Overview of Models for Automated Process Control

Overview of Models for Automated Process Control James B. Rawlings Department of Chemical and Biological Engineering April 29, 29 Utilization of Process Modeling and Advanced Process Control in QbD based Drug Development and Manufacturing Workshop Baltimore, MD Rawlings Models for automated process control / 46

Outline Introduction and Motivation 2 Modeling Nonlinear differential equations Linear time invariant models Discrete time models Input-output models Constraints Modeling the noise 3 From Models to Automated Process Control Regulation State estimation Disturbance modeling and zero offset 4 Future Developments 5 Further Reading Rawlings Models for automated process control 2 / 46

The power of abstraction process dx = f (x, u) dt y = g(x, u) actuators sensors Rawlings Models for automated process control 3 / 46

Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Rawlings Models for automated process control 4 / 46

Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Rawlings Models for automated process control 4 / 46

Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity 3-5 M$/year increased profit due to increased throughput (28) Rawlings Models for automated process control 4 / 46

Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity 3-5 M$/year increased profit due to increased throughput (28) Praxair experience with MPC Praxair currently has more than 5 MPC installations Rawlings Models for automated process control 4 / 46

Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity 3-5 M$/year increased profit due to increased throughput (28) Praxair experience with MPC Praxair currently has more than 5 MPC installations 6 M$/year increased profit (28) Rawlings Models for automated process control 4 / 46

Broader industrial impact (Qin and Badgwell, 23) Area Aspen Honeywell Adersa PCL MDC Total Technology Hi-Spec Refining 2 48 28 25 985 Petrochemicals 45 8-2 55 Chemicals 2 3 2 44 Pulp and Paper 8 5 - - 68 Air & Gas - - - Utility - - 4 4 Mining/Metallurgy 8 6 7 6 37 Food Processing - - 4 5 Polymer 7 - - - 7 Furnaces - - 42 3 45 Aerospace/Defense - - 3-3 Automotive - - 7-7 Unclassified 4 4 45 26 45 6 Total 833 696 438 25 45 4542 First App. DMC:985 PCT:984 IDCOM:973 PCL: SMOC: IDCOM-M:987 RMPCT:99 HIECON:986 984 988 OPC:987 Largest App 63x283 225x85-3x2 - Rawlings Models for automated process control 5 / 46

Input, output, and state variables input output process u y Input/output description Rawlings Models for automated process control 6 / 46

Input, output, and state variables input output process u y Input/output description input u R m state x R n output y R p process sensor State description Rawlings Models for automated process control 6 / 46

Nonlinear models Nonlinear differential equations dx = f (x, u, t) dt y = h(x, u, t) x(t ) = x x is the state u is the input y is the output t is time Rawlings Models for automated process control 7 / 46

Linear time invariant models Linear differential equations dx = Ax + Bu dt y = Cx + Du x() = x Rawlings Models for automated process control 8 / 46

Continuous time and discrete time models Continuous time x dx = Ax + Bu dt u t t Rawlings Models for automated process control 9 / 46

Continuous time and discrete time models Continuous time Discrete time x dx = Ax + Bu dt x x(k + ) = Ax + Bu u t u sample time t = k t t = k Rawlings Models for automated process control 9 / 46

Discrete time models Linear difference equations x(k + ) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) x() = x Rawlings Models for automated process control / 46

Discrete time models Linear difference equations x(k + ) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) x() = x Easier notation Linear difference equations x + = Ax + Bu y = Cx + Du x() = x in which x(k) + denotes x(k + ). Rawlings Models for automated process control / 46

Input-output models input output process u y Input/output description input u R m state x R n output y R p process sensor State description Rawlings Models for automated process control / 46

Input-output to state space Current output y(t) is modeled as a time series in past inputs and outputs, and noise y(t) = a y(t ) + a 2 y(t 2) + b }{{} u(t ) + b 2 u(t 2) +v(t) }{{} past outputs past inputs We estimate the model parameters from (u, y) data (Ljung, 2) θ = [ a a 2 b b 2 ] Rawlings Models for automated process control 2 / 46

