EXPLICIT PREDICTIVE CONTROL LAWS

On the Geometry of Feasible Domains and the Presence of Nonlinearities

Sorin Olaru, Didier Dumur

Automatic Control Department, Sup

elec, 3 rue Joliot Curie, Gif-sur-Yvette, France

Simona Dobre

CRAN, Nancy - Universit

e, CNRS UMR 7039, BP 239, F-54506 Vandœuvre-l

es-Nancy Cedex, France

Keywords:

Predictive control, parameterized polyhedra, explicit control laws.

Abstract:

This paper proposes a geometrical analysis of the polyhedral feasible domains for the predictive control laws

under constraints. The fact that the system dynamics inﬂuence the topology of such polyhedral domains is

well known from the studies dedicated to the feasibility of the control laws. Formally the system state acts as a

vector of parameters for the optimization problem to be solved on-line and its inﬂuence can be fully described

by the use of parameterized polyhedra and their dual constraints/generators representation.

Problems like the constraints redundancy or the construction of the associated explicit control laws at least for

linear or quadratic cost functions can thus receive fully geometrical solutions. Convex nonlinear constraints

can be approximated using a description based on the parameterized vertices. In the case of nonconvex re-

gions the explicit solutions can be obtain by constructing Voronoi partitions based on a collection of points

distributed over the borders of the feasible domain.

1 INTRODUCTION

The philosophy behind Model-based Predictive Con-

trol (MPC) is to exploit in a ”receding horizon” man-

ner the simplicity of the Euler-Lagrange approach

for the optimal control. The control action u

for a

given state x

is obtained from the control sequence

∗

= [u

,...,u

t+N−1

]

as a result of the optimization

problem:

min

ϕ(x

t+N

) +

N−1

∑

k=0

l(x

t+k

)

subj.to : x

t+1

= f(x

) + g(x

;

h(x

) ≤ 0

(1)

constructed for a ﬁnite prediction horizon N, cost per

stage l(.), terminal weight ϕ(.), the system dynam-

ics described by f(.), g(.) and the constraints written

in a compact form using elementwise inequalities on

functions linking the states and the control actions,

h(.).

Unfortunately, the control sequence k

∗

is optimal

only for a single initial condition - x

and produces

an open-loop trajectory which contrasts with the need

for a feedback control law. This drawback is over-

come by solving the local optimization (1) for ev-

ery encountered (measured) state, thus indirectly pro-

ducing a state feedback law. The overall method-

ology is based on computationally tractable optimal

control problems for the states found along the cur-

rent trajectory. However, two important directions

are to be studied in order to enlarge the class of sys-

tems which can take advantage of the MPC method-

ology. One is related to the fact that the measure-

ments can be available faster than the optimal con-

trol sequence becomes available (as output of the op-

timization solver) and thus important information can

be lost with irreversible consequences on the close-

loop performances. Secondly, the lack of a closed

form expression for the feedback law notiﬁes about

the difﬁculties that can be encountered when consid-

ering properties such as stability, typically established

for regions in the state space.

For the optimization problem (1) within MPC, the

current state serves as an initial condition and in-

ﬂuences both the objective function and the feasible

domain. Globally, from the optimization point of

view, the system state can be interpreted as a vec-

tor of parameters, and the problems to be solved

Olaru S., Dumur D. and Dobre S. (2007).

EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 70-77

DOI: 10.5220/0001633200700077

 SciTePress

are part of the multiparametric optimization program-

ming family. From the cost function point of view,

the parametrization is somehow easier to deal with

and eventually can be entirely translated towards the

set of constraints to be satisﬁed (the MPC literature

contains references to schemes based on suboptimal-

ity or even to algorithms restraining the demands to

feasible solution of the receding horizon optimization

(Scokaert et al., 1999)). Unfortunately, similar obser-

vation cannot be made about the feasible domain and

its adjustment with respect to the parameters evolu-

tion.

The optimal solution is often inﬂuenced by the

limitations, the process being forced to operate at the

designed constraints for best performance. The dis-

tortion of the feasible domain during the parameters

evolution will consequently affect the structure of the

optimal solution. Starting from this observation the

present paper focuses on the topological analysis of

the domains described by the MPC constraints.

The structure of the feasible domain is depending

on the model and the set of constraints taken into con-

sideration in (1). If the model is a linear system, the

presence of linear constraints on inputs and states can

be easily expressed by a system of linear inequalities.

