Symbolic models for nonlinear control systems affected by disturbances

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by  Alessandro Borri

SYMBOLIC MODELS FOR NONLINEAR CONTROL SYSTEMS
AFFECTED BY DISTURBANCES
ALESSANDROBORRI1,GIORDANOPOLA1 ANDMARIADOMENICADIBENEDETTO1
2 Abstract. In the last few years there has been a growing interest in the use of symbolic models for the
1 formal verification and control design of purely continuous or hybrid systems. Symbolic models are abstract
0 descriptions of continuous systems where one symbol corresponds to an ”aggregate” of continuous states.
2 In this paper we face the problem of deriving symbolic models for nonlinear control systems affected by
disturbances. The main contribution of this paper is in proposing symbolic models that can be effectively
n
constructedandthatapproximatenonlinearcontrolsystemsaffectedbydisturbancesinthesenseofalternating
a
approximatebisimulation.
J

6
2

C] 1. Introduction
O
Anemergingtrendinthecontrolsystemsandcomputersciencecommunitiesistheuseofsymbolicmodelsfor
.
h theanalysisandcontroldesignofpurelycontinuousorhybridsystems[EFP06]. Symbolicmodelsareabstract
at descriptions of continuous systems where each symbol corresponds to an ”aggregate” of continuous states
m [Tab09]. The use of symbolic models provides a formal approach to solve control problems in which software
and hardware interact with the physical world. Moreover, it provides the designer with a systematic method
[
toaddressawidespectrumofnovelspecificationsthataredifficulttoenforcebymeansofconventionalcontrol

1
design paradigms. Examples of such specifications include logic specifications expressed in linear temporal
v
logic or automata on infinite strings.
4
1 The literature on symbolic models is very broad and includes results on timed automata [AD90], rectangular
6 hybridautomata[HKPV98]ando-minimalhybridsystems[LPS00,BM05]. Earlyresultsforclassesofcontrol
5 systems were based on dynamical consistency properties [CW98], natural invariants of the control system
.
1 [KASL00], l-complete approximations [MRO02] and quantized inputs and states [FJL02, BMP02]. Recent
0 results include work on piecewise-affine and multi-affine systems [HCS06, BH06], set-oriented discretization
2 approach for discrete-time nonlinear optimal control problem [Jun04] and abstractions based on convexity of
1
reachable sets for sufficiently small sampling time [Rei09]. Symbolic models for nonlinear control systems,
:
v time–delay systems and switched systems based on the notions of approximate bisimulation [GP07] and in-
Xi cremental stability [Ang02] have been studied in [PGT08, PT09], [PPDT10, PPDB10] and [GPT10].
Inthispaperwefacetheproblemofderivingsymbolicmodelsfornonlinearcontrolsystemsaffectedbydistur-
r
a bances. The presence of disturbances requires us to replace the notion of approximate bisimulation employed
in [PGT08, GPT10, PPDT10] with the notion of alternating approximate bisimulation introduced in [PT09]
and inspired by Alur and coworkers’ alternating bisimulation [AHKV98]. As discussed in [PT09, Tab09] this
notion is a key ingredient when constructing symbolic models of systems affected by disturbances because
it guarantees that control strategies synthesized on the symbolic models can be readily transferred to the
originalmodel. Theexistenceofalternatingapproximatelybisimilarsymbolicmodelsforincrementallystable
nonlinear control systems affected by disturbances has been proven in [PT09]. However, the results of [PT09]
cannot be easily used for the construction of symbolic models because they rely on the computation of sets of
reachablestateswhichisadifficulttaskingeneral. Inthisworkweproposealternativesymbolicmodelstothe
ones proposed in [PT09] which are proven to be effectively computable. The key ingredient in our results is
thederivationoffiniteapproximationsofthedisturbanceinputfunctionalspacebyresortingtosplineanalysis
ThisworkhasbeenpartiallysupportedbytheCenterofExcellenceforResearchDEWS,UniversityofL’Aquila,Italy.
1
2 ALESSANDROBORRI,GIORDANOPOLAANDMARIADOMENICADIBENEDETTO
[Sch73]. Spline analysis has been also employed in [PPDT10, PPDB10] for deriving symbolic models of time–
delay systems. As discussed in the paper, the approximation scheme proposed in [PPDT10, PPDB10] cannot
be used in this framework because it would lead to symbolic models that cannot be effectively constructed.
Forthisreasoninthispaperweelaboratealternativespline–basedapproximationschemesforthedisturbance
input functional space which instead guarantee the effective computation of the proposed symbolic models.