Input-output to state space Current output y(t) is modeled as a time series in past inputs and outputs, and noise y(t) = a y(t ) + a 2 y(t 2) + b }{{} u(t ) + b 2 u(t 2) +v(t) }{{} past outputs past inputs We estimate the model parameters from (u, y) data (Ljung, 2) θ = [ a a 2 b b 2 ] Given θ we can obtain A, B, C Rawlings Models for automated process control 2 / 46

Converting input-output to state space Define x(t) to hold the past inputs and outputs. 2 x(t) = 4 y(t) 3 y(t ) 5 u(t ) Rawlings Models for automated process control 3 / 46

Converting input-output to state space Define x(t) to hold the past inputs and outputs. The time series model is 2 x(t) = 4 y(t) 3 y(t ) 5 u(t ) 2 3 2 y(t + ) 4 y(t) 5 = 4 a 3 2 a 2 b 2 5 4 y(t) 3 2 y(t ) 5 + 4 b 3 5 u(t) u(t) u(t ) {z } {z } {z } {z } x + A x B 2 y(t) = ˆ 4 y(t) 3 y(t ) 5 {z } u(t ) C {z } x Rawlings Models for automated process control 3 / 46

Converting input-output to state space Define x(t) to hold the past inputs and outputs. The time series model is 2 x(t) = 4 y(t) 3 y(t ) 5 u(t ) 2 3 2 y(t + ) 4 y(t) 5 = 4 a 3 2 a 2 b 2 5 4 y(t) 3 2 y(t ) 5 + 4 b 3 5 u(t) u(t) u(t ) {z } {z } {z } {z } x + A x B 2 y(t) = ˆ 4 y(t) 3 y(t ) 5 {z } u(t ) C {z } x The equivalent state space model is x + = Ax + Bu y = Cx Rawlings Models for automated process control 3 / 46

Constraints Physical bounds on inputs (valves full open, full closed) u 2 u u(k) u k in which E = Eu(k) e k [ ] I I e = [ ] u u x 2 u x Rawlings Models for automated process control 4 / 46

Constraints Physical bounds on inputs (valves full open, full closed) u 2 u u(k) u k in which E = Eu(k) e k [ ] I I e = [ ] u u x 2 u Constraints on states or outputs for reasons of safety, operability, product quality, etc. Fx(k) f k x Rawlings Models for automated process control 4 / 46

Rate of change constraints Important in some applications to limit the rate of change of the input, u(k) u(k ) Rawlings Models for automated process control 5 / 46

Rate of change constraints Important in some applications to limit the rate of change of the input, u(k) u(k ) Augment the state x(k) = [ ] x(k) u(k ) Rawlings Models for automated process control 5 / 46

Rate of change constraints Important in some applications to limit the rate of change of the input, u(k) u(k ) Augment the state x(k) = [ ] x(k) u(k ) Augmented system model [ ] [ ] [ ] [ ] x(k + ) A x(k) B = + u(k) u(k) u(k ) I }{{}}{{}}{{}}{{} ex(k+) ea ex(k) eb y(k) = [ C ] [ ] x(k) }{{} u(k ) }{{} ec ex(k) Rawlings Models for automated process control 5 / 46

Rate of change constraints The rate of change constraint u(k) u(k ) k Rawlings Models for automated process control 6 / 46

Rate of change constraints The rate of change constraint u(k) u(k ) k is stated as F x(k) + Eu(k) e F = [ ] I I E = [ ] I I e = [ ] Rawlings Models for automated process control 6 / 46

Rate of change constraints The rate of change constraint u(k) u(k ) k is stated as F x(k) + Eu(k) e F = [ ] I I E = [ ] I I e = [ ] General linear constraints subsumes all previous forms Fx(k) + Eu(k) e k Rawlings Models for automated process control 6 / 46

