Optimal control of discrete-time switched
linear systems via continuous
parameterization
Jérémie Kreiss ∗ Laurent Bako ∗∗ Eric Blanco ∗∗
arXiv:1704.06985v1 [cs.SY] 23 Apr 2017
∗
Laboratoire Ampère, INSA de Lyon, Université de Lyon
20, Avenue Albert Einstein, 69100 Villeurbanne, France
E-mail: jeremie.kreiss@insa-lyon.fr
∗∗
Laboratoire Ampère – Ecole Centrale de Lyon – Université de Lyon
36, Avenue Guy de Collongue, 69134 Ecully, France
Abstract: The paper presents a novel method for designing an optimal controller for discretetime switched linear systems. The problem is formulated as one of computing the discrete mode
sequence and the continuous input sequence that jointly minimize a quadratic performance
index. State-of- art methods for solving such a control problem suffer in general from a high
computational requirement due to the fact that an exponential number of switching sequences
must be explored. The method of this paper addresses the challenge of the switching law design
by introducing auxiliary continuous input variables and then solving a non-smooth block-sparsity
inducing optimization problem.
Keywords: optimal control, switched linear systems, quadratic control, dynamic programming.
1. INTRODUCTION
Switched systems constitute a class of dynamic systems
consisting of a finite number of subsystems which are activated one after another over time by a switching signal [12],
[8], [7]. In many cases, the switching signal is an external
input which, together with the continuous input vector,
can be used to control the behavior of the switched system.
Examples of real processes that can be represented as
switched systems with external discrete and/or continuous
inputs are: autonomous vehicles, chemical processes, electrical circuits, etc. (see e.g., [12] and [8] for a background).
The problem of controlling switched systems of this type
therefore involves designing a continuous control law along
with a switching sequence to achieve some performance
specifications. We will be interested more specifically in
optimal control design for discrete-time switched linear
systems (SLS).
The existing approaches to the problem of optimal control
for switched systems can be classified into two groups:
the ones addressing the continuous-time case [13, 5, 9,
10, 11] and those pertaining to the discrete-time case
[14], [6]. Other classifications can be done with respect
to the length of the control horizon (finite or infinite), the
simplifying assumptions posed a priori on the structure
of the discrete or the continuous input or the conceptual
nature of the methods. One observation that can be made
is the following. Considering the quadratic optimal control
of SLS in continuous-time where no structure is imposed
in advance on the discrete and continuous control policies,
there is so far no exact solution for both finite and
infinite control horizons. In discrete-time an exact solution
has been derived in [14] for the case of finite control
horizon. However direct numerical implementation proves
to be so expensive that it is not affordable in practice.
Therefore a relevant question of major importance is how
to develop some suboptimal strategies that would be much
less expensive while still being close to optimality. Some
relaxations have been discussed in [14, 6] for this purpose.
However the resulting control algorithms still suffer from
an exponential demand in storage capacity.
In this paper, we focus on the quadratic optimal control of
switched linear systems. The discussion here is restricted
to finite-horizon problems but the envisioned ultimate goal
is to extend it to infinite horizon. We first observe that the
origin of the huge complexity associated with computing
the solution to this optimization problem is the presence
of discrete variables. Therefore a key idea in our approach
is to parameterize the discrete inputs by auxiliary control
variables which are continuous. With this reformulation,
the SLS optimal control problem becomes completely
continuous but nonconvex. We then discuss some convex
relaxation strategies. More precisely, we propose solving
a sequence of convex problems in order to estimate the
auxiliary control variables. The whole process yields an
implementation which is shown in simulation to coincide
statistically very often with the true optimal control. The
advantage of the proposed computational scheme is that
it has only a polynomial complexity. Moreover it requires
the same storing capacity (for the sequence of positive
semidefinite matrices generated by the Riccati recursion)
as the solution to the simpler linear quadratic optimal
problem.
