https://doi.org/10.1007/s10626-020-00315-z
Refinements of behaviour al abstractions
for the super visory control of hybrid systems
Jung-Min Yang 1 · Thomas Moor 2 · J ¨
org Raisch 3
Received: 7 December 2018 / Accepted: 4 March 2020 /
© The Author(s) 2020
Abstract
A common approach to controller synthesis for hybrid systems is to first establish a discrete-
e vent abstraction and then to use methods from supervisory control theory to synthesise a
controller . In this paper , we consider behavioural abstractions of hybrid systems with a pre-
scribed discrete-e vent input/output interface. W e discuss a family of abstractions based on
so called experiments which consist of samples from the external beha viour of the hybrid
system. The special feature of our setting is that the accuracy of the abstraction can be care-
fully adapted to suit the particular control problem at hand. T echnically , this is implemented
as an iteration in which we alternate trial control synthesis with abstraction refinement.
While localising refinement to where it is intuiti vely needed, we can still formally establish
that the ov erall iteration will solv e the control problem, provided that an abstraction-based
solution exists at all.
Keywords Hybrid systems · Behavioural abstractions · l -complete approximations ·
Local refinement · Supervisory control
This article belongs to the T opical Collection: T opical Collection on Theory-2020
Guest Editors: Francesco Basile, Jan K omenda, and Christoforos Hadjicostis
The research of J.–M. Y ang was supported in part by Basic Science Research Program through the
National Research Foundation of K orea (NRF) funded by the Ministry of Education
(NRF–2018R1D1A1A09082016), and in part by the National Research Foundation of K orea (NRF)
grant funded by the K orea gov ernment (MSIT) (NRF–2018R1A5A1025137 and
NRF–2015M3A9A7067220).
J ¨
org Raisch
[email protected] berlin.de
Jung–Min Y ang
[email protected]
Thomas Moor
[email protected]
1 School of Electronics Engineering, Kyungpook National Uni v ersity , 80 Daehakro,
Bukgu, Daegu, 41566, Republic of K orea
2 Lehrstuhl f ¨ ur Regelungstechnik, Friedrich-Ale xander -Univ ersit ¨
at Erlangen-N ¨ urnberg,
91058, Erlangen, Germany
3 Fachgebiet Re gelungssysteme, T echnische Uni versit ¨
at Berlin, 10587, Berlin, Germany
Published online: 22 April 2020
Discrete Event Dynamic Systems (2020) 30:533–560
1 Introduction
The analysis and the control of hybrid systems ha ve become an important subject in modern
control theory; see, e.g., Alur et al. ( 2000 )a n dT a b u a d a( 2009 ) and the references cited
therein. A common approach is to construct a finite-state abstraction of the hybrid system
under consideration and then to apply methods kno wn from the domain of discrete-e vent
systems, most notably model checking, reacti ve synthesis, or supervisory control.
A well established frame work to obtain a finite-state abstraction is to strate gically con-
struct a finite partition or a finite cov er on the continuous state space and to thereby define
symbolic dynamics associated with the hybrid system; see e.g. Sti ver et al. ( 1995 ), T abuada
( 2009 ), Reissig et al. ( 2017 ), Pola and T ab uada ( 2009 ) ,G o le ta l .( 2014 ), Zamani et al.
( 2012 ), and Liu and Ozay ( 2016 ). For controller synthesis, this approach is particularly
suited when the design of a discrete-e vent interface is considered part of the synthesis prob-
lem. In contrast, if the hybrid plant is equipped with a prescribed discrete-ev ent interface, so
called behavioural abstr actions are an adequate alternati v e. In this approach, one seeks to
deri ve a finite-state abstraction directly in terms of the e xternal signals. This is the situation
we study in the present paper . 1
The beha vioural abstractions proposed by Moor and Raisch ( 1999 ) and Moor et al.
( 2002 ) are based on the notion of l -completeness from W illems’ beha vioural systems theory;
see, e.g., W illems ( 1991 ). By definition, a discrete-time system is l -complete if its infinite-
time behaviour can be exactly reco v ered from all length l samples, l ∈ N 0 , taken from
all infinite-length signals. For a system which does not e xhibit this property , the str ongest
l -complete appr oximation is then introduced as the tightest behavioural o ver -approxima-
tion that is l -complete. In the follo wing, whene ver clear from the conte xt, we simply refer
to l -complete approximations when we mean strongest l -complete approximations. An l -
complete approximation can be obtained by exhausti vely taking samples of length l from the
original beha viour and generating the abstraction by superposition of these samples. For the
control of hybrid systems with discrete-v alued input- and output-signals, l -complete approx-
imations can be used to synthesise controllers that address inclusion-type specifications in
these signals. In this situation, the finite external signal range of the hybrid system leads
to a finite-state realisation of the l -complete approximation and a v ariant of Ramadge and
W onham’ s supervisory control theory (Ramadge and W onham 1987 , 1989 ) is subsequently
applied to synthesise a supervisory controller . It is sho wn by Moor and Raisch ( 1999 )a n d
Moor et al. ( 2002 ) that if the supervisor suitably restricts the behaviour of the l -complete
approximation, it also accomplishes the control objecti v e for the underlying hybrid system.
The applicability of this approach has been demonstrated with case studies in the area of
process engineering, including the start-up of a distillation column (Moor and Raisch 2002 ).
Recent methodological extensions ha v e been reported by Schmuck and Raisch ( 2014 ), Park
and Raisch ( 2015 ), and Moor and G ¨
otz ( 2018 ) to address time-v ariant systems and partial
observ ation.
The existence of an appropriate supervisor depends on the approximation accurac y ,
namely , if the abstract model is too coar se , no supervisor may exist that meets the speci-
fication. This leads to an iteration of trial synthesis and abstraction refinement, until either
a solution to the control problem is established or computational resources are e xhausted.
Considering l -complete approximations, the construction of a finer abstraction effecti vely
amounts to incrementing the length l of samples taken from the original hybrid system.
1 A technical comparison of abstractions by symbolic dynamics and behavioural abstractions is gi v en in
Schmuck et al. ( 2015 ).
534 Discrete Event Dynamic Systems (2020) 30:533–560
This can be done uniformly for all samples as, e.g., proposed by Moor and Raisch ( 1999 )
or , more efficiently , in a non-uniform way tailored for the particular control problem at
hand. For abstractions by symbolic dynamics, Clark e et al. ( 2003 ) introduce counter e xam-
ple guided r efinement for the verification of h ybrid systems, with a further de v elopment
to address synthesis by Stursber g ( 2006 ). For beha vioural abstractions, Moor et al. ( 2006 )
introduce the notion of an e xperiment as a set of non-uniform length samples taken from the
original beha viour , with a subsequent discussion that leads to abstractions obtained from
experiments. T echnically , the resulting abstractions are still l -complete and, hence, they can
be safely utilised in an abstraction-based design.
In the present paper , we further de velop abstraction-based synthesis by e xperiments
on beha viours. While the study by Moor et al. ( 2006 ) is entirely set within W illems’
beha vioural systems theory , we no w mak e use of explicit state machine realisations reported
by Moor and G ¨
otz ( 2018 ). By a more detailed discussion, we gain some rele v ant benefits.
First, we can literally refer to supervisory control of sequential behaviours with a synthesis
algorithm gi ven by Thistle and W onham ( 1994a ). As a consequence, we can address more
general li veness specifications with e ventual task completion as a prototypical example.
This extends the results from Moor et al. ( 2006 ), which are restricted to l -complete safety
specifications. Second, the technically in volv ed temporal decomposition of the control prob-
lem used by Moor et al. ( 2006 ) to guided abstraction refinement can no w be replaced by
the contr ollability pr efix introduced by Thistle and W onham ( 1994b ). The latter is a n inter -
mediate result of the synthesis procedure and characterises winning states , from which on
the control objecti ve can be accomplished. In the case that the synthesis procedure fails
and, thus, a refinement of the abstraction is required, intuiti vely , such a refinement does not
need to address any winning states. Lik e wise, we can identify failing states , from which on
no supervisor can possibly satisfy the specification. Again intuiti vely , this status is retained
under refinement and, thus, no refinement should address the beha viour after a failing state.
For the iteration of trial controller synthesis and abstraction refinement, it is therefore pro-
posed to refine the experiment only by addressing states that are neither winning nor failing.
Since this iteration intentionally generates only specific experiments, it may fail to generate
a particular experiment for which controller synthesis w ould succeed. Here, our main tech-
nical result, Theorem 17, guarantees that a successful experiment will be generated pro vided
that such a one exists.
A predecessor of this paper has been presented at the W orkshop on Discrete Event Sys-
tems; see Y ang et al. ( 2018 ). The present version has been e xtended (a) to address iterati v e
refinements in contrast to a single local refinement, (b) to allo w for specifications with a
B¨ uchi acceptance condition, and (c) to include a formal proof of our main technical result.
The remainder of the paper is or ganised as follo ws. After introducing elementary notation
in Section 2 , we summarise the concept of experiments and the beha vioural abstractions
obtained therefrom in Section 3 . Realisations of the hybrid plant and its abstractions are dis-
cussed in Section 4 . In Section 5 we present the control problem under consideration and
deri ve an abstraction-based solution procedure. Finally , we discuss the proposed abstraction
refinement scheme in Section 6 . A simple example is used throughout the entire discussion,
illustrating the suggested ideas and demonstrating the applicability of the proposed strategy .
2N o t a t i o n
W e denote the positiv e, respecti vely non-ne gati v e, inte gers by N , respecti vely N 0 .T h e
cardinality of a finite set A is denoted by | A |∈ N 0 .
535 Discrete Event Dynamic Systems (2020) 30:533–560
Gi ven a set W , referred to as a signal r ange ,a n d l ∈ N ,w ed e n o t eb y W l :=
{ w 1 ,...,w
l |∀ k, 1 ≤ k ≤ l : w k ∈ W } the set of sequences over W of length l and we
let W + := ∪{ W l | l ∈ N } . Introducing the empty sequence ∈ W , we formally define
W 0 := { } and write W ∗ := ∪{ W l | l ∈ N 0 }={ }∪ W + for the set of finite sequences ov er
W . For a sequence s ∈ W l ⊆ W ∗ , its length l is denoted | s | . The set of all countably infinite
sequences over W is denoted W N 0 , with w ∈ W N 0 commonly interpreted as a discr ete time
signal w : N 0 → W . Subsets S ⊆ W ∗ are referred to as ∗ - languag es ,o r languag es of finite
wor ds , in contrast to ω - langua ges B ⊆ W N 0 ,o r langua ges of infinite wor ds .
For tw o finite sequences s = w 1 ,...,w
l ∈ W l and r = u 1 ,...,u
n ∈ W n ,
the concatenation is defined by s, r : = w 1 ,...,w
l ,u
1 ,...,u
n ∈ W l + n .F o rt h e
empty sequence, let s, : = s =: , s . The concatenation of the finite sequence
s = w 1 ,...,w
l ∈ W l with a signal w ∈ W N 0 is denoted v = s, w ∈ W N 0 , with
v (k ) = w k + 1 for 0 ≤ k< l and v (k ) = w (k − l) for k ≥ l . Again, for the empty sequence
let , w : = w . For notational con venience, we write · , · , · for · , · , · .