Nonlinear constraints For nonlinear models, linear constraints are not required u(k) U x(k) X k u 2 x 2 U u X x Rawlings Models for automated process control 7 / 46

Nonlinear constraints For nonlinear models, linear constraints are not required u(k) U x(k) X k Even more general (u(k), x(k)) Z k u 2 x 2 U u X x Rawlings Models for automated process control 7 / 46

Typical process data 2 8 6 y 4 2-2 -4-6 5 5 2 time Rawlings Models for automated process control 8 / 46

Modeling the noise Random disturbances x + = Ax + Bu + Gw y = Cx + v w is the random variable affecting the process Rawlings Models for automated process control 9 / 46

Modeling the noise Random disturbances x + = Ax + Bu + Gw y = Cx + v w is the random variable affecting the process v is the random variable affecting the measurement Rawlings Models for automated process control 9 / 46

Modeling the noise Random disturbances x + = Ax + Bu + Gw y = Cx + v w is the random variable affecting the process v is the random variable affecting the measurement So v models measurement noise and w models process disturbance Rawlings Models for automated process control 9 / 46

Modeling the noise Random disturbances x + = Ax + Bu + Gw y = Cx + v w is the random variable affecting the process v is the random variable affecting the measurement So v models measurement noise and w models process disturbance The controller needs to know the relative amounts of each disturbance Rawlings Models for automated process control 9 / 46

The model predictive control framework Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u Rawlings Models for automated process control 2 / 46

Predictive control The future influences the present just as much as the past does. Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u Rawlings Models for automated process control 2 / 46

Predictive control Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u min u(t) T y sp g(x, u) 2 Q + u sp u 2 R dt ẋ = f (x, u) x() = x (given) y = g(x, u) Rawlings Models for automated process control 2 / 46

State estimation When I want to understand what is happening today or try to decide what will happen tomorrow, I look back. Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u Rawlings Models for automated process control 22 / 46

State estimation Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u min x,w(t) T y g(x, u) 2 R + ẋ f (x, u) 2 Q dt ẋ = f (x, u) + w (process noise) y = g(x, u) + v (measurement noise) Rawlings Models for automated process control 22 / 46

Separation of the control problem input output process u y Input/output description Rawlings Models for automated process control 23 / 46

Separation of the control problem input output process u y Input/output description input u R m state x R n output y R p process sensor State description Rawlings Models for automated process control 23 / 46

Separation of the control problem input output process u y Input/output description input u R m state x R n output y R p process sensor State description input u process state x Regulation problem regulator Rawlings Models for automated process control 23 / 46

Separation of the control problem input output process u y estimator Estimation problem estimate ˆx Rawlings Models for automated process control 24 / 46

Separation of the control problem input output process u y estimator Estimation problem estimate ˆx input u process output y regulator estimator Control problem ˆx Rawlings Models for automated process control 24 / 46

Controller Design for Zero Offset Often want zero tracking error and rejection of constant disturbances Rawlings Models for automated process control 25 / 46

Controller Design for Zero Offset Often want zero tracking error and rejection of constant disturbances Augment the model with a constant disturbances (Davison and Smith, 97, Kwakernaak and Sivan, 972, p. 278). [ ] x(k + ) = d(k + ) y(k) = [ C [ A Bd I ] [ ] x(k) + d(k) ] + v(k) C d ] [ x(k) d(k) [ ] B u(k) + w(k) Rawlings Models for automated process control 25 / 46

