Model Predictive Control

1. MPC
MODEL PREDICTIVE CONTROL

2. CONTENTS
• Advantages & Drawbacks
• MPC Concept
• Terminology
• Applications
• Prediction Models
• State Space Model
• Optimization Window
• Closed-loop Control System
• State Estimate Predictive Control
• Constraints
• Numerical Solutions
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3. MPC ADVANTAGES
• Very intuitive concepts
• Relatively easy tuning
• Requires little computation
• Control a great variety of processes
• The multivariable case can easily be dealt with
• The treatment of constraints is conceptually simple
• Very useful when future references (robotics or batch processes) are known
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4. MPC DRAWBACKS
• Derivation of control law is more complex than the classical PID
• In adaptive control case all computation has to be carried out at every
sampling time
• Need for an appropriate model of the process to be available
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5. MPC CONCEPT
ANALOGYTOCHESS
I
(Controller)
Opponent
(Plant)
My move
Opponent’s move
New State
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6. MPC TERMINOLOGY
• Moving horizon window
• Prediction horizon
• Receding horizon control
• The information at time ti in order to predict the future is denoted as x(ti)
• Cost function is denoted as J
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7. APPLICATIONS
Chemical Process
Control
more than 4500 different
chemical processes
area-wide
application
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8. APPLICATIONS
Variable-pitch wind
turbines
stochastic
uncertainty
fatigue
constraints
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9. APPLICATIONS
Autonomous racing
reference tracking
short sampling intervals
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10. PREDICTION MODELS
Models
Linear
Non-
Linear
Discrete
Continuous
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11. STATE SPACE MODEL
𝑥 𝑚 𝑘 + 1 = 𝐴 𝑚 𝑥 𝑚 𝑘 + 𝐵 𝑚 𝑢 𝑘
𝑦 𝑘 = 𝐶 𝑚 𝑥 𝑚 𝑘
𝑥 𝑚 𝑘 + 1 − 𝑥 𝑚 𝑘 = 𝐴 𝑚(𝑥 𝑚 𝑘 − 𝑥 𝑚 𝑘 − 1 ) + 𝐵 𝑚(𝑢 𝑘 − 𝑢 𝑘 − 1 )
Δxm(k + 1) = AmΔxm(k) + BmΔu(k)
y(k + 1) − y(k) = Cm(xm(k + 1) − xm(k)) = CmΔxm(k + 1)
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= CmAmΔxm(k) + CmBmΔu(k)

12. STATE SPACE MODEL
Δxm(k + 1)
𝑦(𝑘+1)
= 𝐴 𝑚 𝑂 𝑚
𝑇
𝐶 𝑚 𝐴 𝑚 1
Δxm(k)
𝑦(𝑘)
+ Bm
𝐶 𝑚 𝐵 𝑚
Δu(k)
𝑦 𝑘 = 𝑂 𝑚 1
Δxm(k)
𝑦(𝑘)
𝑥 𝑘 + 1 𝑥 𝑘𝐴 𝐵
𝐶
𝑂 𝑚 =zeros(n1,1)
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13. MATLAB EXAMPLE 1
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14. PREDICTIVE CONTROL WITHIN ONE
OPTIMIZATION WINDOW
• Current Time: ki
• Prediction horizon (Np): Number of prediction samples
• Control horizon (Nc): dictating number of parameters used to capture the
future control trajectory
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15. PREDICTIVE CONTROL WITHIN ONE
OPTIMIZATION WINDOW
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16. PREDICTION OF STATE AND OUTPUT VARIABLES
x(ki + 1 | ki) = Ax(ki) + BΔu(ki)
x(ki + 2 | ki) = Ax(ki+1|ki) + BΔu(ki+1)
= A2x(ki) + ABΔu(ki) + BΔu(ki+1)
⋮ ⋮
x(ki + Np | ki) = ANpx(ki) + ANp-1BΔu(ki) + ANp-2BΔu(ki+1)+...
+ ANp-NcBΔu(ki+Nc-1)
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17. PREDICTION OF STATE AND OUTPUT VARIABLES
y(ki+1|ki) = CAx(ki) + CBΔu(ki)
y(ki+2|ki) = CA2x(ki) + CABΔu(ki) + CBΔu(ki+1)
⋮ ⋮
y(ki+Np|ki) = CANpx(ki) + CANp-1BΔu(ki) + CANp-2BΔu(ki+1)+
. . . + CANp-NcBΔu(ki+Nc-1)
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18. PREDICTION OF STATE AND OUTPUT VARIABLES
𝑌 =
𝑦(𝑘𝑖 + 1|𝑘𝑖)
𝑦(𝑘𝑖 + 2|𝑘𝑖)
𝑦(𝑘𝑖 + 3|𝑘𝑖)
⋮
𝑦(𝑘𝑖 + 𝑁 𝑝|𝑘𝑖)
, 𝑈 =
𝛥𝑢(𝑘𝑖)
𝛥𝑢(𝑘𝑖 + 1)
𝛥𝑢(𝑘𝑖 + 2)
⋮
𝛥𝑢(𝑘𝑖 + 𝑁𝑐 − 1)
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19. PREDICTION OF STATE AND OUTPUT VARIABLES
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20. OPTIMIZATION
Cost function
𝑅 𝑠 = 𝑜𝑛𝑒𝑠 𝑁 𝑝 ,
1 ∗ 𝑟 𝑘𝑖 = 𝑅 𝑠 𝑟 𝑘𝑖
𝑅 = 𝑟 𝜔 𝐼 𝑁 𝑐×𝑁𝑐 (𝑟 𝜔 ≥ 0)
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• First term is linked to the objective of minimizing the errors between
the predicted output and the set-point signal.
• Second term reflects the consideration given to impact of ΔU when the
objective function J is made to be as small as possible.

