Feedforward for Stabilization of an Ammonia Synthesis Reactor Erik Holter Master of Science in Engineering Cybernetics Submission date: June 2010 Supervisor: Morten Hovd, ITK Norwegian University of Science and Technology Department of Engineering Cybernetics Problem Description Feedback control is necessary for stabilization of unstable systems. Feedforward on the other hand, is normally used to improve control performance at high frequencies, beyond the achievable bandwidth for stable closed-loop control. Recently, it has been showed that feedforward can be used to avoid input constraints which would otherwise cause the system to go unstable. The aim for the assignment will be to illustrate this for a model of an ammonia synthesis reactor with heat integration. The assignment will be based on an existing MATLAB/Fortran77 model. The project will include: 1. Extensions and modifications of the original model, to include a manipulated variable for control, disturbances and input constraints, 2. design of feedforward and feedback controllers, 3. verification of results using simulation. Supervisor: Professor Morten Hovd Assignment given: 11. January 2010 Supervisor: Morten Hovd, ITK Preface As a part of the two year MSc program in Engineering Cybernetics- here at Nor- wegian University of Science and Technology (NTNU)- a compulsory master thesis has to be written during the fourth semester. I changed assignment between the third and fourth semester, so this master thesis is not based on the student project which was carried out in the third semester. There are a number of people who should be thanked for helping me in this period. First, for guiding me through this thesis, I would like to give special thank to my supervisor Professor Morten Hovd, at the Department of Cybernetics, for invaluable input during the semester. Second, for reading through my thesis, great thanks to Annette Aubert. Third, I also wish to thank my fellow students Haavard Pershon and Inge Brat- bakken for all the helpful discussions. Last, but not least, I wish to thank all my fellow students for making my stay here at NTNU and Trondheim a real pleasure. i Abstract This thesis illustrates different control structures and tries to demonstrate how feed- forward control can be used in stabilizing an unstable ammonia reactor with heat integration. The demonstration of feedforward is done under very special circum- stances. While feedback control is necessary for stabilization of the reactor system, feedforward control can be used to avoid input constraints which would otherwise make the input saturate and thereby make the system unstable. It turned out that the ammonia reactor was not the best system to apply the feed- forward strategies in question. The main reason is a combination of; the existence of the lower (undesired) steady-state operating point (corresponds to extinction of reaction), positive feedback from the heat exchanger and the manipulated variable range of actuation. The reason is that there are trade offs between making more of the (cold stream) mass flow go through the heat exchanger and making the cold stream mass flow get mixed with the reactor flow between the beds at the quench points. Letting more mass flow entering the heat exchanger will reduce the heat exchanger efficiency. Lowering the efficiency means that the hot stream mass flow through the heat exchanger can not liberate enough heat to the cold stream mass flow entering the heat exchanger. As a result of this, the reactor inlet temperature will decrease because of the positive feedback from the heat exchanger. Thus, it does not exist a range of actuation where the system can be stabilized when influenced by disturbance. iii Contents Preface................................... i Abstract.................................. iii Abbreviations............................... ix ListofFigures............................... xi 1 Introduction 1 1.1BasisfortheProject........................... 1 1.2AimofWork............................... 1 1.3ThesisOutlines.............................. 1 2 Mathematical Model of the System 2 2.1Introduction................................ 2 2.2ReactorModel.............................. 4 2.2.1 MassandEnergyBalance.................... 6 2.2.2 AmmoniaSynthesis........................ 7 2.2.3 ModelingofQuenchPoints.................... 7 2.3ModelofHeatExchanger......................... 8 2.4‘BypassValve’............................... 9 2.5State-spaceModel............................. 12 3 Control of Reactor 14 3.1Introduction................................ 14 3.2Open-loopSimulations.......................... 15 3.3PID-controller............................... 18 3.4 P-controller and the Reactor Inlet Temperature as Measurement . 