Industrial Control Systems - PID Controllers

Industrial Control

Behzad Samadi

Department of Electrical EngineeringAmirkabir University of Technology

Winter 2010Tehran, Iran

Behzad Samadi (Amirkabir University) Industrial Control 1 / 95

Feedback Control Loop

r : reference signal

y : process (controlled) variable

u: manipulated (control) variable

e: control error

d : load disturbance signal

n: measurement noise signal

F : feedforward filter

C : controller

P: plant

[Visioli, 2006]


On-Off Control

One of the simplest control laws:

u =

{umax if e > 0umin if e < 0

Disadvantage: persistent oscillation of the process variable

P =1

10s + 1e−2s , umax = 2, umin = 0

[Visioli, 2006]


On-Off Control


u =



P =1

10s + 1e−2s , umax = 2, umin = 0

[Visioli, 2006]


On-Off Control


u =



P =1

10s + 1e−2s , umax = 2, umin = 0

[Visioli, 2006]


On-Off Control


u =



P =1

10s + 1e−2s , umax = 2, umin = 0

[Visioli, 2006]


On-Off Control

a) Ideal on-off controller

b) Modified with a dead zone

c) Modified with hysteresys

[Visioli, 2006]


PID Control

1 Proportional action

2 Integral action

3 Derivative action

[Visioli, 2006]


Proportional Action

Proportional control action:

u(t) = Kpe(t) = Kp(r(t)− y(t)),

Kp: proportional gain

Controller transfer function:

C (s) = Kp

Advantage: small control signal for a small error signal

Disadvantage: steady state error

[Visioli, 2006]


Proportional Action


u(t) = Kpe(t) = Kp(r(t)− y(t)),



C (s) = Kp



[Visioli, 2006]


Proportional Action


u(t) = Kpe(t) = Kp(r(t)− y(t)),



C (s) = Kp



[Visioli, 2006]


Proportional Action


u(t) = Kpe(t) = Kp(r(t)− y(t)),



C (s) = Kp



[Visioli, 2006]


Proportional Action


u(t) = Kpe(t) = Kp(r(t)− y(t)),



C (s) = Kp



[Visioli, 2006]


Proportional Action

Steady state error occurs even if the process presents an integratingdynamics, in case a constant load disturbance occurs.

Adding a bias (or reset) term:

u(t) = Kpe + ub

The value of ub can be fixed or can be adjusted manually until thesteady state error is zero.

[Visioli, 2006]


Proportional Action

Steady state error occurs even if the process presents an integratingdynamics, in case a constant load disturbance occurs.

Adding a bias (or reset) term:

u(t) = Kpe + ub

The value of ub can be fixed or can be adjusted manually until thesteady state error is zero.

[Visioli, 2006]


Proportional Action

Proportional Band (PB):

PB =100%

Kp

[Astrom and Hagglund, 1995]


Integral Action

Integral control action:

u(t) = Ki

∫ t

0e(�)d�,

Ki : integral gain


C (s) =Ki

s

Advantage: zero steady state error

Disadvantage: integrator windup in the presence of saturation

[Visioli, 2006]


Integral Action


u(t) = Ki

∫ t

0e(�)d�,

Ki : integral gain


C (s) =Ki

s



[Visioli, 2006]


Integral Action


u(t) = Ki

∫ t

0e(�)d�,

Ki : integral gain


C (s) =Ki

s



[Visioli, 2006]


Integral Action


u(t) = Ki

∫ t

0e(�)d�,

Ki : integral gain


C (s) =Ki

s



[Visioli, 2006]


Integral Action


u(t) = Ki

∫ t

0e(�)d�,

Ki : integral gain


C (s) =Ki

s



[Visioli, 2006]


PI Controller

Proportional Integrator Controller:

Transfer function:

C (s) = Kp(1 +1

Ti s)

Integral action is able to set automatically the value of ub.

The integral action is also called automatic reset.

[Visioli, 2006]


PI Controller


Transfer function:

C (s) = Kp(1 +1

Ti s)



[Visioli, 2006]


PI Controller


Transfer function:

C (s) = Kp(1 +1

Ti s)



[Visioli, 2006]


Derivative Action

Derivative control action:

u(t) = Kdde(t)

dt,

Kd : derivative gain


C (s) = Kds

Advantage: Derivative action is an instance of predictive control.

Disadvantage: Sensitive to the measurement noise in the manipulatedvariable

[Visioli, 2006]


Derivative Action


u(t) = Kdde(t)

dt,



C (s) = Kds



[Visioli, 2006]


Derivative Action


u(t) = Kdde(t)

dt,



C (s) = Kds



[Visioli, 2006]


Derivative Action


u(t) = Kdde(t)

dt,



C (s) = Kds



[Visioli, 2006]


Derivative Action


u(t) = Kdde(t)

dt,



C (s) = Kds



[Visioli, 2006]


Derivative Action

Derivative action is an instance of predictive control.

Taylor series expansion of the control error at time Td ahead:

e(t + Td ) ≈ e(t) + Tdde(t)

dt

A control law proportional to e(t + Td )

u(t) = Kp

(e(t) + Td

de(t)

dt

)Derivative action is also called anticipatory control, or rate action, orpre-act.

[Visioli, 2006]


Derivative Action




dt


u(t) = Kp

(e(t) + Td

de(t)

dt


[Visioli, 2006]


Derivative Action




dt


u(t) = Kp

(e(t) + Td

de(t)

dt

)

Derivative action is also called anticipatory control, or rate action, orpre-act.

[Visioli, 2006]


Derivative Action




dt


u(t) = Kp

(e(t) + Td

de(t)

dt


[Visioli, 2006]


PID Controller

Transfer function:

C (s) = Kp

(1 +

1

Ti s+ Tds

)Time windows:

Proportional action responds to current error.Integrator action responds to accumulated past error.Derivative action anticipated future error.

Peter Woolf umich.edu


PID Controller

Transfer function:

C (s) = Kp +Ki

s+ Kds

Frequency band:

Proportional action: all-bandIntegrator action: low passDerivative action: high pass

[Li et al., 2006]


PID Controller

Transfer function:

C (s) = Kp +Ki

s+ Kds

[Li et al., 2006]


PID Controller

Implementation methods:

PneumaticHydraulicElectronicDigital


PID Controller

Structures of PID controllers:

Ideal or noninteracting form:

Ci (s) = Kp

(1 +

1

Ti s+ Tds

)

Series or interacting from:

Cs(s) = K′p

(1 +

1

T′i s

)(1 + T

′ds)

Parallel form:

Ci (s) = Kp +Ki

s+ Kds

[Visioli, 2006] and [Astrom and Hagglund, 1995]


PID Controller



Ci (s) = Kp

(1 +

1

Ti s+ Tds

)Series or interacting from:

Cs(s) = K′p

(1 +

1

T′i s

)(1 + T

′ds)

Parallel form:

Ci (s) = Kp +Ki

s+ Kds



PID Controller



Ci (s) = Kp

(1 +

1

Ti s+ Tds

)Series or interacting from:

Cs(s) = K′p

(1 +

1

T′i s

)(1 + T

′ds)

Parallel form:

Ci (s) = Kp +Ki

s+ Kds



PID Controller




PID Controller


Series to ideal form conversion:

Kp =K′p

T′i + T

′d

T′i

Ti =T′i + Td

′

Td =T′i T′d

T′i + T

′d


PID Controller


Ideal to series form conversion: Only if Ti ≥ 4Td

K′p =

Kp

2

(1 +

√1− 4

Td

Ti

)

T′i =

Ti

2

(1 +

√1− 4

Td

Ti

)

T′d =

Ti

2

(1−

√1− 4

Td

Ti

)[Visioli, 2006]


PID Controller

Alternative series form:

Cs(s) = Kp(1 +1

�Ti s)(� + Tds)

Ideal to alternative series form conversion: Only if Ti ≥ 4Td

� =1±

√1− 4 Td

Ti

2> 0

[Li et al., 2006]


PID Controller

A PID controller has two zeros and one pole at the origin.

