CC BY
78
ISSN 1810-0198. Вестник Тамбовского университета. Серия: естественные и технические науки
Том 23, № 123
2018
DOI: 10.20310/1810-0198-2018-23-123-331-352
ON STABILIZATION OF DIFFERENTIAL SYSTEMS WITH HYBRID FEEDBACK CONTROL
ac M.S. Alves, M.J. Alves
Eduardo Mondlane University PO. Box 257, Main Campus, Maputo, Mozambique E-mail: [email protected], [email protected]
Abstract. In this paper two-dimensional systems of differential equations are considered together with their stabilization by a hybrid feedback control. A stabilizing hybrid control for an arbitrary controlled system that belongs to a certain category within two-dimensional systems is constructed as a result of this study and some stabilization proprieties of the system with the obtained hybrid control are presented.
Keywords: stabilization; hybrid feedback control; linear hybrid control; upper Lya-punov exponent
1. Notations
We will use the following notations: C(M") is the set of all continuous functions u : [0, e ) => M", CS(R") is the set of all piecewise continuous functions such that u : [0, €E ) R™, the euclidean norm 11 xj | in the spacc M" will be denoted by |jr ||, the set of all matrices with real entries of dimension m 0n we denote by M(m, n, R), 77®" , Rm) is the set of all linear operators from R" to Rm and (t(A) is the set of all eigenvalues of a square matrix A, called the spectrum of A.
2. Formulation of the problem
Let us consider a controlled system
1 x = Ax + Bu
} V = Cx ' ^
where x / M" is the state vector, y / Mra is the output vector, u / R^ is the control vector. The system (2.1) is completely defined by the triple of matrices [A. B,C), where A/M(n,n,R), B / M(n,£, R) and C / M(m, n,R).
The work is partially supported by Linnaeus Palme project and SI DA/S A RE C Global Research Programme in Mathematics and Statistics.
In this paper wc will consider the system (2.1) together with the so called hybrid feedback control. The notion of hybrid feedback control was given in several papers such as [1—3].
Definition 2.1. A hybrid automaton is a set of six objects A = (Q, I. M, U g0), where
1. Q is a finite set of all the automaton's states;
2. I is a finite set called the input alphabet;
3. M : Q OI =>■ Q is an function that, determines a new state of the automaton based on its previous state q and a element from the alphabet i / I that corresponds to the switching moment of the state;
4. U : Q (0, € ) is a function that establishes the time period U (q) between two
switching moments, satisfying inf U (q) > 0;
qeQ
5. j : Mm =;> J is a function that corresponds to the output vector y / Mm and the element j(y) of I;
6. go = q(0) is the automaton's initial state.
Each hybrid automaton A = (Q, M, U, j, q0) is associated to an operator Fa : F(Mm) =$> P( Q) called the hybrid operator. Such that P(X) is a set of functions v : [0, € ) X. Let us present the recursive definition of Fa-
Definition 2.2. For any : [0, e ) =>■ Km, the function = (FAy)() : [0, e ) =i- Q is defined by:
1. q(0) = qo, h=U(qo), q(t) = q0 (A / [0,ti));
2. q(t1) = M(q0,j(y(t1))), t2 = t1+U(q(t1)), q(t) = q(tj), (A / [tut2)) ;
3. Let k / }2,3, . Suppose that. £0 = 0, ij and that the values of q{t) for t / [0, ifej were already defined. Then, and q(t) for t / [it, ¿jt+i) are defined by:
q{tk) = M(q(tk-1), j(y(tk))), tk+1=tk+U(q(tk)\ q(t) = q(tk)
(A / [tk7tk+1)).
Definition 2.3. A pair u = (A,$), where A = (Q,I,M,U,j,go) is a hybrid automaton and $ : Mra QQ =i> is a function, is called hybrid feedback control (HFC).
The hybrid control operator Wu : C(Km) Cs(M'), associated to the control u = (A, <£), is defined by
(H^)(t) = $(y(i),(FAy)(i)), t/[ o,e), where Fa is the operator that was recursively defined above.
Remark 2.1. According to the Definition 2.3, the linear system (2.1) with the hybrid control u = (A, <&) is equivalent to a functional differential equation [4]
x(t) = Ax(t) + B$(Cx(t), (FACx)(t)),t / [0, e ). (2.2)
Definition 2.4. Let u = (A, be a hybrid control of the system (2.1), where
A = (Q,/,M,U,j,g0).
The HFC u is called linear hybrid control (LHFC) if it satisfies the following conditions:
(a) the function j : Mm =>■ /, satisfies the condition j(\y) = j(y) for any y / Mra and A >0;
(b) the function q) is linear in relation to y.
We will denote the LHFC class by TC = TC(£,m).
It is convenient to represent the LHFC u in the following manner : u = (A, }Gq\ g where Gq / M(£, m) (q / Q).
Therefore the hybrid control operator Wu : 0(Mra) CS(R£) associated with u = (A, jjpq) has the form of the following linear dependence:
(Wuy)(t) = G{FAymy(t),t/[0,e).
Definition 2.5. Let (2.1) be a system with the triple fi = (A,B,C) and with a control u / TC. The infimum of A / M with which for every solution of the system it holds:
||r(t)||>Jlfe*||r(0)|l £/[0,e). (2.3)
with M positive and independent from the solution constant is called upper Lyapunov exponent of the system (2.1) with the control u and is denoted by A(Q, u).
Definition 2.6. Upper exponent of the system (2.1) with linear hybrid feedback control is the value A(£1.7X) defined by
A(ii, TC) = inf
ueCH
Surely, the upper exponent is important because it characterizes the asymptotic behaviour of the solutions.
If the upper exponent A(£l. TC) < 0, then existis u / TC such that the solution of the controllable system (2.1) exponentially stable which means that the system is stabilizable by LHFC.
