Extending Affine Control Policies to Robust Control of ... · Linear Robust MPC Finite horizon,...

Extending Affine Control Policiesto Robust Control of Hybrid Systems

(DC-DC Buck Converter Example)

Robin Vujanic, Marius Schmitt, Joe WarringtonManfred Morari

Institut fur Automatik (IfA)Department of Electrical Engineering

Swiss Federal Institute of Technology (ETHZ)

May 22, 2015

http://control.ee.ethz.ch/∼ vujanicr

Outline

Introduction

Robust MPC for Linear Systems

Extension to Hybrid Systems

Example: DC-DC Buck ConverterLinear ModelHybrid Model

Conclusions


Introduction

In this talk:

1. (Review of) MPC-based robust controllers for linear systems

2. New robust controller for some hybrid models

3. Example: DC-DC buck converter


Outline

Introduction




Conclusions


Linear Robust MPCFinite horizon, optimal control problem formulation

min

[N−1∑k=0

(xk − xref )TP(xk − xref ) + uTk Quk

]s.t. x0 = x0,

xk+1 = Axk + Bukxk ∈ Xk ,uk ∈ Uk

I ideally, obtain optimal control policy using e.g. DPI ”optimal decision is taken at each stage in the horizon”I uk = π(xk)I often intractable

I conservative approximation: open-loop control policyI ”decide now for the entire horizon” (plan can’t be modifed)I uk = vkI poor performance, infeasibility issues

I middle ground: affine recourseI ”decisions are affinely adjusted once disturb. are meas.”I http://control.ee.ethz.ch/∼ vujanicr

Linear Robust MPCFinite horizon, robust optimal control problem formulation

min E

[N−1∑k=0

(xk − xref )TP(xk − xref ) + uTk Quk

]s.t. x0 = x0,

xk+1 = Axk + Buk + Gwk ,xk ∈ Xk ,uk ∈ Uk

∀w ∈ WI ideally, obtain optimal control policy using e.g. DP

I ”optimal decision is taken at each stage in the horizon”I uk = π(xk)I often intractable


I middle ground: affine recourseI ”decisions are affinely adjusted once disturb. are meas.”I http://control.ee.ethz.ch/∼ vujanicr

Linear Robust MPC

Finite horizon, optimal control problem formulation

min E[(x− xref)TP(x− xref) + uTQu

]s.t.

x = Ax0 + Bu + Gw,Exx + Euu ≤ e

}∀w ∈ W



I middle ground: affine recourseI ”decisions are affinely adjusted once disturb. are meas.”I


Linear Robust MPC



]s.t.


}∀w ∈ W



I middle ground: affine recourseI ”decisions are affinely adjusted once disturb. are meas.”I uk = vk +

∑kj=0 Kkjxj


Linear Robust MPC



]s.t.


}∀w ∈ W



I middle ground: affine recourseI ”decisions are affinely adjusted once disturb. are meas.”I uk = vk +

∑kj=0 Mkjwj


Explicit Robust Counterpart

I can show that the robust counterpart is equivalent to

minv,M,Λ E[(x− xref)

TP(x− xref) + uTQu]

s.t. ExAx0 + (ExB + Eu)v − e ≤ −ΛT · hΛT · S = ExG + ExBM + EuMΛ ≥ 0

(R-MPCavg )where wk is bounded by S · wk ≤ h (Goulart 2006)

I finite convex QP


Outline

Introduction




Conclusions


Robust Control of Hybrid Systems

I hybrid system dynamics (MLD)

xk+1 = Axk + Buk + B2δk + B3zk + Gwk

Exxk + Euuk + Eδδk + Ezzk ≤ ekδk ∈ {0, 1}nδ , zk ∈ Rnz

I δk , zk characterize hybrid behavior,I in the dynamics, e.g. switching between modesI in the constr., e.g. logic conditions on the inputs

