jMPC Toolbox v3.11 User`s Guide
Contents
1. and the output vector y as y Load Angle frad ya Load Torque Nm and input u as ui Motor Voltage V 20 Control Objectives The control objective is to control the Load Angle yi to a desired setpoint by adjusting the Motor Input Voltage u1 We require the Load Angle to settle at the setpoint within around 10 seconds System Constraints The Motor Voltage Constraint is set as 220V lt ui lt 220V And the Torque Constraint on the shaft is 78 5Nm lt y2 lt 78 5Nm It is important to note in this example that we cannot directly measure the Load Torque however we must constrain it This is a perfect example of the power of model based control where we can use the internal model outputs ym in order to predict unmeasured outputs The predicted outputs can then be controlled and or constrained as per a normal output For this example we will constrain the Load Torque only and not control it to a setpoint Linear MPC Simulation Step 1 Create The Plant The first step is to implement the model equations above into MATLAB as a jSS object The parameters are first defined then the model A B C and D matrices are created and passed to the jSS constructor Parameters k0 1280 2 Torisional Rigidity kT 10 Motor Constant JM 0 5 Motor Inertia JL 50 JM Load Inertia p 20 Gear Ratio BM 0 1 Motor viscous friction BL 25 Load viscous fraction R 20 Armature Resistance Plant2. 0 1 0 271 ini in2 ywt 0 1 out1 con u 5 5 2 5 5 5 2 5 umin umax del umax con y 5 5 lymin ymax setp 1 setp1 Ts 0 1 4 6 Loading and Saving Settings In order to be able to save and load an MPC configuration three methods are provided Import from Export to Workspace When you wish to use a setup locally i e within the MATLAB workspace only this option allows the use of GUI objects which are specific to this tool They contain all options specified by the GUI with the exception of simulation data which is only read during an active simulation Both of these options are available vai the respective File Load File Save menus and each will launch a separate window for loading or saving Load From Workspace This window is almost identical to the Load Example window with the exception that this window scans the base MATLAB workspace for GUI objects rather than loading a pre saved example Quanser 3DOF Helicopter Np 15 Nc 6 Limit Input to 24 Y Limit Elevation to 27 Sdeg Limit Pitch to 60deg Limit Rotation to 180deg No Model Plant Mismatch Out1 Elevation Out2 Pitch Figure 15 Load from workspace interface Save To Workspace 17 Use this method to save the current settings into a jGUI object which will be saved to the base MATLAB workspace Not you must have a working setup for this to work as this option will rebuild a jMPC obje
3. 0 0 50 100 150 200 0 50 100 150 200 Fraction 0 5 Au Change in Reflux Ratio Us Reflux Ratio 4 0 2 w 0 3 25 4 M 1 0 2 4 m 1 0 50 100 150 200 0 50 100 150 200 Sample Sample Figure 35 Distillation Column Simulation Inputs 50 References 1 J Akesson MPCtools 1 0 Reference Manual 2006 2 Hahn J and Edgar T F An improved method for nonlinear model reduction using balancing of empirical gramians Computers and Chemical Engineering 26 3 Henson M and Seborg D Nonlinear Process Control Prentice Hall P TR 1997 4 Intel Intel Math Kernel Library 10 3 2011 5 Quanser Coupled Water Tanks User Manual 2003 6 Quanser 3 DOF Helicopter Reference Manual 2007 7 The Mathworks MPC Toolbox v3 2 1 2010 51
4. 42 mdist 220 260 2 Tf linspace 0 20 41 Slow cooling of Tf mdist 261 end 2 Tf 20 Tf final Set Initial values at linearization point Plant x0 xop Model x0 xop Step 7 Set MPC Options For advanced settings of the MPC controller or simulation you can use the jMPCset function to create the required options structure For this example we wish to set the plot titles and axis as well as setting the initial control input at k 0 to the linearization point Set Options opts jMPCset InitialU u0 Set initial control input at linearization point InputNames Feed Concentration Feed Temperature Jacket Temperature InputUnits mol m 3 K K QutputNames Reactor Concentration Reactor Temperature QutputUnits mol m 3 K Step 8 Build the MPC Controller and Simulation Options Now we have specified all we need to build an MPC controller and Simulation environment we call the two required constructor functions 4 Build MPC amp Simulation MPC1 jMPC Model Np Nc uwt ywt con Kest opts simopts jSIM MPC1 Plant T setp mdist MPC1 is created using the jMPC constructor where the variables we have declared previously are passed as initialization parameters Simulation options simopts is created similarly except using the jSIM constructor Step 9 Run the MPC Simulation and Plot Results With the con
5. a linear time invariant 1ti model for the controller as the jMPC object does not currently support nonlinear models Plant Model Model Plant Plant Model Non Linear Model States 4 Fen nl pend Inputs 1 States 4 Outputs 2 Ts D 05 s Dead Time 0 00 s Figure 14 Use of NL objects in the GUI Due to increased computation of the simulation an ODE solver on older machines you may experience a noticeable performance slow down in the GUI 4 5 Auto Setup To speed up users familiar with the jMPC object and using scripts to run the simulation this button allows the GUI to check the workspace to find setup options with default variable names e Np Prediction Horizon integer e Nc Control Horizon integer or Blocking Moves vector e uwt Input tuning weights vector e ywt Output tuning weights vector e con Constraint structure in the form con u is Umin Umax Delumax matrix con y is Ymin Ymax matrix e setp Setpoints matrix Note only the first row of this is used first setpoint e Ts 16 Sampling time of the controller scalar All variables must defined exactly as above Note you must have a model loaded first before you can use this feature An example setup is listed below Plant Gip tf 1 1 1 217 411 11 1 4 51 Model Gim tf 1 2 1 1 1 5 2 21 1 4 1 7 1 2 4 4 5 11 Np 10 Nc 5 uwt
6. be required to be used in order to gain the desired result The properties are e State Estimation To account for Model Plant mismatch and the affect of noise state estimation allows the plant states to be estimated The estimate is applied to the model states via a gain matrix of which one is designed using a gain supplied by the slider bar 10 Setup PAE Prediction Horizon Default Settings B NN INI 15 1 E Control Horizon or 2 0 0500 10 Blocking Moves QP Solver 3 0 100 6 MPC Wright IP Constraints Umin Umax Umax Rate Ymin Ymax 1 24 24 15 0 4800 0 4800 2 24 24 15 1 0500 1 0500 3 0 0 0 3 1416 31416 _ Figure 6 Setup Tab Controller Sampling Period This is a highly important period and must be chosen with consideration Models with different sampling periods will be re sampled at this frequency e Window Length Determines the length of time visible on the past and present windows Note a maximum of 100 samples can be displayed in the window so choose a window length appropriate to the sample time Soft Output Constraints Enables a penalty weighting on the output constraints such that breaking the output con straints due to for example a disturbance does not render the QP problem infeasible Defaults to a small penalty term 100 for example use Initial Setpoints To prevent initial instability or to start
7. but not smaller then disable this option This option is enabled by default Auto Calculate Ts The variable Ts is the sampling time of the controller and must be chosen with consideration of the speed of the model To aid amateur users from selecting sample times too fast or slow for a given process 18 this option will perform a step test of the model and extract the sample time and length of simulation It will then use these values for choosing the Controller Sampling Period Ts and Window Length Disable this option to manually select these values This option is enabled by default Note You must have the Control Systems Toolbox to use this option Data Logging If you wish to log the data displayed by the GUI then enable this option This can be useful for later analysis as you can view the entire simulation not just the window size Note that not just the controller inputs and plant outputs are returned but also the model output plant and model states and the rate of change of the inputs With this option enabled three objects are automatically created in the base workspace e plotvec his is a structure containing all the elements listed above such that it can be accessed by a user s plotting function easily e simresult This is a SIM object which contains the same information as plotvec but within one of the properties of the object The advantage of this is the plot command is overloaded for thi