Ti > 4Td : two real zeros

Ti = 4Td : two coincident zeros

Ti < 4Td : two complex conjugate zeros

[Visioli, 2006]


Problems with the Derivative Action

Noise:n(t) = A sin(!t)

Derivative action:u(t) = A! cos(!t)

u(t) is large for high frequencies.

In practice, a (very) noisy control signal might lead to a damage ofthe actuator.

[Visioli, 2006]







[Visioli, 2006]







[Visioli, 2006]







[Visioli, 2006]


Modified Derivative Action

Modified ideal form:

Ci1a(s) = Kp

(1 +

1

Ti s+

TdsTdN s + 1

)

Gerry and Shinskey, 2005:

Ci1b(s) = Kp

⎛⎜⎝1 +1

Ti s+

Tds

1 + TdN s + 0.5

(TdN s)2

⎞⎟⎠Modified series form:

Cs(s) = K′p

(1 +

1

T′i s

)⎛⎝T′ds + 1

T′d

N s + 1

⎞⎠N generally assumes a value between 1 and 33, although in themajority of the practical cases its setting falls between 8 and 16 (Anget al., 2005).

[Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 24 / 95



Ci1a(s) = Kp

(1 +

1

Ti s+

TdsTdN s + 1

)Gerry and Shinskey, 2005:

Ci1b(s) = Kp

⎛⎜⎝1 +1

Ti s+

Tds

1 + TdN s + 0.5

(TdN s)2

⎞⎟⎠

Modified series form:

Cs(s) = K′p

(1 +

1

T′i s

)⎛⎝T′ds + 1

T′d

N s + 1





Ci1a(s) = Kp

(1 +

1

Ti s+

TdsTdN s + 1


Ci1b(s) = Kp

⎛⎜⎝1 +1

Ti s+

Tds

1 + TdN s + 0.5

(TdN s)2


Cs(s) = K′p

(1 +

1

T′i s

)⎛⎝T′ds + 1

T′d

N s + 1

⎞⎠

N generally assumes a value between 1 and 33, although in themajority of the practical cases its setting falls between 8 and 16 (Anget al., 2005).




Ci1a(s) = Kp

(1 +

1

Ti s+

TdsTdN s + 1


Ci1b(s) = Kp

⎛⎜⎝1 +1

Ti s+

Tds

1 + TdN s + 0.5

(TdN s)2


Cs(s) = K′p

(1 +

1

T′i s

)⎛⎝T′ds + 1

T′d

N s + 1




Alternative modified ideal form:

Ci2a(s) = Kp

(1 +

1

Ti s+ Tds

)1

Tf s + 1

Astrom and Hagglund, 2004:

Ci2b(s) = Kp

(1 +

1

Ti s+ Tds

)1

(Tf s + 1)2

Derivative kick: A spike in the control signal due to an abrupt(stepwise) change of the set-point signal.

If the set-point is constant, the derivative action can be applied onlyto the process variable:

u(t) = −Kddy(t)

dt

[Visioli, 2006]




Ci2a(s) = Kp

(1 +

1

Ti s+ Tds

)1

Tf s + 1


Ci2b(s) = Kp

(1 +

1

Ti s+ Tds

)1

(Tf s + 1)2



u(t) = −Kddy(t)

dt

[Visioli, 2006]




Ci2a(s) = Kp

(1 +

1

Ti s+ Tds

)1

Tf s + 1


Ci2b(s) = Kp

(1 +

1

Ti s+ Tds

)1

(Tf s + 1)2



u(t) = −Kddy(t)

dt

[Visioli, 2006]




Ci2a(s) = Kp

(1 +

1

Ti s+ Tds

)1

Tf s + 1


Ci2b(s) = Kp

(1 +

1

Ti s+ Tds

)1

(Tf s + 1)2



u(t) = −Kddy(t)

dt

[Visioli, 2006]


Derivative Action

80% of the employed PID controllers have the derivative partswitched-off (Ang et al., 2005).

Derivative action is the most difficult to tune, why?

Consider a first-order-plus-dead-time (FOPDT) plant:

P(s) =K

Ts + 1e−Ls

and a PD controller:

C (s) = Kp(1 + Tds)

Open loop frequency response:

∣C (j!)P(j!)∣ = KKp

√1 + T 2

d!2

1 + T 2!2

[Visioli, 2006]


Derivative Action




P(s) =K

Ts + 1e−Ls


C (s) = Kp(1 + Tds)


∣C (j!)P(j!)∣ = KKp

√1 + T 2

d!2

1 + T 2!2

[Visioli, 2006]


Derivative Action




P(s) =K

Ts + 1e−Ls


C (s) = Kp(1 + Tds)


∣C (j!)P(j!)∣ = KKp

√1 + T 2

d!2

1 + T 2!2

[Visioli, 2006]


Derivative Action




P(s) =K

Ts + 1e−Ls


C (s) = Kp(1 + Tds)


∣C (j!)P(j!)∣ = KKp

√1 + T 2

d!2

1 + T 2!2

[Visioli, 2006]


Derivative Action


KKp

√1 + T 2

d!2

1 + T 2!2≥ KKp min

(1,

Td

T

)

Td ≥ T ⇒ min(

1, TdT

)= 1

If Td ≥ T and KKp > 1, then

∣C (j!)P(j!)∣ ≥ 1

Td ≤ T ⇒ min(

1, TdT

)= Td

T

If Td ≤ T and KKpTdT > 1, then

∣C (j!)P(j!)∣ ≥ 1

[Visioli, 2006]


Derivative Action


KKp

√1 + T 2

d!2

1 + T 2!2≥ KKp min

(1,

Td

T

)Td ≥ T ⇒ min

(1, Td

T

)= 1


∣C (j!)P(j!)∣ ≥ 1

Td ≤ T ⇒ min(

1, TdT

)= Td

T


∣C (j!)P(j!)∣ ≥ 1

[Visioli, 2006]


Derivative Action


KKp

√1 + T 2

d!2

1 + T 2!2≥ KKp min

(1,

Td

T

)Td ≥ T ⇒ min

(1, Td

T

)= 1


∣C (j!)P(j!)∣ ≥ 1

Td ≤ T ⇒ min(

1, TdT

)= Td

T


∣C (j!)P(j!)∣ ≥ 1

[Visioli, 2006]


