Milno (Itly) August 8 - September, 011 Time Optiml Control of the Brockett Integrtor S. Sinh Deprtment of Mthemtics, IIT Bomby, Mumbi, Indi (emil : sunnysphs4891@gmil.com) Abstrct: The Brockett integrtor is 3-dimensionl system with non-holonomic constrint nd the non-holonomic constrint mkes the stbiliztion nd control of the system chllenging. The 3-dimensionl non-holonomic integrtor cn be generlized in more thn one wy nd this work looks into the time-optiml control of the generlized Brockett integrtor which evolves on R m Skew m (R). The problem hs been cst in Hmiltonin frmework nd the Pontrygin s Mximum Principle hs been used to obtin cndidte control inputs for the time-optiml stte trnsfer. The existence of solution to optiml control problem hs been proved nd complete procedure is presented to chieve the stte trnsfer in time-optiml wy. Abnorml extremls re shown to correspond to sttionry curve on the cotngent bundle which proves the non-existence of the bnorml extremls. Simultion results, which hve been performed in MATLAB, re presented nd the simultion results vlidte the results obtined theoreticlly. 1. INTRODUCTION One of the most intriguing exmple of system with nonholonomic constrint tht is of utmost interest to the control engineers is tht of the non-holonomic integrtor, which is lso known s Brockett integrtor. The system is of importnce becuse it is closely relted to the current-fed induction motor, Heisenberg flywheel nd vriety of robotic nd mobile non-holonomic systems. Mthemticlly, the Brockett integrtor cn be expressed s 1 = u 1 = u 3 = x 1 u x u 1 (1) where u i re controls. See Brockett [1981]. The lst eqution is the constrint nd non-integrble one. The non-holonomicity of the constrint cn be esily proved by Frobenius theorem (for Frobenius theorem see Boothby [003]). The non-holonomic integrtor cn be generlized or extended in severl wys. This is the min subject of Brockett, Di [1993], in which minly two ides for the generliztion of the three dimensionl cse re considered. One is to enlrge both the number of controls nd the dimension of the stte spce. The second generl possibility considered in Brockett, Di [1993] is to enlrge the stte spce in order to ccount for higher non-liner effects, where controllbility is chieved by tking higher order Lie brckets nd eventully enlrging lso the number of controls used. In this pper, first the problem of time-optiml control hs been defined. Then in the next section the Hmiltonin hs been defined for the generlized Brockett integrtor. Then the existence of solution to the optiml control problem hs been proved using the Filipov s theorem. In the next section, the complete procedure for the timeoptiml stte trnsfer hs been given. Next, the bnorml extremls hve been proved to correspond to sttionry point on the co-tngent bundle generting geometriclly irrelevnt solutions. Finlly, simultion results hve been presented nd they perfectly vlidte the results obtined theoreticlly.. GENERALIZED BROCKETT INTEGRATOR In this work the generlized model tht hs been considered is obtined from the 3-dim Brockett integrtor by incresing the number of controls nd the dimension of the stte spce. The model obtined by generlizing in this wy is ẋ = u(t) Ż = xu T (t) u(t)x T, () where x nd u(t) re curves in R m, with m >. The vectoril function u(t) is the control of the system nd is ssumed to be mesurble. The superscript T denotes mtrix trnsposition nd Z is m m skew-symmetric mtrix. This problem hs lso been discussed in Brockett [1981]. In this work, the time-optiml control of this generlized Brockett integrtor hs been considered. Problem Sttement :Given the system of eqution () nd > 0, steer the stte trjectory from (0, 0,, 0, 0 m m ) to (0, 0,, 0, P m m ) (where P m m = [ ij ] such tht ij = ji ), subject to u i (t) 1, such tht the time in reching the finl stte is minimum. P m m is chosen s Copyright by the Interntionl Federtion of Automtic Control (IFAC) 3509
