Optimal control of linear systems with limited control actions: threshold-based event-triggered control

0.75 MB·

by  Burak Demirel

DRAFT. 1
Optimal control of linear systems with limited control actions:
threshold-based event-triggered control
Burak Demirel, Euhanna Ghadimi, Daniel E. Quevedo and Mikael Johansson
Abstract—We consider a finite-horizon linear-quadratic opti- problem from a quadratic programming (QP) to a mixed-
mal control problem where only a limited number of control integer quadratic programming (MIQP) problem. As it has
messages are allowed for sending from the controller to the
beenprovedin[12],itisanNP-completeproblem.Therefore,
actuator. To restrict the number of control actions computed
Gao and Lie [13] provided an efficient branch and bound
and transmitted by the controller, we employ a threshold-based
event-triggeringmechanismthatdecideswhetherornotacontrol algorithm to compute the optimal control sequence. Since the
messageneedstobecalculatedanddelivered.Duetothenatureof cardinality constraint is highly non-convex, a convex relax-
7
1 threshold-based event-triggering algorithms, finding the optimal ationofthisproblem,basedon(cid:96)1-regularized(cid:96)2 optimization,
control sequence requires minimizing a quadratic cost function
0 wasconsideredintheliterature;see,e.g.,[14]–[17].Although
over a non-convex domain. In this paper, we firstly provide
2 (cid:96) -normregularizationofthecostfunctionisaneffectiveway
an exact solution to the non-convex problem mentioned above 1

n by solving an exponential number of quadratic programs. To of promoting sparse solutions, no performance guarantees can
a reducecomputationalcomplexity,we,then,proposetwoefficient be provided in terms of the original cost function due to the
J heuristicalgorithmsbasedongreedysearchandtheAlternating transformed cost function.

7 Direction Method of Multipliers (ADMM) method. Later, we
Event-andself-triggeredcontrolsystemshavebeenbroadly
1 consider a receding horizon control strategy for linear systems
used in the literature to reduce the amount of communication
controlled by event-triggered controllers, and we also provide
] a complete stability analysis of receding horizon control that between the controller and the actuator while guaranteeing an
Y usesfinitehorizonoptimizationintheproposedclass.Numerical attainablecontrolperformance;see,e.g.,[18]–[22].Asdistinct
S examplestestifytotheviabilityofthepresenteddesigntechnique. fromsparsecontroltechniquesasmentionedearlier,theevent-
. andself-triggeredcontrolrequirethedesignofbothafeedback
s
c Index terms — Optimal control; Linear systems; Event- controller that computes control actions and a triggering
[
triggered control; Receding horizon control mechanism that determines when the control input has to be

1 updated. A vast majority of the works in the literature first
v I. INTRODUCTION designed a controller without considering sparsity constraint,
1 and then, in the subsequent design phase, they developed
7 The problem of determining optimal control policies for
the triggering mechanism for a fixed controller. In contrast,
8 discrete-time linear systems has been extensively investigated
anotherlineofresearchconcentratesonthedesignoffeedback
4 intheliterature;see,e.g.,[1]–[3].Thestandardoptimalcontrol
0 controllawwhilerespectingapredefinedtriggeringcondition.
problem assumes that an unlimited number of control actions
. The design of event- and self-triggered algorithms can
1 is available at the actuator. Although this assumption is valid
0 for many applications, it does not hold for some specific be extremely beneficial in the context of model predictive
7 control(MPC)strategies;see,e.g.,[23]–[31].MPCisacontrol
scenarios where communication and computation resources
1 scheme that solves a finite-horizon optimal control problem
become scarce; for example, (a) control systems with con-
:
v strained actuation resources [4]–[6], (b) control systems with at each sampling instant and only applies the first element
i of the resulting optimal control input trajectories. The use of
X shared processor resources [7], or (c) information exchange
event-triggered algorithms, therefore, reduces the frequency
over a shared communication channel [8].
r
of solving optimization problems and transmitting control
a Toreducethecommunicationburdenbetweenthecontroller
actions from the controller to the actuator, and, consequently,
and the actuator, Imer and Bas¸ar [9] introduced a constraint
saves computational and communication resources. Lehmann
on the number of control actions while designing optimal
et al. [26] proposed an event-based strategy for discrete-time
controlpoliciesforlinearscalarsystems.Later,Bommannavar
systems under additive, bounded disturbances. The controller
and Bas¸ar [10] and Shi et al. [11] extended the work of [9]
only computes a new control command whenever the dif-
to a class of higher-order systems. The problem, proposed
ference between actual state and predictive state exceeds a
in [9], can be also formulated as a cardinality-constrained
threshold.Sijsetal.[23]combinedstateestimationwithMPC
linear-quadratic control problem. The introduction of cardi-
in order to design an event-based estimation framework. The
nality constraint changes the standard linear-quadratic control
authors of [24], [25] combined MPC with event-triggered
B. Demirel and D. E. Quevedo are with the Faculty of Electrical En- sampling strategies based on the ISS concept. As different
gineering and Information Technology, The University of Paderborn, War- from the aforecited works, the authors of [19]–[21] studied
burger Str. 100, 33098 Paderborn, Germany (e-mail: [email protected];
infinite horizon quadratic cost. All works discussed above
[email protected]).
E. Ghadimi is with Huawei Technologies Sweden AB, Skalhogatan 9-11 focus on discrete-time linear/non-linear systems; however,
box54,SE-16494,Kista,Sweden(e-mail:[email protected]). there are also a substantial number of works, which considers
M. Johansson is with ACCESS Linnaeus Center, School of Electrical
continuous-time systems, in the literature; see, e.g., [30]–
Engineering,KTHRoyalInstituteofTechnology,Osquldasväg10,SE10044
Stockholm,Sweden(e-mail:[email protected]). [32]. For instance, the authors of [30], [32] investigated the
DRAFT. 2
stability of event-based MPC algorithms for continuous-time II. FINITEHORIZONOPTIMALEVENT-TRIGGERED
nonlinearsystemsyettheydidnotconsiderdisturbance.Later, CONTROL
LiandShi[31]studiedtheMPCproblemforcontinuous-time
We consider the feedback control loop, depicted in Fig. 1.
nonlinear systems subject to bounded disturbances.