A sequence r ∈ W ∗ is a pr efix of s ∈ W ∗ if there e xists t ∈ W ∗ such that r, t = s ;
we then write r ≤ s . If, in addition, r = s , we say that r is a strict pr efix of s and write
r< s . Like wise, r ∈ W ∗ is a prefix of a signal w ∈ W N 0 , if there e xits v ∈ W N 0 such that
r, v = w ; we then write r< w . The set of all prefixes of a gi v en sequence s ∈ W ∗ or
a gi ven signal w ∈ W N 0 is denoted pre s ⊆ W ∗ or pre w ⊆ W ∗ , respecti vely . A sequence
t ∈ W ∗ is a suffix of s ∈ W ∗ if there exists r ∈ W ∗ such that r, t = s .
The left-shift operator σ l , l ∈ N 0 , is defined for signals w ∈ W N 0 by σ l w ∈ W N 0 with
(σ l w )(k ) := w (k + l) for all k ∈ N 0 ,a n dw el e t σ := σ 1 . F or a signal w ∈ W N 0 ,t h e
r estriction to a finite inte ger interv al D ⊆ N 0 is denoted w | D with, e.g., D =[ k 1 ,k
2 ) :=
{ k ∈ N 0 | k 1 ≤ k< k
2 } and left-open and/or right-closed interv als defined like wise. When
taking restrictions, we drop absolute time and reinterpret w | D as finite sequence, i.e., we
identify w | [ k 1 ,k 2 ) with w (k 1 ) ,..., w (k 2 − 1 ) ∈ W k 2 − k 1 .
T aking point-wise images, all operators and maps in this paper are identified with their
respecti ve e xtension to set-v alued ar guments; e.g., we write σ B for the image { σ w | w ∈ B }
of the ω -language B ⊆ W N 0 under the operator σ with domain W N 0 , and, like wise, pre B
for the image { pre w | w ∈ B } of B under the operator pre.
3 Behavioural abstractions
W e present a general scheme of system abstraction that allows for a guided refinement,
and we do so within W illems’ beha vioural systems theory . In this frame work, a dynamical
system is defined as a triple = (T , W , B ) ,w h e r e T is the time axis , W is the external
signal rang e ,a n d B ⊆ W T := { w | w : T → W } is the behaviour , i.e., the set of all
external signals that the system may generate. It is then proposed to discuss and to cate gorise
dynamical systems in terms of their behaviours. For the present paper , we focus attention on
time-in variant systems with an e xternal discrete-e vent interface. T echnically , we consider
the discrete time axis T = N 0 and the finite external signal range W , | W |∈ N ,a n dw e
interpret ω -languages B ⊆ W N 0 as beha viours. Regarding time in variance, we refer to the
follo wing definition from W illems ( 1991 ).
Definition 1 A beha viour B ⊆ W N 0 is time in variant if σ B ⊆ B . A system =
( N 0 ,W , B ) is time in variant if its beha viour B is time in v ariant.
536 Discrete Event Dynamic Systems (2020) 30:533–560
Gi ven a system = ( N 0 ,W , B ) ,a behavioural abstr action i sas y s t e m =
( N 0 ,W , B ) with B ⊆ B , i.e., provided that the original system accounts for all possible
trajectories the actual phenomenon modelled by can generate, then so does the abstrac-
tion . Beha vioural abstractions are commonly used for the verification and the synthesis
of safety properties, with optional li veness properties being addressed by additional struc-
tural requirements. Considering a time in variant system, we ask for a time in v ariant system
abstraction. W e refer to Moor et al. ( 2006 ) for the following notion of an e xperiment which
we will use to construct a rich family of time-in v ariant beha vioural abstractions.
Definition 2 An e xperiment over W is a ∗ -language S ⊆ W ∗ . If there e xists a uniform upper
bound on the length of all sequences in S , we say that S is of bounded length . M ore over,
S ⊆ W ∗ is an e xperiment on a behaviour B ⊆ W N 0 if S accounts for each signal from B in
terms of a prefix, i.e., if ( pre w ) ∩ S =∅ for all w ∈ B .
Gi ven an e xperiment S on some beha viour B , Moor et al. ( 2006 ) discuss abstractions B S ,
B ⊆ B S , that can be obtained exclusi vely from S . T o this end, we consider the candidate
B S := { w : N 0 → W |∀ k ∈ N 0 ∃ l ∈ N 0 : w | [ k, k + l) ∈ S } , (1)
i.e., B S consists of all those trajectories which at any instance of time continue to e v olve on
some finite future sequence that matches S ;s e eF i g . 1 .
It is immediate from the construction that B S is time in v ariant and that S is an e xperiment
on B S . Moreov er , for any beha viour ˜
B ⊆ W N 0 we ha ve the follo wing implication:
if ˜
B is time-in variant and if S is an experiment on ˜
B then ˜
B ⊆ B S .( 2 )
It is sho wn in Moor et al. ( 2006 ), Proposition 7, that B S is the unique smallest behaviour for
which the abov e implication holds. Assuming that the original beha viour B is indeed time
in variant, B S is the smallest superset of B that can be characterised exclusi vely in terms of
Fig. 1 Experiment S on B with associated abstraction B S . Here, the prefixes o f behaviours are interpreted
as infinite computational trees with the empty string as the root at the very left and progress of discrete time
to wards the right. In the sketch, the trees are abstractly sho wn as cones; i.e., pre B in dark yello w and pre B S
in light green. The bounded-length ∗ -language S is shown as a dark green stripe. In particular , the sk etch
indicates that any signal from B must pass S , as required by Definition 3. Reg arding the abstraction, the
signal w ∈ B S sho wn in the sketch at e v ery instance of time k must hav e a finite-length future that matches
S ;s e eE q . 1 . This is illustrated by the transparent grey cop y of S shifted such that the root matches w (k ) to
indicate that w | [ k, k + l) = w | [ k, k + l − 1 ] ∈ S
537 Discrete Event Dynamic Systems (2020) 30:533–560
the experiment S . Therefore the system S = ( N 0 ,W , B S ) is referred to as the behavioural
abstraction obtained fr om S under the assumption of time in variance , or in short as the
abstraction obtained fr om S .
Moor et al. ( 2006 ) and Moor and G ¨
otz ( 2018 ) provide a comprehensi ve discussion
regarding algebraic properties of e xperiments and the abstractions obtained therefrom. F or
the present paper , we pragmatically refer to Eq. 1 as the defining equation and point out
some technical consequences rele v ant for the subsequent discussion. It is immediate from
Eq. 1 that, reg arding the associated abstraction, we may without loss of generality restrict
our considerations to pr efix-fr ee experiments, i.e., to e xperiments that satisfy
∀ s, t ∈ S : s ≤ t ⇒ s = t .( 3 )
A prefix-free experiment S is commonly interpreted as a tree with r oot ∈ pre S , nodes
pre S and leaves S . Like wise, it is not restricti v e to assume that the e xperiment is trim in the
sense that e very sequence s ∈ S contrib utes to the abstraction, i.e.,
∀ s ∈ S ∃ k, l ∈ N 0 : s ∈ B S | [ k, k + l) , (4)
or , equiv alently , S ⊆ pre B S . Thus, whene ver con venient, we may restrict our discussion to
trim and prefix-free experiments.
For the special case of the experiment S = B | [ 0 ,l ] with samples with uniform
length l + 1, l ∈ N 0 , the abstraction B S obtained from S amounts to the str ongest l -
complete appr oximation proposed by Moor and Raisch ( 1999 ) and Moor et al. ( 2002 ).
The approximation-refinement scheme kno wn from l -complete approximations amounts to
incrementing the sample length. This concept of refinement generalises to experiments as
follo ws.
Definition 3 Gi ven tw o experiments S and S over W , we say that S is a refinement of S if
( ∀ s ∈ S ∃ s ∈ S : s ≤ s ) and ( ∀ s ∈ S ∃ s ∈ S : s ≤ s ) .( 5 )
W e then write S ≤ S .
The first conjunct in Eq. 5 ensures that the refinement accounts for each sample s ∈ S
from the original experiment by an e xtended sample s ∈ S , s ≤ s , including the tri vial
case of s = s . The second conjunct ensures that no other samples are in the refinement
than those obtained by (possibly tri vially) extending samples from the original e xperiment.
Figure 2 illustrates a prefix-free experiment S on B where t he leav es s ∈ S form a “barrier”
through which each trajectory from w ∈ B must pass. In this vie w , a prefix-free refinement
S of S is obtained by pushing the “barrier” to the right. The figure also indicates that a
refinement is expected to lead to a tighter abstraction in that it more accurately encodes
which trajectories are not within B ,e . g , v ∈ B is possibly in B S but cannot be in B S .
T echnically , we obtain for two experiments S and S on B with S ≤ S that
B ⊆ B S ⊆ B S (6)
as an immediate consequence of Eq. 1 and the second conjunct in Eq. 5 ; i.e., it is guaranteed
that the abstraction does not become worse and we may optimistically expect it to become
better .
For the systematic construction of refinements, we propose to nominate a set R ⊆ S of
r efinement candidates and observ e that
S := { s ∈ S | s ∈ R }∪{ s, w | s ∈ R, w ∈ W, s, w ∈ pre B } (7)
538 Discrete Event Dynamic Systems (2020) 30:533–560
Fig. 2 Prefix-free experiment S on B with prefix-free refinement S , S ≤ S .A si nF i g . 1 ,p r e B is interpreted
as an infinite computational tree with the empty string as the root at the very left, graphically represented
as a dark yello w cone. Both bounded length experiments S and S are sho wn as “barriers” and, as indicated
in the sketch, an y signal from B must pass both S and S . Being a prefix-free refinement, the sequences in
S are obtained by extending specific sequences from S , i.e., pushing the boundary to the right. Referring
exclusi v ely to the experiment S , both signals w and v could possibly belong to some beha viour on which S
was conducted. Hence, we may expect v ∈ B S although v ∈ B . In contrast, we have v | [ 0 ,l) ∈ S for all l ∈ N 0
and, hence, v ∈ B S
is indeed an experiment on B with S ≤ S . F or the special case of S = B | [ 0 ,l ] and R = S we
obtain S = B | [ 0 ,l + 1 ] , which coincides with the refinement of l -complete approximations.
4 State machine realisations
W e further elaborate the proposed scheme of behavioural abstractions in the conte xt of state
realisations. Here, we account for a class of transition systems referred to as state mac hines ,
which we will utilise for realisations of the plant model, finite state abstractions, and the
specification in the control problem under consideration.
Definition 4 A state machine is a quadruple P = ( X ,W ,δ ,X
0 ) ,w h e r e X is the state set ,
W is the e xternal signal rang e , δ ⊆ X × W × X is the transition r elation ,a n d X 0 ⊆ X is
the set of initial states . The state machine P is called a finite state machine if | X |∈ N .
W e use the following terminology .
• Whene ver con venient, we reinterpret the transition relation with a set-v alued map,
recursi vely defined for (x , s ) ∈ X × W ∗ and w ∈ W by (a) δ( x , ) := { x } and (b)
δ( x, s w ) := { x |∃ x ∈ δ( x, s) : (x ,w ,x ) ∈ δ } ; i.e., δ( x, s) denotes the set of states
reachable from x by taking | s | transitions with labels as specified by s .