The structure of this paper is as follows. In Section 2,
we formulate the switched quadratic control problem. We
discuss the expression of the optimal solution and the
associated complexity issue. In Section 3 we present the
new continuous parameterization of the discrete control
variable and develop a four-steps algorithm for solving
it approximately. A numerical illustration is provided in
Section 4. Finally, some concluding remarks are given in
section 5.
[4]. However a direct implementation of the optimal control
suffers from an exponential complexity in both storing
and computational resources. The goal of this paper is to
discuss some alternative formulations and corresponding
solutions of the quadratic optimal control problem for SLS
so as to yield more efficient implementations.
2. PROBLEM FORMULATION
Let us first characterize the solution to problem (3). For
this purpose, let Jk (x, u(·), σ(·)) denote the performance
index corresponding to the situation when the system
starts in an arbitrary state x ∈ Rn at time k and evolves
under the action of the inputs u(·) and σ(·). Introduce the
function Jk∗ : Rn → R defined by
Jk∗ (x) =
min
Jk (x, u(·), σ(·))
2.1 Switched linear systems
We consider a discrete-time switched linear system (SLS)
described by
x(k + 1) = Aσ(k) x(k) + Bσ(k) u(k), x(0) = x0
(1)
where k ∈ Z+ is a time index, x(k) ∈ Rn and u(k) ∈ Rnu
are respectively the state and the continuous input at time
k, x0 is the initial state; σ(k) ∈ Ω , {1, . . . , q} refers to
the value taken by the switching signal (also called here
the discrete input) at time k. Ω is a finite set collecting
the indices of the different subsystems of the SLS (1). For
any i ∈ Ω, the pair (Ai , Bi ) ∈ Rn×n × Rn×nu of matrices
is associated with the subsystem i. Throughout the paper,
we use notations of the type u(·) to designate the entire
sequence {u(k)}.
It is important to note that the switching signal σ(·) ,
{σ(k)} is viewed here as an external input. For simplicity,
we assume that σ(k) can be selected freely in Ω without
any constraint. The control problem of interest is that of
designing jointly a continuous control sequence u(·) and a
discrete control sequence σ(·) so as to minimize a certain
performance index over a finite time horizon. For this
purpose it will be assumed throughout the paper that the
SLS (1) is stabilizable.
2.2 Switched optimal quadratic problem
We consider a quadratic performance index associated
with system (1) in the form
1
J0 (x0 , u(·), σ(·)) = xT (N )Ψx(N )
2
N −1
(2)
1 X T
x (k)Q(k)x(k) + uT (k)R(k)u(k)
+
2
k=0
N −1
where N denotes the control horizon, {R(k)}k=0 ⊂
Rnu ×nu is a given sequence of symmetric positive definite
N
matrices; {Q(k)}k=0 ⊂ Rn×n with Q(N ) = Ψ represents
a sequence of positive semidefinite matrices. The problem
of interest in this paper is stated as follows.
Problem 1. Given the performance matrices {Q(k)}N
k=0
N −1
and {R(k)}k=0 , find a continuous input sequence u(·)
and a discrete input sequence σ(·) that minimize the
performance index (2) subject to the switched system
equation (1). In more formal terms, this is equivalent to
solving the optimization problem
min J0 x0 , u(·), σ(·)
u(·),σ(·)
(3)
s.t. Eq. (1).
We start by observing that the solution to problem (3) can
be mathematically characterized in a quite straightforward
way using, for example, the Bellman optimality principle
u(k),··· ,u(N −1)
σ(k),··· ,σ(N −1)
(4)
s.t. x(t + 1) = Aσ(t) x(t) + Bσ(t) u(t), x(k) = x
t = k, . . . , N − 1.
∗
Jk (x) is called the value function or the cost-to-go. It is
∗
interesting to note from (2) that JN
(x) = xT Ψx for all
n
x ∈ R . Equipped with the notation (4), the Bellman
principle of optimality [4] can be expressed as
∗
ℓ(x, u, k) + Jk+1
(Ai x + Bi u) (5)
Jk∗ (x) =
min
nu
(u,i)∈R
×Ω
1 T
where ℓ(x, u, k) =
x Q(k)x + uT R(k)u is the running
2
cost.
