CC BY
6672 Probl. Anal. Issues Anal. Vol. 8 (26), No 1, 2019, pp. 72-83
DOI: 10.15393/j3.art.2019.4610
UDC 517.977
A. N. Kirillov
THE METHOD OF NORMAL LOCAL STABILIZATION
Abstract. A problem of nonlinear systems stabilization is studied. Admissible controls are piecewise constant. The notion of normal local stabilizability is proposed. A point P (not necessary equilibrium) is normally locally stabilizable if for any t > 0 there exists such neighborhood D(P; < t) of P that any point x £ D(P; < t) can be steered, in a time less than t, to any neighborhood of P and remains there. The constructive method of normal local stabilization of nonlinear autonomous systems is presented. This method involves a special sequence of contracting cylinders containing a trajectory. A domain of attraction of a given point is constructed.
Key words: dynamical system, positive basis, normal stabilization
2010 Mathematical Subject Classification: 34H15
1. Introduction. The notion of normal stabilization, introduced in this article, is motivated by the concept of normal local controllability, introduced by N. N. Petrov [6].
Definition 1. A system x = f (x,u),x £ Rn, is normally locally controllable at a point P if for each t > 0 there exists a neighborhood D(P; < t) of P such that any point x £ D(P; < t) can be steered to P, in a time less than t, by an admissible control u.
In case of n = 2, for analytical f and piece-wise constant control N. N. Petrov obtained the necessary and sufficient conditions of the normal local controllability in terms of serial expansion coefficients for f [7]. For n > 2 such conditions are not yet found.
Definition 2. A system x = f (x,u),x £ Rn, is normally locally stabilizable at a point P if for each t > 0 there exists a neighborhood D(P; < t) of P such that any point x £ D(P; < t) can be steered by an admissible control to any neighborhood of P, in a time less than t, and remains there.
©Petrozavodsk State University, 2019
This notion does not deal necessarily with the equilibrium state stabilization and, therefore, it differs from the finite-time stability notion [4]. The concept of normal stabilization is useful for the control of hybrid systems, which structure may be time variant and the controller has to be able to stabilize the desired state in some constrained time. Some ideas of normal stabilization appeared in [5], though this notion was not formulated there.
The admissible control is supposed to be a piecewise constant function of time with values in {ui, i = 1,... ,n +1}, where ui E Rm are constant vectors. Given a control system x = f (x,u), x E Rn, we prove that if the vectors f (0, ui), i = 1,..., n +1, affinely generate Rn, then the point x = 0 can be normally stabilized by admissible control. The sufficient condition for normal stabilization means that {f (0, ui),i = 1,..., n + 1} is the positive basis of Rn [1], i.e. any x E Rn may be represented as x = ^n+i1 Af (0,ui) with nonnegative Ai E R.
The main result of the article is the constructive method of normal local stabilization. It is worth to note that the result of N. N. Petrov [6], [7] is not constructive. He proved the existence of control in the problem of normal local controllability, using his implicit positive function theorem.
The method of normal stabilization, presented in this article, may be used in problems of stabilization via sliding mode [2], [3]. But, unlike [3], where the method of stabilization is constructed under the crucial obtuse condition, i. e. the angles between vectors of a special positive basis are obtuse, in this article the angles between vectors f (0, ui), i = 1,..., n, are arbitrary.
The structure of the article is organized as follows. In Section 2 we give some definitions, present properties of the positive basis and introduce some geometric objects. Section 3 is devoted to the construction of the main instrument of the normal stabilization method which is the cylinder of stabilization. The method of normal stabilization, based on the sequence of cylinders of stabilization, is presented in Section 4. Some examples are given in Section 5.
2. Positive basis. Consider the nonlinear control system
x = f (x,u) (1)
where x(t) E Rn is the state, u(t) E U C Rm is the control input. Let f: Rn x Rm ^ Rn be Lipschitz with respect to x. Denote by xTy the scalar product, || where T is the symbol of transposition. The
admissible control u(t) is a piecewise constant function.
Definition 3. The set {aj}, j = 1,... ,k, aj £ Rn, is called a positive basis of Rn if for each x £ Rn there exist such Xj £ R, Xj ^ 0, j = 1,... ,k, that x = Y1 j=\ Xj aj.