It is clear, from the point of view of the stabilization of controllable systems, that it is good when A(f1,TC) = € .
Consider the linear differential system with control:
};:%:+Bou ^ (IH1101
(2.4)
this is, the system
¡^ ¿1 = fix 1 + x2
¿2 = X1 + flX 2 + U
y = x i
{
called the generalized harmonic oscillator. Note that the triple ii^j = (A^. B0,C0) of the system (2.4) is the canonical triple of the equivalence classes H(2,0, /i) where ц / } 1. 0. 11 . As in [3] and [5] we will not limit the study of the system to these three values of the parameter fi but will consider the system with an arbitrary parameter /i / R.
We have categories of systems that can be stabilized by hybrid control and a hybrid control was already constructed for the canonical cases of these categories [3, 5]. Specifically, the category ii(2,0, which contains all the triples (Л, B7 C) that satisfy ВС = 0, CAB V: 0 will be examined. This category consists of three equivalence classes corresponding to cases when fi / } 1,0,1| and the characteristic propriety of each of these classes is С В = 0, CAB 0 and sign tr Л = fi, tr A = an + a2 2 + >ooc+ ann is the trace of matrix A. The canonical form of these classes is
НЧ
In (2) and (6) a class of hybrid controls was presented. It stabilizes the system
}x = Ax + Bu y = Cx
with the canonical triple Пдо.
Let S = Af(2,2,R)0(JW(2,l,R) }0|) 0(M(1,2,R) }0\), this means, E is the set of all the triples of matrices (А, ЩС) where A / M(% 2,R), В / M(2,1,M) and С / M(1,2,M), so that В and С are non-zero matrices. Let us denote by GL{2) the multiplicative group of the square non-singular real matrices of order 2.
Definition 2.7. We define the applications T[ (D), Т2{тл, m.2l m3) and T3(a) from E to E by the formulas:
TV{D){A,B,C) = (DAD-^DB. CD-1), D / GL{2); Т2(т1,т2,™3)(Д В,С) = (miA7m2B,m3C),
rrii > 0, m2, m3 / R }0| ; T3(a)(A,B,C) = (A + aBC,B,C), а/Ж. ^
Let us consider the set of all the applications defined above:
GT0 = }Ti(D): D / GL(2)\ {
}T2(mlym2ym3): mi > 0; ra2, ra3 / R }0| | { }T3(a): at / R| .
V
It is clear that any element in T / GT0 is a bijective function T : E =>> E, this means, is a transformation of the set E. Therefore, GT0 E) where -B(E) is the group of all transformations on E with the binary operation that is the composition of transformations. In that way we defined the transformation's group GT, generated by the set GT0.
By having an arbitrary triple fi that satisfies ВС = 0, CAB 0 the goal is to construct a hybrid control with the triple fi for the corresponding system, using the theorem from the next section. This means, to construct a hybrid control for an arbitrary system that
belongs to the category in question. For that it is necessary to determine the parameters of the transformation T from GT so that T(fi) = ii^j and with the aid on the inverse transformation T-1, find the linear hybrid control that stabilizes the system £1 with any upper Lyapunov exponent.
This paper contains the solution for the problem described above. This is the main problem and the results presented are new.
3. Relation between hybrid trajectories of equivalent systems
Proposition 3.1. Let the transformation T / GT be given and represented in the following form :
T = Ti(D) <T2(mi, m2, m3) <Хз(°0
for some matrix D / GL(2) and some constants m\ > 0, т2,гпз / M }0| and a / M. Then, the inverse transformation T~l of T is defined by
T1 = T3( a)<T2 m^m^m^^ir1).
Theorem 3.1. Let the triples fij = (Aj, B^Ci) / E (i = 1,2) be given, such that Q2 = T(Qi), T / GT can be written as:
T = T3(a) <r2{mum^,m3) ^(D), (3.1)
with some matrix D / GL(2) and some constants mi > 0, m2: m3 / K }0| and a / K. Let us consider two controllable systems (S\) and (S2) ;
} with hybrid control
llcT ' ^(ArJ^W/TZtM),
where Aj = (Q, /, M, ju q0),
} with hybrid control
T=Z+ ' "2 = (A2, }<42)| / TC (i, i),
where A2 = (Q, I, M, I^ J2, q0), such that the components of the hybrid automatons A, are the same and
Life) = mr'ufa) (A/Q), h(y) = h(y^m3) {4, / IK),
Consider the hybrid trajectories hi(t) = (x^ (t), qi(t), Ti(t)), (t / [0, e )) ofthesystems (Si) (i = 1,2), such that the initial conditions of the components x^ of these trajectories satisfy the relation = Then, the following relations take place: Jt / [0, €E )
x{2\t) = DxW(mit), q2(t) = ql(m11), r2{t) = m^T^t).
The results of the theorem above follow naturally from the results that are found in [2], however, some changes were necessary because of some inaccuracy found in it.
Corollary 3.1. Let us consider the same systems with hybrid controls (Si) and (S2) as in Theorem 3.1. For any solution x'-1' of the system (S^) the exponential estimate is satisfied:
|trW(£)||>MieAf|(rW(0)|t t/[ 0,e) (3.3)
such that the constants A / K and Mi > 0 that do not depend on the solutions if and only if for any solution xof system (S2) the exponential estimate is satisfied:
||r<2>(i)||> M2em*xt\\r™(OH t / [0,e ) (3.4)
such that M2 > 0 do not depend on the solution and the constant mi > 0 is the same as in the transformation (3.1).