The main contribution of this paper lies in showing that:
If the control system is incrementally stable and the disturbance input signals are bounded and Lipschitz
continuousthensymbolicmodelscanbeeffectivelyconstructedwhichareshowntobealternatingapproximately
bisimilar to the original control systems with any desired accuracy.
Apreliminaryversionofthisworkappearedintheconferencepublication[BPD11]. Thispaperisorganizedas
follows. PreliminarydefinitionsarerecalledinSection2. InSection3weproposeaspline–basedapproximation
scheme for the disturbance input functional space. In Section 4 we show how to construct symbolic models
that approximate nonlinear control systems affected by disturbances in the sense of alternating approximate
bisimulation. Section 5 shows an illustrative example. Finally Section 6 offers some concluding remarks.
2. Preliminary definitions
2.1. Notation. A singleton is a set containing exactly one element. The identity map on a set A is denoted
by 1 . Given two sets A and B, if A is a subset of B we denote by 1 : A (cid:44)→ B or simply by ı the natural
A A
inclusion map taking any a ∈ A to ı(a) = a ∈ B. Given a function f : A → B the symbol f(A) denotes the
image of A through f, i.e. f(A):={b∈B :∃a∈A s.t. b=f(a)}; if C ⊂A we denote by f| the restriction
C
of f to C, i.e. f| (x) := f(x) for any x ∈ C. Given a relation R ⊆ A×B, the symbol R−1 denotes the
C
inverse relation of R, i.e. R−1 :={(b,a)∈B×A:(a,b)∈R}; we set R(A)={b∈B|∃a∈A s.t. (a,b)∈R}
and R−1(B) = {a ∈ A|∃b ∈ B s.t. (a,b) ∈ R}. The symbols N, Z, R, R+ and R+ denote the set of natural,
0
integer, real, positive real, and nonnegative real numbers, respectively. Given a vector x ∈ Rn, we denote
by (cid:107)x(cid:107) the infinity norm of x. Given a measurable function f :R+ →Rn, the (essential) supremum of f is
0
denoted by (cid:107)f(cid:107) . Given µ ∈ R+ and A ⊆ Rn, we denote by µA the set {b ∈ Rn|∃a ∈ A s.t. b = µa}. A
∞
continuous function γ :R+ →R+ is said to belong to class K if it is strictly increasing and γ(0)=0; function
0 0
γ is said to belong to class K if γ ∈K and γ(r)→∞ as r →∞. A continuous function β :R+×R+ →R+
∞ 0 0 0
is said to belong to class KL if, for each fixed s, the map β(r,s) belongs to class K with respect to r and,
∞
for each fixed r, the map β(r,s) is decreasing with respect to s and β(r,s) → 0 as s→∞. The symbol
C0([0,τ];Y) denotes the set of continuous functions from a closed interval of the form [0,τ] with τ ∈ R+ to
a set Y ⊆ Rm. Consider a bounded set A ⊆ Rn with interior. Let H = [a ,b ]×[a ,b ]×···×[a ,b ] be
1 1 2 2 n n
the smallest hyperrectangle containing A and set µˆ = min (b −a ). It is readily seen that for any
A i=1,2,...,n i i
µ≤µˆ and any a∈A there always exists b∈(2µZn)∩A such that (cid:107)a−b(cid:107)≤µ.
A
2.2. Controlsystemsandincrementalstability. Inthispaperweconsiderthefollowingnonlinearcontrol
system:
(2.1) x˙ =f(x,u,d),
where x ∈ X ⊆ Rn is the state, u ∈ U ⊆ Rm and d ∈ D ⊆ Rl are the control and disturbance inputs. We
suppose that f(0,0,0) = 0, the set X is convex with the origin as an interior point and the sets U and D
are compact, convex, with the origin as an interior point. Control input functions are supposed to belong to
the set U of piecewise–constant functions of time from intervals of the form ]a,b[⊆R to U. Disturbance input
functions are supposed to belong to the set D of continuous functions of time of the form d:]a,b[⊆R→D
satisfying the following Lipschitz assumption: there exists κ ∈R+ such that:
d
(2.2) (cid:107)d(t )−d(t )(cid:107)≤κ |t −t |,
2 1 d 2 1
SYMBOLIC MODELS FOR NONLINEAR CONTROL SYSTEMS AFFECTED BY DISTURBANCES 3
foranyd∈Dandt ,t ∈]a,b[. Functionf :Rn×U×D →Rn iscontinuousandenjoysthefollowingLipschitz
1 2
assumption: for every compact set K ⊂Rn, there exists a constant k ∈R+ such that
(cid:107)f(x,u,d)−f(y,u,d)(cid:107)≤k(cid:107)x−y(cid:107),
for all x,y ∈ K, u ∈ U and d ∈ D. In the sequel, we refer to the nonlinear control system in (2.1) by means
of the tuple:
(2.3) Σ=(X,U,D,f),
where each entity has been defined above. Since control inputs are piecewise–constant, system Σ is often
referred to in the literature as a nonlinear sample–data control system, see e.g. [NT01].