8. from using that input and penalizing outputs will encourage the controller to move that output to the setpoint faster A special function in this toolbox is the ability to set a weight to 0 as done in ys This tells the MPC constructor this is an uncontrolled output such that it is not fed back and it will not have a setpoint however you may still place constraints on it as is done in this example The final line creates a discrete time observer using an overloaded version of the MATLAB dlge function assuming identity Q and R covariance matrices Step 5 Setup Simulation Options Next we must setup the simulation environment for our controller Simulation Length 22 T 300 ASetpoint setp ones T 1 setp 1 50 0 Initial values Plant x0 0 0 0 Model x0 0 0 0 01 01 The simulation length is always defined in samples such that in this example it is set to 30 seconds 300 samples x 0 1s The setpoint must be defined as T samples long and is a user specified trajectory for the controller to follow throughout the entire simulation When their are multiple controlled outputs the setpoint must a matrix with each column representing a different output Finally the initial states of the Plant and Model can be set individually such that simulations where the controller starts a state mismatch can be tested Step 6 Set MPC Options For advanced settings of the MPC controller or simulation you can use
9. jMPC Model States setp Controller Xp Plant States Yo Model Output y Figure 1 MPC Toolbox Block Diagram The above figure illustrates the functionality provided by the jMPC Toolbox within an example applica tion The Plant represents a dynamic industrial process which is to be controlled to a setpoint by the MPC controller Two forms of disturbances can be added to the simulation a load disturbance on the input and or measurement noise on the plant output The measured output is then fed back to the MPC controller in order to calculate the next control move Also viewable within this toolbox are controller model states and outputs for post analysis of model plant mismatch The MPC controller is created as a jMPC object while the Plant can be a SS object for linear simulations or a jNL object for nonlinear simulations These classes are described in detail within this document together with several examples illustrating their use 1 3 Nomenclature Several standard typographical and variable name conventions are used within this document and the toolbox itself Typographical Conventions t Italic Scalar a Lowercase Bold Vector A Uppercase Bold Matrix Standard Variable Names u Input vector Manipulated Variable x State vector y Output vector Measured Outputs MPC Variable Names umdist Unmeasured input disturbances mdist Measured input disturbances noise Output disturbances Up Plant input Div Mo
10. 10 for both 32bit and 64bit systems and should work out of the box If you do experience problems and wish to rebuild the C MEX files supplied you can run the supplied function jMPC MEX Install m located within the Utilities directory You will require the Intel Math Kernel Library MKL 4 as well as the default BLAS amp LAPACK libraries supplied with MATLAB to rebuild all files This should only be conducted if you require changes to the original source files 2 4 Locating Documentation The jMPC Toolbox is fully documented within the normal MATLAB help browser Simply type doc at the command prompt and navigate to the jMPC Toolbox section Supplementary help is also available by typing help x where x is the name of the jMPC function for example help jMPC This PDF can also be used as reference guide or as a class handout 2 5 MPC Simulation GUI A nice way to get a feel for automatic control using MPC is using the supplied Graphical User Interface GUI This allows real time interaction with an animated process such that tuning parameters con straints and model plant mismatch among other parameters can all be investigated To start using the GUI type the following at the MATLAB command line jMPCtool To load an example MPC setup click File Load Example Models and select a model and setup from the list Further details of the jMPC GUI will be presented later in this manual 2 6 Command Line Exam
11. A 0 1 0 0 k0 JL BL JL k0 p JL 0 0001 kO p JM O k0 p 2 JM BM kT 2 R JM B 0 0 kT R JM C 1 0 0 0 Plant Output Cm 1 0 0 0 Model contains unmeasured outputs kO O k0 p 01 D 0 Create jSS Object Plant jSS A B C D Step 2 Discretize the Plant Object Plant is now a continuous time state space model but in order to use it with the jMPC Toolbox it must be first discretized A suitable sampling time should be chosen based on the plant dynamics in this case use 0 1 seconds Discretize Plant Ts 0 1 Plant c2d Plant Ts 21 Step 3 Create Controller Model Now we have a model of the system for the internal calculation of the controller we also need a controller model for the internal calculation of the controller The plant does not have to be the same as the controller model and in real life will not be However for ease of this simulation we will define the plant and model to be the same Create jSS Model Model jSS A B Cm D Model c2d Model Ts In order to assign the second model output Torque as unmeasured we use the following method Set Unmeasured Outputs in model Model SetUnmeasuredOut Model 2 Step 4 Setup MPC Specifications Following the specifications from this example s reference enter the following MPC specifications Horizons Np 10 Prediction Horizon Nc 2 Control Horizon Constraints con u 220 220 1e2 ini umin umax
12. C Controller in MATLAB import it to SIMULINK and then run a SIMULINK Simulation 26 5 2 Quad Tank Example The completed example can be found in Examples Documentation Examples QuadTank Example m Reference J Akesson 2006 1 and Quanser 5 1 y1 Figure 22 Quad Tank Schematic Linear System Model The continuous time state space equation of the 4 tank system is as follows y k too te NEC Dub 0 A 0 wie Ax 0 o 4 i Ax 0 173 ka Au T3 1 1 31 k 3 0 0 0 7 Be 1 0 ke 0 0 0 0 ke 0 0 AYS io 5 ke 0a 0 0 0 ke where we have defined the state vector x as x Tank 1 Level z Tank 2 Level r3 Tank 3 Level z4 Tank 4 Level and the output vector y as 27 y Y2 Y3 Ya Tank 1 Level cm Tank 2 Level cm Tank 3 Level cm Tank 4 Level cm and input vector u as u Pump 1 Voltage V u2 Pump 2 Voltage V Co ntrol Objectives Due to the non square nature of the system 2 inputs and 4 outputs we will control two outputs namely the bottom two tank levels yi y2 to a specified setpoint This will be achieved by controlling the pump voltages via the inputs u1 u2 The top two tank levels are measured and are also constrained within safe operating limits but are not directly controlled System Constraints Both Pump Voltages are constrained as 12V lt U 1 2 lt 12V The Bottom Two Tank Levels are constrained as Oem lt y 2 lt 25cm And The Top Two Ta
13. Ca mol m z Temperature of reactor T K and the output vector y as y Concentration of A in reactor Ca mol m y Temperature of reactor T K and input vector u as ui Concentration of A in feed Ca f Measured Disturbance mol m uz Temperature of feed Tp Measured Disturbance K ug Temperature of cooling jacket Te K 40 Control Objectives The control objective is to control the temperature of the reactor by adjusting the temperature of the cooling jacket Both the concentration and temperature of the feed are measured disturbances and cannot be modified by the controller The reactor concentration is constrained but not controlled to a setpoint System Constraints The jacket cooling temperature is set as 278 15 lt uj3 X 450K And the reactor is constrained as 0 X y 3 mol m 278 15 lt ya lt 450K Linear MPC with Nonlinear Simulation Step 1 Build Nonlinear ODE Callback Function As detailed in creating a NL object the first step is to write an m file which contains the above expres sions suitable for use with a MATLAB integrator function xdot nl cstr t x u param Nonlinear CSTR model Assign Parameters q V KO E R H Cp rho UA param r k0xexp E R x 2 x 1 xdot 1 1 g V u 1 x 1 r xdot 2 1 end q V u 2 x 2 H Cp rho r UA Cp rho V u 3 x 2 The file n1 cstr m is saved in a suitable fol
14. INDUSTRIAL INFORMATION amp CONTROL CENTRE IC INDUSTRIAL INFORMATION amp CONTROL CENTRE jMPC Toolbox v3 11 User s Guide Jonathan Currie AUT University email jonathan currie aut ac nz July 5 2011 Contents 1 Introduction IMMEOS TUI APP 12 MPC Block Diagram 2 2 2m woe e mane ck A OX dre OC Royce ee Re t3 Nomenclatures usce eX GERE Rk we ek A CURA URL we one E Getting Started 2 Pre Heg ist S 4 42 4 vee ku EXER EORR RW wd a 2 2 Installing the Toolbox 4 44 5 96 koh ck k xo 9e xo kx v x x 9k a eo ee s 2 3 Compiling C MEX Files een 2 4 Locating Documentation 2 5 MPC simulation GUI sae s 4444 on n OUR ROC Roo Reb alas a RE Red 2 6 Command Line Examples les Model Predictive Control Overview 3 1 Jntroducuoh esea 6a eee eek A mRSST SA A deu oe ER X 3 2 Linear MPC Algorithm i ke bk Re e EE ee ee Ehe e ye y dU 3 3 The jMPC Controller sea co to sema eaa a ek EO REOR SR RE Graphical User Interface GUI 4 1 GUI Ov tview uon ee 2 2 BoR A Pew a Ph A OR ORE eRe AR RR OR RO RD de s 4 2 Running An Example 4 3 Using MATLAB LII Models e o 44 Using NL Models s oh oh oh o m o ee 4 5 Auto Setup iaa Ba X a RUE ROG P RUE ONU a ORG ee E E e UR AR 4 6 Loading and Saving Settings ee ee AST CODLUODS 5s d revera pg ln BO eedem d duh ep REP REX RUP e pla eas a eee Case Studies 5 1 Servomechanism Example