Definition 4. The set {ai}, i = 1,... ,m, is called a positive basis of a hyperplane nn-i £ Rn if for each x £ nn-i there exist such Xj £ R, Xi ^ 0, i = 1,... ,m, that x = mi Xiai.
It is easy to see that that the minimum positive basis of Rn consists of (n + 1) vectors and for nn-i it consists of n vectors.
Lemma 1. The set {ai,..., an+i} is the positive basis of Rn if and only if the following both statements are true.
1. The origin O £ Rn is the interior point of the convex hull of ai,..., an+i: O £ int(co(ai,..., an+i)).
2. Any n vectors among ai,..., an+i are linear independent.
Lemma 2. If {ai,..., an+i} is the positive basis of Rn then any ai, i = = 1,... ,n +1 cannot be expressed as the nonnegative linear combination of the other vectors of the positive basis.
The proofs of Lemmas 1, 2 simply follow from the definition [1]. Let ai, i = 1,..., n, be linear independent. Denote by A = (ai,..., an) the matrix with the coordinate columns ai,... ,an in some basis. We put a = (ai,..., a,n)T > 0 (> 0), if ai ^ 0 (> 0), i = 1,...,n. Denote by K and K* the open cone and the conjugate open cone, respectively, spanned by ai,...,an
K = {x £ Rn : x = Aa, a > 0}, K* = {y £ Rn : ATy < 0}, (2) and let K, K* be the closures of K and K*, respectively
K = {x : x = Aa, a ^ 0}, K* = {y : ATy ^ 0}. Denote K = Rn \ (K U (-K)).
Lemma 3. Assume that ai,..., an, ai £ Rn, n ^ 2, are linear independent and —K* C K with K, K* defined in (2). Then for any y £ K* there exists a hyperplane nn-i(y) C K such that zTy = 0 for each z £ nn-i(y).
Proof. Since —y £ —K* C K then there exists a > 0 such that —y = Aa and AT(—y) > 0. Hence ATAa > 0. The linear independence of ai,... ,an implies the existence of such 7 £ Rn that z = Ay. Then we obtain
zT(—y) = (AY)TAa = YTATAa = Yi Pi + ... + YnPn,
where Yi are the components of vectors ATAa, 7, respectively, $ > 0. In order to satisfy zTy = 0 we can put, for example, 7^ = — 7^Yi = 0, s = i = /, = 0 for any fixed /, s E {1,..., n}. □
Remark. If — K C K* we assume K = Rn \ (K* U (—K*)), and Lemma 3 is true in this case for any y E K.
Definition 5. Vector y is called a normal to plane nn-1(y).
Lemma 4. Assume that a1,..., an are linear independent and —K* C K. Let bi = bi(y) be the orthogonal vector projection of ai onto nn-1(y) with y E K*. Then {b1,..., bn} is the positive basis of nn-1(y).
Proof. First, we prove that the origin O is the interior point of the convex hull of bi,i = 1,..., n. Let /(O) be the straight line with directing vector y E K*, containing O. Denote by ei the orthogonal projection of ai onto /(O): ei = (aT(—e))(—e) = —kiy, where e = y/||y||, ki = = (aT(—e))/||y|| > 0. Then ai = bi + ei and
ai = bi — kiy, i = 1,..., n. (3)
Multiplying equalities (3) by ai > 0, adding them and taking into account that —y = a1a1 + ... + anan with ai > 0, we obtain (k — 1)y = a1b1 + + ... + anbn, wherejc = a1k1 + ... + ankn. Multiplying the latter equality by yT, we obtain (k — 1)yTy = 0. Since y = 0 then k = 1 and therefore a1b1 + ... + anbn = 0, where ai > 0, which means that O is the interior point of the convex hull of bi, i = 1,..., n.
