Tutorial: Algorithms & Hardware for Embedded Optimization

14:00 What Is Different about Embedded Optimization? Eric Kerrigan

14:40 Survey of Industrial Applications of Embedded MPC Alexander Domahidi

15:00 Efficient QP Frameworks for Industrial Embedded MPC Giorgio Kufoalor

15:20 Implicit vs Explicit MPC Martin Klauco (on behalf of Michal Kvasnica)

15:40 Robustness of Explicit MPC Pedro Ayerbe

What can I expect from this session?

What is Different About Embedded Optimization?Eric Kerrigan, Bulat Khusainov, George Constantinides

Communication

ProcessingStorage

Computing System Physical

System

On time & on budget in an uncertain world

+

+

+

Applications of Embedded Optimization

A Fundamental Problem

dt

dt= 1

Correctness should be a function of time

Process + Embedded Optimizer = Uncertain Cyber-Physical System

Time + Uncertainty ⇒ Approximate today might be better than accurate tomorrow

u⇤(y) := argminu

f(u, y)

s.t. g(u, y) = 0

h(u, y) 0

yu⇤(y)measurementsoptimal inputs

disturbances

numerical errors

physical system

computing system

(p⇤, c⇤) := argminp,c

�(p, c)

s.t. ↵(p, c) = 0

�(p, c) 0

for physical system

for computing system

p⇤

c⇤

co-designer

cyber-physical system

optimal design parameters

optimal design parameters

Ordering is important:

Discretized Optimal Control/Estimation Problems

minq2Q,s2S

N�1X

i=0

`(qi, si, si+1, i, d, t)

f(qi, si, si+1, i, d, t) = 0, i = 0, . . . , N � 1

g(qi, si, si+1, i, d, t) 0, i = 0, . . . , N � 1

x := (s0, q0, s1, q1, . . . , sN�1, qN�1, sN )

Nonlinear)Programming)Strategies)On)High3Performance)Computers)

Victor)M.)Zavala)Scalable)Systems)Laboratory)Department)of)Chemical)&)Biological)Engineering)University)of)Wisconsin3Madison!

With:)Carl)Laird)(Purdue))))))Jia)Kang)(Sabre)CorporaMon)))))))Nai3Yuan)Chiang)(United)Technologies))

Acknowledgement:)Yankai)Cao)(Purdue))

Sparse & structured matrices ⇒ small (and full) not always better

Exploit Structure: Interior Point Method

Time Space

Proposed = [Cantoni, Farokhi, Kerrigan, Shames, AuCC16]

Control / Automation

SYSTEM

MODEL

ALGORITHM

Unknown Inputs

Known Inputs

Known Outputs

Unknown Outputs

Estimates of Unknown Outputs

Estimates of Known Outputs

CONTROLLER

Corrections

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

3.Implement first parttime

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

3.Implement first parttime

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

3.Implement first part

4.Go to step 1

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

3.Implement first part

4.Go to step 1

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

3.Implement first part

4.Go to step 1

time

time

output

input

Real-time Optimal/Predictive Control (e.g. Receding Horizon)

1.Take measurement

2.Solve optimal control problem

3.Implement first part

4.Go to step 1

time

time

output

input

Feedback Algorithm

Signal Processing / Estimation / Learning

SYSTEM

MODEL

ALGORITHM

Unknown Inputs

Known Inputs

Known Outputs

Unknown Outputs

Estimates of Unknown Outputs

Estimates of Known Outputs

ESTIMATOR

Corrections

Process + Embedded Optimizer = Uncertain Cyber-Physical System

HC

Pz

vy

w

euUnknowns Behaviour

Optimal closed-loop system?

u(t) 2 Proj argmin

x

{J(x, d, t) | x 2 X(d, t), d = D(y, v, t)}

When is Sub-optimal Optimal? Real-time Dynamic Optimization

latency/computational delay

objec

tive

func

tion

value

monotonic & sub-optimal

optimal

non-monotic & sub-optimal

When is Sub-optimal Optimal? Real-time Dynamic Optimization

u⇤(t, �) =

(0 8t 2 [0, �)

1/(1� �) 8t 2 [�, 1)

y(0) = 0, y(1) � 1

u(t) = 0, 8t 2 [0, �)

y(t) = u(t), 8t 2 [0, 1)

s.t.

V (u⇤(·, 0.2)) = 1.25 < V (1.1u⇤(·, 0.2)) < V (u⇤(·, 0.4)) ⇡ 1.7

(u⇤(·, �), y⇤(·, �)) := arg min(u,y)

V (u), V (u) :=

Z 1

0u(t)2dt

Precision, Accuracy and Latency: Fast Gradient Method

Bounds for i ! 1 [Jerez et al., ECC 2013, IEEE TAC 2014]

0 20 40 60 80 10010−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1||z∗(x)−

zi||2

Number of fast gradient iterations i

doubleb = 18b = 24b = 32

Quantization Errors in Communication: Distributed First-Order Optimization

[Pu, Zeilinger, Jones, arXiv 2015]

Hardware or Algorithm?

