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

Introduction to Model Predictive Control (MPC)

Oscar Mauricio Agudelo Mañozca

Computergestuurde regeltechniek Course :

Bart De Moor

ESAT - KU Leuven May 11th, 2017

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

2

Basic Concepts

Control method for handling input and state constraints within an optimal control setting.

• It handles multivariable interactions
• It handles input and state constraints
• It can push the plants to their limits
• f performance.
• It is easy to explain to operators and

engineers

Principle of predictive control

1 k  2 k  3 k 

Prediction horizon

k N  k 1 k  2 k 

( ) u k ( ) y k

Prediction of

( ) y k

Reference

Future Past

measurement

ref

y 

 

2 ref ( ), , ( 1) 1

min ( )

N u k u k N i

y y k i

  

 



subject to

• model of the process
• input constraints
• output / state constraints

Why to use MPC ?

time

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

3

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

4

Some applications of MPC

Control of synthesis section of a urea plant

Results of a preliminary study done by

Throughput increment of 11.81 t/h thanks to the MPC controller

2NH3 + CO2  NH2COONH4

Ammonia Carbon dioxide Ammonium carbamate

Reaction 1: Fast and Exothermic

NH2COONH4  NH2CONH2 + H2O

Ammonium carbamate Urea Water

Reaction 2: Slow and Endothermic

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

5

Some applications of MPC

Flood Control: The Demer

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 the province of Limburg Control Strategy: PLC logic (e.g., three-pos controller) 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.

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

6

Some applications of MPC

Flood Control: The Demer

Flooded area during the flood event of 1998. Control Strategy: PLC logic (e.g., three-position controller)

DIEST HASSELT

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

7

Some applications of MPC

Flood Control: The Demer

Upstream part of the Demer that is modelled and controlled in a preliminary study carried

• ut by STADIUS

Maximal water levels for the five reaches for the current three-pos. controller and the MPC controller together with their flood levels (Flood event 2002) .

Notice: The MPC controller takes rain predictions into account!

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

8

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

Basic Concepts

9

Kinds of MPC

• Linear MPC : it uses a linear model of the plant

 Convex optimization problem

• Nonlinear MPC: it uses a nonlinear model of the plant

 Non-convex optimization problem

( 1) ( ) ( ) k k k    x Ax Bu

 

( 1) ( ), ( ) k k k   x f x u

Linear MPC formulation (Classical MPC)

Remark: Since linear MPC includes constraints, it is a non-linear control strategy !!!

       

1 ref re ref ref 1 f ,

( ) ( ) min ( ) ( ) ( ( ( ) ) ( ) )

N N

N N T i i T

k i k i k i k k i k i k i k i i

  

            

 

x u

x x Q x x u u R u u

subject to

( 1 ) ( ) ( ), 0,1, , 1, k i k i k i i N         x Ax Bu

max

( ) , 0,1, , 1, k i i N     u u

max

( ) , 1,2, , , k i i N    x x

Model of the plant Input constraints State constraints

 

( 1); ( 2); ; ( ) ,

N

k k k N     x x x x

 

( ); ( 1); ; ( 1)

N

k k k N     u u u u

with:

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC Algorithm

10

Typical MPC control Loop MPC Algorithm

At every sampling time:

MPC Plant Observer

( ) k u ( ) k y ˆ( ) k x

ref ( )

k x

ref ( )

k u

ZOH

( ) t u ( ) t y

s

T

Digital system

• Read the current state of the process, x(k)
• Compute an optimal control sequence by solving the MPC optimization problem
• Apply to the plant ONLY the first element of such a sequence  u(k)

( ), ( 1), ( 2), , ( 1) k k k k N     u u u u

Solution 

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

LQR and Classical MPC

11 For simplicity, Let’s assume that the references are set to zero.

LQR CLassical Linear MPC

, 1

( ) min ( ) ( ( ) )

T T k

k k k k

 

 





x u

x Q R u x u

subject to

( 1) ( ) ( ), 0,1, , k k k k      x Ax Bu

subject to

( 1) ( ) ( ), 0,1, , 1, k k k k N      x Ax Bu

max

( ) , 0,1, , 1, k k N    u u

max

( ) , 1,2, , , k k N   x x

• Constraints are not taken into account
• No predictive capacity

The optimal solution has the form:

( ) ( ) k k   u Kx

PRO

• Explicit, Linear solution
• Low online computational burden

CON PRO

• Takes constraints into account
• Proactive behavior

CON

• High online computational burden
• No explicit solution
• feasibility? Stability?

