2. Single-Loop Control Enhancements - Åbo Akademi · 2. Single-Loop Control Enhancements 2.1 PID...

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Laboratory 2. Single-Loop Control Enhancements 2.1 PID controller enhancements 2.1.1 The ideal PID controller 2.1.2 Derivative filter 2.1.3 Setpoint weighting 2.1.4 Handling integrator windup 2.1.5 Industrial PID controllers

2.2 Model-based controllers 2.2.1 Time-delay compensation 2.2.2 Internal model control (IMC) 2.2.3 Gain scheduling 2.2.4 Sampled-data control

2.3 Use of extra measurements 2.3.1 Cascade control 2.3.2 Feedforward control 2.3.3 Selective control

KEH Plantwide Control 2–1

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2. Single-Loop Control Enhancements

2.4 Use of extra actuators 2.4.1 Split-range control 2.4.2 Coordinated control 2.4.3 Valve position control


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2. Single-Loop Control Enhancements The basic feedback control loop

[WÅJ]

𝑦 = measured output (= controlled variable) 𝑑 = disturbance (acting on output) 𝑢 = control (manipulated) variable 𝑟 = setpoint (desired value of controlled variable) 𝑒 = 𝑟 − 𝑦 = control error


r ed

yu

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Control-loop enhancements

In single-loop control, the objective is to control the measured output 𝑦 (a single variable) well according to some criterion, using a control signal 𝑢 affecting a single actuator.

In this chapter we are considering ways of improving the control of 𝑦 when basic single-loop control is insufficient (because of the process properties).

The methods include

enhancements to the ideal PID controller

model-based controllers

use of extra measurements

use of extra actuators


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2.1 PID controller enhancements PID controller (”pee-i-dee”) is a generic name for a controller containing a linear combination of proportional (P) integral (I) derivative (D)

terms acting on a control error (or sometimes the process output). All parts need not be present. Frequently I and/or D action is missing, giving a controller like P, PI, or PD controller

It has been estimated that of all controllers in the world 95 % are PID controllers


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2. Single-Loop Control Enhancements 2.1 PID controller enhancements

2.1.1 The ideal PID controller Control law 𝑢 𝑡 = 𝐾c 𝑒 𝑡 + 1

𝑇i∫ 𝑒 𝜏 𝑑𝜏𝑡0 + 𝑇d

d𝑒(𝑡)d𝑡

+ 𝑢0

= 𝐾c𝑒 𝑡 + 𝐾i ∫ 𝑒 𝜏 𝑑𝜏𝑡0 + 𝐾d

d𝑒(𝑡)d𝑡

+ 𝑢0

Transfer function

𝐺 𝑠 = 𝐾c 1 + 1𝑇i𝑠

+ 𝑇d𝑠 = 𝐾c𝑇i𝑠

1 + 𝑇i𝑠 + 𝑇i𝑇d𝑠2

Block diagrams [ÅM]

A possible way of implementing a PI controller


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1iK s−

dK s

cK

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2.1.2 Derivative filter Problems with derivative term measurement noise setpoint changes cannot be implemented exactly

A solution [Silva et al.: PID Controllers for Time- include derivative filter Delay Systems, Birkhäuser, 2005] do not differentiate setpoint

𝑢 𝑡 = 𝐾c 𝑒 𝑡 + 1𝑇i∫ 𝑒 𝜏 𝑑𝜏𝑡0 − 𝑇d

d𝑦f(𝑡)d𝑡

+ 𝑢0, 𝑇fd𝑦fd𝑡

+ 𝑦f = 𝑦

𝑈 𝑠 = 𝐾c 1 + 1𝑇i𝑠

𝐸 𝑠 − 𝐾c𝑇d𝑠𝑇f𝑠+1

𝑌(𝑠)

An easy way of implementing a differentiator + filter exactly [ÅM]


f

11 T s−+

cK ′

dc c

f

TK KT

=′

y

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2.1.3 Setpoint weighting A controller has (or can have) two main tasks disturbance rejection setpoint tracking

A problem is that the optimal tuning of a standard PID controller for one of the tasks is seldom (even close to) optimal for the other task.