15. MPC Model Np Nc uwt ywt con Kest opts simopts jSIM MPC1 Plant T setp mdist MPC1 is created using the jMPC constructor where the variables we have declared previously are passed as initialization parameters Simulation options simopts is created similarly except using the jSIM constructor Step 9 Run the MPC Simulation and Plot Results With the controller and environment built we can run the simulation and plot the results We use SIMULINK as the evaluation environment as it runs significantly faster than MATLAB for nonlinear sim ulations 4 Simulate amp Plot Result simresult sim MPCi simopts Simulink plot MPC1 simresult detail As shown in Figures 34 and 35 the linear MPC controller retains good control of the distillation column even when faced with a large nonlinear system with multiple disturbances 49 y4 Liquid Mole Fraction in Reflux Drum 2 E E 0 20 40 60 80 100 120 140 160 180 200 y Liquid Mole Fraction in Feed Tray 5 E E 0 20 40 60 80 100 120 140 160 180 200 ys Liquid Mole Fraction in Reboiler 0 08 E 0 06 m E 0 04 0 20 40 60 80 100 120 140 160 180 200 Sample Figure 34 Distillation Column Simulation Output Responses u Feed Flowrate MeasDist Au Change in Feed Flowrate MeasDist 2 1 0 0 50 100 150 200 0 50 100 150 200 mol min 0 u Feed Mole Fraction MeasDist Au Change in Feed Mole Fraction MeasDist 0 54 0 02 LL L1
16. Simulation MPC1 jMPC Model Np Nc uwt ywt con Kest opts simopts jSIM MPC1 Plant T setp 35 MPC1 is created using the jMPC constructor where the variables we have declared previously are passed as initialization parameters Simulation options simopts is created similarly except using the jSIM constructor Both constructors contain appropriate error checking thus the user will be informed of any mistakes when creating either object Step 8 Run the MPC Simulation and Plot Results With the controller and environment built we can run the simulation and plot the results 4 Simulate amp Plot Result simresult sim MPC1 simopts Matlab plot MPC1 simresult summary The plot produced is shown below Outputs yy States xy X Amplitude Amplitude 0 100 200 300 Input u k Change in Input u k Amplitude ce Amplitude 0 100 200 300 0 100 200 300 Sample Sample Figure 26 Tuned MPC 3DOF helicopter response We can also plot a detail plot using the detail switch to better view the output responses Viewable in the output responses the elevation angle and rotation angle both appear controlled with minimal overshoot and fast rise and fall times Note the pitch angle is constrained as required at 1 rad when moving the helicopter around the rotation axis as was specified Linear MPC with Nonlinear Simulation In order to more accurately verify the controlle
17. States 6 nots 2 Oupts 3 Ts 0 1 s Dead Time 0 00 s Setup EPA Trovos Detout Setings Time EE E GP Soler MPC Wright P v Past amp Present Inputs O Future Inputs Aindow Length Set to 10 8 Quanser 3DOF Helicopter Samping Period Set to 0 100 2 Np 15 Nc 6 Puneing Lmt input to 24V Stopped Umt Elevation to 27 Sdeg Limt Pech to 60de3 Limt Rotation to 1604e5 No Moder Plant Mismatch Out Elevation Lexa mina Figure 3 jMPC GUI Main Window Past and Present Windows These windows display the current and past data measured by the process The top window displays the process outputs while the bottom window displays the process inputs Using a custom Scrolling Object these plots scroll as the data is received allowing only the most recent data to be viewed The window size is customizable through the advanced tab There are typically 3 lines plotted for each output the output line itself solid the desired setpoint dash dot and the constraints dotted For the input the same rules apply but without the setpoint Prediction Windows One of the main advantages of MPC is the predictive component This window allows you to view in Real Time the actual predicted response as calculated using the supplied Model The upper window is the predicted process output while the lower window is calculated future input sequence of which only the first is applied to the pl
18. ace model Initial U fFeed 24 60 Feed Flowrate mol min aFeed 0 5 Feed Mole Fraction RR 3 Reflux Ratio Linearize Plant u0 fFeed aFeed RR x0 Zunknown operating point Model xop linearize Plant u0 x0 0dei5s Note in this instance we have specified to use ode15s for linearization As we do not know the states at our linearization point the linearize function will also solve for the steady state using ode15s if one exists This will be returned as xop which can be used for initial states later on Step 4 Create Controller Model Returned from linearize is a MATLAB 1ti object containing the linearized model of our system This must be converted to a jSS object and discretized to be used to build an MPC controller Build jSS object amp discretize Model jSS Mode1 Ts 1 Model c2d Model Ts In order to assign the measured disturbances in the model we use the following method Set Measured Disturbances fFeed aFeed Model SetMeasuredDist Model 1 21 Provide index of which inputs are mdist Step 5 Setup MPC Specifications The MPC Controller is specified as follows 4Horizons Np 25 Prediction Horizon Nc 10 Blocking moves Constraints con u O 10 1 con y 0 92 1 0 48 0 52 0 0 1 Weighting uwt 1 48 ywt 10 0 0 ZEstimator Gain Kest dlqe Model Step 6 Setup Simulation Options Next we must setup the simulation environm
19. ain appropriate error checking thus the user will be informed of any mistakes when creating either object Step 7 Run the MPC Simulation and Plot Results With the controller and environment built we can run the simulation and plot the results 4 Simulate amp Plot Result simresult sim MPC1 simopts Matlab plot MPC1 simresult summary The plot produced is shown below Outputs yy States x k o w ge pi 2 2 a a E E X X 0 50 100 150 200 250 Change in Input u k 6 4 H Bo X X 2 4 0 50 100 150 200 250 0 50 100 150 200 250 Sample Sample Figure 23 Tuned MPC quad tank response As shown in the top left plot Outputs the final step change at k 200 causes one of the top tanks to hit a constraint limiting the flow into both bottom tanks This results in the system being unable to meet the setpoints which is the desired response obey constraints first Linear MPC Simulation with Unmeasured Disturbances Step 1 Add an Unmeasured Input Disturbance and Measurement Noise This next example will show how to add an unmeasured input load disturbance as well as measurement noise to the system Add the following code in a new cell block 30 Disturbances umdist zeros T 2 umdist 80 end 1 1 noise randn T 4 30 The variable umdist now contains a 2 column matrix of input disturbance values which in this case u has a 1V bias subtracted from it for samples 80 to 100 The va
20. ant Predicted Outputs Past amp Present Outputs O Figure 4 Left Past and Present Outputs Right Predicted Outputs Plant and Model Selection When running an MPC simulation you must artificially select a Plant to act as the real process In real life the plant is the real continuous process and is sampled discretely by the controller In this software you load the Plant and Model separately where the Model is used by the controller exclusively and the plant for simulation purposes exclusively The two lists will only display jSS jNL and 1ti ss zpk tf objects as these are the only compatible models These models are sourced from the base workspace If you have added new models to the workspace and wish for these lists to be updated click Refresh Variables to perform a new search of the workspace By default this software will automatically select models called Plant and Model for each list if they exist Plant Model Model Plant Plant Dead Time 0 00 s Plant Model States 6 States 6 Inputs 2 Inputs 2 Outputs 3 Outputs 3 Ts 0 1 s Ts 0 1 s Dead Time 0 00 s Figure 5 Plant and Model Selection Window Tabs Due to space constraints 3 tabs have been implemented on the GUI using the user generated function tabpanel Acknowledgements to Elmar Tarajan for creating this functionality Note that the supplied tabpanel function is covered by a BSD licens
21. aoaaa 5 2 Quad Tank Example s rae 22 aem bee o pm ER a e ee e a 5 3 3DOF Helicopter Example 0 0020 2 ee 5 4 CSTR Example 3 444 445 6644 8 54 eae edb ba oe Webs RO do wot eum Cg 5 5 Distillation Column Example es 1 Introduction 1 1 Overview This manual details the key functionality provided by the jMPC Toolbox The toolbox is a collection of m files SIMULINK models and compiled C MEX functions for use in simulating and controlling dynamic systems using Model Predictive Control Key functionality offered by this toolbox includes Easily build linear MPC controllers Simulating controllers within linear and nonlinear environments Applying linear inequality constraints to inputs and outputs Various Quadratic Programming QP solvers are provided to solve the quadratic cost function e Implement advanced functionality such as state estimation blocking and soft constraints View real time MPC control using the supplied Graphical User Interface GUI Connect the SIMULINK MPC block to real world dynamic systems using an A D and D A This work is the result of my research into embedded MPC during my time at AUT University It is a work in progress over the last two years and will continue to be upgraded as time allows 1 2 MPC Block Diagram Unmeasured Measurement Disturbance Noise Measured Disturbance umdist noise Manipulated Variable u Y Measured Output Setpoint