Proof. By the Theorem 3.1, a function a''1' : [0, € ) =>■ M2 is a system's solution (Sy) if and only if the function x^ : [0, € ) => M2 defined by
x<2>(f) = Dx^imJ), t / [0,e ), which is the solution of the system (S2). So, from the estimate (3.3) we have: ||r<2>(t)|| = p^^tmji)!!^ \D\\\c^(m1t)\\>\D\M1em^t\\c^(0)\\ = \D\M1tr*x\p-1x^Q)\\>M2fr**\\№(Q)\[ t / [0,e )
where M2 = Mi \D\\D~l\. Reciprocally, from the estimate (3.4) we have:
|tci1>(i)||=|p-V2)(m^1i)||>\i?-1\|tc(2Hm^1£)||>\i?-1\M2emimrlAi|tc{2>(0)||
= \JD-1\M2eAi|px(1>(0)||>M1eAi|^1)(0)H t / [0, e )
where Mi = M2\D~1\\D\.
Corollary 3.2. Let us consider the same systems with the hybrid control (Si) and (S2) as in the Theorem 3.1, which means, the systems with the triples £1; = (Ai, Bt. Ct) such that i}2 = T(Qi) where T is defined by (3.1) with controls u,, / TC. connected by (3.2). Then the upper Lyapunov exponents of (Si) satisfy the relation:
A(02, u2) = mi A(ii1)Ul).
The corollary's 3.2 proof follows from the Corollary 3.1.
4. Transformation of the triple (A, B, C) incase BC = 0, CAB 0
into canonical form
In this section the transformation T / GT will be determined in the form of a composition of the transformations (i = 1, 2, 3) defined above that transform a triple fi that satisfies BC = 0. CAB V= 0, in the canonical triple
li 1 1 fi
[1 0]
, fi/} 1,0,1| . (4.1)
Let the initial triple fi be given and defined by Q = (A, B, C) =
an «21
«12 a22
{■] 11
[C2 C2]
such that CB = b\Ci + b2c2 = 0, CAB 0. Let fi = sign(trA). According to the classification, there exists only one transformation T / GT such that T(iY) = ii^]. The goal now is to find the representation of this transformation T in terms of elements of matrices A, B and C. The problem is solved in some steps, described bellow.
1) First, the transformation T3(/3J is applied, where I 2det A t.r 2A
ß =
\ i
2 CAB det A 1 CAB
if trA¥=0
if tr A = 0
det A |tr 2A + CAB
1
(4.2)
We get a new triple
T3(P)(U) =T3(P)(A,B,C) = (A + PBC,B,C) = (A^B^CJ = Q1.
As it can be noted, the only matrix that suffers some transformations is the matrix A, such that in the triple fi^ the matrices B1 and C\ are the same to the matrices B and C, respectively, from the initial triple fi. Now the form of the matrix Ay will be determined:
Ai =
On
a21
a12
a22
+ß
«11 + ßb\Cx «12 + ßblC2{
a21 + ßb2ci a2 2 + ßb2c2\
The goal of applying the transformation T^(/3) is to obtain the matrix Ay with two complex eigenvalues which have the same real and imaginary parts by modulo. More precisely, we
I \trA
"(A) =
siie
tr A t.r A t.r A
i x—, ——h i x——-
if tr A V= 0 if trA = 0
2 ' 2 } h i\ ,
Note that the idea of usihg the transformation T3(/3) with the described propriety of the spectrum of Ay can be found in [6, p. 33], however, some changes were necessary due to some inaccuracy in the expressions of (3 and
2) Next, the transformation T2(v, 1,1) is applied to the triple fi^ with
„= ^ ¿¡I' if 1)11 . (4.3)
1, if fi = 0
The triple fi2 = (A2, B2: C2) = T2[i/, 1, l)(Aj, A2, A3) is obtained. Being that the two of the last parameters of T2 are equal to 1, the matrices B and C remain the same. Thus, B2 and C2 are the same as By and C\, that are the matrices B and C from the initial triple fi. The matrix A2 has the following form:
A2 = vAl =
v(aYl + ßbycy) v(a12 + ßb^) f v{a 2i + ßb2ci) v{a22 + ßb2c2)\
The goal of applying the given transformation T2 (y, 1,1) is to obtain the spectrum a(A2) =
i,fi + i\ (4*/} i,o,i|)-
3) The goal of this third step is to obtain the canonical matrix A^j, defined by (4.1) from the matrix A2. This transformation was obtained from the theorem 9 in [7, p. 299].
Let us determine an eigenvector v of the matrix A2 associated to the eigenvalue A = fj, + % :
{A2 (fi + i)I)v = 0 oo
}
i/(au + ßWci) (fj, + i)[vi + f(ai2 + ßb\c2) v2 =0
v(a21 + ßb2cl) Vi + v(a22 + ßb2c2) (fi +1)[ v2 =0
00
v =
{
v(ai2 + ßbic2)
v{a\2 +ßb\c2)
fi v(a\i + ßb\Ci) + i
and define a real matrix V by
V = [Re v Im u] =
1
fi f(an + ßbiCi)
0 1
¡/(«12 + ,3i>ic2) v(a12 + 0bic2) Let us now apply the transformation (D) for the triple 02 where
1 0 ^(«11 + №£1) ¡i v(a12 + /?i>i
We obtain the triple fi3 = (A3l B3, C3) = Ti(D)(fi2), such that, (see [7, p. 299]),
D = V~1 =
wi'
(4.4)
A3 = DAoD'1 = V^AnV =
1 H-
1 /4
Note that the matrices B3 and C3 are:
w
B-i = DB =
C3 = CD~[ =CV =
v(anbi + a12b2) iibx \
c2(pi v(an + ßbic,]))
ci +
C2
{
¡/(«12 + fibic2) v(ai2 + f3bi
So, by the steps 1), 2) and 3) the matrix A3 = A^ is obtained from the canonical triple ii^]- The goal of the next two steps in to find the transformations from the group GT that transform B3 and C3, to Bq = [0 1]T and Co = [1 0], conserving the matrix A3 = A^j.
