Feedforward and ratio control

Feedforward and ratio control Structure of discussion: • Feedforward control – summary • Feedforward control – details - Advantages of feedback control

• Statistical Process Control

• Feedforward and ratio control

- Introduction to feedforward control - Case study: control of a continuous caster - Feedforward control design based on dynamic models - Closed loop stability issues – feedforward controllers

• Cascade control • Split range and selective control • Control of MIMO processes

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1. Feedforward control - summary

• Ratio control – summary • Ratio control – details • Lifelong learning

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2. Feedforward control - details

Used to anticipate and compensate for disturbances.

2.1 Advantages of feedback control

The feedrate has been measured upstream, and then fed forward and used to adjust the heat input into the process. The objective is to ensure that the process temperature is always maintained within present quality control limits, even when product flow changes.

Previously, is was emphasized that feedback control is an important technique that is widely used in the process industries. Its main advantages are as follows. 1. Corrective action occurs as soon as the controlled variable deviates from the set point, regardless of the source and type of disturbance. 2. Feedback control requires minimal knowledge about the process to be controlled; it particular, a mathematical model of the process is not required, although it can be very useful for control system design. 3. The ubiquitous PID controller is both versatile and robust. If process conditions change, retuning the controller usually produces satisfactory control. 4

Feedback corrects for disturbances after they occur; feedforward corrects for disturbances before they affect the process (though it is normally used with feedback control). To set up a feedforward controller, it is important to know how the process responds in steady state and dynamically. This will allow correct set-up of the lead-lag element. Process knowledge can be determined mathematically or by experiments on the process (or both).

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Reference: Seborg, D.E. et al. (2004). Process dynamics and Control, 2nd edition, Chapter 15.

However, feedback control also has certain inherent disadvantages: 1. No corrective action is taken until after a deviation in the controlled variable occurs. Thus, perfect control, where the controlled variable does not deviate from the set point during disturbance or set-point changes, is theoretically impossible. 2. Feedback control does not provide predictive control action to compensate for the effects of known or measurable disturbances. 3. It may not be satisfactory for processes with large time constants and/or long time delays. If large and frequent disturbances occur, the process may operate continuously in a transient state and never attain the desired steady state. 4. In some situations, the controlled variable cannot be measured online, and, consequently, feedback control is not feasible.

2.2 Introduction to Feedforward Control The basic concept of feedforward control is to measure important disturbance variables and take corrective action before they upset the process. Feedforward control has some disadvantages: 1. The disturbance variables must be measured on-line. In many applications, this is not feasible. 2. To make effective use of feedforward control, at least a crude process model should be available. In particular, we need to know how the controlled variable responds to changes in both the disturbance and manipulated variables. The quality of feedforward control depends on the accuracy of the process model. 3. Ideal feedforward controllers that are theoretically capable of achieving perfect control may not be physically realizable. Fortunately, practical approximations of these ideal controllers often provide very effective control.

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• A boiler drum with a conventional feedback control system is shown in the figure. The level of the boiling liquid is measured and used to adjust the feedwater flow rate. • This control system tends to be quite sensitive to rapid changes in the disturbance variable, steam flow rate, as a result of the small liquid capacity of the boiler drum. • Rapid disturbance changes can occur as a result of steam demands made by downstream processing units.

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The feedforward control scheme in the figure can provide better control of the liquid level. Here the steam flow rate is measured, and the feedforward controller adjusts the feedwater flow rate.

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• In practical applications, feedforward control is normally used in combination with feedback control. • Feedforward control is used to reduce the effects of measurable disturbances, while feedback trim compensates for inaccuracies in the process model, measurement error, and unmeasured disturbances.

2.3 Case study: Control of a continuous caster One of the products of the steel industry is a so-called bloom, which is a rectangular slab of steel. Blooms are produced in a process called a continuous caster. A diagram of an industrial bloom caster is given in the figure.

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This work is taken from the real life case studies at http://csd.newcastle.edu.au/control

Molten steel flows into a mould (at a rate controlled by a valve), which is open from above and below. Through cooling, the steel in the mould is transformed into a semi-solid state and the steel comes out in a continuous slab. The principal components of such a system relevant to the discussion here are shown in the figure.

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The tundish can be thought of as a large container that acts as a reservoir for molten steel. A control valve regulates the rate of flow of steel that enters the mould mounted under the tundish. The mould, whose cross-sectional area equals the cross-sectional area of the desired bloom, is open from above and below. By intense cooling, steel in the mould is cooled to a semi-solid state. In this state, it is sufficiently firm so that the strand can be withdrawn continuously from the mould by rolls. The resulting continuous strand is then subjected to further cooling and finally cut into blooms. This photo shows the continuous caster at BHP in Newcastle, Australia. 12

The cast strip in the secondary cooling chamber

Operators viewing the mould

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Objective: To control the level of the molten steel in the mould to prevent it from overflowing or emptying. The level is controlled by adjusting the control valve, based on measurements of the casting speed and the valve position.