22. ase of this simulation we will define the plant and model to be the same Model Plant no model plant mismatch Step 4 Setup MPC Specifications Following the specifications from this example s reference enter the following MPC specifications Horizons Np 10 Prediction Horizon Nc 3 Control Horizon Constraints con u O 12 6 0 12 6 con y O 25 0 25 o 10 o 101 Weighting uwt 8 8 ywt 10 10 0 0 ZEstimator Gain Kest dlqe Model In this case we have chosen a prediction horizon of 10 samples 30 seconds at Ts 3s and a control horizon of 3 samples 6 seconds Step 5 Setup Simulation Options Next we must setup the simulation environment for our controller Simulation Length T 250 Setpoint setp 15 ones T 2 setp 100 end 1 2 20 setp 200 end 2 15 Initial values Plant x0 0 0 0 0 Model xO 0 0 0 0 29 Step 6 Build the MPC Controller and Simulation Options Now we have specified all we need to build an MPC controller and Simulation environment we call the two required constructor functions 4 Build MPC amp Simulation MPC1 jMPC Model Np Nc uwt ywt con Kest simopts jSIM MPC1 Plant T setp MPC1 is created using the jMPC constructor where the variables we have declared previously are passed as initialization parameters Simulation options simopts is created similarly except using the jSIM constructor Both constructors cont
23. ck run time messages being displayed in the command window a console is used within the GUI to alert the user of actions being undertaken on their behalf It is important to identify each message as they will often lead to undesirable results if ignored 12 Console indow Length Set to 10 3 Sampling Period Set to 0 100 3 Running Stopped Figure 9 Console Window Description This control is optional and provided simply for adding a description to the current setup This descrip tion is saved as part of the GUI object and thus can help you identify objects at a later stage Note this only updates when an example is loaded or the user enters information Description Quanser 3DOF Helicopter a Np 15 Nc 6 Limit Input to 24V Limit Elevation to 27 5deg Limit Pitch to 60deg Limit Rotation to 180deg n No Model Plant Mismatch Out1 Elevation T Die kD ID uS Figure 10 Description Window Buttons e Start Stop Simply starts or stops the simulation Note the same button changes function depending on whether the simulation is running or stopped e Pause Pause execution of the simulation without losing the current data e Refresh Variables When you add new or change models objects to the workspace this button allows the GUI to rescan the base workspace for changes Any new models or changes will be identified and populated within the respective lists
24. controller we must linearize the ODE function and generate the required linear state space model Linearize Plant u0 Tg 2 Kf la Tg 2 Kf 1a Model linearize Plant u0 In this instance we have chosen the linearize the ODE expression above the motor voltages of 0 5V We have not specified a state to linearize about such that function will also automatically determine a steady state to linearize about Step 4 Rebuild jSS Object The model returned from the linearize function is a MATLAB 1ti object and can be automatically converted to a jSS object and then converted to a discrete model Build jSS object Model jSS Model Discretize model Model c2d Model Ts Step 5 Rebuild and Re Simulate the MPC Controller The last step is to rebuild the MPC controller and simulation environment and then re run the simulation Rebuild MPC amp Simulation MPC1 jMPC Model Np Nc uwt ywt con Kest simopts jSIM MPC1 Plant T setp Re Simulate amp Plot simresult sim MPCi simopts Matlab plot MPC1 simresult summary The output result is shown below As viewable in the plot above the controlled response of the nonlinear system is quite similar to the linear system however viewing the plots side by side shows the differences specifically the nonlinearities From the nonlinear plot it is now evident there is significantly more overshoot on the rotation setpoint This is attributed to the nonlinear equa
25. ct from the current settings to save To Workspace nm Variable Name MPC GUI Figure 16 Save to workspace interface Load From Save To File When you wish to use a permanent setup or permanently save your MPC configuration use the Load From Save To File option to do so Both of these options will open a standard file Window and allow the user to specify a file name and location Files will be saved as a mat file and thus only these can be loaded Note as above only a working setup can be saved Export MPC Controller To use a controller created within the GUI i e within SIMULINK or a standard jMPC simulation you can export just the jMPC controller using this feature A standard Save To Workspace window will be presented to choose a variable name for the controller 4 7 Options To accommodate different user preferences the following options are available from the options menu Auto Scaling Auto scaling refers to the ability of the scrolling plots to resize in real time When this option is enabled and the process is at steady state the response windows will begin to rescale such that window will zoom back to the response lines This is an advantage when a process has become unstable and you wish to zoom back to the detail of the plot Performing multiple small step changes will encourage the plot to rescale to smaller sizes If you wish to keep the plot size constant i e it will get bigger
26. ction of A Measured Disturbance mol u3 Reflux Ratio Control Objectives The control objective is to control the mole fraction of A in the reflux drum by manipulating the reflux ratio Both the feed mole fraction and flow rate are measured disturbances and cannot be modified by the controller The mole fractions on the other trays are not controlled but constrained within sensible limits System Constraints The reflux ratio is constrained as 0 uj lt 10 And the tray mole fractions are constrained as 0 98 lt y 1 mol 0 48 lt yo lt 0 52 mol 0 ya lt 0 1 mol Linear MPC with Nonlinear Simulation Step 1 Build Nonlinear ODE Callback Function As detailed in creating a NL object the first step is to write an m file which contains the above expres sions suitable for use with a MATLAB integrator function xdot nl distil t x u param Nonlinear Distillation model Assign Parameters FT rVol aTray aCond aReb param xdot NaN size x fFeed u 1 aFeed u 2 RR u 3 Additional Eqs D 0 5 fFeed Distillate Flowrate mol min L RR D Flowrate of the Liquid in the Rectification Section mol min V L D Vapor Flowrate in the Column mol min FL fFeed L Flowrate of the Liquid in the Stripping Section mol min Determine Vapour Flows Delta Flows y x rVol i rVol 1 x Assume constant relative volatility to determine vapour flows dx diff x dy d
27. del states Lp Plant states Ym Model output Up Plant output 2 Getting Started 2 1 Pre Requisites The jMPC Toolbox is designed to run independent of any other MATLAB Toolbox however several toolboxes will significantly increase functionality These include e Control Systems Toolbox Transfer Functions amp Manipulations Highly Recommended e Optimization Toolbox Using quadprog Not Critical e Simulink 3D Animation For 3D MPC Demos gt Not Critical To run the supplied C MEX functions you must also install the Microsoft VC 2010 runtime available from the Microsoft website Ensure you download the version x86 or x64 suitable for your platform This runtime s required to run this toolbox 2 2 Installing the Toolbox Installing the jMPC Toolbox is simple with the supplied automated MATLAB install script Copy the jMPC directory to a suitable permanent location and using MATLAB browse to the contents of the jMPC directory and run the install script gt gt jMPC Install m The install script will perform the following steps based on user responses 1 Uninstall previous versions of the jMPC Toolbox 2 Save the required jMPC Toolbox paths to the MATLAB path 3 Run a post installation self check 4 Display the read me file Ensure you do not rename the jMPC directory from jMPC as the name is used in automated MEX file compilation 2 3 Compiling C MEX Files All MEX files supplied are built using VC 20
