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REVIEW 3 major objections 3 minor 86 references

Disturbance Decoupling and Instantaneous Fault Detection in Boolean Control Networks

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In Boolean control networks, a single state-feedback controller can decouple disturbances and detect faults in the next step exactly when every fault variable is reflected in the output and every disturbance variable is redundant.

desk verdict The paper's headline necessary-and-sufficient claim is false: Lemma 14's reflectivity/rank equivalence fails in both directions, and Theorems 15 and 17 inherit the error. read the letter →

arxiv 1908.05235 v1 pith:4FKDRWTV submitted 2019-08-08 eess.SY cs.SYmath.LOmath.OC

S Sutavani , K Sonam , S Wagh , N Singh This is my paper · ORCID
classification eess.SYcs.SYmath.LOmath.OC MSC 93B5293C5594C10
keywords Booleancontrolnetworksdisturbancedecouplinginstantaneousfaultdetectionsemi-tensorproductstatefeedbackreflectivityredundancyreachability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies Boolean control networks, models in which states, inputs, and outputs are binary and update in discrete time. It argues that the usual way of defining disturbance decoupling is too strict, because it forces the system onto a trajectory that ignores disturbances rather than letting the system evolve naturally while feedback cancels their effect. The central claim is that a single state-feedback controller can decouple disturbances and detect faults in the very next time step exactly when the output map has a block structure: every fault variable must be reflective (flipping it changes every output component for every setting of the other variables) and every disturbance variable must be redundant (flipping it leaves the output unchanged). This matters because full observability of Boolean control networks is NP-hard, and the paper's rank-based conditions give a structural test that avoids expensive state reconstruction.

What carries the argument

The machinery is the semi-tensor product representation, which rewrites Boolean dynamics as ordinary matrix equations, together with two rank conditions on the output map $M_O$. Lemma 14 identifies reflective variables with full column rank of the corresponding sub-block of the structure matrix; Lemma 16 adds redundant disturbance variables as identical sub-blocks. Algorithm A.7 turns these conditions into a state-feedback matrix $M_x$ by selecting, for each state index, an input block whose sub-block has the required identical-full-rank structure, and the number of valid controllers is the product $\prod_j |C_j|$. The reachability-graph constructions in Section V extend the same logic to decoupling in mapping, in finitely many iterations, and invariant-output decoupling.

What would settle it

Take two fault variables $a,b$ and two outputs both equal to $a \oplus b$. Each single variable is reflective by definition (28), yet the map $(a,b)\mapsto(a\oplus b,a\oplus b)$ is not injective and the corresponding $2\times 4$ block of $M_O$ has rank 2, not 4. Checking such a system against Theorem 17 would settle whether the stated full-column-rank condition is actually necessary.

Watch

Extended reading notes

Core claim

The paper's main result, Theorem 17, states that for a Boolean control network $x^+ = L u x \xi_d \xi_f$, $y = H x$, with state feedback $u = M_x x$, instantaneous fault detection together with disturbance decoupling is achievable exactly when the composed output map $M_O = H L M_x \psi_n$ satisfies Lemma (Fault + Disturbance): split $M_O$ into $2^n$ blocks of size $2^p \times 2^{d+f}$; each block must split into $2^d$ identical sub-blocks, and each sub-block must have full column rank. In words, disturbances must be invisible to the output and each fault must change the output in a distinguishable way. The paper also widens the notion of disturbance decoupling itself, introducing decoupling in mapping, in finitely many iterations, and invariant-output decoupling, so that systems that fail the classical rank test can still be decoupled in a weaker, useful sense.

Load-bearing premise

The proof leans on the claim that a variable being reflective—flipping it changes every output component for every setting of the other variables—is exactly the same as the corresponding block of the output map having full column rank; for two or more variables together that equivalence can fail, and the main theorem's necessity argument depends on it.

