Discrete state observability of hybrid systems

TL;DR: In this paper, the authors propose a definition of observability, motivated by safety critical applications, given with respect to a subset of critical discrete states that model unsafe or unallowed behaviors.

Abstract: SUMMARY We propose a novel definition of observability, motivated by safety critical applications, given with respect to a subset of critical discrete states that model unsafe or unallowed behaviors. For the class of discrete event systems, we address the problem in the setting of formal (regular) languages and propose a novel observability verification algorithm. For the class of switching systems, we characterize the minimal set of extra output information to be provided by the continuous signals in order to satisfy observability conditions, and propose a milder observability notion that allows a bounded delay in state observation. For the class of hidden Markov models, we analyze decidability and complexity of the verification problem. Copyright q 2009 John Wiley & Sons, Ltd.

Summary

1. INTRODUCTION

• Estimation methods and observer design techniques are essential in this regard, for the design of a control strategy for error propagation avoidance and/or error recovery.
• The main contribution with respect to the results of [9] consists in the analysis of the computational complexity for the observability verification.

2. DISCRETE EVENT SYSTEMS

• The authors analyze the verification problem using the discrete output of the system and propose a novel verification procedure that can be executed in polynomial time.
• E→ is the output function, that associates with each edge a discrete output symbol, also known as .
• The associated observation P( ) is obtained erasing all unobservable outputs from the output string.
• Since two distinct executions can generate the same observation, the intersection set PQ1 ∩PQ2 is not necessarily empty for Q1∩Q2=∅.
• The definitions of nondeterministic finite automaton (NFA), DFA, regular language.

Proof

• The initial discrete state is q̂0, and the initial condition of is given by the initial probability distribution (0)=.
• Since any i (k+nc) is upper bounded by one and lower bounded by zero, then limn→∞ (k+nc) is a vector of zeros and ones.

3. SWITCHING SYSTEMS

• Called switching systems, where a continuous dynamical system is associated with each discrete state.the authors.
• When the information given by the discrete output are not sufficient to build an observer, the authors provide an algorithm to compute the minimum set of extra information they need in order to make the system observable.
• {Eq}q∈Q associates with each discrete state q∈Q the continuous time-invariant dynamics Eq : ẋ= fq(x,u) (3) with output y=gq(x).
• The authors consider nonblocking switching systems, i.e. systems such that all hybrid executions are defined for all time instants [19].
• This optimal solution can be computed in exponential time (with respect to the cardinality |Q| of the discrete state space) using the following algorithm.

4. HIDDEN MARKOV MODELS

• The authors extend their results to the class of hidden Markov models [21].
• Notice that the output function P :L→P defined in Section 2 is not invertible.
• With the assumption that their observer generates as output the most likely current discrete state according to the Viterbi algorithm estimate, the authors formalize an observability definition that requires a bound in the probability of estimation error.
• The authors can formalize the above properties as follows.
• The definition above characterizes a structural property of the hidden Markov model M. When m= M=1 the observer provides correct estimate with probability 1.

5. EXAMPLE

• Consider the discrete event systemD described in Figure 2.
• The authors use the theoretical results discussed above to analyze the discrete state observability.
• By detecting if the system visited q2 or q3, the authors anticipate the uncertainty between q4,q6,q7 and they use only two extra outputs.

6. CONCLUSIONS

• For discrete event systems, the authors exploited properties of regular languages to propose an algorithm for checking observability in polynomial time.
• The authors extended their result to switching systems: they proposed an algorithm to find the minimum set of extra output information, retrieved from the continuous observations, to satisfy the observability condition, and discussed a notion of observability with bounded delay.
• The authors then extended their results to hidden Markov models: they proposed an observability definition that requires a bound in the probability of observation reliability, and they showed that the verification problem is decidable and belongs to the complexity class EXPTIME.
• The framework proposed in this paper can be used for the simulation of real safety critical procedures, and verification of the detection of dangerous operations, as shown in [2, 3].

Figures

Figure 1. Discrete states of S are split by Algorithm 4, to consider the generation time of h(q).

Figure 2. Discrete layers of D and Oq7 .