The dynamics of the physical plant can be described by the
G
Contributions: In this paper, we formulate a finite-horizon discrete-time linear time-invariant system:
optimal event-triggered control problem where a threshold-
: x(t+1)=Ax(t)+Bu(t), x(0)=x , (1)
0
based event-triggering algorithm dictates the communication G
between the controller and the actuator. Then, we propose wherex(t) Rn isthestatevariableattimeinstantt,u(t)
∈ ∈
various algorithms to compute a control action sequence that Rm is the control input at time instant t, A Rn×n and
∈
provides optimal and sub-optimal solutions for this problem, B Rn×m are system matrices of appropriate dimensions,
∈
whichis,ingeneral,hardtosolveduetotwomainreasons:(a) and x0 Rn is the given initial condition. The pair (A,B)
∈
it has the combinatorial nature since the decisions are binary is assumed to be stabilizable, and A is not necessarily Schur
variable (i.e., transmit or not transmit), and (b) introduction stable.
of a threshold-based triggering condition leads to optimizing In this paper, we assume that the sensor S takes periodic
a convex cost function over a non-convex domain. The main noise-free samples of the plant state x(t) and transmits these
contributions of this paper are threefold: samples to the controller node. The controller K is event-
triggered, and it computes new control commands and trans-
(i) weshowthattheoptimalsolutionofthecontrolproblem, mits them to the actuator A only at times when x(t) (cid:44)
∈ S
mentioned above, can be determined via solving a set of Rn with
\C0
quadratic programming problems;
(ii) we provide an efficient heuristic algorithm based on C0 (cid:44){x∈Rn :(cid:107)x(cid:107)∞<ε}, (2)
ADMMtodesignacontrolinputsequencethatprovidesa for a given threshold ε>0. In case x(t) , the controller
sub-optimal solution for the finite-horizon optimal event- does not compute any new control action∈s,Ca0nd the actuator
triggered control problem; input to the plant is set to zero:
(iii) we describe the receding horizon implementation of the
event-triggered control algorithm, including a proof of u(t)=0m, ∀t∈T, (3)
practical stability.
where
Outline: The remainder of this paper is organized as fol- (cid:44) t N ;t<N :x(t) . (4)
T { ∈ 0 ∈C0}
lows. Section II formulates the event-triggered finite-horizon
Itisworthnotingthat denotesasetoftimeinstantsatwhich
LQ control problem and introduces assumptions. Section III T
nocontrolcomputationsareneededand,inturn,theplantruns
providesasimpleprocedurefordesigningoptimalcontrollaws
open loop. Notice that is not a given set of variables; in
tominimizethelinear-quadraticcostfunctionwhileSectionIV T
contrast,itisgeneratedbytheinitialstatex andthetentative
presents two heuristic methods that usually achieves tolera- 0
control actions u(t) for all t 0, ,T 1 . Hence, to
ble sub-optimal performance while significantly reducing the ∈ { ··· − }
determine the set , it is necessary to compute the tentative
computational complexity. In Section V, a receding horizon T
control actions u(t). In other words, the set of time instants
control scheme with event-triggered algorithm is presented.
is strongly coupled to the tentative control inputs u(t).
Section VI demonstrates the effectiveness and advantages of T
Throughout this paper, we aim at designing an admissible
thepresentedapproachwhileSectionVIIconcludesthepaper.
optimalcontrolsequenceπ = u(0),u(1), ,u(N 1) to
Notation: We write N for the positive integers, N0 for minimize the quadratic cost fu{nction: ··· − }
N ∪ {0}, and R for the real numbers. Let Rn(cid:23)0 denote the (cid:124)
set of non-negative real vectors of dimension n, and Rn be J(x(0),π)=x (N)Px(N)
the set of real vectors of dimension n. Vectors are written N−1
(cid:88) (cid:0) (cid:124) (cid:124) (cid:1)
in bold lower case letters (e.g., u and v) and matrices in + x (t)Qx(t)+u (t)Ru(t) , (5)
capital letters (e.g., A and B). If u and v are two vectors t=0
in Rn, the notation u v corresponds to component-wise wherethematricesQandP aresymmetricandpositivesemi-
≤
inequality. The set of all real symmetric positive semi-definite definite while R is symmetric and positive definite. Then, we
matrices of dimension n is denoted by Sn(cid:23)0. We let 0n be the consider the constrained finite-time optimal control problem:
n–dimensional column vectors of all zeros, 1 be the vectors
n J(cid:63)(x(0)): minimize J(x(0),π)
of all ones. The Kronecker product of two matrices (e.g., A
π
and B) is denoted by A B. For any given x Rn, the (cid:96)∞– subject to x(t+1)=Ax(t)+Bu(t),
⊗ ∈
norm is defined by x = max x . For a square matrix t 0, ,N 1 ,
(cid:107) (cid:107)∞ 1≤i≤n| i| x(0)=∀x ∈, { ··· − }
A, λ (A) denotes its maximum eigenvalue in terms of 0
max u(t)=0 , t .
magnitude. The notation xk k∈K stands for x(k) : k , m ∀ ∈T
{ } { ∈ K} (6)
where N0. The power set of any set , written P( ),
K ⊆ N N
is the set of all subsets of , including the empty set and Recall that the set of time instants, at which no computations
N N
itself. The cardinality of a set denoted by . (and also transmissions) are needed, i.e., , is defined in (4).
|N| T
DRAFT. 3
x
A G S 0 0 x(t)2Rn\C0
x(t)
2C0
u(t) x(t)
1
K 2
3
K
Fig. 1. Event-triggered control system with the process , the actuator A, t
thesensorSandthecontroller . G
K Fig.2. Adecisiontreediagram.DenoteT (cid:44){t∈N0;t<N :x(t)∈C0}.
Thismeansthat representsasetoftimeinstancesatwhichanewcontrol
Remark 1 Note that the event-triggered control problem, T
signal does not need to be computed and transmitted from the controller to
describedinSectionII,isaparticularclassofhybridsystems. theactuator.Inthefigure,darkcoloredcirclesrepresentthecasex(t)
Onemay,therefore,convertitintoanequivalentmixedlogical foranyt∈T whereaslightcoloredcirclesrepresentthecasex(t)∈R∈nC\0
for any t / . For instance, when N = 4, the set gets values in
dynamical(MLD)systemthatrepresentsthesystembyusinga PC0( 1,2,3 )=∈(cid:8)T, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , 1T,2,3 (cid:9).
