Design of PID Controllers:H∞ Region Approach

*****

An introduction to PID Hinf Designerwww.pidlab.com/pidhinf

Miloš Schlegel, Michal Brabec

REX Controls s.r.o.

Pilsen, Czech Republic

PID design method

• For a long time, the development of PID controller design methods has been the goal of the control community. Despite that manual model-free tuning of controllers is still the most commonly used PID design method in industry.

• Tuning rules (Ziegler-Nichols, Lambda tuning, AMIGO method [1], Internal model control, Skogestad’s SIMC method [2], … )

Universal relations between model and controller parameters.

• Optimization-based method (MIGO [3], SWORD [4], MATLAB pidTuner )

Treats each process model individually.

[1] Astrom, K.J. and T. Hagglund: Advanced PID Control. ISA, 2006, ISBN 1-55617-942-1[2] Sigurd Skogestad and Chriss Grimholt. “The SIMC Method for Smooth PIDController Tuning”. PIDControl in the Third Millennium.Springer. 2012[3] Astrom, K.J., Panagopoulos, H., Hagglund, T.: Design of PI Controllers based on Non-Convex Optimalization. Automatica, Vol. 34, No. 5, pp. 585-601, 1998.[4] Garpinger O. Analysis and Design of Software-Based Optimal PID Controllers. PhD Thesis, Department of Automatic Control Lund University, 2015.

There exists no generally accepted design method for PID controller

High Order Plant High Order Controller

Low Order Plant Low Order Controller

LQG or H∞Design

LQG or H∞Design

ModelReduction

ControllerReduction

DirectDesign

The design procedures associated with modern control theory (Hinf, LQR) provide high order controllers. Practice prefers simple controllers.

Anderson, B.D.O.: Controller Design Moving from Theory to Practice. 1992 Bode Prize Lecture.

Requirements for effective design method

• It should be applicable to a wide range of systems (i.e. stable/unstable/non minimal phase/oscillatory process transfer functions)

• It should have the possibility to introduce specifications that capture the essence of real control problems (i.e. robustness/performance trade-off, servo/regulator problem)

• The method should be robust in the sense that it provides controller parameters if they exist, or if the specifications cannot be meet an appropriate diagnosis should be presented

PID Hinf Designer(www.pidlab.com/pidhinf)

• PID Hinf Designer is the first advanced easy to used web design tool for the analysis and design of optimal PI(D) controllers with respect to performance integral criteria IE, IAE, ITAE, ISE and Hinf robustness constraints.

• PID Hinf Designer can be used for a wide range of proces models (unstable, non-minimal phase, oscillating, time-delayed systems, systems of any order, …) and also for so-called model sets created from any number of process transfer functions.

• Supported design specifications reflect the essence of real control problems. Optimization of integral criteria IE, ISE, IAE, ITAE under Hinf constraints is supported for both load disturbance attenuation and set-point tracking problems).

• Designing of PI(D) controller with typical specifications using PID Hinf Designer is a routine procedure that does not require deeper knowledge of control theory from the user.

• With more skills and efforts from the designer it should be possible using PID Hinf Designer to design high performance PID controllers extended with a suitable linear compensator (Cascade Controller, Resonant Controller, Smith Predictor, Repetitive Control, …).

• PID Hinf Designer also supports simple process models obtained from popular identification experiments. Specifically, two- or three-parameter models obtained from the step response of the process are supported, as well as models obtained from the relay experiment (based on the knowledge of one point of the frequency response). Moreover, the non-standard moment model set provided by the PIDMA-autotuner from the company REX Controls is also supported.

PID Hinf Designer Options(www.pidlab.com/pidhinf)

PID Controller

1DOF PI Controller

2DOF PI Controller

1DOF PID Controller

2DOF PID Controller

( )PI D−

Compensator Process

Static or Integrating Process

(Arbitrary Rational Transfer Function (RTF)

without poles on the imaginary axis except

one for integrating processes)

RTF +

Assembled Mode

Time Delay (RTFTD)

Model Set (

l

Set o

( )

f RTFTD)

- One Frequency Point Process Model

- , , Process Model

- Model with Parametric Uncertainty

Arbitrary Rational Transfer

Function (RTF) without

poles on the imaginary axis

- second order compensator

- lead-lag compensator

- Load D

isturbance Attenuation

PID Con

trol

Advanced PID Control

Cascade Control, Resonant Control, Repetitive Control, PID Control f

- Set-point T

or Complex Sy

ra

st

cking

ems,

Problems

...