Full text

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int. J. Robust Nonlinear Control 2009; 19:1564–1580
Published online 27 March 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1432
Discrete state observability of hybrid systems
Maria D. Di Benedetto
‡
, Stefano Di Gennaro
§
and Alessandro D’Innocenzo
∗, †
Department of Electrical and Information Engineering, Center of Excellence DEWS, University of L’Aquila,
Poggio di Roio, 67040 L’Aquila, Italy
SUMMARY
We propose a novel definition of observability, motivated by safety critical applications, given with respect
to a subset of critical discrete states that model unsafe or unallowed behaviors. For the class of discrete
event systems, we address the problem in the setting of formal (regular) languages and propose a novel
observability verification algorithm. For the class of switching systems, we characterize the minimal set
of extra output information to be provided by the continuous signals in order to satisfy observability
conditions, and propose a milder observability notion that allows a bounded delay in state observation. For
the class of hidden Markov models, we analyze decidability and complexity of the verification problem.
Copyright q 2009 John Wiley & Sons, Ltd.
Received 21 December 2007; Revised 13 November 2008; Accepted 9 December 2008
KEY WORDS
: observability; discrete event system; switching system; hidden Markov model; automatic
verification; computational complexity
1. INTRODUCTION
In many safety critical applications, e.g. in air traffic management procedures [1–3], it is often
required to detect if the current behavior of the system is associated with a dangerous or unallowed
operation. Estimation methods and observer design techniques are essential in this regard, for the
design of a control strategy for error propagation avoidance and/or error recovery. Discrete event
and hybrid systems are a powerful tool for the analysis and control of multi-agent systems, since
it is convenient to model undesired or dangerous behaviors by means of discrete states that we call
critical states. Then, the possibility of detecting dangerous situations depends on the observability
properties of the system with respect to the critical states.
∗
Correspondence to: Alessandro D’Innocenzo, Department of Electrical and Information Engineering, Center of
Excellence DEWS, University of L’Aquila, Poggio di Roio, 67040 L’Aquila, Italy.
†
E-mail: alessandro.dinnocenzo@ing.univaq.it
‡
E-mail: mariadomenica.dibenedetto@univaq.it
§
E-mail: stefano.digennaro@univaq.it
Contract/grant sponsor: European Commission under Project IST NoE HyCON; contract/grant number: 511368
Contract/grant sponsor: European Commission under Project iFly; contract/grant number: TREN/07/FP6AE/
S07.71574/037180
Copyright q 2009 John Wiley & Sons, Ltd.

DISCRETE STATE OBSERVABILITY OF HYBRID SYSTEMS 1565
Various notions of observability have been introduced in the literature for discrete event systems
[4–8] and hybrid systems [2, 9–13]. We focus in this paper on the observability of the discrete
state, and propose a definition of observability with respect to a subset of discrete critical states.
We first formulate our problem in the setting of discrete event systems, then we extend our results
to switching systems and hidden Markov models.
We first consider discrete event systems and propose our definition of discrete state observability.
Observability conditions can be checked on the structure of the discrete state observer [2, 4, 5, 9],
which can be constructed in exponential time with respect to the cardinality of the discrete state
space: this implies that the complexity of the verification algorithm is exponential as well. We
address the observability verification problem in the setting of formal (regular) languages [14],and
propose a new verification algorithm, executable in polynomial time, which exploits properties of
operations on regular languages. The main contribution with respect to the results of [9] consists in
the analysis o f the computational complexity for the observability verification. We prove that our
observability conditions can be checked efficiently in polynomial time, instead of exponential time.
Moreover, our algorithms provide (i) the minimum number of steps K after which the critical states
can be observed and (ii) the minimum set of the extra signals needed to satisfy the observability
conditions.
We then consider a subclass of hybrid systems, called switching systems, where a continuous
dynamical system is associated with each discrete state. When the information given by the discrete
output are not sufficient to build an observer, the continuous dynamics can be exploited as proposed
in [9] to generate some discrete signals that provide additional information useful to discriminate
the discrete states. This can be done by using fault detection techniques [15, 16], as for example in
[9, 17] where a bank of Luenberger observers is used to identify the discrete state. However, the
choice of the extra signals needed to satisfy the observability conditions is not unique. We propose
an algorithm to compute the minimum extra information needed to achieve observability. Since
the generation of these extra output symbols requires a nonzero generation time, a milder notion
of observability, which allows a bounded delay in the observation, and a verification algorithm are
proposed.
Finally, we consider hidden Markov models. We propose an observability definition similar to
that given in [18] for the continuous states of jump linear systems, which allows a bound in the
probability of estimation uncertainty. As one of the main results of the paper, we show that the
addressed observability verification problem is decidable, and we characterize its computational
complexity.
The organization of the paper is as follows. In Section 2, we analyze discrete state observability
for discrete event systems. In Section 3, we extend our results to switching systems. In Section
4, we address the observability verification problem for hidden Markov models. In Section 5, an
illustrative example is presented. Finally, in Section 6, we offer some concluding remarks.
2. DISCRETE EVENT SYSTEMS
In this section we propose a formal definition of observability of a subset of discrete states for
discrete event systems. We analyze the verification problem using the discrete output of the system
and propose a novel verification procedure that can be executed in polynomial time.
Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1564–1580
DOI: 10.1002/rnc