blendoflinearandbinaryconstraintsontheoriginalvariables { } ∅ { } { } { } { } { } { } { }
and some auxiliary variables; see, e.g., [33], [34]. In this
optimization problems of the form
paper, instead of using the MLD system formulation, we will
exploit geometric properties of the problem at hand, which JT(cid:63)(x(0)): minimize J(x(0),π)
π
allows us to propose an optimization problem for optimal subject to x(t+1)=Ax(t)+Bu(t),
finite-horizon control and establish stability result for its t 0,1, ,N 1 ,
∀ ∈{ ··· − }
receding horizon implementation. x(t) , t ,
∈C0 ∀ ∈T
x(t) Rn , t / ,
∈ \C0 ∀ ∈T
u(t)=0 , t ,
Discussion: Although the problem (6) resembles the m ∀ ∈T
x(0)=x ,
cardinality-constrained optimal control problem (proposed 0
(7)
in [9]–[17]), there is a fundamental difference between them,
which stems from the question whether scheduling is an where P( ) with = 1,2, ,N 1 . To obtain the
T ∈ N N { ··· − }
exogenous input or autonomously generated? While solving optimalsolutionoftheproblem(6),itisnecessarytosolvethe
the cardinality-constrained control problem, one needs to problem (7) for all subsets of the power set of , i.e., P( ),
N N
optimize both control action and scheduling sequence, which and select the one providing the lowest cost; see Fig. 2. More
arestronglycoupledtoeachotherintheoptimizationprocess. precisely, this can be written as:
On the other hand, the event-triggered control systems are
J(cid:63)(x(0)): minimize J(cid:63)(x(0)).
switched systems with internally forced switchings. In these T (8)
T∈P(N)
problems,schedulingsequencesaregeneratedimplicitlybased
The number of sub-problems, which are needed to be
on the evolution of the state x(t) and the control signal
solved, grows exponentially with the control horizon N (i.e.,
u(t). The major difficulty of this problem is that scheduling
P( ) =2N−1). Bear in mind that since these sub-problems
dependsontheparticularinitialconditionx andthetentative
0 | N |
are independent of each other, they can be also solved in
control input u(t), and cannot be explicitly determined unless
parallel.Inadditiontoitscombinatorialnature,theproblem(7)
a specific control signal was given; see the survey paper [35]
requires the optimization of a convex function over a non-
and references therein.
convex domain. In the literature, there exist some available
tools to solve the convex objective non-convex optimization;
see, e.g., [36]–[38].
III. COMPUTATIONOFOPTIMALCONTROLACTIONS
As the feasible region can be expressed as a finite
S
unionofpolyhedra,thedisjunctiveformulationcanbeapplied.
In this section, we concentrate on finding the control input In particular, assuming n-dimensional problem instance (6),
sequenceπthatsolvesthefinite-horizonoptimalcontrolprob- its feasibility region can be divided into 2n+1 disjunctive
lem,proposedin(6).Therefore,wepresentaframeworkwhich polyhedra1. Following the construction above, we write:
is based on dividing the non-convex domain into convex sub-
2n
domains. This is a simple yet effective procedure to follow; (cid:44) (cid:91) , (9)
however, the number of convex optimization problems, which S Cp
p=1
needs to be solved, grows exponentially in the length of the
control horizon. where n is the state dimension, and Cp ⊂ Rn are full-
The optimal control problem (6) is hard to solve due to the
restriction on the control space. Nevertheless, without loss of
1See Fig. 3 for an example of two dimensional problem where each
generality, it is possible to convert this problem to a set of disjunctivesetisshownindifferentcolors.
DRAFT. 4
dimensional polyhedra, i.e., 1
C2
(cid:44)(cid:8)x Rn :T x d(cid:9), C1
Cp ∈ p ≤
0.5
for some Tp ∈R(2n−1)×n and d ∈R2n−1. x2
We define a piece-wise constant function of time σ(t), e
bl
pwhdicehtertmakineess,onatveaalcuhestiimne{0t,1,··0·,,2n},Tand 1wh,otsheevsatlautee varia 0 C0
e
variable x(t) to belong to the i∈nte{rior··o·f the−poly}hedron Cp. Stat 0.5
We divide the non-convex set into convex subsets with
p −
S C
p 0,1, ,2n and, at each time step t 0, ,T 1 ,
∈{ ··· } ∈{ ··· − } C3
we choose only one active set. The switching signal, 1 C4
(cid:40) −
0 if t , 1 0.5 0 0.5 1
σ(t)= ∈T (10) − −
p otherwise, Statevariablex1
gives a sequence Σ= σ(0), ,σ(N 1) . Then, for given rFeiggi.o3n.s, wThhiechnoanr-ecoinllvuesxtrasteetdSb(cid:44)y dRi2ffe\reCn0t ccaonlobrse.dFivoirdeadgiinvteonfosucrhecdounlvinexg
{ ··· − }
and Σ, we rewrite the optimization problem (7) as sequence, i.e., = 6,7 , the optimal control inputs can be computed by
T solving the optTimizat{ion p}roblem (11) for all possible combinations of five
J(cid:63)(x(0)): minimize J(x(0),π) convex sets. For N = 7, finding optimal control actions requires solving
Σ π 15,625 problems, and selecting the minimum control cost among 2,650
subject to x(t+1)=Ax(t)+Bu(t),
feasible solutions. In Fig. 3, the blue curve represents the state trajectory
t 0,1, ,N 1 , forΣ = 4,4,4,1,1,0,0 ,theredcurverepresentsthestatetrajectoryfor
x(t)∀∈∈Cσ{(t), ∀·σ·(·t)∈−Σ, } fΣo2rΣ=1{=4,{14,1,1,2,2,2,2,0,3,0,0},,0a}n.dthemagentacurvedenotesthestatetrajectory
u(t)=0 , t , 3 { }
m ∀ ∈T
x(0)=x0. optimal control inputs, it is necessary to solve 5N−1 convex
(11) quadratic programming problems.
It is necessary to solve all possible combination of prob-
lem(11)todeterminetheglobaloptimalsolution.Particularly, Example 2 Let us consider the second-order system:
the problem (6) can be solved as a sequence of (2n+1)N−1 (cid:20) (cid:21) (cid:20) (cid:21)
0.9 0.2 0.6
quadratic programming (QP) problems. Hence, we have the x(t+1)= x(t)+ u(t), (13)
0.8 1.5 0.8
following series of optimization problems:
(cid:124)
with the initial condition x = [0−1] . The event-triggered
J(cid:63)(x(0)): minimize J(cid:63)(x(0)). 0
Σ (12) controller transmits control messages if the (cid:96) -norm of the
Σ∈{0,···,2n}N−1 ∞
statevariablex(t)isgreaterthanε=0.25.Theperformance
It is worth noting that even though the number of QP
indices are given by Q = diag 2,2 and R = 5. The
sub-problems grows exponentially with N and n, all these { }
control horizon is chosen as N = 7. The state trajectories,
problems are independent of each other; therefore, one can
depicted in Fig. 3, can be obtained via solving the optimiza-
parallelize the problems and reduce the computation time.
