Model Predictive Diesel Engine Control - LIRMM

EECI19 M22: MPC for Diesel Engines

Model Predictive Diesel

Engine Control

1


Components:

1. Compressor

2. Variable Geometry Turbine (VGT)

3. EGR Throttle

4. EGR Valve

5. Intake Manifold

6. Exhaust Manifold

7. Pre-Throttle Volume

8. Pre-Intercooler Volume

9. Combustion

10. EGR cooler

11. Intercooler

Diesel engine schematics

2

High speed diesel engine


Diesel air path control

Tracking objectives:

• Control intake manifold pressure and EGR rate (= EGR valve flow

/ Cylinder Flow) to specified set-points

• Engine speed and fuel flow are measured disturbances

• Fast response desired to reduce turbo-lag

Constraints:

• Input constraints: EGR valve, EGR throttle, and VGT lift have limited

ranges

• Output constraints are imposed on intake manifold pressure

overshoot, exhaust manifold pressure overshoot, turbocharger

speed, EGR rate, etc.

3


Model Predictive Control

4


Low complexity explicit LQ-MPC

5

• Nonlinear inversion to make plant to which MPC is applied “more linear”

• Rate-based model at a single linearization point

• Intermittent tightened constraint enforcement

• Constraint re-modeling (e.g., constraint on turbocharger speed is remodeled as a constraint on intake manifold pressure)

• Special MPC formulation:

• Control horizon = 1

• Prediction horizon = 1

• LQ-based terminal cost

• Longer constraint horizon, with intermittent enforcement

• Low complexity gain scheduling


Low complexity explicit LQ-MPC

6


Diesel air path controller structure

7


Combined EGR valve – EGR throttle actuator

• Two actuators are “ganged” together to deliver EGR flow

8


• Define state and control increments and an augmented state:

• The rate-based model has the following form,

Δ𝑢𝑘 = 𝑢𝑘 − 𝑢𝑘−1Δ𝑥𝑘 = 𝑥𝑘 − 𝑥𝑘−1𝑒𝑘 = 𝑦𝑘−1 − 𝑟𝑘−1𝜉𝑘 = Δ𝑥𝑘 𝑒𝑘

𝑇 .

𝜉𝑘+1 = ҧ𝐴𝜉𝑘 + ത𝐵Δ𝑢𝑘𝑒𝑘 = ҧ𝐶𝜉𝑘

ҧ𝐴 =𝐴 0𝐶 𝐼

, ത𝐵 =𝐵𝐷

, ҧ𝐶 = 0 𝐼 .

Rate-based model

(assumes 𝑟𝑘 = 𝑐𝑜𝑛𝑠𝑡)


The MPC optimization problem then becomes,

min 𝜉𝑁𝑇𝑃𝜉𝑁 +

𝑘=1

𝑁−1

𝑒𝑘𝑇𝑄𝑒𝑘 + Δ𝑢𝑘

𝑇𝑅Δ𝑢𝑘

subject to:𝜉𝑘+1 = ҧ𝐴𝜉𝑘 + ത𝐵Δ𝑢𝑘

𝑒𝑘 = ҧ𝐶𝜉𝑘𝑢𝑘 = 𝑢𝑘−1 + Δ𝑢𝑘

𝑒𝑘 + 𝑟 ∈ 𝑌𝑘 ,𝑢𝑘 ∈ 𝑈𝑘 .

Rate-based MPC (cont’d)


Linear model through system identification

• A local 7th order linear model was obtained through system identification

• Nominal operating condition chosen as 1600 rpm engine speed and 30 mm3/stroke fuel rate. Engine operates between 800 and 4000 rpm and 0 and 60 mm3/stroke

• Model order reduction applied to obtain a 4th order linear model

𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘,

𝑦𝑘 = 𝐶𝑥𝑘 + 𝐷𝑢𝑘 .

