Explicit MPC: Basics, Fast Implementations, Advantages and ...

Explicit MPC: 

Basics, Fast Implementations, 

Advantages and Limitations 

Alberto Bemporad 

http://cse.lab.imtlucca.it/~bemporad 

A. Bemporad Explicit MPC -‐ ECC’13 -‐ Zurich, July 17, 2013 1

Outline 

• Formulation of linear MPC design framework 

• Multiparametric quadratic programming (explicit MPC) 

• An automotive control example 

• Fast circuit implementations 

• Final conclusions (explicit vs implicit MPC) 

A. Bemporad Explicit MPC -‐ ECC’13 -‐ Zurich, July 17, 2013 

2

Finite-‐time optimal control of linear systems 

linear model 

 

xk+1 = Ax k + Bu k 

performance index 

y k = Cx k 

z = 

min x u 0 ,...,u N−1 

N Px N + 

N−1 

k=0 

x k Qx k + u k Ru k 

min 

2 z Hz + x 0 F z + 1 2 x 0 Yx 0 

s.t. Gz ≤ W + Sx 0 

z 

1 

H 0 

⎡ 

⎢ 

⎣ 

⎤ 

u 0 

u 1 

⎥ 

. ⎦ 

u N−1 

constraints 

 

umin ≤ u k ≤ u max 

y min ≤ Cx k ≤ y max 

(convex) Quadratic Program (QP) 


3

Linear MPC algorithm 

@ each sampling step t: 

past 

future 

y k 

r(t) 

predicted outputs 

u k 

manipulated 

inputs 

t t+k t+N 

•Measure (or estimate) the current state x(t) 

feedback ! 

• Get the solution z * ={u 0 

*,...,u N-1 * } of the QP 

min z 

s.t. 

1 

2 

z Hz + x (t)F z 

Gz ≤ W + Sx(t) 

• Apply only u(t)= u 0 

*, discard remaining optimal inputs u 1 

*,...,u N-1 * 

Routinely used in process control, now spreading in automotive/aerospace 


Requirements for embedded MPC 

1. Speed: fast enough to provide a solution within 

short sampling intervals (such as 10-‐100 ms) 

2.Require simple/cheap hardware (microcontroller, 

microprocessor, FPGA) and little memory to store 

problem data and code 

3.Code simple enough to be verifiable/certifiable 

4.Worst-‐case execution time must be (tightly) estimated for 

embedding MPC in a real-‐time platform 


5

Computing the MPC command action 

Algorithms for solving QP on-‐line given x(t): 

min z 

1 

2 

z Hz + x (t)F z 

• active set (AS) methods 

• interior point (IP) methods 

• gradient projection (GP) methods 

• conjugate gradient (CG) methods 

• alternating direction method of multipliers (ADMM) 

s.t. Gz ≤ W + Sx(t) 

Quadratic Program (QP) 

(the control law u=u(x) is implicitly defined by the QP solver) 

Alternative: solve the QP off-‐line for all x(t) to find the control law 

u=u(x) explicitly via multiparametric programming 


6

Offline multiparametric QP algorithm 

Fix a point x 0 2 X in the parameter space 

solve QP to find 

z ∗ (x 0 ), λ ∗ (x 0 ) 

identify active constraints at 

form matrices 

active constraints: 

˜G, ˜W , ˜S 

z ∗ (x 0 ) 

by collecting 

˜Gz ∗ (x 0 ) − ˜W − ˜Sx 0 =0 

(Bemporad, Morari, Dua, Pistikopoulos, 2002) 

min z 

s.t. 

1 

2 

z Hz + x F z 

Gz ≤ W + Sx 

KKT 

optimality 

conditions: 

(1) Hz + Fx+ G λ = 0 (2) ˜Gz − ˜W − ˜Sx =0 

(3) λ i (G i z − W i − S i x) = 0 (4) Ĝz ≤ Ŵ + Ŝx 

(5) ˜λ i ≥ 0, ˆλ i =0 

From (1) : 

z = −H −1 (Fx+ ˜G ˜λ) 

From (2) : 

z(x) =H −1 ˜G ( ˜GH −1 ˜G ) −1 ( ˜W +(˜S + ˜GH −1 F )x) − Fx 

In some neighborhood of x 0 , λ and z are explicit affine functions of x 

(Zafiriou, 1990) 


7

Multiparametric QP algorithm 

• Impose primal and dual feasibility: 

Ĝz(x) ≤ Ŵ + Ŝx 

˜λ(x) ≥ 0 

from (4) 

from (5) 