Theorem 1. Consider the control problem (3) and let the
functions Jk∗ be defined as in (4). Denote with u∗ (·), σ ∗ (·),
x∗ (·) respectively the optimal continuous input, discrete
input and continuous state. Then the following statements
hold.
1. The value function Jk∗ is quadratic and can be written
as
Jk∗ (x) = xT P ∗ (k)x ∀x ∈ Rn , ∀k ∈ {0, . . . , N } , (6)
where {P ∗ (k)} is a sequence of matrices generated
recursively backward in time according to the Ricatti
recursion defined by
P ∗ (k) = ρσ∗ (k),k (P ∗ (k + 1)), P ∗ (N ) = Ψ
(7)
with
ρi,k (P ) =Q(k) + AT
i P Ai
−1 T
T
Bi P Ai .
− Ai P Bi R(k) + BiT P Bi
(8)
2. The optimal continuous input is given by
u∗ (k) = −Kσ∗ (k),k P ∗ (k + 1) x∗ (k)
(9)
−1
BiT P Ai .
where Ki,k (P ) = R(k) + BiT P Bi
∗
3. The optimal discrete input σ (·) is given by
σ ∗ (k) ∈ argmin x∗ (k)T ρi,k (P ∗ (k + 1))x∗ (k) . (10)
i∈Ω
Proof. The proof follows by a simple backward induction
exploiting the Bellman optimality equation (5). It is therefore omitted.
2.3 Implementation of the optimal control
In order to implement the control law (9)-(10), we need
to fully compute offline the sequence {P ∗ (k)}N
k=0 by the
Ricatti recursion (7). Note that this recursion depends on
the optimal switching signal σ ∗ (·) which in turn depends
on the optimal continuous state x∗ (·) as can be seen from
Eq. (10). Unfortunately the optimal state is not available
offline. This is a source of a major difficulty. To cope with
this challenge an elegant solution has been developed in
[14]. The authors first showed that the value function given
in (6) can be re-expressed as
Jk∗ (x)
• A nonsmooth block-sparsity-inducing optimization
involving the auxiliary control variables. The purpose
of this is to enforce their expected structure as will
be described next.
3.1 Continuous parameterization of the discrete input
T
= min x P x
P ∈Hk
where {Hk } is a sequence of sets of symmetric positive
semidefinite matrices defined by 1
HN = {Ψ}
Hk = ρi,k (P ) : P ∈ Hk+1 , i ∈ Ω , k = N − 1, . . . , 0.
(11)
The sets {Hk } collect indeed all the possible values of the
Ricatti sequence (7) for all admissible switching signals.
Since the definition of the sets {Hk } is now freed from the
dependence on the continuous state, they can be computed
offline and stored. Once this is done the optimal control
law can be obtained online by applying
σ ∗ (k) ∈ arg min x∗ (k)⊤ ρi,k (P )x∗ (k)
(12)
i∈Ω,P ∈Hk+1
along with (9).
It turns out that the trick of [14] makes the optimal
control (9)-(10) implementable. This is done however at
the price of a huge complexity. In effect, the cardinality
of the sets Hk grows exponentially fast with respect to
the control horizon N . For example, the cardinality of Hk
is about q N −k . Indeed the above implementation requires
an exponential load in terms of both computational and
storage resources. This complexity affects all the steps
of the implementation: offline computation and storage
of the sets {Hk } and online reading and search for the
optimal discrete control by Eq. (12) over the sets {Hk }. To
reduce the complexity, a suboptimal solution is discussed
in [14]. But it is fair to observe that some shortcomings still
persist. First, the proposed procedure does not alleviate
the off-line computational load ; it only allows for a saving
of the necessary storage capacity. Moreover the cardinality
reduction algorithm has still to test all the elements in the
sets Hk hence resulting in an exponential complexity.