A curve ξ :]a,b[→Rn is said to be a trajectory of Σ if there exist u∈U and d∈D satisfying
ξ˙(t)=f(ξ(t),u(t),d(t)),
foralmostallt∈]a,b[. Althoughwehavedefinedtrajectoriesoveropendomains,weshallrefertotrajectories
ξ :[0,τ]→Rn defined on closed domains [0,τ], τ ∈R+ with the understanding of the existence of a trajectory
ξ(cid:48) :]a,b[→Rn such that ξ =ξ(cid:48)| . We also write ξ (t) to denote the point reached at time t under the
[0,τ] xud
control input u and disturbance input d from initial condition x; this point is uniquely determined, since the
assumptions on f ensure existence and uniqueness of trajectories [Son98]. A control system Σ is said to be
forward complete if every trajectory is defined on an interval of the form ]a,∞[. Sufficient and necessary
conditions for a system to be forward complete can be found in [AS99]. In the sequel, we will make use of the
following stability notion.
Definition 2.1. [Ang02] A control system Σ is incrementally input–to–state stable (δ–ISS) if it is forward
complete and there exist a KL function β and two K functions γ and γ such that for any t ∈ R+, any
∞ u d 0
x ,x ∈Rn, any u ,u ∈U and any d ,d ∈D, the following inequality is satisfied:
1 2 1 2 1 2
(cid:107)ξ (t)−ξ (t)(cid:107)≤β((cid:107)x −x (cid:107),t)+γ ((cid:107)u −u (cid:107) )+γ ((cid:107)d −d (cid:107) ).
x1u1d1 x2u2d2 1 2 u 1 2 ∞ d 1 2 ∞
The above incremental stability notion can be characterized in terms of dissipation inequalities, as follows.
Definition 2.2. [Ang02] A smooth function V : Rn ×Rn → R is called a δ–ISS Lyapunov function for a
control system Σ = (X,U,D,f) if there exist λ ∈ R+ and K functions α, α, σ and σ such that for any
∞ u d
x ,x ∈X, any u ,u ∈U and any d ,d ∈D the following conditions hold true:
1 2 1 2 1 2
(i) α((cid:107)x −x (cid:107))≤V(x ,x )≤α((cid:107)x −x (cid:107)),
1 2 1 2 1 2
(ii) ∂V f(x ,u ,d )+ ∂V f(x ,u ,d )≤−λV(x ,x )+σ ((cid:107)u −u (cid:107))+σ ((cid:107)d −d (cid:107)).
∂x1 1 1 1 ∂x2 2 2 2 1 2 u 1 2 d 1 2
The following result adapted from [Ang02] completely characterizes δ–ISS in terms of existence of δ–ISS
Lyapunov functions.
Theorem 2.3. The control system Σ in (2.3) is δ–ISS if and only if it admits a δ–ISS Lyapunov function.
2.3. Transition systems and approximate equivalence notions. We will use alternating transition sys-
tems [AHKV98] to describe both control systems as well as their symbolic models.
Definition 2.4. An (alternating) transition system T is a quintuple:
(cid:45)
T =(Q,L, ,O,H),
consisting of:
• a set of states Q;
• a set of labels L=A×B, where:
– A is the set of control labels,
– B is the set of disturbance labels;
(cid:45)
• a transition relation ⊆Q×L×Q;
• a set of outputs O;
4 ALESSANDROBORRI,GIORDANOPOLAANDMARIADOMENICADIBENEDETTO
• an output function H :Q→O.
A transition (q,(a,b),q(cid:48))∈ (cid:45) is denoted by q (a,(cid:45)b) q(cid:48). A state run of T is a sequence of transitions:
(2.4) q (a1,b(cid:45)1) q (a2,b(cid:45)2) ... (aN−1,bN−(cid:45)1) q .
