# 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.
- Alessandro D’Innocenzo, Department of Electrical and Information Engineering, Center of Excellence DEWS, University of L’Aquila, Poggio di Roio, 67040 L’Aquila, Italy, also known as ∗Correspondence to.
- 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. Various notions of observability have been introduced in the literature for discrete event systems [4–8] and hybrid systems [2, 9–13].
- The main contribution with respect to the results of [9] consists in the analysis of the computational complexity for the observability verification.
- In Section 4, the authors address the observability verification problem for hidden Markov models.

### 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

- As first step of the proof, the authors remark that condition (4) can be rewritten as ∀p∈PQc ∩PQ\QcP[ ∈LQc |P( )= p]∈[0,1− m]∪[ M ,1] (5) In fact, for any given m, M , for all p∈P (PQc ∩PQ\Qc) and for all ∈ P−1(p), then 1. either ∈LQc , and thus P[ ∈LQc |P( )=p]=1.
- The initial discrete state is q̂0, and the initial condition of is given by the initial probability distribution (0)=.
- If for all those executions c(|p|) does not reach the set (1− m, M ), then for all executions with more than one cycle c(|p|) does not reach the set (1− m, M ) as well, and condition (5) is satisfied.
- The algorithm consists of four iterations.
- 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.
- Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1564–1580 DOI: 10.1002/rnc.
- 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.
- Thus, the authors can state that ∀p∈PQc ∩PQ\Qc, P[ ∈Lq7 |P( )= p]∈{0}∪[0.9,1] hence the critical set {q7} is observable for M with reliability ( maxm =1, maxM =0.9).

### 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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##### References

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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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