Hybrid Model Predictive Control

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SYSTEM bouncing_ball {
INTERFACE {
/* Description of variables and constants */
STATE { REAL height [-10,10];
REAL velocity [-100,100]; }
}
IMPLEMENTATION {
AUX {
PARAMETER {
REAL g;
REAL alpha; /* 0=elastic, 1=completely anelastic */
REAL Ts; }
REAL hnext;
REAL vnext;
BOOL negative; }
AD { negative = height CBD$&=&'(,83;+,+E&FG&5H53+&#IFF #] !"#
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>>Ts=0.05;
>>g=9.8;
>>alpha=0.3;
>>S=mld('bouncing_ball',Ts);
>>N=150;
>>U=zeros(N,0);
>>x0=[5 0]';
>>[X,T,D]=sim(S,x0,U);
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u 0
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x N = x r
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r = Cx r + D 1 u r + D 2 δ r + D 3 z r
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lim
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lim
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lim
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1./"1**"3&.,%01).%,"10+"C$*S**+/"1%"+13A"%)'+"t≥06
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Ū t+1
V ∗ (x(t)) −y(t) − r 2 Q −u − u r 2 R
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∗ (x(t))
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y(t) − r 2 Q , u(t) − u r 2 R
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4C`6&N+,)5-83+;&+N&46:
R-ba"Q&0'$*1%)&."&C"Ra4
!@17A$

012,3.&46:&(Q-)*5(
5>(0)/"Ra4"Z"781'2*+
ai#",>,%+'9
&:+;L8,-3;8b
\2+."*&&2"(+A1B)&0
:&"%&"/+'&""@demos/hybrid/bm99sim.m
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF PF !"#
012,3.&46:&(Q-)*5(
5>(0)/"Ra4"Z"781'2*+
5]I^7_
'&/+*
/* 2x2 PWA system - Example from the paper
A. Bemporad and M. Morari, ``Control of systems integrating logic, dynamics,
and constraints,'' Automatica, vol. 35, no. 3, pp. 407-427, 1999.
(C) 2003 by A. Bemporad, 2003 */
SYSTEM pwa {
INTERFACE {
STATE { REAL x1 [-10,10];
REAL x2 [-10,10];}
INPUT { REAL u [-1.1,1.1];}
OUTPUT{ REAL y;}
PARAMETER {
REAL alpha = 1.0472; /* 60 deg in radiants */
REAL C = cos(alpha);
REAL S = sin(alpha);}
}
IMPLEMENTATION {
AUX { REAL z1,z2;
BOOL sign; }
AD { sign = x1

012,3.&46:&(Q-)*5(
5>(0)/"Ra4"Z"781'2*+
4*&,+/Z*&&29
&6(,N+,)-;7(&3;.(Qb
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF PM!"#
012,3.&46:&e&J()*(,-8,(&7+;8,+5
>>refs.x=2; % just weight state #2
>>Q.x=1;
>>Q.rho=Inf; % hard constraints
>>Q.norm=2; % quadratic costs
>>N=2; % optimization horizon
>>limits.xmin=[25;-Inf];
>>C=hybcon(S,Q,N,limits,refs);
>> C
Hybrid controller based on MLD model S
2 state measurement(s)
0 output reference(s)
0 input reference(s)
1 state reference(s)
0 reference(s) on auxiliary continuous z-variables
20 optimization variable(s) (8 continuous, 12 binary)
46 mixed-integer linear inequalities
sampling time = 0.5, MILP solver = 'glpk'
Type "struct(C)" for more details.
>>
>>[XX,UU,DD,ZZ,TT]=sim(C,S,r,x0,Tstop);
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF PP!"#

012,3.&46:&e&J()*(,-8,(&7+;8,+5
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF P]!"#
$&>3)*5(&OQ-)*5(&3;&>**51&:K-3;&4-;-H()(;8
-1)+1$'+%&%(L
");&)'/%,(5
%&'1"0&%(5
-1)+1$'+%&%(M
");&)'/%,(7
-1)+1$'+%&%(K
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF PG!"#