This complexity restricts the applicability of the solution
of [14] to the control of switched systems with small
number of subsystems and small control horizons. Noting
that exponential complexity is generated by the presence
of discrete variables in the optimization problem, we
discuss here a new approach which relies on a continuous
parameterization of the switching sequence.
3. PROPOSED SOLUTION
The proposed design method relies on two main ideas:
• A continuous parameterization of the discrete control
variable σ(·). This consists in replacing the discrete
input in the SLS (1) with continuous variables called
auxiliary control variables. Consequently, problem (3)
can be written as a constrained optimization problem
in only continuous variables hence getting rid of its
combinatorial feature.
1 Note that in the setting of [14] the weighting matrices Q and R
depend on the subsystem index, not on time. As a consequence ρ is
indexed there only by i.
A starting point is to notice that the SLS dynamics in (1)
can be written in the form
x(k + 1) = Ai x(k) + Bi u(k) + fi (k), ∀i ∈ Ω
(13)
where fi (k) = (Aσ(k) − Ai )x(k) + (Bσ(k) − Bi )u(k). Each
fi (k) can be interpreted as the difference between the state
of the SLS at time k + 1 under σ(k) and the state that
would have been obtained if σ(k) was set equal to i. It
follows that for any time instant k, fσ(k) (k) = 0.
From now on let us forget about the explicit expressions
of the fi (k)’s and view them just as unknown control
variables satisfying the following constraint: for all k, there
exists j = j(k) ∈ Ω such that fj (k) = 0n,1 where 0n,1
denotes a n-dimensional vector with all entries equal to
zero. By letting f¯(k) = [f1 (k)T · · · fq (k)T ]T ∈ Rqn , the
above constraint can be written as f¯(k) ∈ S with S being
a subset of Rqn defined by
S = v = (v1T , · · · , vqT )T ∈ Rqn , with vi ∈ Rn , i ∈ Ω :
∃j ∈ Ω : vj = 0n,1 .
(14)
There are many other equivalent ways of representing the
set S. One of those which are smooth is the following
n
S = v = (v1T , · · · , vqT )T ∈ Rqn , with vi ∈ Rn , i ∈ Ω :
q
Y
kvi k = 0
i=1
o
for any vector norm k·k on Rn . The variable f¯(k) is then
regarded as an auxiliary control variable to be computed
along with the continuous input variable. Based on this
parameterization we now restate the control problem as
follows.
N
N −1
Problem 2. Given the matrices {Q(k)}k=0 and {R(k)}k=0 ,
find a continuous input sequence u(·) and an auxiliary
input sequence f¯(·) that minimize the performance index
N
−1
X
1 T
ℓ(x(k), u(k), k)
x (N )Ψx(N ) +
2
k=0
(15)
subject to system (13) written in expanded form as
B1
A1
x(k + 1)
.
..
..
= . x(k) + .. u(k) + f¯(k) (16)
.
Bq
x(k + 1)
Aq
and the constraint f¯(k) ∈ S.
V (x0 ,u(·), f¯(·)) =
To write this in a more compact form, define
T
T T
, B̄ = B1T · · · BqT ,
Ā = AT
1 · · · Aq
T
Ln,q = [In · · · In ] = 1q ⊗ In ,
where 1q = [1 · · · 1]T is a vector of q ones, In is the ndimensional identity matrix and ⊗ denotes the Kronecker
product. Then problem (2) can be written as a constrained
optimization problem as follows
min V x0 , u(·), f¯(·)
¯
u(·),f(·)
s.t. Ln,q x(k + 1) = Āx(k) + B̄u(k) + f¯(k) (17)
f¯(k) ∈ S, k = 0, . . . , N − 1
x(0) = x0 .
It is important to note that problems (3) and (17) are
equivalent. But contrary to (3), problem (17) is an optimization problem in which all the decision variables
are continuous. One can therefore hope for designing an
algorithm that solves it in polynomial time. A remaining
challenge to deal with is the non convexity of (17).