Editorial extensions

If this is right

  • Systems that fail the classical row-wise rank test for disturbance decoupling may still be decouplable, because the paper's condition is imposed on the composed output map $M_O$ rather than on every row of the transition matrix.
  • A single state-feedback controller $M_x$ can achieve both disturbance decoupling and instantaneous fault detection, so no observer or state reconstruction is needed during the fault-detection phase.
  • The number of controllers that achieve the combined goal is finite and computable as $\prod_j |C_j|$, allowing the designer to enumerate all valid state-feedback laws.
  • The same block condition makes the system fault-output invertible, so the occurrence of a fault is uniquely identifiable from the next output.
  • For $s$ disturbances and $t$ faults, the criterion becomes $2^s$ identical divisions of each sub-block with an invertible or pseudo-invertible fault-to-output map.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper treats reflectivity one variable at a time; a natural strengthening would require every simultaneous fault combination to have its own output signature, which would make the condition more robust when two faults occur in the same step.
  • If the full-column-rank characterization is repaired for multi-variable blocks, the same block-decomposition idea could be applied to asynchronous or probabilistic Boolean networks, where 'instantaneous' would mean the output distribution changes immediately after a fault.
  • The decoupling-in-iteration notion is a testable practical extension: one could count how many published Boolean gene regulatory networks become decouplable under Algorithm A.8 even though they fail the classical conditions.
  • An alternative to the rank test is a direct injectivity check on the columns of each sub-block, which is a weaker premise and would make the necessary-and-sufficient claim easier to verify by hand on small networks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper addresses disturbance decoupling (DD) and instantaneous fault detection (IFD) in Boolean control networks. It proposes relaxed notions of DD (in mapping, in iterations, invariant in output), algorithms for controller synthesis, and claims necessary and sufficient conditions for IFD and for combined DD+IFD. The central theoretical results are Lemma 14, which equates "reflective" fault variables with a full-column-rank block condition on the output structure matrix, and Lemma 16, which extends this to simultaneous redundant disturbance variables and reflective fault variables. Theorems 15 and 17 then state that separate reflectivity of each fault variable, together with redundancy of each disturbance variable, is necessary and sufficient for IFD and for combined DD+IFD with a single state-feedback controller.

Significance. If the central conditions were correct, the paper would offer a practically useful, checkable characterization for fault detection and disturbance rejection in BCNs without requiring full observability, and it would provide explicit controller construction algorithms with finite termination. The paper contains several worked examples and algorithmic procedures (A.2, A.4, A.6, A.7, A.8). However, the load-bearing Lemma 14 is false, and the main necessary-and-sufficient claims therefore are not established. The paper also introduces interesting conceptual distinctions among reachability notions and between DD in mapping versus in iterations, though these are not the main technical contribution.

major comments (3)
  1. [Section VII-C, Lemma 14, Eq. (28)] The above counterexample respects the lemma's premise 0 < r ≤ n, so the error is not an artifact of an excluded boundary case.
  2. [Section VII-D, Theorem 15] The necessity direction of Theorem 15 also relies on the false Lemma 14, so the error is load-bearing for both directions of the claimed characterization.
  3. [Section VIII-B, Lemma 16, and Section VIII-C, Theorem 17] The redundancy part of Lemma 16 (all disturbance sub-blocks identical) is not affected by this counterexample, but the reflective part is essential to the combined theorem, so the combined result is not supported.
minor comments (3)
  1. [Section V-B, Algorithm A.2 and Theorem 11] The same observation applies to Theorems 12 and 13, whose proofs say the statements follow from the reachability definitions; these results are presented as characterizations but are essentially definitions of the constructed sets.
  2. [Throughout] These do not affect the mathematical content but make the manuscript difficult to read.
  3. [Section V-D, Eq. (22) and surrounding text] A reader cannot easily verify the claimed equivalence in this section without reconstructing the argument from scratch.

Circularity Check

1 steps flagged · score 6.0 of 10

Central DD+IFD theorem restates disturbance redundancy as its own sufficient condition; no self-citation loop, but the headline necessary-and-sufficient claim is partially definitional.

