The explicit linear quadratic regulator for constrained systems * Alberto Bemporad 夽

Automatica 38 (2002) 3}20
The explicit linear quadratic regulator for constrained systems夽
Alberto Bemporad *, Manfred Morari, Vivek Dua, Efstratios N. Pistikopoulos
Dip. Ingegneria dell'Informazione, Universita% di Siena, Via Roma 56, 53100 Siena, Italy
Automatic Control Laboratory, ETH Zentrum, ETL I 26, 8092 Zurich, Switzerland
Centre for Process Systems Engineering, Imperial College, London SW7 2BY, UK
Received 24 September 1999; revised 9 October 2000; received in "nal form 16 June 2001
We present a technique to compute the explicit state-feedback solution to both the xnite and inxnite
horizon linear quadratic optimal control problem subject to state and input constraints. We show that
this closed form solution is piecewise linear and continuous. As a practical consequence of the result,
constrained linear quadratic regulation becomes attractive also for systems with high sampling rates,
as on-line quadratic programming solvers are no more required for the implementation.
Abstract
For discrete-time linear time invariant systems with constraints on inputs and states, we develop an algorithm to determine
explicitly, the state feedback control law which minimizes a quadratic performance criterion. We show that the control law is
piece-wise linear and continuous for both the "nite horizon problem (model predictive control) and the usual in"nite time measure
(constrained linear quadratic regulation). Thus, the on-line control computation reduces to the simple evaluation of an explicitly
de"ned piecewise linear function. By computing the inherent underlying controller structure, we also solve the equivalent of the
Hamilton}Jacobi}Bellman equation for discrete-time linear constrained systems. Control based on on-line optimization has long
been recognized as a superior alternative for constrained systems. The technique proposed in this paper is attractive for a wide range
of practical problems where the computational complexity of on-line optimization is prohibitive. It also provides an insight into the
structure underlying optimization-based controllers. 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Piecewise linear controllers; Linear quadratic regulators; Constraints; Predictive control
1. Introduction
As we extend the class of system descriptions beyond
the class of linear systems, linear systems with constraints
are probably the most important class in practice and the
most studied. It is well accepted that for these systems, in
general, stability and good performance can only be
夽
This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Associate Editor
Per-Olof Gutman under the direction of Editor Tamer Basar.
* Corresponding author. Dip. Ingegneria dell'Informazione, Universita% di Siena, Via Roma 56, 53100 Siena, Italy.
E-mail
addresses:
[email protected]
(A.
Bemporad),
[email protected] (M. Morari), [email protected] (V. Dua), [email protected] (E. N. Pistikopoulos).
achieved with a non-linear control law. The most popular approaches for designing non-linear controllers for
linear systems with constraints fall into two categories:
anti-windup and model predictive control.
Anti-windup schemes assume that a well functioning
linear controller is available for small excursions from the
nominal operating point. This controller is augmented by
the anti-windup scheme in a somewhat ad hoc fashion, to
take care of situations when constraints are met. Kothare
et al. (1994) reviewed numerous apparently di!erent
anti-windup schemes and showed that they di!er only in
their choice of two static matrix parameters. The least
conservative stability test for these schemes can be formulated in terms of a linear matrix inequality (LMI)
(Kothare & Morari, 1999). The systematic and automatic
synthesis of anti-windup schemes which guarantee
closed-loop stability and achieve some kind of optimal
0005-1098/02/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 7 4 - 1
4
A. Bemporad et al. / Automatica 38 (2002) 3}20
performance, has remained largely elusive, though
some promising steps were achieved recently (Mulder,
Kothare, & Morari, 1999; Teel & Kapoor, 1997). Despite
these drawbacks, anti-windup schemes are widely used in
practice because in most SISO situations, they are simple
to design and work adequately.
Model predictive control (MPC) has become the accepted standard for complex constrained multivariable
control problems in the process industries. Here, at each
sampling time, starting at the current state, an open-loop
optimal control problem is solved over a "nite horizon.
At the next time step, the computation is repeated starting from the new state and over a shifted horizon, leading
to a moving horizon policy. The solution relies on a linear dynamic model, respects all input and output constraints, and optimizes a quadratic performance index.
Thus, as much as a quadratic performance index together
with various constraints can be used to express true
performance objectives, the performance of MPC is excellent. Over the last decade, a solid theoretical foundation for MPC has emerged so that in real life, large-scale
MIMO applications controllers with non-conservative
stability guarantees can be designed routinely and with
ease (Qin & Badgwell, 1997). The big drawback of the
MPC is the relatively formidable on-line computational
e!ort, which limits its applicability to relatively slow
and/or small problems.
In this paper, we show how to move all the computations necessary for the implementation of MPC o!line, while preserving all its other characteristics.
This should largely increase the range of applicability of
MPC to problems where anti-windup schemes and other
ad hoc techniques dominated up to now. Moreover, such
an explicit form of the controller provides additional
insight for better understanding of the control policy
of MPC.
We also show how to solve the equivalent of the
Hamilton}Jacobi}Bellman equation for discrete-time
linear constrained systems. Rather than gridding the
state space in some ad hoc fashion, we discover the
inherent underlying controller structure and provide its
most e$cient parameterization.
The paper is organized as follows. The basics of MPC
are reviewed "rst to derive the quadratic program which
needs to be solved to determine the optimal control
action. We proceed to show that the form of this quadratic program is maintained for various practical extensions of the basic setup, for example, trajectory following,
suppression of disturbances, time-varying constraints
and also the output feedback problem. As the coe$cients
of the linear term in the cost function and the right-hand
side of the constraints depend linearly on the current
state, the quadratic program can be viewed as a multiparametric quadratic program (mp-QP). We analyze the
properties of mp-QP, develop an e$cient algorithm to
solve it, and show that the optimal solution is a piecewise
a$ne function of the state (con"rming other investigations on the form of MPC laws (Chiou & Za"riou,
1994; Tan, 1991; Za"riou, 1990; Johansen, Petersen, &
Slupphaug, 2000).
The problem of synthesizing stabilizing feedback controllers for linear discrete-time systems, subject to input
and state constraints, was also addressed in Gutman
(1986) and Gutman and Cwikel (1986). The authors obtain a piecewise linear feedback law de"ned over a partition of the set of states X into simplicial cones, by
computing a feasible input sequence for each vertex via
linear programming (this technique was later extended in
Blanchini, 1994). Our approach provides a piecewise
a$ne control law which not only ensures feasibility
and stability, but is also optimal with respect to LQR
performance.
The paper concludes with a series of examples which
illustrate the di!erent features of the method. Other
material related to the contents of this paper can be
found at http://control.ethz.ch/ &bemporad/
explicit.
2. Model predictive control
Consider the problem of regulating to the origin the
discrete-time linear time invariant system
x(t#1)"Ax(t)#Bu(t),
(1)
y(t)"Cx(t),
while ful"lling the constraints
y )y(t))y , u )u(t))u
(2)
at all time instants t*0. In (1)}(2), x(t)31L, u(t)31K, and
y(t)31N are the state, input, and output vectors, respectively, y )y
(u )u ) are p(m)-dimensional
vectors, and the pair (A, B) is stabilizable.
Model predictive control (MPC) solves such a
constrained regulation problem in the following way.
Assume that a full measurement of the state x(t) is
available at the current time t. Then, the optimization
problem
,W \
J(;, x(t))"x W Px W # [x
min
R>, R R>, R
R>IR
I
3OSR 2SR>,S\ Qx
#u Ru ] ,
R>IR
R>I R>I
The results of this paper hold also for the more general mixed
constraints D x(t)#D u(t))d.
A. Bemporad et al. / Automatica 38 (2002) 3}20
s.t. y )y
)y , k"1,2, N ,
R>IR
u )u )u , k"0,1,2, N ,
R>I
x "x(t),
RR
x
"Ax
#Bu , k*0,
R>I>R
R>IR
R>I
y
"Cx
, k*0,
R>IR
R>IR
u "Kx
, N )k(N ,
R>I
R>IR
S
W
(3)
is solved at each time t, where x
denotes the predicted
R>IR
state vector at time t#k, obtained by applying the input
sequence u ,2, u
to model (1) starting from the
R
R>I\
state x(t). In (3), we assume that Q"Q_0, R"RY0,
P_0, (Q, A) detectable (for instance, Q"CC with
(C, A) detectable), K is some feedback gain, N , N , N
W S are the output, input, and constraint horizons, respectively, with N )N and N )N !1 (the results of this
S
W
W
paper also apply for N *N ).