4) As it was deducted in [6, p. 32], the matrix A3 commutes with any matrix of form
Lfa e) =
if £ £ <p
{
such that L((p, £)A3(L(ip, e)) 1 = A3. Let us now find the values of cp and £ such that. L(ip,£)B3 = BQ = [0 1 T. Solving the linear system L({p,£)B3 = B0. this means
)
b1tp+ ¡/(aii&i + a12b2) /.tb1 [ £ =0 i/(an&i + ai2b2) fifn [ (p b[ £ = 1
in respect of cp and we obtain
"{cinbx + cn2b2) fibi
<P =
bl + (/y(aii£>i + ai2b2) fibi)2 '
£ =
h
bl + {yiciubi + aub2) yhif
(4.5)
Let us now apply the transformation T\ (L), where L = L(ip, e) with <p and £ defined by (4.5), that means
1
v (aii&i + ai2b2) fibi bi
bi f (an&i + aub2) ßbi
bf + + aub2) M&i)2.
The triple = (A4, B4, C4) = Ti(L)(iï3) is obtained, where
(4.6)
Ai = LA3L 1 = A3 =
' H
1 4'
B4, — LB3 — Bq —
CU — C3L 1 —
]S o(,
where
5 = ) A + «12^2) ßbi
X)Ci +
c2(n v{aii+ßbici) u(au + ßbic2)
bic2
i/(a12 +ßWc2y
Simplifying the expression of 5, according to (4.2), (4.3) and CB = bici + b2c2 = 0, we obtain
5 = u xlet [B AB] C), (4.7)
where
ll if b2^0 i »2
u)(B,C) =
c2
I *
^ 0,
if bi ^ 0.
Note that. ci/b2 = c2/bi in case of bib2 0, because CB = 0. The constant U)(B,C) has the following geometric: interpretation: if consider B and CT as vectors in R2, then we have uj(B,C) = pT||/||B| if the angle between the vectors B and CT are equal to tt/2, and lj(B.C) = \\/\\B\\ if the angle between the vectors B and C is equal to 7r/2.
5) At last, we apply the transformation T2( 1,1,obtaining the canonical triple ii^j defined by (4.1).
6) Thus, a resultant transformation is presented:
T = T2( 1, M"1) <Ti(L) <Ti(D) <T2(v, 1,1) <T3(/?),
such that T(Çl) = iî^]. By applying the propositions of the lemma 2.6 from the article [1], the transformation T can be presented in a much compact form:
T = T1(LD) <T2(u,l,5-1)<r3(^),
such that the matrices L, D and the real constants v, 6 and 8 are defined in (4.6), (4.4), (4.3), (4.7) and (4.2), respectively. To conclude the formalization of T, we compute the matrix LD and simplify the expressions of its entries.
Thus, the following theorem has been proved: Theorem 4.1. Let be given a triple of matrices
ii = (A,B,C) = where CB = 0 and CAB 0 and the triple
iVi = (Am,B0,C0) =
an a12 f J* jjC2 c2]
«21 «22 b2 \
V 1 /10 /
Mb V
[1 0]
where fi = sign(trzl). Therefore there exists a unique transformation T / GT such that T(Q) = and that transformation can be represented as following:
T=T1(P) <Ta(„,l,i-1)<T3(JS),
where
V
rp lf p/} Ml
det A kr 2 A +
P =-
* = Ik
l 1, if ¡1 = 0
5 = vxlet[B AB] Xjj(B,C) such that u{B,C) =
CAB
Pi P2
P3 Pi
{
{
If ¿2^0
»2
if 61^0
»1
and the elements of the matrix P =
v(al2b2 fiblcx)
are defined by
Pi =
P2 =
P3 =
Pa =
b\ + (v(aubi + al2b2) fibi)2 '
_biv(ai2 +/3bic2)
b\ + (v(anbi + ai2&2) phi)2 ''
+ (y(on&i + «12^2) pbl)(iy(an + fibici) fj) bj + (f(an&i + «12^2) pbi)2
v(v(anbi + a12b2) /J&i)(a12 + PWc2) b\ + (i/(aub1 + a12&2) i-tbi)2
Let us now present three examples of the triples £1 = (A, B, C) / S from the category with the invariant CB = 0, CAB 0 that belong to the three different equivalence classes H(2,0,fi) for fi = 1, fi = 1 and ¡i = 0, and construct for each of the triples, basing ourselves on the Theorem 4.1, the transformation T that maps this triple into the canonical triple
Example 4.1. Consider the triple of matrices
0 = (A№)]1 l{]
4]
Of course that CB = 0, CAB ^ 0 and p = signftrA) = sign4 = 1. So, Çl / H(2,0; 1). Also note that &(A) = }2 3i, 2 + 3i . The transformation T that maps £1 to the canonical form
11
n[i] =
1 1
:t
[1 0]
(T / GT, such that T(O) = (il[ij)) is defined by the formula:
T = Ti
/ . 74 37 . 7 L 7
Example 4.2. Let us consider the triple of matrices Q = (A, B, C) =
1 2
5 2
; • ■ h
CB = 0, CAB 0 and ¡i = sign(trA) = sign ( 1) = 1. Therefore ii / #(2,0, 1). Also note that cr(A) = } 4, 3| . The transformation T that maps fi into a canonical form
o]
is defined by:
/ J io L 7
T = Tl
Example 4.3. Consider the triple
n = {A, B, C) =
2
25
<T3( 2).