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Pictorial basis of the model Tundish

Molten Steel

Modelling: To make progress on the control-system design problem, it is first necessary to gain an understanding of how the process operates. This understanding is typically expressed in the form of a mathematical model that describes the steady-state and the dynamic behaviour of the process. To construct such a model, we first define relevant process variables. Thus, we introduce the following: h * = desired level of steel in the mould; h ( t ) = actual level of steel in the mould; v( t ) = valve position; σ( t ) = casting speed; q in ( t ) = inflow of matter into the mould; q out ( t ) = outflow of matter 15 from the mould.

Valve Mould Level Cooling Water

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Now, assume q in ( t ) = k.v( t ) ; furthermore, assume k = 1. Also, q out ( t ) = A.σ( t ) , A = cross-sectional area of the mould; again, for simplicity assume A = 1. Our engineering knowledge suggests that the mould level will be proportional to the integral of the difference between inflow and t outflow: h ( t ) = ∫ [q in (τ) − q out (τ)]dτ −∞

Hence, the process model becomes h ( t ) =

A block diagram of the overall process model and the measurements is shown in the figure.

t

∫ [v(τ) − σ(τ)]dτ

−∞

The casting speed can be measured fairly accurately, but mouldlevel sensors are typically prone to high-frequency measurement noise, which we take into account by introducing an additive spurious signal, n(t): h m (t) = h(t) + n(t) where h m ( t ) is the measurement of h(t) that is corrupted by noise. 17

We observe that a deviation between set-point and measurement must first occur before the controller can react. We know from the nature of the system, however, that a change in casting speed requires a modified operating point for the valve. Thus, rather than letting a change in casting speed occur, which then leads to an error in the mould level to which the feedback controller reacts, we can improve the strategy by changing the valve-position proactively (feedforward). This leads to a final controller of the form: 1   v( t ) = K {h * − h m ( t )} + σ( t ) K   Note that this controller features joint feedback and a pre-emptive action (feedforward). In particular, the second (feedforward) term gives the predictable action necessary to compensate for the casting speed changes, while the first (feedback) term reacts to the remaining error. 19

The simplest feedback controller is a constant gain, K, driving the valve proportional to the error between the desired mould level, h * , and the measurement of the actual mould level, h m ( t ) :

v( t ) = K[h * − h m ( t )]

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A block diagram for the final control system is shown in the figure.

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Java Applet simulation:

Now there are several things you should notice.

A JAVA applet (see http://www.csd.newcastle.edu.au/ … go to Interactive Case Studies, Continuous Casting) is a simulation of this model, which allows you to alter the various system parameters and see the results. Edit the value for the controller gain, K, and the level of noise (set this to zero to remove noise completely). The set-point type can also be altered between square and sinusoidal (note that you must press the "Update" button to see the results of your changes). The simulation can be stopped and restarted by pressing the start and stop buttons.

• For small controller gains (e.g. K = 1), the system does not respond very quickly, and is not very susceptible to noise. • For larger controller gains (e.g. K = 5), the system responds more quickly, but is much more susceptible to noise; actuator wear would also be increased. • This hints at a trade-off between response speed and noise sensitivity.

0.8 0.6

K=1

0.4 0.2

Change the controller gain to K = 1

The response is very slow.

Change the controller gain to K = 5

The response is faster, but jumps around a lot due to the influence of measurement noise

There might be a trade-off Experiment with the controller gain to try and get a fast response between speed and noise attenuation with little noise.

K=5

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Mould level

Things to notice

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2.4 Feedforward Control Design based on Dynamic Models As a starting point for the discussion, consider the block diagram shown.