11


Computational complexity

Table: Comparison of MPC strategies. 𝑁𝑧 denotes the number of zonespartitioning the engine operating range, 𝑁𝑦 denotes the number of outputconstraints considered, 𝑁𝑐 denotes the number of constraints in the MPCoptimization problem (inputs and outputs with intermittent constraintenforcement), σ𝑁𝑟 is the total number of regions in the explicit solution, ROMand Time are estimated memory and computation time usage for explicitMPC on a ~150 MHz processor. The sampling/update period is 32 ms.

12


Low complexity gain scheduling1


Simulation results – nonlinear model

14


Engine experiments

1200 rpm 1600 rpm

15


2400 rpm

Engine experiments

16


Experimental results over NEDC and WLTC

drive cycles

17


Robust MPC for air path control


Robust MPC for air path control: Experimental results


From LQ MPC to Nonlinear MPC


From LQ MPC to Nonlinear MPC


Nonlinear MPC


Basic NMPC computational strategy










Towards more advanced NMPC


Modeling for Nonlinear MPC


NMPC based on move blocked rate-based model


MPC for feedforward and feedback


Experimental implementation of NMPC


Experimental implementation of NMPC

Supervisory MPC


minz

𝐽 𝑧, 𝜌 =

𝑘=0

𝑁

𝑙(𝑥𝑘 , 𝑢𝑘 , 𝑢𝑘−1, 𝑠, 𝜌)

𝑠. 𝑡 𝑥𝑘+1 = 𝑓 𝑥𝑘 , 𝑢𝑘 , 𝜌, Δ𝜏𝑘 , 𝑘 = 0, … 𝑁 − 1,𝜙 𝑥𝑘 , 𝑢𝑘 , 𝜌 − 𝜙𝑙(𝜌, 𝑤𝑐) ≤ 𝑠, 𝑘 = 0,…𝑁 − 1 ,

0 ≤ 𝜒𝑘𝑡𝑟𝑔

≤ ҧ𝜒(𝜌), 𝑘 = 0…𝑁 − 1,

0 ≤ 𝑞𝑘 ≤ 𝑞𝑡𝑟𝑔 𝜌 , 𝑘 = 0…𝑁 − 1,𝑠 ≥ 0

where

𝑙 = 𝛾 𝜒𝑘𝑡𝑟𝑔

− ҧ𝜒 𝜌2+ 𝑢𝑘 − 𝑢𝑘−1 𝑅

2 + 𝛼 𝑞𝑡𝑟𝑔 − 𝑞𝑘 + 𝛽𝑠

Fuel trackingDampingEGR tracking

𝑧 = 𝑢0𝑇 𝑥1

𝑇 … 𝑢𝑁−1𝑇 𝑥𝑁

𝑇 𝑇

FAR slack penalty

Dynamics

Softened FAR constraint

EGR rate constraint

Fuel bounds

𝑝𝑖𝑚,𝑘𝑡𝑟𝑔

= ҧ𝑝𝑖𝑚(𝜌𝑘)

𝑢 =𝜒𝑡𝑟𝑔

𝑞

𝑥 = 𝑝𝑖𝑚 𝑝𝑒𝑥 𝑤𝑐 𝐹1 𝐹2𝑇

MPC formulation

Experimental results (WLTC)


Experimental results

FBRS method for convex quadratic programming


Results of computational experiments


Symbolic code optimization

Towards economic MPC (eMPC)

eMPC as an outer loop supervisor

Performance outputs:

𝑦𝑝 𝑡 =𝜙𝑁𝑂𝑥𝜂

Measured outputs:

𝑦𝑚 𝑡 =𝑝𝑖𝑚𝑤𝑐

Targets for inner loop:

𝑟 𝑡 =𝑝𝑖𝑚𝑡𝑟𝑔

𝜒𝑡𝑟𝑔

Actuators:

