Given a bimodal system de¯ned by the equations
½
x_ (t) = A1x(t) + Bu(t) if ctx(t) · 0
x_ (t) = A2x(t) + Bu(t) if ctx(t) ¸ 0 (1)
where B 2Mn;m and Ai 2Mn, i = 1; 2, are such that A1;A2 coincide on the hyper-
plane V =Kerct. We consider in the set of matrices de¯ning the above systems the
simultaneous feedback equivalence de¯ned by ([A1;B]; [A2;B]) » ([A0
1;B0]; [A0
2;B0]) if
[A0
i B0] = S¡1[Ai B]
·
S 0
R T
¸
i = 1; 2 with S(V) = V
This equivalent relation corresponds to the action of a Lie group. Under this action we
obtain, in the case m · 1, the semiuniversal deformation, following Arnold's technique.
Then the problem of structural stability is studied.Postprint (published version

Puerta Coll, Xavier

Puerta Sales, Ferran

English

Given a bimodal system de¯ned by the equations

½

x_ (t) = A1x(t) + Bu(t) if ctx(t) · 0

x_ (t) = A2x(t) + Bu(t) if ctx(t) ¸ 0 (1)

where B 2Mn;m and Ai 2Mn, i = 1; 2, are such that A1;A2 coincide on the hyper-

plane V =Kerct. We consider in the set of matrices de¯ning the above systems the

simultaneous feedback equivalence de¯ned by ([A1;B]; [A2;B]) » ([A0

1;B0]; [A0

2;B0]) if

[A0

i B0] = S¡1[Ai B]

·

S 0

R T

¸

i = 1; 2 with S(V) = V

This equivalent relation corresponds to the action of a Lie group. Under this action we

obtain, in the case m · 1, the semiuniversal deformation, following Arnold's technique.

Then the problem of structural stability is studied

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On the perturbation of bimodal systems