One solution to this problem is to use setpoint weighting

𝑒p(t) ≡ 𝑏𝑟(𝑡) − 𝑦(𝑡), 𝑒(𝑡) ≡ 𝑟(𝑡) − 𝑦(𝑡), 𝑒d(𝑡) ≡ 𝑐𝑟(𝑡) − 𝑦f(𝑡)

𝑢 𝑡 = 𝐾c 𝑒p 𝑡 + 1𝑇i∫ 𝑒 𝜏 𝑑𝜏𝑡0 + 𝑇d

d𝑒d(𝑡)d𝑡

+ 𝑢0, 𝑇fd𝑦fd𝑡

+ 𝑦f = 𝑦

𝑈 𝑠 = 𝐾c 𝑏 + 1𝑇i𝑠

+ 𝑐𝑇d𝑠 𝑅 𝑠 − 𝐾c 1 + 1𝑇i𝑠

+ 𝑇d𝑠𝑇f𝑠+1

𝑌 𝑠

= 𝐺vPID𝑅 𝑠 − 𝐺PIDf𝑌(𝑠)


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2.1 PID controller enhancements 2.1.3 Setpoint weighting

The controller is a special case of a two-degrees-of-freedom (2DOF) controller

[Visioli]

An equivalent structure ( 𝐹 = 𝐺vPID/𝐺PIDf )

[Visioli]


PIDfG

vPIDG

G

G

u

PIDfG

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2.1 PID controller enhancements 2.1.3 Setpoint weighting

Control illustration [WÅJ]

Response of system to a setpoint change and a load disturbance for controller with different values of 𝛽 (= 𝑏 in equations).


u

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2.1.4 Handling integrator windup A practical problem with integral action in a controller (a PI or PID controller) is integrator windup. This can occur when the control signal reaches a constraint, e.g., a valve fully open or closed.

Control illustration of integrator windup [ÅH]

PI control of an integrating process, when the control signal is limited as −0.1 ≤ 𝑢 ≤ 0.1. The setpoint change causes the control signal to saturate momentarily (because of the value of 𝐾c). Control law: 𝑢 𝑡 = 𝐾c𝑒 + 𝐼

𝐼 = 𝐾c𝑇i∫ 𝑒 𝜏 𝑑𝜏𝑡0


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2.1 PID controller enhancements 2.1.4 Handling integrator windup

There are several ways to avoid integrator windup. One possibility is to stop updating the integral when the actuator saturates; this is called

conditional integration. A more advanced method is to recompute the integral term so that its new value gives an output at

the saturation limit; this is called back-calculation (or tracking). However, it is advantageous not to reset the integrator immediately, but dynamically, as illustrated in the block diagram.


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tK

iK

cK

dK sf

11T s +

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2.1 PID controller enhancements 2.1.4 Handling integrator windup

cKFor a PI controller, a simplified anti-windup scheme, corre- sponding to 𝐾t = 𝐾i/𝐾c, is shown in the block diagram. [Visioli] Control illustration with anti-windup technique [ÅH]

The process is the same as in the integrator windup illustration.

Here anti-windup using back-calculation is applied.

Notice that the output of the integrator is quickly reset to a value such that the controller output is at the saturation limit, giving the integral a negative value during the initial phase.


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2.1.5 Industrial PID controllers Industrial PID controllers in automation systems contain many more special functions than those presented above.

The following illustration is not even an extreme case.

PID block with the inputs and outputs required for various forms of tracking. [Smith]


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2.1 PID controller enhancements 2.1.5 Industrial PID controllers

Inputs and outputs for the PID block. [Smith]


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2.2 Model-based controllers In spite of its simplicity, a PID controller can be tuned to perform surprisingly well for processes with “simple” dynamics.

The structure of the PID controller is not general enough to give good control of time-delay processes processes with complex

dynamics (e.g., inverse response)

The figure [WÅJ] illustrates PI control of a first-order time-delay process for both setpoint and load changes. A larger controller gain in order to make the response faster would give a more oscillatory response, which is not desired (in this case).

Better control can be achieved by a model-based controller, whose structure depends the process model


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2. Single-Loop Control Enhancements 2.2 Model-based controllers

2.2.1 Time-delay compensation Consider a time-delay system [WÅJ]

𝐺p 𝑠 = 𝐺0(𝑠)e−𝐿𝑠 where 𝐿 is the time delay and 𝐺0(𝑠) is the time-delay-free part of the system. Assume that a (PID) controller 𝐶(𝑠)

is designed for control of 𝐺0(𝑠). For the fictitious system 𝐺0(𝑠), this gives the closed-loop system

𝐺0r(𝑠) = 𝐺0 𝑠 𝐶(𝑠)1+𝐺0 𝑠 𝐶(𝑠)

Assume now that a controller 𝐺c(𝑠) is used to control the true time-delay system 𝐺p 𝑠 . The closed-loop system is then

𝐺r 𝑠 = 𝐺p 𝑠 𝐺c 𝑠1+𝐺p 𝑠 𝐺c 𝑠

= e−𝐿𝐿𝐺0 𝑠 𝐺c 𝑠1+e−𝐿𝐿𝐺0 𝑠 𝐺c 𝑠


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2.2 Model-based controllers 2.2.1 Time-delay compensation