Derivative Action


KKp

√1 + T 2

d!2

1 + T 2!2≥ KKp min

(1,

Td

T

)Td ≥ T ⇒ min

(1, Td

T

)= 1


∣C (j!)P(j!)∣ ≥ 1

Td ≤ T ⇒ min(

1, TdT

)= Td

T


∣C (j!)P(j!)∣ ≥ 1

[Visioli, 2006]


Derivative Action


KKp

√1 + T 2

d!2

1 + T 2!2≥ KKp min

(1,

Td

T

)Td ≥ T ⇒ min

(1, Td

T

)= 1


∣C (j!)P(j!)∣ ≥ 1

Td ≤ T ⇒ min(

1, TdT

)= Td

T


∣C (j!)P(j!)∣ ≥ 1

[Visioli, 2006]


Frequency Response

The magnitude of the open-looptransfer function is not less than0 dB. As a consequence, sincethe phase decreases when thefrequency increases because ofthe time delay, the closed-loopsystem will be unstable.

[Visioli, 2006]


Derivative Action

Consider the following process:

P(s) =2

s + 1e−0.2s

controlled by a PID controller in series form with Kp = 1 and Ti = 1.

If Td = 0.01, GM=12.3dB, PM=68.2 deg.

If Td = 0.05, GM=13.2dB, PM=72.7 deg.

If Td = 0.5, the system stability is lost!

In summary:

Sensitive to noise

Hard to tune (4 parameters)

[Visioli, 2006]


Derivative Action


P(s) =2

s + 1e−0.2s


If Td = 0.01, GM=12.3dB, PM=68.2 deg.

If Td = 0.05, GM=13.2dB, PM=72.7 deg.


In summary:

Sensitive to noise


[Visioli, 2006]


Derivative Action


P(s) =2

s + 1e−0.2s


If Td = 0.01, GM=12.3dB, PM=68.2 deg.

If Td = 0.05, GM=13.2dB, PM=72.7 deg.


In summary:

Sensitive to noise


[Visioli, 2006]


Derivative Action


P(s) =2

s + 1e−0.2s


If Td = 0.01, GM=12.3dB, PM=68.2 deg.

If Td = 0.05, GM=13.2dB, PM=72.7 deg.


In summary:

Sensitive to noise


[Visioli, 2006]


Derivative Action


P(s) =2

s + 1e−0.2s


If Td = 0.01, GM=12.3dB, PM=68.2 deg.

If Td = 0.05, GM=13.2dB, PM=72.7 deg.


In summary:

Sensitive to noise


[Visioli, 2006]


Derivative Action


P(s) =2

s + 1e−0.2s


If Td = 0.01, GM=12.3dB, PM=68.2 deg.

If Td = 0.05, GM=13.2dB, PM=72.7 deg.


In summary:

Sensitive to noise


[Visioli, 2006]


Integral Windup

Solid:process output,Dashed:process input,Dotted:integral term

[Visioli, 2006]


Integral Windup

Solid:process output,Dashed:process input,Dotted:integral term

[Visioli, 2006]


Conditional Integration

The integral term is limited to a predefined value.

The integration is stopped when the error is greater than a predefinedthreshold, namely, when the process variable value is far from thesetpoint value.

The integration is stopped when the control variable saturates, i.e.,when u

′ ∕= u.

The integration is stopped when the control variable saturates andthe control error and the control variable have the same sign (i.e.,when ue > 0).

[Visioli, 2006]






′ ∕= u.


[Visioli, 2006]






′ ∕= u.


[Visioli, 2006]






′ ∕= u.


[Visioli, 2006]


Anti-windup for Automatic Reset Configuration

Automatic reset:

Limiting the controller output:

Limiting the integrator output:

[Visioli, 2006]



Automatic reset:



[Visioli, 2006]



Automatic reset:



[Visioli, 2006]


Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)

Tuning rule for Tt (Astrom and Hagglund 1995):

Tt =√

TdTi

Not useful for a PI controller

[Visioli, 2006]


Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)


Tt =√TdTi


[Visioli, 2006]


Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)


Tt =√TdTi


[Visioli, 2006]


Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)


Tt =√TdTi



Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)

Bohn and Atherton, 1995 suggest Tt = Ti .

Conditioning technique (Hanus et al., 1987;Walgama et al., 1991):This is a tracking rule (u tracks u

′). In this framework:

Tt = Kp

[Visioli, 2006]


Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)




Tt = Kp

[Visioli, 2006]


Back-calculation

Integrator input:

ei =Kp

Tie +

1

Tt(u′ − u)




Tt = Kp

[Visioli, 2006]


PID with Tracking Input

SP: Setpoint, MV: Manipulated Variable, TR: Tracking Input[Astrom and Hagglund, 1995]


Bumpless Transfer

Bumpless Transfer between Manual (M) and Automatic (A)

A is to track M in this case.

[Visioli, 2006]


Bumpless Transfer


Incremental manual input



Bumpless Transfer


Incremental manual input



Bumpless Transfer

Manual Control Module:



Bumpless Transfer

PID with Manual Switch:



PID Controller Design

In process industries, more than 97% of the regulatory controllers areof the PID type.

Most loops are actually under PI control (as a result of the largenumber of flow loops).

Pulp and paper industry over 2000 loops:

Only 20% of loops worked well (i.e. less variability in the automaticmode over the manual mode).30% gave poor performance due to poor controller tuning.30% gave poor performance due to control valve problems (e.g. controlvalve stick-slip, dead band, backlash).20% gave poor performance due to process and/or control systemdesign problems.

[Yu, 2007]







[Yu, 2007]







[Yu, 2007]






Only 20% of loops worked well (i.e. less variability in the automaticmode over the manual mode).

30% gave poor performance due to poor controller tuning.30% gave poor performance due to control valve problems (e.g. controlvalve stick-slip, dead band, backlash).20% gave poor performance due to process and/or control systemdesign problems.

[Yu, 2007]






Only 20% of loops worked well (i.e. less variability in the automaticmode over the manual mode).30% gave poor performance due to poor controller tuning.

30% gave poor performance due to control valve problems (e.g. controlvalve stick-slip, dead band, backlash).20% gave poor performance due to process and/or control systemdesign problems.

[Yu, 2007]






Only 20% of loops worked well (i.e. less variability in the automaticmode over the manual mode).30% gave poor performance due to poor controller tuning.30% gave poor performance due to control valve problems (e.g. controlvalve stick-slip, dead band, backlash).

20% gave poor performance due to process and/or control systemdesign problems.

[Yu, 2007]







[Yu, 2007]



Process industries:

30% of loops operated on manual mode.20% of controllers used factory tuning.30% gave poor performance due to sensor and control valve problems.

Chemical process industry:

Half of the control valves needed to be fixed (results of the Fisher diagnosticvalve package).Most poor tuning was due to control valve problems.

Refining, chemicals, and pulp and paper industries over 26,000 controllers:

Only 32% of loops were classified as excellent or acceptable.32% of controllers were classified as fair or poor, which indicatesunacceptably sluggish or oscillatory responses.36% of controllers were on open- loop, which implies that the controllerswere either in manual or virtually saturated.PID algorithms are used in vast majority of applications (97%). For the rarecases of complex dynamics or significant dead time, other algorithms areused.