21. OPTIMIZATION
Hessian matrix
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22. MATLAB EXAMPLE 2
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23. CLOSED-LOOP CONTROL SYSTEM
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24. BLOCK DIAGRAM OF DISCRETE-TIME PREDICTIVE
CONTROL SYSTEM
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25. STATE ESTIMATE PREDICTIVE CONTROL
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26. MPC DESIGN WITH CONSTRAINTS
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With sampling interval Δt = 0.1

27. MPC DESIGN WITH CONSTRAINTS
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The prediction horizon Np = 10 and the control horizon Nc = 3. There is
no weight on the control signal, i.e., 𝑅 = 0. Examine what happens if the
control amplitude is limited to ±25 by saturation.

28. MPC DESIGN WITH CONSTRAINTS
CASE A. WITHOUT CONTROL SATURATION
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29. MPC DESIGN WITH CONSTRAINTS
CASE B. WITH CONTROL SATURATION
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30. MPC DESIGN WITH CONSTRAINTS
CASE C. WITH MODIFIED CONTROL SATURATION
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31. MPC DESIGN WITH CONSTRAINTS
CASE C. WITH MODIFIED CONTROL SATURATION
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32. MPC DESIGN WITH CONSTRAINTS
CASE COMPARISON
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33. FREQUENTLY USED OPERATIONAL CONSTRAINTS
Control Variable Incremental Variation
• These are hard constraints on the size of the control signal movements, i.e., on the
rate of change of the control variables (Δu(k))
Amplitude of the Control Variable
• These are the most commonly encountered constraints among all constraint types.
• These are the physical hard constraints on the system.
Output Constraints
• Output constraints are often implemented as ‘soft’ constraints.
• Output constraints often cause large changes in both the control and incremental
control variables when they are enforced.
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34. NUMERICAL SOLUTIONS USING QUADRATIC
PROGRAMMING
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Quadratic
Programming
Inequality
Constraints
Equality
Constraints

35. QUADRATIC PROGRAMMING FOR EQUALITY CONSTRAINTS
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36. QUADRATIC PROGRAMMING FOR EQUALITY CONSTRAINTS
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𝑥1 = 0.5 → 𝑥2 = 1 − 𝑥1 = 0.5

37. LAGRANGE MULTIPLIERS
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Constraint equation: 𝑀𝑥 = 𝛾
The procedure of minimization is to take the first partial derivatives with
respect to the vectors x and λ, and then equate these derivatives to zero:

38. LAGRANGE MULTIPLIERS
EXAMPLE
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39. LAGRANGE MULTIPLIERS
EXAMPLE
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40. ACTIVE SET METHOD
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λi ≥ 0
λi < 0
The point is a local solution
The objective function value can
be decreased by relaxing the
constraint i

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MINIMIZATION WITH INEQUALITY CONSTRAINTS
EXAMPLE

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Clearly the third element in λ is negative, therefore, the third constraint is an
inactive constraint and will be dropped from the constrained equation
λ =
5
3
→ x =
0.3333
1.3333
−0.6667
MINIMIZATION WITH INEQUALITY CONSTRAINTS
EXAMPLE
Inactive
constraint
Inactive
constraint

43. REFERENCES
1. Model Predictive Control System Design and
Implementation Using MATLAB, Liuping
Wang
2. Model Predictive Control, 2nd edition, E.F.
Camacho
3. A Lecture on Model Predictive Control, Jay
H. Lee
4. Model Predictive Control: Basic Concepts, A.
Bemporad
5. Lecture 14 - Model Predictive Control Part
1: The Concept, Gorinevsky
6. Principles of Optimal Control, Lecture 16
Model Predictive Control
7. Model Predictive Control, 4 Lectures 2016,
Mark Cannon
8. Model Predictive Control, S. Boyd

44. PRESENTED BYThanks for your
attention! Pooyan Nayyeri
Faraz AbedAzad
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