28. delumax con y inf inf outi ymin ymax 78 5 78 5 out2 ymin ymax Weighting uwt 0 1 ywt 1 0 do not control the second output Torque Estimator Gain Kest dlqe Model In this case we have chosen a prediction horizon of 10 samples 1 second at Ts 0 1s and a control horizon of 2 samples 0 2 seconds The control horizon must always be less than or equal to the prediction horizon based on the formulation of the MPC problem The prediction horizon should be chosen based on the dynamics of the system such that it can predict past any system dead time and sufficient enough for the controller to predict required constraint handling The constraints are specified exactly as above noting a default rate of change of the input voltage has been chosen as 100V sample This variable must be defined if the controller is constrained however if you are not concerned about the rate of change you can use a suitably large value such as done here Also note the output load angle y1 is not constrained such that minimum and maximum are defined as inf and inf which removes that output from constraint handling in the controller The weighting is the user s main handle for tuning the controller performance A larger number represents a larger penalty for that input output where inputs are penalized based on a zero input no plant input and outputs based on setpoint deviation Penalizing inputs will discourage the controller
29. der on the MATLAB path Step 2 Build the NL Object The next step is to build the NL object passing the function above as a function handle AParameters q 100 Volumetric flow rate m 3 min 100 Volume in reactor m 3 V k E R H 5e4 Heat of Reaction J mol Cp 239 Heat capacity J g K 1000 Density g m 3 UA 5e4 Heat Transfer Area J min K Output Matrix C eye 2 Nonlinear Plant param Plant jNL nl_cstr C param O 7 2e10 Pre exponential nonthermal factor 1 min 7 2752e4 Activation energy in the Arrhenius Equation J mol 8 31451 Universal Gas Constant J mol K Je la Kf Fg Tg Jp lh Jt parameter cell array Note we have passed the parameters as a cell array and also retained the linear C output matrix for now This could also be a function handle to a nonlinear output function 41 Step 3 Linearize the jNL Object In order to use the nonlinear plant with our linear MPC controller we must linearize the ODE function and generate the required linear state space model The system is linearized about an unsteady operating point as specified in the original reference Initial U CAf 1 Feed Concentration mol m 3 Tf 350 Feed Temperature K Tc 300 Coolant Temperature K Linearize Plant ud CAf Tf Tc xop 0 5 350 unstable operating point Ca Tr Model linearize Plant u0 xop ode15s Note in this instance we hav
30. e Each of the tabs are described below Setup Tab This is the main Tab used for setting up the MPC controller It has the following elements e Prediction Horizon Length of the prediction of the model output Control Horizon or Blocking Moves Control Horizon scalar Length of samples calculated as the future input sequence or Blocking Moves vector Vector of samples calculated as blocking moves i e 4 2 2 2 Tuning Weights These are used by the MPC Quadratic Program cost function Placing a higher number will increase the penalty cost as this variable deviates from the setpoint or for using that input Weights are squared in calculation Constraints The fundamental advantage of MPC is the ability to add constraints on the process This software allows constraints on the min and max input max input change and the min and max output Note all input constraints are hard constraints QP Solver n order to compare implementations 5 QP solvers are available for use including MATLAB s built in quadprog provided the Optimization Toolbox is installed Default When starting out this button automatically fills in default values within all settings allowing almost single click simulation Note these values are dependent on the Model size only and no heuristics are used in choosing values Advanced Tab These settings although listed as advanced will likely
31. e Auto Load This button is covered in detail in the Auto Setup section 13 4 2 Running An Example Provided with this software are several example setups which allow for new users to get familiar with the software To load an example click File Load gt Example Models Fite Options Help Load From File Save Import From Workspace Exit Example Models Present 0 9 0 8 Figure 11 Locating examples from the File menu You will be presented with a new window containing a list of currently saved examples Rossiter SISO Quanser 3DOF Helicopter Rossiter MIMO Np 15 Nc 6 Maciejowski Cessna Limit Input to 24 Quanser 3DOF Helicopter Limit Elevation to 4 27 5deg Wood Berry Limit Pitch to 60deg Lund NL Helicopter Limit Rotation to 180deg DIW NL Pendulum a No Model Plant Mismatch Figure 12 Examples list Each example is complete with a functional description displayed as each example is selected As this software matures expect this list to change Click OK to select an example which will be automatically loaded into the GUI At this point you can run the example as is or modify existing settings as required 4 3 Using MATLAB LTI Models When you have the MATLAB Control Systems Toolbox installed this tool can also load 1ti models for use within MPC simulation This is the intended purpose of this software however jSS objects are also provided for use without the Control Sy
32. e specified to use ode15s for linearization While this is not used in this particular scenario we are not solving for a steady state here it will be used for all future simulations of this jNL plant instead of the default ode45 Step 4 Create Controller Model Returned from linearize is MATLAB 1ti object containing the linearized model of our system This must be converted to a jSS object and discretized to be used to build an MPC controller Build jSS object amp discretize Model jSS Model Ts 0 05 Model c2d Model Ts In order to assign the measured disturbances in the model we use the following method 4Set Measured Disturbances Caf Tf Model SetMeasuredDist Model 1 2 Provide index of which inputs are mdist Step 5 Setup MPC Specifications The MPC Controller is specified as follows AHorizons Np 30 Prediction Horizon Nc 10 10 10 Blocking moves Constraints con u 278 15 450 20 con y 0 3 278 15 450 AWeighting uwt 1 ywt 0 5 Estimator Gain Kest dlqe Model Step 6 Setup Simulation Options Next we must setup the simulation environment for our controller Simulation Length T 300 Setpoint CA setp zeros T 1 setp 1 xop 2 setp 50 end 1 xop 2 25 setp 200 end 1 xop 2 25 Measured Disturbances Caf Tf mdist zeros T 2 mdist 1 CAf mdist 2 Tf mdist 130 140 1 CAf 0 1 Step disturbance of Caf
33. ent for our controller Simulation Length T 200 Setpoint Reflux Drum setp zeros T 1 setp 1 xop 1 setp 10 end 1 xop 1 0 02 Measured Disturbances mdist zeros T 2 mdist 1 fFeed mdist 2 aFeed mdist 60 end 2 aFeed 0 01 Small increase in feed mole fraction mdist 140 end 1 fFeed 1 Small increase in feed flow rate Set Initial values at linearization point Plant x0 xop Model x0 xop Step 7 Set MPC Options For advanced settings of the MPC controller or simulation you can use the jMPCset function to create the required options structure For this example we wish to set the plot titles and axis as well as setting the initial control input at k 0 to the linearization point Set Options opts jMPCset InitialU u0 Set initial control input at linearization point InputNames Feed Flowrate Feed Mole Fraction Reflux Ratio InputUnits mol min Fraction Ratio OutputNames Liquid Mole Fraction in Reflux Drum Liquid Mole Fraction in Feed Tray Liquid Mole Fraction in Reboiler QutputUnits Fraction Fraction Fraction Step 8 Build the MPC Controller and Simulation Options Now we have specified all we need to build an MPC controller and Simulation environment we call the two required constructor functions 4 Build MPC amp Simulation MPC1 j
34. f the Kalman Filter and long plant dynamics For those interested re run the simulation without state estimation and note how the control performance substantially degrades after the input bias is applied as the plant and model states deviate enable the plot model option 32 5 3 3DOF Helicopter Example The completed example can be found in Examples Documentation Examples Heli Example m Reference J Akesson 2006 1 and Quanser 6 Figure 25 3DOF Helicopter Schematic 1 Linear System Model The continuous time state space equation of the 3DOF helicopter is as follows 0100 0 0 2o 0000 0 0 a dem 0 001 0 0 0 0 Flo 00 e 9 9 o T 0000 0 1 d Ud 0000 0 0 ge n 100000 y 0 0 1 0 0 0 x 000010 where we have defined the state vector x as z Elevation Angle rad z Elevation Angular Velocity rad s 13 Rotation Angle rad X4 Rotation Angular Velocity rad s x5 Pitch Angle rad xe Pitch Angular Velocity rad s and the output vector y as y Elevation Angle rad y2 Rotation Angle rad y3 Pitch Angle rad and input vector u as ur Motor 1 Voltage V ug Motor 2 Voltage V 33 Control Objectives The control objective is to control the Elevation and Rotation angles yi y2 to their respective setpoints by adjusting Motor 1 and 2 Voltages u1 u2 The Pitch Angle y3 is constrained but not controlled to a setpoint System Constraints Both Motor Voltages are constra