[Yu, 2007]



Process industries:

30% of loops operated on manual mode.

20% of controllers used factory tuning.30% gave poor performance due to sensor and control valve problems.





[Yu, 2007]



Process industries:

30% of loops operated on manual mode.20% of controllers used factory tuning.

30% gave poor performance due to sensor and control valve problems.





[Yu, 2007]



Process industries:






[Yu, 2007]



Process industries:






[Yu, 2007]



Process industries:



Half of the control valves needed to be fixed (results of the Fisher diagnosticvalve package).

Most poor tuning was due to control valve problems.



[Yu, 2007]



Process industries:






[Yu, 2007]



Process industries:






[Yu, 2007]



Process industries:





Only 32% of loops were classified as excellent or acceptable.

32% of controllers were classified as fair or poor, which indicatesunacceptably sluggish or oscillatory responses.36% of controllers were on open- loop, which implies that the controllerswere either in manual or virtually saturated.PID algorithms are used in vast majority of applications (97%). For the rarecases of complex dynamics or significant dead time, other algorithms areused.

[Yu, 2007]



Process industries:





Only 32% of loops were classified as excellent or acceptable.32% of controllers were classified as fair or poor, which indicatesunacceptably sluggish or oscillatory responses.

36% of controllers were on open- loop, which implies that the controllerswere either in manual or virtually saturated.PID algorithms are used in vast majority of applications (97%). For the rarecases of complex dynamics or significant dead time, other algorithms areused.

[Yu, 2007]



Process industries:





Only 32% of loops were classified as excellent or acceptable.32% of controllers were classified as fair or poor, which indicatesunacceptably sluggish or oscillatory responses.36% of controllers were on open- loop, which implies that the controllerswere either in manual or virtually saturated.

PID algorithms are used in vast majority of applications (97%). For the rarecases of complex dynamics or significant dead time, other algorithms areused.

[Yu, 2007]



Process industries:






[Yu, 2007]



Ziegler-Nichols PID Tuning Method:

Ziegler-Nichols closed-loop tuning method

Ziegler-Nichols open-loop tuning method

Ziegler, Nichols, “Optimum settings for automatic controllers”, Trans.ASME, 64, pp. 759-768, 1942[Woolf, 2007]



Ziegler-Nichols closed-loop tuning method:1 Remove integral and derivative action.

2 Create a small disturbance in the loop by changing the set point.Adjust the proportional, increasing and/or decreasing, the gain untilthe oscillations have constant amplitude.

3 Record the gain value (Ku) and period of oscillation (Pu).

[Woolf, 2007]



Ziegler-Nichols closed-loop tuning method:1 Remove integral and derivative action.2 Create a small disturbance in the loop by changing the set point.

Adjust the proportional, increasing and/or decreasing, the gain untilthe oscillations have constant amplitude.


[Woolf, 2007]






[Woolf, 2007]






[Woolf, 2007]






[Woolf, 2007]



Ziegler-Nichols closed-loop tuning method:

[Love, 2007]



Ziegler-Nichols closed-loop tuning method (ultimate cycle method):

C (s) = Kp(1 +1

Ti s+ Tds)

Kp Ti Td

P Controller Ku/2 - -

PI Controller Ku/2.2 Pu/1.2 -

PID Controller Ku/1.7 Pu/2 Pu/8

[Woolf, 2007]




Advantages:

Easy experiment; only need to change the P controllerIncludes dynamics of whole process, which gives a more accuratepicture of how the system is behaving

Disadvantages:

Experiment can be time consumingCan venture into unstable regions while testing the P controller, whichcould cause the system to become out of controlIt does not hold for I , D and PD controllers.

[Woolf, 2007]




Advantages:

Easy experiment; only need to change the P controller

Includes dynamics of whole process, which gives a more accuratepicture of how the system is behaving

Disadvantages:


[Woolf, 2007]




Advantages:


Disadvantages:


[Woolf, 2007]




Advantages:


Disadvantages:


[Woolf, 2007]




Advantages:


Disadvantages:

Experiment can be time consuming

Can venture into unstable regions while testing the P controller, whichcould cause the system to become out of controlIt does not hold for I , D and PD controllers.

[Woolf, 2007]




Advantages:


Disadvantages:

Experiment can be time consumingCan venture into unstable regions while testing the P controller, whichcould cause the system to become out of control

It does not hold for I , D and PD controllers.

[Woolf, 2007]




Advantages:


Disadvantages:


[Woolf, 2007]



Process Reaction Curve:

P: the size of the step disturbance in the setpoint

L: the time taken from the moment the disturbance was introducedto the first sign of change in the output signal

ΔCp: the change in output signal in response to the initial stepdisturbance

T : the time taken for this change to occur

[Woolf, 2007]



Process Reaction Curve:

N =ΔCp

T : reaction rate

[Woolf, 2007]



Ziegler-Nichols open-loop tuning method (process reaction method):

C (s) = Kp(1 +1

Ti s+ Tds)

Kp Ti Td

P Controller K - -

PI Controller 0.9K L/0.3 -

PID Controller 1.2K 2L 0.5L

K =P

NL[Woolf, 2007]



Ziegler-Nichols open-loop tuning method:

Advantages:

Quick and easier to use than other methodsIt is a robust and popular methodOf these two techniques, the Process Reaction Method is the easiestand least disruptive to implement

Disadvantages:

It depends upon purely proportional measurement to estimate I and Dcontrollers.Approximations for the Kc , Ti , and Td values might not be entirelyaccurate for different systems.It does not hold for I , D and PD controllers.

[Woolf, 2007]




Advantages:

Quick and easier to use than other methods

It is a robust and popular methodOf these two techniques, the Process Reaction Method is the easiestand least disruptive to implement

Disadvantages:


[Woolf, 2007]




Advantages:

Quick and easier to use than other methodsIt is a robust and popular method

Of these two techniques, the Process Reaction Method is the easiestand least disruptive to implement

Disadvantages:


[Woolf, 2007]




Advantages:


Disadvantages:


[Woolf, 2007]




Advantages:


Disadvantages:


[Woolf, 2007]




Advantages:


Disadvantages:

It depends upon purely proportional measurement to estimate I and Dcontrollers.

Approximations for the Kc , Ti , and Td values might not be entirelyaccurate for different systems.It does not hold for I , D and PD controllers.

[Woolf, 2007]




Advantages:


Disadvantages:

It depends upon purely proportional measurement to estimate I and Dcontrollers.Approximations for the Kc , Ti , and Td values might not be entirelyaccurate for different systems.

It does not hold for I , D and PD controllers.