  1. self definitional [Section VIII-C, Theorem 17 (with Lemma 16 and Algorithm A.7)]
    "Section V-D: 'for disturbance to not affect the output→ all the elements in any block should be identical.' Theorem 17: 'In the BCN described by above mentioned dynamics, instantaneous fault detection and disturbance decoupling can be achieved if every fault variable is reflected in output separately and every disturbance variable is redundant, i.e. MO satisfies lemma (Fault + Disturbance) for every element of ξf and ξd.'"

    Lemma 16 defines the 'redundant variable' condition by requiring the 2^s sub-blocks M_i^j to be identical (rank(M_i^j)=2^{n-r-s} and M_i^j=M_i^k). Identical sub-blocks mean the output map is independent of those variables, which is exactly the paper's own criterion for disturbance not to affect the output ('all the elements in any block should be identical'). Algorithm A.7 likewise uses 'all divisions are identical' as its DD test and 'full rank/invertible Fm' as its IFD test. Theorem 17's necessary-and-sufficient condition therefore does not derive the DD half from independent structure: it restates the definition of disturbance redundancy as the desired decoupling property, and the sufficiency proof invokes an algorithm whose pass condition is the theorem's own conclusion.

full rationale

The paper has no fitted constants, no parameter estimation, and no load-bearing self-citation chain; the only self-citation ([2]) is contextual. The central circularity is in the headline DD+IFD claim. In the paper's own terms, 'disturbance not to affect the output' is exactly the condition that all elements in each relevant block of the output map are identical, and Lemma 16 defines a 'redundant disturbance variable' by precisely that block-identity condition. Theorem 17 then announces that DD+IFD is achieved iff every fault is reflective and every disturbance is redundant; the disturbance half is thus the target property restated rather than a derived necessary-and-sufficient condition. Algorithm A.7's success criterion is the same pair of conditions, so the sufficiency argument is an algorithmic check of the theorem's own hypothesis. The IFD half has independent mathematical content through Lemma 14, which equates individual reflectivity with full-column-rank blocks; that lemma is false (e.g., two fault variables with outputs y1=y2=ξ1⊕ξ2 are individually reflective but the pair map is not injective), but that is a correctness issue rather than a circularity. Propositions 8-10 are explicitly trivial graph encodings of their reachability definitions and are not the paper's central claim. Overall: partial definitional circularity in the central DD+IFD theorem, with no self-citation or fitting circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical fitting and no invented physical entities. The load-bearing assumptions are the semi-tensor-product framework, the multiplicative Boolean model of disturbance and fault, static feedback, and the one-step definition of instantaneous detection.

assumptions (4)
  • standard math Semi-tensor product representation represents every Boolean function as a logical matrix multiplication.
    Invoked in Corollaries 5-6 and throughout; sourced to Cheng and Qi [3].
  • domain assumption Disturbances and faults enter the dynamics as appended Boolean vector factors: x(t+1)=L u x ξ_d ξ_f.
    Section V Eq. (19) and Section VIII Eqs. (31)-(32); this is a modeling choice, not derived.
  • domain assumption Control is restricted to static state or output feedback, u=M_x ψ_n x or u=M_y H x.
    Section V and Section VIII; the whole DD+FD analysis assumes a fixed feedback law.
  • domain assumption Instantaneous fault detection means the fault is visible in the very next output y(t+1), with the state known under state feedback.
    Section VII definition; it defines what the theorem is about rather than deriving it.

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Cite this review

Pith. "Pith review of Disturbance Decoupling and Instantaneous Fault Detection in Boolean Control Networks." pith.science (2026). https://pith.science/paper/4FKDRWTV

@misc{pith2026190805235,
  author       = {Pith},
  title        = {Pith review of: Disturbance Decoupling and Instantaneous Fault Detection in Boolean Control Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4FKDRWTV}},
  note         = {Machine review of arXiv:1908.05235}
}
read the original abstract