35. icated by the progress bar In this instance we have specified we wish to run a MATLAB simulation default rather than a SIMULINK or other MPC implementation The final command plot is once again overloaded for the jMPC Class and allows us to pass the controller required for constraint listing the simulation results and specify the type of plot we wish to view In this instance just a summary will do The plot produced is shown below Viewable in the top left axis Output the controlled response of the load angle blue line is poor and the value oscillates for some time before we would expect to see it settle The next task is to tune the controller such that we achieve a settling period of the load angle within around 10 seconds Noting the physical constraints on the system we can see the input voltage is less 23 Outputs yy amp Unmeasured Outputs y k States x k 40 gt N c eC oo coc Amplitude Amplitude 0 100 200 300 Input u k Change in Input A u k Amplitude Amplitude 50 2 0 100 200 300 0 100 200 300 Sample Sample Figure 18 Untuned MPC servo response than a third of its maximum such that we can reduce the weight on this input to encourage the controller to use more voltage for a faster response The torque is also much lower than the constrained value such that increasing the voltage shouldn t be a problem Step 8 Re Tune Controller Weights Decrease the input weight and increase
36. iff y Difference molar flow rates Calculate flowrates on each tray xdot 1 1 1 aCond V y 2 x 1 Reflux Drum xdot 2 FT 1 1 L dx 1 FT 2 V dy 2 FT 1 aTray Top section xdot FT 1 1 aTray fFeed aFeed L x FT 1 FL x FT V y FT y FT 1 Feed tray xdot FT 1 end 1 FL dx FT end 1 V dy FT 1 end aTray Bottom section xdot end FL x end 1 fFeed D x end V y end aReb Reboiler end The file nl_distil m is saved in a suitable folder on the MATLAB path Step 2 Build the jNL Object 47 The next step is to build the jNL object passing the function above as a function handle AParameters nTrays 32 Number of Trays feedTray 17 Feed tray location rVol 1 6 Relative Volatility aTray 0 25 Total Molar Holdup on each Tray aCond 0 5 Total Molar Holdup in the Condensor aReb 1 0 4 Total Molar Holdup in the Reboiler Output Matrix C 7 zeros 3 nTrays C 1 1 1 C 2 feedTray 1 C 3 end 1 Nonlinear Plant param feedTray rVol aTray aCond aReb aFeed parameter cell array Plant jNL nl_distil C param Note we have passed the parameters as a cell array and also retained the linear C output matrix for now This could also be a function handle to a nonlinear output function Step 3 Linearize the jNL Object In order to use the nonlinear plant with our linear MPC controller we must linearize the ODE function and generate the required linear state sp
37. ined as 3V lt ug lt 3V Each axis rotation angle is constrained as follows 0 6 lt y lt 0 6 pi lt ya pi 1 lt ya lt 1 Linear MPC Simulation Step 1 Create The Plant The first step is to implement the model equations above into MATLAB as a jSS object The parameters are first defined then the model A B C and D matrices are created and passed to the jSS constructor Parameters Je 0 91 Moment of inertia about elevation axis la 0 66 Arm length from elevation axis to helicopter body Kf 0 5 Motor Force Constant Fg 0 5 Differential force due to gravity and counter weight Tg la Fg Differential torque Jp 0 0364 Moment of inertia about pitch axis lh 0 177 Distance from pitch axis to either motor Jt Je 4 Moment of inertia about travel axis Plant A 010000 000000 000100 0000 Fg 1la Jt 0 000001 00000901 B 0 0 Kf la Je Kf la Je 00 00 00 Kf lh Jp Kf 1lh Jpl C 100000 001000 0000101 D 0 Create jSS Object Plant jSS A B C D Step 2 Discretize the Plant Object Plant is now a continuous time state space model but in order to use it with the jMPC Toolbox it must be first discretized A suitable sampling time should be chosen based on the plant dynamics in this case use 0 12 seconds 34 Discretize Plant Ts 0 12 Plant c2d Plant Ts Step 3 Create Controller Model For ease of this simulation we will define the Plant and M
38. nk Levels are constrained as Ocm lt y a 4j 10cm Linear MPC Simulation Ste p 1 Create The Plant The first step is to implement the model equations above into MATLAB as a jSS object The parameters are first defined then the model A B C and D matrices are created and passed to the jSS constructor Parameters Tank cross sectional area A1 15 5179 A2 15 5179 A3 15 5179 A4 15 5179 Dutlet cross sectional area al 0 1781 a2 0 1781 a3 0 1781 a4 0 1781 Gravity g 981 Pump coefficients k1 4 35 k2 4 39 Ratio of allocated pump capacity between lower and upper tank gi 0 36 g2 0 36 Steady state values x0 15 0751 15 0036 6 2151 6 1003 ud 7 7 Constants Ti Ai ai sqrt 2 x0 1 g T2 A2 a2 sqrt 2 x0 2 8 T3 A3 a3 sqrt 2 x0 3 g T4 A4 a4 sqrt 2 x0 4 g 28 Plant A 1 T1 O A3 A1 T3 0 O 1 T2 0 A4 A2 T4 0 0 1 T3 0 000 1 T4 B gi k1 A1 0 O g2 k2 A2 O 1 g2 k2 A3 1 g1 k1 A4 0 C eye 4 D 0 Create jSS Object Plant jSS A B C D Step 2 Discretize the Plant Object Plant is now a continuous time state space model but in order to use it with the jMPC Toolbox it must be first discretized A suitable sampling time should be chosen based on the plant dynamics in this case use 3 seconds Discretize Plant Ts 3 Plant c2d Plant Ts Step 3 Create Controller Model For e
39. odel to be the same Model Plant no model plant mismatch Step 4 Setup MPC Specifications Following the specifications from this example s reference enter the following MPC specifications 4Horizons Np 30 Prediction Horizon Nc 10 Control Horizon Constraints con u 3 3 1e6 3 3 1e6 con y 0 6 0 6 pi pi 1 1 Weighting uwt 1 1 ywt 10 10 1 Estimator Gain Kest dlqe Model Step 5 Setup Simulation Options Next we must setup the simulation environment for our controller Simulation Length T 300 ASetpoint setp zeros T 3 setp 1 0 3 setp 125 250 1 0 3 setp 2 2 setp 200 end 2 0 Initial values Plant x0 00 00 0 Model x0 00000 0 Step 6 Set MPC Options For advanced settings of the MPC controller or simulation you can use the MPCset function to create the required options structure For this example we just wish to set the plot titles and axis Set Options opts jMPCset InputNames Motor 1 Voltage Motor 2 Voltage InputUnits V V OutputNames Elevation Angle Rotation Angle Pitch Angle QutputUnits rad rad rad Step 7 Build the MPC Controller and Simulation Options Now we have specified all we need to build an MPC controller and Simulation environment we call the two required constructor functions 4 Build MPC amp
40. ogether with the current plant output measurement yp to calculate the optimal control input u For simulation purposes you can also add an unmeasured disturbance umdist and measurement noise noise to the plant to test how the controller responds For analysis purposes you can also read the model states Xm and the model output ym which can be compared to the plant states xp and plant output yp When running a simulation the plant is represented by either a jSS State Space linear plant or a collection of nonlinear or linear but this is unusual as it should be converted to state space Ordi nary Differential Equations ODEs in jNL plant This enables you to perform linear and nonlinear simulations using the same jMPC controller and just changing the plant object k Past j Future 4 Ymax o setp A ia aia id e Measured 9 Predicted Prediction Horizon Np Umax Aus T e Past Inputs e Planned Inputs Umin 43 2 1 k 1 2 3 4 5 6 7 8 9 10 Sample Figure 2 MPC Prediction Overview 4 Graphical User Interface GUI The jMPC Toolbox also contains a GUI to aid in understanding plant dynamics running live demon strations of MPC and classroom use The GUI retains many of the features of the jMPC Class and users with find the GUI intuitive to migrate to 4 1 GUI Overview jMPC Simulation Tool Past amp Present Outputs a Predicted Outputs Plant
41. origin which is the effective dead time In order to avoid this make sure you either supply discrete models with dead time at the correct sampling rate or try reselecting the model after changing T s Models with Dead Time As the original jMPC class does not handle dead time naturally models supplied with dead time are handled differently from those above The Control Systems Toolbox function delay2z is used to convert all dead time into leading zeros in the dynamics which effectively adds dead time at integer multiples of the sampling time For continuous models they are first discretized as above before this function is applied This avoids using an approximation such as the function pade The downside to this method is greater computational load especially as the dead time increases beyond 10 sampling instants as the state space matrices are increased in size Note the supplied Model object class jSS does not have a field for dead time thus only 1ti models can be used for dead time simulations 4 4 Using jNL Models In order to more accurately simulate a given process you can use a jNL object as the Plant This allows you to use nonlinear Ordinary Differential Equations ODEs to represent the plant dynamics while 15 retaining a simpler linear state space model as the internal controller model See the help section on jNL objects on how to create the plant and the limit of the associated object Note you must still supply