Can we design 𝐺c(𝑠) to achieve 𝐺r 𝑠 = 𝐺0r(𝑠) ? No, because – 𝐺c 𝑠 = 𝐶(𝑠)e+𝐿𝑠 is not realizable (unless 𝐶(𝑠) contains a time delay ≥ 𝐿) – if the uncontrolled system has a time delay 𝐿, the controlled system

cannot have a time delay < 𝐿

Can we achieve 𝐺r 𝑠 = 𝐺0r(𝑠)e−𝐿𝑠, i.e., a closed-loop system 𝐺r 𝑠 that behaves exactly as 𝐺0r(𝑠), except for the time-delay 𝐿 ? Yes, if

𝐺0 𝑠 𝐺c 𝑠1+e−𝐿𝐿𝐺0 𝑠 𝐺c 𝑠

= 𝐺0 𝑠 𝐶(𝑠)1+𝐺0 𝑠 𝐶(𝑠)

can be achieved with a realizable controller 𝐺c(𝑠)

Solving gives

𝐺c 𝑠 = 𝐶 𝑠1+(1−e−𝐿𝐿)𝐺0 𝑠 𝐶 𝑠

= 𝐶 𝑠1+[𝐺0 𝑠 −𝐺p(𝑠)]𝐶 𝑠

𝐺c 𝑠 is realizable as shown in the block diagram! [WÅJ]

It is called a Smith predictor


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r

0G

cG

C

e Ls−

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Control illustration with Smith predictor [WÅJ]

The system is the same as was controlled with a standard PI controller previously. Because of the time-delay compensation, it was possible to use a five

time larger controller gain. This resulted in faster control responses, both for setpoint changes

and load disturbances, without oscillations and with small overshoot.


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An alternative way of implementing a Smith predictor 𝐺p 𝑠 = 𝐺0(𝑠)e−𝐿𝑠 is the true system 𝐺�p 𝑠 = 𝐺�0(𝑠)e−𝐿�𝑠 is a model of the system

However, the Smith predictor also has drawbacks – it does not work for unstable (or even integrating) systems 𝐺0(𝑠) – a disturbance 𝑛 is amplified for certain frequencies


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0G− ˆe Ls−

cGr

y−

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2.2.2 Internal model control (IMC) Consider the block diagram, where

– 𝐺IMC is a controller – 𝐺�p is a model of the true system 𝐺p

The dashed rectangle corresponds to a controller 𝐺c in a standard feedback loop. It can be expressed as

𝐺c 𝑠 = 𝐺IMC(𝑠)1−𝐺IMC(𝑠)𝐺�p(𝑠)

The closed-loop transfer function is

𝐺r 𝑠 = 𝐺p 𝑠 𝐺c 𝑠1+𝐺p 𝑠 𝐺c 𝑠

= 𝐺p 𝑠 𝐺IMC(𝑠)1+ [𝐺p 𝑠 −𝐺�p 𝑠 ]𝐺IMC 𝑠

≈ 𝐺p 𝑠 𝐺IMC(𝑠)

To obtain a desired closed-loop system 𝐺r 𝑠 , 𝐺IMC(𝑠) should be chosen as

𝐺IMC 𝑠 ≈ 𝐺p−1 𝑠 𝐺r(𝑠) ≈ 𝐺�p−1 𝑠 𝐺r(𝑠)


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2.2 Model-based controllers 2.2.2 Internal model control (IMC)

Is this always possible? No, it does not work if – 𝐺p(𝑠) (or 𝐺�p 𝑠 ) is unstable – 𝐺�p 𝑠 contains a zero in the right-half plane (which is not cancelled by an

identical zero in 𝐺r(𝑠)), because it gives an unstable controller 𝐺IMC(𝑠) – 𝐺�p 𝑠 contains a time delay (not cancelled by a time delay in 𝐺r(𝑠))

An improved design procedure The model is factored as 𝐺�p 𝑠 = 𝐺�p

⊖(𝑠)𝐺�p⊕(𝑠), where

– 𝐺�p⊕(𝑠) contains all non-minimum phase factors of 𝐺�p 𝑠 (i.e., RHP zeros

and time delays), but no minimum phase factors, and normalized so that 𝐺�p⊕ 0 = 1 (i.e., the static gain is 1)

– 𝐺�p⊖(𝑠) contains all minimum phase factors of 𝐺�p 𝑠

𝐺IMC(𝑠) is calculated as 𝐺IMC 𝑠 = 𝐺f(𝑠)[𝐺�p⊖(𝑠)]−1, where

– 𝐺f(𝑠) is a filter usually of the form 𝐺f 𝑠 = 1/(𝑇r𝑠 + 1)𝑛 with 𝑛 large enough to make 𝐺IMC(𝑠) proper

If the model is perfect (i.e., 𝐺�p 𝑠 =𝐺p(𝑠)), the closed-loop transfer function 𝐺r 𝑠 = 𝐺f(𝑠), which gives an indication how to choose 𝐺f(𝑠)


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Illustration of internal model control [WÅJ]

The system is the same as was previously controlled with a standard PI controller and a Smith predictor. The overshoot in the step response has disappeared. The step response is slightly slower than with pure time-delay

compensation — it could be tuned to become faster, but then an imperfect model might worsen the control performance.