1566 M. D. DI BENEDETTO, S. DI GENNARO AND A. D’INNOCENZO
Definition 1 (Discrete event system)
A discrete event system is a tuple D =(Q, Q
0
, E, , ) such that
• Q is a finite set of N discrete states.
• Q
0
⊆ Q is the set of initial conditions.
• E ⊆ Q × Q is a collection of edges; each edge e ∈ E is an ordered pair of discrete states, the
first component of which is the source and is denoted by s(e), while the second is the target
and is denoted by t (e).
•  is the finite set of discrete output symbols. It includes the empty string  that corresponds
to unobservable output.
• : E →  is the output function, that associates with each edge a discrete output symbol.
The executions of discrete states of D are the sequences ={q
k
}
||
k=0
such that q
0
∈ Q
0
,(q
k
,q
k+1
) ∈
E, k = 0,1,...,||−1, with ||0 the length of the execution.
From this definition, it is not possible that a system has two edges e
1
, e
2
with the same source
s(e
1
) = s(e
2
) and target t (e
1
) = t(e
2
). There is no loss of generality since it is always possible to
construct an equivalent system that complies our model by splitting the source or the target state,
where ‘splitting’ a state q
i
means creating two states q

i
, q

i
, keeping the incoming and outgoing
edges.
Definition 2 (Formal language of executions)
The formal language of the executions of discrete states of D is given by
L{={q
k
}
||
k=0
:q
0
∈ Q
0
,(q
k
,q
k+1
) ∈ E,
k =0, 1,...,||−1}
Given a subset of discrete states Q
∗
⊆ Q,wedefine
L
Q
∗
{∈ L :||<∞,q
||
∈ Q
∗
}
the language of executions with finite cardinality, such that the last visited discrete state belongs
to Q
∗
.Forq ∈ Q, we use for simplicity the notation L
q
instead of L
{q}
. Given an execution
={q
k
}
||
k=0
, the associated output string is {((q
k
,q
k+1
))}
||−1
k=0
. The associated observation P()
is obtained erasing all unobservable outputs from the output string.
Definition 3 (Formal language of observations)
The formal language of the observations of D is given by
P{P() : ∈L}
Given a subset of discrete states Q
∗
⊆ Q,wedefineP
Q
∗
the language of the observations
generated by executions whose last visited state belongs to Q
∗
P
Q
∗
{P() : ∈L
Q
∗
}
Since two distinct executions can generate the same observation, the intersection set P
Q
1
∩P
Q
2
is not necessarily empty for Q
1
∩ Q
2
=∅. This is a crucial issue for observability of the discrete
state, as we will show in the following.
Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1564–1580
DOI: 10.1002/rnc

DISCRETE STATE OBSERVABILITY OF HYBRID SYSTEMS 1567
Let Q
c
⊂ Q be the set of critical states of D, i.e. the set of discrete states associated with unsafe
or unallowed behaviors of D. We say that Q
c
is observable for D if it is possible to construct a
system that, on the basis of the observations, is able to detect whether the current discrete state of
D belongs to Q
c
or not. A necessary and sufficient condition can be given in terms of observations.
Definition 4
Given a discrete event system D,thesetQ
c
is observable if and only if
P
Q
c
∩P
Q\Q
c
=∅ (1)
Intuitively, each observation can be generated either only by executions whose last visited state
belongs to Q
c
, o r only by executions whose last visited state does not belong to Q
c
.
In the following we address the observability verification problem in the setting of regular
languages [14]. Given a discrete event system D =(Q, Q
0
, E, , ), one of the algorithms
proposed in [2, 4, 5, 9] can be used to construct the discrete state observer O
Q
c
=(
ˆ
Q ⊆ 2
Q
, ˆq
0
=
{Q
0
},
ˆ
Q
c
,
ˆ
E,
ˆ
=\{}, ˆ). O
Q
c
is a deterministic finite automaton (DFA), where each discrete
state ˆq ∈
ˆ
Q is a subset of Q and the final set
ˆ
Q
c
{ˆq ∈
ˆ
Q: ˆq ∩ Q
c
=∅∧ˆq ∩ Q\Q
c
=∅}
is induced by the critical set Q
c
. The definitions of nondeterministic finite automaton (NFA), DFA,
regular language, and an algorithm to construct the discrete state observer O
Q
c
are recalled in the
Appendix.
The DFA O
Q
c
accepts the language P
Q
c
∩P
Q\Q
c
and it is therefore possible to verify observ-
ability conditions directly on O
Q
c
checking if the accepted language is empty, i.e. if
ˆ
Q
c
=∅. Hence,
the observability verification can be done in time exponential in N =|Q| by constructing the
observer. However, there exists an NFA having a discrete state space cardinality polynomial in N ,
which accepts the same language as O
Q
c
. This implies that it is possible to construct an observer
that consists of a set of concurrent DFAs, and whose output is given by a logical operation on
the outputs of the DFAs. We exploit this property of regular languages to define an observability
verification procedure that can be executed in time polynomial in N, on a discrete event system
D. The main idea of the algorithm is to use operations on regular languages to check condition
(1) without constructing the observer.
Algorithm 1
Given a discrete event system D and a critical set Q
c
1. Construct the NFA N
Q
c
that accepts P
Q
c
.
2. Construct the NFA N
Q\Q
c
that accepts P
Q\Q
c
.
3. Construct the NFA N
∩
that accepts P
Q
c
∩P
Q\Q
c
.
4. Q
c
is observable for D if and only if the language accepted by N
∩
is empty.
Theorem 1
Algorithm 1 can be executed in O(N
4
).
Proof
The first and second steps require N
2
iterations each, since P
Q
c
, P
Q\Q
c
are finite unions of
the regular languages |Q
c
|, |Q\Q
c
|, respectively. The third step requires N
4
iterations, since the
intersection of the two regular languages P
Q
c
, P
Q\Q
c
is accepted by a NFA with state space
Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1564–1580
DOI: 10.1002/rnc