W
One possibility is to choose K"0 (Rawlings &
Muske, 1993), and P as the solution of the Lyapunov
equation
P"APA#Q.
(4)
The choice is only meaningful for open-loop stable systems, as it implies that after N time steps, the control
S
is turned o!.
Alternatively, one can set K"K
(Chmielewski
*/
& Manousiouthakis, 1996; Scokaert & Rawlings, 1998),
where K and P are, the solutions of the unconstrained
*/
in"nite horizon LQR problem with weights Q, R
K "!(R#BPB)\BPA,
*/
(5a)
P"(A#BK )P(A#BK )#K RK #Q.
*/
*/
*/
*/
(5b)
This choice of K implies that, after N time steps, the
S
control is switched to the unconstrained LQR. With
P obtained from (5), J(;, x(t)) measures the settling cost
of the system from the present time t to in"nity under this
control assumption. In addition, if one guarantees that
the predicted input and output evolutions do not violate
the constraints also at later time steps t#k,
k"N #1,2,#R, this strategy is indeed the optimal
S
in"nite horizon constrained LQ policy. This point will be
addressed in Section 3.
The MPC control law is based on the following idea:
At time t, compute the optimal solution ;H(t)"
uH,2, uH S to problem (3), apply
R
R>, \
u(t)"uH
R
5
The two main issues regarding this policy are feasibility of the optimization problem (3) and stability of the
resulting closed-loop system. When N (R, there is no
guarantee that the optimization problem (3) will remain
feasible at all future time steps t, as the system might
enter `blind alleysa where no solution to problem (3)
exists. On the other hand, setting N "R leads to an
optimization problem with an in"nite number of constraints, which is impossible to handle. If the set of
feasible state#input vectors is bounded and contains
the origin in its interior, by using arguments from maximal output admissible set theory (Gilbert & Tin Tan,
1991), for the case K"0 Bemporad (1998) showed that
a "nite constraint horizon, N , can be chosen without
loss of guarantees of constraint ful"llment, and a similar
argument can be repeated for the case K"K . Similar
*/
results about the choice of the smallest N ensuring
feasibility of the MPC problem (3) at all time instants
were also proved in Gutman and Cwikel (1986) and have
been extended recently in Kerrigan and Maciejowski
(2000).
The stability of MPC feedback loops was investigated
by numerous researchers. Stability is, in general, a complex function of the various tuning parameters
N , N , N , P, Q, and R. For applications, it is most
S W useful to impose some conditions on N , N and P, so
W that stability is guaranteed for all Q_0, RY0. Then,
Q and R can be freely chosen as tuning parameters to
a!ect performance. Sometimes, the optimization problem (3) is augmented with a so-called `stability constrainta (see Bemporad & Morari (1999) for a survey of
di!erent constraints proposed in the literature). This additional constraint imposed over the prediction horizon,
explicitly forces the state vector either to shrink in some
norm or to reach an invariant set at the end of the
prediction horizon.
Most approaches for proving stability follow in spirit
the arguments of Keerthi and Gilbert (1988) who establish the fact that under some conditions, the value
function <(t)"J(;H(t), t) attained at the minimizer
;H(t) is a Lyapunov function for the system.
Below, we recall a simple stability result based on such
a Lyapunov argument (see also Bemporad, Chisci, &
Mosca, 1994).
Theorem 1. Let N "R, K"0 or K"K , and
W
*/
N (R be suzciently large for guaranteeing existence of
feasible input sequences at each time step. Then, the MPC
law (3)}(6) asymptotically stabilizes system (1) while
(6)
as input to system (1), and repeat the optimization (3) at
time t#1, based on the new state x(t#1). Such a control strategy is also referred to as moving or receding
horizon.
As discussed above, several techniques exist to compute constraint
horizons N which guarantee feasibility, see Bemporad (1998), Gilbert
and Tin Tan (1991), Gutman and Cwikel (1986) and Kerrigan and
Maciejowski (2000).
6
A. Bemporad et al. / Automatica 38 (2002) 3}20
enforcing the fulxllment of the constraints (2) from all initial
states x(0) such that (3) is feasible at t"0.
Theorem 1 ensures stability, provided that the optimization problem (3) is feasible at time t"0. The problem
of determining the set of initial conditions x(0) for which
(3) is feasible has been addressed in Gutman and Cwikel
(1986), and more recently in Kerrigan and Maciejowski
(2000).
(Acevedo & Pistikopoulos, 1999; Dua & Pistikopoulos,
1999, 2000; Fiacco, 1983; Gal, 1995). In Section 4, we will
describe an algorithm to solve mp-QP problems. To the
authors' knowledge, no algorithm for solving mp-QP
problems has appeared in the literature. Once the multiparametric problem (7) has been solved o! line, i.e. the
solution ;H";H(x(t)) of (7) has been found, the model
R
predictive controller (3) is available explicitly, as the
optimal input u(t) consists simply of the "rst m components of ;H(x(t))
2.1. MPC computation
u(t)"[I 0 2 0];H(x(t)).
By substituting x
"AIx(t)#I\AHBu
,
R>IR
H
R>I\\H
Eq. (3) can be rewritten as
1
1
;H;#x(t)F;,
<(x(t))" x(t)>x(t)#min
2
2
3
s.t. G;)=#Ex(t) ,
(7)
where the column vector ;O[u ,2, u S ]31Q,
R
R>, \
sOmN , is the optimization vector, H"HY0, and
S
H, F, >, G, =, E are easily obtained from Q, R, and (3) (as
only the optimizer ; is needed, the term involving > is
usually removed from (7)).
The optimization problem (7) is a quadratic program
(QP). Since the problem depends on the current state x(t),
the implementation of MPC requires the on-line solution
of a QP at each time step. Although e$cient QP solvers
based on active-set methods and interior point methods
are available, computing the input u(t) demands signi"cant on-line computation e!ort. For this reason, the
application of MPC has been limited to `slowa and/or
`smalla processes.
In this paper, we propose a new approach to implement MPC, where the computation e!ort is moved
o!-line. The MPC formulation described in Section 2
provides the control action u(t) as a function of x(t)
implicitly de"ned by (7). By treating x(t) as a vector of
parameters, our goal is to solve (7) o!-line, with respect to
all the values of x(t) of interest, and make this dependence
explicit.
The operations research community has addressed
parameter variations in mathematical programs at two
levels: sensitivity analysis, which characterizes the change
of the solution with respect to small perturbations of the
parameters, and parametric programming, where the
characterization of the solution for a full range of parameter values is sought. In the jargon of operations research,
programs which depend only on one scalar parameter
are referred to as parametric programs, while problems
depending on a vector of parameters as multi-parametric
programs. According to this terminology, (7) is a multiparametric quadratic program (mp-QP). Most of the
literature deals with parametric problems, but some
authors have addressed the multi-parametric case
(8)
In Section 4, we will show that the solution ;H(x) of the
mp-QP problem is a continuous and piecewise a$ne
function of x. Clearly, because of (8), the same properties
are inherited by the controller.
Section 3 is devoted to investigate the regulation problem over an in"nite prediction horizon N "#R,
W
leading to solve explicitly the so-called constrained linear
quadratic regulation (C-LQR) problem.
3. Piecewise linear solution to the constrained linear
quadratic regulation problem
In their pioneering work, Sznaier and Damborg (1987),
showed that "nite horizon optimization (3), (6), with
P satisfying (5), also provides the solution to the in"nitehorizon linear quadratic regulation problem with constraints
< (x(0))" min
x(t)Qx(t)#u(t)Ru(t) ,
SS2 R
s.t. y )Cx(t))y ,
u )u(t))u ,
x(0)"x ,
x(t#1)"Ax(t)#Bu(t), t*0
(9)
with Q, R, A, and B as in (3). The equivalence holds for
a certain set of initial conditions, which depends on the
length of the "nite horizon. This idea has been reconsidered later by Chmielewski and Manousiouthakis
(1996) independently by Scokaert and Rawlings (1998),
and recently also in Chisci and Zappa (1999). In Scokaert
and Rawlings (1998), the authors extend the idea of
We refer to (3) as xnite horizon also in the case N "R. In fact,
W
such a problem can be transformed into a "nite-horizon problem by
choosing P as in (4) or (5). Therefore `"nitenessa should refer to the
input and constraint horizons, as these are indeed the parameters which
a!ect the complexity of the optimization problem (3).