• Referring to the set-v alued map interpretation of δ ,t h es e t δ( X 0 ,W ∗ ) are the r eac hable
states ,a n d P is said to be r eachable if δ( X 0 ,W ∗ ) = X .I f δ( x, W ) is non-empty for
e very reachable state x ∈ δ( X 0 ,W ∗ ) ,t h e n P is termed deadlock-fr ee .
• The state machine P induces the full behaviour
B full := { ( w , x ) |∀ k ∈ N 0 : ( x (k ), w (k ), x (k + 1 )) ∈ δ and x ( 0 ) ∈ X 0 } (8)
and the state space system full := ( N 0 ,W × X, B full ) .
539 Discrete Event Dynamic Systems (2020) 30:533–560
• The e xternal behaviour B ex of full is the projection of B full onto W N 0 , i.e.,
B ex := P W B full := { w |∃ x : ( w , x ) ∈ B full } .( 9 )
If a state machine P induces the external beha viour B of a system , P is termed a
realization of , denoted ∼
= P .
• If | δ( X 0 ,s ) |≤ 1f o re v e r y s ∈ W ∗ ,t h e n P is said to be past-induced ; in automata
theory , this is also referred to as deterministic . W e then write δ 0 (s ) ∈ X for s ∈ W ∗ with
| δ( X 0 ,s ) |= 1 to denote the unique state reachable from X 0 via the external sequence
s .
In our subsequent discussion of controller synthesis, we assume that the plant is gi ven
as a state machine P = ( X ,W ,δ ,X
0 ) with unrestricted initial conditions, i.e., X 0 = X ,
and we note that this assumption implies time in variance for the induced full beha viour as
well as for the induced external beha viour . Moreov er , we consider the product W = U × Y
as external signal space, where U and Y denote the ranges of input symbols and output
symbols , respecti v ely , and where we assume that
∀ x ∈ X, u ∈ U ∃∈ Y, x ∈ X : (x , (u, y ), x ) ∈ δ . (10)
State machines with this property are called I/S/- machines . F or the induced e xternal
beha viour , it can be seen that the input is fr ee and the output does not anticipate the input ,
both technical terms defined within W illems’ behavioural frame w ork; see Proposition 24 in
Moor and Raisch ( 1999 ), which refers to Definitions VIII.1 and VIII.4 in W illems ( 1991 ).
In particular , we will consider supervisory controllers which at any instance of time disable
specific input symbols and which in turn accept any output symbol. T echnically , all exter -
nal symbols are organised as pairs w = (u, y ) ∈ U × Y = W , with only the U -component
considered controllable; this will be follo wed up in Section 5 , including a formal definition
of the corresponding control patterns ( 26 ). T o this end, we address two remarks on ho w the
requirement of a time-in variant I/S/- plant model can be relax ed by a preprocessing stage
applicable in the context of controller synthesis with upper -bound beha vioural-inclusion
specifications.
Remark 5 Formally , Eq. 10 requires that ev ery input symbol can be applied regardless of the
current state of the plant. Ne v ertheless, if we are provided with a state machine P = (X, U ×
Y, δ, X) which fails to satisfy ( 10 ), we consider the substitute model P = (X, U × Y ,δ ,X )
with Y = Y ˙
∪{ ‡ } where we define the transition relation δ to issue the distinguished output
symbol ‡ ∈ Y whenev er an in valid input symbol was applied:
δ := δ ∪{ (x , (u, ‡ ), x ) |∀ x ∈ X, y ∈ Y : (x , (u, y ), x ) ∈ δ } . (11)
Clearly , P satisfies ( 10 ). The considered upper-bound beha vioural-inclusion specification
can then be used to pre vent the distinguished output symbol ‡ ∈ Y from occurring in the
closed-loop configuration. No w assume that a controller that has been designed for the plant
substitute P such that ‡ does not indeed occur in any e xternal closed-loop signal. Whene v er
P attains a state x ∈ X such that there exists a transition (x , (u, ‡ ), x ) ∈ δ for some u ∈ U ,
this transition will be pre vented by the controller . In our specific setting of W = U × Y ,
the controller can only directly restrict the U -component of the external symbol. Hence, the
controller must at least effecti vely disable the e xternal symbols
ρ x := { (u, y ) ∈ U × Y | u ∈ U, y ∈ Y s.t. (x , (u, ‡ ), x ) ∈ δ } (12)
540 Discrete Event Dynamic Systems (2020) 30:533–560
whene ver the plant attains the state x ∈ X . For the remaining transitions, ho we v er , δ
matches the original transition relation δ ; i.e.,
{ (x , (u, y ), x ) ∈ δ | (u, y ) ∈ ρ x }={ (x , (u, y ), x ) ∈ δ | (u, y ) ∈ ρ x } . (13)
Therefore, the supervisor designed for the substitute P will implement exactly the same
closed-loop beha viour when applied to the actual plant P .
Remark 6 The situation of restricted initial states is addressed in a similar fashion. Gi ven
P = (X, U × Y, δ, X 0 ) with X 0 X , we consider the substitute model P = (X, U ×
Y ,δ ,X ) , U = U ˙
∪{ † } , Y = Y ˙
∪{ ‡ } . Here, the distinguished input symbol † is introduced
to test whether the state is within the original set of initial states, and, if so, this is confirmed
by the distinguished output symbol ‡. T echnically , we define the transition relation by
δ := δ ∪{ (x , ( † , ‡ ), x ) | x ∈ X 0 }∪{ (x , ( † ,y ) ,x ) | x ∈ X − X 0 ,y ∈ Y } . (14)
W e then design a controller for P with a s pecification that requires (a) that † appears e xclu-
si vely as the unique first input symbol and (b) that an y further closed-loop requirements are
only imposed conditionally , subject to ‡ being generated as the very first output symbol.
In order to apply the resulting controller to the actual plant, we need an additional de vice
that generates an adequate output symbol when the distinguished input symbol † is applied,
i.e., we need to implement the additional transitions introduced by δ .H o w e v e r ,b yc l a u s e
(a), this is only necessary for the very first transition tak en and, since P has the initial state
restricted to X 0 , the additional de vice is a-priori kno wn to generate ‡ and it does so without
affecting any state. From a practical perspecti ve, this can be implemented by intercepting
the closed-loop interconnection to hide the very first input symbol † from the plant and by
injecting a fake response ‡ to the controller . For an y subsequent transitions, the actual plant
P matches the substitute P as in our pre vious Remark 5. Hence, the supervisor will enforce
the conditional specification (b) in the adapted closed-loop configuration with the actual
plant.
For a discrete-time model of hybrid plant dynamics with a discrete e xternal interface, we
consider an I/S/- machine P = (X, U × Y, δ , X) with a state set X ⊆ D × R n , | D |∈ N ,a n d
with finite input- and output-ranges, i.e., | U | , | Y |∈ N . This is a rather general setting and,
for practical applications, one needs to formally deri v e the transition relation δ from a more
detailed model. Since the literature provides a rich v ariety of models for hybrid dynamics,
we demonstrate this step by example.
Example 1 Consider a physical system with linear continuous dynamics and a finite number
of linear controllers to implement indi vidual modes of operation. Discretising time by a
regular sampling period, we obtain the switched affine system
x (k + 1 ) = A d (k ) x (k ) + B d (k ) , (15)
where k ∈ N 0 denotes the discrete time; x : N 0 → R n is the sampled continuous state
trajectory; the discrete signal d : N 0 → D selects the mode of operation d (k ) at time k ;a n d
the square matrix A d ∈ R n × n and the column vector B d ∈ R n are obtained by sampling the
closed-loop configuration for mode of operation d ∈ D . W e can either directly interpret d
as our input signal, or encode additional discrete dynamics by
d (k + 1 ) = f( d (k ), u (k )), (16)
541 Discrete Event Dynamic Systems (2020) 30:533–560
where f : D × U → D is a complete transition function and u : N 0 → U is the dis-
crete input signal. As discrete output, we propose a mode dependant finite partition of the
continuous state space, i.e.,
y (k ) = g( d (k ), x (k )), (17)
with g : D × R n → Y . The transition relation δ is then formally defined by
δ := { ((x , d ), (u, y ), (x ,d )) | x = A d x + B d ,d = f (d , u), y = g( d , x ) } . (18)
W ith W := U × Y , X := D × R n and unrestricted initial states X 0 := X , the latter
completes the construction of an I/S/- machine P = ( X ,W ,δ ,X
0 ) with time-in variant
induced external and internal beha viour , respecti vely .
Example 2 In a similar way to the abo ve e xample, hybrid automata address the situation of
a finite number of modes of operation, each with specific continuous dynamics. Ho we ver ,
and in contrast to the abov e e xample, the generation of e v ents is org anised in dependency of
the e v olution of the continuous state and by referring to so called mode in variants and guar d
r elations . A general and formal definition of hybrid automata semantics is quite in volv ed
and interested readers are referred to the literature; see, e.g., Henzinger ( 2000 ) or Chapter 7
in T abuada ( 2009 ).
In the present study , we provide a simple practical e xample with hybrid automata seman-
tics to which we will refer later in the context of abstraction based controller design.
T o this end, consider a vehicle which we shall navigate within a rectangular area A :=
[ 0 ,w ]×[ 0 ,h ]⊆ R 2 ;s e eF i g . 3 .
The vehicle is equipped with lo w-le v el continuous controllers which implement the
modes of operation U := { u nw , u ne , u sw , u se } to dri v e the vehicle in the respecti v e
direction, i.e., north-west, north-east, south-west or south-east. W ith each mode u ∈ U ,
we associate a differential inclusion d
d t ϕ( t ) ∈ F u , where the constant right-hand-side
F u ⊆ R 2 is set up as the sum of the respecti ve nominal v elocity ( − v, v) , (v , v ) , ( − v, − v)
or (v , − v) ∈ R 2 and a square with diameter d ∈ R , d> 0, as a bounded additiv e distur -
bance. Each mode of operation is associated with the entire rectangular area I u := A as the
mode in variant .
Fig. 3 V ehicle navigation example with rectangular area A =[ 0 ,w ]×[ 0 ,h ]⊆ R 2
542 Discrete Event Dynamic Systems (2020) 30:533–560
An output e vent y ∈ Y := { y n , y s , y w , y e } will be generated while the v ehicle is
inside a guar d r e gion , denoted G y n , G y s , G y w , G y e ⊆ R 2 , respecti vely . T o av oid tri v-
ial Zeno-beha viour , guards are enabled and disabled according to the mode of operation,
e.g., when dri ving north-west, only the north guard G y n and the west guard G y w are
enabled.
For a discrete-time model of the v ehicle, we consider the o verall state set X := A .O nt h i s
state set, we define the transition relation δ ⊆ X × (U × Y) × X by (x , (u, y ), x ) ∈ δ ,i fa n d
only if (a) there exists τ ≥ 0 and a continuous state trajectory ϕ :[ 0 ,τ ]→ I u , differentiable
on ( 0 ,τ ) with d
d t ϕ( t ) ∈ F u for all t ∈ ( 0 ,τ ) , with the initial s tate ϕ( 0 ) = x and the final
state x = ϕ( τ ) ;( b ) x is within the guard G y ; and (c) the guard G y is enabled by mode
u ∈ U . T o practically test whether a tuple (x , (u, y ), x ) satisfies the abov e conditions (a)–
(c), we observe that the rele v ant sets I u = A , F u and G y are con ve x closed polyhedra. This
implies that the positions reachable by some qualifying continuous trajectory ϕ amount to
the con v e x closed polyhedron V ={ x + λv | v ∈ F u ,λ ≥ 0 }∩ A ; see, e.g., Halbwachs et al.