1L8()&i-,3-25(L
!"3&.%).$&$,",%1%+,9
x ij WkX&k&-)+;8&+N&j&K+5.&3;&3;9(;8+,1&i
-8&83)( k&WikFE#E&jkFE#X
!"3&.%).$&$,"&$%2$%,9
y j WkX&k&-)+;8&+N&j&L+5.&-8&83)(&k
WjkFE#X
!"3&.%).$&$,").2$%,9
u ij WkX&k&-)+;8&+N&j&8-_(;&N,+)&3;9(;8+,1&i&
&&&&&&&&&&&-8&83)(&k WikFE#E&jkFE#X
!"().10>").2$%,9
U Xij WkX&k&F&3N&)-;N-78,(,&X&*,+.7(L&-;.&L(;.&j&8+&3;9(;8+,1&i&
&&&&&&&&&&&&-8&83)(&k
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF P"!"#
:+;L8,-3;8L
!"R18"31213)%>"&C").B+.%&0>"i9"
""
""P·"∑ j x ij WkX"·"x Mi
@)(,37-5&9-5(Lb&&
x MF kFIE&x M# kFI
!"R18"%01.,2&0%1%)&."C0&'").B+.%&0)+,9"
""
""P·"u ij WkX"·"u M
!"#"20&/$3%"31."&.*>"(+",+.%"%&"&.+").B+.%&0>9
l#NNEkG"1./"l#WNEkG"31..&%"(+"MN"1%"%A+",1'+"%)'+
lTNNEkG"1./"lTWNEkG"31..&%"(+"MN"1%"%A+",1'+"%)'+"lTNWEkG"1./"lTWWEkG"
31..&%"(+"MN"1%"%A+",1'+"%)'+
l4NWEkG"1./"l4WWEkG"31..&%"(+"MN"1%"%A+",1'+"%)'+
!"#"'1.$C13%$0+0"31."&.*>"20&/$3+"&.+"%>2+"&C"20&/$3%"1%"&.+"%)'+9
LlTNNEkGMN"&0"lTWNEkGMNO"1./"LlTNWEkGMN"&0"lTWWEkGMNO"31..&%"(+"%0$+
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF P[!"#

B1;-)37L
@)(,37-5&9-5(Lb
P AF kPE&P BF kGE&P B# k"E&P C# kM
!"_+B+*"&C").B+.%&0)+,9"
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF P\!"#
012,3.&B1;-)37-5&4+.(5
SYSTEM supply_chain{
INTERFACE {
STATE { REAL x11 [0,10];
REAL x12 [0,10];
REAL x21 [0,10];
REAL x22 [0,10]; }
INPUT { REAL u11 [0,10];
REAL u12 [0,10];
REAL u21 [0,10];
REAL u22 [0,10];
BOOL UA11,UA21,UB11,UB12,UB21,UB22,UC12,UC22; }
OUTPUT {REAL y1,y2;}
PARAMETER { REAL PA1,PB1,PB2,PC2,xM1,xM2;}
}
IMPLEMENTATION {
AUX { REAL zA11, zB11, zB12, zC12, zA21, zB21, zB22, zC22;}
DA { zA11 = {IF UA11 THEN PA1 ELSE 0};
zB11 = {IF UB11 THEN PB1 ELSE 0};
zB12 = {IF UB12 THEN PB2 ELSE 0};
zC12 = {IF UC12 THEN PC2 ELSE 0};
zA21 = {IF UA21 THEN PA1 ELSE 0};
zB21 = {IF UB21 THEN PB1 ELSE 0};
zB22 = {IF UB22 THEN PB2 ELSE 0};
zC22 = {IF UC22 THEN PC2 ELSE 0}; }
!demos/hybrid/supply_chain.m
} }
CONTINUOUS {x11 = x11 + zA11 + zB11 - u11;
x12 = x12 + zB12 + zC12 - u12;
x21 = x21 + zA21 + zB21 - u21;
x22 = x22 + zB22 + zC22 - u22; }
OUTPUT { y1 = u11 + u21;
y2 = u12 + u22; }
MUST { ~(UA11 & UA21);
~(UC12 & UC22);
~((UB11 | UB21) & (UB12 | UB22));
~(UB11 & UB21);
~(UB12 & UB22);
x11+x12 =0;
x21+x22 =0; }
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]I!"#

