Introduction to Model Predictive Control (MPC) Oscar Mauricio - PowerPoint PPT Presentation

Introduction to Model Predictive Control (MPC) Oscar Mauricio Agudelo Mañozca Bart De Moor Course : Computergestuurde regeltechniek ESAT - KU Leuven May 11th, 2017

Basic Concepts Control method for handling input and state constraints within an optimal control setting. Principle of predictive control N      2 min y y k ( i ) ref   u k ( ), , ( u k N 1)  i 1 Future Past subject to   model of the process y Reference ref  input constraints  output / state constraints y k ( ) Prediction of measurement Why to use MPC ? y k ( )  It handles multivariable interactions u k ( ) time  It handles input and state constraints k  k  k  k  k   2 1 k 1 2 3 k N  It can push the plants to their limits of performance. Prediction horizon  It is easy to explain to operators and engineers Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 2

Some applications of MPC Control of synthesis section of a urea plant MPC strategies have been used for stabilizing and maximizing the throughput of the synthesis section of a urea plant, while satisfying all the process constraints. Urea plant of Yara at Brunsbüttel (Germany), where a MPC control system has been set by IPCOS Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 3

Some applications of MPC Control of synthesis section of a urea plant Reaction 1: Fast and Exothermic 2NH 3 + CO 2  NH 2 COONH 4 Ammonia Carbon Ammonium dioxide carbamate Reaction 2: Slow and Endothermic NH 2 COONH 4  NH 2 CONH 2 + H 2 O Ammonium Urea Water carbamate Throughput increment of 11.81 t/h thanks to the MPC controller Results of a preliminary study done by Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 4

Some applications of MPC Flood Control: The Demer A Nonlinear MPC control strategy has been implemented (2016) for avoiding future floodings of the Demer river in Belgium. Partners: STADIUS, Dept. Civil Engineering of KU Leuven, IPCOS, IMDC, Antea Group, and Cofely Fabricom. The Demer in Hasselt Flooding events due to heavy rainfall: 1905, 1926, 1965, 1966, 1993-1994, 1995, 1998, 2002 and 2010. The Demer and its tributaries in the south of Control Strategy: PLC logic the province of Limburg (e.g., three-pos controller) Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 5

Some applications of MPC Flood Control: The Demer DIEST HASSELT Flooded area during the flood event of 1998. Control Strategy: PLC logic (e.g., three-position controller) Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 6

Some applications of MPC Flood Control: The Demer Upstream part of the Demer that is modelled and controlled in a preliminary study carried out by STADIUS Maximal water levels for the five reaches for the current Notice: The MPC controller takes rain three-pos. controller and the MPC controller together with predictions into account! their flood levels (Flood event 2002) . Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 7

Some applications of MPC In addition MPC, has been used in all sort of petrochemical and chemical plants,  in food processing,  in automotive industry,  in the control of tubular chemical reactors,  in the normalization of the blood glucose level of  critical ill patients, in power converters,  for the control of power generating kites under  changing wind conditions, in mechatronic systems (e.g., mobile robots),  in power generation,  in aerospace,  in HVAC systems (building control)  …  Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 8

Basic Concepts Kinds of MPC     Linear MPC : it uses a linear model of the plant x ( k 1) Ax ( ) k Bu ( ) k  Convex optimization problem      Nonlinear MPC: it uses a nonlinear model of the plant x ( k 1) f x ( ), ( ) k u k  Non-convex optimization problem Remark: Since linear MPC includes constraints, it is a non-linear control strategy !!! Linear MPC formulation (Classical MPC)  N N 1              T           T min x ( k i ) x ( k i ) Q x ( k i ) x ( k i ) u ( k i ) u ( k i ) R u ( k i ) u ( k i ) ref ref ref re f x , u   N N i 1 i 0 subject to          Model of the plant x ( k 1 i ) Ax ( k i ) Bu ( k i ), i 0,1, , N 1,      Input constraints u ( k i ) u , i 0,1, , N 1, max     State constraints x ( k i ) x , i 1,2, , N , max             x x x x u u u u with: ( k 1); ( k 2); ; ( k N ) , ( ); ( k k 1); ; ( k N 1) N N Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 9

MPC Algorithm Typical MPC control Loop x ref ( ) k u ( ) t u ( ) k MPC Plant y ( ) t ZOH u ref ( ) k ˆ ( ) x k Observer y ( ) k T s Digital system MPC Algorithm At every sampling time :  Read the current state of the process, x ( k )  Compute an optimal control sequence by solving the MPC optimization problem     Solution  u ( ), ( k u k 1), ( u k 2), , ( u k N 1)  Apply to the plant ONLY the first element of such a sequence  u ( k ) Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 10

LQR and Classical MPC For simplicity, Let’s assume that the references are set to zero. CLassical Linear MPC LQR  N N 1      T T  min x ( ) k Q x ( k ) u ( ) k R u ( k ) T T min x ( ) k Q x ( ) k u ( ) k R u ( k ) x , u   x , u N N k 1 k = 0 i 0    k 1 subject to subject to      x ( k 1) Ax ( ) k Bu ( ), k k 0,1, , N 1,      x ( k 1) Ax ( ) k Bu ( ), k k 0,1, ,   x ( ) k x , k 1,2, , N , max    u ( ) k u , k 0,1, , N 1, max   The optimal solution has the form: u ( ) k Kx ( ) k PRO PRO  Explicit, Linear solution  Takes constraints into account  Low online computational burden  Proactive behavior CON CON  High online computational burden  Constraints are not taken into account  No explicit solution  No predictive capacity  feasibility? Stability? If N  ∞ , and constraints are not considered  the MPC and LQR give the same solution Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 11

MPC optimization problem – Implementation details The following optimization problem,  N N 1              T           T min x ( k i ) x ( k i ) Q x ( k i ) x ( k i ) u ( k i ) u ( k i ) R u ( k i ) u ( k i ) ref ref ref re f x , u   N N i 1 i 0 subject to         x ( k 1 i ) Ax ( k i ) Bu ( k i ), i 0,1, , N 1,     u ( k i ) u , i 0,1, , N 1, max    x x ( k i ) , i 1,2, , N , max                 n N n N with: x x ( k 1); ( x k 2); ; ( x k N ) , u u ( ); ( k u k 1); ; ( u k N 1) x u N N can be rewritten as a LCQP (Linearly Constrained Quadratic Program) problem in as follows: x 1  T T min 2 x Hx f x x with: subject to         N n n x x ; u x u  A x b N N e e  A x b i i Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 12

MPC optimization problem – Implementation Details where N times     x (1) Q ref             Q x ( N )                         N n n N n n N n n ref     H x u x u 2 f x u H u (0) R     ref                u ( N 1) R ref N times   x max     I  n N     2 N times x     I      x             n N      2 N n n N n n    2 N n n x max  A x u x u  b x u i i I  u     n N max u      I 2 N times      n N u    u  max n x = number of states , n u = number of inputs, N = prediction horizon Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 13