tion problem (11) for given sets Σ = 4,4,4,1,1,0,0 ,
1 { }
Σ = 4,1,1,2,2,0,0 , and Σ = 4,1,2,2,2,0,0 . Fig. 3
Corollary 1 The global optimal solution of the control prob- 2 { } 3 { }
shows the state trajectories for two different sets. The set Σ
lem (6) can be computed by solving a set of convex quadratic leads to the global optimal solution. 1
programming problems of the form (11), in parallel, and
selecting the solution that provides the lowest control cost Example1demonstratesthatobtainingtheoptimalsolution
among all feasible solutions. to the finite-horizon optimal control problem, mentioned ear-
lier,requiressolvinganexponentialnumberofQPs.Therefore,
The following examples should give the reader a better
in the next section, we propose heuristic algorithms that sig-
understanding of the problem.
nificantly reduce computational complexity, by compromising
the optimality guarantees slightly.
Example 1 Letn=2and =R2 with = (x ,x )
S \C0 C0 { 1 2 ∈
R2 : ε<x <ε, ε<x <ε . The non-convex set can
be di−vided in1to four−convex2subse}ts: S IV. COMPUTATIONOFSUB-OPTIMALCONTROLACTIONS
Inthissection,weproposetwoheuristicalgorithmsthatusu-
= (x ,x ) R2 :x x ,x x ,x ε ,
C1 { 1 2 ∈ 1 ≥ 2 1 ≥− 2 1 ≥ } ally achieve a low control loss while significantly decreasing
= (x ,x ) R2 :x x ,x x ,x ε , the computation time. We, firstly, develop a simple algorithm
C2 { 1 2 ∈ 1 ≤ 2 1 ≥− 2 2 ≥ }
C3 ={(x1,x2)∈R2 :x1 ≤x2,x1 ≤−x2,x1 ≤−ε}, tsheaqtuetankceesΣtheininaitigarleesdtaytemxa0nnaesr ibnypuitncarnedasifinngdsthae shuoirtiazbolne
= (x ,x ) R2 :x x ,x x ,x ε ,
C4 { 1 2 ∈ 1 ≥ 2 1 ≤− 2 2 ≤− } of the problem from 1 to N. Secondly, we apply an algo-
whichareillustratedbydifferentcolorsinFig.3.Theoptimal rithm based on ADMM to compute the heuristic scheduling
controlinputscanbecomputedbysolvingquadraticprograms sequenceΣ.Thissequenceisthenusedtosolvethedisjunctive
for different combinations of convex sets. In order to compute problem(11)andobtainagoodapproximatecontrolsequence
DRAFT. 5
Data: x0, n, N; of random initial starts and a number of local search methods
1 Compute σ(0) based on x0; applied to ADMM solutions.
2 for Iteration count k =2,...,N do
3 Obtain x˜ =x(cid:63) and u˜ =u(cid:63) by solving (11) with
Σ= σ(0), ,σ(k 2),p for all In this paper, using some of the techniques introduced
{ ··· − }
p 0,1, ,2n ; above, we first apply ADMM to original non-convex problem
∈{ ··· }
4 Compute p(cid:63) =argminΣJΣ(cid:63)(x(0)); to achieve an intermediate solution. This algorithm, for a
5 Set σ(k 1)=p(cid:63); general non-convex problem, does not necessarily converge;
−
6 end however, when it does, the corresponding solution gives a
7 return Σ, x˜, and u˜; good initial guess for finding the optimal solution to this
Algorithm 1: Greedy heuristic problem. In a second phase of our heuristic method, we
perform a polishing technique with the aim of achieving a
π. Both techniques are of polynomial time complexity, hence, feasible and hopefully closer to the optimal solution (see [38]
reducing the computational efforts significantly. Numerical for an overview of polishing techniques). Our polishing idea
investigations (see Section VI) indicate that these heuristic is based on disjunctive programming problem (11) with the
algorithms usually provide close to optimal solutions. fixed event index and trajectory sequence Σ.
T
A. Greedy heuristic method
Initially developed to solve large-scale convex structured
Weconstructanintuitivegreedyalgorithmtosolvetheopti- problems, ADMM has been recently advocated, to a great
malcontrolproblem,proposedin(6),efficiently.Asdiscussed extent, for being effective approximation technique to solve
earlier,givenafeasibleswitchingsequenceΣ,acorresponding non-convex optimization problems (see [38]–[40] and refer-
optimal control problem can be formulated as a quadratic ences therein). We start with casting (6) to the ADMM form.
programming problem of the form (11). One, then, solves Essentially, we have
the combinatorial problem (12) to find the optimal switch
1
sequenceΣ(cid:63),therebysolvingtheoptimalcontrolproblem(6). minimize z(cid:62)Fz+ (w)
z 2 I
A good way to deal with the combinatorial nature of this (14)
Gz =h,
typeofalgorithmsandreducethecomputationalcomplexityis
z =y,
to employ a greedy algorithm, which solves a global problem
where z collects the state and control components; i.e.,
by making a series of locally optimal decisions. Greedy algo-
z (cid:44)[x(0)(cid:62),...,x(N)(cid:62),u(0)(cid:62),...,u(N 1)(cid:62)](cid:62), F repre-
rithms do not necessarily provide optimal solutions; however,
−
sentsthepositivedefiniteHessianofthequadraticcost.Thatis
they might determine local optimal solutions, which can be
F (cid:44)blkdiag(Q ,R )withQ (cid:44)I QandR (cid:44)I R.
used to estimate a globally optimal solution, in a reasonable s s s N+1 s N
⊗ ⊗
Moreover, (y) is the indicator function enforcing the event-
time.
I
Our proposed greedy algorithm works as follows. Since x triggering control law; that is
0
is known, we can compute corresponding switching signal (cid:26) if(16)holds,
σ(0). Then, we set Σ = σ(0),p for all p 0,1,...,2n I(y)= ∞0 otherwise, (15)
{ } ∈ { }
and solve 2n + 1 number of QPs of the form (11) to
obtain the optimal switch signal σ(1). The same procedure
t 0,...,N 1 y(t+1) andy(t+N +2)=0 ,
0 m
then repeats for k = 2,...,N 1, where at each step k, {∃ ∈{ − }| ∈C (cid:54) }
(16)
−
we solve 2n + 1 times the QP problem (11) to find the
best σ(k) keeping the previously found scheduling sequence where (16) detects the constraint violation in (6) in terms
Σ= σ(0),...,σ(k 1) unchanged. of having x(t) 0 while corresponding u(t) = 0m.