1 2 N
An output run is a sequence {oi}i∈N of outputs such that there exists a state run of the form (2.4) with
o =H(q ), i=1,2,...,N. Transition system T is said to be:
i i
• countable, if Q and L are countable sets;
• symbolic, if Q and L are finite sets;
• metric, if the output set O is equipped with a metric d:O×O →R+.
0
In the sequel we consider bisimulation relations [Mil89, Par81] to relate properties of control systems and
symbolicmodels. Intuitively,abisimulationrelationbetweenapairoftransitionsystemsT andT isarelation
1 2
betweenthecorrespondingstatesetsexplaininghowastaterunr ofT canbetransformedintoastaterunr
1 1 2
ofT ,andviceversa. Whiletypicalbisimulationrelationsrequirethatr andr havethesameoutputrun,i.e.
2 1 2
H (r )=H (r ),thenotionofapproximatebisimulationrelation,introducedin[GP07],relaxesthiscondition
1 1 2 2
and require that H (r ) is simply close to H (r ), where closeness is measured with respect to a metric on
1 1 2 2
the set of outputs. In this work we consider a generalization of approximate bisimulation, called alternating
approximate bisimulation, that has been introduced in [PT09] to relate properties of control systems affected
by disturbances and their symbolic models.
(cid:45)
Definition 2.5. Consider a pair of metric transition systems T = (Q ,A ×B , ,O ,H ) and T =
1 1 1 1 1 1 2
1
(cid:45)
(Q ,A ×B , ,O ,H ) with the same set of outputs O = O and metric d and consider a precision
2 2 2 2 2 1 2
2
ε∈R+. A relation
0
R⊆Q ×Q
1 2
issaidtobeanalternatingε–approximate(AεA)bisimulationrelationbetweenT andT ifforall(q ,q )∈R
1 2 1 2
the following conditions are satisfied:
(i) d(H (q ),H (q ))≤ε;
1 1 2 2
(ii) ∀a ∈A ∃a ∈A ∀b ∈B ∃b ∈B such that q (a1,b(cid:45)1) q(cid:48), q (a2,b(cid:45)2) q(cid:48) and (q(cid:48),q(cid:48))∈R;
1 1 2 2 2 2 1 1 1 1 1 2 2 2 1 2
(iii) ∀a ∈A ∃a ∈A ∀b ∈B ∃b ∈B such that q (a1,b(cid:45)1) q(cid:48), q (a2,b(cid:45)2) q(cid:48) and (q(cid:48),q(cid:48))∈R.
2 2 1 1 1 1 2 2 1 1 1 2 2 2 1 2
Transition systems T and T are alternating ε–approximately (AεA) bisimilar if there exists an AεA bisimu-
1 2
lation relation such that R(Q )=Q and R−1(Q )=Q .
1 2 2 1
As discussed in [PT09], the notion of alternating approximate bisimulation guarantees that control strategies
synthesized on symbolic models, based on alternating approximate bisimulations, can be readily transferred
totheoriginalmodel,independentlyoftheparticularrealizationofthedisturbanceinputs. WhensetsB and
1
B are singletons, the above notion boils down to approximate bisimulation [GP07]. When ε = 0, the above
2
notion can be viewed as the two-player version of alternating bisimulation [AHKV98].
3. Spline approximation of the disturbance space
Oneofthekeyingredientsintheresultspresentedinthispaperistheapproximationofthedisturbanceinput
functionalspacethroughsplineanalysis[Sch73]. Inthissectionwedescribethisapproximationscheme. Given
a time parameter τ ∈R+, define
D :={d∈D| the domain of d is [0,τ]},
τ
SYMBOLIC MODELS FOR NONLINEAR CONTROL SYSTEMS AFFECTED BY DISTURBANCES 5
and set
(3.1) M = sup (cid:107)d(cid:107) .
∞
d∈Dτ
In the sequel we propose an approximation of the functional space D in the sense of the following definition.
τ
Definition 3.1. A map
A:R+ →2C0([0,τ];D)
is a finite inner approximation of D if for any desired precision θ ∈R+ the following properties hold:
τ
(i) A(θ) is a finite set;
(ii) A(θ)⊆D ;
τ
(iii) for any d∈ D there exists z ∈A(θ) such that (cid:107)y−z(cid:107) ≤θ.