CR 0 

linear inequalities in x ! 

x-‐space 

• Remove redundant constraints (this requires solving LP’s): 

critical region CR 0 

• x 

0 

X 

• CR 0 is the set of all and only parameters x for which 

combination of active constraints at the optimizer 

is the optimal 


8

Multiparametric QP solvers 

Method #1: Split and proceed iteratively 


Method #2: Split and proceed iteratively 

(Tøndel, Johansen, Bemporad, 2003) 

x-‐space 

R 2 

R • x 1 0 

CR 0 

R 3 

CR 1 

R N R 4 

X 

add constraint #i 

CR 0 

CR 0 

CR 0 

Ĝ i z(x) ≤ Ŵ i + Ŝ i x 

˜λ j (x) ≥ 0 

remove constraint #j 

Method #3: exploit the facet-‐to-‐facet property 

(Spjøtvold, Kerrigan, Jones, Tøndel, Johansen, 2006) (Spjøtvold, 2008) 

Step out ² outside each facet, solve QP, get new region, iterate. (Baotic, 2002) 


9

Properties of multiparametric QP 

z ∗ : X ∗ → R s 


z ∗ (x) =argmin z 

s.t. 

1 

2 

x Hz + x F z 

Gz ≤ W + Sx 

continuous, 

piecewise affine 

V ∗ (x) = 1 2 x Yx+min z 

s.t. 

1 

2 

x Hz + x F z 

Gz ≤ W + Sx 

convex, continuous, 

piecewise quadratic, 

C 1 (if no degeneracy) 

Corollary: The linear MPC controller is a continuous piecewise affine 

function of the state 

z ∗ = 

⎡ 

⎢ 

⎣ 

u ∗ 0 

u ∗ 1 

. 

u ∗ N−1 

⎤ 

⎥ 

⎦ 


10

Applicability of explicit MPC approach 

• Consider the following general MPC formulation 

min z 

N−1 

k=0 

1 

2 (y k − r(t + k) S(y k − r(t + k))+ 1 2 ∆u k T ∆u k 

+(u k − u r (t + k)) V (u k − u r (t + k)) + ρ 2 

subj. to x k+1 = Ax k + Bu k + B v v(t + k), k =0,...,N − 1 

y k = Cx k + Du k + D v v(t + k), k =0,...,N − 1 

u min (t + k) ≤ u k ≤ u max (t + k), k =0,...,N − 1 

∆u min (t + k) ≤ ∆u k ≤ ∆u max (t + k), k =0,...,N − 1 

y min (t + k) − V min ≤ y k ≤ y max (t + k) + V max , k =1,...,N 

• Everything marked in red can be time-‐varying in explicit MPC 

• Not applicable to time-‐varying problems (weights and/or system matrices) 

• Can be extended to hybrid systems (continuity of control law may be lost) 

(Bemporad, Borrelli, Morari, 2001) (Mayne, Rakovic, 2002) (Mayne, ECC 2001) 

(Borrelli, Baotic, Bemporad, Morari, 2005) (Alessio, Bemporad, 2006) 


11

Hybrid Toolbox for MATLAB 

Features: 

• Explicit MPC control (via multi-‐parametric programming) 

and C-‐code generation 

(Bemporad, 2003-‐2013) 

Supported by 

• Hybrid models: design, simulation, verification 

5000+ download requests 

since October 2004 

http://cse.lab.imtlucca.it/~bemporad/hybrid/toolbox/ 

Other good tools: Multiparametric Toolbox (see next talk !) 


12

Explicit MPC for idle speed control 

(Di Cairano, Yanakiev, Bemporad, Kolmanovsky, Hrovat, 2011) 

• Ford pickup truck, V8 4.6L gasoline engine 

• Process: 

-‐ 1 output (engine speed) to regulate 

-‐ 2 inputs (airflow, spark advance) 

-‐ input delays 

• Objectives and specs: 

-‐ regulate engine speed at constant rpm 

-‐ saturation limits on airflow and spark 

-‐ lower bound on engine speed (≥450 rpm) 


13

Explicit MPC for idle speed -‐ Experiments 

(Di Cairano, Yanakiev, Bemporad, Kolmanovsky, Hrovat, 2011) 

• Sampling time = 30 ms 

• Explicit MPC implemented in dSPACE 

MicroAutoBox rapid prototyping unit 

Load torque (power steering) 

• Observer tuning as much important 

as tuning of MPC weights ! 