∈ C (cid:54)
{ − } The constraints in (14) are twofold. The first constraint
Algorithm1providesthedetailedstepsofdiscussedgreedy
heuristic method. Algorithm 1 executes (2n + 1)(N 1) in (14) describes the LTI system dynamics (1). Here G
∈
numberofQPscomparedto(2n+1)N−1 asinoptimalco−ntrol RnN×(n(N+1)+mN) have the following i = 1,...,N , j =
{ }
1,...2N +1 - blocks,
procedure. As a result, it runs in polynomial time.
{ }

Ai−1 for i=1...N,j =1,
B. A heuristic method based on ADMM Gij = −AiI+nN−jB ijf=j iaNnd+j2>a1n,di+N j,
In this section, we develop a computationally efficient  0 othe≥rwise. ≥
methodforsolving(6).Ourapproachisbasedon,Alternating
Direction Method of Multipliers (ADMM), a well developed and h Rn(N+1)+mN (cid:44) [x(0)(cid:62),0(cid:62),...,0(cid:62)](cid:62). Finally, the
technique to solve large-scale disciplined problems [39]. The last con∈straint in (14) is of consensuns-type tomensure that non-
idea of utilizing ADMM as a heuristic to solve non-convex convex constraint in (6) is asymptotically satisfied.
problems has been recently considered in literature [38]–[41].
Inparticular,[38],[40]employADMMtoapproximatelysolve
problems with convex costs and non-convex constraints. The After formulating the problem, each iteration of ADMM
approximate ADMM solutions are improved by multiplicity algorithmconsistsof(see[39]foracompletetreatmentofthe
DRAFT. 6
subject): Data: (cid:15)tol, z0 (0,σ2I), zbest = , and fbest = ;
∼N ∅ ∞
(cid:18) 1 for Iteration count k =1,2,...,K do
1
zk+1/2 =argmin z(cid:62)Fz 2 update z from (17);
2
+ρ(cid:13)(cid:13)(cid:13)z(cid:20) G (cid:21)z (cid:20) 0 (cid:21)zk (cid:20) h (cid:21)+uk(cid:13)(cid:13)(cid:13)2(cid:33) 43 if (cid:107)Gzbzeskt−=hz(cid:107)k;≤(cid:15)tol and (1/2)zk(cid:62)Fzk <fbest then
2(cid:13) I − I − 0 (cid:13) (17) 5 end
zk+1 =Π(zk+1/2+(cid:2) 0 I (cid:3)uk) 6 end
uk+1 =uk+(cid:20) G (cid:21)zk+1/2 (cid:20) 0 (cid:21)zk (cid:20) h (cid:21). 87 if fbOesbtta<in∞zbetshtenfrom disjunctive problem (11);
I − I − 0 9 end
Here, k denotes the iteration counter, ρ > 0 is the step-size 10 return zbest;
Algorithm 2: ADMM-Disjunctive heuristic
(penalty) parameter, and Π denotes the projection onto non-
convex constraint and is given by
doesnotimprovethepreviousoneintermsofconstraint
(cid:26)
0 ifi N +2 and x(i N 1) violations and quadratic cost.
Πi(x)= x(i) othe≥rwise. − − ∈C0 3) The variable z0 is initialized randomly using a normal
(18) distribution. In general, repeating the ADMM algorithm
When the cost function in hand is convex and the con- for a multiple of initial points may improve the ap-
straints set is closed and convex then the ADMM algo- proximate solution at the cost of extra computational
rithm converges to the optimal solution of the problem. complexity (as suggested in e.g., [40]). However, in
In our case, since (y) is non-convex the ADMM algo- ourapplicationwedidnotexperiencesignificantperfor-
I
rithm (17) may not converge to optimal point. Moreover, manceimprovementbyrepeatingtheADMMalgorithm
examples can be found in which (17) does not even con- (steps1 6inAlgorithm2)withdifferentinitializations.
−
verge into a feasible point. To further improve the ADMM OnereasoncouldbethatinAlgorithm2,weemploythe
solution, we utilize an additional convex problem to pol- solution trajectories Σ(cid:101) and (cid:101) based on the ADMM so-
T
ish the intermediate results. That is, we restrict the search lution z˜and not z˜itself in the disjunctive problem (11).
space to a convex set that includes the ADMM solution pat- 4) Using caching and LDL(cid:62) matrix factorization tech-
tern. Let z˜(cid:44)[x˜(0)(cid:62),...,x˜(N)(cid:62),u˜(0)(cid:62),...,u˜(N 1)(cid:62)](cid:62) niquesinthefirst-stepof(17)[39],eachz-updateinAl-
− (cid:0) (cid:1)
be the output of the ADMM algorithm and the sets Σ(cid:101) gorithm2costsontheorderofO (n(N +1)+mN)2
and (cid:101) (cid:44) t N ;t<N :x˜(t) denote the trajectory for dense P matrices. Furthermore, if we assume N
sequeTnce a{ss∈ocia0ted with [x˜(cid:62),.∈..C,x0˜}(cid:62)](cid:62) and the restriction max n,m then each ADMM iteration costs on th≥e
1 N
{ }
index of x˜ into 0, respectively. Then our heuristic polishing order of O(N2). This means that overall cost of the
C
procedure formulates (11) with = (cid:101) and Σ = Σ(cid:101). Algo- ADMM algorithm is of O(KN2) where K is the
T T
rithm2providesasummaryofourheuristicmethod.Theinput number of ADMM iterations. In our application, with
variablesincludetheerror-tolerancethreshold(cid:15)tol,randomized proper algorithm parameter selection, the ADMM algo-
initial state z0 as well as zbest and fbest to store the final rithm converges almost always within a few hundreds
solution. of iterations (see the results presented in Section VI).