Secondly, let us prove that any n — 1 vectors among b1,..., bn are linearly independent. Assume, without loss of generality, that b1,..., bn-1 are linearly dependent. Then there exist , j = 1,..., n — 1, such that 51b1 + ... + 5n-1bn-1 = 0 for ^ E R, + ... + ^n-1 = 0. Using (3) we obtain that (¿1 — ka1)a1 + ... + (¿n-1 — kan-1)an-1 — kanan = 0, where k = ¿1k1 + ... + ¿n-1kn-1. Since ai,i = 1,..,n, are linearly independent and ai > 0, we obtain that k = = ... = = 0. Therefore b1,... ,bn are linearly independent. The conclusion of lemma follows from Lemma 1. □
Definition 6. Let us call b1,..., bn a projective positive basis of nn-1(y) corresponding to a basis a1,..., an.
Remark. The similar result remains valid in the case —K C K*, y E K.
Denote ai(M) = f (M, ui), where ui C U are any constant vectors, i = 1,..., n + 1, and denote by K(M; j) a cone with vertex M, spanned
by vectors ai(M), i = 1,... ,n +1, i = j,
n+i
K(M; j) = {x £ Rn : x = ^ aiai(M), ai > 0}.
i=i,i=j
Assume that there exists a neighborhood D(O), a ball of center O, such that for any M £ D(O): {ai(M),.. .,an+i(M)} is the positive basis of Rn.
Lemma 5. If M £ D(O) then
1. K (M j) n K (M; s) = 0 if j = s, j,s = 1,...,n + 1;
2. Un+i K(M; j) = Rn;
3. nS K(M; j) = M.
The proof of this lemma, with slight modification, see in [3]. 3. Cylinder of Stabilization. In what follows, without loss of generality, we consider the normal stabilization of the origin. The following assumptions are hold valid throughout the paper
M £ D(O), O £ K(M; j), (4)
where D(O), K(M; j) are described above. Next, assume that
\\ai(M)\\ ^ A, i = 1,...,n +1 (5)
for some A > 0.
Let K*(M; j) be a conjugate open cone with respect to K(M; j) (2). Consider the case —K *(M; j) C K (M; j) for some j £ I = {1,... ,n +1}. Then for any y(M) £ K*(M; j)
aT(M)y(M) < 0, i = 1,...,n + 1, i = j.
In what follows, y(M) is a unit vector: \\y(M)\\ = 1. The finiteness of the set {ai(M), i = 1,... ,n +1} implies the existence of constant d(M) > 0 such that
aT(M)y(M) ^ —d(M), i =1,...,n +1, i = j.
Assume that there exist y £ K*(M; j) which does not depend on M (but not on j) and d > 0 which does not depend on M and j such that for each M £ D(O)
aT(M)y ^ —d,i =1,...,n + 1, i = j. (6)
For instance, the continuity of ai(M) implies that if {ai (O), i = 1,..., n+ 1} is the positive basis then such y and d > 0 exist. Naturally, such y and d are not unique. Denote by /(M; y) the straight line through M with directing vector y, and let nn-1(M; y) be a plane through M with normal vector y, i.e. nn-1(M; y) is a plane orthogonal to /(M; y). For i = j let ei(M) be a vector projection of ai(M) on /(M; y). Then
ai(M) = bi (y) + ei(M),
where {bi(y),i = 1,...,n + 1, i = j} is a projective positive basis of nn-1(M; y) corresponding to a basis {ai(M), i = 1,..., n +1, i = j}.
Lemma 6. If y satisfies (6) then
min ||ei(M)|| ^ d. ie/,i=j
Proof. Since
ei(M) = (aT(M)(—y))(—y),
then ||ei(M)|| = |aT(M)y| ^ d. □
Let us construct the stabilization cylinder C. Denote by nn-k (M; y) a plane through M of codimension k, k E {1,..., n — 1}, with normal vector y. Denote N = /(M; y) if nn-1(O; y). In what follows we need N = O. If N = O we can take another y E K*(M; j), which is not unique. Introduce the balls
Dn(O; N) = {x E Rn : ||x|| ^ ||ON||},
Dn-1(O; N) = Dn(O; N) f nn-1(O; y). Let [MN] be the segment with M and N as the endpoints. Definition 7. Let us call the set
C = Dn-1(O; N) x [MN] C D(O) (7)
cylinder of stabilization, with the bottom base Dn-1 = Dn-1(O; N) and the top base D-1 = C P| nn-1(M; y).
