DOI: 10.15393/j3.art.2014.2589 Issues of Analysis. Vol. 3(21), No. 2, 2014

A. N. KlRILLQY

ON THE STABILIZATION OF THE LINEAR HYBRID SYSTEM STRUCTURE1

Abstract. The linear control hybrid system, consisting of a finite set of subsystems (modes) having different dimensions, is considered. The moments of reset time are determined by some complementary function - evolutionary time. This function satisfies the special complementary ordinary differential equation. The mode stabilization problem is solved for some class of piece-wise linear controls. The method of stabilization relies on the set of invariant planes, the existence of which is due to the special form of the hybrid system.

Key words: stabilization, variable structure, hybrid system.. 2010 Mathematical Subject Classification: 93B12.

1. Introduction

The variable structure linear dynamical systems, i. e. the systems changing, while functioning, their dimensions and coefficient matrices, belong to the class of hybrid systems. The hybrid systems are dynamical systems demonstrating interacting of continuous and discrete dynamics [1], [2]. Their applications involve mathematical modeling in robotics, biochemistry, chemical technology, air and ground transportation and so on.

Roughly speaking, the main components of these systems are the partition of the phase space into a finite set of subspaces, the set of subsystems (modes) with continuous dynamics (each subsystem is defined in its subspace), and the set of mappings realizing, in some sense, a transfer from one subspace to another. The moments of reset time may be determined, as it was proposed in [3], [4], by the solutions of complementary system of differential equations. In this paper the problem of stabilizing of modes is

1 This work was supported by the Programm of strategic development of the PetrSU.

© Kirillov A. N., 2014

solved for the linear control hybrid system. The method of stabilization is developed. Otherwise speaking, this method permits to achieve and then to preserve the desired structure of the system.

2. The controllable evolutionary time

Let us consider the hybrid linear system which consists of a set of linear subsystems, or modes, Sk

Xk = Ak Xk, (1)

which dimensions k depend on time, k = k(t), k = 1, ...,n, and the coefficient matrices are piecewise constant. Suppose that a function y = y(t), evolutionary time (in our terminology), satisfying the following conditions

y = BT Xk + u, (2)

y(t) e Ak = [yk, yk+i ], (3)

is responsible for a changing of modes: the mode undergoes a changing when y(t) becomes equal to yk or yk+1 (the procedure of mode changing will be described below). Here Rk D XT = (x1, ...,xk) - a system state, T - the symbol of transposition, R 9 y(t) - an evolutionary time, Ak = = {aij} - a square constant matrix of order k, BT = (&1, ...,bk) e Rk -a constant vector, u - a control, yk - some given constants (thresholds), yk < yk+1, y1 = , yn+1 = We say that the system S, described by equations (1)-(3), is in the state Sk.

It is essential that the system (1), (2) has invariant sets [5], namely the integral planes nk = {(x1 ,...,xk ,y) : a1 x1 + ... + ak Xk + Pk y = c} with constant c, where (a1, ...,ak, Pk) is a nonzero solution of the system

a1ia1 + ... + aki ak + (bi + Pi )Pk =0, i = 1,...,k.

The procedure of mode changing: when the trajectory of the system (1), (2) attains the plane y = yk at some moment of time tk, the transfer from the state Sk to the state Sk-1 occurs, while hitting the plane y = = yk+1 at some moment of time tk+1 means the transfer from Sk to Sk+1. The transfer maps pk, k-1 (Pk,k+1) from Sk to Sk-1 (from Sk to Sk+1) have the following form

Pk,k-1 : ^k ^ C(k - 1, k)Zk + Ek-i(-£),

(4)

where Zk = (xi ,...,xk ,yk )T, Ek-i(-e) = (0,..., 0, -e)T, (-e) is at k-th place, with

0 < e < mink (yk+i - yk ),k = 1,...,n - 1,

C(k,k - 1) - the k x (k +1) constant matrix with elements cij, where Ck,k+i = 1, Ck,j =0, j = 1,..., k, Ci,k+i =0, i = 1, ...,k - 1,

Pk,k+i : Zk ^ D(k, k + 1)Zk + Ek+2(e), (5)

where D(k, k + 1) - the (k + 2) x (k + 1) constant matrix with elements dij, dk+2,k+i = 1, dk+2j =0, j = 1,..., k, di,k+i = 0, i = 1,..., k +1. We suppose that pk,k-i(Zk), (pk,k+i(Zk)) are the initial data for the states Sk (Sk+i) and the switching occurs instantaneously.

Remark 1. The transfer matrices C(k,k - 1), D(k, k +1) are supposed to be constant. It is not essential for the obtained results which remain valid for the elements of C(k, k - 1), D(k, k + 1) dependent on time or k.