42. output 1 weight as follows then rebuild the controller Weighting uwt 0 05 ywt 2 0 Rebuild MPC MPC1 jMPC Model Np Nc uwt ywt con Kest opts Step 9 Re Tune Controller Weights With the new controller run the simulation again using the existing simulation environment and plot the results using the detail switch Re Simulate amp Plot simresult sim MPC1 simopts Matlab plot MPC1 simresult detail The detailed output result is shown below The top plot shows the Load Angle with respect to setpoint and with the new tuning we can see we achieve the desired settling time while meeting the constraints on the Torque bottom plot Linear SIMULINK Simulation Step 1 Re Run MPC Simulation with SIMULINK Switch 24 y Load Angle 1 5 1 E o5 0 T 50 100 150 200 250 300 yz Load Torque Unmeasured E 0 50 100 150 200 250 300 Sample Figure 19 Tuned MPC servo response In order to use the supplied SIMULINK MPC components the simplest way to get started is to simply use the Simulink switch Re Simulate amp Plot simresult sim MPC1 simopts Simulink plot MPC1 simresult detail which all going to plan will give exactly the same results as the Matlab simulation This mode populates the supplied jMPC simulink mdl SIMULINK Model with the auto generated MPC components and calls SIMULINK to simulate it The difference here is that the simulation is no
43. ples The Examples folder within the jMPC directory contains a number of worked linear and non linear MPC simulation examples using the jMPC Toolbox via objects For example to open the linear MPC examples supplied type gt gt edit jMPC linear examples These examples are described in detail within the MPC documentation and later in this manual 3 Model Predictive Control Overview 3 1 Introduction Model Predictive Control MPC is an Advanced Process Control APC technology which is used commonly in the Chemical Engineering and Process Engineering fields MPC allows control of all plants from Single Input Single Output SISO through to Multiple Input Multiple Output MIMO and non square systems different numbers of inputs and outputs However its real advantage is the ability to explicitly handle constraints such that they are included when calculating the optimal control input s to the plant Among other attractive features this has enabled MPC to be the preferred APC technology for a number of years and continues to this day Several variants of the MPC algorithm exist including the original Dynamic Matrix Control DMC Predicted Functional Control PFC Generalized Predictive Control GPC as well as nonlinear and linear MPC However all retain a common functionality a model of the system is used to predict the future output of the plant so that they controller can predict dead time constraint violations futu
44. r tuning we can also create the nonlinear model of the plant using the nonlinear differential equations of the helicopter system 36 y4 Elevation Angle 0 5 rad 0 5 0 50 100 150 200 250 300 Yo Rotation Angle rad 0 50 100 150 200 250 300 ys Pitch Angle rad 0 50 100 150 200 250 300 Vy Vp T Step 1 Build Nonlinear ODE Callback Function As detailed in creating a NL object the first step is to write an m file which contains the above expres sions suitable for use with a MATLAB integrator function xdot nl heli t x u param Nonlinear Helicopter Model Assign Parameters Je la Kf Fg Tg Jp lh Jt param xdot 1 1 xdot 2 1 xdot 3 1 xdot 4 1 xdot 5 1 xdot 6 1 end x 2 K 1a Je u 1 u 2 Tg Je x 4 Fg 1la Jt sin x 5 x 6 Kf 1h Jp u 1 u 2 The file n1 heli m is saved in a suitable folder on the MATLAB path Step 2 Build the jNL Object The next step is to build the NL object passing the function above as a function handle 37 Non Linear Plant param Je la Kf Fg Tg Jp lh Jt parameter cell array Plant jNL nl_heli C param Note we have passed the parameters as a cell array and also retained the linear C output matrix for now This could also be a function handle to a nonlinear output function Step 3 Linearize the jNL Object In order to use the nonlinear plant with our linear MPC
45. re setpoints and plant dynamics when calculating the next control move This enables robust control which has proved its use in industry and academia for over 20 years 3 2 Linear MPC Algorithm The MPC algorithm used within the jMPC Toolbox is a linear implementation where a linear state space model is used for the system prediction This enables the controller to formulate the control problem as a Quadratic Program QP which when setup correctly results in a convex QP ensuring an optimal input global minimum The fundamental algorithm characteristic is the system prediction and subsequent control move optimization shown below Viewable in Figure 2 is the standard MPC prediction plot where the top plot shows the predicted plant output for the prediction horizon and the bottom plot shows the planned control inputs for the control horizon The linear model which is used to build the controller is used for the plant prediction viewable as the blue dots With a good model of your system this prediction can be quite accurate enabling the control moves to optimally move the system to the setpoint Also shown in the above plots are the constraints available to MPC including maximum and minimum inputs and outputs as well maximum rate of change of the input 3 3 The jMPC Controller Referring to Figure 1 shows how the jMPC controller is included in a system The jMPC controller uses the current setpoint setp and measured disturbance mdist t
46. riable noise now contains a 4 column matrix of normal distribution noise with a mean of 0 and a low power Step 2 Rebuild Simulation Rebuild the simulation object as follows this time including the disturbance matrices Rebuild Simulation simopts jSIM MPC1 Plant T setp umdist noise Step 3 Re Simulate the MPC Controller Finally re simulate the MPC controller with the rebuilt jSIM object Re Simulate amp Plot simresult sim MPC1 simopts Matlab plot MPC1 simresult summary Which produces the following plot Outputs yy States x k A Mna 25 0 2 25 20 0 1 EET Z5 3 B Eo 2 10 lt lt 5 5 0 1 0 0 0 100 200 0 100 200 0 100 200 Input u k Change in Input u k Unmeasured Input Disturbance 6 4 0 2 w 10 e e 3 E 2 5 0 4 E Eo Ens X 5 X MES 2 0 8 0 4 1 0 100 200 0 100 200 0 100 200 Sample Sample Sample Figure 24 Tuned MPC quad tank response with measurement noise and load disturbance Note the addition of two new axes which are automatically created when the simulation contains any disturbance Viewable in the Output axis at sample 80 when the input bias is applied although the outputs deviate briefly from their setpoints they quickly return to normal as the controller compensates 31 with a higher input voltage viewable in the Input axis Also note the measurement noise has a minimal effect on the controlled response in due partly to the filtering affect o
47. s object so plotting MPC responses is simple e MPCI This is a jMPC object used for plotting the results above To plot the response using the SIM object use the following command gt gt plot MPC1 simresult You can also call the function with an optional argument to obtain detailed plots gt gt plot MPC1 simresult detail Demo Mode A visual feature only demo mode allows the GUI to automatically vary the setpoints every 100 samples such that it provides a live demonstration The algorithm will randomly select an output to modify and randomly choose a new setpoint The new setpoint will be on a normal distribution with a mean of 1 and a standard deviation of 2 The algorithm does not respect any constraints on the system 19 5 Case Studies The following examples are exercises to aid new users becoming familiar with the jMPC Toolbox 5 1 Servomechanism Example The completed example can be found in Examples Documentation Examples Servo Example m Reference MATLAB MPC Toolbox 2010 7 R Figure 17 Servomechanism Schematic Linear System Model The continuous time state space equation of the servomechanism is as follows 0 1 0 0 0 ko Br ko 0 Jr Jr pJr 0 ES 3 0 1 mem u k k Bm k2 R kr pi 0 p der Tar RJm where we have defined the state vector x as z Load Angle rad z2 Load Angular Velocity rad s z3 Motor Angle rad z4 Motor Angular Velocity rad s
