Discrete state observability of hybrid systems
Summary (2 min read)
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.
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].
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Citations
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Cites background or methods from "Discrete state observability of hyb..."
...(Rudie, Lafortune, & Lin, 2003) and the notion of critical observability in Di Benedetto et al. (2008) and in the present paper....
[...]
...(Rudie, Lafortune, & Lin, 2003) and the notion of critical observability in Di Benedetto et al. (2008) and in the present paper. Many papers mentioned above consider deterministic FSMs with partially observable transitions while in our paper we consider nondeterministic FSMs with observable transitions. Connectionswith the concept of states disambiguation in e.g. (Rudie et al., 2003) which is in (S)–(D)– (A) as our paper, follow. The work (Rudie et al., 2003) uses states disambiguation as a mean to address communication problems without the need to consider communication and coobservability issues together. States disambiguation deals with studying conditions under which any pair of states of a DES can be distinguished on the basis of the traces associated to state runs of the DES and ending in the two states. It is readily seen that an FSM where all states are disambiguated is also critically observable, while the converse implication is not true in general. Apart from technical differences between the concepts of states disambiguation and critical observability, the present paper approaches the efficient design of decentralized critical observers by extending techniques based on formal methods, a new approach that was not explored before in the literature with the only exception of Zad et al. (2003). However, while Zad et al....
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...setting; within (L)–(D)–(CD) the notion of co-observability (Rudie &Wonham, 1989) extending the one of language observability to a decentralized setting, and thework (Wang, Girard, Lafortune, & Lin, 2011) proposing an extension of the notion of co-observability to dynamic observations in controlledDESs and results for translating co-observability problems into co-diagnosability problems;within (S)–(C)–(A) the notions of diagnosability with a state space approach in Zad et al. (2003), detectability e....
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...…algorithms for the computation of indistinguishable states in e.g. (Wang, Lafortune, & Lin, 2007); within (S)–(D)–(A) the concept of states disambiguation in e.g. (Rudie, Lafortune, & Lin, 2003) and the notion of critical observability in Di Benedetto et al. (2008) and in the present paper....
[...]
...(Rudie, Lafortune, & Lin, 2003) and the notion of critical observability in Di Benedetto et al. (2008) and in the present paper. Many papers mentioned above consider deterministic FSMs with partially observable transitions while in our paper we consider nondeterministic FSMs with observable transitions. Connectionswith the concept of states disambiguation in e.g. (Rudie et al., 2003) which is in (S)–(D)– (A) as our paper, follow. The work (Rudie et al., 2003) uses states disambiguation as a mean to address communication problems without the need to consider communication and coobservability issues together. States disambiguation deals with studying conditions under which any pair of states of a DES can be distinguished on the basis of the traces associated to state runs of the DES and ending in the two states. It is readily seen that an FSM where all states are disambiguated is also critically observable, while the converse implication is not true in general. Apart from technical differences between the concepts of states disambiguation and critical observability, the present paper approaches the efficient design of decentralized critical observers by extending techniques based on formal methods, a new approach that was not explored before in the literature with the only exception of Zad et al. (2003). However, while Zad et al. (2003) use bisimulation-based reduction for checking diagnosability of single FSM, our approach proposes bisimulation-based reduction for analyzing critical observability of networks of FSMs....
[...]
18 citations
Cites background from "Discrete state observability of hyb..."
...…unique and intriguing peculiarities, not supported by their integer-order counterpart, which raise exciting challenges and opportunities related to the development of discontinuous control and estimation methodologies involving fractional order dynamics affected by uncertainties and disturbances....
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14 citations
10 citations
Cites background from "Discrete state observability of hyb..."
...…(see, for instance, Vidal et al., 2003; Babaali and Pappas, 2005; Alessandri et al., 2005; Baglietto et al., 2007; Alessandri et al., 2007; Di Benedetto et al., 2009; Baglietto et al., 2009, and the references therein), such results have yet to be fully exploited in the context of adaptive…...
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References
21,819 citations
13,779 citations
"Discrete state observability of hyb..." refers background in this paper
...Let cl (Q∗) be the -closure [14] of a set of states Q∗ ⊆Q, namely the set of states that can be reached from Q∗ via a path of edges whose outputs are unobservable....
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...It is possible to define the languages of observations for each discrete state by means of regular expression [14]: Pq1 = { }, Pq2 =a(aa+bb)∗ Pq3 = a(bb)∗, Pq4 =a(aa+bb)∗b Pq5 = a(aa+bb)∗b, Pq6 =a(bb)∗b Pq7 = a(bb)∗b...
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...In the following we address the observability verification problem in the setting of regular languages [14]....
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...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....
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6,804 citations
5,995 citations
"Discrete state observability of hyb..." refers methods in this paper
...Since we have defined on a hidden Markov model M a probability measure in the target and output of a discrete transition, one can use the discrete observations to compute (using the Viterbi algorithm [22, 23]) the conditional probability distribution of the current discrete state given the measured observation....
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4,330 citations
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Frequently Asked Questions (4)
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