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5 Time [s]

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4 3 2 K=1

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Valve command

Things to try

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The closed-loop transfer function for disturbance changes is: Y(s) G d + G t G f G v G p = D(s) 1 + G cG v G p G m Ideally, we would like the control system to produce perfect control where the controlled variable remains exactly at the set point despite arbitrary changes in the disturbance variable, D. Thus, if the set point is constant (Ysp(s) = 0), we want Y(s) = 0, even though D(s) is not equal to 0. Put numerator of Y(s)/D(s) equal to zero and solve for Gf: Gd Gf = − G t G vG p

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Thus, a disturbance upsets the process via the disturbance transfer function, Gd; however, a corrective action is generated via the path through GtGfGvGp. Ideally, the corrective action compensates exactly for the upset so that signals Yd and Yu cancel each other and Y(s) = 0. 24

Examples Example 1: Suppose that G d =

Kd G = Kp , p Tps + 1 Tds + 1

G d , the ideal feedforward controller is G t G vG p (assuming G t (s) = K t , G v (s) = K v ): Then from G f = −

 K d  Tps + 1   G f = −  K K K  t v p  Tds + 1 − sτ Ke p K Example 2: Suppose that G d = d , G p = p Tds + 1 Tps + 1  K d  Tps + 1 + sτ p  e Then the ideal feedforward controller is: G f = −   K t K v K p  Tds + 1  K d  (Tp + τp )s + 1  A realizable approximation is: G f = − 25  K K K  T s + 1   d  t v p 

2.5 Closed loop stability issues – feedforward controllers • To analyze the stability of the closed-loop system in the figure, the closed-loop transfer function is considered: Y(s) G d + G t G f G v G p = D(s) 1 + G cG v G p G m • Setting the denominator equal to zero gives the characteristic equation, 1 + G cG v G p G m = 0 • The roots of the characteristic equation (i.e. closed loop system poles) completely determine the stability of the closed-loop 27 system.

Example 3: If G d =

Kp Kd , Gp = Tds + 1 (Tp1s + 1)(Tp 2s + 1)

then the ideal feedforward controller is

 K d  (Tp1s + 1)(Tp 2s + 1)  G f = −  K K K (Tds + 1)  t v p This controller is physically unrealizable because the numerator is a higher order polynomial in s than the denominator. Again, the controller could be approximated as follows:

 K d  (Tp1 + Tp 2 )s + 1   G f = −  Tds + 1   K t K v K p  These three examples demonstrate that a reasonable approximation to an ideal feedforward controllers is (T s + 1) Gf = Kf 1 26 (T2s + 1)

• Because Gf does not appear in the characteristic equation, the feedforward controller has no effect on the stability of the feedback control system. • This is a desirable situation that allows the feedback and feedforward controllers to be tuned individually.

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Block diagram for feedforward/feedback Block diagram for feedback control – control – regulator response (change in wi) regulator response (change in wi)

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Block diagram for feedforward/feedback control – regulator response (change in Ti)

Block diagram for feedback control – regulator response (change in Ti)

Feedforward control makes no difference to the servo response Block diagram for one feedforward/feedback control arrangement – servo response

Block diagram for feedback control – servo response

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In summary ….

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Some formative questions

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• Why is the feedback controller retained when the feedforward controller is implemented ? • When would a feedforward controller give zero steady state offset ? • What is the additional cost of feedforward/feedback control over the feedback controller arrangement ? • What procedure is used to tune feedforward controllers? True or false ? • The feedback controller tuning does not change when combined with a feedforward controller. • The feedforward controller has no tuning parameter. • The feedforward controller should react immediately the disturbance is measured. • Feedforward could be applied for a set-point change. 36

Virtual laboratory – feedforward control

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Tutorial question

3. Ratio control - summary Used to keep one or more flow rates in proportion to another flow as conditions change. Example: Waste incinerator controlling feed rate and fuel flow rate. The ratio is set (or biased) by the temperature controller (TC) to ensure that the minimum allowable exhaust temperature is met under all conditions.

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Without ratio control, a simple feedback control loop would sense that change is needed and make the necessary corrections. Adjustment could well be delayed, leading to constant violations of minimum allowable exhaust temperature and/or excessive energy use. 46

Ratio control – Method 1

4. Ratio control - details • Ratio control is a special type of feedforward control that has had widespread application in the process industries. • The objective is to maintain the ratio of two process variables as a specified value. • The two variables are usually flow rates, a manipulated variable u, and a disturbance variable d. U • Thus, the ratio R = is controlled rather than the individual D variables.

Typical applications of ratio control 1. Setting the relative amounts of components in blending operations. 2. Holding the fuel-air ratio to a furnace at the optimum value. 47

• The main advantage of Method 1 is that the actual ratio R is calculated. • A key disadvantage is that a divider element must be included in the loop, and this element makes the process gain vary in a nonlinear fashion with d.  ∂R  1 Kp =   =  ∂u d d 48

Ratio control – Method 2 (preferred)

Tutorial question

• Regardless of how ratio control is implemented, the process variables must be scaled appropriately. • For example, in Method 2 the gain setting for the ratio station Kd must take into account the spans of the two flow transmitters. • Thus, the correct gain for the ratio station is

KR = Rd

Sd Su

where Rd is the desired ratio, Su and Sd are the spans of the flow transmitters for the manipulated and disturbance streams, respectively. 49

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Question

Which of the following statements appear in the feedforward design criteria?