PID Hinf Designer GUI(www.pidlab.com/pidhinf)

Entering transferfunctions of processes

Selection of a model set

Enter H∞ limitations

Select the controller type

1

2

Create a closed-loop assembled transfer function(e.g. for cascade control)

The resulting controller

H∞ region selection

Manual tuning of the controller

Enter weighting functions and compensators

Select weighting functions,compensators, systems,sensitivity functions and values of H∞ limitations

Estimate kd (PID) :ManuallyAutomatically

Estimate tau (PID) :Manually

Selection of the design criterion

1

2

Parameter Plane Formulation of Basic PI-Controller Design Problem

rp ik k s+

S

T

CS n

ueid

PS

( )P s

Sod

y

* *( , [ , ]) ( ) ( , ), , , , weighting sensitivity function

, : ( , ) sup ( , ) , the closed-loop is stable - region in the parameter plane

1) Find the - regi

p i C P

p i

H s k k k W s S s k S S T S S

k k H s k H j k H

H

=

0

on in the - plane. (

2) Find the optimal PI-controller in the - region with respect to the criterion ( )

for the step in the reference value (

See Appendix A for details.)

servo p

i pk k

H IAE e t dt

r

roblem) or load disturbance (regulator problem).id

PI-controller Process

Parameter Plane Formulation of Basic PID-Controller Design Problem

* *

[ , ]

( , [ , , , ]) ( ) ( , ), , , , weighting sensitivity function

, : ( , ) sup ( , ) , the closed-loop is stable - region in the parameter plane - ford

p i d C P

k p i i p

H s k k k k W s S s k S S T S S

k k H s k H j k H k k

=

[ , ]

Choose the derivative gain and the time constant manually or with the help of a built-in function. See Appendix

the fixed and

1) (

2) Find the - region in the - p

B for details.)

d

d

p

d

k i

k

k

H k k

[ , ]0

lane.

3) Find the optimal PID-controller in the - region with respect to the criterion ( )

for the step in the reference value (servo problem) or load disturbance (regulator

dk

i

H IAE e t dt

r d

problem).

r( 1)p i dk k s k s s+ + +

S

T

CS n

ue

idPS

( )P s

Sod

yPID-controller Process

H∞ limitations supported

Servo problem

(set-point tracking)

:

:

:

S S

T T

C C C

r e W S M

r y W T M

n u W S M

→

→

→

Regulator problem

(load disturbance rejection)

:

:

:

i P P P

o S T

C C C

d y W S M

d e W S M

n u W S M

→

→

→

Sensitivity functions (

1, , ,

1

Weighting functions

( ), ( ), ( ), ( ),

gang of four)

C P

S T C P

S T C PS S C S S PSC P

W s W s W s W s

= = = =+

r( )PI D

ne

id

( )P s

od

yCompenzator

SW CW

u

PW

e u

y

TWy

( )C s

−

)( ) ( ) , 0,S s M S j MS S

𝐻∞-Region in the Parametric Plane 𝑘𝑖 − 𝑘𝑝(It contains all PI controllers that meet the specified H∞ limitations)

( )

Finding the - region

is generally a very difficult problem. PID Hinf Designer www.pidlab.com/pidhinf is the first

software tool availabl

, : ( , ) , the closed-loop is s

e to fully a

tablep ik k k k

H

H s

=

ddress this issue.