48. stems Toolbox Any MATLAB 1ti model can be loaded into the GUI including transfer function tf state space ss and zero pole gain zpk models All of these will appear in the plant and model lists automatically 14 Plant Model Model Plant Plant Plant Model States B States B Inputs 2 Inputs 2 Outputs 3 Outputs 3 Ts D0 1 s Ts D0 1 s Dead Time 0 00 s Dead Time 0 00 s Figure 13 Plant and Model list Continuous Models As the underlying Model Predictive Controller is discrete all models used will be converted to discrete If Auto Calculate Ts is selected then the routine described here will calculate a suitable sampling rate and the model will be discretized using the Control Systems Toolbox function c2d If Auto Calculate Ts is not selected then the sampling period entered in the Controller Sampling Period edit box will be used by c2d Discrete Models When Auto Calculate Ts is selected the Model s sampling period will be used as the Controller Sampling Period If this option is not selected and the Plant or Model s sampling time is not the same as the value in the Controller Sampling Period edit box then it will be re sampled using the Control Systems Toolbox function d2d The same applies if the Plant and Model sampling periods are different Problems can occur using d2d on Models which have already been approximated using delay2z This is due to multiple poles at the
49. the MPCset function to create the required options structure For this example we just wish to set the plot titles and axis Set Options opts jMPCset InputNames Motor Voltage InputUnits V OutputNames Load Angle Load Torque QutputUnits rad Nm Step 7 Build the MPC Controller and Simulation Options Now we have specified all we need to build an MPC controller and simulation environment we call the two required constructor functions 4 Build MPC Simulation MPC1 jMPC Model Np Nc uwt ywt con Kest opts simopts jSIM MPC1 Plant T setp MPC1 is created using the jMPC constructor where the variables we have declared previously are passed as initialization parameters Simulation options simopts is created similarly except using the jSIM constructor Both constructors contain appropriate error checking thus the user will be informed of any mistakes when creating either object Step 8 Run the MPC Simulation and Plot Results With the controller and environment built we can run the simulation and plot the results 4 Simulate amp Plot Result simresult sim MPC1 simopts Matlab plot MPC1 simresult summary The sim function will be familiar to most MATLAB users and is overloaded here for the jMPC Class Passing the controller and the simulation environment will allow a full MPC simulation to run with progress ind
50. tion governing the relationship between the rotation angle and the pitch angle sine which cannot be fully defined a single linearization point However the control is still reasonable and would expect to function sufficiently well on a real plant Note also that further tuning may improve this response which is left to the reader to implement 38 Amplitude Outputs yy States x k Amplitude Amplitude 0 100 200 300 0 100 200 300 Input u k Change in Input u k Amplitude Amplitude e 5 0 100 200 300 0 100 200 300 Sample Sample Figure 28 Tuned MPC 3DOF helicopter response with nonlinear plant Linear Outputs yy Nonlinear Outputs yy 100 150 200 250 300 50 100 150 200 250 300 50 Figure 29 Comparison of MPC 3DOF linear and nonlinear simulation responses 39 5 4 CSTR Example The completed example can be found in Examples Nonlinear Examples m in the Henson CSTR cell block Reference M Henson amp D Seborg 1997 3 Car Tt Tc Figure 30 CSTR Schematic Nonlinear System Model The continuous time ordinary differential equations of the CSTR model are as follows E o kger Ca dC q a y ee a d H UA zd gne T T ag po a te The equations represent a continuously stirred tank reactor with a single reaction from A B and a complete mass and energy balance We have defined the state vector x as zi Concentration of A in reactor
51. troller and environment built we can run the simulation and plot the results We use SIMULINK as the evaluation environment as it runs significantly faster than MATLAB for nonlinear sim ulations 4 Simulate amp Plot Result simresult sim MPC1 simopts Simulink plot MPC1 simresult detail As shown in Figures 31 and 32 the system shows good control with minimal overshoot even for large step changes This particular problem presents a considerable challenge to linear MPC based on the unstable operating point and significant nonlinearities of the system However correct tuning can give good results even when responding to disturbances 43 y4 Reactor Concentration of A mol m o o 9 o o N E e co 200 250 350 o 100 15 o en o yz Reactor Temperature 380 340 320 350 IK 2 3 50 100 150 200 250 Sample Figure 31 CSTR Simulation Output Responses Au Change in Feed Concentration MeasDist u Feed Concentration MeasDist 02 mol m J un wo e 0 2 0 100 200 300 0 100 200 300 u Feed Temperature MeasDist Au Change in Feed Temperature MeasDist 350 0 pt my i 330 1 0 100 200 300 0 100 200 300 uz Jacket Temperature Au Change in Jacket Temperature 10 310 300 Y 0 290 280 10 0 100 200 300 0 100 200 300 Sample Sample Figure 32 CSTR Simulation Inputs 5 5 Distillation Column Example The completed example can be found in E
52. w running the jMPC S Function written in C and supplied as a SIMULINK block This code is optimized for general MPC use and thus will run much faster than the MATLAB Simulation code However the m file algorithm is still useful for those wishing to learn how to create their own MPC implementations like me and as such is still used Step 2 Build a SIMULINK Model To Test the Controller Fire up SIMULINK and build the following SIMULINK model using the jMPC Toolbox and Standard SIMULINK blocks ym jMPC Controller mr Discrete MPC Controller Figure 20 MPC control of a Servo Simulink Model 25 Set the SIMULINK Solver configuration as follows Fixed Step Discrete 0 1s Sampling Time 0 gt 50s run time Set the pulse generator configuration as follows Sample based with a sample time of 0 1 Period 400 samples Pulse width 200 samples Amplitude of 1 Phase delay of O Set the Selector block as follows Input port size of 2 Index as 2 By default the MPC Controller and Plant will have variable names already used in the example above but you can customize these as you require You can also use the completed model located in Examples Servo Simulink mdl Step 3 Running the jMPC SIMULINK Simulation Click Run to generate the following plot in the scope Load Torque Figure 21 Tuned MPC servo response in SIMULINK This example shows just how easy it is to create a jMPC MP
53. with a transient use this table to set the setpoints at sample k 0 Simulation Speed To slow down the execution of the simulation you can enable Slow Mode which will run the simulation at approximately 1 2 speed Simulation Tab This tab is automatically selected when Start is pushed and allows editing of Real Time Setpoints and Disturbances The properties are e Setpoint With the correct output selected you can modify in real time the setpoint value Note due to this being a real time implementation setpoint look ahead cannot be used The setpoint ranges from 10 to 10 thus the system must be normalised for values outside this range e Measurement Noise 11 Advanced EEA State Estimation Y Enable Estimation Gain 0 8 Figure 7 Advanced Tab This adds a normally distributed noise signal to the selected output The power of the noise is selected by the slider Note the slider value ranges from 5 to 5 and the value displayed and added to the output is 10 to the power of this value e Load Disturbance This adds a bias term to the selected input The value of the bias is selected by the slider Note the slider ranges from 5 to 5 Simulation Setpoint Measurement Er Load Disturbance F Enable Noise iows e Value Figure 8 Simulation Tab Console To prevent the need for the user to che
54. xamples Nonlinear Examples m in the Hahn Distillation cellblock Reference J Hahn and T F Edgar 2002 2 Condenser Reboiler Bottoms Figure 33 Distillationmn Column Schematic 45 Nonlinear System Model The continuous time ordinary differential equations of the distillation column model are as follows Condenser DN V y2 21 Rectification Trays Se cn Up saec cep dt Atray Feed Tray dz 1 E o ee Lxp 1 FLxp V ys ys Stripping Trays e EE fi V yi Yi dt Atray Reboiler dEn 1 F Laos Feed D Tn Vyn dt Areboiler Further Equations D 0 5Feed L Reflux D V L D FL Feed L xQ aoe a 1 x The equations represent a binary distillation column with n trays and a feed tray which can be located at any tray f The model assumes a constant relative volatility to determine vapour flows We have defined the state vector x as 21 Reflux Drum Liquid Mole Fraction of A T2 Tray 1 Liquid Mole Fraction of A Tf Feed Tray Liquid Mole Fraction of A In 1 Tray n Liquid Mole Fraction of A Za Reboiler Liquid Mole Fraction of A and the output vector y as y Reflux Drum Liquid Mole Fraction of A ya Feed Tray Liquid Mole Fraction of A ys Reboiler Liquid Mole Fraction of A and input vector u as mol mol mol mol mol mol mol mol 46 uj Feed Flowrate Measured Disturbance mol min ug Feed Mole Fra
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