Computing Systems and Resources

Communication

Storage

Processing

Space

Energy

Time

Possible Design Parameters

Computer Hardware Algorithm

cost, space, energy, power accuracy, termination tolerances

# processors/cores/arithmetic units # iterations in each loop

pipeline depth step length parameters

clock frequency and supply voltage amount of data/results to store

memory architecture, latency, size time horizon

communication architecture, bandwidth complexity of physical model

number representation, word length coarseness of discretization

actuation and sampling schedule/rate scheduling/communication strategy

Co-Design as Multi-objective Optimization

perform

ance

lim

it

perform

ance

constraint

computational constraint

Pareto frontier

com

puting

resources

performance cost

H(c)

C(c)

Pz

vy

w

eu

minc{F(H(c),c) | (H(c),c) 2 G}

[Khusainov, Kerrigan, Constantinides, ECC2016]

Explicit Constrained LQR in Fixed Point

10-4 10-3 10-2 10-1 100

1

1.1

1.2

1.3

1.4

1.5

1.6

Computing Power [W]

Clos

ed-lo

op C

ost

[Suardi et al., ECC 2013]

H(c)

C(c)

Pz

vy

w

eu

minc{F(H(c),c) | (H(c),c) 2 G}

c := #bits

Size is Very Important in Microprocessor Design

Die area = 1 Working = 64

Die area = 4 Working = 4

Cost per die = f(areax), x∈[2,4]

FPGA Resources for an Optimal Controller

5 23 520

2

4

6

x 105

# of bits

#of

FFs

requ

ired

5 23 520

2

4

6

x 105

# of bits

#of

LU

Ts

requ

ired

5 23 520

2000

4000

# of bits

#of

DSP

sre

quir

ed

5 23 520

5

x 10−6

# of bits

com

puta

tion

dela

y[s

]

Interior-point [Rao et al., 1998] on Xilinx Virtex 6 [Longo, Kerrigan, Constantinides, Automatica 2014]

Optimal Control in Low Precision Floating-Point

0 5 10 15 20−2

0

2

4

time [s]

stat

e5

5 bits, shift

5 bits, delta

52 bits, shift

0 5 10 15 20

−0.5

0

0.5

time [s]

inpu

t2

[Longo, Kerrigan, Constantinides, Automatica 2014]

Computational Resources for an Adder

Number representation

Registers/Flip-Flops (FFs)

Lookup-Tables (LUTs)

Latency/delay (clock cycles)

double floating-point 52-bit mantissa 1035 852 12

single floating-point 23-bit mantissa 542 445 12

fixed-point 53 bits 53 53 1

fixed-point 24 bits 24 24 1

Xilinx Virtex-7 XT 1140 FPGA:

Cheap and low power processors often only have fixed-point

Floating-Point Arithmetic

Fixed-Point Arithmetic

Computational Resources for an Adder

Given a fixed amount of silicon (£/$/€):

200x more fixed point additions

• per second

• per Joule

than in floating point

Fast Gradient Method in Fixed Point Arithmetic

Atomic force microscope Actuation rate > 1 MHz

y

d

r

cantilever

sample

Piezo plate actuatoru

[Jerez et al., ECC 2013 & IEEE TAC 2014]

QP solver on FPGA: latency < 1 µs power < 1 W within 0.1% of optimal

minu

u0Hu+ u0D(y, v, t)

s.t. u u u

Latency, Precision and Silicon: MINRES

Xilinx Virtex-7 XT 1140 FPGA

float52

float23

more parallelism

0 20 40 60 80 1000

200

400

600

800

1000

late

ncy

(cycl

esper

iter

ation)

% Registers (FFs)

fixed53

fixed29

faster

[Jerez, Constantinides, Kerrigan, IEEE TC 2015]

Fixed point (FPGA) vs floating point (GPU): MINRES

10−15 10−10 10−5 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 x 1012

oper

ations

per

seco

nd

error tolerance for >90% of problems (η)

k = 41P = 4

k = 23P = 11

k = 17P = 21

k = 58P = 2

actual sustained

FPGA

theoretical peak GPU

NVIDIA C2050 1.03 TFLOP/s

1.15GHz, 100W 10 GFLOP/s/W

Xilinx Virtex-7 XT 1140 400MHz, 22W >180 GOP/s/W

[Jerez, Constantinides, Kerrigan, IEEE TC 2015]

What is Different About Embedded Optimization?

Optimal?

HC

Pz

vy

w

eu

Nonlinear)Programming)Strategies)On)High3Performance)Computers)

Victor)M.)Zavala)Scalable)Systems)Laboratory)Department)of)Chemical)&)Biological)Engineering)University)of)Wisconsin3Madison!

With:)Carl)Laird)(Purdue))))))Jia)Kang)(Sabre)CorporaMon)))))))Nai3Yuan)Chiang)(United)Technologies))

Acknowledgement:)Yankai)Cao)(Purdue))

Feedback

Structure

Hardware ⇔ Algorithm

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