comma;omegan2&comma;&apos;output&apos;&equals;samplepoints&comma;&apos;controller&apos;&equals;P

solution:=K&equals;14&comma;K&equals;19

(7)

FeasibleGainssys2&comma;&zeta;2&comma;omegan2&comma;&apos;output&apos;&equals;region&comma;&apos;controller&apos;&equals;P

10.14252948<KandK<18.59166305or18.59166305<K

(8)

Example 3

sys3:=DynamicSystems:-NewSystems&plus;12s3&plus;123s2&plus;12s&plus;20&colon;

&zeta;3:=0.7&semi;omegan3:=2

&zeta;3:=0.7

omegan3:=2

(9)

solution:=FeasibleGainssys3&comma;&zeta;3&comma;omegan3&comma;&apos;output&apos;&equals;samplepoints&comma;&apos;controller&apos;&equals;P

solution:=K&equals;3780&comma;K&equals;4709&comma;K&equals;6170

(10)

FeasibleGainssys3&comma;&zeta;3&comma;omegan3&comma;&apos;output&apos;&equals;region&comma;&apos;controller&apos;&equals;P

2929.002531<KandK<4630.529993or4630.529993<KandK<4787.368518or4787.368518<KandK<7553.341051

(11)

solution:=FeasibleGainssys3&comma;&zeta;3&comma;omegan3&comma;&apos;output&apos;&equals;samplepoints&comma;&apos;controller&apos;&equals;&Pi;

solution:=K&equals;3694&comma;Ki&equals;5711&comma;K&equals;4130&comma;Ki&equals;6880&comma;K&equals;4874&comma;Ki&equals;6880&comma;K&equals;4216&comma;Ki&equals;8759&comma;K&equals;4941&comma;Ki&equals;8759&comma;K&equals;5601&comma;Ki&equals;8759&comma;K&equals;4434&comma;Ki&equals;13910&comma;K&equals;5073&comma;Ki&equals;13910&comma;K&equals;6306&comma;Ki&equals;13910&comma;K&equals;6349&comma;Ki&equals;49511

(12)

FeasibleGainssys3&comma;&zeta;3&comma;omegan3&comma;&apos;output&apos;&equals;region&comma;&apos;condition&apos;&equals;damping&comma;&apos;controller&apos;&equals;&Pi;

FeasibleGainssys3&comma;&zeta;3&comma;omegan3&comma;&apos;output&apos;&equals;region&comma;&apos;condition&apos;&equals;all&comma;&apos;controller&apos;&equals;&Pi;

See Also

ControlDesign, ControlDesign[Characterize], ControlDesign[CohenCoon], ControlDesign[GainPhaseMargin], ControlDesign[RegionPoles], ControlDesign[ZNFreq], ControlDesign[ZNTimeModified]


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