[Woolf, 2007]




Advantages:


Disadvantages:


[Woolf, 2007]



Summary:

Ziegler-Nichols closed-loop tuning method: Gain margin of 2

Ziegler-Nichols open-loop tuning method: Decay ratio of 0.25



Ziegler-Nichols closed-loop tuning methods:

G (s) =Ke−Ds

�s + 1

[Yu, 2007]



Ziegler-Nichols closed-loop tuning methods:

G (s) =Ke−td s

�s + 1

Quarter decay ration for 0 < td� < 1 [O’Dwyer, 2002]

[Chau, 2002]



Quantities to characterize the error:

Maximum error: max e(t)

Integrated Absolute Error: IAE =∫∞

0 ∣e(t)∣dtIntegrated Error for non-oscillatory processes: IE =

∫∞0 e(t)dt

Integrated Squared Error: ISE =∫∞

0 e2(t)dt

Integrated Time Absolute Error: ITAE =∫∞

0 t∣e(t)∣dtIntegrated Time Error: ITE =

∫∞0 te(t)dt

Integrated Time Squared Error: ITSE =∫∞

0 te2(t)dt

Integrated Squared Time Error: ISTE =∫∞

0 t2e2(t)dt







0 ∣e(t)∣dt

Integrated Error for non-oscillatory processes: IE =∫∞

0 e(t)dt


0 e2(t)dt



∫∞0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt








∫∞0 e(t)dt


0 e2(t)dt



∫∞0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt








∫∞0 e(t)dt


0 e2(t)dt



∫∞0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt








∫∞0 e(t)dt


0 e2(t)dt


0 t∣e(t)∣dt

Integrated Time Error: ITE =∫∞

0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt








∫∞0 e(t)dt


0 e2(t)dt



∫∞0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt








∫∞0 e(t)dt


0 e2(t)dt



∫∞0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt








∫∞0 e(t)dt


0 e2(t)dt



∫∞0 te(t)dt


0 te2(t)dt


0 t2e2(t)dt




ITAE optimization:

G (s) =Ke−td s

�s + 1

[Chau, 2002]



Ciancone and Marlin Tuning:

G (s) =Ke−td s

�s + 1

Fractional dead time: Tf = tdtd +�

Using Tf , compute �CM and �CM :

Tf 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

�CM 1.1 1.1 1.8 1.1 1.0 0.8 0.59 0.42 0.32

�CM 0.23 0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53

Compute the controller gains:

Kp =�CM

K, Ti = �CM(td + �)

Minimizing IAE or ISE [Chau, 2002]



Ciancone and Marlin PI Tuning:

[Chau, 2002]



Ciancone and Marlin PID Tuning:

[Chau, 2002]



Direct Synthesis:

We have:C

R=

GcGp

1 + GcGp

Therefore:

Gc =1

Gp

C/R

1− C/R

[Chau, 2002]



Direct Synthesis:

C

R=

1

�cs + 1⇒ Gc =

1

Gp

(1

�cs

)

Example:

Gp =Kp

�ps + 1⇒ Gc =

�p

Kp�c

(1 +

1

�ps

)[Chau, 2002]



Direct Synthesis:

C

R=

1

�cs + 1⇒ Gc =

1

Gp

(1

�cs

)Example:

Gp =Kp

�ps + 1⇒ Gc =

�p

Kp�c

(1 +

1

�ps

)[Chau, 2002]



Direct Synthesis:

For delayed systems:

C

R=

e−�s

�cs + 1⇒ Gc =

1

Gp

(e−�s

(�cs + 1)− e−�s

)

Considering e−�s ≈ 1− �s:

Gc ≈1

Gp

(e−�s

(�c + �)s

)

Example:

Gp =Kpe

−td s

�ps + 1⇒ Gc =

�p

Kp(�c + �)

(1 +

1

�ps

)for � = td

[Chau, 2002]



Direct Synthesis:


C

R=

e−�s

�cs + 1⇒ Gc =

1

Gp

(e−�s

(�cs + 1)− e−�s

)Considering e−�s ≈ 1− �s:

Gc ≈1

Gp

(e−�s

(�c + �)s

)

Example:

Gp =Kpe

−td s

�ps + 1⇒ Gc =

�p

Kp(�c + �)

(1 +

1

�ps

)for � = td

[Chau, 2002]



Direct Synthesis:


C

R=

e−�s

�cs + 1⇒ Gc =

1

Gp

(e−�s

(�cs + 1)− e−�s


Gc ≈1

Gp

(e−�s

(�c + �)s

)

Example:

Gp =Kpe

−td s

�ps + 1⇒ Gc =

�p

Kp(�c + �)

(1 +

1

�ps

)for � = td

[Chau, 2002]



Direct Synthesis:


C

R=

e−�s

�cs + 1⇒ Gc =

1

Gp

(e−�s

(�cs + 1)− e−�s


Gc ≈1

Gp

(e−�s

(�c + �)s

)

Example:

Gp =Kpe

−td s

�ps + 1⇒ Gc =

�p

Kp(�c + �)

(1 +

1

�ps

)for � = td

[Chau, 2002]



Direct Synthesis:

Second order underdamped desired response:

C

R=

1

�2s2 + 2��s + 1⇒ Gc =

1

Gp

(1

�2s2 + 2��s

)

Example:

Gp =Kp

(�1s + 1)(�2s + 1)⇒ Gc =

(�1s + 1)(�2s + 1)

Kp�s(�s + 2�)

Assuming �2 > �1 and � = 2��2

Gc =�1

4Kp�2�2

(1 +

1

�1s

)[Chau, 2002]



Direct Synthesis:


C

R=

1

�2s2 + 2��s + 1⇒ Gc =

1

Gp

(1

�2s2 + 2��s

)

Example:

Gp =Kp

(�1s + 1)(�2s + 1)⇒ Gc =

(�1s + 1)(�2s + 1)

Kp�s(�s + 2�)

Assuming �2 > �1 and � = 2��2

Gc =�1

4Kp�2�2

(1 +

1

�1s

)[Chau, 2002]



Direct Synthesis:


C

R=

1

�2s2 + 2��s + 1⇒ Gc =

1

Gp

(1

�2s2 + 2��s

)

Example:

Gp =Kp

(�1s + 1)(�2s + 1)⇒ Gc =

(�1s + 1)(�2s + 1)

Kp�s(�s + 2�)

Assuming �2 > �1 and � = 2��2

Gc =�1

4Kp�2�2

(1 +

1

�1s

)[Chau, 2002]



Direct Synthesis:


C

R=

1

�2s2 + 2��s + 1⇒ Gc =

1

Gp

(1

�2s2 + 2��s

)

Example:

Gp =Kp

(�1s + 1)(�2s + 1)⇒ Gc =

(�1s + 1)(�2s + 1)

Kp�s(�s + 2�)

Assuming �2 > �1 and � = 2��2

Gc =�1

4Kp�2�2

(1 +

1

�1s

)[Chau, 2002]



Internal Model Control: Assume that we have an approximate model Gp ofthe process Gp

Open loop:

Gc = G−1p

Closed loop:

[Chau, 2002]



Internal Model Control: Assume that we have an approximate model Gp ofthe process Gp

Open loop:

Gc = G−1p

Closed loop:

[Chau, 2002]



Internal Model Control:

P = G ★c (R − C + C )

= G ★c (R − C + GpP)

Therefore:

P =G ★

c

1− G ★c Gp

(R−C )

P = Gc (R − C )

Therefore:

Gc = G★c1−G★c Gp

[Chau, 2002]




P = G ★c (R − C + C )

= G ★c (R − C + GpP)

Therefore:

P =G ★

c

1− G ★c Gp

(R−C )

P = Gc (R − C )

Therefore:


[Chau, 2002]




P = G ★c (R − C + C )

= G ★c (R − C + GpP)

Therefore:

P =G ★

c

1− G ★c Gp

(R−C )

P = Gc (R − C )

Therefore:


[Chau, 2002]Behzad Samadi (Amirkabir University) Industrial Control 64 / 95



Closed loop transfer function:

C =

[(1− G ★

c Gp)GL

1 + G ★c (Gp − Gp)

]L +

[GpG

★c

1 + G ★c (Gp − Gp)

]R

How to choose G ★c :

Gp = Gp+Gp−

Gp+ contains all positive zeros if any.