Offset free control system x + = A x + Bũ (Q, R) regulator u plant y x s u s ˆx estimator y sp, u sp, z sp (Q s, R s ) target selector ˆx ˆd [ ] + x = d [ ] [ A Bd x I d] y = [ ] [ x C C d d (Q w, R v) [ B + u + w ] MPC controller consisting of: receding horizon regulator, state estimator, and target selector. ] + v Rawlings Models for automated process control 26 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c Reaction A B T c T, c r F h Rawlings Models for automated process control 27 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c T c r T, c F h Reaction A B Controlled variables: level of the tank (h) Rawlings Models for automated process control 27 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c T c r T, c F h Reaction A B Controlled variables: level of the tank (h) concentration of species A (c) Rawlings Models for automated process control 27 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c T c r T, c F h Reaction A B Controlled variables: level of the tank (h) concentration of species A (c) State variable: reactor temp (T ) Rawlings Models for automated process control 27 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c T c r T, c F h Reaction A B Controlled variables: level of the tank (h) concentration of species A (c) State variable: reactor temp (T ) Manipulated variables: coolant liquid temp (T c ) Rawlings Models for automated process control 27 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c T c r T, c F h Reaction A B Controlled variables: level of the tank (h) concentration of species A (c) State variable: reactor temp (T ) Manipulated variables: coolant liquid temp (T c ) outlet flowrate (F ) Rawlings Models for automated process control 27 / 46

Design example. (Pannocchia and Rawlings, 23) F, T, c T c r T, c F h Reaction A B Controlled variables: level of the tank (h) concentration of species A (c) State variable: reactor temp (T ) Manipulated variables: coolant liquid temp (T c ) outlet flowrate (F ) Disturbance: inlet flowrate (F ) Rawlings Models for automated process control 27 / 46

More measured outputs than inputs and zero offset Component mass and energy balances Rawlings Models for automated process control 28 / 46

More measured outputs than inputs and zero offset Component mass and energy balances Nonlinear differential equation model: dc dt = F ( (c c) Ur 2 k c exp E h RT dt dt = F (T T ) Ur 2 h dh dt = F F Ur 2 + H k c exp ρc p ) ( E ) + 2U (T c T ) RT rρc p Rawlings Models for automated process control 28 / 46

Well-stirred reactor example: parameters Parameter Nominal value Units F. m 3 /min T 35 K c kmol/m 3 r.29 m k 7.2 min E/R 875 K U 54.94 kj/min m 2 K ρ kg/m 3 C p.239 kj/kg K H 5 4 kj/kmol Rawlings Models for automated process control 29 / 46

Well-stirred reactor example: linearized model Using = min, the linearized model is x + = Ax + Bu + B p p y = Cx in which all the states are measured c c s [ ] x = T T s Tc T u = s c c s c h h s F F s y = T T s p = F F s h h s Rawlings Models for automated process control 3 / 46

Well-stirred reactor example: linearized model Using = min, the linearized model is x + = Ax + Bu + B p p y = Cx in which all the states are measured c c s [ ] x = T T s Tc T u = s c c s c h h s F F s y = T T s p = F F s h h s and.268.338.728 A = 9.73.3279 25.44 C =.537.655.75 B =.297 97.9 B p = 69.74 6.637 6.637 Rawlings Models for automated process control 3 / 46

Reactor case study - two integrating disturbances We have two inputs, T c and F Rawlings Models for automated process control 3 / 46

Reactor case study - two integrating disturbances We have two inputs, T c and F We try to remove offset in two controlled variables, c and h Rawlings Models for automated process control 3 / 46

Reactor case study - two integrating disturbances We have two inputs, T c and F We try to remove offset in two controlled variables, c and h Model the disturbance with two integrating output disturbances on first and third controlled variables by choosing: [ ] C d = B d = Simulate the response of the controlled system after a % increase in F at time t = min using the nonlinear differential equations for the plant model Rawlings Models for automated process control 3 / 46

Reactor case study - two integrating disturbances We have two inputs, T c and F We try to remove offset in two controlled variables, c and h Model the disturbance with two integrating output disturbances on first and third controlled variables by choosing: [ ] C d = B d = Simulate the response of the controlled system after a % increase in F at time t = min using the nonlinear differential equations for the plant model Is there steady offset in any of the outputs? Which ones? Rawlings Models for automated process control 3 / 46

Two integrating disturbances: output results c (kmol/m 3 ).93.92.9.9.89.88.87 5 5 2 25 3 35 4 45 5 326 324 T (K) 322 32 38 36 5 5 2 25 3 35 4 45 5.8.76 h (m).72.68.64 5 5 2 25 3 35 4 45 5 time (min) Three measured outputs versus time after a step change in F at t = min Rawlings Models for automated process control 32 / 46