Let us formulate the problem: we need to construct a control u which transfers the system S from a state Sk to a state Sm, k, m e {1,..., n}, k = m, in a finite time and then preserves the terminal state Sm.

Assume that the admissible control is u = pixi + ... + Pk(t)Xk(t) = = Pr[Xk, where pi = pi (t) are the piecewise constant coefficients. Here k(t) is the integer valued function: k(t) e {1,...,n}. Thus u belongs to a class of the piecewise linear functions of Xk(t) •

Remark 2. This problem differs from the traditional controllability or stabilizability problems. The proposed system is described by a continuous state (Xk ,y) as well as by a discrete state k, and we control the latter. It may be said that a discrete state represents the structure of the system. Thus it is apposite to designate the problem formulated above as the problem of the structure stabilization.

Below we propose the method of the structure stabilization. Assume that for k = 1, ...,n the systems (1) are asymptotically stable. The method consists of three steps. Consider the transfer Sk ^ Sk+i.

1. Let tk(Xko,5) be such moment of time that ||X(t, Xko)|| < 8 (|| • || is an euclidean norm) for t > tk(Xk0,5) and some 5 > 0, where X(t,Xk0) is the trajectory of the system (1), satisfying the initial condition X(0, Xk0) = Xk0. It is clear that tk (Xk0, 5) exists due to the asymptotic stability of (1). Let us take pi = -bi, i = 1, ...,k, while t < tk(Xk0, 5).

As a result the trajectory, remaining in Rk x Ak, gets into the cylinder Us(0) = {Xk E Rk : \ \Xk\\ < S] and remains there.

2. At the moment t = tk(Xko, S) we take u = pixi + ... + pk(t)xk(t) = = PTXk, where pi,i = 1...,k, satisfy the following inequality

kk

J^Pj " S Ak-i),i(j)xi0 > j=i i=i

kk

> detAk(yk+i - yo) bj A(k-i),i(j)xio,

j=i i=i

if detAk > 0 (for detAk < 0 the sign of inequality is opposite), where Ai(k-i),i(j) is a cofactor of the j-th row and the i-th column elements of Aki - the matrix obtained from Ak via substituting of it's i-th column by -(bi + pi,bk + Pk)T• The existence of the inequality solution was proved in [6].

The main idea of this step is that the vector (pi,...,pk), satisfying inequality, given above, provides such position of the integral plane nk C C Rk x R, for which nk intersects the Y axis with y > yk+i [6]. Then, in view of asymptotical stability of the equilibrium of the system (1) and taking into account that \\X(t,Xk0)\\ < S for t > tk(Xk0,S) , we obtain that the trajectory intersects the plane y = yk+i in a finite time. As a result we obtain the transition to the state Sk+i.

3. Then let us put pi = -bi,i = 1, ...,k + 1, at the moment of time when y = yk+i + £, and the system preserves its state Sk+i.

Using the presented procedure, providing the transfer Sk ^ Sk+i, we can obtain the transfer from any state Sk to any state Sm, k,m E E {1,..., n], k = m.

The method of structure stabilization essentially relies on the asymptotic stability of the system (1). It means that each matrix Ak has all eigenvalues with negative real parts. Let us consider the general hyperbolic case, e. g. the spectrum of Ak has not common points with the imaginary axis. It is known that in this case we have the direct decomposition Rk = Es ® Eu, where Es, Eu are such invariant subspaces of the operator Lk, corresponding to Ak, that the eigenvalues of the restrictions Lk\Es have negative real parts and the eigenvalues of Lk\Eu have positive real parts. If Xk0 E Es then the method of structure stabilization remains the same, as it was presented above.

Therefore we consider the case Xk0 £ Es. Then there exists such component Xj (t) of X(t,Xk0) that Xj (t) — to as t — +to. Thus we can take pj in the following form: pj = -bj + Cj.sign Xj, where Cj > 0 -some constant, while pi = —bi for i = i = j. As a result (2)

implies that the trajectory attains the plane y = yk+i in a finite time. Then, similarly to the third step of the method presented above, we put pi = —bi, i = 1, ...,k + 1. Thus we obtain the transfer Sk — Sk+1.

3. The controllable continuous state

Now instead of (1), (2) we consider the controllable system *k

Xk = Ak Xk + Gk vk, y = BT Xk, (6)

together with (3), where vk £ Rl - a control, Gk is a constant matrix. Then we can use the classical method of linear stabilization [7] in order to make the origin Xk = 0 asymptotically stable. It is sufficient to find such matrix Mk that the system Xk = (Ak + GkMk)Xk = NkXk be asymptotically stable, where

vk = Mk Xk.

Without loss of generality, but keeping in view the method of structural stabilization, we consider the case for which vk £ R, Gk £ Rk, Mk £ Rk.

If ai xi + ... + ak Xk + fik y = ai xio + ... + ak Xko + yo is the integral plane of the system

* * T

Xk = Nk Xk, y = Bk Xk,

then, analogously to [6], it is easy to show that aT = (ai, ...,ak) satisfies the linear system

NT a = —fa Bk, (7)

where Nk = Ak + GkMT. Let y be the coordinate of the intersection

point of an integral plane with the axis Y and assume that

1k

y = -T- ^2aiXi0 >yk+i. (8)

It is well known, that if

rankLk = rank(Gk Ak Gk A2k Gk ...Akk~i Gk) = k, (9)

then we can find such Mk that the matrix Nk has any desired eigenvalues [7]. Let all eigenvalues be with the negative real parts, which provides the asymptotic stability of the system Xk = NkXk. In addition [7]

Mk = dT P-'L-1, (10)

where Pk = (Ikp + ek,I%p + ek-i,...,Ijkp + ei), p = —L-1AJkGk, Ik = = (0, e',..., ek-1), 0 E Rk is the zero vector, ei E Rk - the vector with zero components except the i-th component which equals 1. The components di of d E Rk equal di = pi-qi, i = 0,1,..., k — 1, where pi - the components of p, qi - the coefficients of the polynomial Qk (A) which roots equal to the eigenvalues of the matrix Nk.

The result, presented below, provides only local stabilization. Assume that Xko E Ug (0), where Ug (0) is the cylinder from section 2. Then for vk, constructed above, the trajectory X(t,Xk0) remains in Ug(0). The previous arguments imply the following theorem.

Theorem 1. Let Xk0 E Ug(0), Xk0 = 0 and assumptions (7)-(10) are valid. Then the control Vk = Mk Xk, stabilizing the system Xk = Ak Xk + +GkVk, provides the transfer of the system (6), (3) from Sk to Sk+1.

Remark 3. It is naturally to formulate the following problem: for which maximal set of Xk0 the presented theorem is true?

4. Conclusion

The method of the linear hybrid system structure stabilization is proposed in this paper. The feedback piecewise linear control of the discrete state is constructed.

The hybrid systems, considered in this paper, have different applications in the problems of mathematical modeling, particularly in the economic dynamics. Thus, in [8] the linear model of the economic growth dynamics with the structural variations was proposed. The control growth problem was solved for the industrial group, consisting of several enterprises, the number of which was not constant, but depended on the economic efficiency and changed in time.

Future investigations will be dedicated to some extensions of the proposed methods of the structure stabilization, especially the constraints on the admissible controls and phase variables, specific for applications, will be taken into account.

References

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[2] Handbook of hybrid systems control. Theory, tools, applications. Ed. by J. Lunze. Cambridge University Press., 2009, 565 p.

[3] Roger W. Brockett. Hybrid models for motion control systems. Essays in Control: Perspectives in the Theory and its Applications. Ed. by H. L. Trentelman and J. C. Willems, Boston, Birkhauser, 1993, pp. 29-53.

[4] Kirillov A. N. The systems with variable fhase space in modeling of biological wastewater treatment processes. The Russian conference "Mathematical problems of ecology". The conference abstracts. Dushanbe, 1991, p. 44.

[5] Kirillov A. N. The control of multistage technological processes. Vestnik Sankt-Peterburgskogo Universiteta. Seriya 10. Prikladnaya Matemati-ka. Informatika. Protsessy Upravleniya, 2006, vol. 4, pp. 127-131.

[6] Galakhova M. E., Kirillov A. N. The linear variable structure system control. Proceedings of the Karelian Research Centre of the RAS. No. 5. Mathematical modeling and information technologies, 2012, vol.3, pp. 18-21.

[7] Smirnov E. Y. The stabilization of programmed movements. Saint-Petersburg, SPbSU, 1997, 308 p.

[8] Kirillov A. N. The method of dynamical decomposition in modeling of variable structure systems. Information and Control Systems, 2009, no. 1, pp. 20-24.

Received August 20, 2014.

Institute of Applied Mathematical Research,

Karelian Research Centre, Russian Academy of Sciences,

Pushkinskaya St., 11, 185910 Petrozavodsk, Russia.

Petrozavodsk State University,

Lenin Avenue, 33, 185910 Petrozavodsk, Russia.

E-mail: kirillov@krc.karelia.ru