τ ∞
We start by recalling from [Sch73] the notion of spline. Given N ∈N consider the following functions:
(cid:26)
1−t/h, t∈[0,h],
s (t)=
0 0, otherwise,

1−i+t/h, t∈[(i−1)h,ih],

s (t)= 1+i−t/h, t∈[ih,(i+1)h], i=1,2,...,N,
i
 0, otherwise,
(cid:26)
1+(t−τ)/h, t∈[τ −h,τ],
s (t)=
N+1 0, otherwise,
where h = τ/(N +1). Functions s called splines are used to approximate D . More precisely, the approxi-
i τ
mation scheme that we propose is based on the following three steps:
• We first scale function d∈D (Figure 1; first panel) to get the function d =ρd with:
τ 1
(cid:26) (cid:27)
µ 2µ(N +1)
ρ=1−max , ,
M κ τ
d
where M is as in (3.1), κ is as in (2.2) and µ ∈ R+ is a suitable quantization parameter whose role
d
will be clear in the sequel.
• We then approximate function d ∈ D (Figure 1; second panel) by means of the piecewise–linear
1 τ
functiond (Figure1; thirdpanel)obtainedbythelinearcombinationoftheN+2spliness centered
2 i
at times t=ih with amplitudes1 d (ih).
1
• We finally approximate function d by means of function d (Figure 1; fourth panel) obtained by the
2 3
linear combination of the N +2 splines s centered at times t=ih with amplitudes di chosen in the
i 3
lattice [D] =(2µZl)∩D and minimizing the distance from2 d (ih), i.e.
2µ 2
di =arg min (cid:107)d−d (ih)(cid:107).
3 2
d∈[D]2µ
Given N ∈N and µ∈R+ define the following functions:
(cid:26) (cid:27)
µ 2µ
(3.2) ρ (N,µ)=1−max , ,
κd,τ,M M κ h
d
(3.3) Θ (N,µ)=(1−ρ (N,µ))M +(1+ρ (N,µ))κ h+µ,
κd,τ,M κd,τ,M κd,τ,M d
where we recall h=τ/(N+1). Function Θ will be shown to be an upper bound of the error associated to the
approximation scheme that we propose for D . The following technical result will be useful in the sequel.
τ
1Thissecondstepallowsustoapproximatetheinfinite-dimensionalspaceDτ bymeansofthefinite-dimensionalspaceDN+2.
2Thisthirdstepallowsustoapproximatethefinite-dimensionalspaceDN+2 bymeansofthefinite set([D]2µ)N+2.
6 ALESSANDROBORRI,GIORDANOPOLAANDMARIADOMENICADIBENEDETTO
Figure 1. Spline–based inner approximation scheme of the disturbance input functional space.
Lemma 3.2. For any θ ∈R+ there exist N ∈N and µ∈R+ such that
(3.4) Θ (N,µ)≤θ, ρ (N,µ)>0.
κd,τ,M κd,τ,M
Proof. Choose µ= 1 , N ∈N. Function ρ (N,µ) in Eq. (3.2) rewrites as
(N+1)2 κd,τ,M
(cid:18) (cid:19) (cid:26) (cid:27)
1 1 2
ρ N, =1−max , .
κd,τ,M (N +1)2 M(N +1)2 κ τ(N +1)
d
The right-hand side of the previous equality is increasing with N, and it converges to 1 as N goes to infinity;
(cid:16) (cid:17)
then it is clear that for a sufficiently large N one gets ρ N, 1 > 0. Furthermore, one can write
κd,τ,M (N+1)2
the following upper-bound for the function Θ (N,µ) in (3.3):
κd,τ,M
SYMBOLIC MODELS FOR NONLINEAR CONTROL SYSTEMS AFFECTED BY DISTURBANCES 7
(cid:18) (cid:19) (cid:26) (cid:27)
1 1 2M
Θ N, =max ,
κd,τ,M (N +1)2 (N +1)2 κ τ(N +1)
d
(cid:18) (cid:26) (cid:27)(cid:19)
1 2 κ τ 1
+ 2−max , d +
M(N +1)2 κ τ(N +1) N +1 (N +1)2
d
(cid:18) (cid:26) (cid:27) (cid:19)
1 1 2M 1
≤ max , +2κ τ + .
N +1 N +1 κ τ d N +1
d
The right-hand side of the previous inequality is decreasing with N, and goes to zero as N goes to infinity.
Hence, the result follows. (cid:3)
We are now ready to formally introduce the approximation scheme of the disturbance input functional space.
Definition 3.3. Consider the map
A :R+ →2C0([0,τ];D)
Dτ
that associates to any precision θ ∈R+ the set A (θ) consisting of the collection of all functions:
Dτ
N(cid:88)θ+1
(3.5) z(t):= z s (t), t∈[0,τ],
i i
i=0
satisfying the following conditions:
(i) z ∈(2µ Zl)∩D for any i=0,1,...,N +1,
i θˆ θ
(ii) (cid:107)z −z (cid:107)≤κ τ/(N +1) for any i=0,1,...,N ,
i+1 i d θ θ
with θˆ=min{θ,µˆ } where µˆ is defined in Section 2.1.