1568 M. D. DI BENEDETTO, S. DI GENNARO AND A. D’INNOCENZO
cardinality N
2
×N
2
. The last step can be executed during step 3. Hence, the overall complexity is
given by 2N
2
+N
4
∼O(N
4
). 
The previous result can be extended to the case of state observability after a transient of K
transitions.
Definition 5
Given a discrete event system D,thesetQ
c
is observable in K -steps if and only if
∀: P() ∈P
Q
c
∩P
Q\Q
c
, ||<K (2)
In order to verify condition (2), Algorithm 1 can be used with line 4 replaced by:
4

. Q
c
is observable in K -steps for D if and only if the final states of N
∩
can only be reached
by finite paths that contain less than K transitions.
The minimum value K
min
such that Q
c
is observable in K
min
-steps can be computed in polynomial
time by searching for the maximum length of all paths that reach a final state of the system N
∩
.
3. SWITCHING SYSTEMS
In this section we extend our results to a subclass of hybrid systems, called switching systems,
where a continuous dynamical system is associated with each discrete state. When the information
given by the discrete output are not sufficient to build an observer, we provide an algorithm to
compute the minimum set of extra information we need in order to make the system observable.
These extra information are determined from the continuous input and output signals and cannot be
generated instantaneously. We propose an algorithm to construct an abstract model that formalizes
the generation of extra information by means of discrete output symbols. We then introduce a
milder observability definition that allows bounded delay in the observation of the current discrete
state and give a procedure to verify this property on the abstract system.
Definition 6 (Switching system)
A switching system is a tuple S = (D, X, X
0
,U, Y, E) such that:
• D = (Q, Q
0
, E, , ) is a discrete event system as in Definition 1.
• X ⊆ R
n
is the continuous state space.
• X
0
⊆ X is the set of initial continuous conditions.
• U ⊆R
m
, Y ⊆ R
p
are the sets of continuous control input and observable output.
•{E
q
}
q∈Q
associates with each discrete state q ∈ Q the continuous time-invariant dynamics
E
q
:˙x = f
q
(x, u) (3)
with output y = g
q
(x).
It is worth noting that a solution of Equation (3) exists and is unique under the assumption that
f
q
is continuous with respect to time and Lipschitz continuous with respect to x, and the control
input is piecewise continuous from the right and with left limit.
This class of switching systems is nondeterministic, in general. The continuous state evolves
following deterministic dynamics, and the discrete state performs nondeterministic transitions.
We recall in the Appendix, the definitions of a hybrid time basis {I
k
}
0k||
with cardinality
Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1564–1580
DOI: 10.1002/rnc

Frequently asked questions

Q1. What are the contributions in "Discrete state observability of hybrid systems" ?

The authors propose a novel definition of observability, motivated by safety critical applications, given with respect to a subset of critical discrete states that model unsafe or unallowed behaviors. For the class of discrete event systems, the authors address the problem in the setting of formal ( regular ) languages and propose a novel observability verification algorithm. For the class of switching systems, the authors characterize the minimal set of extra output information to be provided by the continuous signals in order to satisfy observability conditions, and propose a milder observability notion that allows a bounded delay in state observation. For the class of hidden Markov models, the authors analyze decidability and complexity of the verification problem. 

Q2. What are the future works mentioned in the paper "Discrete state observability of hybrid systems" ?

Future work will focus on the extension of their results to continuous time hidden Markov models. 

Q3. What is the definition of observability for discrete event systems?

For the class of switching systems, the authors characterize the minimal set of extra output information to be provided by the continuous signals in order to satisfy observability conditions, and propose a milder observability notion that allows a bounded delay in state observation. 

Q4. What are the key words in the definition of observability?

WORDS: observability; discrete event system; switching system; hidden Markov model; automatic verification; computational complexity