( 1997 ). Provided that the guard G y is enabled by mode u ,w eh a v et h a t (x , (u, y ), x ) ∈ δ if
and only if x ∈ V ∩ G y . In this context, the transitions in δ are also referred to as lo gic-time
transitions to contrast with the e volution of the continuous state with respect to physical
time.
This completes the construction of the I/S/- machine P = ( X ,W ,δ ,X
0 ) with W :=
U × Y and X 0 = X = A .
Referring back to Definition 2, recall that an experiment S must account for all trajecto-
ries w ∈ B by some finite prefix s ∈ ( pre w ) ∩ S . Hence, the construction of an experiment
practically amounts to the inspection of specific finite prefixes in pre B . F or example, to set
up an initial experiment by S := B | [ 0 ,l ] ⊆ pre B for some l ∈ N we need an implementable
test for whether or not s ∈ pre B for all finite sequences s of length l + 1. Like wise, refer -
ring to Eq. 7 , a refinement of an experiment S w .r .t. the candidates R ⊆ S requires us to
test whether or not s, w ∈ pre B for all s ∈ R and w ∈ W . Gi v en an I/S/- machine
P = ( X ,W ,δ ,X ) with the external beha viour B , Moor et al. ( 2002 ) propose to base the
required test on the follo wing recursi v ely defined sets of compatible states :
X ( ) := X, (19)
X ( r , (u, y ) ) := { x |∃ x ∈ X (r ) : (x , (u, y ), x ) ∈ δ } , (20)
where r ∈ W ∗ , u ∈ U , y ∈ Y , and the right hand side of Eq. 20 is a one-step forward
reachability operator applied to X (r ) with (u, y ) as a constraint for the e xternal symbols. By
construction, X (s ) ⊆ X consists of all states the I/S/- machine can attain after generating
the finite sequence of e xternal symbols s ∈ W ∗ . Since I/S/- machines do not deadlock, this
implies that s ∈ pre B if and only if X (s ) =∅ . Hence, for beha viours realised by I/S/-
machines, setting up and/or refining experiments effecti vely amounts to a finite iteration of
the one-step forward-reachability operator in Eq. 20 . The latter type of reachability operator
has been intensi vely in vestigated o v er the past two decades and the literature pro vides a
v ariety of efficient computational methods addressing specific classes of hybrid systems;
see, e.g., Alur et al. ( 2000 ), Alur et al. ( 1996 ), and Lafferriere et al. ( 2000 ) for the exact
computation of sets of reachable states for a restricted class of continuous dynamics, or ,
e.g., Althoff et al. ( 2010 ), Chutinan and Krogh ( 1998 ), Frehse ( 2008 ), Henzinger et al.
( 2000 ), Maler and Dang ( 1998 ), Mitchell et al. ( 2005 ), and Reissig ( 2011 ) f or s af e over-
approximations for richer classes of continuous dynamics.
543 Discrete Event Dynamic Systems (2020) 30:533–560
Example 3 (cont.) For our v ehicle na vigation example, all rele vant sets are con ve x closed
polyhedra and the differential inclusions ha ve a constant polyhedral right-hand-side. Sets of
states reachable by one logic-time transition can hence be computed exactly , e.g., using the
software P arma Polyhedra Library (PPL); see Bagnara et al. ( 2008 ). W e outline the ov erall
computational procedure that is used to construct an initial experiment and a refinement
thereof.
Consider u nw as the first input symbol being applied in the vehicle na vigation e xam-
ple. Then the subsequent output symbol can be either y n or y w since the initial states
are not restricted and since G y n and G y w are the only enabled guards. This implies
X (( u nw , y s )) =∅ and X (( u nw , y e )) =∅ . Under the additional hypothesis that we
actually observe y w , we conclude that the attained state must be within G y w . Hence,
we obtain X (( u nw , y w )) = G y w as compatible states; see Fig. 4 . Like wise, we obtain
X (( u nw , y n )) = G y n . Repeating the abov e reasoning for all possible first input symbols,
we establish that
B | [ 0 , 0 ] ={ ( u ne , y n ), ( u ne , y e ), ( u nw , y n ), ( u nw , y w ),
( u se , y s ), ( u se , y e ), ( u sw , y s ), ( u sw , y w ) } , (21)
and obtain our initial experiment S := B | [ 0 , 0 ] . This e xperiment encodes the fact that for
our vehicle na vigation e xample, only rele v ant guards are enabled depending on the input
symbol. This is expected to be insufficient for the design of a supervisor that solves typical
na vigation tasks like, e.g., to visit a specific guard region.
For illustration purposes, we consider R := { ( u nw , y w ) }⊆ S as our refinement candi-
date. Since X (( u nw , y w )) = G y w from the foregoing discussion, we no w need to compute
X ( ( u nw , y w ), (u, y ) ) for all u ∈ U and y ∈ Y by applying Eq. 20 . Consider the case of
u = u ne , i.e., we apply the input symbol u ne when the vehicle position x is initially in
X (( u nw , y w )) . The continuous time motion of the vehicle is then modelled by a trajectory
ϕ :[ 0 ,τ ]→ I u ne = A with ϕ( 0 ) = x and d
d t ϕ( t ) ∈ F u ne for all t ∈ ( 0 ,τ ) . Recall that
in our example, all rele vant sets are con ve x closed polyhedra. This implies that all positions
reachable by the vehicle are gi v en by
V := { x + λv | x ∈ X (( u nw , y w )), v ∈ F u ne ,λ ≥ 0 }∩ A ; (22)
see also Fig. 4 . Compatible states are then obtained by X ( ( u nw , y w ), ( u ne ,y ) ) =
V ∩ G y with y ∈ Y and are again con v e x closed polyhedra. More specifically , we hav e
X ( ( u nw , y w ), ( u ne , y n ) ) as sho wn in Fig. 4 and X ( ( u nw , y w ), ( u ne ,y ) ) =∅ for
Fig. 4 Sets of compatible states for the na vigation example
544 Discrete Event Dynamic Systems (2020) 30:533–560
y = y n . This procedure can be applied repeatedly for subsequent input symbols. T o this
end, Fig. 4 sho ws the sets of compatible states for ( u nw , y w ), ( u ne , y n ), ( u se , y s )
and ( u nw , y w ), ( u ne , y n ), ( u se , y e ) . Coming back to the refinement, we apply the
same analysis to X (( u nw , y w )) as abov e, b ut no w for the remaining choices of the input
symbol, i.e., u = u ne , u = u sw and u = u se . As it turns out, the only extensions of
our refinement candidate ( u nw , y w ) by one more pair of an input symbol and an output
symbol with a non-empty set of compatible states are the follo wing length-two sequences:
( u nw , y w ), ( u ne , y n ) , ( u nw , y w ), ( u se , y s ) ,
( u nw , y w ), ( u nw , y n ) , ( u nw , y w ), ( u nw , y w ) ,
( u nw , y w ), ( u sw , y s ) , ( u nw , y w ), ( u sw , y w ) . (23)
Referring to Eq. 7 , the refined experiment S is obtained from S := B | [ 0 , 0 ] ,E q . 21 ,b y
removing the refinement candidate ( u nw , y w ) and by including the abo ve six e xtensions;
for a tree representation of S see Fig. 5 .
Once an experiment S on the e xternal beha viour B of the plant P has been obtained, it
can be used to set up a finite-state realisation of the corresponding abstraction S . Roughly
speaking, the realisation tracks the longest suffix of the signal generated so far that matches
some prefix within S .
Theor em 7 (See Moor and G ¨
otz ( 2018 ), Lemma 14) Given a pr efix-fr ee trim experiment
S ⊆ W ∗ , ∈ S =∅ , of bounded length, the abstraction S = ( N 0 ,W , B S ) obtained
under the assumption of time in variance (see Eq. 1 ) is r ealised by the state mac hine P S =
(Z S ,W ,δ
S , { } ) ,w h e r e Z S := pre S consists of the pr efixes of S and wher e the transition
r elation δ S is defined as follows: ( z ,w ,z
) ∈ Z S × W × Z S is in δ S if and only if z = z ,w
with z the longest suffix of z that is in Z S b ut not in S ; i.e., if and only if z is the unique
longest sequence in { r ∈ W ∗ |∃ t ∈ W ∗ : t, r = z }∩{ r ∈ Z S | r ∈ S } . Mor eover , P S is
r eac hable, deadloc k-fr ee and past-induced. Pr ovided that W is a finite set, P S is a finite
state machine .
Note that for the degenerated cases of ∈ S or S =∅ we ha ve B S = W N 0 or B S =∅ ,
respecti vely , with well known realisations. T o provide some intuition re garding the transi-
tion relation gi ven by the abov e theorem, we consider the initial state z 0 := and a signal
Fig. 5 Refined experiment S on the e xternal beha viour B for the navigation e xample
545 Discrete Event Dynamic Systems (2020) 30:533–560
w ∈ B S . In particular , there exists l ∈ N 0 such that w | [ 0 ,l ) ∈ S .B y ∈ S , this implies
l ≥ 1. Moreov er , since S is prefix-free, we hav e w | [ 0 ,k ) ∈ S and w | [ 0 ,k ) ∈ pre S = Z S for
all k< l .N o wl e t z k := w | [ 0 ,k) for all k ≤ l and observe, for all k< l ,t h a t z k is its o wn
longest suffix z qualifying for z ∈ S and z ∈ Z S , and, hence, (z k , w (k ), z k + 1 ) ∈ δ S .I n
other words, the state records all past e xternal symbols until z l ∈ S . Howe ver , once in state
z l ∈ S , no additional external symbols can be recorded with the gi v en state set unless one
first drops a sufficient amount of symbols recorded earlier . T echnically , we are asking for a
suffix z of z l such that z , w (l ) ∈ Z S . T o see that such a suffix exists, we consider the
shortest suffix of z l and we refer to time in v ariance to obtain , w (l ) = w (l ) ∈ Z S .F o r
the transition relation proposed in the theorem, we take the longest qualifying z to drop as
little as possible of the recorded symbols and, as a conjecture, obtain (z l , w (l ), z l + 1 ) ∈ δ S
with z l + 1 := z , w (l ) . If we can continue this construction indefinitely and if the conjec-
ture holds true at each stage, we obtain a state trajectory z : N 0 → Z S , z (k ) := z k for all k ,
such that ( w , z ) is in the full behaviour induced by P S . Note that it is neither ob vious that
the construction actually can be continued indefinitely , nor , for the con verse beha vioural
inclusion, that any trajectory generated by P S is within B S . The cited reference in Theorem
7 provides technical proofs for both claims and the abo ve theorem.