\1 5 2 f 1 _
6 2 2
/ 0 5 V. 3 VJ
CB = 0,CAB ^ 0 and fi = sign(trA) = signO = 0. So, ii / #(2,0,0). Note, c(A) = } 5,5| . T that transforms fi into the canonical form
iî[o] =
0 1 1 0
H
[1 0]
this means, T / GT such that T(O) = (0[0j), is defined by:
T = T1
i _
3+1 Ov^ 3(3+10 v'^) 5 91+15^2
3+10^2
573
<T2
V,1,—1—=i <r3^ / 18 + 60 2. /
13
9 + 30 2
5. Inverse Transformation
Let ii = (A: B: C) be an arbitrary triple, such that CB = 0, CAB 0. Having the transformation
Td = T1(P) <T2(i/)l)r1)<r3(^),
such that T(il) = O^j where fi = sign (tr A) (see the Theorem 4.1), let us now determine the inverse transformation of T^, this is, the transformation T = T^ 1 such that T(fi^]) = O.
According to the Proposition 3.1 the transformation T can be represented in the following form:
T = T3(a) <T2(a, b1 c) <Ti(D),
where
D = P~\ a = 6=1, c = 5, a= /3.
1
a = -, v
Using the formulas of the Theorem 4.1, by rewriting the parameters of T in function of the matrices of the triple 17, we get the following theorem:
Theorem 5.1. Let the triple of matrices tt = (A,B.C) = be given, where CB = 0, CAB 0 and the triple ^N = (A\p\,BQ,C0) =
all °12
a21 a22
{■] t {■
[c2 c2]
y. i f I o r
Mb V
[1 0]
where fi = sign (tr A). There exists a unique transformation T / GT such that Tffi^]) = Q and that transformation can be represented in the following form:
where
a =
|tr2A det A + 1
T = T3(a) <T2(a, i), c) <Ti(D),
Ik A
CAB
a=iL_i'+1 H b=L if b2^0
c=^det [B AB]cj(B,C) with w(B, C) = \ b'2
D =
(au 0-22)^1 + 2ai2fe2 2a
2 ÎI21&1 (an a22)b2 2a
\
e-2
f, if fei^O
b 1
(5.1)
w
For each triple from the examples 4.1, 4.2 and 4.3 let us present a transformation T that, maps the canonical triple to these triples. The transformation T can be obtained from the Theorem 5.1 or by inverting the transformation that was obtained in each of the examples in the Section 4 with the use of the Proposition 3.1.
Example 5.1. Consider the triple of matrices
n = (wj = )]J ;{.] i{,n
4]
in which CB = 0, CAB 0 and fi = sign(trzl) = 1. The transformation T / GT such that T(ii[!j) = Q is defined by the formula
T = 7i
Example 5.2. Consider the triple
(\
14/ \ 5
( <T2 (2,1,37) <T3 ) Yi
°=<w>=)]; i{] i{.]-\{
such that CB = 0. CAB V= 0 and fi = sign (tr A) = 1. The transformation T / GT such that T(ii[_!]) = £1 is defined by the formula
3 1 f
10 0\
T = T1
Example 5.3. Consider the triple
n = (A, B, C) =
25
~2~
<TU 2).
\1 5 11} 2 f 1
0 „ L 6 2 2
J. 5 V. 3 \'J
such that CB = 0, CAB 0 and fi = sign (tr A) = 0. The transformation T / GT such that T(f}[0]) = Q is defined by the formula
T = TL
3 5 2 2 15 3
( <T2)l,1,18 + 60 2^<T3)
13
9 + 30 2
6. Construction of a stabilizing hybrid control for case CB = 0, CAB V= 0
Consider the controllable differential linear two-dimensional system:
[ ¿1= «nil + aux2 + byu
X2 = «21^1 + + b2U
^ y = ClXl+C2X 2
(6.1)
where : [0, € ) R depends only from the output ii(>} : [0, e ) R by a linear hybrid control. Suppose that the real parameters an, a\2, a2\, a22, b2, ci, c2 of the system that satisfy the conditions:
fcici + b2c2 = 0, an&ici + a\2b2ci + a2\bic2 + a22b2c2 0.
(6.2)
This section contains the main results of this paper: the control u / TC that stabilizes the system (6.1), satisfying (6.2), such that the solution's norm decreases exponentially with any Lyapunov exponent.
Note that the system (6.1) with the conditions (6.2) in its vectorial form is:
)
x = Ax + Bu y = Cx
(6.3)
in which the triple of matrices
Ü = (A, B, C) =
satisfies CB = 0 and CAB ¥= 0. Thus, we have the triple from the class H(2,0, fj.) where ¿/ = sign(trA) /} 1,0,1|. The canonical form of the class ii(2,0.^i) is
«11 «12 f f1 \AC2 C2]
021 «22 \\ £>2 L
^M = (^M^o, Co) =
fi If" 1 fi \\
,[1 0]
According to the Theorem 5.1 the transformation T / GT exists and is unique and Tfiifri]) = ii- This transformation can be presented as following:
T = T3(a)<T2(a,b7c)<T1(D) (6.4)
such that the constants a, a, fe, c and the matrix D are defined by the formulas (5.1).
Let us generalize the results concerning the stabilization of the system ii^] by a control 7-L(R, m) / TC , for the system with an arbitrary triple fi such that CB = 0, CAB 0. The generalization is based on the theorems 3.1 and 5.1.
Firstly, let us define the LHFC £ (ii, R, S, m) / TC such that R > 0, S > 0 and m / }0,1 in the following way. If (5V) is the system with the triple f2[M] and control ui = H(R,m) and (S'2) is the system with the triple fi and control u2 = £ (£1, ii, m), then the parameters of the control u2 can be expressed by the parameters of the control Ui using the formulas (3.2) from the Theorem 3.1 with the use of the expressions (5.1) from the Theorem 5.1 for the transformation parameters T (T has the form (6.4) such that
T(ilw) = il).