g2l(7839(L
! "R++%"3$,%&'+0"/+'1./"1,"'$3A"1,"2&,,)(*+9"""""y 5( $"r 5 ;"y 7( $"r 7
!"R).)')=+"%01.,2&0%1%)&."3&,%,
!"Q$*S**"1**"3&.,%01).%,
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]F !"#
6(,N+,)-;7(&>*(7L
penalty on demand tracking error
cost for shipping
from inv.#1 to
market
cost for
shipping
from inv.#2 to
market cost from A to
inventories
cost from B to
inventories
cost from C to
inventories
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]# !"#

3)5-83+;&L(8*
>>refs.y=[1 2]; % weights output2 #1,#2
>>Q.y=diag([10 10]); % output weights
…
>>Q.norm=Inf; % infinity norms
>>N=2;
% optimization horizon
>>limits.umin=umin; % constraints
>>limits.umax=umax;
>>limits.xmin=xmin;
>>limits.xmax=xmax;
>>C=hybcon(S,Q,N,limits,refs);
>> C
Hybrid controller based on MLD model S [Inf-norm]
4 state measurement(s)
2 output reference(s)
12 input reference(s)
0 state reference(s)
0 reference(s) on auxiliary continuous z-variables
44 optimization variable(s) (28 continuous, 16 binary)
176 mixed-integer linear inequalities
sampling time = 1, MILP solver = 'glpk'
Type "struct(C)" for more details.
>>
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]M !"#
>3)5-83+;&,(L58L
>>x0=[0;0;0;0];
>>r.y=[6+2*sin((0:Tstop-1)'/5)
5+3*cos((0:Tstop-1)'/3)];
% Initial condition
% Reference trajectories
>>[XX,UU,DD,ZZ,TT]=sim(C,S,r,x0,Tstop);
4al"%)'+9"$"No"',"2+0"%)'+",%+2"E$,).:"X_an"&."%A),"'13A).+G
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]P!"#

OQ*53738&K12,3.&46:
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]] !"#
OQ*53738&K12,3.&46:&W4`B&N+,)5-83+;X
min ξ J(ξ,x(t)) =
subject to
⎧
⎪⎨
⎪⎩
T
−1
k=0
Q(y k ) ∞ + Ru k ∞
x k+1 = Ax k + B 1 u k + B 2 δ k + B 3 z k + B 5
y k = Cx k + D 1 u k + D 2 δ k + D 3 z k + D 5
E 2 δ k + E 3 z k ≤ E 4 x k + E 1 u k + E 5
x 0 = x(t)
!"\.Z*).+"&2%)')=1%)&.9",&*B+"%A+"20&(*+'"C&0!1":)B+.",%1%+"xEtG
R)8+/Z-.%+:+0"_).+10"a0&:01'"ER-_aG
!"\[Z*).+"&2%)')=1%)&.9",&*B+"%A+"R-_a")."1/B1.3+"H9$'1;;"xEtG
min ξ
s.t.
T −1
k=0
x k + u k
Gξ ≤ W + Sx(t)
+8;)%NCBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]G!"#