In the case of nonlinear systems, these properties are

lost and the domains are in general difﬁcult to han-

dle. However, there are several approaches to trans-

form the dynamics to those of a linear system over

the operating range as for example by piecewise lin-

ear approximation, feedback linearisation or the use

of time-varying linear models.

As a consequence, speciﬁc attention for the lin-

ear constraints and the associated polyhedral feasi-

ble domains may be proliﬁc. More than that, the

use of polyhedral domains between the convex sets is

not hazardous since they offer important advantages,

like the closeness over the intersection or the fact that

the polyhedral invariant sets (largely used for enforc-

ing stability) are less conservative than the ellipsoidal

ones for example. In the current paper, these polyhe-

dral feasible domains will be analyzed with a focus on

the parametrization leading to the concept of parame-

terized polyhedra (Olaru and Dumur, ):

min

F(x

)

subj.to :







≤ b

+ B

= b

+ B

h(x

) ≤ 0

(2)

where the objective function F(x

) is usually lin-

ear or quadratic.

Secondly it will be shown that the optimization

problem may take advantage during the real-time im-

plementation either from the possible alleviation of

the set of constraints for the on-line optimization rou-

tines either from the construction of the explicit solu-

tion on geometrical basis when possible. With these

two aspects, one can consider that MPC awareness

is improved both from the theoretical (insight on the

global control law) and practical (computational as-

pects) point of view.

An important remark concerning the presence of

nonlinearities in the constraints is that if the feasi-

ble domain remains convex, then an approximation in

terms of parameterized polyhedra can lead to an ap-

proximate explicit solution in terms of piecewise lin-

ear control laws. But, if the feasible domain becomes

non-convex due to the presence of nonlinearities, then

in order to obtain the explicit solution some assump-

tion have to be relaxed. A special role, in ﬁnding the

explicit control laws in the nonlinear case, is played

by the Voronoi partition.

In the following, Section 2 introduces the basic

concepts related to the parameterized polyhedra and

interprets the feasible domains of the receding hori-

zon optimization problems (2) in this context. Sec-

tion 3 presents the use of the feasible domain analysis

for the construction of the explicit solution for linear

and quadratic objective functions. In Section 4 an ex-

tension to nonlinear type of constraints is addressed,

simple examples illustrating the construction of ex-

plicit solutions.

2 PARAMETRIZATION OF

POLYHEDRAL DOMAINS

2.1 Double Representation

A mixed system of linear equalities and inequalities

deﬁnes a polyhedron (Motzkin and R.M., 1953). In

the parameter free case, it is represented by the equiv-

alent dual (Minkowski) formulation:

P =



∈ R



= b

≤ b



⇐⇒

P = conv.hullV+ coneR+ lin.spaceL

{z }

generators

(3)

where conv.hullV denotes the set of convex combi-

nations of vertices V =

{

,...,v

}

, coneR denotes

nonnegative combinations of unidirectional rays

in R =



,...,r



and lin.spaceL =

{

,...,l

}

represents a linear combination of bidirectional rays

(with ϑ, ρ and λ the cardinals of the related sets).

This dual representation (Schrijver, 1986) in terms of

generators can be rewritten as:

EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities

P =



∈ R

∑

i=1

∑

i=1

∑

i=1

;

0 ≤ α

≤ 1,

∑

i=1

= 1 , β

≥ 0 , ∀γ



(4)

with α

, β

, γ

the coefﬁcients describing the convex,

non-negative and linear combinations in (3).

Numerical methods like the Chernikova algorithm

(Leverge, 1994) are implemented for constructing the

double description, either starting from constraints (3)

either from the generators (4) representation.

2.2 The Parametrization

A parameterized polyhedron (Loechner and Wilde,

1997) is deﬁned in the implicit form by a ﬁnite num-

ber of inequalities and equalities with the note that the

afﬁne part depends linearly on a vector of parameters

x ∈R

for both equalities and inequalities:

P (x) =



(x) ∈ R



= B

x+ b

;

≤ B

x+ b

}



(x)

(x) =

∑

i=1

(x)v

(x)

∑

i=1

∑

i=1



0 ≤ α

(x) ≤ 1,

∑

i=1

(x) = 1 , β

≥ 0 , ∀γ

(5)

This dual representation of the parameterized poly-

hedral domain reveals the fact that only the vertices

are concerned by the parametrization (resulting the

so-called parameterized vertices - v

(x)), whereas the

rays and the lines do not change with the parame-

ters’ variation. In order to effectively use the gener-

ators representation in (5), several aspects have to be

clariﬁed regarding the parametrization of the vertices

(see for exemple (Loechner and Wilde, 1997) and the

geometrical toolboxes like POLYLIB (Wilde, 1993)).