Remark. The cylinder of stabilization C is not unique, because y is not unique. Moreover, we assume that D(O) is so large that C C D(O).
Denote by -2, Sn-2 the spheres which are the boundaries of the balls Dn-1,Dn-1, respectively. Let nn-2 (M) be the (n — 2)-dimensional
plane through M, tangent to Sin-2. Obviously, nn-2(M) C nn-i(M; y). Let x(t,t0, M,u) be the trajectory of the system (1), corresponding to a control u, such that x(t0,t0, M, u) = M.
Lemma 7. There exists such constant control u = us £ U, s £ I, that the trajectory x(t,t0, M,us) of the system (1) enters C at the moment t = t0, i. e. M is the ingress point.
Proof. Let {bi(y)} be the projective basis corresponding to a,i(M), i = j, i = 1,...,n + 1, i.e. the positive basis of nn-i(M;y). Hence the vectors bi(y) cannot be directed from M into one half plane of nn-i(M; y) with the boundary nn-2(M) [1], then there exists bs(y) for some s £ I, which is transversal to Sin-2 and is directed into the interior of Dn-i. The corresponding us is the required control. Really, let nn-i([MN]) be the hyperplane tangent to C such that [MN] C nn-i([MN]). Denote by p(M) a vector orthogonal to nn-i([MN]) and such that bs(y)Tp(M) < 0. Multiplying scalarly the equality as(M) = bs(y) + es(M) by p(M) we obtain that as(M)Tp(M) < 0. Taking in account that a]:(M)y ^ —d, we obtain that the trajectory x(t, t0, M, us) enters the cylinder C when t = t0, i.e. at point M. □
Lemma 8. There exists the admissible (piecewise constant) control u £ U such that the trajectory x(t,t0, M,u), remaining in C, reaches the bottom base i of C in a finite time.
Proof. Let us note that the trajectory x(t,t0, M,us), where us is the control from Lemma 7, intersects any hyperplane parallel to nn-i(O;y) in the direction of the half-space containing the origin O. It follows from Lemma 6. If x(t, t0, M, us) reaches i, remaining in C, then this lemma is proved. Suppose that Mi is the next point (after M) at which the trajectory x(t,t0, M,us) hits the boundary of C before attaining Dbn-i. Then, similarly to the proof of Lemma 7, it can be proved that there exists a constant control usi £ U such that the trajectory x(t,t0, Mi,us) enters C at a point Mi. Let us switch the control from us to usi at the moment when the trajectory hits the boundary of C at Mi. Use this control till the trajectory hits boundary of C at some point M2 before attaining Dbn_ i. Then switch the control from usi to us2 with which the trajectory enters C at a point M2 and so on. We obtain the control sequence us,usi,us2,.... Lemma 6 implies that there exists usi £ U with which the trajectory reaches i in a finite time. □
Remark. In the case —K(M; j) C K*(M; j) the proof is analogous to the above one.
4. Method of normal stabilization.
Definition 8. A point M is t-reducible if there exists an admissible control u E U such that M can be steered by u, along a trajectory of the system (1), to any neighborhood of O, in a time less than t, and remains there.
Definition 9. The domain of t-reducibility of the origin is the set of all t-reducible points.
Theorem 1. Assume that conditions (4), (5), (6) are fulfilled. If
||OM|| <Td(1 — 6), (8)
where 6 E (0,1) is a constant determined in the proof, then M is a t—reducible point.
Proof. Let us construct the cylinder C as it was described in the previous section. Introduce the radius r0 = ||ON|| of the base and the height h0 = ||NM || of the cylinder C. As it was proved (Lemmas 7, 8) there exists a piecewise constant u E U such that the trajectory x(t,t0,M,u) reaches the base D-1 of C at some point M1 E D-1 in finite time t0: x(t0 + T0,t0,M,u) = M1. Then, considering M1 as the initial point, we can construct the cylinder C1, the construction of which is analogous to C. In addition, suppose that O E K(M1; j4), j E {1,...,n + 1}. The trajectory x(t,t0 + t0,M1,u) reaches the low base of C1 in some finite time t1. Introduce the radius r1 = ||ON1| of the base and the height h1 = ||N1M1| of the cylinder C1, where N1 is the point belonging to the bottom base of C1 and the vector N1M1 is orthogonal to the base of C1. Proceeding this procedure we obtain the sequences Tk, rk = ||ONk||, hfc = ||Nfc Mfc ||, O E K (Mfc; jfc), jfc E I, k = 0,1, 2,..., Mo = M, No = N, j0 = j. Let us prove that Mk ^ O, when k ^ ro. Since
OMk = Nfc Mfc + ONfc, (9)
where NkMk is a vector orthogonal to ONk, then ||OMk+1|| ^ ||ONk||. Thus, denoting dk = ||OMk || = \J+ r2, we obtain
dfc+i ^ rfc < dfc.