Example 4 (cont.) For the e xperiment S from Fig. 5 , the realisation P S is obtained by the
follo wing construction. The nodes s ∈ pre S in Fig. 5 become the states of P S and the
edges in Fig. 5 become transitions with an ev ent label to match the most recent e v ent of
the respecti ve t arget node; see Fig. 6 , transitions gi v en in black color . Then, per leaf z ∈ S ,
we: (a) drop the minimum prefix from the node label z ∈ S to obtain z ∈{ r ∈ Z S | r ∈
S } ;a n d( b )f o ra l l w ∈ W such that z := z ,w ∈ Z S insert a transition from z to z
with label w . For the node z = ( u nw , y w ), u nw , y w ) ,w eh a v e z = ( u nw , y w )
with out-going transitions indicated in green color in Fig. 6 . For all other nodes z ∈ S ,
z = ( u nw , y w ), u nw , y w ) , we obtain z = and insert per w ∈ W the transition
(z, w , w ) ∈ δ S . This amounts to 12 × 8 = 96 transitions, which are omitted in Fig. 6 .
W e observe for the vehicle na vigation e xample that not only the actual plant P bu t al so
the realisation P S of the abstraction is an I/S/- machine. This can always be achie v ed by suit-
able trimming, and we can without loss of generality restrict the discussion to experiments
such that P S is an I/S/- machine.
Fig. 6 Fragment of the realisation P S of the experiment S from Fig. 5
546 Discrete Event Dynamic Systems (2020) 30:533–560
Pr oposition 8 Consider an experiment S ⊆ W ∗ , W = U × Y , on an e xternal behaviour
B ⊆ W N 0 r ealised by an I/S/- mac hine P with input rang e U and output rang e Y . W ith
P S = (Z S ,W ,δ
S , { } ) fr om Theor em 7, let
Z io := { z ∈ Z S |∀ u ∈ U ∃ y ∈ Y, z ∈ Z S : (z, (u, y ), z ) ∈ δ S } (24)
and trim S by
S := { s ∈ S | pre s ⊆ Z io } . (25)
Then S ⊆ S is an experiment on B .
Pr oof To s h o w t h a t S is an e xperiment on B , we pick an arbitrary w ∈ B .S i n c e S is
an experiment on B , there e xists s ∈ S with s< w . T o show that s ∈ S ,w ep i c ka n
arbitrary prefix z ≤ s and an arbitrary u ∈ U and establish the existence of y ∈ Y and
z ∈ Z S such that (z, (u, y ), z ) ∈ δ S . For our choice, we refer to an y state trajectory x
such that ( w , x ) is in the full behaviour induced by P .S i n c e P is an I/S/- machine, there
exists another trajectory ( w , x ) from the full beha viour such that ( w , x ) | [ 0 ,l ) = ( w , x ) | [ 0 ,l ) ,
w (l ) = (u, y ) and x (l ) = x (l ) for l =| z | and some y ∈ Y .B y B ⊆ B S we obtain the
unique state trajectory z such that ( w , z ) is in the full beha viour induced by P S .B yt h e
definition of δ S we hav e z (k ) = w | [ 0 ,k ) for all k ≤ l . In particular , we ha ve z (l ) = z and,
hence, (z, (u, y ), z (l + 1 )) ∈ δ S .
Applying the abov e trimming procedure repeatedly , it generates a monotonously decreas-
ing sequence S ⊇ S ⊇ S ⊇ ··· of experiments. F or our situation of a finite e xternal
signal range and experiments of bounded length, S is a finite set. Hence, a fixpoint S ∗ is
attained after finitely many stages of trimming. Then, the definition of Z io implies that the
S ∗ is indeed realised by an I/S/- machine. Moreov er , the realisation by an I/S/- machine is
retained under refinement by Eq. 7 .
5 Supervisory Control
Gi ven a plant model with discrete-e vent interface, we seek to design a controller that
restricts the behaviour to satisfy a prescribed upper bound specification. This type of con-
trol problem is addressed by supervisory contr ol theory , as introduced by Ramadge and
W onham ( 1987 , 1989 ), howe v er , using regular ∗ -languages and finite automata realisations
as base models. For the present paper , we refer to an adaption of supervisory control to ω -
languages to address infinite-length signals as discussed by (Thistle and W onham 1994a ;
1994b ), and we propose to substitute the actual plant by a finite state abstraction obtained
from an experiment.
Formally , a supervisor is defined as a causal feedback map f with domain W ∗ that, at
any instance of time k ∈ N 0 , maps the present prefix s = w | [ 0 ,k) generated by the plant to
a contr ol pattern γ = f( s ) ⊆ W , with the effect that the subsequently generated symbol
must satisfy w (k ) ∈ γ , i.e., the plant is restricted to only generate symbols that match the
respecti ve control pattern. Most commonly the range Γ of all admissible control patterns
is deri ved from partitioning W into controllable and uncontrollable e v ents. Ho we v er , to
address I/S/- machines as plants, we refer to the product W = U × Y and define
Γ := { γ ⊆ U × Y |∅ = γ and ∀ (u, y ) ∈ γ, y ∈ Y : (u, y ) ∈ γ } (26)
as the range of the supervisor , i.e., f : W ∗ → Γ . By this choice, the supervisor imposes its
restriction on the input symbol only and accepts any output symbol generated by the plant.
547 Discrete Event Dynamic Systems (2020) 30:533–560
Note also that, by construction, Γ is closed under unions of control patterns and, thus, our
setting here is formally cov ered by the rele v ant references (Thistle and W onham 1994a , b ).
In the more common setting with a partition into controllable and uncontrollable e vents,
a supervisor could apply a control pattern such that the plant in its current state cannot
generate any of the enabled symbols. This form of temporal blocking is undesired and,
in general, needs to be addressed by the synthesis procedure. Howe ver , for the specific
situation of I/S/- machines and our tailored choice of Γ in Eq. 26 , temporal blocking is not
an issue.
Definition 9 Giv en a deadlock-free state machine P = ( X ,W ,δ ,X
0 ) , a supervisor
f : W ∗ → Γ pr eserves liveness in closed-loop configuration with P if for all signals
( w , x ) ∈ B full from the induced full beha viour that comply with f up to some time k ∈ N 0 ,
i.e., w (κ ) ∈ f( w | [ 0 ,κ ) ) for all κ< k ,t h e r ee x i s t w ∈ f( w [ 0 ,k ) ) and ( w , x ) ∈ B full such
that x | [ 0 ,k ] = x | [ 0 ,k ] , w | [ 0 ,k ) = w | [ 0 ,k ) and w (k ) = w .
Pr oposition 10 Given an I/S/- machine P = ( X ,W ,δ ,X
0 ) wher e W = U × Y with input
rang e U and output rang e Y , any supervisor f : W ∗ → Γ pr eserves liveness in closed-loop
configuration with P .
Pr oof Consider any ( w , x ) ∈ B full compliant with f up to time k ∈ N 0 . Then at time
k ∈ N 0 the supervisor applies the control pattern γ := f( w [ 0 ,k ) ) and P is in state x = x (k ) .
By Eq. 26 , γ =∅ and we can pick a symbol w := (u, y ) ∈ γ .S i n c e P is an I/S/- machine,
there exist y ∈ Y and x ∈ X such that (x , (u, y ), x ) ∈ δ . Referring again to Eq. 26 ,
we obtain w := (u, y ) ∈ γ . T o construct the signals x and w ,l e t x (κ ) := x (κ ) for
0 ≤ κ ≤ k , x (k + 1 ) = x , w (κ ) := w (κ ) for 0 ≤ κ< k and w (k ) = w . Observe that
( x (κ ), w (κ ), x (κ + 1 )) ∈ δ for 0 ≤ κ ≤ k .S i n c e P itself is deadlock-free, the signals can
be extended to the entire time axis by taking arbitrary transitions from δ . W e then obtain
( w , x ) ∈ B full as required.
Restricting consideration to I/S/- machines and the corresponding choice of Γ ,E q . 26 ,
the problem of supervisory control is stated as follo ws.
Definition 11 Consider a plant = ( N 0 ,W , B ) realised by an I/S/- machine P =
( X ,W ,δ ,X
0 ) with input range U and output range Y , and a specification spec =
( N 0 ,W , B spec ) . For a supervisor f : W ∗ → Γ with Γ from Eq. 26 ,t h e closed-loop
behaviour is defined by
B f := { w ∈ B |∀ k ∈ N 0 : w (k ) ∈ f( w | [ 0 ,k ) ) } . (27)
A supervisor f : W ∗ → Γ solves the contr ol pr oblem if it enfor ces the specification , i.e., if
B f ⊆ B spec .
The provided references (Thistle and W onham 1994a , b ) present an algorithmic solution
to the abov e problem for the case that the rele v ant beha viours are ω -re gular and realised by
past-induced finite state machines, extended by an acceptance condition. In the conte xt of
the present paper , we will substitute the actual plant by the abstraction S obtained from
some experiment S with past-induced finite state realisation P S = (Z S ,W ,δ
S , { } ) .R e g a r d -
ing the specification, we account for past-induced finite realisations with B ¨
uchi acceptance
condition , i.e., we consider a state machine P spec = (X spec ,W ,δ
spec , { x spec0 } ) with a set of
548 Discrete Event Dynamic Systems (2020) 30:533–560
accepting states X specM ⊆ X spec and require that signals in the full beha viour visit accepting
states infinitely often. The specification spec = ( N 0 ,W , B spec ) is then formally defined by
B spec := { w : N 0 → W |∃ x : N 0 → X spec :
( w , x ) is in the full behaviour of P spec ,
and x (k ) ∈ X specM for infinitely many k ∈ N 0 } . (28)
As a technical consequence of introducing an acceptance condition, it is not restricti v e to
assume that the transition relation δ spec is full , i.e., for all χ ∈ X spec and all w ∈ W there
exists χ ∈ X spec such that (χ , w , χ ) ∈ δ spec . For e xample, assume that we wish to exclude
all closed loop trajectories that exhibit a certain string of symbols from W . W e can encode
this in a specification state machine with full transition relation where the occurrence of
such a string leads into a “dump state”, from where no other state can be reached. The
assumption of a full transition relation is common in automata theory and it simplifies the
subsequent discussion.
Supervisory controller synthesis is conducted in the follo wing tw o steps. First, we extend
the abstraction state set to also encode the specification state by the product composition
P × := P S × P spec := ( Q ,W ,λ , { q 0 } ) ,w h e r e Q = Z S × X spec , q 0 = ( , x spec0 ) ,a n dw h e r e
λ ⊆ Q × W × Q is defined by ((z, χ ), w , (z ,χ )) ∈ λ if and only if ( z ,w ,z
) ∈ δ S and
(χ , w , χ ) ∈ δ spec .S i n c e δ spec is full, the induced beha viours of P × equal the respecti v e
beha viours induced by P S . Moreov er , past-inducedness of both components P S and P spec
implies past-inducedness of the product P × . W ith Q M := Z S × X specM ⊆ Q we lift the
acceptance condition of the specification accordingly . Note also that, since δ spec is full and
since we assume P S to be an I/S/- machine, P × is also an I/S/- machine and, hence, is
deadlock-free. Regarding the acceptance condition, ho wev er , there can be liv e-locks; i.e.,
reachable states with no ex ecution path to attain a state in Q M thereafter . This is addressed
by the second step of the synthesis approach, where we refer to the iteration proposed by
Thistle and W onham ( 1994a , b ) in order to identify states in P × which can be controlled to
e ventually visit some accepting states of Q M and to do so infinitely often. The resulting state
set Q win is referred to as the set of winning states — once the closed-loop has generated
a prefix that corresponds to a winning state q ∈ Q win , a supervisor can be employed to
enforce the specification from then on. In particular , the supervisory control problem has
a solution if and only if the initial state of P × is a winning state. Addressing more general
acceptance conditions for both the plant and the specification, Thistle and W onham ( 1994a ,
b ) obtain the set of winning states by a fi v e-nested fixpoint iteration, which for the specific
situation in the present paper collapses to the follo wing simplified algorithm.
