Hyrid Systems Modeling, Anlysis nd Control Rdu Grosu Vienn University of Tehnology Leture 5
Finite Automt s Liner Systems Oservility, Rehility nd More
Miniml DFA re Not Miniml NFA (Arnold, Diky nd Nivt s Exmple) x 2 x 4 x 2 L = (* + *)
Miniml NFA: How re they Relted? (Arnold, Diky nd Nivt s Exmple) x 2 x 4 x 5 x 2 x 4 x 5 L = + + + + + No homomorphism of either utomton onto the other.
Miniml NFA: How re they Relted? (Arnold, Diky nd Nivt s Exmple) x 2 x 4 x 8 x 5 x 6 x 7 x 8 Crrez s solution: Tke oth in terminl NFA. Is this the est one n do? No! One n use use liner (similrity) trnsformtions.
Convergene of Theories Hyrid Systems Computtion nd Control: - onvergene etween ontrol nd utomt theory. Hyrid Automt: n outome of this onvergene - modeling formlism for systems exhiiting oth disrete nd ontinuous ehvior, - suessfully used to model nd nlyze emedded nd iologil systems.
Lk of Common Foundtion for HA v V E x& = Ax+ Bu v = Cx v V R Mode dynmis: - Liner system (LS) voltge(mv) v V U Stimulted s / di = t v V / R s v t < V U = Mode swithing: - Finite utomton (FA) Different tehniques: - LS redution - FA minimiztion time(ms) LS & FA tught seprtely: No ommon foundtion!
Min Conjeture of this Tlk Finite utomt n e onveniently regrded s time invrint liner systems over semimodules: - liner systems tehniques generlize to utomt Exmples of suh tehniques inlude: - liner trnsformtions of utomt, - minimiztion nd determiniztion of utomt s oservility nd rehility redutions - Z-trnsform of utomt to ompute ssoited regulr expression through Gussin elimintion.
Finite Automt s Liner Systems Consider finite utomton M = (X, Σδ,, S,F) with: - finite set of sttes X, finite input lphet - trnsition reltion δ X Σ X, - strting nd finl sets of sttes S, F X Σ,
Finite Automt s Liner Systems Consider finite utomton M = (X, Σδ,,S,F) with: - finite set of sttes X, finite input lphet Σ, - trnsition reltion δ X Σ X, - strting nd finl sets of sttes S,F X For eh input letter Σ: - represent δ() X X s oolen mtrix A(), - write A = A() whe re (x) = Σ { if x = } otherwise
Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x(n+) = x(n)a, x y(n) = S(ε)ε = x(n)c, C = F(ε)ε x nd y re row vetors
Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x(n+) = x(n)a, x = S(ε)ε y(n) = x(n)c, C = F(ε)ε Exmple: onsider following utomton: L x x 2 A() = A() =, x (ε)' =, C(ε) =
Polynomils nd their Opertions A, C, x(n) nd y(n) re polynomils with: - powers: strings in Σ * (the input strings) - oeffiient s: mtries nd vetors over B
Polynomils nd their Opertions A, C, x(n) nd y(n) re polynomils - powers: strings in Σ (the input strings) - oeffiients: mtries nd vetors over B Addition nd multiplition: 2 (A() + A()) = * with: done over polynomils A()A() + A()A() + A()A() + A()A( ) = ˆ A() + A() + A() + A()
Boolen Semimodules B is douly idempotent, ommuttive semiring: - (B,+,) is ommuttive idempotent monoid (or), - (B,,) is ommuttive idempotent monoid (nd), - multiplition distriutes over ddition, - is n nnihiltor: =
Boolen Semimodules B is douly idempotent, ommuttive semiring: - (B,+,) is ommuttive idempotent monoid (or), - (B,,) is ommuttive idempotent monoid (nd), - multiplition distriutes over ddition, - is n nnihiltor: = n B is semimodule over slrs in B: - r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = - x = x, x = r(sx),
Boolen Semimodules B is douly idempotent, ommuttive semiring: - (B,+,) is ommuttive idempotent monoid (or), - (B,,) is ommuttive idempotent monoid (nd), - multiplition distriutes over ddition, - is n nnihiltor: = n B is semimodule over slrs in B: - r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = r(sx), - x = x, x = Note: No dditive nd multiplitive inver ses!
Divergene of Clssi/Disrete Mth Cnonil prtil order in semirings: + iff!. + = iff!. =
Divergene of Clssi/Disrete Mth Cnonil prtil order in semirings: + iff!. + = iff!. = Exmple of nonil PO for Nturl numers: + 5 iff!4. + 4 = 5
Divergene of Clssi/Disrete Mth Cnonil prtil order in semirings: + iff!. + = iff!. = Exmple: Cnonil PO for Nturl numers: + 5 iff!4. + 4 = 5 Exmple: Cnonil PO for Integer numers: 5 + - iff!(-6). 5 + (-6) = Semiring: Either inverses or prtil order!
Oservility Let L = [S,A,C] e n n-stte utomton. It's output: [y() y()... y(n-)] = x [C AC... A n- C] = x O () L is oservle if x is uniquely determined y ().