The literature available on disturbance decoupling (DD) of Boolean control network (BCN) is built on a restrictive notion of what constitutes as disturbance decoupling. The results available on necessary and sufficient conditions are of limited applicability because of their stringent requirements. This work tries to expand the notion of DD in BCN to incorporate a larger number of systems deemed unsuitable for DD. The methods available are further restrictive in the sense that system is forced to follow trajectory unaffected by the disturbances rather than decoupling disturbances while the system follows its natural course. Some sufficient conditions are provided under which the problem can be addressed. This work tries to establish the notion of disturbance decoupling via feedback control,analogous to the classical control theory. This approach though, is not limited to DD problems and can be extended to the general control problems of BCNs. Determination of observability, which is sufficient for the fault detection, is proven to be NP-hard for Boolean Control Network. Algorithms based on reconstructability, a necessary condition, of BCN turn out to be of exponential complexity in general.In such cases it makes sense to search for the availability of some special structure in BCN that could be utilized for fault detection with minimal computational efforts. An attempt is made to address this problem by introducing instantaneous fault detection (IFD) and providing necessary and sufficient conditions for the same. Later necessary and sufficient conditions are proposed for solving the problem of instantaneous fault detection along with disturbance decoupling using a single controller.

Figures

Figures reproduced from arXiv: 1908.05235 by the authors.

Figure 1
Figure 1. Equivalence in a Nutshell HLD HL˜, where LD : LEdξEdu for all inputs. i.e. for every input, disturbances are effectively benign. There￾fore the disturbance is decoupled from BCN implying that L˜ can be divided into blocks of rank one. For state feedback, the BN-BCN equivalence output sequence requires OS(ColiL) OS(ColiL˜ x) and state transition equivalence requires Coli(L) Col(BlkiL˜ x). For output feedback, the BN-… view at source ↗
Figure 2
Figure 2. Clean reachability of output sets [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Definite reachability of output sets i ∈ {1, 2}, k ∈ {1, 2, 3, 4}. x i+ sk :⇒ X 1+ 1 {2, 4} X 2+ 1 {3, 4} X 1+ 2 {2, 4} X 2+ 2 {1, 4} X 1+ 3 {2, 4} X 2+ 3 {3, 4} X 1+ 4 {2, 4} X 2+ 4 {3, 4} Oi k+ : O1 1+ {1} O2 1+ {1, 2} O1 2+ {1} O2 2+ {1, 2} O1 3+ {1} O2 3+ {1, 2} O1 4+ {1} O2 4+ {1, 2} Reachability to OS1 : y1 1 X1 : 1 ∈ O1 1+& O1 1+ is singleton, ∴ CX1 {1} hence cleanly reachable X2 : 1 ∈ O1 2+& O1 2+ is singlet… view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: Uncertain reachability of states with control [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Disturbance decoupling in mapping • x i+ sk : X1−s X1 δ 1 4 , X2 δ 2 4 , X3 δ 3 4 , X4 δ 4 4 i → 1 δ 1 4 2 δ 2 4 3 δ 3 4 4 δ 4 4 X i+ 1 {{1, 2} {2, 3} {1, 2, 4} {2, 4}} X i+ 2 {{3, 4} {2, 4} {3, 4} {1, 4}} X i+ 3 {{2, 4} {1, 3, 4} {1, 3} {1, 2, 3}} X i+ 4 {{1, 2, 3} {1…
Figure 7
Figure 7. Figure 7: Output set digrap based behaviour in analysis and control design. If, a digraph is constructed with all possible output_sets as verices numbered according to Boolean output vectors, i.e. V : {δ 1 2 p , . . . , δ2 p 2 p } or V : {1, . . . , 2 p } for simplicity of notat…
Figure 8
Figure 8. Figure 8: Control invariant subset of a set and is equivalent to indefinite reachability with restriction. If no states of the sub-system remain unclassified then the disturbance can be decoupled in at-most SD steps, where SD is maximum of the number of evolutions required by al…
Figure 9
Figure 9. Figure 9: Clean reachability of output sets set of the output sub-system. The algorithm A.4 can be modified as Algorithm A.8 to work on the state transition matrix L 0 of the output sub-system, instead of the set of the output sub-system states.5 Theorem 13: If algorithm A.4 cla…
Figure 10
Figure 10. Figure 10: Reconstructibility with output feedback is 2 × 2 × · · · × 2 |¨¨¨¨¨¨¨¨¨¨¨{z¨¨¨¨¨¨¨¨¨¨¨} 16 time s 2 16 65536, for n 5, m 2 the number reaches to 8 × 8 × · · · × 8 |¨¨¨¨¨¨¨¨¨¨¨{z¨¨¨¨¨¨¨¨¨¨¨} 32 time s ' 8×1028. Therefore not a feasible approach to check for reconstruct…
Figure 12
Figure 12. Figure 12: Observer aided fault detection: Reference track [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