The design is based on Gp− only:

G ★c =

1

Gp−

[1

�cs + 1

]r

r = 1, 2

[Chau, 2002]





C =

[(1− G ★

c Gp)GL

1 + G ★c (Gp − Gp)

]L +

[GpG

★c

1 + G ★c (Gp − Gp)

]R


Gp = Gp+Gp−



G ★c =

1

Gp−

[1

�cs + 1

]r

r = 1, 2

[Chau, 2002]





C =

[(1− G ★

c Gp)GL

1 + G ★c (Gp − Gp)

]L +

[GpG

★c

1 + G ★c (Gp − Gp)

]R


Gp = Gp+Gp−



G ★c =

1

Gp−

[1

�cs + 1

]r

r = 1, 2

[Chau, 2002]



Pade Approximation of Delays:

e−td s ≈ Nd (s)Dd (s)

Gd1/0(s) = 1− tds

Gd0/1(s) = 11+td s

Gd1/1(s) =− td

2s+1

td2

s+1

Gd2/2(s) =t2d

12s2− td

2s+1

t2d

12s2+

td2

s+1

http://mathworld.wolfram.com/PadeApproximant.html






Gd1/0(s) = 1− tds

Gd0/1(s) = 11+td s

Gd1/1(s) =− td

2s+1

td2

s+1

Gd2/2(s) =t2d

12s2− td

2s+1

t2d

12s2+

td2

s+1







Gd1/0(s) = 1− tds

Gd0/1(s) = 11+td s

Gd1/1(s) =− td

2s+1

td2

s+1

Gd2/2(s) =t2d

12s2− td

2s+1

t2d

12s2+

td2

s+1







Gd1/0(s) = 1− tds

Gd0/1(s) = 11+td s

Gd1/1(s) =− td

2s+1

td2

s+1

Gd2/2(s) =t2d

12s2− td

2s+1

t2d

12s2+

td2

s+1







Gd1/0(s) = 1− tds

Gd0/1(s) = 11+td s

Gd1/1(s) =− td

2s+1

td2

s+1

Gd2/2(s) =t2d

12s2− td

2s+1

t2d

12s2+

td2

s+1






Example: Gp =Kpe−td s

�ps+1

Gp ≈Kp

(�ps + 1)( td2 s + 1)

(− td

2s + 1

)

Gp− =Kp

(�ps + 1)( td2 s + 1)

Gp+ = − td

2s + 1

[Chau, 2002]




Example: Gp =Kpe−td s

�ps+1

Gp ≈Kp

(�ps + 1)( td2 s + 1)

(− td

2s + 1

)

Gp− =Kp

(�ps + 1)( td2 s + 1)

Gp+ = − td

2s + 1

[Chau, 2002]




Example:

G ★c = G−1

p−1

�cs + 1=

(�ps + 1)( td2 s + 1)

Kp

1

�cs + 1

Gc =G ★

c

1− G ★c Gp

= Kp(1 +1

Ti s+ Tds)

where:

Kc =1

Kp

2�p

td+ 1

2 �ctd

+ 1; Ti = �p +

td

2; Td =

�p

2�p

td+ 1

[Chau, 2002]




Example:

G ★c = G−1

p−1

�cs + 1=

(�ps + 1)( td2 s + 1)

Kp

1

�cs + 1

Gc =G ★

c

1− G ★c Gp

= Kp(1 +1

Ti s+ Tds)

where:

Kc =1

Kp

2�p

td+ 1

2 �ctd

+ 1; Ti = �p +

td

2; Td =

�p

2�p

td+ 1

[Chau, 2002]



[Chau, 2002]Behzad Samadi (Amirkabir University) Industrial Control 69 / 95


Skogestad PID Tuning Method:

S. Skogestad, “Probably the best simple PID tuning rules in theworld” Presented at AIChE Annual meeting, Reno, NV, USA, 04-09Nov. 2001, Paper no. 276h

[Skogestad, 2001]



Skogestad PID Tuning Method:

g(s) = ke−�s

(�1s + 1)(�2s + 1)

[Skogestad, 2001]



Skogestad Internal Model Control (SIMC):(y

ys

)desired

=1

�cs + 1e−�t

c(s) =(�1s + 1)(�2s + 1)

k

1

(�c + �)s

c(s) = Kc(1 +1

�I s)(�Ds + 1)

Tuning Rule

Kc =1

k

�1

�c + �, �I = �1. �D = �2

[Skogestad, 2001]




ys

)desired

=1

�cs + 1e−�t

c(s) =(�1s + 1)(�2s + 1)

k

1

(�c + �)s

c(s) = Kc(1 +1

�I s)(�Ds + 1)

Tuning Rule

Kc =1

k

�1

�c + �, �I = �1. �D = �2

[Skogestad, 2001]




ys

)desired

=1

�cs + 1e−�t

c(s) =(�1s + 1)(�2s + 1)

k

1

(�c + �)s

c(s) = Kc(1 +1

�I s)(�Ds + 1)

Tuning Rule

Kc =1

k

�1

�c + �, �I = �1. �D = �2

[Skogestad, 2001]




ys

)desired

=1

�cs + 1e−�t

c(s) =(�1s + 1)(�2s + 1)

k

1

(�c + �)s

c(s) = Kc(1 +1

�I s)(�Ds + 1)

Tuning Rule

Kc =1

k

�1

�c + �, �I = �1. �D = �2

[Skogestad, 2001]



Skogestad Internal Model Control (SIMC):

Tuning Rule

Kc =1

k

�1

�c + �, �I = �1, �D = �2

Effective cancellation of the first order dynamics by �I = �1

For processes with large �1, this choice results in a long settling timefor disturbance response.