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• The process must be linear

Answer

No, although the controller was designed assuming a linear plant, it will provide adequate compensation for many nonlinear plants. Naturally, we have to evaluate the performance of feedforward on a case-by-case basis, because the nonlinearities and the performance requirements are different.

• There must not be a causal relationship between the manipulated variable and the feedforward disturbance variable Yes.

• The disturbance dynamics must not be significantly faster than the manipulated variable dynamics Yes.

• The process must be linear • There must not be a causal relationship between the manipulated variable and the feedforward disturbance variable • The disturbance dynamics must not be significantly faster than the manipulated variable dynamics. • The measured feedforward variable must measure a significant 54 disturbance.

Question True or false? • Feedforward can be applied to compensate for any measured disturbance • Only one measured variable can be used for feedforward control. • Ratio control can be achieved by feedforward principles, but not feedback only. • All measured disturbances should be used for feedforward control

• The measured feedforward variable must measure a significant disturbance. Yes. 55

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• Feedforward can be applied to compensate for any measured disturbance.

Answer

No, the application of feedforward must conform to the design criteria. Specifically, the disturbance should not be much faster than the correction path. Also, there should be no causal relationship between the valve and the disturbance. This situation is shown in the figure.

Answer - continued • All measured disturbances should be used for feedforward control.

• Only one measured variable can be used for feedforward control. No, many disturbances in one process can be compensated using feedforward. Remember, we have assumed that the plant behaves (approximately) linearly; therefore, we add all feedforward compensations together.

No, feedforward should not be used whenever it is possible. We want to achieve a significant improvement in dynamic performance. The figure shows a case where feedforward is not needed, because the control of a first order process (without dead time) is easy, and the performance of feedback-only is likely to be good.

• Ratio control can be achieved by feedforward principles, but not feedback only. No, ratio control is not required to use feedforward concepts. However, it can be implemented using feedforward concepts, as shown. A feedback design is also shown.

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Question

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• The lack of a tuning parameter

Answer

No, we can adjust the parameters in the feedforward controller to improve its performance.

• The occurrence of steady-state offset

A major deficiency of feedforward control is

Yes, feedforward alone will not return the controlled variable to its desired value, because of inevitable model errors and unmeasured disturbances. Even careful model building using fundamental or empirical methods will result in such modelling errors.

• The propagation of measurement noise No, we can tune the feedforward controller to reduce the propagation of high frequency measurement noise. This is not a fundamental deficiency of feedforward control

• The lack of a tuning parameter • The occurrence of steady-state offset • The propagation of measurement noise

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Question

• Include A100 in a feedforward control strategy

Answer

No, a causal relationship exists between the manipulated flow and A100. Therefore, feedforward is not possible.

If the blended product quality were measured by A100 how could this be used to improve the control performance for the initial design shown ?

• Remove the ratio controller and use feedback instead No, the ratio controller improves the performance and should be retained.

• Have feedback from A100 manipulate the ratio Yes, the feedback can function in conjunction with the ratio controller. See the following figure.

• Include A100 in a feedforward control strategy • Remove the ratio controller and use feedback instead • Have feedback from A100 manipulate the ratio.

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Question and Awnser

5. Lifelong learning Books:

How are the results of feedforward and feedback controllers combined in linear designs?

• • • •

The two The two The two The two

signals are added signals are multiplied signals are subtracted signals are not combined.

Answer: The feedback and feedforward adjustments are added, because they are responding to different deviations. Also, we assume that the plant is (approximately) linear. 63

1. Seborg, D.E. et al. (2004). Process dynamics and control, 2nd edition, Chapter 15. 2. Marlin, T.E. (2000). Process control, Chapter 15.

Trade magazines (e.g. Control Engineering) often have web-accessible tutorial articles on aspects of feedforward and ratio control. Examples of these articles are: 1. Pettetier, B. and VanDoren, V. (2003). “Advanced control saves energy”, Control Engineering, p. 41-44, January, http://www.controleng.com/article/CA269797.html 2. Harrold, D. (2001). “Push the limits”, Control Engineering, February, http://www.controleng.com/article/CA190205.html 3. Gordon, L. (2005). “Advanced Regulatory control – Adaptation and Feedforward”, Control Engineering, March, http://www.controleng.com/article/CA509772.html

Finally, there are many web based tutorials and discussions on the basics of feedforward and ratio control. One example: Ratio Control, http://www.jashaw.com/pid/ratio.html 64