A. Bemporad et al. / Automatica 38 (2002) 3}20
Sznaier and Damborg (1987) by showing that the controller is stabilizing, and that the C-LQR problem is
solved by a "nite-dimensional QP problem. Unfortunately, the dimension of the QP depends on the initial
state x(0), and no upper-bound on the horizon (and
therefore on the QP size) is given. On the other hand,
Chmielewski and Manousiouthakis (1996) describe an
algorithm which provides a semi-global upper-bound.
Namely, for any given compact set of initial conditions,
their algorithm provides the horizons N "N such that
S
the "nite horizon controller (3), (6) solves the in"nite
horizon problem (9).
We brie#y recall the approach given by Chmielewski
and Manousiouthakis (1996) and propose some
modi"cations, which will allow us to compute the
closed-form of the constrained linear quadratic
controller (9)
u(t)"f (x(t)), ∀t*0
(10)
for a compact set of initial conditions (0), and show that
it is piecewise a$ne.
Assume that (i) QY0, (ii) the sets , U of feasible states
and inputs, respectively, are bounded (e.g. u , u ,
y , y (R, and C"I) and contain the origin in
their interior, and (iii) that for all the initial states
x(0)3(0), there exists an input sequence driving the
system to the origin under constraints. We also assume
that ,U are polytopes, as this is the case at hand in our
paper, to simplify the exposition. Denote by P,, problem
(3) with NON "N , N "R, K"K (or, equivaS
W
*/
lently, N "N, and P solving the Riccati equation (5b))
W
and by P , problem (9). Note that in view of Bellman's
principle of optimality (Bellman, 1961), (10) can be
reformulated by substituting x
,u
for x(t), u(t),
R>IR R>I
respectively, as predicted and actual trajectories
coincide.
Chmielewski and Manousiouthakis (1996) prove that,
for a given x(0), there exists a "nite N such that P, and
P are equivalent, and that the associated controller is
exponentially stabilizing. The same result was proved in
Scokaert and Rawlings (1998). In addition, by exploiting
the convexity and continuity of the value function
<
with respect to the initial condition x(0),
Chmielewski and Manousiouthakis (1996) provide the
tools necessary to compute an upper-bound on N for
every given set (0) of initial conditions. Such an
upper-bound is computed by modifying the algorithm
in Chmielewski and Manousiouthakis (1996), as
follows.
(1) Let XM "x3: K x3U the set of states such that
*/
the unconstrained LQ gain K is feasible. As ,U
*/
are polytopes, XM is a polytope, de"ned by a set of
linear inequalities of the form AM Gx)bM G, i"1,2, n ,
where AM ,bM ,n depend on the de"nition of ,U.
7
(2) Compute Z "x31L: xPx)c, where the largest
A
c such that Z is an invariant subset of XM , is deterA
mined analytically (Bemporad, 1998) as
bM G
c" min
.
GP\(A
A
M
M G)
2
G L
(11)
(3) Let q"c (P\Q), where denotes the min
imum eigenvalue.
(4) Let (0) be a compact set of initial conditions. Without loss of generality, assume that (0) is a polytope,
and let x l
be the set of its vertices. For each
G G
x compute < (x ). This is done by computing the
G
G
"nite horizon value function <(x ) for increasing
G
values of N until x
3Z (this computation is
R>I,
A
equivalent to the one described in Scokaert and
Rawlings (1998), where a ball B is used instead of Z ).
P
A
< (x ), which is an upper
(5) Let ;"max
G2l G
bound to < (x) on (0), as < and (0) are convex
(in Chmielewski & Manousiouthakis, 1996), a di!erent approach is proposed where ; depends on x ).
(6) Choose N as the minimum integer such that
N*(;!c)/q.
In Section 4, we will show that P, has a continuous
and piecewise a$ne solution. Therefore, on a compact set
of initial conditions (0), the solution (10) to the CLQR
problem P is also piecewise a$ne. Note that if .
then the solution is also global because ,U are bounded.
4. Multi-parametric quadratic programming
In this section, we investigate multi-parametric quadratic programs (mp-QP) of form (7). We want to derive
an algorithm to express the solution ;H(x) and the minimum value <(x)"J(;H(x)) as an explicit function of the
parameters x, and to characterize the analytical properties of these functions. In particular, we will prove that
the solution ;H(x) is a continuous piecewise a$ne function of x, in the following sense.
De5nition 1. A function z(x) : XC1Q, where X-1L is
a polyhedral set, is piecewise a$ne if it is possible to
partition X into convex polyhedral regions, CR , and
G
z(x)"HGx#kG, ∀x3CR .
G
Piecewise quadraticity is de"ned analogously by letting z(x) be a quadratic function x=Gx#HGx#kG.
4.1. Fundamentals of the algorithm
Before proceeding further, it is useful to de"ne
zO;#H\Fx(t),
(12)
8
A. Bemporad et al. / Automatica 38 (2002) 3}20
z31Q, and to transform (7) by completing squares to
obtain the equivalent problem
1
< (x)"min zHz
X
2
X
s.t. Gz)=#Sx(t),
(13)
where
SOE#GH\F,
and
< (x)"<(x)!x
X
(>!FH\F)x. In the transformed problem, the parameter vector x appears only on the rhs of the constraints.
In order to start solving the mp-QP problem, we need
an initial vector x
inside the polyhedral set
X"x: ¹x)Z of parameters over which we want to
solve the problem, such that the QP problem (13) is
feasible for x"x . A good choice for x is the center of
the largest ball contained in X for which a feasible
z exists, determined by solving the LP
max
VXC
s.t.
,
¹Gx#¹G)ZG,
(14)
Gz!Sx)=
(in particular, x will be the Chebychev center of X when
the QP problem (13) is feasible for such an x ). If )0,
then the QP problem (13) is infeasible for all x in the
interior of X. Otherwise, we "x x"x and solve the QP
problem (13), in order to obtain the corresponding optimal solution z . Such a solution is unique, because HY0,
and therefore uniquely determines a set of active constraints GI z "SI x #=
I out of the constraints in (13).
We can then prove the following result:
Theorem 2. Let HY0. Consider a combination of active
constraints GI , SI , =
I , and assume that the rows of GI are
linearly independent. Let CR be the set of all vectors x, for
which such a combination is active at the optimum (CR is
referred to as critical region). Then, the optimal z and the
associated vector of Lagrange multipliers are uniquely
dexned azne functions of x over CR .
Proof. The "rst-order Karush}Kuhn}Tucker (KKT)
optimality conditions (Bazaraa et al., 1993) for the
mp-QP are given by
Hz#G"0, 31O,
(15a)
(GGz!=G!SGx)"0, i"1,2, q,
G
(15b)
*0,
(15c)
Gz)=#Sx,
(15d)
where the superscript i denotes the ith row. We solve
(15a) for z,
z"!H\G
(16)
and substitute the result into (15b) to obtain the complementary slackness condition (!GH\G!
=!Sx)"0. Let x and I denote the Lagrange multipliers corresponding to inactive and active constraints,
respectively. For inactive constraints, x "0. For active
constraints, !GI H\GI I !=
I !SI x"0, and therefore,
I "!(GI H\GI )\(=
I #SI x),
(17)
where GI ,=
I ,SI correspond to the set of active constraints,
and (GI H\GI )\ exists because the rows of GI are linearly
independent. Thus, is an a$ne function of x. We can
substitute I from (17) into (16) to obtain
z"H\GI (GI H\GI )\(=
I #SI x)
(18)
and note that z is also an a$ne function of x. 䊐
A similar result was obtained by Za"riou (Chiou
& Za"riou, 1994; Za"riou, 1990) based on the optimality
conditions for QP problems reported in Fletcher (1981).
Although his formulation is for "nite impulse response
models realized in a particular space form, where the
state includes past inputs, his arguments can be adapted
directly to the state-space formulation (3)}(6). However,
his result does not make the piecewise linear dependence
of u on x explicit, as the domains over which the di!erent
linear laws are de"ned are not characterized. We show
next, that such domains are indeed polyhedral regions of
the state space.
Theorem 2 characterizes the solution only locally in
the neighborhood of a speci"c x , as it does not provide
the construction of the set CR where this characteriza
tion remains valid. On the other hand, this region can be
characterized immediately. The variable z from (16) must
satisfy the constraints in (13):
GH\GI (GI H\GI )\(=
I #SI x))=#Sx
(19)
and by (15c), the Lagrange multipliers in (17) must remain non-negative:
!(GI H\GI )\(=
I #SI x)*0,
(20)
as we vary x. After removing the redundant inequalities
from (19) and (20), we obtain a compact representation of
CR . Obviously, CR is a polyhedron in the x-space, and
represents the largest set of x3X such that the combination of active constraints at the minimizer remains unchanged. Once the critical region CR has been de"ned,
the rest of the space CR"X!CR has to be ex
plored and new critical regions generated. An e!ective
approach for partitioning the rest of the space was proposed in Dua and Pistikopoulos (2000). The following
theorem justi"es such a procedure to characterize the rest
of the region CR.
A. Bemporad et al. / Automatica 38 (2002) 3}20
Theorem 3. Let >-1L be a polyhedron, and CR O
x3>: Ax)b a polyhedral subset of >, CR O.
Also let
AGx'bG
R " x3>:
,
G
AHx)bH, ∀j(i
i"1,2, m,
where m"dim(b), and let CROK R . Then
G G
(i) CR
CR ">, (ii) CR R ", R R ", ∀i
G
G
H
Oj, i.e. CR , R ,2, R is a partition of >.
K
Proof. (i) We want to prove that given an x3>, x either
belongs to CR or to R for some i. If x3CR , we are
G
done. Otherwise, there exists an index i such that
AGx'bG. Let iH"min i: AGx'bG. Then, x3R H , beGWK
G
cause AGHx'bGH and AHx)bH, ∀j(iH, by de"nition of iH.
(ii) Let x3CR . Then xi such that AGx'bG, which
implies that x,R , ∀i)m. Let x3R and take i'j. Since
G
G
x3R , by de"nition of R (i'j) AHx)bH, which implies
G
G
that x,R . 䊐
H
Example 4.1. In order to exemplify the procedure proposed in Theorem 3 for partitioning the set of parameters
X, consider the case when only two parameters x and
x are present. As shown in Fig. 1(a), X is de"ned by the
inequalities x\)x )x>, x\)x )x>, and CR
by the inequalities C1)0,2, C5)0 where
C1,2, C5 are a$ne functions of x. The procedure consists of considering, one by one, the inequalities which
de"ne CR . Considering, for example, the inequality
C1)0, the "rst set of the rest of the region
9
CROXCR is given by R "C1*0, x *x\,
x\)x )x>, which is obtained by reversing the sign
of the inequality C1)0 and removing redundant constraints (see Fig. 1(b)). Thus, by considering the rest of
the inequalities, the complete rest of the region is
CR" R , where R ,2, R are graphically reG G
ported in Fig. 1(d). Note that the partition strategy suggested by Theorem 3 can be also applied also when X is
unbounded.
Theorem 3 provides a way of partitioning the nonconvex set, XCR , into polyhedral subsets R . For each
G
R , a new vector x is determined by solving the LP (14),
G
G
and, correspondingly, an optimum z , a set of active
G
constraints GI G, SI G, =
I G, and a critical region CR . Theorem
G
3 is then applied to partition R CR into polyhedral
G
G
subsets, and the algorithm proceeds iteratively. The
complexity of the algorithm will be fully discussed in
Section 4.3.
Note that Theorem 3 introduces cuts in the x-space
which might split critical regions into subsets. Therefore,
after the whole x-space has been covered, those polyhedral regions CRG are determined where the function z(x) is
the same. If their union is a convex set, it is computed to
permit a more compact description of the solution
(Bemporad, Fukuda, & Torrisi, 2001). Alternatively, in
Borrelli, Bemporad, and Morari (2000), the authors
propose not to intersect (19)}(20) with the partition generated by Theorem 3, and simply use Theorem 3 to guide
the exploration. As a result, some critical regions may
appear more than once. Duplicates can be easily eliminated by recognizing regions where the combination of
active constraints is the same. In the sequel, we will
denote by N , the "nal number of polyhedral cells de"n
ing the mp-QP solution (i.e., after the union of neighboring cells or removal of duplicates, respectively).
4.1.1. Degeneracy
So far, we have assumed that the rows of GI are linearly
independent. It can happen, however, that by solving the
QP (13), one determines a set of active constraints for
which this assumption is violated. For instance, this
happens when more than s constraints are active at the
optimizer z 31Q, i.e., in a case of primal degeneracy. In
this case, the vector of Lagrange multipliers might not
be uniquely de"ned, as the dual problem of (13) is not
strictly convex (instead, dual degeneracy and non-uniqueness of z cannot occur, as HY0). Let GI 31l"Q, and
let r"rank GI , r(l. In order to characterize such a
degenerate situation, consider the QR decomposition
GI "[0 ]Q of GI , and rewrite the active constraints in the
form
Fig. 1. Partition of CROXCR ; (b) partition of CR step 1; (c)
partition of CR Step 2; and (d) "nal partition of CR.
R z"= #S x,
(21a)
0"= #S x,
(21b)
10
A. Bemporad et al. / Automatica 38 (2002) 3}20
where [1 ]"Q\SI , [5 ]"Q\=
I . If S is non-zero,
1
5
because of the equalities (21b), CR is a lower-dimen
sional region, which, in general, corresponds to a common boundary between two or more full-dimensional
regions. Therefore, it is not worth to explore this combination GI , SI , =
I . On the other hand, if both = and S are
zero, the KKT conditions do not lead directly to (19) and
(20), but only to a polyhedron expressed in the (, x)
space. In this case, a full-dimensional critical region can
be obtained by a projection algorithm (Fukuda, 1997),
which, however, is computationally expensive (the case
= O0, S "0 is not possible, since the LP (14) was
feasible).
In this paper, we suggest a simpler way to handle such
a degenerate situation, which consists of collecting r constraints arbitrarily chosen, and proceed with the new
reduced set, therefore avoiding the computation of projections. Due to the recursive nature of the algorithm, the
remaining other possible subsets of combinations of constraints leading to full-dimensional critical regions will
automatically be explored later.
4.2. Continuity and convexity properties
Continuity of the value function < (x) and the solution
X
z(x), can be shown as simple corollaries of the linearity
result of Theorem 2. This fact, together with the convexity of the set of feasible parameters X -X (i.e. the set of
parameters x3X such that a feasible solution z(x) exists
to the optimization problem (13)), and of the value function < (x), is proved in next Theorem.
X
Theorem 4. Consider the multi-parametric quadratic program (13) and let HY0, X convex. Then the set of feasible
parameters X -X is convex, the optimizer z(x) : X C1Q is
continuous and piecewise azne, and the optimal solution
< (x) : X C1 is continuous, convex and piecewise
X
quadratic.
Proof. We "rst prove convexity of X and < (x). Take
X
generic x , x 3X , and let < (x ), < (x ) and z , z the
X X corresponding optimal values and minimizers. Let
3[0,1], and de"ne z Oz #(1!)z , x Ox #
?
?
(1!)x . By feasibility, z , z satisfy the constraints
Gz )=#Sx , Gz )=#Sx . These inequalities
can be linearly combined to obtain Gz )=#Sx , and
?
?
therefore, z is feasible for the optimization problem (13),
?
In general, the set of feasible parameters x can be a lower-dimensional subset of X (Filippi, 1997). However, when in the MPC formulation (3), u (u , y (y , the mp-QP problem has always
a full-dimensional solution in the x-space, as the critical region corresponding to the unconstrained solution contains a full-dimensional ball
around the origin.
where x(t)"x . This proves that z(x ) exists, and there?
?
fore, convexity of X " CR . In particular, X
G
G
is connected. Moreover, by optimality of < (x ), < (x ))
X ?
X ?
z Hz , and hence < (x )![z Hz #(1!)z Hz ])
? ?
X ?
z Hz ![z Hz #(1!)z Hz ]"[z Hz #
? ? (1!)z Hz #2(1!)z Hz !z Hz !(1!)z
Hz ]"!(1!)(z !z )H(z !z ))0, i.e. <
X
(x #(1!)x ))< (x )#(1!)< (x ),
X X ∀x , x 3X, ∀3[0,1], which proves the convexity of
< (x) on X . Within the closed polyhedral regions CR in
X
G
X , the solution z(x) is a$ne (18). The boundary between
two regions belongs to both closed regions. Since HY0,
the optimum is unique, and hence, the solution must be
continuous across the boundary. The fact that < (x)
X
is continuous and piecewise quadratic, follows
trivially. 䊐
In order to prove the convexity of the value function
<(x) of the MPC problems (3) and (7), we need the
following lemma.
Lemma 1. Let J(;, x)";H;#xF;#x>x, and
let >F F
_0.
Then
<(x)Omin
J(x,
;)
subject to
H
3
G;)=#Ex is a convex function of x.
Proof. By Theorem 4, the value function < (x) of the
X
optimization problem < (x)Omin zHz subject to
X
X
Gz)=#Sx is a convex function of x. Let zH(x), ;H(x)
be the optimizers of < (x) and <(x), respectively, where
X
zH(x)";H(x)#H\Fx. Then, <(x)";H(x)H;H(x)#
xF;H(x)#x>x"[zH(x)!H\Fx]H[zH(x)!H\
Fx]#xF[zH(x)!H\Fx]#x>x"< (x)!xFH\
X
F
Fx#x>x"< (x)#x(>!FH\F)x. As >
F H
X
_0, its Schur's complement >!FH\F_0, and therefore, <(x) is a convex function, being the sum of convex
functions. 䊐
Corollary 1. The value function <(x) dexned by the optimization problem (3), (7) is continuous and piecewise quadratic.
Proof. By (3), J(;, x)*0, ∀x, ;, being the sum of nonY F
negative terms. Therefore, J(;, x)"[V]
[V]*0
S F H S
for all [V], and the proof easily follows by Lemma 1. 䊐
S
A simple consequence of Corollary 1 is that the
Lyapunov function used to prove Theorem 1 is continuous, convex, and piecewise quadratic. Finally, we can
establish the analytical properties of the controller (3),
(6) through the following corollary of Theorem 4.
A. Bemporad et al. / Automatica 38 (2002) 3}20
Corollary 2. The control law u(t)"f (x(t)), f : 1LC1K, dexned by the optimization problem (3) and (6) is continuous
and piece-wise azne
1.5
(22)
1.6
1.7
1.8
where the polyhedral sets HGx)kG, i"1,2, N )
N are a partition of the given set of states X.
1.9
end
f (x)"FGx#gG if HGx)kG, i"1,2, N
Proof. By (12), ;(x)"z(x)!H\Fx is a linear function
of x in each region CR "x : HGx)KG, i"1,2, N .
G
By (8), u"f (x) is a combination of linear functions, and
therefore, is linear on CR . Also, u is a combination of
G
continuous functions, and therefore, is continuous. 䊐
Multiparametric quadratic programming problems
can also be addressed by employing the principles of
parametric non-linear programming, exploiting the Basic
Sensitivity Theorem (Fiacco, 1976, 1983) (a direct consequence of the KKT conditions and the Implicit Function
Theorem). In this paper, we opted for a more direct
approach, which exploits the linearity of the constraints
and the fact that the function to be minimized is
quadratic.
We remark that the command actions provided by the
(explicit) feedback control law (22) and the (implicit)
feedback control law (3)}(6) are exactly equal. Therefore,
the control designer is allowed to tune the controller by
using standard MPC tools (i.e., based on on-line QP
solvers) and software simulation packages, and "nally
run Algorithm 1 to obtain the explicit solution (22) to
e$ciently implement the controller.
4.3. Ow-line algorithm for mp-QP and explicit MPC
Based on the above discussion and results, the main
steps of the o!-line mp-QP solver are outlined in the
following algorithm.
Algorithm 1
1 Let X-1L be the set of parameters (states);
2 execute partition(X);
3 for all regions where z(x) is the same and whose union
is a convex set, compute such a union as described by
Bemporad, Fukuda, and Torrisi (2001);
4 end.
procedure partition(>)
1.1 let x 3> and the solution to the LP (14);
1.2 if )0 then exit; (no full dimensional CR is
in >)
1.3 For x"x , compute the optimal solution
(z , ) of the QP (13);
1.4 Determine the set of active constraints when
z"z , x"x , and build GI , =
I , SI ;
11
If r"rank GI is less than the number l of rows of
GI , take a subset of r linearly independent rows,
and rede"ne GI , =
I , SI accordingly;
Determine I (x), z(x) from (17) and (18);
Characterize the CR from (19) and (20);
De"ne and partition the rest of the region as in
Theorem 3;
For each new sub-region R , partition(R );
G
G
procedure.
The algorithm explores the set X of parameters recursively: Partition the rest of the region as in Theorem
3 into polyhedral sets R , use the same method to partiG
tion each set R further, and so on. This can be representG
ed as a search tree, with a maximum depth equal to the
number of combinations of active constraints (see Section 4.4 below).
The algorithm solves the mp-QP problem by partitioning the given parameter set X into N convex poly
hedral regions. For the characterization of the MPC
controller, in step 3 the union of regions is computed
where the "rst N components of the solution z(x) are the
S
same, by using the algorithm developed by Bemporad,
Fukuda, and Torrisi (2001). This reduces the total number of regions in the partition for the MPC controller
from N to N .
4.4. Complexity analysis
The number N of regions in the mp-QP solution
depends on the dimension n of the state, and on the
number of degrees of freedom s"mN and constraints
S
q in the optimization problem (13). As the number of
combinations of l constraints out of a set of q is
(Ol)"q!/(q!l)!l!, the number of possible combinations
of active constraints at the solution of a QP is at most
Ol (Ol)"2O. This number represents an upper bound
on the number of di!erent linear feedback gains which
describe the controller. In practice, far fewer combinations are usually generated as x spans X. Furthermore,
the gains for the future input moves u ,2, u S
are
R>
R>, \
not relevant for the control law. Thus, several di!erent
combinations of active constraints may lead to the same
"rst m components uH(x) of the solution. On the other
R
hand, the number N of regions of the piecewise
a$ne solution is, in general, larger than the number
of feedback gains, because non-convex critical regions
are split into several convex sets. For instance, the
example reported in Fig. 6(d) involves 13 feedback gains,
distributed over 57 regions of the state space.
A worst-case estimate of N can be computed from the
way Algorithm 1 generates critical regions. The "rst
critical region CR is de"ned by the constraints (x)*0
(q constraints) and Gz(x))=#Sx (q constraints). If the
strict complementary slackness condition holds, only
12
A. Bemporad et al. / Automatica 38 (2002) 3}20
q constraints can be active, and hence, every CR is
de"ned by q constraints. From Theorem 3, CR consists of q convex polyhedra R , de"ned by at most q inG
equalities. For each R , a new CR is determined which
G
G
consists of 2q inequalities (the additional q inequalities
come from the condition CR -R ), and therefore, the
G
G
corresponding CR partition includes 2q sets de"ned by
2q inequalities. As mentioned above, this way of generating regions can be associated with a search tree. By
induction, it is easy to prove that at the tree level k#1,
there are k!mI regions de"ned by (k#1)q constraints. As
observed earlier, each CR is the largest set corresponding
to a certain combination of active constraints. Therefore,
the search tree has a maximum depth of 2O, as at each
level, there is one admissible combination less. In conclusion, the number of regions in the solution to the
mp-QP problem is N )O\k!qI, each one de"ned by
I
at most q2O linear inequalities. Note that the above analysis is largely overestimating the complexity, as it does not
take into account: (i) the elimination of redundant constraints when a CR is generated, and (ii) that empty sets
are not partitioned further.
Table 1
O!-line computation time to solve the mp-QP problem and, in parentheses, number of regions N
in the MPC controller (N "num
S
ber of control moves, n"number of states)
N
S
n"2
n"3
n"4
n"5
2
3
4
0.44 s (7)
1.15 s (13)
2.31 s (21)
0.49 s (7)
2.08 s (15)
5.87 s (29)
0.55 s (7)
1.75 s (15)
3.68 s (29)
1.43 s (7)
5.06 s (17)
15.93 s (37)
problems on random SISO plants subject to input constraints. In the comparison, we vary the number of free
moves N and the number of states of the open-loop
S
system. Computation times were evaluated by running
Algorithm 1 in Matlab 5.3 on a Pentium III-650 MHz
machine. No attempts were made to optimize the e$ciency of the algorithm and its implementation.
4.7. On-line computation time
4.5. Dependence on n, m, p, N , N
S Let q Orank S, q )q. For n'q the number of polyQ
Q
Q
hedral regions N remains constant. To see this, consider
the linear transformation x "Sx, x 31O. Clearly, x and
x de"ne the same set of active constraints, and therefore
the number of partitions in the x - and x-space are the
same. Therefore, the number of partitions, N , of the
x-space de"ning the optimal controller is insensitive to
the dimension n of the state x for all n*q , i.e. to the
Q
number of parameters involved in the mp-QP. In particular, the additional parameters that we will introduce
in Section 6 to extend MPC to reference tracking, disturbance rejection, soft constraints, variable constraints, and
output feedback, do not a!ect signi"cantly, the number
of polyhedral regions N (i.e., the complexity of the mp
QP), and hence, the number N
of regions in the MPC
controller (22).
The number q of constraints increases with N and, in
the case of input constraints, with N . For instance,
S
q"2s"2mN for control problems with input conS
straints only. From the analysis above, the larger
N , N , m, p, the larger q, and therefore N . Note that
S
many control problems involve input constraints only,
and typically horizons N "2,3 or blocking of control
S
moves are adopted, which reduces the number of constraints q.
4.6. Ow-line computation time
In Table 1, we report the computation time and the
number of regions obtained by solving a few test MPC
The simplest way to implement the piecewise a$ne
feedback law (22) is to store the polyhedral cells
HGx)kG, perform an on-line linear search through
them to locate the one which contains x(t), then lookup
the corresponding feedback gain (FG, gG). This procedure
can be easily parallelized (while for a QP solver, the
parallelization is less obvious). However, more e$cient
on-line implementation techniques which avoid the storage and the evaluation of the polyhedral cells are currently under development.
5. State-feedback solution to constrained linear quadratic
control
For t"0, the explicit solution to (3) provides the
optimal input pro"le u(0),2, u(N !1) as a function of
W
the state x(0). The equivalent state-feedback form
u( j)"f (x( j)), j"0,2, N !1 can be obtained by solvH
W
ing N mp-QPs. In fact, clearly, f (x)"Kx for all
S
H
j"N ,2, N !1, where f : X C1K, and X O1L for
S
W
H H
H
j"N #1,2, N , X Ox: y )C(A#BK)Fx)y ,
W H
u )K(A#BK)Fx)u , h"0,2, N !j for j"
N ,2, N . For 0)j)N !1, f (x) is obtained by solvS
S
H
ing the mp-QP problem
F (x)O
H
min
J(;, x)"x(N )Px(N )
W
W
3OS H2S,S \
,W \
# [x(k)Qx(k)#u(k)Ru(k)]
IH
A. Bemporad et al. / Automatica 38 (2002) 3}20
s.t. y )y(k))y , k"j,2, N ,
u )u(k))u , k"j,2, N ,
x( j)"x,
x(k#1)"Ax(k)#Bu(k), k*j,
13
sider the MPC problem
(23)
y(k)"Cx(k), k*j,
u(k)"Kx(k), N )k(N
S
W
and setting f (x)"[I 0 2 0]F (x) (note that, compared
H
H
to (3), for j"0, (23) includes the extra constraint
y )y(0))y . However, this may only restrict
the set of x(0) for which (23) is feasible, but does
not change the control function f (x), as u(0) does not
a!ect y(0)).
Similar to the unconstrained "nite-time LQ problem,
where the state-feedback solution is linear timevarying, the explicit state-feedback solution to (3) is the
time-varying piecewise a$ne function f : 1KC1L,
H
j"0,2, N !1. Note that while in the unconstrained
W
case dynamic programming nicely provides the statefeedback solution through Riccati iterations, because of
the constraints here, dynamic programming would lead
to solving a sequence of multiparametric piecewise quadratic programs, instead of the mp-QPs (23).
The in"nite horizon-constrained linear quadratic
regulator can also be obtained in state-feedback form by
choosing N "N "N "N, where N is de"ned acS
W
cording to the results of Section 3.
6. Reference tracking, disturbances, and other
extensions
The basic formulation (3) can be extended naturally to
situations where the control task is more demanding. As
long as the control task can be expressed as an mp-QP,
a piecewise a$ne controller results, which can be easily
designed and implemented. In this section, we will mention only a few extensions to illustrate the potential. To
our knowledge, these types of problems are di$cult to
formulate from the point of view of anti-windup or other
techniques not related to MPC.
6.1. Reference tracking
The controller can be extended to provide o!set-free
tracking of asymptotically constant reference signals. Future values of the reference trajectory can be taken into
account explicitly, by the controller, so that the control
action is optimal for the future trajectory in the presence
of constraints.
Let the goal be to have the output vector y(t) track r(t),
where r(t)31N is the reference signal. To this aim, con-
,W \
min
[y
!r(t)]Q[y
!r(t)]
R>IR
R>IR
3OBSR 2BSR>,S\ I
#u
Ru
R>IR
R>IR
s.t. y )y
)y , k"1,2, N ,
R>IR
u )u )u , k"0,1,2, N ,
R>I
u )u )u , k"0,1,2, N !1,
R>I
S
x
"Ax
#Bu , k*0,
R>I>R
R>IR
R>I
y
"Cx
, k*0,
R>IR
R>IR
u "u
#u , k*0,
R>I
R>I\
R>I
u "0, k*N .
R>I
S
(24)
Note that the u-formulation (24) introduces m new
states in the predictive model, namely, the last input
u(t!1) (this corresponds to adding an integrator in the
control loop). Just like the regulation problem (3), we can
transform the tracking problem (24) into the form
1
min ;H;#[x(t) u(t!1) r(t)]F;
2
3
x(t)
s.t. G;)=#E u(t!1) ,
(25)
r(t)
where r(t) lies in a given (possibly unbounded) polyhedral set. Thus, the same mp-QP algorithm can be
used to obtain an explicit piecewise a$ne solution
u(t)"F(x(t), u(t!1), r(t)). In case the reference r(t) is
known in advance, one can replace r(t) by r(t#k) in (24)
and similarly get a piecewise a$ne anticipative controller
u(t)"F(x(t), u(t!1), r(t),2, r(t#N !1)).
W
6.2. Disturbances
We distinguish between measured and unmeasured
disturbances. Measured disturbances v(t) can be included
in the prediction model
x
"Ax
#Bu #<v(t#kt),
R>I>R
R>IR
R>I
(26)
where v(t#kt) is the prediction of the disturbance at
time t#k based on the measured value v(t). Usually,
v(t#kt) is a linear function of v(t), for instance
v(t#kt),v(t) where it is assumed that the disturbance
is constant over the prediction horizon. Then v(t) appears
as a vector of additional parameters in the mp-QP,
and the piecewise a$ne control law becomes
u(t)"F(x(t), v(t)). Alternatively, as for reference tracking,
when v(t) is known in advance one can replace v(t#kt)
by v(t#k) in (26) and get an anticipative controller
u(t)"F(x(t), u(t!1), v(t),2, v(t#N !1)).
W
14
A. Bemporad et al. / Automatica 38 (2002) 3}20
Usually unmeasured disturbances are modeled as the
output of a linear system driven by a white Gaussian
noise. The state vector x(t) of the linear prediction model
(1) is augmented by the state x (t) of such a linear disturL
bance model, and the mp-QP provides a control law of
the form u(t)"F(x(t), x (t)) within a certain range of
L
states of the plant and of the disturbance model. Clearly,
x (t) is estimated on line from output measurements by
L
a linear observer.
6.3. Soft constraints
Fig. 2. Example 7.1: (a) closed-loop MPC; (b) state-space partition and
closed-loop MPC trajectories.
State and output constraints can lead to feasibility
problems. For example, a disturbance may push the
output outside the feasible region where no allowed control input may exist which brings the output back inside
at the next time step. Therefore, in practice, the output
constraints (2) are relaxed or softened (Zheng & Morari,
1995) as y !M)y(t))y #M, where M31N is
a constant vector (MG*0 is related to the `concerna for
the violation of the ith output constraint), and the term
is added to the objective to penalize constraint violations ( is a suitably large scalar). The variable plays the
role of an independent optimization variable in the mpQP and is adjoined to z. The solution u(t)"F(x(t)) is
again a piecewise a$ne controller, which aims at keeping
the states in the constrained region without ever running
into feasibility problems.
6.4. Variable constraints
The bounds y , y ,u ,u , u , u
may
change depending on the operating conditions, or in the
case of a stuck actuator, the constraints become
u "u "0. This possibility can again be
built into the control law. The bounds can be treated
as parameters in the mp-QP and added to the
vector x. The control law will have the form
u(t)"F(x(t), y , y ,u ,u , u , u ).
7. Examples
7.1. A simple SISO system
Consider the second order system
0.7326 !0.0861
0.1722
y(t)"[0 1.4142]x(t).
To this aim, we design an MPC controller based on the
optimization problem
x
Px
# [x
x
# 0.01u ]
R>R R>R
R>IR R>IR
R>I
I
!2)u )2, k"0,1,
(29)
R>I
x "x(t)
RR
min
SR SR>
s.t.
where P solves the Lyapunov equation P"APA#Q
(in this example, Q"I, R"0.01, N "N "2, N "1).
W
S
Note that this choice of P corresponds to setting u "0
R>I
for k*2 and minimize
x
x
#0.01u .
R>IR R>IR
R>I
I
0.9909
x(t)#
0.0609
0.0064
u(t).
(27)
(30)
The MPC controller (29) is globally asymptotically
stabilizing. In fact, it is easy to show that the value
function is a Lyapunov function of the system. The
closed-loop response from the initial condition
x(0)"[1 1] is shown in Fig. 2(a).
The mp-QP problem associated with the MPC law has
the form (7) with
sample the dynamics with ¹"0.1 s, and obtain the
state-space representation
(28)
!2)u(t))2.
H"
2
y(t)"
u(t),
s#3s#2
x(t#1)"
The task is to regulate the system to the origin while
ful"lling the input constraint
1.5064 0.4838
0.4838 1.5258
,
F"
9.6652
5.2115
7.0732 !7.0879
,
Let ;H"[uH, uH] be the optimal solution at time t. Then
R
;"[uH,0] is feasible at time t#1. The cost associated with ; is
J(t#1, ;)"J(t, ;H)!x(t)x(t)!0.01u(t)*J(t#1, ;H ),
which
R
R>
implies that J(t, ;H) is a converging sequence. Therefore,
R
x(t)x(t)#0.01u(t))J(t, ;H)!J(t#1, ;H )P0, which shows stabR
R>
ility of the system.
A. Bemporad et al. / Automatica 38 (2002) 3}20
saturated controller, and regions C6 and C9 are
transition regions between the unconstrained and the
saturated controller. Note that the mp-QP solver provides three di!erent regions C2,C3,C4, although in all
of them, u"uH"2. The reason for this is that the
R
second component of the optimal solution, uH , is di!erR>
ent, in that uH "[!3.4155 4.6452]x(t)!0.6341 in reR>
gion C2, uH "2 in region C3, and uH "!2 in
R>
R>
region C4. Moreover, note that regions C2 and C4 are
joined, as their union is a convex set, but the same cannot
1
0
!1
0
G"
0
1
0
!1
2
,
="
2
2
0 0
,
E"
2
0 0
.
0 0
0 0
CFFFFFFFFFFFFFFFFFFFFFDFFFFFFFFFFFFFFFFFFFFFE
The solution was computed by Algorithm 1 in 0.66 s (15
regions examined), and the corresponding polyhedral
partition of the state-space into N "9 polyhedral cells is
depicted in Fig. 2(b). The MPC law is
[!5.9220 !6.8883]x
5.9220
6.8883
x)
!1.5379 6.8291
1.5379
2.0000
,
2.0000
(Region C2, C4)
!3.4155 4.6452
if 0.1044
0.1259
if
2.0000
2.0000
!6.8291
(Region C1)
2.0000
2.0000
!5.9220 !6.8883
if
15
2.6341
0.1215 x) !0.0353 ,
0.0922
!0.0267
0.0679 !0.0924
0.1259 0.0922
!0.0524
x)
,
!0.0519
(Region C3)
!0.1259 !0.0922
!2.0000
if
u"
!0.0679 0.0924
!0.0519
x)
,
!0.0524
(Region C5)
[!6.4159 !4.6953]x#0.6423
if !0.0275 0.1220
6.4159
4.6953
(Region C6)
!2.0000
[!6.4159 !4.6953]x!0.6423
!6.4159 !4.6953
3.4155
!4.6452
1.3577
x) !0.0357 ,
2.6423
2.6341
if !0.1044 !0.1215 x) !0.0353 ,
!0.1259 !0.0922
(Region C7, C8)
6.4159
if 0.0275
4.6953
1.3577
!0.1220 x) !0.0357
!6.4159 !4.6953
(Region C9)
and consists of N "7 regions. Region C1 corres
ponds to the unconstrained linear controller, regions
C2, C3, C4 and C5, C7, C8 correspond to the
!0.0267
2.6423
be done with region C3, as their union would not be
a convex set, and therefore cannot be expressed as one set
of linear inequalities.
16
A. Bemporad et al. / Automatica 38 (2002) 3}20
Fig. 3. Example 7.1. Additional constraint x
*!0.5: (a) closedR>IR
loop MPC; (b) state-space partition and closed-loop MPC trajectories.
The same example is repeated with the additional state
constraint x
*x ,
R>IR
!0.5
x O
,
!0.5
k"1. The closed-loop behavior from the initial condition x(0)"[1 1] is depicted in Fig. 3(a). The MPC
controller was computed in 0.99 s (22 regions examined).
The polyhedral partition of the state space corresponding
to the modi"ed MPC controller is depicted in Fig. 3(b).
The partition provided by the mp-QP algorithm consists
now of N "11 regions (as regions C1, C2 and C3, C9
can be joined, the MPC controller consists of N "9
regions). Note that there are feasible states smaller
than x , and vice versa, infeasible states x*x . This
is not surprising. For instance, the initial state
x(0)"[!0.6,0] is feasible for the MPC controller
(which checks state constraints at time t#k, k"1),
because there exists a feasible input such that x(1)
is within the limits. On the contrary, for
x(0)"[!0.47,!0.47] no feasible input is able to produce a feasible x(1). Moreover, the union of the regions
depicted in Fig. 3(b) should not be confused with the
region of attraction of the MPC closed-loop. For instance, by starting at x(0)"[46.0829,!7.0175] (for
which a feasible solution exists), the MPC controller runs
into infeasibility after t"9 time steps.
7.2. Reference tracking for a MIMO system
Consider the plant
(31)
which was studied by Mulder, Kothare, and Morari
(1999) and Zheng, Kothare and Morari (1994) and by
other authors as an example for anti-windup control
synthesis. The input u(t) is subject to the saturation
constraints
i"1, 2.
Mulder et al. (1999) use a decoupler and two identical PI
controllers
1
K(s)" 1#
100s
2
2.5
1.5
2
.
For the set-point change r"[0.63,0.79] they show that
very large oscillations result during the transient when
the output of the PI controller saturates.
We sample the dynamics (31) with ¹"2 s, and design
an MPC law (24) with N "20, N "1, N "0,
W
S
Q"I, R"0.1I. The closed-loop behavior starting from
zero initial conditions is depicted in Fig. 4(a). It is similar
to the result reported by Mulder et al. (1999), where an
anti-windup scheme is used on top of the linear controller
K(s).
The mp-QP problem associated with the MPC law has
the form (7) with two optimization variables (two inputs
over the one-step control horizon), and six parameters
(two states of the original system, two states to memorize
the last input u(t!¹ ), and two reference signals),
Q
with
H"
0.7578
!0.9699
!0.9699
1.2428
,
F"
4
!5
10
u(t),
y(t)"
100s#1 !3
4
!1)u (t))1,
G
Fig. 4. MIMO example: (a) closed-loop MPC: output y(t) (left), input
u(t) (right); (b) state-space partition obtained by setting u"[0 0] and
r"[0.63 0.79].
0.1111
!0.1422
!0.0711
0.0911
0.7577
!0.9699
!0.9699
1.2426
!0.1010
0.1262
0.0757
!0.1010
,
G"
1
0
!1
0
0
1
0
!1
,
="
1
1
1
1
,
A. Bemporad et al. / Automatica 38 (2002) 3}20
0 0 !1
E"
0
0 0
0 0
0
0 0
1
0
0 0
0 0
0
1
0 0
!1 0 0
.
The solution was computed in 1.15 s (13 regions examined) with Algorithm 1, and the explicit MPC controller
is de"ned over N "9 regions.
A section of the x-space of the piecewise a$ne solution
obtained by setting u"[] and r"[] is depicted in
Fig. 4(b). The solution can be interpreted as follows.
Region C1 corresponds to the unconstrained optimal
controller. Regions C4,C5,C6,C9 correspond to the
saturation of both inputs. For instance in region C4 the
optimal input variation u(t)"[]!u(t!1). In regions
C2,C3,C7,C8 only one component of the input vector
saturates. Note that the component which does not saturate depends linearly on the state x(t), past input u(t!1),
and the reference r(t), but with di!erent gains than the
unconstrained controller of region C1. Thus, the optimal
piecewise a$ne controller is not just the simple saturated
version of a linear controller.
In summary, for this example, the mp-QP solver determines: (i) a two-degree of freedom optimal controller for
this MIMO system (31); and (ii) a set-point dependent
optimal anti-windup scheme, a nontrivial task as the
regions in the (x, r, u)-space where the controller should
be switched must be determined. We stress the fact
that optimality refers precisely to the design performance
requirement (24).
7.3. Inxnite horizon LQR for the double integrator
Consider the double integrator
1
y(t)" u(t)
s
Fig. 5. Double integrator example (N "2, N "7, computation
S
time 0.77 s): (a) closed-loop MPC; (b) polyhedral partition of the statespace and closed-loop MPC trajectories.
This task is addressed by using the MPC algorithm (3)
where N "2, N "2,
W
S
1 0
Q"
,
0 0
R"0.01, and K, P solve the Riccati equation (5). For
a certain set of initial conditions (0), this choice of
P corresponds to setting u "K x
"
R>I
*/ R>IR
[!0.81662 !1.7499]x
and minimizes
R>IR
1
y
y
# u .
(35)
R>IR R>IR 10 R>I
I
On (0), the MPC controller coincides with the constrained linear quadratic regulator, and is therefore
stabilizing. Its domain of attraction is, however, larger
than (0). We test the closed-loop behavior from the
initial condition x(0)"[10!5], which is depicted in
Fig. 5(a).
The mp-QP problem associated with the MPC law has
form (7) with
H"
and its equivalent discrete-time state-space representation
x(t#1)"
1 1
0 1
x(t)#
0
1
(32)
obtained by setting yK (t)+(y (t#¹)!y (t))/¹, y (t)+
(y(t#¹)!y(t))/¹, ¹"1 s.
We want to regulate the system to the origin while
minimizing the quadratic performance measure
1
y(t)y(t)# u(t)
10
R
subject to the input constraint
(33)
(34)
1.6730 0.7207
0.7207 0.4118
,
F"
1
0
!1
0
2.5703 1.0570
0
1
0
!1
1
,
="
1
1
1
0.9248 0.3363
G"
u(t),
y(t)"[1 0]x(t),
!1)u(t))1.
17
,
0 0
,
E"
0 0
0 0
0 0
and was computed in 0.77 s (16 regions examined). The
corresponding polyhedral partition of the state-space is
depicted in Fig. 5(b).
The same example was solved by increasing the number of degrees of freedom N . The corresponding partiS
tions, computation times, and number of regions are
reported in Fig. 6. Note that by increasing the number of
free control moves N , the control law only changes far
S
away from the origin, the more in the periphery the larger
N . This must be expected from the results of Section 3,
S
as the set where the MPC law approximates the
18
A. Bemporad et al. / Automatica 38 (2002) 3}20
Fig. 6. Partition of the state space for the MPC controller: N "numS
ber of control degrees of freedom, N "number of polyhedral re
gions in the controller, o!-line computation time: (a) N "3,
S
N "15, CPU time: 2.63 s; (b) N "4, N "25, CPU time: 560 s;
S
(c) N "5, N "39, CPU time: 9.01 s; (d) N "6, N "57, CPU
S
S
time: 16.48 s.
constrained in"nite-horizon linear quadratic regulation
(C-LQR) problem gets larger when N increases
S
(Chmielewski & Manousiouthakis, 1996; Scokaert
& Rawlings, 1998). By the same arguments, the exact
piecewise a$ne solution of the C-LQR problem can be
obtained for any set of initial conditions by choosing the
"nite horizon as outlined in Section 3.
Preliminary ideas about the shape of the constrained
linear quadratic regulator for the double integrator were
presented in Tan (1991), and are in full agreement with
our results. By extrapolating the plots in Fig. 6, where the
band of unsaturated control actions is partitioned in
2N !1 sets, one may conjecture that as N PR, the
S
S
number of regions of the state-space partition tends to
in"nity as well. Note that although the band of unsaturated control may shrink asymptotically as
xPR, it cannot disappear. In fact, such a gap is
needed to ensure the continuity of the controller proved
in Corollary 2. The fact that more degrees of freedom are
needed as the state gets larger in order to preserve LQR
optimality is also observed by Scokaert and Rawlings
(1998), where for the state-space realization of the double
integrator chosen by the authors the state x"[20,20]
requires N "33 degrees of freedom.
S
8. Conclusions
We showed that the linear quadratic optimal controller for constrained systems is piece-wise a$ne and we
provided an e$cient algorithm to determine its parameters. The controller inherits all the stability and performance properties of model predictive control (MPC)
but can be implemented without any involved on-line
computations. The new technique is not intended to
replace MPC, especially not in some of the larger applications (systems with more than 50 inputs and 150
outputs have been reported from industry). It is expected
to enlarge its scope of applicability to situations which
cannot be covered satisfactorily with anti-windup
schemes or where the on-line computations required for
MPC are prohibitive for technical or cost reasons, such
as those arising in the automotive and aerospace industries. The decision between on-line and o!-line computations must be related also to a tradeo! between CPU (for
computing QP) and memory (for storing the explicit
solution). Moreover, the explicit form of the MPC controller allows to better understand the control action,
and to analyze its performance and stability properties
(Bemporad, Torrisi, & Morari, 2000). Current research is
devoted to develop on-line implementation techniques
which do not require the storage of the polyhedral cells
(Borrelli, Baotic, Bemporad, & Morari, 2001), and to
develop suboptimal methods that allow one to trade o!
between performance loss and controller complexity
(Bemporad & Filippi, 2001).
All the results in this paper can be extended easily to
1-norm and R-norm objective functions instead of the
2-norm employed in here (Bemporad, Borrelli, & Morari,
2000). The resulting multiparametric linear program can
be solved in a similar manner as suggested by Borrelli et
al. (2000) or by Gal (1995). For MPC of hybrid systems,
an extension involving multiparametric mixed-integer
linear programming is also possible (Bemporad, Borrelli,
& Morari, 2000; Dua & Pistikopoulos, 2000). Finally, we
note that the semi-global stabilization problem for discrete-time constrained systems with multiple poles on the
unit circle which has received much attention since the
early paper by Sontag (1984) can be addressed in a completely general manner in the proposed framework.
Acknowledgements
This research was supported by the Swiss National
Science Foundation. We wish to thank Nikolaos
A. Bozinis for his help with the initial implementation
of the algorithm, and Francesco Borrelli and Ali
H. Sayed for suggesting improvements to the original
manuscript.
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20
A. Bemporad et al. / Automatica 38 (2002) 3}20
of the AIChE and was elected to the National Academy of Engineering
(U.S.). Professor Morari has held appointments with Exxon R&E
and ICI and has consulted internationally for a number of major
corporations.
Alberto Bemporad received the master degree in Electrical Engineering in 1993 and
the Ph.D. in Control Engineering in 1997
from the University of Florence, Italy. He
spent the academic year 1996/97 at the
Center for Robotics and Automation,
Dept. Systems Science & Mathematics,
Washington University, St. Louis, as
a visiting researcher. In 1997-1999, he held
a postdoctoral position at the Automatic
Control Lab, ETH, Zurich, Switzerland,
where he is currently a$liated as a senior
researcher. Since 1999, he is assistant professor at the University of
Siena, Italy. He received the IEEE Center and South Italy section `G.
Barzilaia and the AEI (Italian Electrical Association) `R. Mariania
awards. He has published papers in the area of hybrid systems, model
predictive control, computational geometry, and robotics. He is involved in the development of the Model Predictive Control Toolbox for
Matlab. Since 2001, he is an Associate Editor of the IEEE Transactions
on Automatic Control.
Vivek Dua is a Research Associate at the
Centre for Process Systems Engineering,
Imperial College. He obtained B.E.(Honours) in Chemical Engineering from Panjab University, Chandigarh, India in 1993
and M.Tech. in Chemical Engineering
from the Indian Institute of Technology,
Kanpur in 1995. He joined Kinetics Technology India Ltd. as a Process Engineer in
1995 and then Imperial College in 1996
where he obtained PhD in Chemical Engineering in 2000. His research interests
are in the areas of mathematical programming and its application in
process systems engineering.
Manfred Morari was appointed head of
the Automatic Control Laboratory at the
Swiss Federal Institute of Technology
(ETH) in Zurich, in 1994. Before that he
was the McCollum-Corcoran Professor
and Executive O$cer for Control and Dynamical Systems at the California Institute
of Technology. He obtained the diploma
from ETH Zurich and the Ph.D. from the
University of Minnesota. His interests are
in hybrid systems and the control of biomedical systems. In recognition of his research he received numerous awards, among them the Eckman Award
of the AACC, the Colburn Award and the Professional Progress Award
Stratos Pistikopoulos is a Professor in the
Department of Chemical Engineering at
Imperial College. He obtained a Diploma
in Chemical Engineering from the Aristotle University of Thessaloniki, Greece in
1984 and his PhD in Chemical Engineering from Carnegie Mellon University,
USA in 1988. His research interests include
the development of theory, algorithms and
computational tools for continuous and
integer parametric programming. He has
authored or co-authored over 150 research
publications in the area of optimization and process systems engineering applications.