The basic idea is to identify the parameterized polyhe-

dron with a non-parameterized one in an augmented

space:

P =





∈ R

p+n





−B







= b

;

−B

]





≤ b



(6)

The original polyhedron in (5) can be found for

any particular value of the parameters vector x

through P(x) = Proj



P∩H(x)



, for any given hy-

perplane H(x

) =





∈ R

p+n

x = x



and

using Proj

(.) as the projection from R

p+n

to the ﬁrst

p coordinates R

Within the polyhedral domains

P , the correspon-

dent of the parameterized vertices in (5) can be found

among the faces of dimension n. After enumerating

these n-faces:

(

P ),...F

(

P ),...,F

(

P )

, one

can write: ∀i, ∃j ∈

{

1,...,ς

}

s.t.



(x)



∈

(

P ) or equivalently:

(x) = Proj



(

P ) ∩H(x)



(7)

From this relation it can be seen that not all the n-

faces correspond to parameterized vertices. However

it is still easy to identify those which can be ignored

in the process of construction of parameterized ver-

tices based on the relation: Proj



(



< n with

Proj

(.) the projection from R

p+n

to the last n coor-

dinates R

(corresponding to the parameters’ space).

Indeed the projections are to be computed for all the

n-faces, those which are degenerated are to be dis-

carded and all the others are stored as validity do-

mains - D

∈ R

, for the parameterized vertices that

they are identifying:

= Proj



(



(8)

Once the parameterized vertices identiﬁed and their

validity domain stored, the dependence on the param-

eters vector can be found using the supporting hyper-

planes for each n-face:

(x) =





−1









(9)

where

represent the subset of the in-

equalities, satisﬁed by saturation for F

(

P). The in-

version is well deﬁned as long as the faces with de-

generate projections are discarded.

2.3 The Interpretation from the

Predictive Control Point of View

The double representation of the parameterized poly-

hedra offers a complete description of the feasible do-

main for the predictive control law as long as this is

based on a multiparametric optimization with linear

constraints.

Using the generators representation, with simple

difference operations on convex sets one can compute

the region of the parameters space where no parame-

terized vertex is deﬁned:

ℵ = R

{

∪D

; i = 1...ϑ

}

(10)

representing from the MPC point of view, the set of

infeasible states for which no control sequence can be

designed due to the fact that the limitations are overly

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

constraining. As a consequence the complete descrip-

tion of the infeasibility is obtained.

Remark: the presence of rays and lines in the set

of generators doesn’t imply that the infeasibility is

avoided. The feasibility is strictly related with the ex-

istence of valid parameterized vertices for the given

value of the parameter (state) vector.

The vertices of the feasible domain cannot be

expressed as convex combinations of other distinct

points and, due to the fact that from the MPC point

of view, they represent sequences of control actions,

one can interpret them in terms of extremal perfor-

mances of the controlled system (for example in the

tracking applications the maximal/minimal admissi-

ble setpoint (Olaru and Dumur, 2005)).

3 TOWARDS EXPLICIT

SOLUTIONS

In the case of sufﬁciently large memory resources,

construction of the explicit solution for the multipara-

metric optimization problem can be an interesting al-

ternative to the iterative optimization routines. In this

direction recent results where presented at least for

the case of linear and quadratic cost functions (see

(Seron et al., 2003),(Bemporad et al., 2002),(Good-

win et al., 2004),(Borelli, 2003),(Tondel et al., 2003)).

In the following it will be shown that a geometrical

approach based on the parameterized polyhedra can

bring a useful insight as well.

3.1 Linear Cost Function

The linear cost functions are extensively used in con-

nection with model based predictive control and espe-

cially for robust case ((Bemporad et al., 2001), (Ker-

rigan and Maciejowski, 2004)). In a compact form,

the multiparametric optimization problem is:

∗

) = min

subject to A

≤ B

+ b

(11)

The problem deals with a polyhedral feasible do-

main which can be described as previously in a dou-

ble representation. Further the explicit solution can

be constructed based on the relation between the pa-

rameterized vertices and the linear cost function (as in

(Leverge, 1994)). The next result resumes this idea.