I wish to obtain a solution (with some recourse) to

minx,u,δ,z

E[(x− xref)TP(x− xref) + uTQu

]s.t. x = Ax0 + Bu + B2δδδ + B3z + Gw,

Exx + Euu + Eδδδδδδ + Ezz ≤ e , ∀w ∈ Wδδδ ∈ {0, 1}N·nδ

(RHOCP)http://control.ee.ethz.ch/∼ vujanicr

Affine recourse on continuous inputs - R-MPChybMain Idea

Proposed idea:

I we need recourse, but affine functions cannot easily providebinary inputs

I split the inputs into continuous inputs u and binary inputs d

xk+1 = A · xk + Bcontuk + Bbindk + B2 · δk + B3zk + Gwk

I introduce affine recourse on the continuous inputs

u := M ·w + v

Assumption on G :

I disturbances only affect the continuous dynamics


Affine recourse on continuous inputs - R-MPChyb

I can show that the robust counterpart is

minv,M,d,δδδ,z,Λ

f + trace(D · Cw)

s.t. e ≤ −ΛTh ,ΛTS = ExBuM + ExG + EuMΛ ≥ 0 element-wiseδδδ ∈ {0, 1}N·nδ , d ∈ {0, 1}N·nd

(R-MPChyb)

I withx.

= Ax0 + Buv + Bdd + B2δδδ + B3ze.

= Exx + Euv + Edd + Eδδδδδδ + Ezz− e

(variables in case of zero disturbance)

I and appropriate D


Outline

Introduction




Conclusions


Outline

Introduction




Conclusions


Plant: the Buck Converter (BC) (1/2)

RL

RC

id(t)

L

C

RoutVin

iL(t)

VC(t)Vo(t)

Figure: DC-DC buck converter circuit

I BC regulates input voltage Vin down to desired Vo,ref

I operated by switch (controlled input)

I disturbances: |id | ≤ 0.5 · iL,ref

I state constraints: iL ≤ 2 · iL,ref

I sampling frequency: 10 kHz


Averaged Model (standard method)

00.20.40.60.8

1

binary inputcontrol signal

(A)

AVER

AGED

cycle

I replace δ(t) ∈ {0, 1} by duty cycle d(t) ∈ [0, 1]

I average dynamics ”when off” and ”when on” weighted byd(t), obtain

x = Ax + Bu + Gwu ∈ [0, 1]

I linear model


Performance of R-MPCavg

-10123

iL /iL*VC/VC

*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

control signalbinary input

time (ms)


R-MPCavg – Second Experiment

0 1 2 3 4 5 60.8

1

1.2

1.4

1.6

1.8

time (ms)

VC,ref constr.VC/33V (avg)


Outline

Introduction




Conclusions


Hybrid Model [2] (1/2)

I Closer approximationof the switch

I Divide time into cycleswith M samples each

I New binary inputs:switch on δ+

k = 1,switch off δ−k = 1

I New continuous input:uc,k

I New auxiliary state: sk(integration of δ+ − δ−)

I switch position given bysk + uc,k

00.20.40.60.8

1

binary inputcontrol signal

00.20.40.60.8

1

time

(A)

AVER

AGED

(B)

HYBRID

cycle

Figure: PWM in the hybrid model


Hybrid Model [2] (2/2)

I New system equations:(iLVCs

)k+1

=(

A11 A12 B1A21 A22 B2

0 0 1

)·(

iLVCs

)k

+(

B1 0 0B2 0 00 1 −1

)·(

ucδ+δ−

)k

+(

G1G20

)wk

I New constraints:

δ+k , δ

−k ∈ {0, 1} binary inputs

0 ≤ sk ≤ 1 binary state0 ≤ sk + uc,k ≤ 1 limited input−δ−k ≤ uc,k ≤ δ+

k switching time∑M−1i=0 δ+

k+i ≤ 1∑M−1i=0 δ−k+i ≤ 1

}switching constraints

I together with iL,k ≤ 2 · iL,ref they form

xk+1 = A · xk + Bcontuk + Bbindk + B2 · δk + B3zk + Gwk

Exxk + Euuk + Eddk + Eδδk + Ezzk ≤ ek


Performance & Simulation results – R-MPChyb

-1

0

1

2

3 iL/iL*VC/VC

*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

time (ms)

control signal binary inputrange of recourse


R-MPChyb – Second Experiment

0 1 2 3 4 5 60.8

1

1.2

1.4

1.6

1.8

time (ms)