Milno (Itly) August 8 - September, 011 0 0 P m m =, > 0 (3) 0 The configurtion mnifold of the generlized Brockett integrtor is Q = R m Skew m (R), where Skew m (R) is the spce of (m m) rel skew-symmetric mtrices. Hence the dimension of the mnifold is (m + m(m 1) ), tht is m(m+1). Since the mtrix corresponding to Z is skewsymmetric, study of the upper tringulr prt of [Z] gives ll the necessry informtion. So if the sttes of the upper tringulr prt re driven to the desired stte, the sttes of the lower tringulr prt will be driven utomticlly to their desired vlue. Motivted by this fct, the problem is nlyzed on R m(m+1) insted of R m Skew m (R)..1 The Hmiltonin From () the stte equtions re given by i = u i z ij = x i u j x j u i (4) The hmiltonin, which is defined on the cotngent bundle of the configurtion mnifold, is defined s h u (ξ, x) = ξ, f u (x) where ξ re the costte vribles, f u (x) re the velocity vectors nd, is the usul inner product on R n. Hence the hmiltonin is h u (ξ, x) = ξ 1 u 1 + + ξ m u m + ξ 1 (x 1 u x u 1 ) + Now, ξ ij = ξ ji. Hence, +ξ (m 1)m (x m 1 u m x m u m 1 ) h u (ξ, x) = (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m )u 1 + (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m )u + +(ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 )u m (5) The Hmilton s equtions of motion re ( ) h u (ξ, x) = J x ẋ ξ h u (ξ, x) (6) ξ where ( h u (ξ, ) x) is the hmiltonin of the system nd J = 0 In (I I n 0 n is n n identity mtrix). So, when csted in hmiltonin frmework, the equtions of motion for the Brockett integrtor becomes i = u i z ij = x i u j x j u i ξ 1 = (ξ 1 u + ξ 31 u 3 + + ξ m1 u m ) ξ m = (ξ 1m u 1 + ξ m u + + ξ (m 1)m u m 1 ) ξ ij = 0 (7). Existence of solution For studying optiml control problems, the stte spce of the system is extended by one dimension by ppending the cost s the new stte vrible. Once this is done, ny optiml control problem cn be reduced to the study of the ttinble sets of the extended stte spce. It cn be esily proved tht the trjectories of the extended system tht correspond to the optiml trjectories of the originl system, come to the boundry of the ttinble set of the extended system (see Schkov, Agrchev [004]). Hence, the existence of solution to the optiml control problem cn be gurnteed by proving the rechble set to be compct. The sufficient conditions for the compctness of the ttinble set is given by Filipov s theorem (see Schkov, Agrchev [004]). Theorem.1. The solution to the optiml control problem for the generlized Brockett integrtor lwys exists. Proof. The control prmeters u i for the generlized Brockett integrtor re such tht u i 1. Hence the set of control prmeters re compct. If f u (x) = 0, then u i = 0. Hence, in this cse ll the x i s remin sttionry, tht is, constnt. Similrly, z ij = 0 nd hence Z is lso constnt. Hence, for f u (x) = 0, the stte trjectory is just fixed point in R m Skew m (R). Let the fixed point be P = (p 1, p,, p m, P M ), where P M is constnt mtrix in Skew m (R). Filipov s theorem demnds the existence of compct subset K of configurtion mnifold Q, such tht f u (x) = 0 for x / K, u U, where U is the spce of permissible control inputs. Choose subset K such tht K = [ 1, b 1 ] [, b ] [ m, b m ] S M, such tht, p i / [ i, b i ] nd S M is ny constnt mtrix in Skew m (R) not equl to P M. Now, ech of [ i, b i ] is compct nd since the spce of skew-symmetric mtrices is Husdorff, every constnt mtrix in Skew m (R) is compct (since every point set in Husdorff spce is compct. See Munkres [1975]). Hence, K is compct (since product of compct sets is compct. See Munkres [1975]). Agin, the velocity sets cn be esily shown to be convex. Hence, by Filipov s theorem, solution to the optiml control problem exists..3 Time-optiml stte trnsfer In this section complete procedure hs been presented to chieve the given stte trnsfer in time optiml wy. Let, ν be the costte vrible corresponding to the cost. For norml extremls, ν is normlized to 1. Since time is being minimized here, the hmiltonin of the extended system is 3510