Definition 6.1. Given ii / S defined by (6.3) where CB = 0 and CAB ^ 0 and given R > 0, 5 > 0 and m / }0,1| the LHFC C(Sl,R,6,m) / TC is defined by C (fi, R, 6, m) = (A, }aq | qeq) where the components of the hybrid automaton A = (Q, /, M, T, j, g0) are given by
Q = }id, q , I = Kh
M(qd,i+) = M(qd,i_) = M(g_,i_) = q, M(g_,i+) = qd,
3tt
U(qd)=^(R,a) =
2 a 1 + R if vy Ri 0 _ if vy<0 '
, U (g_) = S,
}g_ if m = 0
■ r 1
qd if m = 1
(6.5)
such that
where fi = sign (tr A),
if b2 * 0
c=^det[B AB]u)(B,C) where U)(B,C)= \ 62
a =
|tr2A det A + 1 CAB
I
II V
-, v = sign (c),
P-, if fei * 0
(6.6)
and qeQ = }aq_7aqd\ where aq_ = 0 and a,
Qd —
The families of hybrid controls are introduced:
C (iî, R) = }£ (fi, R, S, m) : 0<6 <--, m /} 0,1
J 4a 1 + R
(R > 0),
It is clear that £ (ii, J2) -»£ (fi) -+T£ .
We define the function A : (0, e ) (0, e ) by
HR)
(6.7)
We remember that in this section we always consider the system (6.1) satisfying the conditions (6.2), or, indeed, the system (6.3) with triple £1 = (A7B7C) satisfying the condition CB = 0, CAB V= 0. For convenience we designate this system for (S).
From the Corollary 2, Theorem 3 and the main results about the stabilization of the system with the triple ii^ from the papers (2) and (6) we get the main result of this paper which is stated in the following theorem:
Theorem 6.1. For any R > 0 , A(fi, £ (ii, R)) = a(fi A(R)), where fi and a are defined in (6.6).
Theorem 6.2. We have the following statements:
1. If tr A > 0 (this is, when fi = 1 ou fi = 0), then JR > 0 the system (S) is stabilizable by a family of hybrid controls C (ii, R).
2. If trA > 0 (this is, when fi = 1), then in case R > A-1(l), the system (S) is stabilizable by a family of hybrid controls L (17, R) and in case R < A_1(l) the system (S) is not stabilizable by a family of hybrid controls C, (ii, R).
Theorem 6.3. For any ii / E, CB = 0, CAB it is valid that A(ii, £ (ii)) = e .
Remark 6.1. According to the Theorem 6.3, the system (S) is stabilizable by the hybrid controls from the family £ (17), such that the negative upper Lyapunov exponent in the solution estimate can be as large by modulo as we define it.
Let us complement the theorems 6.1-6.3 with a result which wording is more convenient for the applications. Also note that exists A-1 : (0, e ) (0, € ) such that lim A-1(s) = 0
and lim A_1(s) = -\-<E . For convenience, let us extend the function A-1 to any set M by
assigning, by definition A 1(s) = 0 when s > 0.
Theorem 6.4. Let N > 0 be an arbitra'ry constant. Then, for any positive number R that satisfies
s—H-oo
-1
(6.8)
where A(R) is defined, in (6.7), exists a 80 = ¿o(tr A. N, R) > 0 such that A / (0, ¿0) and An / }0,1| any solution of the equation x : [0, e ) K2 of the system (S) with LHFC £ (fi, R. 6, m) satisfies the exponential estimate
I II > Af «-""IWO) U */[ 0,€)
where the constant M = R, 6, m) > 0 does not depend on the solution x( >):.
7. Examples of the systems that satisfy CB = 0, CAB 0 and stabilizing
hybrid controls
In this section we will consider three specific systems of type (6.1) that correspond to the triples (A, B, C) considered in the examples from the sections 4 and 5. For these systems, based on the results of the Section 6, linear hybrid controls that stabilize it will be presented. Even more, the chosen control parameters are the ones that decrease the solution's norm as in (6.4) with a given upper Lyapunov exponent N. For convenience, consider that the function A-1 is prolonged to ffi where by definition, A_1(s) = 0 when s > 0.
Example 7.1. Consider the system:
I, ¿i = X\ 2x2 +
x2 = 5^! + 3x2 u (7-1)
^ y = Xl+ 4x2
or, in the vectorial form: x = Ax + Bu
with ii=(A,i?,C) = ) * jjj,
[1 4]
(7.2)
y = Cx
We have CB = 0 and CAB V= 0. Let us compute the constants ¿a, a, c, a by the formulas (6.6):
¡i = sign (tr A) = 1, a = fe^l' + l
c2
|=2, c = —— det [B AB] = 37, ab i
Hi2 A det A + 1 v = sign (c) = 1, ct =-
CAB
5
74'
(7.3)
Consider the hybrid control C (ii, R, 8, m) = ((Q, I, M, T,j, q0), ad\ ) / 7X defined in the Section 6. According to the Definition 6.1 and the expressions (7.3), the control components are given by:
Q = }qd,q-1 , I = }«+,«-! ,
M(qd7i+) = M(qd,i_) = M(g_,i_) = q, M(q_,i+) = qd,
3tt
U(qd)=lAi(R,a) =
4 1 + R' if y m 0
U (q-) = 51
_ if y < 0 '
go =
}
if m = 0 qd if m = 1 '
=0, = ^(5 4 R).
The theorems 6.2 and 6.4 imply the following conclusions about the system (7.1) with linear hybrid control £ (ii, R7 S7 m).
Conclusion 1. For any m / }0,1|, R > A_1(l) c 69.89 where A is defined in (6.7) and for all sufficiently small 6 > 0 the system (7.1) is stabilizable by the hybrid control £(n,R, 6, m).