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IMC-based tuning of PID controllers The IMC methodology is often used for tuning a standard PID controller The IMC-based expression for 𝐺c, i.e., 𝐺c = (1 − 𝐺IMC𝐺�p)−1𝐺IMC , is

converted to the form of a PID controller – if a time delay is present, it has to be approximated;

usually a Padé approximation e−𝐿𝑠 ≈ 1 − 𝐿𝑠, e−𝐿𝑠 ≈ (1 + 𝐿𝑠)−1, or e−𝐿𝑠 ≈ (1 − 1

2𝐿𝑠)(1 + 1

2𝐿𝑠)−1, is used

– other dynamical elements not conforming to the structure of a PID controller also have to be approximated (e.g., by a Taylor series expansion)

The PID controller is implemented in a standard feedback loop – the structural advantage of a true IMC implementation is lost – it is a convenient model based way of tuning a PID controller

A drawback of the IMC design procedure is that it is based on cancel-lation of poles and zeros (this also holds for the true IMC structure) – as a result of this, the response to load disturbances may be poor


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2.2.2 Internal model control (IMC) IMC-based tuning of PID controllers

The following table presents IMC-based PID controller tunings based on a Padé approximation of the time delay. 𝑇r is a design parameter chosen by the user essentially, 𝑇r is the desired time constant of the closed-loop system


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Optimized IMC-based tunings for PI/PID/PIDf controllers The approximation of 𝐺c is optimized to allow a PID controller with a derivative filter (PIDf) 𝑇r is a design

parameter chosen by the user

a PI or PID controller is obtained simply by omitting unwanted parts

negative derivative (Td) or filter (Tf) time constants should not be used!


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2.2.3 Gain scheduling In many situations it is known how the dynamics of a process change with the operating conditions of the process. One source for the change in dynamics may be nonlinearities. If the nonlinearities are known, it is possible to change the parameters

of the controller on-line by monitoring the operating conditions. This strategy is called gain scheduling.

A major difficulty in the design of a gain scheduling mechanism, is to find suitable monitoring or scheduling variables, which give desired information about the operating conditions can be used for adjusting controller parameters

Gain-scheduling can be implemented in various ways. Controller parameters are determined in advance for a number of

operating conditions and selected based on the scheduling variable(s). Controller parameters (mainly the controller gain) are calculated as a

(nonlinear) function of the scheduling variables(s).


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2.2 Model-based controllers 2.2.3 Gain scheduling

Example [WÅJ]

A simple feedback loop with a nonlinear valve is shown in the figure. The static valve characteristic is 𝑣 = 𝑓 𝑢 = 𝑢4, 𝑢 ≥ 0

Since the valve is nonlinear, the control system can perform well at one operating point, but poorly at other operating points.

The simulations show the control performance with fixed controller parameters for a setpoint change at three different operating points.


cKr

y u

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2.2.3 Gain scheduling Example

One way of improving the control performance over the whole operating range is to introduce an inverse of the valve characteristic between the controller and the valve.

The simulations show the control performance obtained with gain scheduling. To make the example more realistic, an approximate inverse of the valve is used.


yu

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2.2.4 Sampled-data control In practice, controllers are implemented in computers. The control algorithm is executed repeatedly with a given frequency

– the time between repetitions is called sampling interval (or period or time); – measurements (“samples”) are available only at the start of a new run; – control actions can be implemented only after the algorithm has been

executed (usually done at the following sampling instant). This introduces a time

delay in the control loop.

Control of a process working in continuous time with a controller working in discrete time is called sampled-data control. Figure from [WÅJ].


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2.2 Model-based controllers 2.2.4 Sampled-data control

Aliasing Aliasing is a phenomenon caused by the sampling mechanism. [WÅJ]

The figure shows two sinusoidals, one with the frequency 0.9 Hz (full

line), the other with the frequency 0.1 Hz (dashed line). When both sinusoidals are sampled with the frequency 1 Hz, they

have the same value at each sampling instant! – Thus, the signals cannot be distinguished from each other (because

information is lost due to sampling). – In practice, the high-frequency signal will be interpreted as a lower-

frequency signal. Can this be avoided?

– prefiltering (a high-frequency signal is probably mainly noise) – a higher sampling frequency


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Selecting a sampling frequency A fundamental sampling property is that a frequency higher than the

Nyquist frequency ωN ≡ 𝜋/ℎ , where ℎ is the sampling interval, cannot be distinguished from a signal in the frequency range [0,𝜔N]. – If 𝜔max is the highest frequency of interest (after prefiltering), the

sampling interval should satisfy ℎ ≤ 𝜋/𝜔max .