Algorithm 12 W inning states Q win of P × = ( Q ,W ,λ , { q 0 } ) w .r .t. the acceptance condition
Q M ⊆ Q and contr ol patterns Γ .
1) Initialise the target restriction with D := Q .
2) Initialise the winning states with Q win := ∅ .
3) Perform the follo wing one-step controlled backward-reachability analysis:
B ={ q ∈ Q |∃ γ ∈ Γ :∅ = λ(q , γ ) ⊆ (Q M ∩ D) ∪ Q win } .
4) If B Q win , then update the winning states by Q win := Q win ∪ B and proceed with
Step 3. Else, proceed with Step 5.
5) If Q M ∩ D Q win , then update the restriction by D := D ∩ Q win and proceed with
Step 2. Else, terminate and report Q win as the result.
549 Discrete Event Dynamic Systems (2020) 30:533–560
W e provide some intuition on the abov e algorithm; see also Fig. 7 . The inner loop o ver
Steps 2–4 begins with Q win =∅ to accumulate in Q win states that can be controlled to
reach Q M ∩ D within a finite number of steps. Since during the inner loop Q win gro ws
monotonously and the reachability analysis in Step 3 is monotone in the iterate Q win , finite-
ness of the state set Q implies that the termination condition B ⊆ Q win is satisfied after
finitely many iterations. When proceeding with Step 5 for the first time, Q win holds the
states that can be controlled to reach Q M at least once and, until then, to remain within Q win .
This is illustrated in Fig. 7 on the left, where the gro wth of Q win occurs counter -clock-wise.
In the figure it is assumed that the top-most transition which does not go to the target Q M
can be disabled by a suitable control pattern. As indicated in the figure, we cannot expect
Q M ⊆ Q win , i.e., so far there may be states Q win that can indeed only be controlled to reach
Q M once. Therefore, Step 5 restricts the effecti ve tar get by letting D = Q win . Repeating
the inner loop again results in a set of winning states, b ut no w they can all be controlled to
reach Q M at least twice. This is illustrated in Fig. 7 on the right, where the target has been
restricted accordingly . Repeating the outer loop by monotonicity leads to a strictly decreas-
ing restriction D and, by finiteness of Q , the termination condition must be satisfied after a
finite number of iterations. At termination in Step 5, any state q ∈ Q win can be controlled
to reach Q M ∩ D , and, by Q M ∩ D ⊆ Q win , can be controlled to do so infinitely often.
A supervisor can be obtained from the abov e algorithm by recording for all q ∈ Q win an
arbitrary successful control pattern from Step 3 of the last run of the inner loop. T echnically ,
this defines a map g : Q win → Γ . By past-inducedness of P × , each prefix s ∈ pre B S
corresponds to exactly one state in P × which we denote λ 0 (s ) ∈ Q . The supervisor f is
then defined for s ∈ W ∗ by f( s ) = g( λ 0 (s )) if λ 0 (s ) ∈ Q win and, else, f( s ) = γ dummy ∈ Γ
with γ dummy := ∪{ γ ∈ Γ } . By construction, this supervisor preserves liv eness in closed-
loop configuration with the abstraction P S (e ven if it w as not an I/S/- machine) it was
designed for , and it conditionally enforces the specification once a prefix s ∈ pre B S with
λ 0 (s ) ∈ Q win has been generated; i.e., we hav e
B S, f ⊆{ w ∈ W N 0 | ( ∃ s ∈ pre w : λ 0 (s ) ∈ Q win ) ⇒ w ∈ B spec } (29)
for the closed-loop behaviour B S, f ; see Definition 11. Since the empty string ∈ pre w
corresponds to the initial state q 0 = λ 0 ( ) , the right-hand-side of the abov e inclusion
collapses to B spec if we hav e q 0 ∈ Q win . In this case, we indeed obtain a solution of
the control problem for the abstraction. This immediately carries ov er to the actual plant
P ∼
= = ( N ,W , B ) : the supervisor f preserves li veness in closed-loop configuration with
P by Proposition 10 and we obtain B f ⊆ B S, f ⊆ B spec as an immediate consequence of
Fig. 7 Synthesis algorithm: first iteration of inner loop (left) and fixpoint of outer loop (right)
550 Discrete Event Dynamic Systems (2020) 30:533–560
B ⊆ B S and Definition 11. If, on the other hand, q 0 ∈ Q win , it follo ws from the detailed
study by Thistle and W onham ( 1994a , b ) that the control problem has no solution for the
abstraction P S at hand. In this case, we interpret Q win as an intermediate result which in an
ov erall synthesis approach can be used to guide a local refinement of the abstraction.
Example 5 (cont.) For the v ehicle na vigation example from the pre vious section, we con-
sider the specification to na vigate the vehicle e ventually to the north guard G y n . This can
be expressed by a state machine P spec with two states, where one is an accepting state and
indicates that y n has occurred at least once. For controller synthesis, we use the abstraction
P S obtained from the experiment S sho wn in Fig. 5 . All states (z, χ ) ∈ Q ,f o rw h i c h z ∈
Z S = pre S includes a y n symbol, are immediately identified as winning states. Also, states
(z, χ ) with z = ( u nw , y w ) turn out to be winning states, because one can apply the con-
trol pattern { ( u ne ,y ) | y ∈ Y } to enforce that the winning state ( u nw , y w ), ( u ne , y n ) is
attained by the next transition. Lik e wise, the initial state is a winning state: by applying the
control pattern { ( u nw ,y ) | y ∈ Y } we either hav e ( u nw , y n ) or ( u nw , y w ) , both kno wn
to be winning states by our pre vious observ ations. Thus, the supervisory control problem
can be solved based on the abstraction. In contrast, if the control objecti ve was to e v entu-
ally visit the west guard G y w , the provided abstraction is too coarse for a positi v e result —
although intuition suggests that the actual plant can be very well controlled accordingly .
6 Guided refinements of experiments
W e now consider the situation where abstraction-based synthesis of a supervisor as dis-
cussed in the pre vious section has failed, i.e., we are gi v en a plant = ( N 0 ,W , B ) realised
by a I/S- machine P and an abstraction S = ( N 0 ,W , B S ) obtained from an experiment S
on B , b ut applying Algorithm 12 sho ws that q 0 ∈ Q win for the winning states Q win ⊆ Q of
the composed state machine P × = ( Q ,W ,λ , { q 0 } ) . Pro vided that we are optimistic about
the control problem with the actual plant to e xhibit a solution, it is proposed to refine the
abstraction and to repeat the synthesis procedure. Referring to Moor and Raisch ( 1999 )
and Moor et al. ( 2002 ), where the abstraction used is an l -complete approximation, i.e.,
S = B | [ 0 ,l ] for some l ∈ N 0 , a refinement can be obtained by substituting l with l + 1.
Effecti vely , this uniformly extends the sampled sequences in length by one more symbol.
Ho we v er , such an e xtension amounts to testing whether or not the e xtended sequence is in
pre B , and this test is implemented as a one-step reachability analysis conducted on the orig-
inal system. Since this is considered computationally e xpensi ve, we seek to identify specific
sequences R ⊆ S that are worth the effort and use Eq. 7 to obtain a refinement of S tai-
lored for the synthesis task at hand. The ov erall abstraction-based approach then becomes
an iteration in which we alternate trial synthesis and abstraction refinement.
Algorithm 13 Iterative pr ocedur e to synthesise a supervisor for an I/ S / − mac hine P =
( X ,W ,δ ,X
0 ) , W = U × Y , X = X 0 , and a past-induced specification P spec =
(X spec ,W ,δ
spec , { x spec0 } ) with accepting states X specM ⊆ X spec .
1) Initialise the experiment S ⊆ W ∗ by S := B | [ 0 , 0 ] ,w h e r e B denotes the external
beha viour induced by P .
2) Referring to Theorem 7, set up P S to realise the abstraction obtained from S .
3) Run Algorithm 12 on the product P × = P S × P spec to obtain the winning states Q win .
551 Discrete Event Dynamic Systems (2020) 30:533–560
4) If the initial state q 0 of P × is within Q win , report the corresponding supervisor and
terminate the iteration.
5) Choose refinement candidates R ⊆ S to obtain a refinement by Eq. 7 to substitute S ,
and proceed with Step 2.
In the case that the procedure terminates at Step 4, we refer to the discussion of the pre-
vious section and recall that the supervisor not only solves the synthesis problem for the
abstraction P S b ut also for the actual plant P . Otherwise, the experiment is refined in Step 5
for the subsequent trial synthesis. The proposed iteration may fail to terminate reg ardless
of the choice for the refinement in Step 5. This is to be expected: since the v erification of
language inclusion is kno wn to be only semi-decidable e v en for restricted classes of hybrid
systems, the synthesis problem cannot be decidable either . Ho we ver , we will propose a
refinement scheme for Step 5 that ensures termination under the hypothesis of the exis-
tence of some experiment for which synthesis succeeds. W e are now left to set up sensible
refinement candidates R to implement Step 5.
For our analysis, we inspect the composed system P × = ( Q ,W ,λ , { q 0 } ) := P S × P spec ,
with lifted marked states Q M ⊆ Q for the specification acceptance condition and winning
states Q win ⊆ Q obtained by the synthesis algorithm. A refinement obtained by extending
specific samples s ∈ S then corresponds to extending the transition relation λ at states
q = (z, χ ) ∈ Z S × X spec = Q with z = s . Hence, our inspection of P × focuses attention
on states in
Q leaf := { (z, χ ) ∈ Q | z ∈ S } , (30)
and we will identify two classes of states that in turn characterise sequences s ∈ S that are
not worth a refinement. F or our formal ar gument, we consider two more e xperiments S and
S on B such that S ≤ S ≤ S . Here, we assume that S is a successful refinement of S
in the sense that there exists a supervisor f such that the closed-loop B S ,f satisfies the
specification. W e then construct S to refine S in the same way as S except for a v oiding
refinement at a specific sequence s ∈ S ,s e eF i g . 8 . T echnically , we let
S := { s }∪{ r ∈ S | s ∩ ( pre r) =∅ } (31)
Fig. 8 Intermediate experiment S with S ≤ S ≤ S by not refining at s ∈ S . Both e xperiments S and S are
sho wn as “barriers” (dark green) in the computational tree pre B (light green). The intermediate e xperiment
S lies between S and S and consists of (a) the sequence s and (b) all sequences r ∈ S except for extensions
of s
552 Discrete Event Dynamic Systems (2020) 30:533–560
to observe that S is indeed an e xperiment on B and that S ≤ S implies S ≤ S ≤ S .W e
then sho w that S is also a successful refinement of S and thereby establish that the synthesis
problem can be solved without refinements at the pre viously identified sequence s ∈ S .
For the remainder of this section, we refer t o the synthesis problems based on the exper -
iments S and S by the same notational con ventions as introduced for S, i.e., we denote
the associated abstractions S = ( N 0 ,W , B S ) and S = ( N 0 ,W , B S ) , the realisa-
tions thereof P S = (Z S ,W ,δ
S , { } ) and P S = (Z S ,W ,δ
S , { } ) , the composed state
machines P
× = (Q ,W ,λ
, { q 0 } ) and P
× = (Q ,W ,λ
, { q 0 } ) , the lifted marked states Q
M
and Q
M , and the winning states Q
win and Q
win as obtained by Algorithm 12, respecti vely .