Conclusion 2. Let N > 0. For any m / }0,1|, R > A_1(l + N/2) and for all sufficiently small 5 > 0, any solution x : [0, t ) M2 of the system (7.1) with control £ (ii, R, 6, m) satisfies the exponential estimate
|(r(i)ll>Me-^|(r(0)|l t / [0,e ),
where the constant M=M(Q,R,5,m)>0 does not depend on the solution x( >t
For example, if a decrease of the solution with N = 2 is needed, then wc can conclude that if R > A-1(2) C 977.35 and 5 > 0 is sufficiently small, then any solution x of the system (7.1) with control £ (17, R. 0) or £ (ii, R, 1) satisfies the condition
IKt)ll>MB-»|Wo)|i */[ o,e)
where M > 0 does not depend on the solution.
Example 7.2. Consider the system:
[, ¿1 = xi + 2x2 X2 = 5^! 2X2
u
{
y = 1^2
or, in its vectorial form:
(7.4)
x = Ax + Bu y = Cx
with n = (^u,o = )] J l{,] J{,]o
(7.5)
We have CB = 0 and CAB V= 0. Let us compute the constants fi, a, c, v, a by the formulas (6.6):
ItrAII „
//=sign (tr A) = 1, a= +1
f ■
v = sign (c) = 1, ct =
\{t^A)2 det A + l CAB
(7.6)
= 2.
Consider the hybrid control £ (ii, ii, m) = ((Q, /, M,T, j, go), «¿1) /T£2 defined in the Section 6. According to the Definition 6.1 and the expressions (7.6), the components of
this control are given by:
Q = hd,q~I, i =}«+,«-!,
M(qd7i+) = M(qd,i-) = M(q-,i-) = q, M(q_,i+) = qd,
3tt
U(qi)=^(Ria) =
1 + R*
U (q-)=S,
j(y) =
1 *+ i J i- i
Og_ = 0,
if y > o _ if y>0 '
R „
^ = — + 2.
go =
}
q- if m = 0 qd if m = 1
25
Theorems 6.2 and 6.4 imply the following about the system (7.4) with linear hybrid feedback control £ (17, J2, 5, m).
Conclusion 1. For any m / }0,1| , R > A_1(l) C 69.89 where A is defined in (6.7) and for all sufficiently small 8 > 0 the system (7.4) is stabilizable by the hybrid control £{il,R, S, m).
Conclusion 2. Let N > 0. For any m / }0,1| , R > A_1(2A 1) and for all small 5 > 0, any solution x : [0, E ) K2 of the system (7.4) with control £ (17, R. 5, m) satisfies the exponential estimate
um^Me^Hm */[o,e)f
where the constant M=M(Q,R,5,m)>0 does not depend on the solution a'(>t
For example, if a decrease of the solution with N = 2 is needed, then we can conclude that if R > A_1(3) C 15545 and 8 > 0 is sufficiently small, then any solution x of the system (7.4) with control £ (17, R. 0) or £ (17, R, 1) satisfies the condition
Ut)W> M e-^m t/[ 0,€)
where M > 0 does not depend on the solution.
Example 7.3. Consider the system
I, £i = 5xi X2+ 2 u
¿2 = 5x2 2X2 + 3 u ^ y = Qxi + 2 2x2
and in its vectorial form
x = Ax + Bu y = Cx
CB = 0 and CAB V= 0. Let us compute the constants a, c, i/, a by the formulas (6.6):
y
J y
with £7 = (A, B, C) =
5 in 2 fl _
0 ~ S, 6 2 2
5 V. 3 \'J
(7.7)
(7.8)
ItrAII
fi = sign(trJ4) = 0, a =
c2
11=1, c = ——det [B AB] = 18 + 60 2, abi
v = sign (c) = 1,
a =
|tr 2 A detA+ 1 CAB
13
9 + 30 2
(7.9)
Consider the hybrid control £ (ii, ii, m) = ((Q, i, M,T, j, q0), ct^| ) /TC2 defined in the Section 6. According to the Definition 6.1 and the expressions (7.9), the components of this control are given by:
Q = hd,g-1 , I = }«+,«-! ,
M(qd7i+) = M(qd,i_) = M(g_,i_) = q, M(q_,i+) = qd,
37T
U (qd) = L\t(R, a) = ===, U (q_) = 5, Z 1 + ri
. , 1 i+ if y i« 0 1
j{y) = S if y < 0 ' 90 = /
n R + 2G
aq_ = U, aqd =
if m = 0 qd if m = 1 '
6(3 + 10 2)
Theorems 6.2 and 6.4 imply the following about the system (7.7) with linear hybrid feedback control £ (0, ii, m).
Conclusion 1. For any m / }0,1| , R > A_1(l) c 69.89 where A is defined in (6.7) and for all sufficiently small 6 > 0 the system (7.7) is stabilizable by the hybrid control £(Q,R, S, to).
Conclusion 2. Let N > 0. For any m / }0,1|, R > A_1(N) for all small 5 > 0 any solution x : [0, € ) =4» K2 of the system (7.7) with control C (fi, R, m) satisfies the exponential estimate
IM^Me-^MIi t/[ 0,e),
where the constant M=M(Q,R,5,m)>0 does not depend on the solution x(
For example, if a decrease of the solution with N = 2 is needed, then wc can conclude that if R > A-1(2) C 977.35 and 5 > 0 is sufficiently small, then any solution x of the system (7.7) with control £ (17, R. 0) or £ (O, R, 1) satisfies the condition
IKt)ll>MB-»|Wo)|i */[ o,e)
where M > 0 does not depend on the solution.