However, selecting the sampling interval according to the sampling theorem is not sufficient to enable good control.

Guidelines from a control perspective [from various sources] – use 10 times higher sampling frequency than 𝜔max (i.e., ℎ ≤ 0.1𝜋/𝜔max) – the sampling interval should be between 10 and 25 % of the rise time of

the controlled system – a full oscillation (of interest) should give 15–45 samples – the sampling frequency should be 10–30 times the bandwidth (which is

the frequency where the closed-loop gain is 1/ 2) Note: ℎ ≡ 2𝜋/𝜔s , where 𝜔s is the sampling frequency expressed in

radians per time unit.


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Intersample ripple A possible problem in sampled-data control is the intersample behaviour. The figure illustrates sample-data control of a step change in the setpoint (in a particular case). if only the sample points of the output 𝑦 are considered, the step

response looks like the response of a first-order system the true continuous-time value of 𝑦 oscillates; this is called

intersample ripple (or ringing) the input signal oscillates at sample points (and continuously)

∘ = sampling point, full line = continuous-time value [SEMD]


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2.2.4 Sampled-data control Intersample ripple

In particular, intersample ripple can occur when a discrete-time controller has been designed based on a discrete-time (sampled) model. When a continuous-time model is sampled using standard techniques,

the transfer function of the sampled model often receives a zero close to (but > than) 𝑧 = −1 even if the continuous-time model has no zero;

– e.g. 𝐺 𝑠 = 𝐾e−𝐿𝐿

(𝑇1𝑠+1)(𝑇2𝑠+1) ⇒ 𝐻 𝑧−1 = 𝑏0+𝑏1𝑧−1

1+𝑎1𝑧−1+𝑎2𝑧−2𝑧−𝑁−1, 𝑏1

𝑏0<≈ 1.

If the controller design is based on the inverse of 𝐻 𝑧−1 , 𝑏0 + 𝑏1𝑧−1 becomes a pole of the controller, i.e., a controller pole close to 𝑧 = −1, which causes intersample ripple.

Two solutions – replace 𝑏0 + 𝑏1𝑧−1 in the model by 𝑏0 + 𝑏1 (Dahlin’s modification) – include 𝑏0 + 𝑏1𝑧−1 as a zero in the desired closed-loop transfer function

(Vogel-Edgar modification); in discrete-time IMC design, 𝑏0 + 𝑏1𝑧−1 is included in the filter


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2.2.4 Sampled-data control Intersample ripple

The figure illustrates sample-data control using the Vogel-Edgar design modification [SEMD]

no intersample ripple no oscillations in the control signal


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2.3 Use of extra measurements Summary of single-loop feedback control Advantages achieves zero steady-state control error (if integral action is used) uses only one measurement algorithm and tuning rules are readily available

Disadvantages control is activated only after an error is detected in the process output control performance can be poor for some combinations of

disturbances and dynamics poor feedback can cause instability

If the control performance is unacceptable, it might be possible to improve it by including extra measurements.


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2. Single-Loop Control Enhancements 2.3 Use of extra measurements

2.3.1 Cascade control If the performance of single-loop feedback control is poor, the difficulty may arise from disturbances with a delayed or “slow” effect on the measured output varying process parameters or nonlinear effects

In both cases, the desired performance improvement on the controlled variable 𝑦 might be achieved by using an extra measurement , called a secondary variable, with the following properties: it is affected by a disturbance or process parameter variation (a non-

linearity), which causes poor control of 𝑦 it is affected by the manipulated variable 𝑢 used to control 𝑦 its dynamics are faster than the dynamics of 𝑦

If these criteria are fulfilled, the control of the primary variable 𝑦 can be improved by cascade control, where the secondary variable is controlled by a secondary (or slave)

controller, which receives its setpoint from a primary (or master) controller controlling the primary variable


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2.3 Use of extra measurements 2.3.1 Cascade control

Block diagram [WÅJ]

𝐺p = 𝐺p1𝐺p2 = process 𝐺c1 = primary controller 𝐺c2 = secondary controller 𝑌1 = measurement of primary variable 𝑌2 = measurement of secondary variable 𝑌r1 = setpoint for primary controller (variable) 𝑌r2 = setpoint for secondary controller (variable) 𝐷1 = disturbance not affecting secondary variable 𝐷2 = disturbance affecting secondary variable KEH Plantwide Control 2–38

Process

Note: The primary variable is the variable we want to control, the secondary variable is controlled only to improve the control.