VC/33V VC,ref constr.(hyb) VC/33V (avg)

Controller RMS deviation of Vo(t) RMS deviation of Vo(t)

1st experiment 2nd experiment

MPCavg 2.09V 15.2[V ]

MPChyb 0.88V 9.74[V ]

MPCopen loophyb 0.89V Infeasibility encountered


Outline

Introduction




Conclusions


Conclusions

I New robust controller R-MPChyb for hybrid systemsI Assessment on the BC

I good performanceI no infeasibility issues

I Future work:I faster computation of solutions,

e.g. using exclusively convex programming (DONE)I other applications?

e.g. dynamic system at the output stage


Conclusions

I Questions?I References:

1. Paul J. Goulart, Eric C. Kerrigan and Jan M. Maciejowski.Optimization over state feedback policies for robust control withconstraints. 2006, Automatica 42 (523 - 533)

2. Claudia Fischer, Sebastian Mariethoz and Manfred Morari.Multisampled Hybrid Model Predictive Control for Pulse-WidthModulated Systems. 50th Conference on Decision and Control andEuropean Control Conference (CDC-ECC), Dec 2011, Orlando, FL,USA

3. Paul J. Goulart and Eric C. Kerrigan. Input-to-state stability ofrobust receding horizon control with an expected value cost. 2008,Autimatica 22 (1171 - 1174)

4. Aharon Ben-Tal, Laurent El Ghaoui and Arkadi Nemirovski. RobustOptimization. 2009, Princeton Series in Applied Mathematics

I Slides of the talk on http://control.ee.ethz.ch/∼ vujanicr/I under ”publications → talks”


Derivation of MPCol/cl

Robust constraints satisfaction

Exx + Euu + Edd + Eδδ ≤ e ∀w ∈ W⇔ e + (ExBuM + ExG + EuM)w ≤ 0, ∀w ∈ W⇔ e + max

w∈W(ExBuM + ExG + EuM)w ≤ 0

withx.

= Ax0 + Buv + Bdd + B2δδδ + B3ze.

= Exx + Euv + Edd + Eδδδδδδ + Ezz− e.

Then

maxw∈W

(ExBuM + ExG + EuM)w s.t. Sw ≤ h

= minΛ≥0

ΛTh s.t. ΛTS = ExBuM + ExG + EuM

hence robust state constraint satisfaction equivalent to

e ≤ −ΛTh ,ΛTS = ExBuM + ExG + EuM ,Λ ≥ 0 element-wise.


Derivation of Objective (Hybrid, OL case)

The output voltage is a linear function of the state xk and thedisturbances wk

Vo =RCRout

RC + RoutiL +

Rout

RC + RoutVC +

RCRout

RC + Routid (1)

Assuming the disturbance is zero mean and iid.

minx,u,δ f (x,u) == minu,δ E [(Ax0 + Bu + B2δ + B3z + Gw − xref)TP...(Ax0 + Bu + B2δ + B3z + Gw − xref) + uTQu]= minu,δ(Ax0 + Bu + B2δ − xref)TP...(Ax0 + Bu + B2δ − xref) + uTQu...

+2 · E [(Ax0 + Bu + B2δ − xref)TPGw]︸︷︷︸=0

+ E [wTGTPGw]

= minu,δ fnom(x,u) + const.


Alternative approach # 1: ignoring disturbances

0 0.5 1 1.5 2 2.5 3

x 10−3

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

time/s

iL/iL*

uC/uC*

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time/s

binary inputcomputed dutycycle


Alternative approach # 2: soft constraints

0 0.5 1 1.5 2 2.5 3

x 10−3

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

time/s

iL/iL*

uC/uC*

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time/s

binary inputcomputed dutycycle


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