D D
Remark 3.4. Since the set D is compact, the set (2µ Zl)∩D is finite. Therefore, the set A (θ) is composed
θˆ Dτ
of a finite number of functions that can be effectively computed.
The following technical result will be used in the sequel.
Lemma 3.5. For any θ ∈R+, A (θ)⊆D .
Dτ τ
Proof. In order to show that any function z in (3.5) is in D , we need to show that z enjoys the Lipschitz
τ
condition (2.2) and (cid:107)z(cid:107) ≤ M. Since z is continuous and defined over the interval [0,τ], by the triangle
∞
inequality it suffices to show that (2.2) holds for any t ,t ∈[ih,(i+1)h], i=0,...,N . By Eq. (3.5) and the
1 2 θ
definitionofspline, thefunctionz ispiecewise-linearandislinearintheinterval[ih,(i+1)h], withz(ih)=z .
i
Hence one can write for any t ,t ∈[ih,(i+1)h]:
1 2
(cid:107)z(t )−z(t )(cid:107) (cid:107)z((i+1)h)−z(ih)(cid:107) (cid:107)z −z )(cid:107)
(3.6) 2 1 = = i+1 i ≤κ ,
(cid:107)t −t (cid:107) h h d
2 1
where the last step holds by condition (ii) in Definition 3.3, concluding the proof of the Lipschitz condi-
tion. We next show that the boundedness condition holds as well. Since z is piecewise-linear, (cid:107)z(cid:107) =
∞
max (cid:107)z (cid:107), hence we just need to show that (cid:107)z (cid:107)≤M for all i. From condition (i) in Definition 3.3,
i=0,...,Nθ+1 i i
z ∈D, implying from (3.1) that (cid:107)z (cid:107)≤M, which concludes the proof. (cid:3)
i i
We are now ready to present the main result of this section.
Theorem 3.6. Map A in Definition 3.3 is a finite inner approximation of D .
Dτ τ
8 ALESSANDROBORRI,GIORDANOPOLAANDMARIADOMENICADIBENEDETTO
Proof. Consider any precision θ ∈ R+. For notational simplicity we set ρ (N ,µ ) = ρ. As discussed
κd,τ,M θ θˆ
in Remark 3.4, the set A (θ) is finite. Hence, condition (i) in Definition 3.1 is satisfied. Condition (ii) in
Dτ
Definition 3.1 is implied by Lemma 3.5. We now show that also condition (iii) in Definition 3.1 is satisfied.
For any function d ∈ D consider a function z as in (3.5) where for any i = 0,1,...,N vectors z are chosen
τ θ i
in the set 2µ Zl such that:
θˆ
(3.7) (cid:107)z −ρd(ih)(cid:107)≤µ .
i θˆ
We first prove that vectors z are in the set D, showing that (cid:107)z (cid:107) ≤ M for all i. From Eq. (3.7), by using
i i ∞
the triangle inequality and the definition of ρ in (3.2), one can write:
(cid:107)z (cid:107) =(cid:107)z −ρd(ih)+ρd(ih)(cid:107)
i ∞ i ∞
≤(cid:107)z −ρd(ih)(cid:107) +(cid:107)ρd(ih)(cid:107)
i ∞ ∞
≤µ +(cid:107)ρd(ih)(cid:107)
θˆ ∞
≤µ +ρM
θˆ
µ
≤µ +(1− θˆ)M
θˆ M
=µ +M −µ =M,
θˆ θˆ
which concludes the proof of the existence of such values z ∈ (2µ Zl)∩D, as in condition (i) of Definition
i θˆ
3.3. We now show that also condition (ii) is satisfied. From (3.7), the following chain of inequalities holds:
(cid:107)z −z (cid:107) ≤ (cid:107)z −ρd((i+1)h)(cid:107)+(cid:107)ρd((i+1)h)−ρd(ih)(cid:107)+(cid:107)ρd(ih)−z (cid:107)
i+1 i i+1 i
≤ ρ(cid:107)d((i+1)h)−d(ih)(cid:107)+2µ
θˆ
≤ ρκdh+2µθˆ≤(1− κ2µdhθˆ)κdh+2µθˆ=κdh,
where h=τ/(N +1) and the last inequality holds by the definition of function ρ in (3.2). Hence, condition
θ
(ii) in Definition 3.3 is satisfied and z ∈A (θ). In order to conclude the proof of condition (iii) in Definition
Dτ
3.1 we need to show that (cid:107)d−z(cid:107) ≤θ. By the assumptions on the disturbance space, the following chain of