3.2 An algorithm in four stages
We now ask the question of how to compute numerically
the solution to problem (17). Because of the non-convex
constraint f¯(k) ∈ S, the problem is not convex. For
efficiency of solving we therefore need to find a convex
relaxation. We will discuss a block-sparsity inducing optimization technique for that. The global computational
procedure can be decomposed into four stages which are
described next.
N −1
Stage (a) In this first stage, the sequence f¯(k) k=0 is
considered as a (unknown) parameter. Hence the criterion
V is minimized with respect to u(·) only, i.e., we solve
min V x0 , u(·), f¯(·)
u(·)
s.t. Ln,q x(k + 1) = Āx(k) + B̄u(k) + f¯(k),
(18)
k = 0, . . . , N − 1
x(0) = x0 ,
where the constraint on f¯(k) has been removed. This is
a convex program. One difference with the classical linear
quadratic control problem is that the dynamic matrix Ā in
(18) is rectangular rather than square. Also the state x(k+
1) is repeated here q times on left hand side of the dynamics equation. Denote with V1 (x0 , f¯(·)) the resulting optimal
functional, i.e., V1 (x0 , f¯(·)) = minu(·) V x0 , u(·), f¯(·) .
Using the method of Lagrange multipliers, construct the
Lagrangian V̄ by embedding the constraints of (18) in the
cost functional. This yields
V̄ = V x0 , u(·), f¯(·) +
N
−1
X
λ̄T (k + 1) Āx(k) + B̄u(k) + f¯(k) − Ln,q x(k + 1)
k=0
(19)
where λ̄(k) , with λ̄(k) = [λ1 (k)T · · · λq (k)T ]T , is a sequence of Lagrange multipliers. Introduce now the discrete
time Hamiltonian H associated with the system (16) and
the performance index (15) defined by
Hk = H x(k), u(k), λ̄(k + 1), k)
= ℓ(x(k), u(k), k) + λT (k + 1) Āx(k) + B̄u(k) + f¯(k)
(20)
With this short-hand notation the extended cost (19) takes
the form
1
V̄ = x(N )T Ψx(N ) − λ̄T (N )Ln,q x(N )
2
N
−1
(21)
X
Hk − λ̄T (k)Ln,q x(k) + H0 .
+
k=1
Considering the minimization of V̄ with respect to u, let
us look at the effect of an elementary variation du of u
and of the induced change dx in the state. These together
induce a variation dV̄ of V̄ , expressible by
T
dx(N )
dV̄ = Ψx(N ) − LT
n,q λ̄(N )
!
N
−1
T
X
∂Hk
∂Hk
T
+
− Ln,q λ̄(k) dx(k) +
du(k)
∂x(k)
∂u(k)
k=1
+
∂H0
∂H0
dx(0) +
du(0).
∂x(0)
∂u(0)
(22)
Note that dx(0) = 0 since the initial state is fixed. At the
optimal value of V̄ we must have dV̄ = 0 as a consequence
of any variation du of the input and any subsequent
variation dx of the state. This implies that all terms in
the previous equation must be set to zero. Thus for all
k = 1, . . . , N − 1, λ̄(k) must satisfy
∂Hk
LT
(23)
, LT
n,q λ̄(k) =
n,q λ̄(N ) = Ψx(N ).
∂x(k)
Similarly, we must impose
∂Hk
= 0.
∂u(k)
It follows that the optimal continuous input can be expressed as
(24)
u(k) = −R−1 B̄ T λ̄(k + 1).
We can rewrite (16) and (23) respectively as follows
Ln,q x(k + 1) = Āx(k) − B̄R−1 B̄ T λ̄(k + 1) + f¯(k) (25)
with initial condition x(0) = x0 and
T
LT
n,q λ̄(k) = Qx(k) + Ā λ̄(k + 1),
with the final constraint
LT
n,q λ̄(N )
(26)
= Ψx(N ).