Consider: g(s) = k e−�s

�1s+1 ≈k�1s for large �1. With a PI controller

c(s) = Kc(1 + 1�I s ), the poles of the closed loop system can be

obtained from the following equation:

�I

k ′Kcs2 + �I s + 1 = 0

with k′

= k�1

[Skogestad, 2001]




Tuning Rule

Kc =1

k

�1

�c + �, �I = �1, �D = �2







�I

k ′Kcs2 + �I s + 1 = 0

with k′

= k�1

[Skogestad, 2001]




Tuning Rule

Kc =1

k

�1

�c + �, �I = �1, �D = �2







�I

k ′Kcs2 + �I s + 1 = 0

with k′

= k�1

[Skogestad, 2001]




Comparing to s2 + 2�!0s + !2:

!0 =

√k ′Kc

�I, � =

1

2

√k ′Kc�I

To avoid slow oscillations, consider: � ≥ 1

�I ≥4

k ′Kc

[Skogestad, 2001]





!0 =

√k ′Kc

�I, � =

1

2

√k ′Kc�I


�I ≥4

k ′Kc

[Skogestad, 2001]





!0 =

√k ′Kc

�I, � =

1

2

√k ′Kc�I


�I ≥4

k ′Kc

[Skogestad, 2001]




Disturbance rejection: Assuming input disturbance (gd (s) = g(s))

∣y(j!)∣ =∣g(j!)∣

∣1 + g(j!)c(j!)∣d ≤ ymax

Assuming ∣gc ∣ >> 1 at low frequencies:

∣c(j!)∣ > d

ymax

For P, PI and PID: c(j!) ≥ Kc

Kc ≥d

ymax

[Skogestad, 2001]





∣y(j!)∣ =∣g(j!)∣

∣1 + g(j!)c(j!)∣d ≤ ymax


∣c(j!)∣ > d

ymax


Kc ≥d

ymax

[Skogestad, 2001]





∣y(j!)∣ =∣g(j!)∣

∣1 + g(j!)c(j!)∣d ≤ ymax


∣c(j!)∣ > d

ymax


Kc ≥d

ymax

[Skogestad, 2001]





∣y(j!)∣ =∣g(j!)∣

∣1 + g(j!)c(j!)∣d ≤ ymax


∣c(j!)∣ > d

ymax


Kc ≥d

ymax

[Skogestad, 2001]




Skogestad Internal Model Control (SIMC)

Kc =1

k

�1

�c + �=

1

k ′1

�c + �

�I = min{�1,4

k ′Kc} = min{�1, 4(�c + �)}

�D = �2

Tuning for fast response with good robustness

�c = �

Tuning for slow response with acceptable disturbance rejection

Kc ≥d

ymax

[Skogestad, 2001]



Rules of thumb:

[McMillan, 2001]


Digital Control

Ts : sampling time

Analog to Digital (A-D) conversion:

y(tk) = y(kTs)

Digital to Analog (D-A) conversion:

u(t) = u(tk) for kTs ≤ t < (k + 1)Ts

[Astrom and Wittenmark, 1996]


Digital Control

Ts : sampling timeAnalog to Digital (A-D) conversion:

y(tk ) = y(kTs)


u(t) = u(tk) for kTs ≤ t < (k + 1)Ts



Digital Control

Ts : sampling timeAnalog to Digital (A-D) conversion:

y(tk ) = y(kTs)


u(t) = u(tk ) for kTs ≤ t < (k + 1)Ts



Digital Control

Difference equation:

x((k + 1)Ts) = ax(kTs) + bu(kTs)

Discrete time:x [k + 1] = ax [k] + bu[k]

Z -transform:

zX (z) = aX (z) + bU(z)⇒ X (z)

U(z)=

b

z − a



Digital Control




Z -transform:

zX (z) = aX (z) + bU(z)⇒ X (z)

U(z)=

b

z − a



Digital Control




Z -transform:

zX (z) = aX (z) + bU(z)⇒ X (z)

U(z)=

b

z − a



Digital Control

Approximating Continuous-Time Controllers:

Euler’s method (forward difference):

z = eTs s ≈ 1 + Tss ⇒ s ≈ z − 1

Ts

x(t) ≈ x(t + Ts)− x(t)

Ts

Backward difference:

z = eTs s ≈ 1

1− Tss⇒ s ≈ 1− z−1

Ts

x(t) ≈ x(t)− x(t − Ts)

Ts



Digital Control


Euler’s method (forward difference):

z = eTs s ≈ 1 + Tss ⇒ s ≈ z − 1

Ts

x(t) ≈ x(t + Ts)− x(t)

Ts

Backward difference:

z = eTs s ≈ 1

1− Tss⇒ s ≈ 1− z−1

Ts

x(t) ≈ x(t)− x(t − Ts)

Ts



Digital Control


Tustin’s approximation:

z = eTs s ≈ 1 + sTs/2

1− sTs/2⇒ s ≈ 2

Ts

z − 1

z + 1

It is also called trapezoidal method or bilinear transformation.



Digital Control


Stability: Images of the left side of the s-plane[Astrom and Wittenmark, 1996]


Digital Control


Re(s) < 0Forward−−−−−−→

DifferenceRe(z) < 1

Stable system in s domainForward−−−−−−→

DifferenceStable or unstable system in z

domain

Re(s) < 0Backward−−−−−−→Difference

(Re(z)− 12 )2 + Im(z)2 < 1

4

Stable or unstable system in s domainBackward−−−−−−→Difference

Stable system in z

domain

Re(s) < 0Tustin−−−−−−−−→

Approximation∣z ∣ < 1

Stable system in s domainTustin−−−−−−−−→

ApproximationStable system in z domain



Digital Control



DifferenceRe(z) < 1



domain


(Re(z)− 12 )2 + Im(z)2 < 1

4


Stable system in z

domain

Re(s) < 0Tustin−−−−−−−−→






Digital Control



DifferenceRe(z) < 1



domain


(Re(z)− 12 )2 + Im(z)2 < 1

4


Stable system in z

domain

Re(s) < 0Tustin−−−−−−−−→






Digital Control



DifferenceRe(z) < 1



domain


(Re(z)− 12 )2 + Im(z)2 < 1

4


Stable system in z

domain

Re(s) < 0Tustin−−−−−−−−→






Digital Control



DifferenceRe(z) < 1



domain


(Re(z)− 12 )2 + Im(z)2 < 1

4


Stable system in z

domain

Re(s) < 0Tustin−−−−−−−−→






Digital Control



DifferenceRe(z) < 1



domain


(Re(z)− 12 )2 + Im(z)2 < 1

4


Stable system in z

domain

Re(s) < 0Tustin−−−−−−−−→






Digital Control

Digital PID:

Proportional: P(t) = Kpe(t)

P[k] = Kpe[k]

Integral: I (t) =Kp

Ti

∫ t0 e(�)d�

I [k + 1] = I [k] +KTs

Tie[k]

Derivative: TdN

dDdt + D = −KpTd

dydt

D[k] =Td

Td + NTsD[k − 1]− KpTdN

Td + NTs(y [k]− y [k − 1])



Digital Control

Digital PID:


P[k] = Kpe[k]

Integral: I (t) =Kp

Ti

∫ t0 e(�)d�

I [k + 1] = I [k] +KTs

Tie[k]

Derivative: TdN

dDdt + D = −KpTd

dydt

D[k] =Td


Td + NTs(y [k]− y [k − 1])



Digital Control

Digital PID:


P[k] = Kpe[k]

Integral: I (t) =Kp

Ti

∫ t0 e(�)d�

I [k + 1] = I [k] +KTs

Tie[k]

Derivative: TdN

dDdt + D = −KpTd

dydt

D[k] =Td


Td + NTs(y [k]− y [k − 1])



Digital Control

Typical sampling time:

Type of variable Sampling time (sec)

Flow 1-3

Level 5-10

Pressure 1-5

Temperature 10-20

Rule of thumb for PI controllers:

Ts

Ti≈ 0.1 to 0.3



Digital Control


Type of variable Sampling time (sec)

Flow 1-3

Level 5-10

Pressure 1-5

Temperature 10-20

Rule of thumb for PI controllers:

Ts

Ti≈ 0.1 to 0.3



Digital Control


Rule of thumb for PID controllers:

NTs

Td≈ 0.2 to 0.6



Digital Control

Inverse Response: When the initial response of a dynamic system is in adirection opposite to the final outcome, it is called an inverse response.