Proposition: The solution for a multiparametric

linear problem is characterized as follows:

a) For the subdomain ℵ ∈R

where the associated

parameterized polyhedron has no valid parameterized

vertex the problem is infeasible;

b) If there exists a bidirectional ray l such that

l 6= 0 or a unidirectional ray r such that f

r ≤ 0,

then the minimum is unbounded;

c) If all bidirectional rays l are such that f

l = 0

and all unidirectional rays r are such that f

r ≥ 0

then there exists a cutting of the parameters in zones

where the parameterized polyhedron has a regular

shape

j=1...ρ

= R

−ℵ. For each region R

the

minimum is computed with respect to the given lin-

ear cost function and for all the valid parameterized

vertices:

(x) = min



(x)|v

(x) vertex of P (x)



(12)

The minimum m

(x) is attained by constant subsets

of parameterized vertices of

P (x) over a ﬁnite num-

ber of polyhedral zones in the parameters space R

(∪R

= R

). The complete optimal solution of the

multiparametric optimization is given for each R

by:

(x) = conv.hull

{

∗

(x),...,v

∗

(x)

}

+ cone

{

∗

,...,r

∗

}

+ lin.spaceP(p)

(13)

where v

∗

are the vertices corresponding to the mini-

mum m

(x) over R

and r

∗

are such that f

∗

= 0



This result provides the entire family of solutions

for the linear multiparametric optimization, even for

the cases where this family is not ﬁnite (for example

there are several vertices attaining the minimum).

Remark: For the regions of the parameters space

characterized by the case (a), the set of constraints

cannot be fulﬁlled and the feasible domain is empty.

Remark: If the solution of the optimization prob-

lem is characterized by the case (b), then the control

law based on such an optimization is not well-posed

as the optimal control action needs an inﬁnite energy

in order to be effectively applied.

Remark: Due to the fact that the parameterized

vertices have a linear dependence on the parameter

vector, the explicit solution will be piecewise lin-

ear. However, the solution is not unique as it can be

seen from the case (c) and equation (13) and thus for

the practical control purposes a continuous piecewise

candidate is preferred, eventually by minimizing the

number of partitions in the parameters space.

3.2 Quadratic Cost Function

The case of a quadratic const function is one of the

most popular at least for the linear MPC. The ex-

plicit solution based on the exploration of the param-

eters space ((Bemporad et al., 2002), (Borelli, 2003),

(Tondel et al., 2003)) is extensively studied lately.

Alternative methods based on geometrical arguments

or dynamical programming ((Goodwin et al., 2004),

EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities

(Seron et al., 2003)) improved also the awareness

of the explicit MPC formulations. The parameter-

ized polyhedra can serve as a base in the construction

of such explicit solution (Olaru and Dumur, ), for a

quadratic multiparametric problem:

∗

) = argmin

+ 2k

subject to A

≤ B

+ b

(14)

In this case the main idea is to consider the uncon-

strained optimum:

) = H

−1

and its position with respect to the feasible domain

given by a parameterized polyhedron as in (5).

If a simple transformation is performed:

= H

1/2

then the isocost curves of the quadratic function are

transformed from ellipsoid into circles centered in

) = H

−1/2

. Further one can use the Eu-

clidean projection in order to retrieve the multipara-

metric quadratic explicit solution.

Indeed if the unconstrained optimum

) is

contained in the feasible domain

P (x

) then it is also

the solution of the constrained case, otherwise exis-

tence and uniqueness are assured as follows:

Proposition: For any exterior point

) /∈

P (x

), there exists an unique point characterized by

a minimal distance with respect to

). This point

satisﬁes:

(

) −

∗

))

(

−

∗

)) 6 0,∀

∈

P (x

)



The construction mechanism uses the parameter-

ized vertices in order to split the regions neighbor-

ing the feasible domain in zones characterized by the

same type of projection.

Remark: The use of these geometrical arguments

makes the construction of explicit solution to deal in

a natural manner with the so-called degeneracy (Be-

mporad et al., 2002). This phenomenon is identiﬁed

by the parameters’ values where the feasible domain

changes its shape (the set of parameterized vertices is

modiﬁed).