Eqs (25) and (26) form a system of linear equations that
characterizes completely the solution to (18).
Stage (b) Now, we seek the minimal value of V1 (x0 , f¯(·))
with respect to f¯(·). Since for any k the solution f¯(k)
is expected to live in the set S defined in (14), the
optimization problem is as follows
min V1 x0 , f¯(·)
f¯(·)
(27)
s.t. f¯(k) ∈ S ∀k ∈ {0, · · · , N − 1}.
As mentioned earlier, this problem is not convex. In order
to keep polynomial complexity, we need to find a convex
relaxation of it.
To do so, let us denote with χS : Rnq → R ∪ {+∞} the
indicator function of S defined from Rnq to real extended
line by χS (z) = 0 if z ∈ S and χS (z) = +∞ otherwise.
Then (27) is equivalent
−1
i
h
NX
χS (f¯(k)) .
min V1 x0 , f¯(·) +
f¯(·)
k=0
To find a convex relaxation of the terms χS (f¯(k)), we view
each vector f¯(k) as being relatively block-sparse in the
sense that it must admit at least one subvector fj (k) which
¯
is equal to zero. With this in mind we
Pqreplace χS (f (k))
with a nonsmooth convex function,
i=1 wi (k) kfi (k)k2
where the wi (k)’s denote some positive weights. As is
suggested by a certain number of results (see e.g., [3, 2]),
minimizing such a function enjoys the nice property that
it is able to promote block sparsity hence yielding a
vector f¯(k) with some of its subvectors fi (k) potentially
equal or close to zero. The role of the weights {wi (k)} is
to discriminate between the different subvectors of f¯(k).
They can be selected, for example, iteratively as follows:
solve the problem with all weights set to one and based
on the resulting solution, retune the weights through a
simple rule of the form wi (k) = 1/(kfˆi (k)k2 + ǫ) for some
small number ǫ > 0. For better numerical stability, one can
consider normalizing a posteriori the weights along the i
dimension.
Finally we formulate the following convex optimization
problem:
q
N
−1 X
h
i
X
¯
(28)
wi (k) kfi (k)k2 .
min γ1 V1 x0 , f (·) + γ2
f¯(·)
k=0 i=1
For writing simplicity, we have not given here the explicit expression of V1 (x0 , f (·)). The derivation of such an
expression follows from straightforward algebraic calculations departing from the system of linear equations (25)(26).
¯(k) = [fˆ⊤ (k) · · · fˆ⊤ (k)]⊤
Stage (c) Let fˆ¯(k) with fˆ
1
q
denote a solution to problem (28). Since this is obtained
under some relaxation of the initial problem, there is no
guarantee that fˆ¯(k) will lie in the set S for all k. Hence,
¯(k) onto S in order to determine
we need to project those fˆ
the switching sequence. The projection we used consists in
selecting the discrete input to be the index i ∈ Ω such that
fˆi has the minimum norm among all, i.e.,
σ̂(k) ∈ argminkfˆi (k)k, k = 0, . . . , N − 1,
(29)
i∈Ω
with k·k denoting the vector 2-norm. In some sense, this
corresponds to forcing to zero the fˆi having the minimum
norm.
Stage (d)
Once we have computed the switching se−1
quence {σ̂(k)}N
k=0 offline, the SLS optimal control problem reduces to that of a linear time-varying system with
matrices defined by (Aσ̂(k) , Bσ̂(k) ). The solution of such
a problem with respect to the quadratic cost (2) can be
determined as in the conventional case [1]. To obtain it, we
can just apply (7)-(8) with σ ∗ (·) replaced by σ̂(·) to generate offline a sequence of matrices {P̂ (k)}. By storing this
single Ricatti sequence, the final continuous and discrete
inputs are then, similarly as in (9)-(10), selected online as
û(k) = −Kσ̂(k),k (P̂ (k + 1))x̂(k)
σ̂(k) ∈ arg min x̂(k)T ρi,k (P̂ (k + 1))x̂(k)
(30)
(31)
i∈Ω
with x̂(·) denoting the associated state trajectory. This
means that the discrete input is recalculated online.