6.1 W inning states
Once the abstraction S has generated a finite sequence s that driv es the composed state
machine P × to a winning state, there exists a supervisor that enforces the specification from
then on. Intuiti vely , for such states, no refinement is necessary .
For a formal ar gument, fix any z = s ∈ S ⊆ Z S such that
{ (z, χ ) | χ ∈ X spec ,s }⊆ Q win , (32)
where
X spec ,s := { χ ∈ X spec |∃ u ∈ W ∗ : u, s ∈ pre B S ,χ ∈ δ spec (X 0 , u, s ) } . (33)
Recall that we ha ve synthesised a supervisor f that enforces the conditional specifica-
tion ( 29 ) with B S as the plant. Moreov er , by hypothesis, there exists a supervisor f for the
refined abstraction B S such that the closed loop satisfies B S ,f ⊆ B spec . In order to estab-
lish the existence of a supervisor that enforces the specification for B S with the relax ed
refinement S in Eq. 31 , we use the candidate f : W ∗ → Γ defined by
f (r ) := ⎧
⎨
⎩
f (r ) if r = u, s , t for all u, t ∈ W ∗ ,o r ,
f( s, t ) if r = u, s , t for some u, t ∈ W ∗
and u chosen to be of minimum length,
(34)
i.e., f applies the same control patterns as f until the sequence s has been observed and,
from then on, behav es as f in ignorance of any symbols generated before s . The intuition
here is that if the closed loop formed by B S and f happens to not generate s ,t h e ni te v o l v e s
within B S and, hence, f enforces the specification. If, on the other hand, s is generated,
this corresponds to a winning state of P × and, hence, f enforces the specification. W e
obtain the follo wing lemma.
Lemma 14 Consider thr ee e xperiments S ≤ S ≤ S over the finite signal r ange W =
U × Y with the r espective associated abstr actions S = ( N 0 ,W , B S ) , S = ( N 0 ,W , B S )
and S = ( N 0 ,W , B S ) , and with r espective past-induced r ealisations P S , P S and P S
given by Theor em 7 . Assume that S r elates to S and S as in Eq. 31 for some s ∈ W ∗ that
complies with Eq. 32 . If ther e e xists a supervisor f such that B S ,f ⊆ B spec , then ther e
also e xists a supervisor f such that B S ,f ⊆ B spec .
Pr oof W e prov e the existence by the candidate supervisor f gi ven in Eq. 34 .T os h o w
B S ,f ⊆ B spec ,p i c ka n y w ∈ B S ,f . W e distinguish two cases.
First, assume that u, s ∈ pre w for all u ∈ W ∗ .W et h e nh a v e f( r ) = f (r ) for all
r ∈ pre w . No w pick arbitrary k ∈ N 0 and refer to w ∈ B S for the choice of l ∈ N 0 such that
(σ k w ) | [ 0 ,l ) ∈ S . Here, the case hypothesis implies (σ k w ) | [ 0 ,l ) = s and, hence (σ k w ) | [ 0 ,l ) ∈
S .S i n c e k ∈ N 0 was arbitrary , we obtain w ∈ B S to conclude with w ∈ B S ,f ⊆ B spec .
553 Discrete Event Dynamic Systems (2020) 30:533–560
For the second case we pick the shortest sequence u ∈ W ∗ such that u, s ∈ pre w .W e
then ha ve (σ | u | w )(k ) = w ( | u |+ k) ∈ f ( w | [ 0 , | u |+ k) ) = f ((σ | u | w ) | [ 0 ,k ) ) for all k ≥| s | .B y
w ∈ B S ,f ⊆ B S there uniquely exist state trajectories z : N 0 → Z S and x : N 0 → X spec
such that ( w ,( z , x )) is in the full behaviour induced by P × . In particular , we hav e that
x ( | u |+| s | ) ∈ X spec ,s . By time in variance, we obtain w := σ k w ∈ σ k B S ⊆ B S and denote
z : N 0 → Z S the unique state trajectory such that ( w , z ) is in the full behaviour induced
by P S . Here, we observe that z ( | s | ) = z = s . W ith x := σ | u | x , we obtain a state trajectory
( z , x ) with (( z (k ), x (k )), w (k ), ( z (k + 1 ), x (k + 1 )) ∈ λ for all k ∈ N 0 . By the choice
of s in Eq. 32 , we observe ( z ( | s | ), x ( | s | )) ∈ Q win and, referring to the case hypothesis,
we also ha ve w (k ) ∈ f( w | [ 0 ,k ) ) for all k> | s | . Therefore, there exist infinitely man y
k> | s | such that ( z (k ), x (k )) ∈ Q M and, hence, x ( | u |+ k) = x (k ) ∈ X specM . This implies
w ∈ B spec and concludes the proof of B S ,f ⊆ B spec .
6.2 F ailing States
Denote Q fail ⊆ Q leaf the set of failing states , i.e., states from which the accepting states
Q M are not reachable:
Q fail := { q ∈ Q leaf | λ(q , W + ) ∩ Q M =∅ } . (35)
Obviously , a state q ∈ Q fail cannot be a winning state and, intuitiv ely , it cannot become a
winning state in any refinement. Therefore, a refinement at a failing state is not e xpected to
be rele v ant for any solution of the control problem.
For our formal ar gument, fix an y z = s ∈ S ⊆ Z S with
{ (z, χ ) | χ ∈ X spec ,s }⊆ Q fail (36)
where X spec ,s is defined in Eq. 33 . By the follo wing proposition, we associate with s as e t
of failing states in the composed state machine P
× based on the refinement S .
Pr oposition 15 Consider two experiments S ≤ S o ver W = U × Y with the r espective
associated abstractions S and S , and with the r espective past-induced r ealisations P S
and P S given by Theor em 7. Referring to the composed state mac hine P
× = P S × P spec ,
let
Q
fail ,s := { ξ ∈ Q |∃ t ∈ W ∗ ,χ ∈ X spec : ξ = ( s, t ,χ ) }⊆ Z S × X spec (37)
for some s ∈ W ∗ that complies with Eqs. 35 and 36 . Then any trajectory ( w , x ) of the full
behaviour induced by P
× that passes Q
fail ,s does not satisfy the specification, i.e., if ther e
e xists k ∈ N 0 with x (k ) ∈ Q
fail ,s then w ∈ B spec .
Pr oof Consider a state ξ ∈ Q
fail ,s , i.e., we ha v e ξ = ( s, t ,χ ) for some t ∈ W ∗ and some
χ ∈ X spec . For a contradiction, assume that there e xists a trajectory ( w , x ) from the full
beha viour induced by P
× that first passes ξ and thereafter passes Q
M . W e can then find u ,
v ∈ W ∗ , v = , such that u, s , t , v ∈ pre w , λ
0 ( u, s , t ) = ξ ,a n d λ
0 ( u, s , t , v ) ∈ Q
M .
W e denote the respectiv e specification components of the state χ u = δ spec , 0 (u) , χ us =
δ spec , 0 ( u, s ) , χ us t = δ spec , 0 ( u, s , t ) = χ and χ us t v = δ spec , 0 ( u, s , t , v ) ∈ X specM .B y
w ∈ B S ⊆ B S , we observe χ us ∈ X spec ,s .B y σ k B S ⊆ B S ⊆ B S ,f o ra n y k ∈ N 0 ,w e
conclude that w := σ | u | w ∈ B S . Hence, we can choose z : N 0 → Z S such that ( w , z ) is
in the full behaviour induced by P S . Since the transition relation δ spec is full, we can use w
to generate a state trajectory for any initial state. In particular , there exists x : N 0 → X spec
such that x ( 0 ) = χ u , x ( | s | ) = χ us , x ( | s |+| t | ) = χ us t , x ( | s |+| t |+| v | ) = χ us t v and
( x (k ), w (k ), x (k + 1 )) ∈ δ spec for all k ∈ N 0 . Therefore, ( z ( | s |+| t |+| v | ), x ( | s |+| t |+| v | )) is
554 Discrete Event Dynamic Systems (2020) 30:533–560
reachable from ( z ( | s | ), x ( | s | )) by transitions from λ . W e observe z ( | s | ) = s = z , and, hence
( z ( | s | ), x ( | s | )) = (z, χ us ) ∈ Q fail . This constitutes a contradiction with x ( | s |+| t |+| v | ) =
χ us t v ∈ X specM . Therefore, no trajectory ( w , x ) from the full behaviour induced by P
×
that passes ξ can pass Q
M thereafter .
Recall that, by hypothesis, there exists a supervisor f that when applied to S enforces
the specification, i.e., B S ,f ⊆ B spec . No w consider for some w ∈ B S ,f the unique state
trajectory x such that ( w , x ) is in the full behaviour induced by P
× . W e then conclude
by the abov e proposition that x does not pass Q
fail ,s . Inspecting the definition of Q
fail ,s
and the relaxed refinement S in Eq. 31 ,t h i si m p l i e st h a t f controls P
× such that the set of
reachable states is within Z S × X spec ⊂ Z S × X spec This suggests that we may apply f
to B S in order to obtain B S ,f = B S ,f ⊆ B spec . W e provide a proof for this conjecture
and obtain the follo wing lemma.
Lemma 16 Consider thr ee e xperiments S ≤ S ≤ S over the finite signal r ange W =
U × Y with the r espective associated abstr actions S = ( N 0 ,W , B S ) , S = ( N 0 ,W , B S )
and S = ( N 0 ,W , B S ) , and with r espective past-induced r ealisations P S , P S and P S
given by Theor em 7. Assume that S r elates to S and S as in Eq. 31 for some s ∈ W ∗ that
complies with Eqs. 35 and 36 .T h e n B S ,f ⊆ B spec for a supervisor f implies B S ,f =
B S ,f .
Pr oof For a preliminary observ ation, denote L s := { r ∈ W ∗ | s ∈ pre r } the set of all
sequences that do not pass s .W et h e nh a v e Q ⊂ Q ⊂ Q
fail ,s ˙
∪ (L s × X spec ) .B yE q . 31 ,
we further obtain S ∩ L s = S ∩ L s , and, referring to the realisations by Theorem 7,
δ S ∩ (L s × W × L s ) = δ S ∩ (L s × W × L s ) , and, hence, λ ∩ ((L s × X spec ) × W × (L s ×
X spec )) = λ ∩ ((L s × X spec ) × W × (L s × X spec )) . In other words, the realisations P
× and
P
× coincide when restricting the respecti ve state set to L s × X spec .
W e always ha v e B S ,f ⊆ B S ,f directly from Definition 11. For the con verse inclusion,
pick any closed-loop trajectory w ∈ B S ,f ,a n dl e t x denote the unique corresponding state
trajectory such that ( w , x ) is in the full behaviour of P
× . For a contradiction, assume that
there exists k ∈ N 0 such that x (k ) ∈ L s × X spec and we pick the smallest such k . In partic-
ular , we hav e x (k ) ∈ Q
fail ,s . Referring to the input-output structure and the corresponding
choice of control patterns, we can then construct a trajectory ( w , x ) in the full beha viour
of P
× such that x | [ 0 ,k ] = x | [ 0 ,k ] and w ∈ B S ,f . Ho we v er , by Proposition 15, we ha v e
that x (k ) ∈ Q
fail ,s implies w ∈ B spec to constitute a contradiction to B S ,f ⊆ B spec .