A few comments related to Algorithm 2 are in order. 5) The computational complexity of the disjunctive
step in our heuristic method falls into the generic
1) In contrast to the ADMM algorithms for convex prob-
complexity of convex QPs on the order of
lemsthatconvergetotheoptimumforallpositiverange
O(( 2n+(N )(2n 1))2) [44].
of the step-size parameter ρ, the stability of the ADMM |T| −|T| −
algorithm for non-convex problems is sensitive to the
choice of ρ. Moreover, similar to the convex case, V. RECEDINGHORIZONCONTROL
the convergence speed is also affected by the choice Except a few special cases, finding a solution to an infinite
of ρ. A variety of techniques including proper step- horizon optimal control problem is not possible. Receding
size selection, over relaxation, constraint matrix pre- horizon control is an alternative scheme to infinite horizon
conditioning and caching can be employed to accelerate problem that repeatedly requires solving a constrained opti-
theADMMprocedure(17)(see[42],[43]forareference mization problem. At each time instant t 0, starting at
≥
on the topic). the current state x(t) (assuming that a full measurement of
2) One can utilize an adaptive iteration count procedure the state x(t) is available at the current time t ∈ N0), the
to improve the probability of finding feasible solution following cost function
to (6). The procedure works as the following. Run the
J(x(t),π(cid:63)N−1)=V (x(t+N t))
ADMM algorithm for a fixed number of iterations and t f |
then check if the disjunctive QP problem has feasible t+N−1
(cid:88)
solution. If not, then increase the number of iterations + (cid:96)(x(k t),u(k t))
| |
(e.g., double it) and repeat the procedure (run ADMM k=t
and disjunctive QP) until finding a feasible solution or subject to system dynamics and constraints involving states
reaching to a point where the current ADMM solution and controls is minimized over a finite horizon N. Here, the
DRAFT. 7
1.2 R = 5 while the prediction horizon is set to N = 6. Fig. 4
showsthestatetrajectoriesoftherecedinghorizonimplemen-
tation of the event-triggered control system, described by (1)
0.6
– (4), for different initial conditions x . As can be seen in
0
2
x the same figure, the number of transmitted control commands
e
bl depends on the initial condition.
aria 0
v
ate Dµ
St VI. NUMERICALEXAMPLES
0.6
− A. Performanceofsub-optimalevent-triggeredcontrolactions
To illustrate performance benefits of the proposed heuristic
1.2 algorithms, we consider a third-order plant with the following
−
1.2 0.6 0 0.6 1.2
− − state-space representation:
Statevariablex
1
   
Fig.4. Evolutionofthestatetrajectoriesofthesystem(13)underthereceding 0.53 2.17 0.62 0.4
−
horizonstrategy(19)andconstraints(2)–(4)forasetofinitialconditionsx0. x(t+1)= 0.22 0.06 0.51x(t)+0.7u(t)
−
0.92 1.01 1.69 0.9
function (cid:96) defines the stage cost, and V defines the terminal − −
f
+B w(t). (20)
cost. Denote the minimizing control sequence, which is a w
function of the current state x(t), by
Assume that there is no disturbance acting on the plant, i.e.,
π(cid:63)N−1 =(cid:2)u(cid:63)(t t)(cid:124), ,u(cid:63)(t+N 1 t)(cid:124)(cid:3), Bw =0.
t | ··· − | Our aim, here, is to minimize (5) with performance indices
then the control applied to the plant at time t 0 is the first Q = P = diag 2,2,2 and R = 5, and the horizon length
≥ { }
element of this sequence, that is, N =8. The initial condition is chosen as
u(t)=(cid:2)I 0 ··· 0(cid:3)πt(cid:63)N−1 . x0 =(cid:2)sinθcosφ sinθsinφ cosθ(cid:3)(cid:124) ,
Time is then stepped forward one instant, and the procedure where 0 θ π and 0 φ π. To perform simulations,
≤ ≤ ≤ ≤
described above is repeated for another N-step ahead opti- we selected 577 different data points, which are equidistant
mization horizon. overahalfsphericalsurface.Weevaluatedtheperformanceof
It is worth noting that, due to the fundamental limitation of the greedy and the ADMM-based heuristic algorithms for all
the optimization problem (6), the state x(t) never converges theseinitialconditions.Foreachdatapointonthehalfsphere,
to the origin if the system is unstable. It can only converge we run 823,543 number of QPs to obtain the global optimal
into a set including the origin. The following result shows the solution. In total, we run more than 475 million number of
existence of this set: QPsinourevaluation.Weusedalinuxmachinewith32cores
in Intel Core i7 Extreme Edition 980X to solve these QPs
Theorem 1 Suppose that Dµ (cid:44){x∈Rn :(cid:107)x(cid:107)∞≤µ} is a in parallel threads. Within this computing resources, it takes
neighborhood of the origin, where approximately 6 days to perform simulations for one problem
(cid:115) setup.
κλ (A(cid:124)PA+Q)nε2
µ(cid:44) max (19) To compare the (true) optimal control performance with
λ2 (Q)
min our heuristic method, we performed Algorithm 2 on the
for some κ > 0. Then, the system (1) with event-triggering same problem set. In particular, we evaluated three sets
constraints (3) and (4) is uniformly practically asymptotically of problem with ε = 0.2, ε = 0.4, and ε=0.6. It is
stable, i.e., lim x(t) µ. generally expected that the control problem becomes more
t→∞(cid:107) (cid:107)∞≤ challenging with increasing the event-threshold ε. We run
Theorem 1 establishes practical stability of the system (1) the ADMM-based heuristic algorithm with adaptive iteration
with event-triggering constraints (3) and (4). It shows that count procedure described in Section IV-B. Moreover, we
if provided conditions are met, then the plant state will be picked ρ = 9.8, ρ = 5.8, and ρ = 6.9 respectively,
ultimatelyboundedinan(cid:96) -normballofradiusµ.Itisworth for the aforementioned problem scenarios. Under this setup,
∞
noting that, as the other stability results, which use Lyapunov in both cases (i.e., ε = 0.2 and ε = 0.4), the ADMM
techniques, this bound will not be tight. Next, we would like method produced always feasible solutions for maximum 300
to exemplify the convergence of the state x(t) to the set number of ADMM-iterations. However, when ε = 0.6 and
Dµ
from a set of initial conditions x . ρ = 6.9, the method created infeasible solutions for two
0
(cid:62)
initial conditions: x = [0.3036, 0.2330, 0.9239] and x =
0 0
Example 3 Consider the second-order system, given by (13), [0.6830, 0.1830, 0.7071](cid:62). To compute feasible solutions, we
(cid:124)
with initial conditions x = 1.2[sin(πk) cos(πk)] for all 0 needed to adjust ρ accordingly for these initial conditions.