Example of Simple Design specification of PI-controller for FOPDT system

Proces transfer function: ( )1

1Controller transfer function: ( ) 1

1Sensitivity function: ( )

1 ( ) ( )

Weighting function:

s

iPI p

i

PI

eP s

s

kC s K k

T s s

S sC s P s

−

=+

= + = +

=+

0

( ) 1

Type of control problem: regulator problem (load step disturbance rejection)

Design specification: = min ( )

PI

S

C

W s

IAE e t dt

=

) subject to ( ) ( ) , 0,S SS s M S j M

PID Hinf Designer

Input:

( )1

PI-controller ( )

1.6

Output:

: 0.463, 0.509

: 0.565, 0.488

: 0.557, 0.492

Optional output:

ZN Ziegler-Nicols (1942, step response)

SIMC Skogestad (2012)

s

iPI p

S

p i

p i

p i

eP s

s

kC s k

s

M

IE k k

IAE k k

ITAE k k

−

=+

= +

=

= =

= =

= =

AMIGO Hagglund and Astrom (2004)

More General Formulation of Design Problem(fully supported by PID Hinf Designer, www.pidlab.com/pidhinf)

1 2, , , model set of transfer functions

process transfer function

, controller transfer function

1, ,

1 1

k

PI PID

P P P

P

C C C

CPS T

CP CP

=

= =+ +

Ρ

P

2

0 0 0 0

, loop sensitivity transfer functions1 1

= , , , design criterion set

( ) , ( ) , ( ) , ( )

C P

CP PS S

CP CP

IE IAE ITAE ISE

IE e t dt IAE e t dt ITAE t e t dt ISE e t dt

I

= =+ +

I

Ι design criterion selected

, , , weightingfunctions

min max

subject to the limitations

: , ,

S T C P

C P

S S T S C

W W W W

I

H

P W S M W T M W S

P

Controller Robust Design Problem

P , .C C P P PM W S M

Example of Design Specification of Robust PI-controller for Process Model Set

0.166 0.166

1 22 2

0.0216 0.0031 0.0174 0.0046Proces model set: ( ) , ( )

0.457 0.0868 0.5978 0.0445

1Controller transfer function: ( ) 1

s s

PI p

i

s sP s e P s e

s s s s

kC s K k

T s

− −− + − + = =

+ + + +

= + = +

P

1Sensitivity functions: ( ) , 1,2

1 ( ) ( )

Weighting functions: ( ) 1, 1,2

Type of control problem: regulator problem (load step disturbance rej

i

i

PI i

i

s

S s iC s P s

W s i

= =+

= =

)

01,2

ection)

Design specification: min max ( )

subject to ( ) , 1, ,2 ( ) , 1,2, 0,

PIi

C i

i S i S

e t dt

S s M i S j M i

= =

PID Hinf Designer

0.166

1 2

0.166

2 2

Input:

0.0216 0.031( )

0.457 0.0868

0.0174 0.0445( )

0.5978 0.0445

( )

1.6

Output:

: 6.313, 0.8704

: 6.313, 0.8704

: 6.313, 0.8704

s

s

iPI p

S

p i

p i

p i

sP s e

s s

sP s e

s s

kC s k

s

M

IE k k

IAE k k

ITAE k k

−

−

− +=

+ +

− +=

+ +

= +

=

= =

= =

= =

Conclusion

• PID Hinf Designer is the first advanced easy to used web design tool for the analysis and design of optimal PI(D) controllers with respect to performance integral criteria IE, IAE, ITAE and 𝐻∞ robustness constraints.

• PID Hinf Designer can be used for a wide range of proces models (unstable, non-minimal phase, oscillating, time-delayed systems, systems of any order, …) and also for so-called model sets created from any number of process transfer functions.

• PID Hinf Designer provide a new explicit algorithm to determine the 𝐻∞- regions in the parameter plane of PI controller for all commonly used 𝐻∞ limitations of the weighted sensitivity functions.

• PID Hinf Designer also supports simple process models obtained from popular identification experiments. Specifically, two- or three-parameter models obtained from the step response of the process are supported, as well as models obtained from the relay experiment (based on the knowledge of one frequency point). Moreover, the non-standard moment model set provided by the PIDMA-autotuner from the company REX Controls is also supported.

• Designing of PI(D) controller with typical specifications using PID Hinf Designer is a routine procedure that does not require deeper knowledge of control theory from the user.

• With more skills and efforts from the designer it should be possible to design high performance PID controllers extended with any linear compensator suitable (Resonant Controller, Smith predictor, Repetitive Control, …).

For more details see: Schlegel M., Medvecová P., Design of PI Controllers: Region Approach.