18 3.5PD-controllerandtheReactorOutletasMeasurement........ 19 4 Feedforward Design 22 4.1Introduction................................ 22 4.2StableDisturbanceModel........................ 23 v 4.3 Unstable Disturbance Model: A”ReferenceGovernor”Approach................... 24 4.3.1 Hankel-Normand-SingularValues............... 26 4.4 H∞-controlandFeedforward....................... 27 5 Simulation Results 30 5.1WiththeReactorInletTemperatureasMeasurement......... 30 5.1.1 Unconstrained Input ....................... 30 5.1.2 Finding an Unstable Operating Point with RHP-poles Further intotheRHP........................... 33 5.2WiththeReactorOutletTemperatureasMeasurement........ 35 5.2.1 Unconstrained Input ....................... 36 5.2.2 Constrained Input ........................ 37 5.2.3 Feedforward............................ 40 6 Discussion 42 6.1ReactorModel.............................. 42 6.2ReactorTemperatureandPressure................... 42 6.3 Extinction and Instability ........................ 43 6.4ControlofReactor............................ 44 6.5Feedforward................................ 44 6.6SimulinkModel.............................. 46 7Conclusion 48 8FurtherWork 50 8.1ControlofAmmoniaReactor...................... 50 8.2Feedforward................................ 50 Appendix: A Numerical Parameters used in the Simulations 52 B Numerical Solution of the Model Equations 54 C MATLAB Files 56 C.1MATLABFunctionsusedineachBed................. 56 C.1.1ReactionRate........................... 58 C.2State-spaceModels............................ 58 C.3NehariExtension............................. 62 C.4 H∞-controller............................... 64 vi C.5MiscellaneousFunctions......................... 65 Bibliography 69 vii viii Abbreviations Here are some terms and abbreviations used through the thesis. To reactor outlet temperature Ti reactor inlet temperature / heat exchanger outlet temperature Tfeed Fresh feed temperature m˙ c cold stream mass flow through heat exchanger m˙ h hot stream mass flow through heat exchanger Q1 mass flow of fresh feed entering quench 1 Q2 mass flow of fresh feed entering quench 2 Q3 mass flow of fresh feed entering quench 3 RHP- Right half plane LHP- Left half plane IMC Internal model control SIMC Skogestad/Simple IMC P-controller Proportional controller PI-controller Proportional-Integral- controller PID-controller Proportional-Integral-Derivative controller 2-DOF Two Degrees Of Freedom MPC Model Predictive Control NMPC Nonlinear Model Predictive Control ix x List of Figures 2.1 Flowsheet of an ammonia synthesis loop with a fixed bed quench cool- ingreactor................................. 3 2.2Ammoniasynthesisreactor........................ 5 2.3Ammoniasynthesisreactorincludingbypassvalve........... 10 2.4Simulinkblockdiagramofreactor.................... 11 3.1 Nonlinear simulation of a drop in fresh feed temperature (Tfeed) ... 16 3.2Nonlinearsimulationofdecreaseinthereactorpressure........ 17 4.1Blockdiagram............................... 23 4.2 Feedforward arrangement for an unstable disturbance transfer function 25 4.3 H∞-controllersynthesisfora2-DOFcontroller............. 28 5.1Nonlinearsimulationwithproportionalcontrol............. 32 5.2Nonlinearsimulationwith(large)decreaseintemperaturefeed.... 34 5.3 Nonlinear simulation with unconstrained input ............. 36 5.4 Nonlinear simulation with unconstrained input and pressure used as disturbance................................. 37 5.5 Nonlinear simulation with constrained input .............. 38 5.6 Nonlinear simulation with constrained input and feedforward ..... 41 xi xii Chapter 1 Introduction 1.1 Basis for the Project This MSc thesis is carried out for, and proposed by, Professor Morten Hovd. The assignment is developed as a result of the scientific paper; Feedforward for stabiliza- tion, written by Morten Hovd and Robert Bitmead[HB07]. This paper demonstrates how feedforward can assist in stabilizing unstable systems. The paper points out, that one fundamental limitation for unstable systems is that the range of actuation for the inputs1 must be sufficiently large to avoid satura- tion. It is well known that when inputs saturate, the feedback path breaks, and the stability of the system is lost. Jens Balchen et.al.[JGBF03] explains feedforward in the traditional setup as: ”Feedforward is used to improve control performance by generating changes in the input which counteracts the influence disturbance has on the system, such that the response does not effect (or has little effect on) the output of the system”. In the paper written by Morten Hovd and Robert Bitmead, feedforward is used to reduce the magnitude of the plant input moves, and hence avoid the instability due to input constraints. 1.2 Aim of Work The main goal for this work is to illustrate