𝑤 𝑡 =

𝑢𝑡ℎ𝑟𝑢𝑣𝑎𝑙𝑢𝑣𝑔𝑡

𝜒 = 𝐸𝐺𝑅 𝑅𝑎𝑡𝑒 =𝐸𝐺𝑅 𝑓𝑙𝑜𝑤

𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟 𝑓𝑙𝑜𝑤=𝑤𝑒𝑔𝑟

𝑤𝑐𝑦𝑙

𝜂 =𝑡𝑜𝑟𝑞𝑢𝑒 ∗ 𝑒𝑛𝑔𝑖𝑛𝑒 𝑠𝑝𝑒𝑒𝑑

𝑓𝑢𝑒𝑙 𝑓𝑙𝑜𝑤 ∗ 𝑙𝑜𝑤𝑒𝑟 ℎ𝑒𝑎𝑡𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒=

𝜏𝑁𝑒𝑤𝑓𝑄𝐿𝐻𝑉

𝜙 =𝑓𝑢𝑒𝑙 𝑓𝑙𝑜𝑤

𝑎𝑖𝑟 𝑓𝑙𝑜𝑤∗

𝐴

𝐹𝑠𝑡𝑜𝑖𝑐ℎ

=𝑤𝑓

𝑤𝑎

𝐴

𝐹𝑠𝑡𝑜𝑖𝑐ℎ

𝜒𝑁 = 𝑁𝑂𝑥 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛𝑝𝑖𝑚 = 𝑖𝑛𝑡𝑎𝑘𝑒 𝑚𝑎𝑛𝑖𝑓𝑜𝑙𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

Exogenous inputs:

𝜌 𝑡 =𝑞𝑡𝑟𝑔

𝑁𝑒Fuel per stroke:

𝑞(𝑡)eMPC control:

𝑢 𝑡 =𝑟𝑞


Efficiency and NOx modeling


eMPC Formulation

minz

𝐽 𝑧, 𝜌 =

𝑘=0

𝑁

𝑙(𝑥𝑘 , 𝑢𝑘 , 𝑢𝑘−1, 𝜌)

𝑠. 𝑡 𝑥𝑘+1 = 𝑓 𝑥𝑘 , 𝑢𝑘 , 𝜌, Δ𝜏𝑘 , 𝑘 = 0, … 𝑁 − 1,

𝜙 𝑥𝑘 , 𝑢𝑘 , 𝜌 ≤ 𝜙lim 𝜌 , 𝑘 = 0,…𝑁 − 1 ,

ln(𝜒𝑁(𝑥𝑘 , 𝑢𝑘 , 𝜌)) ≤ ln(𝜒𝑁lim(𝜌)), 𝑘 = 0,…𝑁 − 1,

𝑟− 𝜌 ≤ 𝑟𝑘 ≤ 𝑟+ 𝜌 , 𝑘 = 0…𝑁 − 1,𝑞− 𝜌 ≤ 𝑞𝑘 ≤ 𝑞𝑡𝑟𝑔 𝜌 , 𝑘 = 0…𝑁 − 1,

where

𝑙 = −𝛽𝜂 𝑥𝑘 , 𝜌 +Δ𝑢𝑘Δ𝜏𝑘 𝑅

2

+ 𝛼 𝑞𝑡𝑟𝑔 − 𝑞𝑘

Dynamics

Emissions constraints

Target constraints

Fuel bounds

Fuel trackingDamping

Efficiency

𝑧 = 𝑢0𝑇 𝑥1

𝑇 … 𝑢𝑁−1𝑇 𝑥𝑁

𝑇 𝑇𝑢 =

𝑝𝑖𝑚𝑡𝑟𝑔

𝜒𝑡𝑟𝑔

𝑞

𝑟 =𝑝𝑖𝑚𝑡𝑟𝑔

𝜒𝑡𝑟𝑔


𝑘 = 0 𝑘 = 1 𝑘 = 2 𝑘 = 3 𝑘 = 4 𝑘 = 5

Non uniform horizon discretization


Numerical solution

➊ Linearize dynamics and non-convex

constraints about the current measured state

and form the single-shooting OCP

➋ Form quadratic programming sub-problem

about the solution at the previous time step

➌ Solve QP and apply the resulting control to

the system

minΔ𝑧

1

2Δ𝑧𝑇𝐻 Δ𝑧 + 𝑓𝑇Δ𝑧

𝑠. 𝑡. 𝐴Δ𝑧 ≤ 𝑏

𝐻 = 𝛻𝑧2𝐹 𝑧𝑘−1, 𝜌𝑘 , 𝑓 = 𝛻𝑧𝐹 𝑧𝑘−1, 𝜌𝑘 ,

𝐴 = 𝛻𝑧𝑐 𝑧𝑘−1, 𝜌𝑘 , 𝑏 = −𝑐(𝑧𝑘−1, 𝜌𝑘)

min𝑧

𝐹(𝑧, 𝜌𝑘)