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Reference graph

Works this paper leans on

86 extracted references · 80 canonical work pages

  1. [1]

    Find all paths that satisfy the proposition

  2. [2]

    For every set of paths (every node appears only ones) identify the (state − input) tuple for all [nodes-edges]

  3. [3]

    List out the paths that follow the output feedback requirement

  4. [4]

    If any such path exists then the output feedback matrixisgivenbythecorresponding (state−input) tuple as, Col state _output _se t(My) /equalxinput E.g. 1. L′ /equalxδ4[ 2 3 4 4 6 7 8 4 1 4 3 5 4 2 3 3 1 1 3 4 5 2 7 8 3 3 4 4 5 5 7 7 ] H /equalxδ4[1 1 2 2 3 3 4 4 ] for My /equalxδ4[1 3 4 2] the resulting state transition matrix is given by ˜L :/equalxδ8[2 3...

  5. [5]

    all the possible next_states(xns) should belong to same output where,i /equalxδi 2m

    Representing elements of X1−s by the correspond- ing ordinal Boolean vectors δk 2s, define Ck :/equalx {se t o f possible control strate gie s f or DD },then i ∈ Ck iff ∀xns ∈ xi+ sk, for xns ∈ Osl for same l∈{ 1,..., 2p i.e. all the possible next_states(xns) should belong to same output where,i /equalxδi 2m

    work page

  6. [6]

    Denoting the same byb C → a

    Clean Reachability: State a is said to be cleanly reachable from stateb if for some input under the system dynamics, stateb transitions to statea directly. Denoting the same byb C → a. i.e. b C → a iff∃i∈{ 1,..., 2m} suchthat Lu i b /equalxa,where L : the state transition matrix,ui : ith input belongingδ2m and a, b : system states i.e.a, b∈δ2n

  7. [7]

    Denoting the same byb d → a, b d → a iff ∃i1, i2,..., ik ∈ {1,..., 2m} such that Lu i1 ui2

    Definite Reachability:State a is said to be definitely reachable from stateb if for some input sequence under the system dynamics, starting from state b the system trajectory reaches statea in a finite (predefined) number of evolutions. Denoting the same byb d → a, b d → a iff ∃i1, i2,..., ik ∈ {1,..., 2m} such that Lu i1 ui2... uik b /equalxa, where k is a po...

    work page

  8. [8]

    Denoting the same byb id → a, b id → a if ∃i1, i2,..., ik ∈ {1,..., 2m} such that Lu i1 ui2

    Indefinite Reachability:State a is said to be indefi- nitely reachable from stateb if for some input sequence of unknown length under the system dynamics, starting from state b the system trajectory may eventually reach state a. Denoting the same byb id → a, b id → a if ∃i1, i2,..., ik ∈ {1,..., 2m} such that Lu i1 ui2... uik b /equalxa, where k is some pos...

    work page

Show all 86 references
  1. [9]

    Therefore, DD in mapping implies removal of uncertain- ties in/from the map

    DD in Mapping:The BCN can be represented as a (state , input)↦→( ne xt_state) and the effect of distur- bance can be incorporated as uncertainties in the map. Therefore, DD in mapping implies removal of uncertain- ties in/from the map. A system is disturbance decoupled in its m...

  2. [10]

    DD in Iterations:The removal of uncertainties could be limited to certain (state , input) tuples. If a set is constructed containing states from all such tuples such that this set contains an invariant set(S1) under some input and if all the remaining states definitely reach th...

  3. [11]

    set of all the sub-states of X1−s that have the same output.⁴ C

    DD Invariant in Output (Y): In addition to DD in mapping (iteration) if the output sets(Os) are controlled invariant under the system dynamics, then it is defined as DD invariant in output(Y) solvable, where OSi :/equalx {xs∈ X1−s|H(xs) /equalxδi 2p}, i.e. set of all the sub-st...

  4. [12]

    Divide the states that appear in the output equa- tions into2p groups corresponding to each possible output, such that every state from a group has the same output

  5. [13]

    Divide state transition matrix Ls into 2m equal blocks, where each block corresponds to one of 2m possible input combinations (L1 su to L2m su of size 2s× 2n+d)

  6. [14]

    Divide each Li su block into 2s sub-blocks, corre- sponding to all possible combinations of output friendly variables (Li1 suo to Li2s suo of size 2s× 2n−s+d)

  7. [15]

    For every index k of Lik suo, list out all the pos- sible next states (entries from Lik suo) as xi+ sk ∀i ∈ {1,..., 2m} (check for all possible next_states)

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    Observability of boolean control net- works: A unified approach based on the theories of finite au- tomata and formal languages,

    K. Zhang and L. Zhang, “Observability of boolean control net- works: A unified approach based on the theories of finite au- tomata and formal languages,” inProceedings of the 33rd Chinese Control Conference. IEEE, 2014, pp. 6854–6861

    2014

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    e2n−s ∈ C1, e2n−s +1 ∈ C2

    Disturbance decoupling is possible if, Ck , {φ} ∀k∈{ 1,..., 2s}; and the state feedback con- troller can be constructed as Mx /equalxδ2m[e1 ∈ C1... e2n−s ∈ C1, e2n−s +1 ∈ C2... e2×2n−s ∈ C2,......, e(2s−1+1)×2n−s ∈ C2s... e2s×2n−s ∈ C2s] i.e. for every xs j ∈ X1− s, for j ∈ {1...

  10. [18]

    Steps (1) to (4): same as Algorithm A.2

  11. [19]

    Let, Ok+ /equalx{Oi k+| i∈{ 1,..., 2m} f or k ∈{ 1,..., 2s}

    Representing elements of X1−s by the correspond- ing ordinal Boolean vectorδk 2s Repeat for k∈{ 1,..., 2s} Repeat for i∈{ 1,..., 2m} Oi k+ /equalx{l| ∃xns∈ xi+ ns such that x ns∈ Osl} Oi k+ is the set of outputs of all the possible next_states of state k with control input i. ...

  12. [20]

    The feedback law is decided as follows: • a

    State feedback control selection: The controller can be constructed as a matrix of2s blocks, a block of size(2m×2n−s), corresponding to every state inX1−s. The feedback law is decided as follows: • a. Control for DD in mapping: For all availablek∈{ 1,..., 2s} Ck /equalx{i | ∀O...

  13. [21]

    S1 /equalx{xs∈ X1−s| ∃i∈{ 1,..., 2m} & L′ui xs∈ S1}

    In the output sub-system construct the largest con- trolled invariant set under the system dynamics with clean reachability, i.e. S1 /equalx{xs∈ X1−s| ∃i∈{ 1,..., 2m} & L′ui xs∈ S1}

  14. [22]

    S2 /equalx{xs∈ X1−s/S1| ∃i∈{ 1,..., 2m} & L′ui xs∈ S1}

    Construct the sub-system states not included in set in step 1, set of all the states that cleanly reach the set in 1, i.e. S2 /equalx{xs∈ X1−s/S1| ∃i∈{ 1,..., 2m} & L′ui xs∈ S1}

  15. [23]

    But there are a few limitations with this approach: • The actual nature of output equation is not consid- ered

    provided an algorithm for DD, which checks for possibility of DD and also provides possible state feedback laws. But there are a few limitations with this approach: • The actual nature of output equation is not consid- ered. • Definition forDDP and conditions for its solution (...

