Acknowledgments The following guide indicates the suggested reading path for each category of reader. Figure 1.1. Fundamental block diagram of predictive control Figure 2.1. a Realigned and b independent models 2.2.2 Independent Model i.e., all past outputs of the process are measured and stored in memory. However, this representation has some limitations. If the order of the system is greater than one, and if the poles are not stable, some numerical imprecision may result and consequently the choice of sampling period becomes critical. ica marae | iy e Syi inability of any subsequently developed regulator to control the unstable plant. Figure 2.6. Decomposition of a process with a single integrator 2.4 Prediction The key to any model-based controller lies in its ability to predict the process response using a model that may be physically realised, i.e., a causal model. We will return to the issues of model and system identification in Chapter 7. In fact, Jetermining the response of a process over time is equivalent to the classical problem of solving differential or difference equations. For convenience, we recall the fundamental concepts used in solving such equations. The solution of a linear differential or finite-difference equation. from the The value can vary as a function of an internal or external process variable (e.g., tracking error). Usually, a future action is sought such that the predicted future response corresponds with several fixed points on the reference trajectory, which are referred to as coincidence points. Figure 3.3. Control of a process without taking the time delay into account (7 = 20) Figure 4.1. Solution with noise present 4.3.2 Computational Requirements Figure 4.9. Controller with compensation: process 7,=30s, disturbance 7, =30s, > This extension was proposed by Chris Vanbaelinghem from EVONIK.DEGUSSA. Figure 4.12. Smoothing procedure of the MV 4.10 Extension to Higher-order System Models A process model of any order may be written in the form: Figure 4.19. Switching from PID or manual modes to PFC control Figure 5.2. Influence of the desired CLTR (CLTRD) on the real CLTR (CLTRR) with a coincidence point h = 30 re The reference trajectory must be re-initialised at each sample point on the actual measured value of the process output. It is possible to implement such a procedure for a first-order process. However, this is not generally the case for processes of a higher order because / must be chosen greater than 1. If the desired and resulting CLTR are similar and, more importantly, change in the same manner, this does not pose any practical problems when tuning. We will use a third- order process with a OLTR = =12s to help us understand Figure 5.6. Harmonic disturbance rejection GM, which renders the closed loop unstable, may be traced for a given coincidence horizon (see Figures 5.7 and 5.8). The OLTR is 65 s and is indicated by the “*” in both diagrams. Figure 5.8. Positive and negative delay margins as a function of the CLTR Figure 5.11. CLTR_R/ (K>/ Km) Figure 5.14. Sensitivity function (“hill curve”): mod(CV/DV)/Disturbance period As discussed in Section 5.4, the CLTR has a strong impact on the sensitivity function. In the case of the CLTR =180s and h=75s we obtain the results shown in Figure 5.14. Note that the shorter the CLTR, the higher the peak of the sensitivity function. But, for low frequencies the disturbance rejection is better. So, it is clear that knowledge of the disturbance spectrum should be exploited in the selection of the CLTR. Figure 5.15. demonstrates the time-domain response of the system to a step disturbance with a step disturbance taken into account. In the common case of a first-order process with pure time delay and step set-point there is only one parameter of regulation, i.e., the CLTR, whose physical significance is intuitive and obvious. Figure 6.5. Respecting the constraint on CV, Figure 6.7. Cascade control with back calculation In the case of cascade control (see Section 7.3) it is crucial that the contraints or the physical MV of the inner controller should be transferred to the outet controller. The inner loop contains a physical actuator (pump, efc.) that may routinely function at its operational limits (e.g., max./min. amplitudes and speed) restricting the regulation performance and the outer regulator’s internal mode’ becomes invalid. Consequently, “transferring” this constraint to the outer regulato1 ensures that the set-point presented to the inner regulator (MV of the outet controller) does not violate any such conditions. The inner controller essentially behaves as though it had been informed of the restriction a priori. Figure 6.9. Cascade control without constraint transfer with MVmax = 6 Figure 7.2. Zone control CLTR = 150 s. Calculation noise at t = 1500 s Figure 7.7. Transparent control without constraint transfer MVax = 40 7.5 Shared Multi-MV Control where MLV, appears to within a constant times OF tne derivative OF MV, . This solution is illustrated in Figure 7.9. A regulator R, which controls tk ohysical output of the process, is introduced. The process is subjected to a slow ar fast actuator that produces a “slow manipulated variable” of value MV, th controls the slow actuator, and a “fast manipulated variable” MV,, generated t lifferentiating the slow manipulated variable using a high-pass filter (HPF) th controls the fast actuator. The two models (slow actuator xX process) and (fa actuator X high-pass filterx process) have free and forced outputs that a1 troduced into the control equation. This results in a manipulated variable that n only acts directly on the slow process but also on the fast process via a high-pa: filter. Figure 7.11. Shared control of two actuators MV, active, MV, =0 potentially large amplitude. The effect of such noise on the reconstructed input signal may be minimised by adapting the CLTR of the reconstruction controller. Figure 7.14. Effect of the estimator and feedforward components in a PFC controller t is also possible to use the estimated input disturbance as a feedforward variable Sut this estimation is just a mathematical operator whose input is the proces utput