Milno (Itly) August 8 - September, 011 h ν u(ξ, x) = (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m )u 1 + (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m )u + +(ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 )u m 1 (8) The cndidte control inputs for time-optiml stte trnsfer of the generlized Brockett integrtor re u i = ±1. (9) This cn be esily deduced by pplying Pontrygin s mximum principle to the hmiltonin of the system. For more generl clss of systems see Kirk [004]. Suppose, just z ij is to be driven from 0 to some p > 0 in time-optiml wy. So, when z ij reches p, x i nd x j should return to zero. The initil position of x i nd x j is zero. Agin, u i = ±1 nd u j = ±1. Proposition.. The given stte trnsfer cn be chieved in time-optiml wy using ny one of the following control sequences : ( 1, 1) (1, 1) (1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (1, 1) (1, 1) (1, 1) (1, 1) ( 1, 1) ( 1, 1) (10) where the first co-ordinte is u i nd second co-ordinte is u j. The number of switchings is three nd the time required to rech the finl stte is p nd the time intervl between consecutive switchings is 1 p. Proof. (Only ide of proof is given due to pge limittions). Since the z ij co-ordinte is independent of x i nd x j coordinte, the stte trjectory is projected on the x i x j plne nd is nlyzed on this plne. It cn be esily shown tht when the projected trjectories move long the lines x i = ±x j, z ij remins constnt. Moreover, on the x i x j plne, the trjectories should originte from the origin nd should come bck to the origin t the end of the stte trnsfer. (m = n), t most ( m C n(n 1)) sttes cn be driven in time-optiml wy. Proof. Suppose z ij (not both i nd j re odd or even t the sme time) is to be driven in time-optiml wy to p > 0. This cn be done by using ny one of the control sequences given in (10). Suppose, is used. At the sme time intervl, z i(i+1) cn be driven in time-optiml wy to p by using the sequence (1, 1) (1, 1) ( 1, 1) ( 1, 1) Similrly, in the sme time intervl, z (i+1)(i+) cn be driven to p by using ( 1, 1) (1, 1) (1, 1) ( 1, 1) Agin, z (i 1)i cn lso be driven to p in the sme time intervl by using z (i )(i 1) cn lso be driven to p in the sme time intervl using the sequence ( 1, 1) ( 1, 1) (1, 1) (1, 1) This is the only scheme which drives z (i )(i 1), z (i 1)i, z i(i+1) nd z (i+1)(i+) ll t the sme time to p in time-optiml wy. Moreover, this is the best tht cn be chieved using the permissible cndidte control inputs. Now, u i = u i+ So, z (i )(i+) remin t zero. So, remin t zero. (z 15, z 59, z 19, ) (z 6, z (10), z 6(10), ) (z 37, z 3(11), z 7(11), ) (z 48, z 4(1), z 8(1), ) Hence, ll the z ij such tht both i nd j re either odd or both re even remin t zero. Let m be odd (m = n + 1). So, number of odd integers present in (1,, (n + 1)) is (n + 1) nd number of even integers present in (1,, (n + 1)) is n. Hence, number of z ij such tht both i nd j re odd is n(n + 1) nd number of z ij such tht both i nd j re even is Fig. 1. One possible x i x j trjectory corresponding to time optiml stte trnsfer Fig 1 depicts the desired x i x j stte trjectories. The switchings occur t, b nd c. This structure is exploited to prove the proposition. Now, not only just one z ij cn be driven time-optimlly to p > 0 in prticulr time intervl, but more thn one stte cn be driven simultneously using vrious combintions of the cndidte control inputs. Proposition.3. For m odd, (m = n+1), t most ( m C n ) cn be driven in time-optiml wy nd for m even, n(n 1) So, number of sttes tht remin t zero is n(n + 1) + n(n 1) = n Agin, let m be even (m = n). Then number of odd integers is n nd number of even integers is n. Hence, number of z ij tht do chnge is n(n 1) n(n 1) + = n(n 1) Now, if we wnt to drive those z ij, such tht i nd j re either both odd or both even, then lso we hve to 3511
Milno (Itly) August 8 - September, 011 dopt similr scheme nd then gin the number of sttes tht remin t zero remins the sme. Hence, in ll cses the number of sttes tht remin t zero is either n or n(n 1), depending on whether m is odd or even respectively. With the dopted scheme, some of the sttes go to zero. But not ll the other sttes go to p > 0. Some of them go to p. The exct number of sttes tht go to p is given by the following proposition. Proposition.4. If, m = 4n + 1, n sttes go to p. If, m = 4n +, n + n sttes go to p. If, m = 4n + 3, n + n sttes go to p. If, m = 4n + 4, n + 3n + 1 sttes go to p. Proof. Suppose the stte z 1 is driven to p > 0 using Then, using the control sequence scheme proposed erlier, z 13 = 0, z 3 = p, z 34 = p, z 14 = p (11) Agin, z 1i = z 1(i+4) (1) So, z 15 = 0, z 16 = p, (13) Let, m = 4n + 1. Then the first row of the Z mtrix fter the ppliction of the first sequence of control input becomes ( 0 p 0 p 0 p 0 p 0 p 0 p 0 ) (14) Now, ny integer (q > 0) is of the form q = 4r + 1 or, q = 4r + or, q = 4r + 3 or, q = 4r + 4, where r 0. Hence, knowing the sttes z 1, z 13, z 14, z 3, z 4, z 34, ll the other sttes cn be clculted. Using the bove fct, the upper tringulr prt of the second row becomes ( p 0 p 0 p 0 p ) (15) Hence, the upper tringulr prt is 0 p 0 p 0 p p 0 p 0 0 p 0 p 0 0 p 0 p (16) 0 p 0 p 0 p 0 0 p 0 The number of sttes tht go to p in the first two rows is n, tht in the next four rows is (n 1), tht in the next four rows is (n ) nd so on. Hence, the totl number of sttes tht go to p is n + 4(n 1) + 4(n ) + + 4(n (n 1)) = n (17) Doing similr clcultions, the other cses re proved esily. Remrk.5. In ll the bove proofs, one prticulr control sequence hs been chosen nd tht hs been used to prove the results. But, even if we strt with ny other control sequence tht drive z ij to p > 0, then lso we will get the sme results becuse u i = u i+4 nd hence things repet. So, the bove propositions lwys hold once we pply the control sequences given by (10). Lemm.6. If, m = 4n + 1, ( 4n+1 C 6n ) sttes go to p. If, m = 4n +, ( 4n+ C 6n 3n) sttes go to p. If, m = 4n + 3, ( 4n+3 C 6n 6n 1) sttes go to p. If, m = 4n + 4, ( 4n+4 C 6n 9n 3) sttes go to p. Proof. For, m = 4n + 1, 4n + 1 = (n) + 1 Hence, from proposition (.3), 4n sttes go to zero nd from proposition (.4), n sttes go to p. Hence, ( 4n+1 C 6n ) sttes go to p. Similrly, the other cses re proved. Theorem.7. The desired finl stte is reched in time m C p, where, 4n+1 C 8n for m = 4n + 1 4n+1 C 8n, 4n for m = 4n + 4n+1 C 8n, 8n 1 for m = 4n + 3 4n+4 C 8n, 1n 4 for m = 4n + 4 Proof. The trnsfer of the sttes z ij s re only considered, becuse the x i s go to zero t the end of ech trnsfer of z ij to p or p. Using the pre-described scheme, when some z ij is driven to p, some of the sttes go to p, some go to p nd others remin t zero. Let, m = 4n + 1. In this cse, when some z ij is driven to p, ( 4n+1 C 6n ) sttes go to p, n sttes go to p nd 4n sttes remin t zero. The finl vlues of the sttes z ij s fter the ppliction of the first set of control inputs which drive ( 4n+1 C 6n ) sttes to p re : z ij z pq z rs z tu z vw z xy p p p p 0 0 Now, there re (4n+1) inputs such tht u i = u i+4. Hence, the totl number of input combintions re (4n + 1)! (n + 1)!n!n!n! Any of these control sequence is such tht ( 4n+1 C 6n ) sttes go to p, n sttes go to p nd 4n sttes remin t zero. The next control sequence is chosen such tht the chnge in the sttes re the following : z ij z lm z rs z gh z vw z ef z xy 0 p p p p 0 0 The next control sequence is such tht the chnge in the sttes re given by : z ij z lm z cd z gh z de z ef z kl z xy 0 0 p p p p 0 0 351
Milno (Itly) August 8 - September, 011 This process is repeted m C times so tht t the end of these m C steps, ll the z ij go to Hence, m C p 6n p n ( m C 8n )p ( m C 8n ) where is the finl vlue of ech z ij. The totl number of steps re m C, tht is 4n+1 C nd t ech step, some z ij go from np to (n + 1)p. Hence, from proposition (.), the time required in ech step is p. Hence the totl time is m C p. The other cses cn be proves similrly. The minimlity of the time required for stte trnsfer cn be proved esily using proposition (.) nd using elementry clculus for finding the minim of function..4 Abnorml extremls The stte spce of the system is extended by ppending the cost s new stte nd the ttinble set of this extended system is studied. Hence the new configurtion spce is R Q where R Q = {ˆx = (y, x) y R, x Q} The co-stte vrible corresponding to the new stte vrible y is ν. The cse when ν = 0 is known s bnorml extreml, otherwise it is known s norml extreml. In cse of norml extremls, since the pir (ν, ξ) cn be multiplied by ny number, ν is normlized to 1. Proposition.8. In cse of bnorml extremls, (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m ) (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m ) (ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 ) go to zero identiclly. Proof. In cse of bnorml extremls, ν = 0. So the Hmiltonin is h ν u(ξ, x) = (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m )u 1 + (18) (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m )u + +(ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 )u m Let, if possible, ech of (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m ) (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m ) (ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 ) be not equl to zero. Now, the Hmiltonin should be mximum. So, the mximized Hmiltonin is h ν u(ξ, x) = (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m ) + (19) (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m ) + (ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 ). Agin, for time optiml control, the mximised Hmiltonin should be zero. But this cnnot hppen becuse of the ssumption tht (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m ) (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m ) (ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 ) re not equl to zero, which contrdicts the Pontrygin s mximum principle. So our ssumption is wrong. Hence, ech of (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m ) = 0 (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m ) = 0 (ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 ) = 0(0) Theorem.9. Abnorml extremls do not exist for the generlized Brockett integrtor. Proof.From proposition (.8), in cse of bnorml extremls, (ξ 1 ξ 1 x ξ 13 x 3 ξ 1m x m ) (ξ ξ 1 x 1 ξ 3 x 3 ξ m x m ) (ξ m ξ m1 x 1 ξ m x ξ m(m 1) x m 1 ) re identiclly zero. Differentiting the bove equtions with respect to time, ξ 1 = (ξ 1 u + ξ 13 u 3 + + ξ 1m u m ) ξ m = (ξ m1 u 1 + ξ m u + + ξ m(m 1) u m 1 ), (1) since ξ ij s re constnt (7) nd i = u i. From (7) nd (1) nd using ξ ij = ξ ji, (ξ 1 u + ξ 13 u 3 + + ξ 1m u m ) = 0 (ξ m1 u 1 + ξ m u + + ξ m(m 1) u m 1 ) = 0 3513
Milno (Itly) August 8 - September, 011 The bove set of equtions re true for ll u i = ±1. Let, in one cse ll u i = 1. Then the first eqution of () becomes ξ 1 + ξ 13 + + ξ 1m = 0 () Let in nother cse u = 1 nd ll the other u i s be one. Then the first eqution of () becomes ξ 1 + ξ 13 + + ξ 1m = 0 (3) Subtrcting (3) from (), we get ξ 1 = 0 Similrly, ll the ξ ij s re identiclly zero. Substituting the vlues of ξ ij s in (0), ξ i = 0 So when ν = 0, tht is, in cse of bnorml extremls, ll the co-stte vribles re identiclly zero. Hence bnorml extremls do not exist. Fig. 5. x 1 trjectories for = 8 3. SIMULATION We tke m = 3 nd = 8. So, 4. From theorem (.7), z 1, z 13 nd z 3 rech in time t = 1 nd x 1, x nd x 3 go to zero t time t = 1. As the following figures depict, t time t = 1, ll the x i s go to zero, while the z ij s go to = 8. The switching times lso mtch s predicted by the theorems. Fig. 6. x 13 trjectories for = 8 Fig.. x 1 trjectories for = 8 Fig. 3. x trjectories for = 8 Fig. 4. x 3 trjectories for = 8 Fig. 7. x 3 trjectories for = 8 REFERENCES A. Rmos. New links nd reduction between the Brockett non-holonomic integrtor nd relted systems. Rend. Sem. Mt. Univ. Pol. Torino Vol. 64 (006), Control Theory nd Stbility., II A. M. Bloch. Nonholonomic Mechnics nd Control. Springer, 003 D. E. Kirk. Optiml Control Theory : An Introduction. Dover Publictions, 004 F. Bullo, A. D. Lewis. Geometric Control of Mechnicl Systems : Modeling, Anlysis nd Design for Simple Mechnicl Control Systems. New York, Springer, 005 J. Munkres. Topology : A First Course. New Delhi, Prentice Hll of Indi, 1975 J. E. Mrsden, T. Rtiu. Introduction to Mechnics nd Symmetry. Springer-Verlg, 1994 L. N. Hnd, J. D. Finch. Anlyticl Mechnics. Cmbridge University Press, 1998 R. W. Brockett. Control theory nd singulr Riemnnin geometry. New Directions in Applied Mthemtics Springer-Verlg, 1981 R. W. Brockett, L. Di. Non-holonomic kinemtics nd the role of elliptic functions in constructive controllbility. Nonholonomic Motion Plnning Kluwer, Norwell, MA 1993 W. M. Boothby, An Introduction to Differentible Mnifolds nd Riemnnin Geometry. Acdemic Press, 003 Y. Schkov, A. Agrchev. Control Theory from the Geometric Viewpoint. Springer, 004 3514