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Analysis To analyse the properties of cascade control, let us consider the two limiting cases no secondary controller is used (i.e., feedback by one controller) the secondary controller is perfect (i.e., 𝑌2 follows 𝑌r2 perfectly)

In the case of no secondary controller (𝐺c2 = 1, 𝐺t2 = 0)

𝑋1 = 𝐺d1𝐷1+𝐺p1𝐺d2𝐷2+𝐺p1𝐺p2𝐺c1𝑌r11+𝐺t1𝐺p1𝐺p2𝐺c1

In the case of perfect secondary control (𝑋2 = 𝐺t2−1𝑌r2)

𝑋1 = 𝐺d1𝐷1+𝐺t2−1𝐺p1𝐺c1𝑌r11+𝐺t2−1𝐺t1𝐺p1𝐺c1

What conclusions can we draw concerning the (possible) advantages of cascade control over single-loop control? the cascade controller is less sensitive disturbances in 𝐷2 the cascade controller is less sensitive to variations in 𝐺p2 the primary controller can be tuned to better handle disturbances in 𝐷1 and setpoint changes in 𝑌r1


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Design and tuning guidelines The choice of secondary measurement is essential, since the secon-

dary loop ideally should enable perfect control of this loop. Therefore – the secondary loop should not contain any time delay or RHP zero – the essential disturbance to be cancelled should enter the secondary loop – the settling time of secondary loop should be significantly faster than the

primary loop dynamics Choice of secondary controller type

– the secondary controller can often be a pure P controller – derivative action may be included to improve the bandwidth of the loop – integral action may be useful mainly if 𝐺p1 contains significant time delay

Controller tuning – the secondary controller is tuned first – the primary controller

can then be tuned, e.g., based on a redrawn block diagram, where the secondary controller is already tuned


2 t2 p2 c2L G G G=

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Control illustration [SEMD]

The figures illustrate the kind of control improvements that can be achieved by cascade control.

Step disturbance in 𝐷1 Step disturbance in 𝐷2


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Multiple cascade levels Because of static and dynamic friction, control valves often stick and do not (easily) achieve the position commanded by the controller. Often a valve positioner, which is a built-in controller in the valve, is used to overcome this problem. This means that a simple flow control loop is usually a cascade loop. If the flowrate setpoint is given by a (primary) controller, it becomes a 3-level cascade. The principle can be extended. If the valve for the cooling water makeup is controlled by a valve positioner and a flow rate controller (not shown in the figure), the two-level temperature control cascade makes this a 4-level cascade.


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Exercise Discuss the advantages and disadvantages of the shown control structures for control of the temperature of an exothermic reactor. [Figures from Smith]


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2.3.2 Feedforward control An obvious limitation of feedback control is that corrective actions can not be taken before a disturbance is noticed in the controlled variable. In many process control applications it is possible to measure the

disturbances affecting the system. – Typical examples are concentration or temperature changes in the feed to a

chemical reactor or distillation column. Measurements of these disturbances can be used to make control

actions before anything is noticed in the output of the process. – It is then possible to make the system respond more quickly than if only

feedback is used.

This is known as feedforward control.

The measured variable, the feedforward variable, must satisfy the following criteria: it must indicate the occurrence of an important disturbance the control variable must not affect the disturbance if feedforward is combined with feedback, the disturbance dynamics

must not be significantly faster than the main input-output dynamics


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2.3 Use of extra measurements 2.3.2 Feedforward control

Block diagram [WÅJ]

𝐺p and 𝐺v are process transfer function 𝐺f is a feedforward controller

The disturbance is totally eliminated from the output if

𝐺f = −𝐺v𝐺p


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Comparison with feedback control

Feedforward Feedback Advantages acts before the effect of a dis-

turbance is seen in the output good for slow systems and

time-delay systems does not cause instability Disadvantages cannot handle unmeasured

disturbances sensitive to variations in

process parameters requires good knowledge of

the process model


Advantages does not require disturbance

identification and measurement insensitive to modelling errors insensitive to variations in

process parameters Disadvantages does not act before the effect of

a disturbance is seen in the output ineffective for slow systems and

time-delay systems may create instability in the

closed-loop system

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Design issues The fact that the optimal feedforward controller is given by 𝐺f = −𝐺v

𝐺p ,

i.e., the ratio between two (usually proper) transfer functions, may give problems if the magnitude of 𝐺v(j𝜔) is large compared to the magnitude of 𝐺p(j𝜔), the required control signal may be too large to be implemented. This can be a problem both at – steady state, causing saturation of the control signal – higher frequencies, causing amplification of measurement noise

if 𝐺p has more poles than 𝐺v (i.e., 𝐺p is of higher order than 𝐺v), the feedforward controller will contain (high-order) derivatives of the measured disturbance

if 𝐺p contains a larger time delay than 𝐺v, the feedforward controller cannot be implemented exactly

For these reasons, an approximate feedforward controller is often implemented.