REFERENCES
1. Litsyn E., Nepomnyashchikh Y., Ponossov A. Classification of linear dynamical systems in the plane admitting a stabilizing hybrid feedback control. Journal on Dynamical and Control Systems, 2000. vol. 6, no. 4, pp. 477-501.
2. Litsyn E., Myasnikova M.. Nepomnyashchikh Y., Ponossov A. Efficient criteria for stabilization of planar linear systems by hybrid feedback controls. Abstract and Applied Analysts, 2004, no. 6, pp. 487-499.
3. Litsyn E., Nepomnyashchikh Y., Ponossov A. Control of the Lyapunov exponents of dynamical systems in the plane with incomplete observation. Nonlinear Analysis. Series A: Theory, Methods & Applications, Special issue: Hybrid Systems and Applications, 2005, vol. 64, no. 2. pp. 329-351.
4. Alves M.J. Singulyarnye funktsional'no-differentsial'nye uravneniya vtorogo poryadka [Singular Second Order Functional Differential Equations]. Perm, Perm State University Publ., 2000, 179 p. (In Russian).
5. Korotaeva I.G. Ehksponentsial'naya stabilizatsiya dvumernykh gibridnykh system [Exponential Stabilization of Two-Dimensional Linear Hybrid Systems]. Perm, 2003, 35 p. (In Russian).
6. Myasnikova M.A. Klassifikatsiya lineynykh dvumernykh gibridnykh system [Classification of Two-Dimensional Linear Hybrid Systems]. Perm, 2003, 51 p. (In Russian).
7. Lay D.C. Linear Algebra and Its Applications. Boston, Pearson Education Inc., 2012, 472 p.
Received 20 April 2018 Reviewed 23 May 2018 Accepted for press 19 June 2018 There is no conflict of interests.
Alves Maria Salomé Manuelevna, Eduardo Mondlane University, Maputo, Mozambique, BSc in Mathematics, Assistant Lecturer, e-mail: [email protected]
Alves Manuel Joaquim, Eduardo Mondlane University, Maputo, Mozambique, PhD in Mathematics, Full Professor, e-mail: [email protected]
For citation: Alves M.S., Alves M.J. On stabilization of differential systems with hybrid feedback control. Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2018, vol. 23, no. 123, pp. 331-352. DOI: 10.20310/1810-0198-2018-23-123-331-352 (In Engl., Abstr. in Russian).
STABILIZATION OF DIFFERENTIAL SYSTEMS WITH HYBRID FEEDBACK CONTROL 351
DOI: 10.20310/1810-0198-2018-23-123-331-352 УДК 517.977.1
О СТАБИЛИЗАЦИИ ДИФФЕРЕНЦИАЛЬНЫХ ГИБРИДНЫХ УПРАВЛЯЕМЫХ СИСТЕМ С ОБРАТНОЙ СВЯЗЬЮ
ас М. С. Алвес, М. Ж. Алвес
Университет Эдуард о Мондлане П.я. 257, Главный кампус, Мапуту, Мозамбик E-mail: [email protected], [email protected]
Аннотация. В данной статье рассматриваются двумерные системы дифференциальных уравнений со стабилизирующим гибридным управлением с помощью обратной связи. В результате исследования для произвольной системы управления, принадлежащей определенному классу двумерных систем, построено стабилизирующее гибридное управление и представлены некоторые стабилизирующие свойства системы с полученным гибридным управлением. Ключевые слова: стабилизация; управление гибридной обратной связью; линейное гибридное управление; верхний показатель Ляпунова
СПИСОК ЛИТЕРАТУРЫ
1. Litsyn Е., Nepomnyashchikh Y., Ponossov A. Classification of linear dynamical systems in the plane admitting a stabilizing hybrid feedback control // Journal on Dynamical and Control Systems. 2000. Vol. 6. № 4. P. 477-501.
2. Litsyn E., Myasnikova M., Nepomnyashchikh Y., Ponossov A. Efficient criteria for stabilization of planar linear systems by hybrid feedback controls // Abstract and Applied Analysis. 2004. № 6. P. 487-499.
3. Litsyn E., Nepomnyashchikh Y., Ponossov A. Control of the Lyapunov exponents of dynamical systems in the plane with incomplete observation // Nonlinear Analysis. Series A: Theory. Methods & Applications. Special issue: Hybrid Systems and Applications. 2005. Vol. 64. № 2. P. 329-351.
4. Alves M.J. Сингулярные функционально-дифференциальные уравнения второго порядка. Пермь: Изд-во Перм. гос. ун-та, 2000. 179 с.
5. Коротаева И.Г. Экспоненциальная стабилизация двумерных гибридных систем. Пермь, 2003. 35 с.
6. Мясникова М.А. Классификация линейных двумерных гибридных систем. Пермь. 2003. 51 с.
7. Lay D.C. Linear Algebra and Its Applications. Boston: Pearson Education Inc., 2012. 472 p.
Работа частично поддерживается проектом Linnaeus Palme и программой по математике и статистике SIDA / SAREC Global Research.
Поступила в редакцию 20 апреля 2018 г. Прошла рецензирование 23 мая 2018 г. Принята в печать 19 июня 2018 г. Конфликт интересов отсутствует.
Алвес Мария Саломье Мануэловна, Университет Эдуардо Мондлане, г. Мапуту, Мозамбик, ассистент, e-mail: [email protected]
Алвес Мануел Жоаким, Университет Эдуардо Мондлане, г. Мапуту, Мозамбик, кандидат физико-математических наук, профессор, e-mail: [email protected]
Для цитирования: Алвес М.С., Алвес М.Ж. О стабилизации дифференциальных гибридных управляемых систем с обратной связью // Вестник Тамбовского университета. Серия: естественные и технические науки. Тамбов, 2018. Т. 23. № 123. С. 331-352. БОТ: 10.20310/1810-0198-2018-23-123-331-352