4 GENERALIZATION FOR

NONLINEAR PROGRAMS

If the feasible domain is described by a mixed lin-

ear/nonlinear set of constraints then the convexity

properties are lost and a procedure for the construc-

tion of exact explicit solutions do not exist for the

general case.

4.1 Nonlinear Constraints Handling

As already mentioned, in dealing with the presence

of nonlinearities constraints, a special case is when

the associated feasible domain is convex. In this case,

the main ideas in ﬁnding the explicit control laws are

the following:

- considering the augmented space (formed by the

extended arguments and parameters space ((x,k

)),

ﬁnd a set of points situated on the borders of the fea-

sible domain (points that will correspond to the pa-

rameterized vertices in the associated linear feasible

domain);

- using this set of extremal points, construct the

dual representation in terms of parameterized polyhe-

dra (as a fact, in the presence of linear constraints, this

set of points could represent the input to the linear al-

gorithm, as well as a set of linear constraints);

- build the corresponding explicit solution by re-

porting the unconstrained optimum to these parame-

terized vertices and their validity domains.

Remark: The solution will be a continuous piece-

wise afﬁne function in the state vector, due to the na-

ture (convexity) of the feasible domain, and it is ob-

tained by projecting the unconstrained optimum on

the linear subset of constraints (associated with the

nonlinear ones). In this nonlinear convex case, the

precision of the solution is directly dependent of the

linearization of the nonlinear constraints using a ﬁ-

nite set of points on the frontier of the feasible do-

main (knowing that the rest of the algorithm retains

the qualities of the linear algorithm).

4.1.1 Simple Example of a Multiparametric

Nonlinear Program

Consider the discrete-time linear system:

t+1



0.9 1

0 1





−1



(15)

and a predictive control law with a prediction hori-

zon of three sampling times and a control horizon of

two steps. A nonlinear set of constraints will be also

considered:







∑

k=0

t+k

≤ 1

∑

k=0

t+k

≤ ln(



0 1



+ 1)



0 1



t+k

≥ 0;k = 0,1,2

(16)

It is obvious that the topology of the feasible do-

main is changing with the system dynamics, which

means that the state vector represents in fact a pa-

rameter. More precisely, in our case only the second

component of the state, x

is inﬂuencing the shape of

the feasible domain and thus one can draw this de-

pendence on the parameter as in ﬁgure 1a. Further

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

this parameterized convex shape can be approximated

with a set of parameterized linear inequalities and ob-

tain a double description of a parameterized polyhe-

dron as in ﬁgure 1b. A precutting in zones with reg-

ular shape (ﬁgure 1c) can help in the development of

explicit solution due to the important degree of redun-

dancy.

(a)

−1

−0.5

0.5

−1

−0.5

0.5

1.5

2.5

(b)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0.5

1.5

2.5

(c)

Figure 1: (a) The nonlinear dependence of the feasible do-

main on the parameters (b) The approximation by a param-

eterized polyhedron; (c) Regions in the parameters’ space

corresponding to redundancy-free constraints sets.

Finally the nonlinear MPC law for the system (15)

and the constraints (16) can be approximated by the

explicit solution found in terms of a piecewise linear

control law as in ﬁgure 2.

−6

−4

−2

−0.6

−0.4

−0.2

0.2

0.4

PWA function over 114 regions

Figure 2: Explicit solution as a piecewise linear function.

4.2 Nonconvex Feasible Domains

In the case when the convexity is lost, one can still

construct the convex hull (and an explicit solution)

associated with the non-convex domain. From the

resulting piecewise afﬁne function one should retain

only those regions where the control law is feasible

w.r.t. the nonlinear constraints. For the violated con-

straints a distribution of points will be obtained. Fi-

nally the missing regions in the explicit solution will

be completed with the Voronoi partition correspond-

ing to these points on the concave border of the feasi-

ble domain.

Algorithm:

1. Obtain a set of points (

V ) on the frontier of the

feasible domain D (based on the nonlinear con-

straints).

2. Considering this set of points, construct the con-

vex

(which will include the non-convex do-

main D and some other infeasible domains from

the MPC point of view).