Oservility Let L = [S,A,C] e n n-stte utomton. It's output: [ O t n- t [y() y()... y(n-)] = x C AC... A C] = x () L is oservle if x is uniquely determined y ( ). Exmple: the oservility mtrix O of L is: O = n AC 3 ε x x2 L x x 2
Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i
Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i - if L is nondeterministi nd hs severl initil sttes, x is not uniquely determined if t here re oolen nd: I,J [..n]. I J = O = i i I i J i O i i i, (2) i
Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i - if L is nondeterministi nd hs severl initil sttes, x is not uniquely determined if there re oolen nd: Line I,J [..n]. I J = O = O r dependene: i i I i J - Def (2) generlizes liner dependene in vetor spes i i i i, (2) (2) for finite I,J nd ny vetor set. i
Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i - if L is nondeterministi nd hs severl initil sttes, x is not uniquely determined if there re oolen nd: Line I,J [..n]. I J = O = O r dependene: i i I i J - Def (2) generlizes liner dependene in vetor - Liner independene is onsequently: I,J [..n]. I J = spn(o ) i I i i i, (2) (2) for finite I,J nd ny vetor set. spes spn(o J) = {} i
Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X
Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X n Theorem (Bsis) If X B hs sis Y then Y is unique.
Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X Theorem (Bsis) If X n B hs sis Y then Y is unique. O = n AC x ε 2 3 L x x 2 [x x2 x3 ]: row sis
Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X n Theorem (Bsis) If X B hs sis Y then Y is uniqu e. O = n AC ε x x2 3 L x x 2 [x x x ]: [C( ε) AC() AC()]: 2 3 row sis, olumn sis.
Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε)
Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5
Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5
Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T
Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T x (n + ) = x (n + )T = x (n)at = x (n)t - AT = x (n)a x (ε) = x (ε)t C(ε) = T - C(ε)
Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T A () A() x (ε) C(ε) A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
Oservility Redution y Columns L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T A () A() x (ε) C(ε) A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
Mixed Oservility Redution L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T A () A() x (ε) C(ε) A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
Originl nd Redued Automt L 2 x 2 L 2, x, 3 x 2 x 4 x 5 NFA L 22 y olumns DFA L 2 y rows x, x 2 NFA L 23 mixed x,, 3 x 2
Originl nd Redued Automt L 2 x 2 L 2 x 4 x 5 NFA L 22 y olumns,, x 2 DFA L 2 y rows x, x 2 NFA L 23 mixed x,, 3 x 2 Let x = x T in L 2 where t Then L 22 = [A 2, x 2,, C 2 ] ' 't ' L 23 = [A 2, x 2,, C 2 ] T = T ' =
Row Bsis ut No Column Bsis 7 L 3 x x 2 x 5 x 6 O ε x x2 x3 x4 x5 x6 x 7 x 4
Row Bsis ut No Column Bsis L 3 x x 5 x 2 x 6 O ε x 2 x 4 x 5 x 6 x 7 x 7 x 4
Row Bsis ut No Column Bsis L 3 x x 2 x 5 x 6 x 7 x 4 O ε x 2 x 4 x 5 x 6 x 7 O ε x x2 x3 x4 x5 x6 7
Row Bsis ut No Column Bsis L 3 x x 2 x 5 x 6 x 7 x 4 O ε x 2 x 4 x 5 x 6 x 7 O ε x x2 x3 x4 x5 x6 7
Oservilty Redution Theorem (Cover): Finding (possily mixed) sis T for O L is equivlent to finding miniml over for O L. - either s its set sis over or s its Krnugh over. Theorem (Complexity): Determining over T for O L is NP-omplete (set sis prolem omplexity). Theorem (Rnk): The row (= olumn) rnk of O L is the size of the set over T (size of Krnugh over).
Rehility: Dul of Oservility Let L = [S,A,C] e n n-stte utomton. It's output: [y() y()... y(n-)] t = C t [x A t x... (A t ) n- x ] = C t R t (3) where x is now olumn vetor. L is rehle if C is uniquely determined y (3).
Rehility: Dul of Oservility Let L = [S,A,C] e n n-stte utomton. It's output: ) R t t t t n- t t [y() y()... y(n-)] = C [x A x... (A x ] = C (3) L is rehle if C is uniquely determined y ( 3. ) Exmple: the rehility mtrix of L is: t R = 2 3 t (A ) x x x n x ε L x x 2 Row sis [x x x [ ε ] = ( ) A () A t t 2 3 x x x ()] ol sis.
Oservilty, Rehility nd More DFA Minimiztion: Is prtiulr se of oservility redution (single initil stte requires distintness only) NFA Determiniztion: Is prtiulr se of rehility trnsformtion (tke ll distint olumns s sis ) Miniml utomt: Are relted y liner mps (ut not y grph isomorphisms!). Better definition of minimlity Other tehniques: Are esily formlized in this setting: Pumping lemm, NFA to RE, Z-trnsforms, et.
Arnold, Diky & Nivt s Exmple Revisited (Oservility Redution) A x 2 x 4 x 5 A x 23 x 24 4 x 5 Define liner trnsf t t x = x T: A(x) = [A(x)T] T t t T = x ( ε) = x ( ε)t C( ε) = [ C( ε)] T
Arnold, Diky & Nivt s Exmple Revisited (Rehility Redution) A x 2 x 4 x 5 A x 23 x 24 4 x 5 Define liner trnsf x = x T: T = A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T