OQ-)*5(&+N&)583*-,-)(8,37&L+583+;
R$*%)2101'+%0)3"_a
GI
PI
#I
x 2 I
=#I
CR { 1,4}
CR { 1,2,3 }
CR { 1,3 }
CR { 2,3 }
=PI
=GI
=GI =PI =#I I #I PI GI
x 1
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]" !"#
4583*-,-)(8,37&4C`6
min
ξ c ,ξ d
s.t.
f cξ c + f d ξ d
G c ξ c + G d ξ d ≤ W + Sx
!"'2ZR-_a"31."(+",&*B+/"E(>"1*%+0.1%).:"R-_a,"1./"'2Z_a,G
!KL'B)M.6&.N$A$L;$6B)*+++"
O'FA*9$*+0"KA+"'$*%)2101'+%0)3",&*$%)&.""""""""""""""),"2)+3+F),+"1J.+
!"FA*'@K4'29:)$9;;*$'%('CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ][!"#

OQ*53738&K12,3.&46:&W6^$&N+,)5-83+;X
min U J(U, x(t))
T
−1
= Q(y k ) ∞ + Ru k ∞
subject to
⎧ k=0
x ⎪⎨ k+1 = A i(k) x k + B i(k) u k + f i(k)
y k = C i(k) x k + D i(k) u k + g i(k)
⎪⎩
i(k) such that H i(k) x k + W i(k) u k ≤ K i(k)
x 0 = x(t)
!"FA*'@K4'29:)$9;;*$'%('".&%"(+"C$**>"2&*>A+/01*"E,++".+8%",*)/+G
h
j
W
N
f
R
o
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF ]\!"#
:+)*8-83+;&+N&(Q*53738&K12,3.&46:&W6^$X
R+%A&/"#9
R+%A&/"T9
W'+,,(553E&'-+837E&'()*+,-.E&4+,-,3E&>-(,3"('+"E&#II]X
mL(&-&7+)23;-83+;&+N&B6&W.1;-)37&*,+H,-))3;HX&-;.&)*`6
WF=;+,)E&%=;+,)XE&+,&)*h6&WS-.,-837&N+,)LX
W'()*+,-.E&%2/'.!@,,$2,8E&#IIMX&W$5(LL3+E&'()*+,-.E&$B0>&#IIGXW4-1;(E&O::&#IIFX
F&=&mL(&2-7_U-,.L&WkB6X&,(-7K-235381&-;-51L3L&N+,&(;)(,-83;H&-55&
&&&&N(-L325(&)+.(&L(S(;7(L&Ik{iWIXEiWFXEnEiWTX}o
#&=&Z+,&(-7K&pQ(.&L(S(;7(&IE&L+59(&8K(&(Q*53738&p;38(=83)(
&&&&&+*83)-5&7+;8,+5&*,+25()&N+,&8K(&7+,,(L*+;.3;H&53;(-,&
&&&&&83)(=9-,13;H&L1L8()&W)*h6&+,&)*`6Xo
M&=&:-L(&F!%=;+,)b&:+)*-,(&9-5(&N;783+;L&-;.&L*538&,(H3+;L%
""""b$1/01%)3"31,+9"".&%"(+"C$**>"
2&*>A+/01* T+%%+0"A+/01
$%&'()*+,-.
/012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF
!"#

012,3.&7+;8,+5&(Q-)*5(L&W,(93L38(.X
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF GF !"#
012,3.&7+;8,+5&(Q-)*5(
ai#",>,%+'9
4*&,+/"*&&29
"4&.,%01).%,9
"\(U+3%)B+9
\2+."*&&2
(+A1B)&09
5>(K(89""@demos/hybrid/bm99sim.m
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF G#!"#

OQ*53738&6^$&7+;8,+55(,
>(783+;&U38K& rkI
x #
5>(K(89""@demos/hybrid/bm99sim.m
3KNO('"-&C(5P25(#G(N&)'"+-(Q(5P9DIR6
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF GM!"#
x F
ai#"*1F"&"Ra4"*1F"e
5>(0)/"3&.%0&*"+81'2*+
012,3.&7+;8,+5&(Q-)*5(
4*&,+/"*&&29
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF GP!"#