∞
inequalities holds:
(cid:107)d−z(cid:107) = max (cid:107)d(ih+t)−z(ih+t)(cid:107)
∞
i=0,1,...,Nθ,t∈[0,h]
≤ max ((cid:107)d(ih+t)−ρd(ih+t)(cid:107)+(cid:107)ρd(ih+t)−ρd(ih)(cid:107)
i=0,1,...,Nθ,t∈[0,h]
+(cid:107)ρd(ih)−z(ih)(cid:107)+(cid:107)z(ih)−z(ih+t)(cid:107))
≤(1−ρ)M +(1+ρ)κ h+µ
d θˆ
=Θ (N ,µ )≤θˆ≤θ,
κd,τ,M θ θˆ
where the last step holds by Eq. (3.3) and by definition of N and µ . From the above chain of inequalities,
θ θ
condition (iii) in Definition 3.1 is satisfied, which concludes the proof. (cid:3)
Remark 3.7. Spline approximation of functional spaces has been also employed in [PPDT10, PPDB10] for
deriving symbolic models of nonlinear time–delay systems. The approximation scheme here proposed is dif-
ferent from the one proposed in [PPDT10, PPDB10] as it can be readily seen by comparing Definition 3.1
and Definition 6 in [PPDT10] (also employed in [PPDB10]). In particular the notion of approximation here
considered is stronger than the one used in [PPDT10, PPDB10], as it can be easily checked by comparing
conditions (ii) in the two definitions. As discussed in the sequel, this notion allows us to provide symbolic
modelsfornonlinearcontrolsystemsaffectedbydisturbanceswhichcanbeeffectively constructed whereasthe
notion employed in [PPDT10, PPDB10] does not.
SYMBOLIC MODELS FOR NONLINEAR CONTROL SYSTEMS AFFECTED BY DISTURBANCES 9
4. Alternating approximately bisimilar symbolic models
In this section, we propose symbolic models that approximate nonlinear control systems with disturbances in
the sense of alternating approximate bisimulation.
Given the control system Σ = (X,U,D,f) in (2.3) and a sampling time parameter τ ∈ R+, consider the
following transition system:
(cid:45)
T (Σ):=(X,U ×D , ,O,H),
τ τ τ
τ
where:
• U ={u∈U| the domain of u is [0,τ] and u(t)=u(0), t∈[0,τ]};
τ
• x (u,(cid:45)d) x(cid:48) if there exists a trajectory ξ :[0,τ]→X of Σ satisfying ξ (τ)=x(cid:48);
xud
τ
• O =X;
• H =1 .
X
TransitionsystemT (Σ)ismetricwhenweregardO =X asbeingequippedwiththemetricd(p,q)=(cid:107)p−q(cid:107).
τ
Transition system T (Σ) can be thought of as the time discretization of the control system Σ. For notational
τ
simplicity,inthefollowingwedenotebyuanyconstantcontrolinputu˜s.t. u˜(t)=uforallt∈[0,τ]. Consider
a vector of quantization parameters
(4.1) P=(τ,µ ,µ ,µ ,N),
x u d
and define the following transition system:
(cid:45)
(4.2) TP(Σ):=(QP,LP, ,OP,HP),
P
where:
• QP =(2µxZn)∩X;
• LP =AP×BP where:
– AP =(2µuZm)∩U;
– BP =ADτ(Θκd,τ,M(N,µd))whereADτ isafiniteinnerapproximationofDτ, asinDefinition3.3,
and function Θ is defined as in (3.3);
(u,(cid:45)d)
• x y if (cid:107)ξ (τ)−y(cid:107)≤µ ;
P xud x
• OP =X;
• HP =ı:QP (cid:44)→OP.
Remark 4.1. It is readily seen that the transition system TP(Σ) is countable and it becomes symbolic when
the set of states X is bounded. As stressed in Remark 3.4, the set of control and disturbance inputs LP can
be effectively computed from which the transition system TP(Σ) can be effectively computed.
We now have all the ingredients to present the main result of this paper.
Theorem 4.2. Consider the control system Σ=(X,U,D,f) in (2.3) and suppose that:
(A1) Thereexistsaδ–ISSLyapunovfunctionsatisfyingtheinequality(ii)inDefinition2.2forsomeλ∈R+.