Two integrating disturbances: input results Tc (K) 3 3 299 298 297 296 295 5 5 2 25 3 35 4 45 5 F (m 3 /min)..8.6.4.2. 5 5 2 25 3 35 4 45 5 time (min) Two manipulated inputs versus time after a step change in F at t = min Rawlings Models for automated process control 33 / 46

Reactor case studies - two and three integrating disturbances Two integrating disturbances - conclusion: Despite integrators in the two controlled variables, c and h, all outputs have nonzero steady offset! Next we try three integrating disturbances: two added to the two controlled variables one added to the second manipulated variable.655 C d = B d = 97.9 6.637 Rawlings Models for automated process control 34 / 46

Three integrating disturbances: output results.88 c (kmol/m 3 ).876.872.868 5 5 2 25 3 35 4 45 5 328 T (K) 326 324 5 5 2 25 3 35 4 45 5.76 h (m).72.68.64 5 5 2 25 3 35 4 45 5 time (min) Three measured outputs versus time after a step change in F at t = min Rawlings Models for automated process control 35 / 46

Three integrating disturbances: input results 3 Tc (K) 3 299 5 5 2 25 3 35 4 45 5 F (m 3 /min).2.6.2.8.4..96 5 5 2 25 3 35 4 45 5 time (min) Two manipulated inputs versus time after a step change in F at t = min Rawlings Models for automated process control 36 / 46

Three integrating disturbances: conclusions We now have zero offset in the two controlled variables, c and h Rawlings Models for automated process control 37 / 46

Three integrating disturbances: conclusions We now have zero offset in the two controlled variables, c and h We have successfully forced the steady-state effect of the disturbance entirely into the second output, T The dynamic behavior of all outputs is superior to that achieved with the model using two integrating disturbances The true disturbance is better represented by including the integrator in the outlet flowrate With a more accurate disturbance model, better overall control is achieved The controller uses smaller manipulated variable action and also achieves better output variable behavior Rawlings Models for automated process control 37 / 46

Three integrating disturbances: conclusions We now have zero offset in the two controlled variables, c and h We have successfully forced the steady-state effect of the disturbance entirely into the second output, T The dynamic behavior of all outputs is superior to that achieved with the model using two integrating disturbances The true disturbance is better represented by including the integrator in the outlet flowrate With a more accurate disturbance model, better overall control is achieved The controller uses smaller manipulated variable action and also achieves better output variable behavior An added bonus is that steady offset is removed in the maximum possible number of outputs Rawlings Models for automated process control 37 / 46

Advanced process control and drug manufacturing Drug manufacturing versus petrochemicals Differences Small volume, high value-added products Rawlings Models for automated process control 38 / 46

Advanced process control and drug manufacturing Drug manufacturing versus petrochemicals Differences Small volume, high value-added products Batch and semi-batch processing Batch-to-batch variability Rawlings Models for automated process control 38 / 46

Advanced process control and drug manufacturing Drug manufacturing versus petrochemicals Differences Small volume, high value-added products Batch and semi-batch processing Batch-to-batch variability Importance of the solid phase Rawlings Models for automated process control 38 / 46

Advanced process control and drug manufacturing Drug manufacturing versus petrochemicals Differences Small volume, high value-added products Batch and semi-batch processing Batch-to-batch variability Importance of the solid phase Novel solid-phase sensors (on-line video imaging) Rawlings Models for automated process control 38 / 46