0 6 6 ≤
k 12. The controller transmits a control message whenever Fig. 5 shows the outcome of the comparison. As it shows,
≤
the event-triggering condition, i.e., x(t) > 0.25, holds. for ε = 0.2, the ADMM-based heuristic algorithm is able
(cid:107) (cid:107)∞
The performance indices are chosen as Q = diag 2,2 and to find near optimal solutions (with relative error less than
{ }
DRAFT. 8
ε=0.2 ε=0.4 ε=0.6 1
1 0.8 0.8
0
0.5 0.4 0.4 x(t)1 1
− exhaustivesearch
ADMM
0 0 0 2
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 − 0 10 20 30 40 50
(Jadmm−J?)/J? (Jadmm−J?)/J? (Jadmm−J?)/J? 1
exhaustivesearch
0.6 0.6 0.6 ADMM
0.5
0.4 0.4 0.4 x(t)2 0
0.2 0.2 0.2
0 0 0 −0.5
-2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 0 10 20 30 40 50
? ? ?
|Tadmm|−|T | |Tadmm|−|T | |Tadmm|−|T | 2
Fig. 5. Relative error (Jadmm −J(cid:63))/J(cid:63) and control-action cardinality 1
difference of the (sub-optimal) ADMM-based heuristic solution from the
optimalcostfor577probleminstancesofthirdorderplantmodel(20). x(t)3 10
ε=0.2 ε=0.4 ε=0.6 − exhaustivesearch
2
− ADMM
0.6 0.4 0.4 −30 10 20 30 40 50
0.4
2
0.2 0.2
0.2
1
0 0 0
0 (J0.g1reedy0−.2J?)/J0?.3 0.4 0 (J0.g1reedy0−.2J?)/J0?.3 0.4 0 (J0.g1reedy0−.2J?)/J0?.3 0.4 u(t) 0
1
0.6 0.4 0.4 −
0 10 20 30 40 50
0.4 2
0.2 0.2
0.2
0 0 0 (t)m 1
-5 -4|Tg-3reedy-2|−|-T1?| 0 1 -5 -4|Tg-3reedy-2|−|-T1?| 0 1 -5 -4 |-T3gree-2dy|−-1|T0?| 1 2 uadm 0
Fig. 6. Relative error (Jgreedy −J(cid:63))/J(cid:63) and control-action cardinality −1
difference of the greedy solution from the optimal cost for 577 problem 0 10 20 30 40 50
instancesofthirdorderplantmodel(20).
Fig.7. Statesandinputversustimefortheevent-triggeredmodelpredictive
controllerpresentedinSectionVI-B.Inthisfigure,x (t),x (t)andx (t)
5%) in almost all cases, and for ε = 0.4 and ε = 0.6, the 1 2 3
representstatevariableswhereasu(t)andu (t)denotetheoptimaland
admm
algorithm finds solutions with relative error less than 5% in theheuristiccontrolinputscomputedbasedontheexhaustivesearchandthe
more than 80% of problem instances. The average optimality ADMMalgorithm,respectively.
gap(J J(cid:63))/J(cid:63) forthethreecaseswere0.0035,0.0237,
admm
−
and0.0716,respectively.Finally,bycomparingthecardinality is chosen to be N = 6 with performance indices Q =
of control event index for the optimal solutions and our P = diag 2,2,2 and R = 5. The event-threshold ε for
T { }
ADMM method, one concludes that the heuristic method in the event-triggered receding horizon implementation is set to
more than 50% of cases finds similar-sparse solution as the 0.4. Fig. 7 demonstrates the time responses of the event-
optimum ones, and in the rest of cases it tends to find more triggered receding horizon control system for the initial con-
sparse control actions. (cid:104) √ √ (cid:105)(cid:62)
dition x = 0, 2, 2 . Here, two different approaches
We also compared the true optimal control performance 0 2 − 2
for computing control input trajectories are considered: the
with a greedy heuristic method, shown in Algorithm 1, for
exhaustive search algorithm and the heuristic ADMM algo-
three sets of problems, mentioned above, with ε = 0.2,
rithm with ρ=4.8. As can be seen in Fig. 7, while the state
ε = 0.4, and ε=0.6. The greedy algorithm detected 22,
trajectories associated to two control approaches deviate from
90, and 135 optimal solutions of the aforementioned 577
one another (except for the few first sampling instants that
problems, respectively.Also, itcomputed the optimalsolution
they match together), the corresponding control inputs follow
withrelativeerrorlessthan5%inmorethan65%ofproblems
a rather similar sparsity pattern. The event-triggered receding
at hand for all three problem setups. It is important to note
horizon control via exhaustive search algorithm uses the com-
that the greedy heuristic method always provided feasible
munication channel 14 times over the 50 sampling instants,
solutions for all problem cases. The average optimality gaps
which results in a reduction of 72% in communication and
(J J(cid:63))/J(cid:63) in all three cases were 0.0514, 0.0780,
greedy
− computational burden, whereas the event-triggered receding
and 0.0826, respectively. As compared to the ADMM-based
horizon control via ADMM algorithm uses the channel 16
heuristic method, the optimality gaps became larger.
timesoverthe50samplinginstants,whichleadstoareduction
of 68% in communication and computational burden. For the
aforementioned initial value, the exhaustive search achieves a
B. Receding horizon control implementation
better control cost: 65.42 compared to 77.72 as of the control
We consider the open-loop unstable discrete-time system cost of the ADMM-based heuristic. However, this conclusion
represented by (20) with B = 0. The prediction horizon does not hold in general. In particular, we experienced cases
w
DRAFT. 9
200 4
programming formulation to obtain the optimal solution via
solving an exponential number of quadratic programs. Then,
175 3
∞
J we proposed a heuristic algorithms to reduce the compu-
loss 150 2 tational efforts while achieving an acceptable performance.
Control 125 1 Later, we provided a complete stability analysis of receding
horizon control that uses a finite-horizon optimization in the
proposed class. Our numerical study confirmed the theory.
100 0
0.6 0.7 0.8 0.9
Communicationrateπ
∞ VIII. APPENDIX
Fig.8. Acomparisonofthecontrolperformanceandthecommunicationfre-
quencyfordifferentevent-thresholdvaluesε>0(showningrayscale).The We begin with the definition of the practical stability of
markedcurvewithisobtainedbyaveraging2,500MonteCarlosimulations control systems.
fthoer tihneitihalorcioznodnitlieonngtxh 5g0e0nesraamtepdlersanwdiothmltyh.e process noise {w(t)}t∈N0and
0 Definition 1 (practical stability [45]) The system (1) is said
wherestartingfromadifferentinitialvalue,therecedinghori- to be uniformly practically asymptotically stable in Rn
A ⊆
if is a positively invariant set for (1) and if there exist a
zon controller derived by exhaustive search performs worse
A
-function β, and a nonnegative constant δ 0 such that
than the one based on ADMM-based heuristic.
KL ≥
x(t) β( x ,t)+δ .
0
(cid:107) (cid:107)≤ (cid:107) (cid:107)
C. Trade-off between communication rate and control perfor-
mance Definition 2 (practical-Lyapunov function [45]) A function
We consider the case where a random disturbance w(t) V : Rn → R≥0 is said to be a practical-Lyapunov function
∈ in for the system (1) if is a positively invariant set and
Rn, which is modeled by a discrete-time zero-mean Gaussian A A
if there exist a compact set, Ω , neighbourhood of the
white process with co-variance Σ , is acting on the plant ⊆ A
w origin,some -functionsα ,α andα ,andsomeconstants
characterizedbytheinjectionmatrixB =diag 1,1,1 .The K∞ 1 2 3
initial condition x is modeled as a ranwdom vari{able ha}ving a d1,d2 ≥0, such that
0
normal distribution with zero mean and co-variance Σ . The V(x) α ( x ), x , (23)
x 1
0 ≥ (cid:107) (cid:107) ∀ ∈A
process noise w(t) is independent of the initial condition x .
0 V(x) α2( x )+d1, x Ω, (24)
The control performance is measured by a quadratic func- ≤ (cid:107) (cid:107) ∀ ∈
V(f(x)) V(x) α ( x )+d , x . (25)
tion: − ≤− 3 (cid:107) (cid:107) 2 ∀ ∈A
499 Proof of Theorem 1. To prove the practical stability of
J = 1 (cid:88)(cid:0)x(cid:124)(t)Qx(t)+u(cid:124)(t)Ru(t)(cid:1), (21) the system (1) with a set of control actions π(cid:63) = u(cid:63)(t
∞ 500 { |
t=0 t), ,u(cid:63)(t + N 1 t) , which fulfills constraints (3)
··· − | }
while the communication rate is measured by and (4), we will analyze the value function,
π = 1 (cid:88)4991 . (22) VN(cid:63)(x(t))(cid:44)mπinJ(x(t),π),
∞ 500 {(cid:107)x(t)(cid:107)∞≥0}
t=0 where J(x(t),π) is as seen in (5). We borrow the shifted
The visualization of the trade-off between the control loss sequence approach, described in [46], and we use a feasible
and the communication rate is demonstrated in Fig. 8. Differ- control sequence, i.e., π˜ = u(cid:63)(t+1 t), ,u(cid:63)(t+N 1
{ | ··· − |
ent data points on the curve correspond to different event- t),uˆ . By the optimality property, we obtain the following
}
threshold values ranging from 0.5 to 4.0, and the control bound (with x(t t)=x(t) and u(cid:63)(t t)=u(t)):
| |
loss (21) and the communication rate (22) are evaluated via V(cid:63)(x(t+1)) V(cid:63)(x(t)) V (x(t+1),π˜) V(cid:63)(x(t))
Monte Carlo simulations. Note that the communication rate N − N ≤ N (cid:0) − N (cid:1)
= (cid:96)(x(t),u(t))+V Ax(t+N t)+Buˆ
decreases dramatically with an increased control loss as the − f |
threshold ε varies between 0 and 2.25 (dark colors). For Vf(x(t+N t))+(cid:96)(x(t+N t),uˆ)
− | |
ε>2.25 (lighter color), both quantities become less sensitive (26)
to changes in the threshold value. We can also identify for all x(t) Rn. We now investigate the quantity of
0 ε 1.5 as a particularly attractive region where a large ∈
≤ ≤
decreaseinthecommunicationratecanbeobtainedforasmall ∆V (t)+(cid:96)(x(t+N t),uˆ),
f |
loss in control performance. where∆V (t)(cid:44)V (cid:0)Ax(t+N t)+Buˆ(cid:1) V (x(t+N t)).
f f | − f |
Based on x(t+N t), there are two possibilities:
VII. CONCLUSION |
• Supposethat(cid:107)x(t+N |t)(cid:107)∞>ε,thenforanyfeasible
In this paper, we considered a feedback control system control law, i.e., uˆ = Kx(t + N t), the following
|
where an event-triggering rule dictates the communication inequality holds:
between the controller and the actuator. We investigated the
∆V (t)+(cid:96)(x(t+N t),uˆ)
optimal control problem with an additional constraint on f |
the event-triggered communication. While this optimization =x(t+N t)(cid:124)(cid:0)A(cid:124) PA P +Q∗(cid:1)x(t+N t) <0,
| K K − |
problem, in general, is non-convex, we used a disjunctive (27)
DRAFT. 10
whereA =A+BK (i.e.,Schur)andQ∗ =Q+K(cid:124)RK. which implies that
K
From (26) and (27), it follows that a η
x(t+1) 2 γ 3 x(t) 2 + . (33)
V(cid:63)(x(t+1)) V(cid:63)(x(t))<0. (cid:107) (cid:107)2≤ a2 (cid:107) (cid:107)2 a2
N − N
Since 0<a a , it follows that γ =1 a2, i.e., 0<γ
• Suppose that (cid:107)x(t+N |t)(cid:107)∞≤ε, then uˆ =0. Hence, 1. Therefore2, b≤y it3erating (33), it is possib−leat3o exponentiall≤y
it follows that:
bound the state evolution via:
∆Vf(t)+(cid:96)(x(t+N |t),u(cid:124)ˆ(cid:0)) (cid:124) (cid:1) x(t) 2 γta3 x(0) 2 +1−γt η .
=x(t+N t) A PA P +Q x(t+N t) (cid:107) (cid:107)2≤ a (cid:107) (cid:107)2 1 γ a
| (cid:124)(cid:0) (cid:124) − (cid:1) | 2 − 2
x(t+N t) A PA+Q x(t+N t)
≤≤≤λλλmmaaxx(cid:0)(cid:0)(cid:0)AAA(cid:124)(cid:124)(cid:124)PPP| AAA+++QQQ(cid:1)(cid:1)(cid:1)nn(cid:107)ε(cid:107)x2x(,t(+t+NN|t|)t)(cid:107)22(cid:107)|2∞ Uansyinvg∈th(cid:107)eRxnno((trs)me(cid:107)e2∞in[4e≤7q]uγ)a,tlniwtyaae3(cid:107)g(cid:107)evtx:(cid:107)(∞0)≤(cid:107)(cid:107)2∞v+(cid:107)121≤−√γγtnaη(cid:107) v. (cid:107)∞ for
≤ max 2 − 2
where we used the norm inequality v √n v Thus, lim x(t) µ. (cid:4)
(cid:107) (cid:107)2≤ (cid:107) (cid:107)∞ t→∞(cid:107) (cid:107)∞≤
for any v Rn; see [47].
∈
As a result, we conclude that, for any x(t+N t) Rn, the REFERENCES
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