3. Split the set

V as V

∪

V ∪

•

V ∈ Int(C

)

•

V ∈ F(C

) and

V \

•

∈ F(

∩

V =

0 and V

saturate at

least one linear constraint in the MPC con-

straints;

•

∈ F(

) and

saturate no linear con-

straint

is described in the dual representation by the

intersection of halfspaces

H (which represent the

faces of the convex hull C

). Split this set in

H ∪

•

H ⊂H such that ∃x ∈ C

with Sat(

H , x) 6=

and B(R

,x) 6=

•

H = H \

5. Compute the unconstrained optimum k

∗

6. Project the unconstrained optimum on

∗

← Proj

{

−c

}

7. If k

∗

saturates a subset of constraints

K ⊂

(a) Retain the set of points:

S =

v ∈

V |∀x ∈C

s.t.Sat(

H , x) = K ;

B(R

,x) = Sat(R

,v)

}

(b) Construct the Voronoi partition for the colec-

tion of points in S

∗

w.r.t. this partition and map the sub-

optimal solution k

∗

← v where v is the vertex

corresponding to the active region

EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities

8. If the quality of the solution is not satisfactory,

improve the distribution of the points

V by aug-

menting the resolution around k

∗

and restart from

the step 2.

The following notations where used:

F(X) The frontier of a compact set X

Int(X) The interior of a compact set X

(D) The set of linear constraints in the deﬁ-

nition of the feasible domain D

(D) The set of nonlinear constraints in the

deﬁnition of the feasible domain D

Sat(R

∗

,x) The subset of constraints in R

∗

(ei-

ther R

either R

) saturated by the vector x

B(R

∗

,x) The subset of constraints in R

∗

. vio-

lated by the vector x

4.2.1 Numerical Example

Consider the MPC problem implemented using the

ﬁrst control action of the optimal sequence:

∗

= argmin

−1

∑

i=0

t+k

+ u

t+k

t+N

(17)

with

Q =



10 0

0 1



;R =



2 0

0 3



;P =



13.73 2.46

2.46 2.99



subject to











t+k+1

}| {



1 1

0 1



t+k

}| {



1 0

2 1



t+k

k > 0



−2



6 u

t+k





0 6 k 6 N

−1

t+k

)



t+k

−2



√

3 0 6 k 6 N

−1

t+k

)



t+k

+ 2



√

3 0 6 k 6 N

−1

t+k



0.59 0.76

- 0.42 - 0.16



| {z }

LQR

t+k

6 k 6 N

−1

One can observe the presence of both linear and

nonlinear constraints. By following the previous al-

gorithm, in the ﬁrst stage, the partition of the state

space is performed by considering only the linear con-

straints (ﬁgure 3).

Each such region correspond with a speciﬁc pro-

jection law. By simply verifying the regions where

this projection law obey the nonlinear constraints, the

exact part of the explicit solution is obtained (ﬁg. 4).

Figure 3: Partition of the arguments space (linear con-

straints only).

Figure 4: Retention of the regions with feasible linear pro-

jections.

Further, a distribution of points on the nonlinear

frontier of the feasible domain has to be obtained

and based on this distribution of points the associated

Voronoi partition constructed. By superposing it to

the regions not covered (white zones in the ﬁgure 4)

at the previous step, one obtain a complete covering

of the arguments space. Figure 5 depicts such a com-

plete partition for distribution of 10 points for each

nonlinear constraint.

By correspondence, the ﬁgure 6 describes the par-

tition of the state space for the explicit solution.

Finally the complete explicit solution is depicted

in ﬁgure 7. The discontinuities are observable in the

regions generated upon the Voronoi partition. In or-

der to give an image of the complexity it can men-

tioned that the explicit solutions contains 31 regions

the computational effort was less than 2s mainly spent

in the construction of the Voronoi partition.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

Figure 5: Partition of the arguments space (nonlinear case)

- 10 points per nonlinear constraint.

Figure 6: Partition of the state space - 10 points per nonlin-

ear constraint.

Figure 7: Explicit control law - 10 points per nonlinear con-

straint.

5 CONCLUSION

The parameterized polyhedra offer a transparent char-

acterization of the MPC degrees of freedom. Once

the complete description of the feasible domain as a

parameterized polyhedron is obtained explicit MPC

laws can be constructed using the projection of the

unconstrained optimum. The topology of the feasible

domain can lead to explicit solution even if nonlinear

constraints are taken into consideration. The price to

be paid is found in the degree of suboptimality.

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EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities