Numerical methods of stabilizer construction via guidance control

Research Article

Received 1 January 2011 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2545MOS subject classification: 93C10; 93C95

Numerical methods of stabilizer constructionvia guidance control

Francisco Mirandaa,b*†

Communicated by A. Kunoth

This work concerns guidance stabilization of non-autonomous control systems. Global stabilization problem is usuallyquite complex and difficult to solve. To overcome this difficulty, guidance control is used. A guidance stabilizer uses a tra-jectory of a globally asymptotically stable auxiliary system as a guide. A local stabilizer keeps the trajectory of the systemin a neighborhood of a solution of the auxiliary system. In this way, the trajectory of the system tends to the equilibriumposition. The main idea of this method is to solve the global stabilization problem by applying local stabilization methods.The presented procedure also yields additional possibilities for the design of a stabilizer that eliminates the peak effect,that is, the large deviation of the solutions from the equilibrium position at the beginning of the stabilization process.This effect represents a serious obstacle to the design of cascade control systems and to guidance stabilization. The opti-mal control problem used in this paper eliminates this effect that we have when we apply known construction methodsof local stabilizers to obtain a high speed of damping of the control systems trajectories. According to this approach andusing �-strategies introduced by Pontryagin in the frame of differential games theory, the stabilizing control is constructedas a function of time defined in a small time interval and not as a feedback. An application to a mechanical stabilizationproblem is provided here. Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: non-autonomous control systems; guidance stabilization; global stabilization and control; global guidance stabilizers

1. Introduction

The main purpose of this work was to construct a control that solves the global stabilization problem of nonlinear systems. Differentaspects of the global stabilization problem were discussed first in [1–15] and in the works referenced therein. The first result, whichstates that the existence of a smooth control-Lyapunov function, that is, control-Lyapunov function of class C1, implies smooth stabi-lization was obtained by Artstein [1] (Theorem 9.3 [2]). The Artstein theorem was obtained as a particular case of more general resultsinvolving relaxed controls. Unfortunately, Artstein’s proof is not constructive. Thus, a question arises whether it is possible to write anexplicit formula for a stabilizing feedback, under the assumptions of Theorem 9.3. Looking for explicit formulas may also help in fac-ing another unclear aspect of Artstein’s theorem, namely, to what extent is it possible to construct a stabilizing feedback with moredesirable smoothness properties. The work of Tsinias [14, 15] focus on the construction of an explicit formula to smooth feedback.An interesting result is obtained by Sontag [10]. Both Sontag’s formula and the second of Tsinias’ formulas can be extended to multi-input systems. In [8], we can see a historical perspective of constructive nonlinear control and numerous references to works relatedto this subject. More recently, we have [16–29] and the works referenced therein. The problem of output-feedback stabilizing con-trol design for a class of nonlinear systems with unmeasured states dependent growth is investigated in [17–19, 21, 23, 28, 29]. In theworks [16, 20, 22] and [24–27], different approaches to prove global stabilization problem of nonlinear systems are used and differenttechniques to stabilizing controls design are presented. This paper suggests an alternative way to construct a stabilizer to global sta-bilization problem. The method is based on guidance control [4], a more general control procedure. Considering an auxiliary globalstabilization problem that we know has an analytic solution, the method consists of using a trajectory of this globally asymptoticallystable auxiliary system as a guide. We construct a local stabilizer that keeps the trajectory of the system in a neighborhood of a solutionof the auxiliary system. In this way, the trajectory of the system tends to the equilibrium position. According to this approach and using�-strategies introduced by Pontryagin in the frame of differential games theory [30], the stabilizing control is constructed as a function

aCenter for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, PortugalbSchool of Technology and Management, Polytechnic Institute of Viana do Castelo, Avenida do Atlântico, 4900-348 Viana do Castelo, Portugal*Correspondence to: Francisco Miranda, Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Viana do Castelo, Avenida do Atlântico, 4900-348 Viana do

Castelo, Portugal.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012

F. MIRANDA

of time defined in a small time interval and not as a feedback. From the practical point of view, �-strategy is similar to a stabilizer, whichdepends on time and position only, because it is usually implemented as a generator of piecewise constant controls. The main contri-bution of this method is to solve the global stabilization problem by applying local stabilization methods that avoid difficulties, whichwe usually have in this problem. This approach overcome serious mathematical difficulties that we have in the proof of the asymptoticstability of the equilibrium position, mainly when we have to use the Krasovskii–LaSalle theorem [31, 32], and eliminates the difficultiesto find a Lyapunov function and a feedback such that the Lyapunov function tends to zero. Then, it is more effective in applications.The numerical implementation of this method is based on the construction of multistep reachability sets [4]. This procedure also yieldsadditional possibilities for the design of a stabilizer that eliminates the peak effect, that is, the large deviation of the solutions fromthe equilibrium position at the beginning of the stabilization process (studies of this effect are presented in [11, 33, 34]). The optimalcontrol problem used in this paper eliminates this effect that we have when we apply known construction methods of local stabilizersto obtain a high speed of damping of the control systems trajectories. The large deviations of control systems trajectories from theequilibrium position during the stabilization process represent a serious obstacle to the design of cascade control systems [11] and toguidance stabilization [4].

Throughout this paper, we denote by R the set of real numbers and by Rn the usual n-dimensional space of vectors .x1, : : : , xn/ , xi 2 R,i D 1, n. The Jacobian and Hessian matrices of the function f will be denoted by rxf and r2

xxf , respectively. The inner product of two

vectors x and y in Rn is expressed by hx, yi D x1y1C : : :Cxnyn. The norm of a vector x 2 Rn is defined by jxj D hx, xi1=2. We denote by Bn

the unit ball in Rn : Bn D fx 2 Rn : jxj � 1g. The space of absolutely continuous functions x.�/ : Œa, b�! Rn is denoted by AC .Œa, b� , Rn/.By SŒa,b� .F, x0/, we denote the set of solutions to the differential inclusion Px.t/ 2 F .t, x.t// , t 2 Œa, b�, with the initial condition x0, thatis, the set of functions x.�/ 2 AC .Œa, b� , Rn/with x.a/D x0 satisfying the differential inclusion almost everywhere. The interior of a set Ais denoted by intA. The support function of a set A is defined as S .x, A/D sup fhx, ai : a 2 Ag .

Let f : R � Rn � U ! Rn be a sufficiently smooth function, that is, there exist all derivatives needed in our considerations, and U anadmissible controls set. Consider a nonlinear control system

Px.t/D f .t, x.t/, u .x.t/// , u .x.t// 2 U, (1)

and g and G functions such that f .t, x.t/, u .x.t///D g .t, x.t//CG .t, x.t// u .x.t//. Let Qu 2 U be such that g .t, Qx/CG .t, Qx/ QuD 0 for all t.The global stabilization problem is to find a map u : Rn ! U such that u .Qx/ D Qu and the equilibrium position x D Qx of the differentialequation Px.t/D g .t, x.t//C G .t, x.t// u .x.t// is globally asymptotically stable. If there is a solution to the problem, we say that system(1) is globally stabilizable. This problem is hard to solve. One of the ways to avoid difficulties is to use a guidance control [4], a moregeneral control procedure. It is defined for system (1) if we have three maps u : Rn � Rm ! U, h : R � Rn � Rm ! Rm and � : Rn ! Rm.The trajectories of system (1), under the guidance control, are defined as solutions to the Cauchy problem:

Px.t/D g .t, x.t//C G .t, x.t// u .x.t/, y.t// , (2)

Py.t/D h .t, x.t/, y.t// , (3)

x .t0/D x0, y .t0/D � .x0/ . (4)

The trajectory y.t/ is considered to be a ‘guide’, and we choose a control u .x.t/, y.t// so that x.t/ tracks y.t/. System (1) will be globallystabilizable by guidance control if nDm, and there exist maps u, h, and � such that u .Qx, Qx/D Qu, h .t, Qx, Qx/D 0, the equilibrium position.x, y/D .Qx, Qx/ of system (2)–(3) is asymptotically stable and any solution .x.�/, y.�// to Cauchy problem (2)–(4) satisfies .x.t/, y.t//! .Qx, Qx/as t!1, for all x0 2 Rn.

To construct the guide, we consider an auxiliary global stabilization problem Py.t/D g .t, y.t//Cw, w 2 �Bn, � > 0. Suppose that weknow an analytic solution to this problem, that is, there exists w� : Rn ! �Bn such that w� .Qx/D Qw and the equilibrium position y D Qxof the differential equation

Py.t/D g .t, y.t//Cw� .y.t// (5)

is globally asymptotically stable. We use a trajectory of system (5) as a guide. If � > 0 is sufficiently small and system (1) satisfies con-ditions of stability or controllability, we can construct a local stabilizer keeping a trajectory of (1) in a neighborhood of the solution ofdifferential equation (5). Following the solution of (5), the trajectory of system (1) tends to the equilibrium position Qx. More precisely,we construct a map u : Rn � Rn! U such that any solution .x.�/, y.�// of the Cauchy problem

Px.t/D g .t, x.t//C G .t, x.t// u .x.t/, y.t// ,Py.t/D g .t, y.t//Cw� .y.t// ,

x .t0/D x0, y .t0/D y0

satisfies jx.t/� y.t/j < �, where � > 0 is sufficiently small, and .x.t/, y.t//! .Qx, Qx/ as t ! 1. Therefore, we obtain a global guidancestabilizer for nonlinear control system (1).

Let z.t/D x.t/�y.t/, u.t/D u .x.t/, y.t// and w.t/D w� .y.t//. If z.t/ is sufficiently small, then z.t/ is a trajectory of the control system

Pz.t/D A.t/z.t/C B.t/u.t/�w.t/C R1 .t, z.t// u.t/C R2 .t, z.t// , u.t/ 2 U, (6)


F. MIRANDA

where A.t/ D rxg .t, y.t// , B.t/ D G .t, y.t// , jR1 .t, z/j D O .jzj/ , z ! 0, and jR2 .t, z/j D O.jzj2/, z ! 0 (see, e.g., [4] or [35]).To stabilize system (6) to zero equilibrium position, the local methods developed for linear control systems can be applied. Then,jz.t/j D jx.t/� y.t/j ! 0, t ! 1. This implies x.t/ ! y.t/, and we have global stabilization of system (1) to globally asymptoti-cally stable equilibrium position Qx. Apparently, the guide tracking property could be achieved through a stabilizer for system (6) thatwould drive z.t/ to the equilibrium position at a very fast rate. But this is not a good idea because of the peak effect, that is, the largedeviation of the solutions from the equilibrium position at the beginning of the stabilization process. Thus, z.t/ is large on initial intervalof time. Then, approximation (6) cannot be used, because Equation (6) is not ‘almost’ linear in z. Studies about this phenomenon canbe seen in [11, 33, 34]. For this reason, we construct the control u.t/ as a solution to the following minimization problem:

maxt2Œt0Ck� ,t0C.kC1/��

jz.t/j �!min,

Pz.t/D A.t/z.t/C B.t/u.t/�w.t/, t 2 Œt0C k� , t0C .kC 1/ �� ,z .t0C k�/D zk ,

z .t0C .kC 1/ �/ 2 ˇ jzkj Bn,

(7)

where 0 < ˇ < 1, � > 0 and k D 0, 1, : : :. In this way, we generate an �-strategy introduced by Pontryagin in the frame of differential

games theory [30], that is, a sequence of controls u defined on the intervalsh

t0CPk

iD1 �i , t0CPkC1

iD1 �i

i, where �i D � . These controls

depend on t and z .t0C k�/ , kD 0, 1, : : :. With this optimal control problem, we gain advantage in relation to other approaches of sta-bilizers construction because we obtain a stabilizer that stabilizes the linearized system to the zero equilibrium point relatively quicklyand eliminating the peak effect. To illustrate this advantage, consider the movement of a point of mass one along the axis x subject toa force u, described by the system �

PxPv

�D A

�xv

�C bu .x, v/ , u .x, v/ 2 R, (8)

where A D

�0 10 0

�, b D

�01

�and v is the velocity of the point. As the vectors b, Ab are linearly independent, then by the pole

assignment theorem [36], there exists a linear feedback

u .x, v/D hc, .x, v/i , c 2 R2,

such that the equilibrium position .x, v/D .0, 0/ of the linear differential equation�PxPv

�D A

�xv

�C b hc, .x, v/i

is asymptotically stable. Moreover, one can generate a linear system with any given set of eigenvalues f�C,��g, where �˙ D �˛˙ iˇwith ˛, ˇ > 0. Therefore, choosing an appropriate linear feedback, one can obtain a closed-loop system with an arbitrary given damp-ing speed. In other words, with the help of a linear control, the system can be stabilized to the zero equilibrium point arbitrarily fast.Choosing different feedbacks u .x, v/D� .�C � ��/ xC .�CC ��/ v, we can obtain the following linear system�

PxPv

�D

�0 1

��C � �� CC ��

��xv

�.

Two trajectories of the closed-loop system are shown by thin lines in Figure 1. The bold lines represent the trajectories of system (8) toa stabilizer obtained by optimal control problem (7). The trajectory shown by thin line in Figure 1(a) corresponds to ˛ D 5 and ˇ D 0,that is, �˙ D�5 (j�˙j � 1). The initial point is .x0, v0/D .1, 0/. Both trajectories in that graphic tend to the zero equilibrium point very

-2

-1.5

-1

-0.5

0

0.5

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

v(t)

x(t)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

v(t)

x(t)

(a) (b)

Figure 1. Trajectories of system (8) to different stabilizers. Thin and bold lines represent the trajectories that correspond to linear feedback ((a) �˙ D �5, (b)

�˙ D�0.2) and stabilizer obtained by (7), respectively.


F. MIRANDA

fast, but the trajectory corresponds to ˛ D 5 and ˇ D 0 has a peak on initial interval of time unlike of the trajectory derived by (7). Thesame phenomenon happens if we choose a slow damping, ˛ D 0.2 and ˇ D 0, that is, �˙ D �0.2 (j�˙j � 1; thin line in Figure 1(b)).The initial point is .x0, v0/D .0, 1/. Moreover, the damping of the trajectory that corresponds to stabilizer obtained by (7) is faster thanthe trajectory corresponds to ˛ D 0.2 and ˇ D 0.

In the following section, we find an estimate for the solution of problem (7) and we show how one can choose the parameters ofthe stabilizer.

2. Stabilizer construction

Consider control system (1) where U� Rk is a compact convex set. Let Qu 2 intU and U� !Bk ,! > 0. Suppose that g .t, Qx/CG .t, Qx/ Qu 0.Moreover, for all � > 0, there exists w� : Rn ! �Bn such that w� .Qx/ D Qw and all trajectories of system (5) tend to Qx when t ! 1.Suppose that jw� .y/j � � for all y.

Let x.�/ and y.�/ be trajectories of systems (1) and (5), respectively. Put z.t/ D x.t/ � y.t/. The existence of a solution to problem (7)for small � > 0 is guaranteed by the following local controllability condition. We denote by ˆ.t1, t2/ the fundamental matrix of theequation Pz.t/D A.t/z.t/. Suppose that there exists a continuous function �� : R� R! R with positive values such that

�� .t1, t2/

Z t2

t1

ˆ.s, t2/ Bnds�

Z t2

t1

ˆ.s, t2/ B.s/Uds, t1 ¤ t2. (9)

Condition (9) implies that the attainable set of the system

Pz.t/D A.t/z.t/C B.t/u.t/, u.t/ 2 U, (10)

contains a ball. This means that starting from zero, we can achieve any point in a neighborhood of zero. Then, system (10) is locallycontrollable in the interval Œt1, t2�. If jzkj, k D 0, 1, : : :, is small, we can achieve points in the ball of the condition of (7), that is, aszk C � .A.s/zk C hBn/ zk C � .h=2/ Bn, h > 0, holds when jzkj is small and s 2 Œt, tC ��, t > 0, for � small, the system can achieve theterminal set ˇ jzkj Bn, with ˇ < 1. Then, there exists a solution to problem (7).

Assume that the guide exists for any � > 0, that is, � can be as small as necessary. The following theorem contains sufficientconditions guaranteeing inclusion (9).

Theorem 1Let y0.�/ be a trajectory of the equation Py0.t/D g .t, y0.t// , t 2 Œa, b�. Set A0.t/D rxg .t, y0.t// and B0.t/D G .t, y0.t//. Letˆ0.t/ be thefundamental matrix of the equation Pz.t/D A0.t/z.t/. If there exists a continuous function �0 : R� R! R with positive values such that

�0 .t1, t2/

Z t2

t1

ˆ0 .s, t2/ Bnds�

Z t2

t1

ˆ0 .s, t2/ B.s/Uds, (11)

for all t1, t2 2 Œa, b� , t1 ¤ t2, then (9) holds for all t1, t2 2 Œa, b�, whenever � is sufficiently small.

ProofFor any � > 0 there exists ı.�/ > 0 such that for all 0 < � < ı.�/ we have supt2Œa,b� jy.t/� y0.t/j < �, where y.�/ is solution of system(5) with y.a/ D y0.a/. Also for any > 0 there exists �./ > 0 such that we have supt1,t22Œa,b� jˆ.t1, t2/�ˆ0 .t1, t2/j < wheneversupt2Œa,b� jy.t/� y0.t/j< �. Thus for > 0 sufficiently small we see that (11) implies (9) for all � 2 �0, ı .�.//Œ and t1, t2 2 Œa, b�. �

Therefore, we can prove the existence of a solution to problem (7) verifying condition (11) that does not depend on � . An expressionfor �0 .t1, t2/ can be written using support function:

�0 .t1, t2/D minf :j jD1g

R t2t1

S . ,ˆ0 .s, t2/ B.s/U/dsR t2t1

S . ,ˆ0 .s, t2/ Bn/ds

and if � � 1, we can take �� .t1, t2/D �0 .t1, t2/.The following result gives us an estimate for the solution of problem (7). Without loss of generality, suppose that t0 D 0.

Theorem 2Set � > 0, � > 0, D supt�0 jA.t/j and � D supt�0

u2UjB.t/u.t/j. The trajectory z.t/ derived by optimal control problem (7) satisfies the

inequalities

maxt2Œk� ,.kC1/��

jz.t/j � e��=2 jzkj C .�C �/e��=2 � 1

, kD 0, 1, : : : , (12)

and

jz0j< .� � �/e�˛ � 1

˛ .e�˛ � ˇ/,

where � > � , ˇ < 1 and ˛ D supt�0 supfp:jpjD1g hA.t/p, pi<C1.


F. MIRANDA

To prove this theorem, we need the following lemma. Consider the differential inclusion

P�.t/ 2 F .t, �.t// , t 2 Œk� , .kC 1/ �� ,

where

F .t, �.t//D A.t/�.t/C �Bn �w.t/ (13)

and jw.t/j< � .

Lemma 1Let q.�/ 2 AC .Œk� , .kC 1/ �� , Rn/. If .�/ 2 AC .Œk� , .kC 1/ �� , R/ satisfies .t/� 0 and

supfp:jpjD1g

.hPq.t/, pi � S .p, F .t, q.t/� .t/p///� P .t/ (14)

for almost all t 2 Œk� , .kC 1/ ��, then for any �0 D �.0/ 2 q.0/C .0/Bn, there exists �.�/ 2 SŒk� ,.kC1/�� .F, �0/ such that �.t/ 2 q.t/C .t/Bn,t 2 Œk� , .kC 1/ ��.

We can see the proof of this result in [37].

Proof of Theorem 2Assume that q.�/ 0. Therefore, from (13) and (14), we obtain

supfp:jpjD1g

.�S .p,�A.t/ .t/pC �Bn �w.t///� P .t/, t 2 Œk� , .kC 1/ �� .

As we have jw.t/j< � , by definition of support function, we obtain

supfp:jpjD1g

hA.t/p, pi .t/� � C � � P .t/, t 2 Œk� , .kC 1/ �� , (15)

and for any �0 2 .0/Bn, there exists �.�/ 2 SŒk� ,.kC1/�� .F, �0/ such that j�.t/j � .t/, t 2 Œk� , .kC 1/ ��. Let ˛ D

supt�0 supfp:jpjD1g hA.t/p, pi<C1. Then,

.t/D e.t�k�/˛ k �

Z t

k�e.t�s/˛ .� � �/dsD e.t�k�/˛ k � .� � �/

e.t�k�/˛ � 1

˛

satisfies (15) and .k�/D k . If � > � and ˇ < 1, then we have

e�˛ k � .� � �/e�˛ � 1

˛< ˇ k

whenever k is sufficiently small. Thus, we have

k < .� � �/e�˛ � 1

˛ .e�˛ � ˇ/(16)

and ..kC 1/ �/ < ˇ k .Let k D jzkj. The function .t/ has to be positive. Therefore, we have

..kC 1/ �/ > 0, k > .� � �/e�˛ � 1

˛e�˛D �0.

From inclusion (9), we know that there exists an admissible trajectory z.�/ for problem (7). Because

d

dtjz.t/j D

hz.t/, Pz.t/i

jz.t/j� jA.t/j jz.t/j C jB.t/u.t/j C jw.t/j � jz.t/j C�C � ,

where D supt�0 jA.t/j and �D supt�0u2UjB.t/u.t/j, we have

jz.t/j � e.t�k�/� jzkj C .�C �/e.t�k�/� � 1

.

So, we obtain the inequality

maxt2Œk� ,.kC1/��

jz.t/j � e��=2 jzkj C .�C �/e��=2 � 1

. (17)


F. MIRANDA

From (16), we have

jz0j< .� � �/e�˛ � 1

˛ .e�˛ � ˇ/.

�In the following, jR1 .t, z.t// u.t/j � c1 jz.t/j ju.t/j and jR2 .t, z.t//j � c2 jz.t/j

2 where c1 D sup jrxG .t, c/j and c2D 1=2 supˇr2

xxg .t, c/ˇ

hold. Let Ou.�/ be the optimal control for problem (7) and Oz.�/ be the respective trajectory. By the following result, we obtain conditionsguaranteeing that this optimal control drives the solutions of system (6) to zero.

Theorem 3The optimal control Ou.�/ drives the solutions of system (6) to zero if � is sufficiently small.

ProofConsider the solution of the Cauchy problem

Pz.t/D A.t/z.t/C B.t/Ou.t/�w.t/C R1 .t, z.t// Ou.t/C R2 .t, z.t// , t 2 Œk� , .kC 1/ �� ,z .k�/D zk .

Obviously, Nz.t/D z.t/� Oz.t/ satisfies

PNz.t/D A.t/Nz.t/C R1 .t, z.t// Ou.t/C R2 .t, z.t// , t 2 Œk� , .kC 1/ �� ,Nz .k�/D 0.

Suppose that R1 and R2 are Lipschitzians with constant l. Then, we have

PNz.t/D A.t/Nz.t/C R1�

t, Oz.t/�Ou.t/C R2

�t, Oz.t/

�C �.t/,

Nz .k�/D 0,

where

j�.t/j Dˇ�

R1 .t, z.t//� R1�

t, Oz.t/��Ou.t/C R2 .t, z.t//� R2

�t, Oz.t/

�ˇ� l jNz.t/j

�ˇOu.t/

ˇC 1

�� l jNz.t/j .! C 1/ .

Thus, we have

ddt jNz.t/j � jNz.t/j C c1!

ˇOz.t/

ˇC c2

ˇOz.t/

ˇ2C l .! C 1/ jNz.t/j

D .C l .! C 1// jNz.t/j C c1!ˇOz.t/

ˇC c2

ˇOz.t/

ˇ2.

Using the Gronwall inequality, we obtain

jNz.t/j � eR t

k� .�Cl.!C1//ds jNz .k�/j CR t

k� eR t

k� .�Cl.!C1//dv�R s

k� .�Cl.!C1//dv�

c1!ˇOz.s/

ˇC c2

ˇOz.s/

ˇ2�ds

DR t

k� e.�Cl.!C1//.t�s/�

c1!ˇOz.s/

ˇC c2

ˇOz.s/

ˇ2�ds, t 2 Œk� , .kC 1/ �� .

Therefore, we have

jNz.t/j �

Z .kC1/�

k�e.�Cl.!C1//..kC1/��s/

�c1!

ˇOz.s/

ˇC c2

ˇOz.s/

ˇ2�ds.

Denote by Mk the optimal value of the functional in problem (7). Then, we obtain

jNz.t/j � .c1! C c2Mk/Mke.�Cl.!C1//� � 1

C l .! C 1/

and the condition to choose � is

.c1! C c2Mk/Mke.�Cl.!C1//� � 1

C l .! C 1/<

1� ˇ

2

for all k. From (16) and (17), we have

Mk � e��=2 jzkj C .�C �/e��=2 � 1

D �k ,

and we conclude that the parameters have to satisfy the inequality

.c1! C c2�0/ �0e.�Cl.!C1//� � 1

C l .! C 1/<

1� ˇ

2.

This condition is satisfied whenever � is sufficiently small. �


F. MIRANDA

These results imply the existence of a trajectory of system (6) that tends to zero, constructed by choosing �-strategy and using asolution to problem (7) satisfying

jx .k�/� y .k�/j �

�1C ˇ

2

�k

jx.0/� y.0/j , kD 1, 2, : : : ,

whenever jx .k�/� y .k�/j > �0. Using this stabilizer, in finite time, we reach �0-neighborhood of zero. Then, we have jx.t/� y.t/j ! 0,that is, x.t/! y.t/. This imply the existence of solution to the global stabilization problem, that is, the existence of a solution to system(1) that tends to the equilibrium position Qx.

These conditions are necessary and sufficient to use numerical computations in the resolution of the global stabilization problemusing guidance control.

3. Application to a mechanical stabilization problem

Consider a satellite moving along a circular orbit. Introduce two Cartesian reference systems X D .X1, X2, X3/ and x D .x1, x2, x3/ (seeFigure 2). The system X D .X1, X2, X3/ is the body reference system. The origin of this system coincides with the satellite mass center, O,and the axes are directed along the principal axes of the inertia. The system x D .x1, x2, x3/ is the orbital reference system. The origin ofthis system coincides with O. The axis x3 is directed along the radius vector of the satellite mass center, and the axis x2 is perpendicularto the orbital plane. The angular position of the satellite with respect to the orbital system is described by the Euler angles .˛,ˇ, �/.The satellite is equipped with three magnetic coils oriented along three orthogonal axes that create the torque by interaction with thegeomagnetic field. Let�D .�1,�2,�3/ be the vector of angular velocity in the body axes.

Consider the rotation of the satellite in the orbital plane. The angles ˇ and � are supposed to be equal to zero. The rotation isdescribed by the equations

J2 P�2 D�3!2.J1 � J3/ sin˛ cos˛C he2, U� Fi, (18)

P D�2 �!, (19)

where J1, J2, and J3 are the diagonal elements of the tensor of inertia J D diag .J1, J2, J3/ , .J3 < J1 � J2/; F is the vector of thegeomagnetic field in the body axes. It is given by F D .f1 cos˛ � f3 sin˛, f2, f1 sin˛C f3 cos˛/ where f1 D �0�mr�3 cos' sin l,f2 D �0�mr�3 cos l, f3 D �2�0�mr�3 sin' sin l and l is the orbital inclination, that is, the angle between the equatorial and orbital

planes. Next, ' D !tC'0, ! D��g=r3

�1=2� 0.001 s�1 is the angular velocity of the orbital motion; r D 7.4� 106 m is the radius of the

orbit; �m D 8.06 � 1022 A m2 is the Earth’s magnetic dipole moment; �g D 3.986 � 1014 m3/c2 is the Earth’s gravitational parameter;�0 D 4��10�7 H/m is the magnetic parameter; '0 describes the initial position of O in the orbit. The vector e2 is the second column ofthe three-dimensional identity matrix. The vector U � F denotes the vector product of the two three-dimensional vectors U and F, andU has the components Ui D Ii�S, i D 1, 2, 3, where Ii stands for the current in the ith coil, � is the number of turns, and S is the area ofa loop. The currents Ii , i D 1, 2, 3, are control parameters. The problem is to find control laws Ii that drive the system to the equilibriumposition .�2,˛/D .!, 0/.

Figure 2. Orbit and satellite coordinate systems.


F. MIRANDA

Let the Lyapunov function derived from the known Jacobi integral

V D1

2J2 .�2 �!/

2C3

2!2�

J1 sin2 ˛C J3 cos2 ˛��

3

2J3!

2. (20)

Obviously, V � 0, and V D 0 at .�2,˛/ D .!, 0/. Then, we have to construct control laws Ii , i D 1, 2, 3, such that (20) tends to zero. It iseasy to see that the derivative of V along a trajectory .�2.t/,˛.t// is given by dV=dtD he2, U�Fi. Therefore, we have to find the controllaws that dV=dt < 0, but it is not easy. An alternative way is to use the method presented above. Consider the auxiliary system

J2PO�2 D�3!2.J1 � J3/ sin O cos O CW , (21)

PO D O�2 �!, (22)

where

W D 3!2.J1 � J3/ sin O cos O � � sin O � ��O�2 �!

�, (23)

for � > 0 and � > 0.

Theorem 4System (21)–(23) possesses an asymptotically stable equilibrium position O�2 D !, O D 0.

ProofConsider the Lyapunov function

V D1

2J2

�O�2 �!

�2C � .1� cos O /

in a neighborhood of�O�2, O

�D .!, 0/. Obviously, V � 0, and V D 0 for the equilibrium position

�O�2, O

�D .!, 0/. Calculating the

derivative of V and taking into account (21)–(23) result in the following:

dV

dtD J2

�O�2 �!

�PO�2C � PO sin O D ��

�O�2 �!

�2< 0,

whenever O�2 ¤ !. If O�2 D !, the following extension of the second Lyapunov method [38] is helpful.Let f : R� Rn! Rn be a sufficiently regular function satisfying f .t, Qx/D 0. Denote the solution to the Cauchy problem

Px D f .t, x/ ,x.0/D x0,

by x .�, x0/ .

Lemma 2Consider a sufficiently smooth function V : Rn ! R satisfying V .Qx/ D 0 and V.x/ > 0, x ¤ Qx. Suppose that there exists K 2 N such thatfor all x0 ¤ Qx, there exists kD k .x0/� K satisfying

dm

dtmV .x .t, x0//

ˇˇ

tD0D 0, m < k,

and

dk

dtkV .x .t, x0//

ˇˇ

tD0

< 0.

Then, the equilibrium position x D Qx is asymptotically stable.

We can see the proof of this lemma in [38].Therefore, as we have

d2V

dt2D�2�

�O�2 �!

�PO�2 D 0

and

d3V

dt3D�2� PO�

2

2 D�2��2

J22

sin2 O < 0,


F. MIRANDA

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25 30

Ω2(

t)-Ω^

2(t)

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25 30

α(t)

-α^ (t)

t t

(a) (b)

Figure 3. Difference between the satellite trajectory and the guide: (a) represents the difference between the angular velocity of the satellite and auxiliary system;

(b) represents the difference between the angular position of the satellite and auxiliary system.

-0.1

-0.05

0

0.05

0.1

-0.1 -0.05 0 0.05 0.1

α(t)

Ω2(t)

Figure 4. Trajectory of the satellite.

whenever sin O ¤ 0, the proof is finished. �

A trajectory of auxiliary system (21)–(23) will be a guide to control system (18)–(19). Let z.t/D .�2.t/,˛.t//��O�2.t/, O .t/

�. Then, we

construct the control U.t/ as a solution to minimization problem (7) where u.t/D U.t/, w.t/D J�12 .W.t/, 0/,

A.t/D

0 �3!2 J1�J3

J2

�cos2 O � sin2 O

�1 0

!

and

B.t/D1

J2

��f1 sin O � f3 cos O 0 f1 cos O � f3 sin O

0 0 0

�.

To simulate the stabilization process, we consider the trajectories of system (18)–(19) and auxiliary system (21)–(23) having the sameinitial position:�2.0/D O�2.0/ and ˛.0/D O .0/, and consider the following parameters: the tensor of inertia JD diag .1.7, 1.8, 1.4/, theorbital inclination lD �=3, the initial position of the mass center in the orbit '0 D��=2, � D 45, � D 0.4 and � D 3.

Implementing an �-strategy in the form of three-step reachability set construction algorithm (see [4]), we obtain the results ofthe numerical simulation that are shown in Figures 3 and 4. This �-strategy consists in the following. First, we divide the inter-vals of time Œt0C k� , t0C .kC 1/ �� in three subsets with the same size Œt0C .3kCm/ �=3, t0C .3kCmC 1/ �=3�, m D 0, 1, 2,k D 0, 1, : : :, and using the Euler’s formula, we obtain approximations to z3kCmC1 D z .t0C .3kCmC 1/ �=3/, which depend onpiecewise constant controls u3kCmC1 on Œt0C .3kCm/ �=3, t0C .3kCmC 1/ �=3�. Finally, using a numerical algorithm to minimizej.z3kC1, z3kC2, z3kC3/j, we obtain these piecewise constant controls. Then, we have U, and therefore, we find the control laws Ii suchthat drive system (18)–(19) to the equilibrium position .�2,˛/ D .!, 0/. Figure 3 represents the difference between the trajecto-ries of satellite (18)–(19) and auxiliary system (21)–(23), and Figure 4 represents the trajectory of satellite (18)–(19) that tends to theequilibrium position.

4. Conclusions

In this work, an alternative approach to solve the global stabilization problem was discussed. There are many works that study thistype of problem, where different methods of construction for a stabilizing feedback are discussed. As this problem is very complexand difficult to solve, we proposed to apply local stabilization methods because they are more simple to use and have many results forlocal and linear stabilization. Considering an auxiliary global stabilization problem that we know has an analytic solution, the method


F. MIRANDA

used in this work consists of using a trajectory of the globally asymptotically stable auxiliary system as a guide. This procedure wasapplied in the mechanical stabilization problem presented in Section 3 to construct a local stabilizer that keeps the trajectory of thesystem in a neighborhood of a solution of the auxiliary system. This way, as we could see in Figure 4, the trajectory easily tended to theequilibrium position showing us the convergence of algorithm. To solve this guidance control problem, we used problem (7) becausein the stabilization of a linear control system using other feedback construction methods, we can have the peak effect, that is, the largedeviation of the solutions from the equilibrium position at the beginning of the stabilization process. Optimal control problem (7) con-structs a stabilizing control as a function of time defined in a small time interval and not as a feedback such that the trajectory tendsto the equilibrium position and eliminates the peak phenomenon. This is very important when the control system is a linearization ofa nonlinear control system. We could compare the result obtained by optimal control problem (7) and another approach for a simplylinear control system in Figure 1. In Section 2, we obtained an estimate for the solution of the optimal control problem and conditionsfor the parameters of the stabilizer to ensure the stabilization of the system.

We conclude that it is possible to use a stabilizer as a function of time defined in a small time interval to globally stabilize controlsystem (1) if the linearized system (6) is controllable, that is, if there exists a solution to problem (7). This procedure can be appliedwith advantage and without limits in complex systems of higher dimension. An example is the three-dimensional rotation motion ofa satellite by using the geomagnetic field. For this case, we use the Burkov’s result [39] to construct an auxiliary system and show thatthe satellite is controllable around the trajectory of the auxiliary system (guide) using the linear control theory. Thus, we construct astabilizer by (7) such that the trajectory of satellite tends to the equilibrium position.

Acknowledgements

The author is grateful to Georgi Smirnov for his valuable comments and suggestions. This work was supported by the PortugueseFoundation for Science and Technology (FCT), the Portuguese Operational Programme for Competitiveness Factors (COMPETE), thePortuguese Strategic Reference Framework (QREN), and the European Regional Development Fund (FEDER).

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Numerical methods of stabilizer construction via guidance control

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