(10)
Multiplying (9) scaralry by ONk, we obtain
{OMT){ONk) = \\ONk\\2 > 0.
Hence
\ONk\\ = (OMT)(ONk)_ (OMT)(ONk)\\OMk\
\\ONk\\ \\ONk\\ • \\OMk\\ '
Thus rk = Skdk, where 5k = ((OMT)(ONk))/(\\ONk\\ • \\OMk\\) > 0. Vectors OMk ,ONk are not collinear. Therefore the Cauchy inequality implies that 5k < 1.
Since O E K(Mk,jk) then there exist such ak ^ 0, i = 1,...,n + 1, that
n+l
MkO = ^ a*ai(Mk), i = jk E I. (11)
ck'
i=l
Assume, on the contrary, that Mk ^ O as k — <x>. Condition (5) and nonnegativeness of a^ imply that there exists ik = jk such that akk ^ n for some n > 0, k = 0,1, 2,.... Multiplying (9) scaralry by (-yk) where yk is a unit normal vector corresponding to a cylinder Ck (like y corresponds to C), taking in account that ai (Mk)(-yk) ^ d (Lemma 6 and assumption (6)), we obtain
n+l n+l
MkO(-yk) = ^ akai(Mk)(-yk) ^ d^ a* > dn,
i=l i=l
or
dk cos /k ^ dn,
where cos¡3k = MkO(-yk)/\\MkO\\ is a cosine of an acute angle /k between MkO and -yk. Since the sequence {dk} is bounded then the latter inequality implies the existence of such /3 that 0 < 3k ^ 3 < n. Then
cos(^n - ^ cos(^n - 3^ = sin3.
Denote 5 = sin 3 < 1. Hence 5k ^ 5 < 1 for any k = 1,2,..., and rk ^ 5dk. Then (10) implies that dk+l ^ 5dk and therefore dk+l ^ 5k+ld0. Thus dk — 0 as k — x>, and Mk — O, which contradicts to assumption.
Now let us prove that for any t > 0 there exists U(O; < t) such that for any point M0 E U(O; < t): Tk < t. Since Tk is the time of moving from Mk-l to Mk, then, according to Lemma 6,
Tk ^ hk-i/d < dk-i/d ^ (do5k-l)/d.
Therefore,
<x <x
^ Tfc ^ (do/d) ^ 6k-1 = do/(d(1 — 6)). fc=1 fc=1
Hence if do = ||OMo|| < Td(1 — 6), then Tfc < t. Thus, Mo = M is a t-reducible point. Thus the point O is normally stabilized if
Mo E U(O; < t) = Ur (O) n U(O; d)
for r < Td(1 — 6), where Ur (O) is a ball of radius r centered at the origin.
If Mk = O for some k then we move a little bit from Mk to some point Mk and then use the procedure described above. □
Remark 1. The procedure described above implies Ck C UyMfco||(O).
Remark 2. The origin may be replaced by any point M with a positive basis {/(M, uj), i = 1,..., n +1}.
5. Examples. In this section we present three examples to illustrate the results discussed above. Example 1. Consider the system
x 1 = g1(x1 ,x2) + u — u3, x 2 = g2(x1,x2) + 1 — u2, (12)
u E { — 2, 0, 2},
where x1, x2 E R, g1(0, 0) = g2(0,0) = 0, g1, g2 are the Lipschitz functions. Consider /(x1,x2,u) = (/1,/2), where /1, /2 are the right-hand sides of equations (12). The vectors
/(0, 0, 0) = (0,1), /(0, 0, 2) = (—6, —3), /(0, 0, —2) = (6, —3)
form the positive basis of R2. Therefore, the origin O is normally locally stabilizable.