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Approximations

Assume that 𝐺p = 𝐾p(𝑇p1s+1)(𝑇p2s+1)

and 𝐺v = 𝐾v𝑇vs+1

.

The optimal feedforward controller is then

𝐺f = −𝐾v 𝑇p1s+1 𝑇p2s+1𝐾p 𝑇vs+1

= −𝐾v 𝑇p1s+1𝐾p 𝑇vs+1

𝑇p2s + 1

which can be implemented as a series connection between an ideal PD controller and a PD controller with filtering.

This might be unsatisfactory (for reasons mentioned previously). When approximations are made, high-frequency components are usually eliminated. This motivates the following kinds of approximations:

𝐺f ≈ −𝐾v[ 𝑇p1+𝑇p2 𝑠+1]𝐾p 𝑇vs+1

, i.e., a PD controller with filtering

𝐺f ≈ − 𝐾v𝐾p

, i.e., a P controller


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Combined feedforward and feedback control Feedforward control is usually used together with feedback control to combine their advantages (and avoid their disadvantages).

The block diagram shows such a control structure [SEMD]. (Note that the notation differs slightly from that in the previous block diagram.)

The feedback controller can essentially be tuned independently of the feedforward controller.


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Control illustration [WÅJ]

This example illustrates control of a tank system. The level is controlled using the input flow. The output flow is a measurable disturbance. It is assumed that there is a calibration error (bias) in the measurement of the output flow.

𝐺p(𝑠) = 1𝑠

𝐺valve 𝑠 = 1𝑠+1

𝐺f 𝑠 = 1𝐺valve 𝑠

= 𝑠 + 1

Approximations to avoid pure differential 𝑠 𝐺f1 𝑠 = 𝑠

𝑇f𝑠+1+ 1

𝐺f2 = 1


r

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2.3.2 Feedforward control Control illustration

Feedforward control Feedforward + feedback control


f1 f( ) , 0.1G s T =

f 2G

f 2G

f1 f( ) , 0.1G s T =

no bias

bias 0.05= −

without feedforward

with feedforward

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Ratio control Ratio control is a simple version of feedforward control, where the effect of a disturbance is counteracted by maintaining a fixed ratio between two variables (one of which is the disturbance). Typical examples are to control the property of blended streams, e.g., the molar ratio of two

feed streams to a reactor to ensure a proper stoichiometric mixture the ratio between the reflux and distillate flow rates in distillation

Consider the mixing of a “hot” stream and a “cold” stream to obtain a given temperature in the mixed stream. Assume that one of the streams is selected as control variable and the other is considered a disturbance. The shown ratio control scheme will keep the ratio 𝑅 = 𝐹b 𝐹a⁄ constant if the RC controller contains integral action.

The control scheme has a clear drawback. The gain between the control error 𝑒𝑅 = 𝑅 − 𝐹b 𝐹a⁄ and 𝐹a is 𝜕𝑒𝑅 𝜕𝐹a =⁄ 𝑅/𝐹a, i.e., inversely proportional to 𝐹a for a given ratio 𝑅. This means that the control performance will vary with 𝐹a. [King: Process Control, Wiley, 2011]


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2.3.2 Feedforward control Ratio control

The ratio control scheme shown here does not have the problem of a varying gain. Now the control error is 𝑒𝐹 = 𝑅𝐹a − 𝐹b and the gain is 𝜕𝑒𝐹 𝜕𝐹a = 𝑅⁄ , i.e., a constant gain. This means that the control performance will not vary with 𝐹a.

If the desired ratio 𝐹b 𝐹a⁄ is very small, or very large, it might be better to define the ratio as 𝑅 = 𝐹b (𝐹a + 𝐹b)⁄ . This ratio scheme is shown to the right. It also has the gain 𝜕𝑒𝐹 𝜕𝐹a = 𝑅⁄ .

In practice, the ratio control scheme is combined with feedback control, where the feedback controller is adjusting 𝑅. KEH Plantwide Control 2–53

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Exercise [Smith]

Discuss the following control schemes for control of a hear exchanger.


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2.3.3 Selective control There are control systems that involve one manipulated variable and more than one possible controlled variable. Since only one output can be controlled by one control variable, the selective control system transfers control from one output to another according to need.

A selective control system is possible if there is one manipulated variable and several potential controlled variables a causal relationship between the manipulated variable and each

controlled variable there is a feasible operating point that satisfies all control objectives

There are many types of selective control systems. Here we consider only two types override control auctioneering control


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2.3 Use of extra measurements 2.3.3 Selective control

Override control During normal operation of a plant, or during its startup or shutdown, it is possible that dangerous situations arise which may lead to destruction of equipment or operating personnel. In such cases, it is necessary to change from the normal control action to one that can prevent a process variable from exceeding an allowable upper or lower limit. This change can be achieved through the use of switches. A high selector switch (HSS, HS, >) selects the higher of two signals. A low selector switch (LSS, LS, <) selects the lower of two signals.