IFAC P

If ( , ) , [ , ], ( ) has no poles on th

apersOnLine 51-

e im

6 (2018), 13-17.

ip p i

kC s k k k k P

s

H

k s

= + =Proposition : aginary axis, and the design

specification is

( , )1( , ) 1,

1 ( , ) ( ) ( , )

then the boundary of the -region is contained in the solutions of the two systems of equations

(i) (

nS

d

n

S s kS s k M

C s k P s S s k

H

S j

= = +

( )

( )

2 2

2

The system of equations i has a solution =0 , i.e. any point on the axis is a solution

of thi

, ) 0

s s

, (ii)

ystem. The solution of the system ii is determ

( , )

in

= ,

( , )( , ) 0, 0.d

i pk k

k S j k

S j kS j k

=

= =

ed by the parametric curves

Appendix A: Isolation of -Region (1)H

)2 2 2 2 2 2 2 2 2

1 1 1 1 1

2 2 2

1 1 1

1 1

1 1

,

0, ,( ) ( ( 2 ) ( )) ( 1)( )

,(2 )

where , , , are the functions of defined by

( ) ( ) ( ) ,

( )( )

ii

S

i i S Sp

S

xk

M

x A B x M A B AA B B B B A B M AA BBk

M ABB A A A B

A A B B

P j A jB A jB

dP jA jB

d

=

+ + − + + + + + − + =+ −

= + +

= +

1 1

4 3 2

2 2 4

2 2 2 2

1

( ) ,

and , 1, , ( ), ( ) 0,2,4 are the frequency dependent real roots of quartic polynomial

0

with the real frequency dependent coefficients

( ) ,

2 ( ) ( 2

i

S

A B

x i l l

a x b x c x d x e

a A B

b M A B B A BAA

+

=

+ + + + =

= +

= + − + 2 2 3

1 1

2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1

2 2 2 2 2 4 2 3 2 2 2 2 3 2

1 1 1 1 1

2 3 2 2 2 2 3 2 2 2 2

1 1 1 1

),

( )(2 4 2 8

2 2 4 ),

2 ( 2

S S S S S

S S S S

S S S S S

B B BA B

c A B A A A BB M A BB A B M A A M A B M AA B M

AA B A B M B M B B B B M B B M

d M A B M A A BM AA B M A BB M

+ + +

= − + + + − − − − +

+ − − + − −

= − − − − − + 2 2 2 4 2 3 2 2

1 1 1

2 2 2 3 2 2 3 2 3 2 2 3 4

1 1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1

3 2 ),

( 1)( 2 ).

S S S

S S S

A B B M B B M B B M

AA B B A BB AA B A A B B B A A B A B B A A B B B

e M A B M B B M AA BB B B A A

− − +

+ − + + + − + + +

= − − − − + + +

Appendix A (2)

Appendix A (3)

3 2

5 4 3 2

Example: region for unstable process:

4 1( ) ,

2 32 14 4 50

( , )

1( , ) 2.8

1 ( , ) ( )

ip

S

H

s s sP s

s s s s s

kC s k k

s

S s k MC s k P s

−

+ − +=

+ + + − +

= +

= =+

( ) ( )The curves representing the solutions of systems i and ii divide the parametric plane into

regions. From them, it is necessary to select those that meet the design specifications. For this

purpose, it is sufficient to test only one point of the respective region.

( , ) 2.8

closed loop is stable

S s k

Appendix A (4)

3 2

5 4 3 2

GUI:

Auxiliary Tools > Multiparametric Analysis

Example: region for unstable process:

4 1( ) ,

2 32 14 4 50

( , )

1( , )

1 ( , ) ( )

2.6,2.7,2.8

ip

S

S

H

s s sP s

s s s s s

kC s k k

s

S s k MC s k P s

M

−

+ − +=

+ + + − +

= +

= +

Appendix B: Selection of kd and tauIf there exists a PI controller for the

given design specification with p

It is recommended to start wi

arameters , , ( = ), then it is recommended to estima

th the ideal PID controller

te

0

( ).

p i i p ik k T k k

=

2 2optimal in the interval 0.2 ,0.3 manually or with the help of GUI build-in function (*).d p i p ik k k k k

(*)