On-line Video Imaging Figure 4. Examples of real in situ images of rod-like crystals. In situ images of rod-like crystals from our crystallization lab (Larsen and Rawlings, 29). ontinued improvements in the design of in situ video sysems will lead to images that are increasingly similar to the pling, but also effective image analysis. For example, Figure Accurate PSD estimation requires not only sufficient sam- dealized, simulated images used in this work. 6 indicates that, assuming perfect image analysis, images provides sufficient samples to achieve a reasonably good estimate of the weight PSD for the given process conditions bsolute PSD measurement Figure 7, on the other hand, shows the PSD estimated from Figures 5 and 6 show, respectively, the number- and image analysis data generated using the SHARC algorithm. 7 eight-based PSDs corresponding to the optimal cooling prole. The continuous curve shows the simulated PSD, and the the image analysis method identifies several false positives This figure indicates that towards the end of the experiment istograms show the Rawlings estimated PSD based on imaging mea- Modelsand for automated also misses process a significant control number of 39 both / 46nucleus-grown

Idealized data from a batch crystallization Figure 2. Examples of images generated at various times during the optimal cooling simulation. Optimal The images cooling correspondprofile. to 6-min intervals Images from h (upper correspond left) to 8 h (lower toright). 6-minute intervals from hours (upper left) to 8 hours (lower right). Rawlings Models for automated process control 4 / 46

Idealized data from a batch crystallization igure 3. Examples of images generated at various times during the linear cooling simulation. The images correspond to 6-min intervals from h (upper left) to 8 h (lower right). Linear cooling profile. Images correspond to 6-min intervals from h (upper left) to 8 h (lower right). DOI.2/aic Published on behalf of the AIChE April 29 Vol. 54, No. 4 AIChE Journal Rawlings Models for automated process control 4 / 46

Inferring size distribution from video images Figure 6. Evolution of measured and estimated weight PSD cooling and perfect image analysis. Measured and Snapshots estimated shown from t 5 minweight to t 5 48 minpsd at 6 intervals. for Bin optimal size 5 lm andcooling N 5. and perfect image analysis. Bin size = µm and bins. analysis algorithms have different tolerances for overlap, with model-based algorithms typically tolerating more overlap than other methods. On the basis of the images in Figure 2, one could reasonably expect improvements in model-based image analysis algorithms to enable effective measurement In previous studies of this system, these measurement difficulties motivated the use of scanning electron and optical microscopy to characterize the PSD qualitatively in terms of habit and maximum size. 5 Quantitative PSD measurement by microscopy could not be achieved because of sampling Rawlings Models for automated process control 42 / 46

Inferring size distribution from video images Figure 7. Evolution of measured and estimated weight PSD cooling and image analysis using SHARC. Measured and Snapshots estimated shown from t 5 minweight to t 5 48 at PSD 6 min intervals. for Bin optimal size 5 lm and cooling N 5. and image analysis using SHARC. Bin size = µm and Z Siþ bins. weight-based c vw (l 5l 3/l 2 4 ) can be measured effectively by imaging. General imaging considerations in which S 5 (S,..., S T) is the vector of breaks between Rawlings Models for automated process control 43 / 46 q i ¼ Si f ðlþdl; i ¼ ;...; T (7)

Further reading I E. J. Davison and H. W. Smith. Pole assignment in linear time-invariant multivariable systems with constant disturbances. Automatica, 7:489 498, 97. H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. John Wiley and Sons, New York, 972. P. A. Larsen and J. B. Rawlings. Assessing the reliability of particle number density measurements obtained by image analysis. Part. Part. Syst. Charact., 25(5 6):42 433, February 29. P. A. Larsen and J. B. Rawlings. The potential of current high-resolution imaging-based particle size distribution measurements for crystallization monitoring. AIChE J., (4):896 95, April 29. L. Ljung. System Identification - Theory for the User. Prentice Hall, New Jersey, 2nd edition, 2. G. Pannocchia and J. B. Rawlings. Disturbance models for offset-free MPC control. AIChE J., 49(2):426 437, 23. Rawlings Models for automated process control 44 / 46

Further reading II S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control Eng. Prac., (7):733 764, 23. J. B. Rawlings and D. Q. Mayne. Model predictive control: Theory and design. Nob Hill Publishing, 29. Rawlings Models for automated process control 45 / 46