The example illustrates override control to protect a boiler system. During normal operation, the steam pressure is controlled by the discharge line. However, if the water level in the boiler falls below an allowable limit, LSS switches the discharge line from [Steph] pressure control to level control.


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2.3 Use of extra measurements 2.3.3 Selective control

Auctioneering control In auctioneering control, the measurement with the highest (or lowest) value of a number of similar measurements is selected as controlled variable. The example illustrates temperature control of a tubular reactor with highly exothermic reactions. The so-called hot spot, which is the highest temperature in the reactor, can be (almost) anywhere in the reactor. It is not allowed to exceed a given limit. The auctioneering system finds the hotspot and allows it to be controlled by [Steph] the temperature controller.


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2.4 Use of extra actuators Another possibility to improve the control performance of single-loop control is to use extra actuators (control signals). Typical situations are to make it possible to achieve a larger range of setpoints than with a

single actuator to speed up the dynamic response of a process during serious upsets

or transitions between setpoints

Often the applications involve a dual mode of operation, e.g., to heat or cool a process to vent or bleed gases for pressure control

Requirements for this kind of control is that there is one controlled variable and more than one manipulated variable a causal relationship between each manipulated variable and the

controlled variable a fixed priority ranking for the proper order of adjusting the

manipulated variables


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2. Single-Loop Control Enhancements 2.4 Use of extra actuators

2.4.1 Split-range control In split-range control, there is one controller the control signal is directed to an actuator depending on the value of

the control signal (i.e., it is split in parts, e.g., 0–50% and 50–100%)

Illustration [Steph] The block diagram illustrates split-range control of the pressure in a gas-phase reactor. The pressure can be reduced by closing valve 𝑉1 and/or opening valve 𝑉2. The operating point is at 6 psi. Pressure disturbances from 3 to 9 psi, are controlled by 𝑉2 with 𝑉1 fully open; above 9 psi, they are controlled by 𝑉1 with 𝑉2 fully open.


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2.4.2 Coordinated control In coordinated control (also called parallel control), there is more than one controller controlling the same output; a given strategy to coordinate the actions of the controllers.

In the case of two controllers, one usually acts fast, but is inaccurate or expensive to use; slowly, but is accurate or inexpensive to use.

Note that only one controller can have integral action when several controllers act simultaneously to control the same output. Why? Illustration [OR] The temperature in a heating tank is controlled. During normal operation, it is controlled by

TC1 using process steam. The steam is cheap, but the temperature response is rather slow.

For setpoint changes, an auxiliary heater and cooler are used. They use expensive elec- trical power, but the effect on temp. is fast.


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2.4.3 Valve position control Valve position control is a special type of coordinated control, where two controllers act together to control the process output using two

manipulated variables one of the controllers uses the output from the other controller as

input to adjust the position of a valve

In most valve position control applications the objective is process optimization, which is achieved when the valve in question operates close to a constraint (i.e., almost fully open or closed) or at some other desired position. This maximizes or minimizes (or optimizes in some other way) the use of the process stream controlled by the valve.

Valve position control resembles cascade control. With this interpretation the valve position controller is the primary controller the process output controller is the secondary controller

Note: A valve position controller is not a “valve positioner”.


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2.4 Use of extra actuators 2.4.3 Valve position control

Illustration: Control of a heat exchanger using a bypass The figure illustrates a heat exchanger with a bypass. Most of the cold liquid is heated in the heat exchanger by steam

(controlled by controller TC2). Part of the cold liquid is used for rapid and accurate control of the

temperature of the heated liquid (controller TC1). To maximize controllability, the bypass valve must not operate near to fully open or closed. Thus, there is a desired value for the valve opening, e.g., 50% open; controller TC2 has this value as setpoint; controller TC2 controls

the steady-state value of the valve opening;

controller TC1 controls the temperature of the heated liquid.


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2.4 Use of extra actuators 2.4.3 Valve position control

Illustration: Maximizing the production rate of a reactor The figure shows a reactor, where an exothermic reaction takes place. Thus, it has to be cooled. In this case it is desired to maximize the feed flow rate to the reactor; the reactor temperature has to be controlled and it must not exceed a

given maximum value. The control system with a valve position controller (VPC) works as follows: the opening of the cooling water valve (CWV) is fed to the VPC; the set point for the VPC is the desired opening of the CWV; the VPC controls the CWV opening by adjusting the feed flow rate; in order to maximize the

feed flow rate, the setpoint for the CWV opening should be at the maximum value, which still allows the temperature to be controlled, e.g., 90 % open.

(Note that the temperature [Smith] control cascade is not shown.) KEH Plantwide Control 2–63

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