peak reduced by 50% 

convergence 10s faster 

explicit MPC 

baseline controller (linear) 

set-‐point 


14

Hardware (ASIC) implementation of explicit MPC 

http://www.mobydic-project.eu/ 


15

continuous functions that are affine over each simplex constitutes an N v -dim 

PWA approximation of MPC over simplices 

P W AS[S] ⊂ PWA[S] ∩ C 0 [S] [19]. Therefore, it is possible to define diffe 

• Approximate 

independent 

a 

functions 

given linear 

belonging 

MPC controller 

to P W 

by 

AS 

using canonical PWA functions 

p [S]. By choosing some (arbitrary 

over simplicial partitions (PWAS) 

(Bemporad, Oliveri, Poggi, Storace , 2011) 

these bases, we can regard them as an N v -length vector, say φ(x). Then a sca 

is defined as a linear combination of the basis functions as follows 

û(x) = 

N v 

k=1 

w k φ k (x) =w φ(x) 

Vector w =[w 1 ... w Nv ] determines û uniquely for each given x ∈ S. 

approximate MPC law 

(Julian, Desages, Agamennoni, 1999) 

Weights wk optimized off-‐line 

to best approximate a given MPC law 

September 29, 2010 

http://www.mobydic-project.eu/ 


16

PWA approximation of MPC over simplices 

• Extremely cheap: PWAS functions can be directly implemented 

on FPGA, or ASIC (Application Specific Integrated Circuits) 

• Extremely fast computations (10-‐100 nanoseconds) 

Computation time [s] Control M p Latency Computation (A - B) time [ns] [s] M Latency online time 

fit criterion 

(A - B) [ns] 

12 0.18187 170 -12 31 

0.1818 to compute 170 - 31 MPC 

36 L 2 0.046131 238 -36 45 

0.0461 238 - 45 

Architecture A: (Xilinx Spartan 3 FPGA) 

170 0.022963 272 -170 46 mainly 0.0229 serial 272 - 46 

11 0.16797 170 -11 31 

0.1679 170 - 31 

34 L ∞ Architecture B: MIMO system dynamics 

0.046131 238 -34 45 

0.0461 238 - 45 

fully parallel 

1.2 1 

160 0.022763 272 -160 46 

0.0227 x k+1 = 272 - 46x k + 

# partitions 

per dimension 

Exact explicit MPC: 52 regions 

383 ns (avg) -‐ 486 ns (max) 

y k = 

0 1.1 

1 1 

x 

1 0 k 

 

0 1 

u 

1 1 k 

subject to saturation on u and y 

• Closed-‐loop stability results are available 

(Bemporad, Oliveri, Poggi, Storace , 2011) 

(Rubagotti, Barcelli, Bemporad, 2012) 

✖Curse of dimensionality (with respect to to state dimension) 


17

Complexity of multiparametric solutions 

•The number of regions depends on the number of 

possible combinations of active constraints 

•Weak dependence on the number of states and references 

•Explicit MPC gets less attractive when number of regions grows, 

too much memory required, too much time to locate state x(t) 

•Fast and simple on-‐line QP may be preferable otherwise 

 


Fast gradient projection for (dual) QP 

• Solve dual QP using fast gradient projection 

(Nesterov, 1983) 

• Main on-‐line operations involve only 

simple linear algebra 

• Using Riccati-‐like iterations, complexity 

per iteration is O(N) [N = prediction horizon] 

(Patrinos, Bemporad, IEEE TAC, 2013) 

• Tight bounds on maximum number of iterations 

• Similar approaches exist 

(Richter, Morari, Jones, 2009/2011/2012) 

(Giselsson, 2012) 

• Dual gradient projection methods for QP work very well in fixed-‐point ! 

(Patrinos, Guiggiani, Bemporad, ECC 2013) 


19

Conclusions 

Explicit or implicit 

CPU time / throughput 

worst-‐case exec estimates 

numerical arithmetics 

off-‐line computations 

solution quality (feas/opt) 

data memory 

control code 

Explicit 

MPC 

Implicit (=on-‐line QP) 

active 

set 

interior 

point 

gradient 

projection 

problem size small medium large medium 

•Explicit typically limited 6÷8 free control moves and 8÷12 parameters 

(states+references) 

•Embedded QP methods are preferable otherwise 


20

Conclusions 

Research on QP solution methods started ~60 years ago ... 

(Beale, 1955) 

... still more research on QP can have an impact in real applications ! 


21

Explicit MPC: Basics, Fast Implementations, Advantages and ...

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