and whose output is to be fed back to MV, by some means. So, from ai aput—output perspective, we just add another feedback loop around the mai ontroller. Thus, this additional loop may affect the global stability of the closed oop system. In practice, an attenuated version of the estimated disturbance is fe ack in the form: Dist” = K.Dist’, where K <1. Example 7.1. Consider Figure 7.14, at time 400 s a step disturbance of amplitude 30 is introduced. The estimator and feedforward components are active at this point. At 600 s, another step disturbance is introduced but, at this point, the estimator and feedforward components are deactivated. TRE aeolian clhunnwmeswwmbionwies Phew Giese bn ce wwliewc een Mie alee sees remem lew 2 It is first necessary to distinguish between functional variables and physical variables. The variable MVFis a functional variable that operates linearly on the dynamic equation of the process. A non-linear relationship exists between MVF and the variable MV@® that physically acts on the process. If this relationship is one-to-one mapping (i.e., a unique mapping exists between each value of MVF and its corresponding MV® and vice versa) and stationary, it is possible to determine a regulator with MVF and subsequently calculate the appropriate value of MV® to apply (see Figure 7.16). The CV should be treated in the same manner. If a one-to-one mapping and stationary relationship exists between the CV® and CVF, the reference trajectory may be initialised using a calculated point CVF according to the measured CVO . Consider the case of a first-order dynamic process, where the relationship between MVF and MV® is generally a polynomial of degree B, where MV® = MVF’. The output is a polynomial of degree 4 CV®=k.CVF“. This simplification facilitates the inversion of these functions. In general, it is possible Figure 7.23. 2x2/A=1 multivariable control Figure 7.25. 2x2/2=0.5 multivariable control Figure 8.1. Heat exchange between a jacket and a vessel A classical academic approximation of the heat-balance equation may bs established by equating the heat-transfer fluid temperature to the outpu temperature of the double jacket: This approximation is valid only when the heat-transfer fluid flow rate is high. A more realistic analysis will be presented in Chapter 10. In most cases, the volume V,, 1s larger than V,, resulting in one real pole dominating the second-order process''. In these circumstances the governing system equation may be approximated using a first-order model. Eliminating 7, from these equations, it may be shown that the system can be represented by a dominant pole resulting in: 9.3 Unstable Zero and a Stable Pole 9.5 Control of an Unstable, Non-minimum Phase Process Figure 10.3. Convexity coefficient as a function of flow rate F, Figure 10.6. A finite convolution representation of 2 infinite sequences The response represents the sum of three terms (one free and two forced response Figure 10.10. Variation of the convexity coefficient [as a function of the Q, and Q, flow Figure 10.11. Example of a control block diagram as implemented on the Emerson DeltaV7 Figure 10.13. Variation of the input product temperature 7.,; Q; fluid expressed in 1/h/5 Figure 10.15. Comparison of the PID and PFC control with pure delay (175 I/h) The initial high-frequency component of MV of the PID regulator has no effect on the output, while the equivalent PFC regulator MV is smoother. A sensor is placed at a distance from the process, resulting in the introduction of a time delay (or transportation lag). Figure 10.15 shows that the resulting behaviour of the PID regulator is marginally stable. On the other hand, the behaviour of the PFC does not vary since the time delay is accounted for in the internal model. Figure 10.14. Comparison of PID and PFC (175 I/h) control The response times of the two regulators were recorded as shown in Figure 10.14. It can be seen that the PID regulator, “tightly” controlled by an experienced independent human operator, is close to instability. Its 95% response time is approximately 82 s; whereas the predictive regulator produces a response time of 36 s. Note, the behaviour of the Q, (MV) in each case is very different. Figure 10.16. Comparison of a PID and PFC regulator with pure delay (80 I/h) Figure 10.18. Continuous casting. Level control of the liquid-steel mould and tundish L0.4.1.2 Control Loops The weight of the steel in the tundish is regulated by a controller whose function is to ensure a relatively constant level while taking any MV constraints into account, i.e., the rotary nozzle. This actuator is a fragile and expensive mechanism consisting of two superimposed, heat-resistant plates; one stationary and the other mobile. Each plate possesses a circular hole; the flow control is achieved by altering the relative positions of the mobile opening. It is important to minimise any rotational movements of the mobile plate in order to limit wear and increase the actuator life span. Figure 10.22. Iso-distance curves PFC Parameters: Figure 10.24. PFC Performance Figure 10.31. Test sequence protocol used to identify the steam temperature 7), Figure 10.32. Controller overview for second exchanger output temperature 7, Figure 10.34. a PID control error and b comparison of PFC and P temperature controllers The regulation of the fluid level in the heating drum is critical. If the ball were to run dry, this would quickly result in the destruction of the evaporation tubes. Conversely, an overflow in the heating drum would result in harmful liquid water entering the turbo-alternators. The flow required by the various steam consumers variables considerably and thus affects the level of the lower steam drum. Figure 10.36. Level control of the steam drum Figure 10.37. Overview of heating drum level control Figure 10.40. a PID error signal, b PID error histogram and ¢ comparison of PFC and P level controllers Figure 10.41. The initial distribution of steam temperature Figure 10.45. Time constant as a function of the flow (normalised units) Figure A.2. PFC control of a first-order integrator process
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