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On the perturbation of bimodal systemsXavier Puerta1, Ferran Puerta21 Institut d’Organitzacio i Control IOC (UPC), E-mail: francisco.javier.puerta@upc.edu2 Departament de Matema`tica Aplicada I, ETSEIB (UPC), E-mail: ferran.puerta@upc.eduAbstractGiven a bimodal system defined by the equations{x˙(t) = A1x(t) +Bu(t) if ctx(t) ≤ 0x˙(t) = A2x(t) +Bu(t) if ctx(t) ≥ 0 (1)where B ∈Mn,m and Ai ∈Mn, i = 1, 2, are such that A1, A2 coincide on the hyper-plane V =Kerct. We consider in the set of matrices defining the above systems thesimultaneous feedback equivalence defined by ([A1, B], [A2, B]) ∼ ([A′1, B′], [A′2, B′]) if[A′iB′] = S−1[AiB][S 0R T]i = 1, 2 with S(V) = VThis equivalent relation corresponds to the action of a Lie group. Under this action weobtain, in the casem ≤ 1, the semiuniversal deformation, following Arnold’s technique.Then the problem of structural stability is studied.1 Preliminaries(1.1) From now on, Mn,m denotes the set of n×m complex matrices. We write Mn,n =Mn. If A ∈ Mn,m, A∗ (resp. At) denotes conjugate transpose of A, (resp. transpose ofA) and tr A the trace of A.(1.2) In [3] the following reduced form is obtained under the above equivalent relation:Let J be a h×h complex Jordan matrix and N be the l×l standard nilpotent matrix. Thenany pair ((A1 b), (A2 b)) with Ai ∈Mn, b ∈Mn,1 and A1|V = A2|V, V = Ker (0, ..., 0, 1)t,is equivalent to a pair ((A10, b0), (A20, b0)) whereA10 = J 0 α10 N 00 α1 0 , withα1 = (0, ..., 0, 1), α1 = (α11, ..., α1h)tA20 = J 0 β110 N β120 α1 β , b0 = 0p²0withβ11 = (β111, ..., β11h)t, β12 = (β121, ..., β12l)t, p = (0, ...0, 1)t.1X. Puerta, F. PuertaWe shall say that this pair is in Kronecker reduced form.(1.3) Let M = {((A1 b), (A2 b));A1|V = A2|V} andG =(S Of t);S ∈ Gl(n), S(V) = V, t 6= 0Notice that S has the form S =(S11 s10 s), so that G can be identified with an openset of Cn2+2.We consider in M the hermitian product defined by< ((A1, b), (A2, b)), ((A′1, b′), (A′2, b′)) >= tr((A1, b), (A2, b))A′∗1b′∗A′2∗b′∗and the action of G on M defined by(S Of t)∗ ((A1, b), (A2, b)) = (S(A1, b)(S−1 Of t), S(A2, b)(S−1 Of t)).We fix a pair ((A10, b0), (A20, b0)) ∈M and let φ : G →M be the map defined byφ(S) = S ∗ ((A10, b0), (A20, b0))with S =(S Of t).Let A0 = ((A10, b0), (A20, b0)) and denote O0 = {S ∗ A0;S ∈ G}. We know that theorbit O0 is a locally closed submanifold of M (see for example [2]). Then if we denoteB = (TA0O0)⊥ and I the unit element in G, we have the following theorem due to Arnold([1]; see also [4]).Theorem 1 The linear variety A0+B has the following universal property. Let ψ : B→M defined by ψ(χ) = A0 + χ. Then for any differentiable map ϕ : CN → M such thatϕ(0) = A0, there exist a neighborhood U of 0 in CN a differentiable map η : U → Bsuch that η(0) = 0 and a differentiable map ξ : U → G with χ(0) = I such that ϕ(µ) =ξ(µ) ? ψ(η(µ)).The linear variety A0+B has the minimum dimension having this universal property.It is called a miniversal deformation of A0.Finally we recall that A0 is said to be structural stable if it is an interior point of itsorbit. Equivalently, if B = 0.2 Construction of a miniversal deformationAs we have said, in order to obtain a miniversal deformation of A0 we have to compute(TA0O0)⊥.Let I be the unit element in G and P =(P Op1 q)∈ TIG with P =(P11 p10 p).Then we have the following lemma.2On the perturbation of bimodal systemsLemma 1dφI(P) = (([P, A10] + b0p1, b0q + Pb0), ([P, A20] + b0p1, b0q + Pb0)).Since TA0O0 = ImdφI one has that (A1, b), (A2, b)) ∈ (TA0O0)⊥ if and only if< (([P, A10] + b0p1, b0q + Pb0), ([P, A20] + b0p1, b0q + Pb0)), ((A1, b), (A2, b)) >= 0for every P ∈ TIG.Let P be as above and introduce the following notation:A10 =(A11 α1α1 α), A20 =(A11 β1α1 β), b0 =(b10²0)andA1 =(B11 δ1δ1 δ), A2 =(B11 γ1δ1 γ), b =(b1²).Then, we have the following result.Theorem 2 A miniversal deformation of A0 is given by the linear varietyA0 + ((A1, b), (A2, b)), where A1, A2 and b are any solution of the following system:(i) 2[A11, B∗11] + α1δ1∗ − 2δ∗1α1 + β1γ1∗ + b10b1∗ = 0(ii) 2α1B∗11 + αδ1∗ − (δ1∗ + γ1∗)A11 − δ¯α1 + βγ1∗ − γ¯α1 + ²0b1∗ = 0(iii) 2α1δ∗1 − δ1∗α1 − γ1∗β1 + ²0²¯ = 0(iv) B∗11b10 + δ∗1²0 = 0(v) (δ1∗ + γ1∗)b10 + (δ¯ + γ¯)²0 = 0(vi) tr(b10b1∗) + ²0²¯ = 0Since the number of unknowns is n2 + 2n and the number of equations is n2 − n+ 4,we have the following result.Proposition 1 There is no pair structural stable in M.Remark 1 If A10, A20 and b0 are real matrices, we can substitute the symbol ? for thesymbol t, corresponding to the transpose matrix.If the pair ((A1, b), (A2, b)) is in Kronecker reduced form, the above equations take asimplified form allowing in many cases the obtention of an explicit solution of a miniuni-versal deformation. In fact, we have in this case,A10 = J 0 α10 N 00 α1 0 , α1 = (0, ..., 0, 1), α1 = (α11, ..., α1h)t(= d(γ)t)3X. Puerta, F. PuertaA20 = J 0 β110 N β120 α1 β , b0 = 0p²0 , p = (0, ...0, 1)t.Then if accordingly with the above notation we writeA1 = B11 B12 δ11B21 B22 δ12δ11 δ12 δ , A2 = B11 B12 γ11B21 B22 γ12δ11 δ12 γ , b = b11b12²the following proposition follows.Proposition 2 ((A1, b), (A2, b)) ∈ (TA0O0)⊥ if and only if(i) 2[J, B11] + α1δ1∗1 + β11γ1∗1 = 0(ii) 2(JB∗21 −B∗21J) + α1δ1∗2 − 2δ∗11α1 + β11γ1∗2 = 0(iii) 2(NB∗12 −B∗12J) + β12γ1∗1 + pb1∗1 = 0(iv) 2[N, B∗22]− 2δ∗12α1 + β12γ1∗2 + pb1∗2 = 0(v) −(δ1∗1 + γ1∗1 )J + βγ1∗1 + ²0b1∗1 + 2α1B∗12 = 0(vi) 2α1B∗22 − (δ1∗2 + γ1∗2 )N − δα1 + βγ1∗2 − γα1 + ²0b1∗2 = 0(vii) 2α1δ∗12 − δ1∗1 α1 − γ1∗1 β11 − γ1∗2 β12 + ²0² = 0(viii) B∗21p+ δ∗11²0 = 0(ix) B∗22p+ δ∗12²0 = 0(x) (δ1∗2 + γ1∗2 )p+ (δ + γ)²0 = 0(xi) tr(0 0pb1∗1 pb1∗2)+ ²0² = 0Notice that l = 0 implies ²0 = 1, α1 = 0 and l > 0 implies ²0 = 0, so that we haveCorollary 1 The above equations reduced to:(1) If l = 0:(i) 2[J, B∗11] + α1δ1∗1 + β11γ1∗1 = 0(ii) −(δ1∗1 + γ1∗1 )J + βγ1∗1 + b1∗1 = 0(iii) −δ1∗1 α1 − γ1∗1 β11 = 0(iv) δ∗11 = 0(v) δ + γ = 0(2) If h = 0:4On the perturbation of bimodal systems(i) 2[N, B∗22]− 2δ∗12α1 + β12γ1∗2 + pb1∗2 = 0(ii) 2α1B∗22 − (δ1∗2 + γ1∗2 )N − δα1 + βγ1∗2 − γα1 = 0(iii) 2α1δ∗12 − γ1∗2 β12 = 0(iv) B∗22p = 0(v) (δ1∗2 + γ1∗2 )p = 0(vi) trpb1∗2 = 03 The case n = 3If n = 3 ( and of course if n = 2) the above equations can be solved easily. We limitourselves to give in the following three examples the dimension of the corresponding orbit.We denote this orbit by O1,O2,O3, respectively.(1) J = (λ), N = (0), so that α1 = 1, α1 = 1, p = 1, ²0 = 0. Then dimO1 = 9.(2) J = (0), N =(0 01 0), so that α1 = (0, 1), α1 = 0, p = (0, 1)t, ²0 = 0. ThendimO2 = 11.(3) J =(λ 01 λ)so that α1 = (0, 0), α1 = (1, 0)t, α = 0, ²0 = 1. Then dimO3 = 11.Notice that, according Proposition 1, any of these pairs is structurally stable.References[1] V.I. Arnold. On matrices depending on parameters, Uspekhi Mat Nauk 26, 1971.[2] James E. Humphreys. Linear Algebraic Groups; Springer-Verlag, 1975.[3] Xavier Puerta. Feedback reduced and canonical forms for switched and bimodal linearsystems; preprint.[4] A. Tannenbaum. Invariance and System Theory: Algebraic and Geometric Aspects;Springer-Verlag; 1981.5

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On the perturbation of bimodal systems

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