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ControlDesign

  

FeasibleGains

  

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

 

Calling Sequence

Parameters

Options

Description

Examples

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

Options

• 

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 KKi 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 KKi 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.

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

sys1DynamicSystems:-NewSystems+3s3+12s2+12s+7:

ζ125;omegan132

ζ1:=25

omegan1:=32

(1)

The following gives sample solutions for the PI controller:

solutionFeasibleGainssys1,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:

solutionFeasibleGainssys1,ζ1,omegan1,'output'=samplepoints,'controller'=P

solution:=K=66

(3)

And the admissible interval for K is obtained by:

solutionFeasibleGainssys1,ζ1,omegan1,'output'=region,'controller'=P

solution:=2.226666667<KandK<129.6601807

(4)

Example 2

The following system is open loop unstable:

sys2DynamicSystems:-NewSystems&plus;3s23s&plus;5&colon;

&zeta;235&semi;omegan22

&zeta;2:=35

omegan2:=2

(5)

solutionFeasibleGainssys2&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;

solutionFeasibleGainssys2&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

sys3DynamicSystems:-NewSystems&plus;12s3&plus;123s2&plus;12s&plus;20&colon;

&zeta;30.7&semi;omegan32

&zeta;3:=0.7

omegan3:=2

(9)

solutionFeasibleGainssys3&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)

solutionFeasibleGainssys3&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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