  16. [24]

    S3 /equalx{xs ∈ X1−s/(S1 Ð S2)|∃i ∈ {1,..., 2m} & L′ui xs∈ S1 Ð S2}

    Construct from states not included in sets in either steps 1 or 2, the set of all the states that cleanly reach[se t in 1 Ð se t in 2], in other words, two sets defined in 1 or 2, i.e. S3 /equalx{xs ∈ X1−s/(S1 Ð S2)|∃i ∈ {1,..., 2m} & L′ui xs∈ S1 Ð S2}

  17. [25]

    Continue till remaining states can not be classified. Fig. 8: Control invariant subset of a set and is equivalent to indefinite reachability with restriction. If no states of the sub-system remain unclassified then the disturbance can be decoupled in at-mostSD steps, where SD is ...

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    The difference is evident from the necessary condition, the method presented here, requires ˜Lir to satisfy the given condition, but the individual rows in the sum need not

    is that it utilizes the output equation of the system. The difference is evident from the necessary condition, the method presented here, requires ˜Lir to satisfy the given condition, but the individual rows in the sum need not. Contrary to this, in [23] every row of˜L individu...

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    Find all possible state feedback controllers

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    ⁷Work presented here is generalization of the existing ideas for more inclusivity

    Out of these, list the ones for which, all the states belonging to one output group have same control action. ⁷Work presented here is generalization of the existing ideas for more inclusivity. Its utility is application dependent. Stricter demands may require a more conservati...

  21. [29]

    If no such controllers, then DD with output feed- back is not possible. Else, the output feedback is given by My, with Col(My)⊆ δ2m and Col j(My) /equalx uk such that uk is the common control input corre- sponding to output_setOs j Remarks: • Output feedback is difficult to cons...

  22. [30]

    Construct from output y→{ possible state s } for y1 to y2p

  23. [31]

    For each yi select an input such thatyi→ X+ yi has least number of possible states

  24. [32]

    Observe the output and apply the corresponding input

  25. [33]

    If|XE| /equalx1, the state is reconstructed

    Observe the next output y+ k; the state is in XE :/equalx {X+ y j∩ yk}. If|XE| /equalx1, the state is reconstructed. If XE /equalx{φ}, fault occurred. If|XE|> 1, Select input according toyk

  26. [34]

    Go to step (4). Possibility of Reconstructibility with Output Feedback: As shown earlier, checking all the possible state feed- backs could be a hopeless task, let alone all the possi- ble arbitrary input sequences. Giving rise to following issues: • For arbitrary input the nu...

  27. [35]

    This could be, depending upon requirements, com- putationally much more efficient

  28. [36]

    Observer can be constructed even for the systems non-reconstrctible in general sense. The second point follows from the fact that any sys- tem, if follows distinguishable trajectory till the state is accurately reconstructed, then from the next evolution onward the system will...

  29. [37]

    Find all the attractors (cycles and fixed points) of the control BN

  30. [38]

    Starting from all the states of the (primary)BCN as initial states, check for only the(x, u) trajecto- ries with control sequence as the attractors of the control BN (secondary). VII. I/n.sc/s.sc/t.sc/a.sc/n.sc/t.sc/a.sc/n.sc/e.sc/o.sc/u.sc/s.sc F/a.sc/u.sc/l.sc/t.sc D/e.sc/t....

  31. [39]

    observer aiding output feedback)

    Some output feedback exits that aids (drives) ob- server (i.e. observer aiding output feedback)

  32. [40]

    Therefore if possible, instantaneous fault detectableBN can be constructed using state feedback fromBCN with the help of observer aiding O/P feedback

    System remains fault free at-least till observer con- verges. Therefore if possible, instantaneous fault detectableBN can be constructed using state feedback fromBCN with the help of observer aiding O/P feedback. B. Algorithm Instantaneous Fault Detection: Let the BCN dynamics...

  33. [41]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.