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ControlDesign[GainPhaseMargin] - PID tuning based on gain-phase margin specifications

Calling Sequence

GainPhaseMargin(sys, GM, PM, opts)

Parameters

sys

-

System; system object

GM

-

realcons; desired gain margin

PM

-

realcons; desired phase margin

opts

-

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

Description

• 

The GainPhaseMargin calculates the controller parameters to meet the specified gain and phase margins. The gain-phase margin specifications correspond to a set of nonlinear equations based on the controller structure. The algorithm first tries to find a numerical solution to the original problem. If no such solution exists, an approximate analytical solution is used.

• 

The numerical solution is found by solving the set of nonlinear equations associated to the selected controller type. For the PI controller, there are four equations (two magnitude equations and two phase equations) and four unknowns (two controller parameters and two crossover frequencies for gain and phase margins) to be solved for. For the PID controller, there are four equations as in the PI case but five unknowns (three controller parameters and the two crossover frequencies). In this case, one extra equation is provided by establishing a ratio between the PID integral and derivative times as Td=αTi. The ratio α is a design parameter whose value can be specified by the user. The default value of this parameter is 14 as suggested in Astrom et al. (1984), Hang et al. (1991) and Ho et al. (1999).

• 

If a numerical solution to the original problem does not exist for the selected controller type and the specified gain-phase margins, then the approximate analytical solutions for the PI and PID controller designs are used as described in Ho et al. (1995). Since the analytical method is based on lower order approximations of the system, the performance of the designed controller must be verified and fine tuned if necessary.

• 

Note: The numerical solution is applied to both stable and unstable systems, but the approximate analytical solution is only applicable to stable systems.

• 

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 GainPhaseMargin command returns a list of the form [Kp, Ki] for a PI controller or [Kp, Ki, Kd] for a PID controller, where Kp is the proportional gain, Ki is the integral gain, and Kd is the derivative gain.

• 

The controller transfer function is then obtained as:

– 

PI: Cs=Kp+Kis

– 

PID: Cs=Kp+Kis+Kds

Examples

withControlDesign:

withDynamicSystems:

Build a transfer function for a SISO system sys1

sys1:=NewSystem2s+15s4+8s3+20s2+21s+6:

GainMarginsys1,'decibels'=false

1.500000001,1.732050808

(1)

PhaseMarginsys1

19.84519766,1.395724232

(2)

Design a PI controller (desired gain margin equal to 25 decibels and desired phase margin equal to 60 degrees)

PIpars:=GainPhaseMarginsys1,25,60,'controller'=Π,'decibels'=true,'radians'=false

PIpars:=0.03833922655,0.01796181549

(3)

PI controller transfer function

Gc:=GainPhaseMarginsys1,25,60,'controller'=Π,'decibels'=true,'radians'=false,'returntype'=system:

PrintSystemGc

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1stf1,1=0.03833922654s+0.01796181549s

(4)

sys_loop:=NewSystemsys1:-tf1,1Gc:-tf1,1:

GainMarginsys_loop,'decibels'=true

28.13997620,1.420339299

(5)

PhaseMarginsys_loop,'radians'=false

83.37149660,0.08945084113

(6)

Design a PID controller with the same gain-phase margin specifications

pid:=GainPhaseMarginsys1,25,60,'controller'=PID,decibels='true','radians'=false,'returntype'=record

pid:=RecordpackedKp=0.05930671873,Ki=0.04693334343,Kd=0.08729712630

(7)

PID controller transfer function

Gc:=GainPhaseMarginsys1,25,60,'controller'=PID,decibels='true','radians'=false,'returntype'=system:

Warning, improper transfer-function, degree(num) > degree(den)

PrintSystemGc

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1stf1,1=0.08729712631s2+0.05930671873s+0.04693334343s

(8)

sys_loop:=NewSystemsys1:-tf1,1Gc:-tf1,1:

GainMarginsys_loop,'decibels'=true

37.51046154,5.276553296

(9)

PhaseMarginsys_loop,'radians'=false

67.31441788,0.2025962852

(10)

Build a transfer function for a SISO system sys2

sys2:=NewSystem120s+25:

GainMarginsys2,'decibels'=false

0.7694501862,1.453085056

(11)

PhaseMarginsys2

19.25926810,1.669405996

(12)