Therefore, we ha ve that x (k ) ∈ L s × X spec for all k ∈ N 0 . Then, ( w , x ) must be in the full
beha viour of P
× and, hence, w ∈ B S . T ogether with w ∈ B S ,f this implies w ∈ B S ,f .
By the arbitrary choice of w ∈ B S ,f , we conclude B S ,f ⊆ B S ,f .
6.3 Main result
Considering the two classes of states identified abo v e, we propose the refinement candidates
R ={ s ∈ S |∃ χ ∈ X spec ,s : (s , χ ) ∈ Q fail ∩ Q win } (38)
for Step 5 of Algorithm 13 and state our main result.
Theor em 17 Given a time in variant plant = ( N 0 ,W , B ) , W = U × Y , r ealised by
an I/S/- machine P = ( X ,W ,δ ,X ) with finite input range U and finite output r ange
555 Discrete Event Dynamic Systems (2020) 30:533–560
Y , consider the supervisory contr oller synthesis pr oblem for a specification spec =
( N 0 ,W , B spec ) r ealised by a past-induced finite state machine P spec = (X spec ,W ,δ , X
spec0 )
with B ¨
uchi acceptance condition X specM ⊆ X spec ;s e e Eq. 28 . Assume that ther e e xists an
e xperiment S ∗ ⊆ W ∗ on B with associated abstraction S ∗ = ( N 0 ,W , B S ∗ ) and a super -
visor f ∗ : W ∗ → Γ such that the closed-loop behaviour B S ∗ ,f ∗ obtained fr om B S ∗ under
supervision f ∗ satisfies the specification; i.e., B S ∗ ,f ∗ ⊆ B spec . Then Algorithm 13 with
r efinement candidates in Eq. 38 terminates with success after finitely many iter ations.
Pr oof For a proof by contradiction, assume the algorithm does not terminate. Denote S j ⊆
W ∗ the experiment in the j -th iteration with refinement candidates R j ⊆ S j identified by
Eq. 38 . W e then hav e that S j ≤ S j + 1 and S j = S j + 1 for all j ∈ N 0 .T h i si m p l i e sp r e S j
pre S j + 1 for all j ∈ N 0 . Since the signal range is finite, we can choose a sufficiently lar ge
j ∈ N 0 such that ( pre S ∗ ) ∩ ( pre S j ) = ( pre S ∗ ) ∩ ( pre S j + 1 ) . This implies ( pre S ∗ ) ∩
S j = ( pre S ∗ ) ∩ S j + 1 . Referring to the general scheme of refinement Eq. 7 , we obtain that
( pre S ∗ ) ∩ R j =∅ . W e no w construct one more experiment on B :
S := { s ∈ S ∗ | ( pre s) ∩ S j =∅ } ∪ { s ∈ S j | ( pre s) ∩ S ∗ =∅ } , (39)
such that S ∗ ≤ S , S j ≤ S and R j ⊆ S ;s e eF i g . 9 .
W e denote the associated behaviour B S ,w h e r e S ∗ ≤ S implies B S ⊆ B S ∗ . Thus, we
can formally interpret B S ∗ as a behavioural abstraction of B S . In p articular , the existence
of a solution to the control problem for S ∗ implies the existence of a solution for S =
( N 0 ,W , B S ) . W e now turn to the ordering S j ≤ S . Since we hav e identified S as a
successful experiment, Lemmata 14 and 16 grant the e xistence of a third e xperiment S such
that (a) S j ≤ S ≤ S ,( b ) S j − S ⊆ R j and (c) the control problem exhibits a solution for
the abstraction S = ( N 0 ,W , B S ) obtained from S . By clause (b) and the construction of
S we conclude S j = S . Hence, S j itself must be a successful experiment. This constitutes
a contradiction and concludes the proof.
Note that the abov e theorem refers to the e xistence of a successful e xperiment S ∗ .H o w -
eve r, S ∗ does not need to be known e xplicitly in order to run Algorithm 13. In other words,
Fig. 9 Experiment S with S ∗ ≤ S and R j ⊆ S . All three experiments are again sho wn as “barriers” in
the computational tree pre B . By construction, S consists of (a) sequences from S ∗ and (b) extensions of
sequences from S ∗ within S j . In particular , S is a refinement of S ∗ . By our choice of a sufficiently lar ge
j ∈ N 0 , all refinement candidates R j ⊆ S j must be on the right from S ∗ and therefore, by construction,
within S
556 Discrete Event Dynamic Systems (2020) 30:533–560
we still hav e the e xpected situation of semi-decidability , b ut we also ha v e the guarantee that
the algorithm terminates with success as long as a successful experiment e xists.
Example 6 (cont.) Recall that for the vehicle na vigation e xample the abstraction obtained
from S in Fig. 5 is too coarse for controller synthesis when the objecti ve is to e ventually visit
the west guard G y w . The worst case for this control objecti v e is an initial state at the very
east of the area A which requires a number of successiv e u nw and u sw control symbols,
where the exact number depends on the with-to-height ratio of the rectangular area. Using
a width of w = 30 units versus a height of h = 10 units, a thickness of o = 1 unit for the
guards, a nominal velocity of v = 10 and a disturbance of diameter d = 2, a numerical
simulation suggests that it can take up to 6 control inputs to reach the guard G y w . Thus,
for a successful experiment we e xpect the longest sequences to be of length 6. On the other
hand, any sequence that contains a y w symbol corresponds to a set of winning states in P ×
and, thus, needs no more refinement.
Running Algorithm 13 with refinement candidates ( 38 ) yields a successful experiment S
with 33262 sequences of length ranging from 1 to 6. W e can also use the a-priori knowledge
that it makes no sense to apply the same input symbol twice in ro w and encode this in the
specification automation P spec . Algorithm 13 then also encounters non-tri vial sets of failing
states and constructs a successful experiment S with only 9020 sequences. Since a winning
state must be a predecessor of a winning state, we can prioritise our refinement candidates
( 38 ) accordingly , i.e., we prefer to refine sequences in the experiment that correspond to a
state q = (z, χ ) ∈ Q with a successor state in Q win . This again reduces the number of
sequences constructed by Algorithm 13 to 164. All three figures compare fa vorable to the
strongest 5-complete approximation, which is constructed from the 59304 sequences found
in B | [ 0 , 5 ] .
7 Conclusion
W e hav e addressed a question arising in abstraction-based control of hybrid systems. More
specifically , the l -complete approximation scheme (Moor and Raisch 1999 ; Moor et al.
2002 ) provides a sequence of finite state abstractions P l , l = 1 , 2 ,..., for a (possibly infi-
nite state) system with discrete-v alued external signals. Here, the parameter l corresponds
to the uniform length of finite sequences S ⊆ W l + 1 sampled from the external beha viour
of the hybrid plant when constructing the finite state abstraction. In this scheme, a refine-
ment amounts to incrementing the parameter l . Although the required number of samples
| S | in most applications is by orders of magnitude less than the worst case | W | l + 1 , it is still
exponential in l . In this paper , we hav e further de veloped the approach proposed by Moor
et al. ( 2006 ) in that we drop the requirement of a uniform length and consider so called
experiments S ⊆ W ∗ , from which we construct the abstraction finite state machine P S .I n
particular , this approach allows us to increase the length of the samples locally as required
for the specific synthesis task at hand. In order to identify the sequences that are worth the
effort of a refinement, we first apply the synthesis procedure provided by Thistle and W on-
ham ( 1994a , b ) on the product P × := P S × P spec ,w h e r e P spec realises the specification
which the supervisor is meant to enforce. If the procedure succeeds and returns a supervi-
sor to control P S , this supervisor is kno wn to also enforce the specification when applied
to the hybrid plant. If, on the other hand, the control problem cannot be completely solv ed
for the abstraction P S , the procedure still establishes a conditional solution, i.e., a supervi-
sor that enforces the specification provided that the state trajectory passes through the set of
557 Discrete Event Dynamic Systems (2020) 30:533–560
so called winning states Q win . Intuiti vely , sequences in S that correspond to winning states
do not need to be increased in length. In contrast to the winning states, we also identify a
set of failing states Q fail , i.e., states such that any trajectory that passes through is known
to violate the specification under any supervisor . Again, sequences in S that correspond to
failing states do not need to be increased in length. T o this end, we refine the abstraction S
by increasing the sample length only for sequences that correspond neither to winning states
nor to failing states to obtain the locally refined experiment S with associated abstraction
P S . Iterating abstraction refinement and trial synthesis, we obtain a sequence of e xperi-
ments S j , j = 1 , 2 ,... . By our main technical result, this iteration will not miss out on
solutions to the control problem: if there exists an e xperiment such that the control prob-
lem can be solved based on the associated abstraction, then the iteration will also produce a
successful experiment S j for some j ∈ N .
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and institutional affiliations.
559 Discrete Event Dynamic Systems (2020) 30:533–560
Jung-Min Yang receiv ed his B.S., M.S., and Ph.D. de grees in elec-
trical engineering from K orea Adv anced Institute of Science and
T echnology (KAIST), Daejeon, K orea, in 1993, 1995, and 1999,
respecti vely . Since September 2013, he has been with the School
of Electronics Engineering, K yungpook National Univ ersity , K orea,
where he is currently a Professor . His research interests include
control of asynchronous sequential machines, systems biology , and
control of hybrid systems.
Thomas Moor receiv ed his PhD degree (Dr .-Ing.) in 1999 from the
Uni versity of the Federal Armed F orces Hamb urg. From 2000 to 2003
he was a research fello w with the Research School of Information
Sciences and Engineering at the Australian National Univ ersity . Since
2003, he holds a professorship at the Lehrstuhl f ¨ ur Regelungstech-
nik, Friedrich-Alexander -Uni versit ¨
at Erlangen-N ¨ urnberg, German y .
His research interests include the control of discrete-e vent systems
and hybrid systems, hierarchical and/or modular control systems,
control system abstraction and faulttolerant control. He serves on
the Editorial Board of the Journal of Discrete Event Dynamic Sys-
tems and Nonlinear Analysis: Hybrid Systems. He is maintainer and
principle de veloper of the discrete-e v ent systems software library
libF A UDES, with a particular focus on supervisory control in an
industrial application context.
J ¨
org Raisch studied Engineering Cybernetics at Stuttgart Univ ersity
and Control Systems at UMIST , Manchester . He receiv ed a PhD and
a Habilitation degree, both from Stuttgart Uni v ersity . He holds the
chair for Control Systems in the EECS Department at TU Berlin,
and he is also an external scientific member of the Max Planck Insti-
tute for Dynamics of Complex T echnical Systems. His main research
interests are hybrid and hierarchical control, distrib uted coopera-
ti ve control, and control of timed discrete e v ent systems in tropical
algebras, with applications in chemical, medical, and power sys-
tems engineering. He was on the editorial boards of the European
Journal of Control, the IEEE T ransactions on Control Systems T ech-
nology , and Automatica. He serves on the editorial boards of Discrete
Event Dynamic Systems and F oundations and T rends in Systems and
Control.
560 Discrete Event Dynamic Systems (2020) 30:533–560
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