If N∞, and constraints are not considered  the MPC and LQR give the same solution

1 , 1

( ) ( min ( ) ( ) )

N N

N T i k N T

k k k k

  





x u

u R u x Q x

k = 0

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC optimization problem – Implementation details

12

The following optimization problem, can be rewritten as a LCQP (Linearly Constrained Quadratic Program) problem in as follows:

       

1 ref re ref ref 1 f ,

( ) ( ) min ( ) ( ) ( ( ( ) ) ( ) )

N N

N N T i i T

k i k i k i k k i k i k i k i i

  

            

 

x u

x x Q x x u u R u u

subject to

( 1 ) ( ) ( ), 0,1, , 1, k i k i k i i N         x Ax Bu

max

( ) , 0,1, , 1, k i i N     u u

max

( ) , 1,2, , , k i i N    x x

with:

 

( 1); ( 2); ; ( ) ,

x

n N N

k k k N



     x x x x

 

( ); ( 1); ; ( 1)

u

n N N

k k k N



     u u u u

x

1 min 2

T T



x

x Hx f x

subject to

e e

 A x b

i i

 A x b

with:

 

 

;

x u

N n n N N  

  x x u

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC optimization problem – Implementation Details

13

where

 

 

 

 

2

x u x u

N n n N n n     

                    Q Q H R R

 

 

ref ref ref ref

(1) ( ) (0) ( 1)

x u

N n n

N N

 

                        x x f H u u

 

 

 

 

2 i

x x x u x u u u

n N n N N n n N n n n N n N         

                  I I A I I

 

max 2 max i max max

x u

N n n  

                      x x b u u

N times N times 2N times 2N times

nx = number of states, nu = number of inputs, N = prediction horizon

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC optimization problem – Implementation Details

14

   

 

e

x x x x u x

n n N n N n n n    

                     I B A I B A A I B

e

( )

x

N n

k



              Ax b

N times N times

nx = number of states, nu = number of inputs, N = prediction horizon Remarks:

• The problem is convex if Q and R are positive semi-definite
• The hessian matrix H, and the matrices Ae and Ai are sparse.
• Number of optimization variables: N(nx + nu). This number can be reduced to N·nu (Condensed

form of the MPC) through elimination of the states x(k) but at the cost of sparsity !!!

Matlab function for solving the LCQP optimization problem

x_tilde = quadprog (H, f, Ai, bi , Ae, be)

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC with terminal cost

15

Short Prediction Horizon Large Prediction Horizon

N

What happens? We have a winner !!!

N

What happens? We have an accident !!!

N should be large enough in order to keep the process under control

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC with terminal cost

16

       

1 ref ref ref ref , 1 1

min ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

N N

N T T i N i

k i k i k i k i k i k i k i k i

   

            

 

x u

x x Q x x u u R u u

subject to

( 1 ) ( ) ( ), 0,1, , 1, k i k i k i i N         x Ax Bu

max

( ) , 0,1, , 1, k i i N     u u

max

( ) , 1,2, , , k i i N    x x

   

ref ref

( ) ( ) ( ) ( )

T

k N k N k N k N        x x S x x

Main goal of the terminal cost: What do we gain?

To include the terms for which i ≥ N in the cost function (To extend the prediction horizon to infinity)

“Stability”

( 1) k  x ( 2) k  x ( ) k N  x

Effect of the stabilizing control law u(k) = -Kx(k)

1( )

x k

2( )

x k

ref ( )

k  x ( ) k x ( ) k   x

steady-state

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC with terminal cost

17

How to calculate S ?

By solving the discrete-time Riccati equation,

How to carry out this calculation in Matlab?

1

( )( ) ( )

T T T T 

     A SA S A SB B SB R B SA Q

where u(k) = -Kx(k) is the stabilizing control law.

   

1

.

T T 

  K B SB R B SA

[K,S] = dlqr (A, B, Q, R )

Implementation details of the MPC with terminal constraint

The only change in the LCQP is the hessian matrix

 

 

 

 

2

x u x u

N n n N n n     

                        Q Q H S R R N - 1 times N times