(A2) There exists a K function γ such that3:
∞
V(x,x(cid:48))−V(x,x(cid:48)(cid:48))≤γ((cid:107)x(cid:48)−x(cid:48)(cid:48)(cid:107)),
for every x,x(cid:48),x(cid:48)(cid:48) ∈X.
3NotethatsinceV issmooth,ifthestatespaceX isbounded,whichisthecaseasinmanyrealapplications,onecanalways
(cid:16) (cid:17)
chooseγ((cid:107)w−z(cid:107))= sup (cid:107)∂V(x,y)(cid:107) (cid:107)w−z(cid:107).
x,y∈X ∂y
10 ALESSANDROBORRI,GIORDANOPOLAANDMARIADOMENICADIBENEDETTO
Then, for any desired precision ε∈R+, any sampling time τ ∈R+, and any choice of quantization parameters
in P satisfying the following inequalities4:
max{σ (µ ),σ (θ )} γ(µ )
(4.3) u u d d + x ≤α(ε),
λ 1−e−λτ
(4.4) µ ≤µˆ ,
x X
(4.5) µ ≤µˆ ,
u U
(4.6) µ ≤µˆ ,
d D
(4.7) Θ (N,µ )≤θ ,
κd,τ,M d d
transition systems Tτ(Σ) and TP(Σ) are alternating ε–approximately bisimilar.
Before giving the proof of the above result we stress that:
Proposition 4.3. For any desired precision ε ∈ R+ and any sampling time τ ∈ R+, there always exists a
choice of the vector P of quantization parameters such that the coupled inequalities in (4.3), (4.4), (4.5), (4.6)
and (4.7) are satisfied.
Proof. It is clear that a choice of sufficiently small parameters µ , µ and θ allows to satisfy the inequalities
x u d
in (4.3)-(4.5), since σ , σ and γ are K functions. Then, for any fixed θ resulting from the previous step,
u d ∞ d
one can choose N and µ such that the inequality in (4.7) is fulfilled (as shown in the proof of Lemma 3.2),
d
and finally µ can be chosen small enough so that the inequality in (4.6) holds. (cid:3)
d
We can now give the proof of Theorem 4.2.
Proof. Consider the relation R ⊆ X ×QP defined by (x,y) ∈ R if and only if V(x,y) ≤ α(ε). Condition (i)
in Definition 2.5 is satisfied by the definition of R and condition (i) in Definition 2.2. Let us now show that
condition(ii)inDefinition2.5holds. Considerany(x,y)∈R. Bycondition(4.5),foranyu ∈U thereexists
1 τ
u2 ∈AP =(2µuZm)∩U such that:
(4.8) (cid:107)u −u (cid:107) ≤µ .
2 1 ∞ u
Moreover by Lemma 3.5 for any d ∈ A (θ ) we can pick d = d ∈ D . Set z = ξ (τ). By condition
2 Dτ d 1 2 τ yu2d2
(4.4) there exists v ∈QP such that:
(4.9) (cid:107)z−v(cid:107)≤µ .
x
Hence, by definition of TP(Σ), the transition y u2,(cid:45)d2 v is in TP(Σ). Consider now the transition x u1,(cid:45)d1 w in
P τ
T (Σ). By Assumption (A1), condition (ii) in Definition 2.2 and the inequality in (4.8), one gets:
τ
∂Vf(w,u ,d )+ ∂Vf(z,u ,d ) ≤ −λV(w,z)+σ ((cid:107)u −u (cid:107))+σ ((cid:107)d −d (cid:107))
∂w 1 2 ∂z 2 2 u 1 2 d 1 2
≤ −λV(w,z)+σ (µ ),
u u
which, by Assumption (A2), the definition of R and the inequality in (4.9), implies:
V(w,v)≤V(w,z)+γ((cid:107)z−v(cid:107))
≤V(w,z)+γ(µ )
x
σ (µ )
≤e−λτV(x,y)+(1−e−λτ) u u +γ(µ )
λ x
σ (µ )
≤e−λτα(ε)+(1−e−λτ) u u +γ(µ ).
λ x
Hence, by the inequality in (4.3), V(w,v)≤α(ε), from which (w,v)∈R and condition (ii) in Definition 2.5 is
proven. Wenowshowcondition(iii)inDefinition2.5. Considerany(x,y)∈R. Foranyu2 ∈AP =(2µuZm)∩U
4SymbolsµˆX,µˆU andµˆD aredefinedasinSection2.1.