As it turns out, the implementation proposed in this paper
has a polynomial complexity. Moreover it requires storing
only a single sequence of matrices {P̂ (k)} just as in the
solution of the linear quadratic problem (single linear subsystem). In comparison with [14], the gain on the memory
demand is significant. However in its current version, the
proposed implementation is not guaranteed to yield the
optimal control. In view of the applied control policy (30)(31), the sole objective of the procedure described above
for solving Problem 2 is the computation of the Ricatti
sequence {P̂ (k)}.
4. RESULTS
Note that for all the following tests, the weights γ1 and γ2
are taken equals to 1.
4.1 A statistical test
In this section we challenge the capability of the proposed
method to obtain the solution to the optimal quadratic
control problem for switched systems of the form (1). For
this purpose 100 examples of two-dimensional switched
systems, with each composed of q = 2 subsystems, are generated at random using the MATLAB function drss. The
initial state is also sampled from a Gaussian distribution
N (0, 20I2 ). For each of these examples, the optimal control
problem is that of finding the discrete and continuous
inputs to minimize a finite horizon performance index
of the form (2) with N = 15 and with Q(k) = I2 and
R(k) = 1 for all k.
For the sake of comparison, we also implement the optimal
control law as described in [14] and recalled above in
Eqs (11)-(12). This is possible here since the number
of subsystems and the control horizon are small. As a
matter of fact the cardinality of H0 in (11) for the current
experiment is about 215 = 32768, which is computationally
affordable on a standard computer. Let Jiopt and Jio ,
i = 1, . . . , 100, denote respectively the optimal index and
the value yielded by our method. For each of the 100
examples, define the following relative error as an empirical
measure of the distance to optimality
J o − J opt
εi = i opti .
Ji
The table below displays the distribution of εi in terms
of probabilities of the type Prob(ε
i ≤ α) with α being
a threshold taking values in 10−10 , 10−8 , 10−7 , 10−5 .
It turns out that the solution achieved by the proposed
method either coincides with the optimal index or lies
generally in a small neighborhood of it. This suggests that
for generic systems, the implemented strategy discussed
in this paper has the potential of recovering the optimal
control and this, at a much affordable price.
Threshold on εi
10−5
10−7
10−8
10−10
0
% examples
100%
98%
97%
96%
83%
Table 1.
Distribution of the relative error εi for the 100 examples of
switched systems: (n, q) = (2, 2).
Repeating a similar experiment as above with this time
examples of switched systems with state dimension n = 3
and number of modes q = 3 yield the results reported
in Table 2. Note that for this last experiment the control
horizon has been reduced to N = 10 in order to alleviate
the computational load associated with the computation
of the exact optimal solution. A little degradation of
the results can be observed in Table 2 in comparison
Threshold
10−2
10−5
10−7
10−8
10−10
0
% examples
100%
96%
93%
92%
90%
81%
2
σ
with the results given in Table 1. This may be due to
numerical artefacts as a result of increased number of
decision variables. The approximate performance index is
still very close to the optimal one.
1
0
1
2
3
4
5
Table 2.
Distribution of the relative error εi for the 100 examples of
switched systems: (n, q) = (3, 3).
5. CONCLUSION
In this paper, we studied the discrete-time quadratic
optimal control problem for switched systems on a finite
time horizon. Based on a continuous parameterization
of the discrete input, we proposed an approach that is
able to yield the optimal solution in polynomial time
with respect to the length of control horizon. Moreover
the proposed algorithm appears to be cheaper than most
existing methods in terms of computational load and
storage resources. Future research will focus on analyzing
the properties of this method.
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4.2 Illustration of performance on a single example
x∗∗1
x
x̂1 2
x̂2
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2
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