𝑠. 𝑡. 𝑐 𝑧, 𝜌𝑘 ≤ 0

𝑧𝑘 = Δ𝑧 + 𝑧𝑘−1𝑢𝑘 = 𝑧 1: 3

1

2

3


➊

➋

➌ 3

2

1

Experimental results


Simulation results

• eMPC is stable over NEDC and WLTC cycles

• Emissions are reduced compared to the benchmark during hard transients ➊

WLTC

1


Simulation results (cont’d)• NOx constraint violations are penalized, leading to soft constraint

enforcement and reducing NOx over the cycles

• Fuel-air-ratio constraint is not always satisfied due to conflict with the

drivability constraint ➊

1


ReferencesHuang, M., Zaseck, K., Butts, K., and Kolmanovsky, I.V., “Rate-based Model Predictive Controller for

diesel engine air path: Design and experimental evaluation,” IEEE Transactions on Control Systems

Technology, vol. 24, no 4, pp. 1922-1955, 2016.

Liao-McPherson, D., Huang, M., and Kolmanovsky, I.V., “A regularized and smoothed Fischer-Burmeister

method for quadratic programming with applications to Model Predictive Control,” IEEE Transactions on

Automatic Control, 2019.

Huang, M., Liao-McPherson, D., Kim, S., Butts, K., and Kolmanovsky, I.V., “Towards real-time automotive

model predictive control: A perspective from a diesel air path control development,” Proceedings of 2018

Annual American Control Conference (ACC), June 27–29, 2018. Wisconsin Center, Milwaukee, USA, pp.

2425-2430.

Walker, K., Samadi, B., Huang, M., Gerhard, J., Butts, K., and Kolmanovsky, I.V., "Design environment for

Nonlinear Model Predictive Control," SAE Technical Paper 2016-01-0627, April, 2016, SAE World

Congress, doi:10.4271/2016-01-0627

Huang., M., Kolmanovsky, I.V., and Butts, K., ``A low complexity gain scheduling strategy for explicit

model predictive control of a diesel air path,’’ Proceedings of 2015 Dynamic Systems and Control

Conference, Columbus, Ohio, 2015, Paper DSCC2015-9654.

Huang, M., Nakada, H., Butts, K., and Kolmanovsky, I.V., “Nonlinear model predictive control of a diesel

engine air path: A comparison of constraint handling and computational strategies,” Proceedings of the 5th

IFAC Symposium on Nonlinear Model Predictive Control, Seville, Spain, September 2015, IFAC-

PapersOnLine, vol. 48, no. 23, 2015, pp. 372-379.


References (cont’d)

Huang, M., Nakada, H., Butts, K., Kolmanovsky, I.V., “Robust rate-based Model Predictive Control of

diesel engine air path,” Proceedings of 2014 American Control Conference, Portland, Oregon, pp. 1505-

1510, 2014.

Huang, M., Nakada, H., Polavarapu, S., Butts, K.R., and Kolmanovsky, I.V., “Rate-based Model Predictive

Control of diesel engines,” Proceedings of the 7th IFAC Symposium on Advances in Automotive Control,

September 4-7, 2013, Tokyo, Japan, pp. 177-182.

Huang, M., Nakada, H., Polavarapu, S., Choroszucha, R., Butts, K.R., and Kolmanovsky, I.V., “Towards

combining nonlinear and predictive control of diesel engines,” Proceedings of 2013 American Control

Conference, Washington DC, pp. 2852-2859.

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Model Predictive Diesel Engine Control - LIRMM