OQ*53738&6^$&,(H5-8+,
"\(U+3%)B+9
6,(.3783+;&K+,3A+;&NkF
6,(.3783+;&K+,3A+;&Nk#
6,(.3783+;&K+,3A+;&NkM
5>(K(89""@demos/hybrid/bm99benchmark.m
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF G]!"#
OQ*53738&46:&e&J()*(,-8,(&7+;8,+5
>>E=expcon(C,range,options);
>> E
Explicit controller (based on hybrid controller C)
3 parameter(s)
1 input(s)
12 partition(s)
sampling time = 0.5
The controller is for hybrid systems (tracking)
This is a state-feedback controller.
Type "struct(E)" for more details.
>>
S&$'"/)(")('>&(3= 5 G= 7 6T#*1$&
/%(= %&( U(8<
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF GG!"#

OQ*53738&46:&e&J()*(,-8,(&7+;8,+5
X+.+01%+/"
4Z3&/+
utils/expcon.h
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF G"!"#
C)*5()(;8-83+;&-L*(78L&+N&K12,3.&46:
!".;)*$:1)%>*(9" ENG",&*B+"R-a"&.Z*).+
""""""""""""""""""" EWG"+B1*$1%+"1"ai#"C$.3%)&.
!"I'1**"20&(*+',"E,A&0%"A&0)=&."NMN;W;"&.+"&0"%F&").2$%,G9"+82*)3)%"ai#"3&.%0&*"*1F"
""""20+C+01(*+
Z"%)'+"%&"+B1*$1%+"%A+"3&.%0&*"*1F"),",A&0%+0"%A1."R-a
Z"3&.%0&*"3&/+"),",)'2*+0"E.&"3&'2*+8",&*B+0"'$,%"(+"
"").3*$/+/")."%A+"3&.%0&*",&C%F10+"eG
Z"'&0+").,):A%")."3&.%0&**+0p,"(+A1B)&0
!"R+/)$'@*10:+"20&(*+',"E*&.:+0"A&0)=&.;"'1.>").2$%,"1./"().10>"B10)1(*+,G9"
""""R-a"20+C+01(*+
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF G[!"#

012,3.&7+;8,+5&.(L3H;&q+U
2A>,)31*
,>,%+'
3&.%0&*"&(U+3%)B+,
m"3&.,%01).%,
A>(0)/
'&/+*
,)'$*1%)&.
1./"B1*)/1%)&.
&2%)'1*"3&.%0&*
,+%$2
3/)T0")&(QVN6
,)'$*1%)&.
1./"B1*)/1%)&.
-+0'"T
*1%1-&'%"$
#/0;&%
NHL($/)'%/0(01W
R1%*1("%&&*9
!"#$%&'F99;#9G
)'2*+'+.%1%)&."
E4"3&/+G"1./"
+82+0)'+.%,
>+,
.&
%&&"
3&'2*+8
c
$%&'()*+,-. /012,3.&4+.(5&6,(.37839(&:+;8,+57+5-&@-A3+;-5(&.3&B+88+,-8+&>CBD$&=&'(,83;+,+E&FG&5H53+&#IFF G\!"#
Y(;(,-5&,()-,_L&-2+8&4C6&)+.(53;H
KA+"3&'2*+8)%>"&C",&*B).:"1"')8+/Z).%+:+0"20&:01'
*10:+*>"/+2+./,"&."%A+".$'(+0"&C").%+:+0"E().10>G"
B10)1(*+,").B&*B+/")."%A+"20&(*+'6
5+.3+C&0%A;"FA+."30+1%).:"1"A>(0)/"'&/+*"&.+"A1,"%&
90)42'(74%):(42)("40;0')3!'(!&/0,)<
=##(";)/-;(+!/)+-",4'!("4,)6,6!//%)20/5,)>>>
X+.+01**>",2+1CBD$&=&'(,83;+,+E&FG&5H53+&#IFF "I!"#

'3253+H,-*K1
`+C+0+.3+,
P*Q))))3J)@17A$