Design a PI controller (desired gain margin equal to 25 decibels and desired phase margin equal to 60 degrees)

Gc:=GainPhaseMarginsys2,25,60,'controller'=Π,decibels='true','radians'=false,'factored'=false,'returntype'=system:

PrintSystemGc

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1stf1,1=0.01875874587s+0.01743451489s

(13)

sys_loop:=NewSystemsys2:-tf1,1Gc:-tf1,1:

GainMarginsys_loop,'decibels'=true

25.82711900,1.049870847

(14)

PhaseMarginsys_loop,'radians'=false

84.66335908,0.06536623173

(15)

Design a PID controller with the same gain-phase margin specifications

Gc:=GainPhaseMarginsys2,25,60,'controller'=PID,decibels='true','radians'=false,'factored'=true,'returntype'=system:

Warning, improper transfer-function, degree(num) > degree(den)

PrintSystemGc

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1stf1,1=0.02389881218s2+0.02529072048s+0.02817486605s

(16)

sys_loop:=NewSystemsys2:-tf1,1Gc:-tf1,1:

GainMarginsys_loop,'decibels'=true

32.12602018,2.176177693

(17)

PhaseMarginsys_loop,'radians'=false

80.45980564,0.1044325480

(18)

Build a transfer function for a SISO system sys3

sys3:=NewSystem120s1s+25:

GainMarginsys3,'decibels'=false

1.812938848,3.142987870

(19)

PhaseMarginsys3

38.06739594,2.454043910

(20)

Design a PI controller (desired gain margin equal to 25 decibels and desired phase margin equal to 60 degrees)

PIpars3:=GainPhaseMarginsys3,25,60,'controller'=Π,decibels='true','radians'=false,'factored'=true,'returntype'=record

PIpars3:=RecordpackedK=0.003864514182,Ti=0.4960037565

(21)

PI controller transfer function

Gc:=GainPhaseMarginsys3,25,60,'controller'=Π,decibels='true','radians'=false,'factored'=true,'returntype'=system

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

(22)

PrintSystemGc

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1stf1,1=0.003864514182s0.007791300230s

(23)

sys_loop:=NewSystemsys3:-tf1,1Gc:-tf1,1:

GainMarginsys_loop,'decibels'=true

25.57023890,0.5471290232

(24)

PhaseMarginsys_loop,'radians'=false

84.97193810,0.02921732225

(25)

Design a PID controller with the same gain-phase margin specifications

params:=GainPhaseMarginsys3,25,60,'controller'=PID,decibels='true','radians'=false,'factored'=false

params:=0.005161515110,0.01070504302,0.002515442705

(26)

Kp:=params1;Ki:=params2;Kd:=params3

Kp:=0.005161515110

Ki:=0.01070504302

Kd:=0.002515442705

(27)

PID controller transfer function

Gc:=NewSystemKp+Kis+Kds:

Warning, improper transfer-function, degree(num) > degree(den)

PrintSystemGc

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1soutputvariable=y1stf1,1=0.002515442705s20.005161515110s0.01070504302s

(28)

sys_loop:=NewSystemsys3:-tf1,1Gc:-tf1,1:

GainMarginsys_loop,'decibels'=true

23.50129588,0.5516567517

(29)

PhaseMarginsys_loop,'radians'=false

83.06374234,0.04012814203

(30)

See Also

ControlDesign, ControlDesign[CohenCoon], ControlDesign[ZNFreq], ControlDesign[ZNTimeModified]

References

  

[1] K. J. Astrom and T. Hagglund, Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins, Automatica, Vol. 20, No. 5, pp. 645-651, 1984.

  

[2] C. C. Hang, K. J. Astrom, and W. K. Ho, Refinements of the Ziegler-Nichols Tuning Formula, IEE Proceedings D, Control Theory and Applications, Vol. 138, No. 2, pp. 111-118, 1991.

  

[3] W. K. Ho, K. W. Lim, C. C. Hang and L. Y. Ni, Getting more Phase Margin and Performance out of PID Controllers, Automatica, Vol. 35, No. 9, pp. 1579-1585, 1999.

  

[4] W. K. Ho, C. C. Hang, L. S. Cao, Tuning of PID Controllers Based on Gain and Phase Margin Specifications Automatica, Vol 31, No. 3, pp. 497-502, 1995.


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