Example 2. Consider the Euler equations for an angular velocity of a rigid body
/1 u 1 = (/2 — /3)^2^3 + u1, /2^2 = (I3 — /1)^1^3 + u2,
/3^3 = (/1 — /2V1W2 + u3,
where u = uj(i) are the components of the angle velocity vector in a fixed coordinate system coinciding with the principal axes, / are the principal inertia moments, i = 1, 2, 3. The applied torque u = (u1,u2,u3) is
a piecewise constant control. If the admissible constant controls uj = = (u1j, u2j, u3j), j = 1, 2, 3, 4, form the positive basis of R3 then the origin O is normally locally stabilizable. For example,
u1 = (1, 0,0), u2 = (0,1, 0), u3 = (0, 0,1), u4 = (—1, —1, —1).
Example 3. Consider the system
x 1 = x1 + cos u, x2 = x1x2 + sinu, u E {n/3; 2n/3; 3n/2}.
Denote x = (x1,x2), O = (0, 0), /1(x) = /(x,n/3), /2(x) = /(x, 2n/3), /3(x) = /(x, 3n/2). The vectors /¿(O) form the positive basis of R2. Therefore, the origin O is normally locally stabilizable.
Let us construct the cylinder C. At first, we find the open set n, O E n C R2, such that for any x E n the vectors /¿(x), i = 1, 2, 3, form the positive basis. The conditions of collinearity for pairs of the vectors /¿(x), /j (x), i = j are as follows:
1. /1(x), /2(x) are collinear if and only if x1x2 = — ^;
2. /1(x), /3(x) are collinear if and only if x2 = — + 2 + v^;
3. /2 (x), /3 (x) are collinear if and only if x2 = — — Therefore, the continuity of /¿(x) implies that
n = |(x1 ,x2) : x1x2 > — ^, x2 < — — (2 + ^3),x2 > — + 2 + v^
2 x1 x1
Let us take, for example, the initial point Mo = (0.1; 0.1) E n and construct C for it. Since O E K(Mo; 1), we consider the hyperplane n(O;y) with normal vector y = (a, b), satisfying yT/i(O) < 0, i = 2, 3. Therefore, 0 < b < . Now, take, for example, (a, b) = (1, 0.25), d = 0.125 and find
the set n(d) D U(O; d), for which yT/(x) < 0, i = 2, 3:
11 ( V3\ 1 1 ( \ 1
x1 — 2 + ^x1 x2 + ^J < — 8, x1 + 4r1x2 — y < — 8,
which leads to II(d) = {x : 8x1 + 2x1x2 < 1}. Then we obtain the hyperplanes, containing the bases of C, e. g. n(Mo; y) : x2 + 4x1 = 0, n(O; y) : x2 + 4x1 = 0.5. Now we can construct the cylinder
C = {x : x2 ^ —4x1, x2 ^ —4x1 + 1, x2 ^ 1 x1 — 1, x2 ^ 1 x1 + -3}.
L 4 8 4 40 J
Instead of C, constructed above, we can take any cylinder (rectangular), containing in C, with basis on straight lines x2 + 4xl = 0, x2 + 4xl = 0.5 and the lateral sides parallel to x2 = 0.25xl — 0.125, x2 = 0.25xl + 0.075.
6. Conclusions. In this article the notion of normal stabilization of control systems is proposed. An admissible control is piece-wise constant. The constructive method of normal stabilization, based on a sequence of cylinders of stabilization, is presented. An extension of the presented approach to non-autonomous systems is the subject of future investigations.
Acknowledgment. This work was supported by RFBR (18-01-00249a).
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Received April 1, 2018.
In revised form, July 10, 2018.
Accepted August 14, 2018.
Published online January 5, 2019.
Institute of Applied Mathematical Research
of the Karelian Research Centre of the Russian Academy of Sciences 11, Pushkinskaya str., Petrozavodsk 185910, Russia; Petrozavodsk State University 33 Lenina pr., Petrozavodsk 185910, Russia E-mail: [email protected]