Obtaining the effective delay (Half rule):

Effective delay =True delay + inverse response time constant(s)

+ half of the largest neglected time constant

+ all smaller higher order time constants

Time constant:

� = the largest time constant +1

2the second largest time constant{

�1 = the largest time constant,�2 = the second largest time constant + 1

2 the third largest time constant

[Skogestad, 2001]


Digital Control

Inverse Response: When the initial response of a dynamic system is in adirection opposite to the final outcome, it is called an inverse response.Obtaining the effective delay (Half rule):




Time constant:





[Skogestad, 2001]


Digital Control

Inverse Response: When the initial response of a dynamic system is in adirection opposite to the final outcome, it is called an inverse response.Obtaining the effective delay (Half rule):




Time constant:





[Skogestad, 2001]


Digital Control

Plant transfer function:

G (s) =

∏j (−Tj0s + 1)∏

j (�i0s + 1)e−td s

First order approximation:

G1(s) =e−�s

�s + 1

� =td +∑

j

Tj0 +�20

2+∑j≥3

�i0 +Ts

2

� =�10 +�20

2

[Skogestad, 2001]


Digital Control

Second order approximation:

G2(s) =e−�s

(�1s + 1)(�2s + 1)

� =td +∑

j

Tj0 +�30

2+∑j≥4

�i0 +Ts

2

�1 =�10

�2 =�20 +�30

2

[Skogestad, 2001]


Digital Control

Series form of the PID controller:

U(s) = Kp(1 + 1�i s

)(1 + �ds)E (s)

U(s) = Kp(1 + 1�i s

)(E (s) + �dsE (s))

U(s) = Kp(1 + 1�i s

)(E (s)− �dsY (s))

U(s) = Kp(1 + 1�i s

)(R(s)− (1 + �ds)Y (s))

U(s) = Kp(1 + 1�i s

)Ed (s)


Digital Control


U(s) = Kp(1 + 1�i s

)(1 + �ds)E (s)

U(s) = Kp(1 + 1�i s

)(E (s) + �dsE (s))

U(s) = Kp(1 + 1�i s

)(E (s)− �dsY (s))

U(s) = Kp(1 + 1�i s

)(R(s)− (1 + �ds)Y (s))

U(s) = Kp(1 + 1�i s

)Ed (s)


Digital Control


U(s) = Kp(1 + 1�i s

)(1 + �ds)E (s)

U(s) = Kp(1 + 1�i s

)(E (s) + �dsE (s))

U(s) = Kp(1 + 1�i s

)(E (s)− �dsY (s))

U(s) = Kp(1 + 1�i s

)(R(s)− (1 + �ds)Y (s))

U(s) = Kp(1 + 1�i s

)Ed (s)


Digital Control


U(s) = Kp(1 + 1�i s

)(1 + �ds)E (s)

U(s) = Kp(1 + 1�i s

)(E (s) + �dsE (s))

U(s) = Kp(1 + 1�i s

)(E (s)− �dsY (s))

U(s) = Kp(1 + 1�i s

)(R(s)− (1 + �ds)Y (s))

U(s) = Kp(1 + 1�i s

)Ed (s)


Digital Control


U(s) = Kp(1 + 1�i s

)(1 + �ds)E (s)

U(s) = Kp(1 + 1�i s

)(E (s) + �dsE (s))

U(s) = Kp(1 + 1�i s

)(E (s)− �dsY (s))

U(s) = Kp(1 + 1�i s

)(R(s)− (1 + �ds)Y (s))

U(s) = Kp(1 + 1�i s

)Ed (s)


Digital Control



Digital Control

Antiwindup digital PID:

public class SimplePID {

private double u,e,v,y;

private double K,Ti,Td,Beta,Tr,N,h;

private double ad,bd;

private double D,I,yOld;

public SimplePID(double nK, double nTi, double NTd,

double nBeta, double nTr, double nN, double nh) {

updateParameters(nK,nTi,nTd,nBeta,nTr,nN,nh);

}

ARTIST Graduate Course on Embedded Control Systems


Digital Control


public void updateParameters(double nK, double nTi,

double NTd, double nBeta, double nTr, double nN,

double nh) {

K = nK; Ti = nTi; Td = nTd; Beta = nBeta;

Tr = nTr

N = nN;

h = nh;

ad = Td / (Td + N*h);

bd = K*ad*N;

}



Digital Control


public double calculateOutput(double yref, double newY) {

y = newY;

e = yref - y;

D = ad*D - bd*(y - yOld);

v = K*(Beta*yref - y) + I + D;

return v;

}

public void updateState(double u) {

I = I + (K*h/Ti)*e + (h/Tr)*(u - v);

yOld = y;

}

}



Digital Control


public class Regul extends Thread {

private SimplePID pid;

public Regul() { pid = new SimplePID(1,10,0,1,10,5,0.1); }

public void run() {

// Other stuff

while (true) {

y = getY();

yref = getYref():

u = pid.calculateOutput(yref,y);

u = limit(u);

setU(u);

pid.updateState(u);

// Timing Code

}}}



Astrom, K. J. and Hagglund, T. (1995).PID Controllers: Theory, Design, and Tuning.Instrument Society of America.

Astrom, K. J. and Wittenmark, B. (1996).Computer-Controlled Systems: Theory and Design.Prentice Hall, 3 edition.

Chau, P. C. (2002).Process Control: A First Course with MATLAB (Cambridge Series inChemical Engineering).Cambridge University Press, 1 edition.

Li, Y., Ang, K. H., and Chong, G. C. Y. (2006).Pid control system analysis and design.IEEE Control Syst. Mag., 26(1):32–41.

Love, J. (2007).Process Automation Handbook: A Guide to Theory and Practice.Springer, 1 edition.


McMillan, G. K. (2001).Good Tuning: A Pocket Guide.The Instrumentation, Systems, and Automation Society (ISA).

O’Dwyer (2002).Handbook of PI & Pid Controller Tuning Rules.World Scientific Publishing.

Skogestad, S. (2001).Probably the best simple pid tuning rules in the world.In AIChE Annual meeting, Reno, NV, USA.

Visioli, A. (2006).Practical PID Control (Advances in Industrial Control).Springer, 1 edition.

Woolf, P., editor (2007).The Michigan Chemical Process Dynamics and Controls Open TextBook.

Yu, C.-C. (2007).


Autotuning of PID Controllers: A Relay Feedback Approach.Springer.


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