Limit Cycle Generation for Multi-Modal and 2-Dimensional Piecewise Affine Control Systems

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1 Lmt Cycle Generaton for Mult-Modal and 2-Dmensonal Pecewse Affne Control Systems atsuya Ka Department of Appled Electroncs Faculty of Industral Scence and echnology okyo Unversty of Scence 6-3- Njuku Katsushka-ku okyo JAPAN Emal: Abstract hs paper consders a lmt cycle control problem of a mult-modal and 2-dmensonal pecewse affne control system. Lmt cycle control means a controller desgn method to generate a lmt cycle for gven pecewse affne control systems. Frst we deal wth a lmt cycle synthess problem and derve a new soluton of the problem. In addton theoretcal analyss on the rotatonal drecton and the perod of a lmt cycle s shown. Next the lmt cycle control problem for pecewse affne control system s formulated. hen we obtan matchng condtons such that the pecewse affne control system wth the state feedback law corresponds to the reference system whch generates a desred lmt cycle. Fnally n order to ndcate the effectveness of the new method a numercal smulaton s llustrated. I. INRODUCION Lmt cycles are known to be qute mportant concept n varous research felds. We can fnd lmt cycles n real world for example stable walkng or gats of humanod robots n robotc engneerng oscllator crcuts n electronc engneerng catalytc hypercycles n chemstry crcadan rhythms n bology boom-bust cycles n economcs and so on. Researches on lmt cycles have been eagerly done from both mathematcal and engneerng perspectves so far Especally some condtons for nonlnear systems that generate perodc solutons and some applcatons were shown n 2 and n 7 a synthess method of hybrd systems whose soluton trajectores converge to desred trajectores was proposed. In these studes t s guaranteed that soluton trajectores of the systems converges to a desred closed curve and the exstence of lmt cycles was confrmed by numercal smulatons However the mathematcal guarantee of the exstence of lmt cycles was not shown. On the other hand the authors proposed a synthess method of mult-modal and 2-dmensonal pecewse affne systems that generate desred lmt cycles n 0 5 and showed a mathematcal proof of the exstence and the unqueness of a lmt cycle for the proposed system. In addton some theoretcal analyss on the rotatonal drecton and the perod of a lmt cycle s derved. In ths study we assume that the whole of a system can be desgned. A method to generate a desred lmt cycle for a gven pecewse affne control system wth tunng some parameters of the system s more useful for a wde varety of stuatons. However such a control method have not been proposed so far. Hence we consder a lmt cycle control problem of multmodal and 2-dmensonal pecewse affne systems n ths paper. he outlne of ths paper s as follows. We frst consder a lmt cycle synthess problem and derve ts new soluton n Secton II. Some theoretcal propertes are also shown. hen n Secton III a formulaton of lmt cycle control problem s presented and necessary and suffcent condtons for the problem whch are called matchng condtons are derved. Fnally a numercal smulaton s shown n order to confrm the effectveness of the new method n Secton IV. II. LIMI CYCLE SYNHESIS OF PIECEWISE AFFINE SYSEMS A. Formulaton of Lmt Cycle Synthess In ths secton we consder a synthess problem of pecewse affne systems whch generate desred lmt cycles. Frst ths subsecton gve the formulaton of the problem. Consder the 2-dmensonal Eucldan space: R 2 ts coordnate: x = x x 2 R 2 and the orgn of R 2 : O. Let us set N (N 3) ponts P O ( = N) n R 2 and denote the vector from O to P by p = p p 2. We also denote the angle between the half lne OP and the x -axs by θ. Now wthout loss of generalty we assume that the N ponts P ( = N) are located n the counterclockwse rotaton from the x -axs that s 0 θ < θ 2 < < θ N holds. Next we defne the sem-nfnte regon D whch s sandwched by the half lnes OP and OP + and the lne segment C jonng P and P + where P N+ = P. Set a polygon that s a unon of C ( = N) as C := C. () Fg. shows an example of a Polygonal Closed Curve for N = 5. We then consder the affne system defned n D : ẋ = a + A x x D (2) where x = x x 2 R 2 s the state varable and a R 2 A = R 2 2 are the affne term and the coeffcent matrx respectvely. Consequently we treat the N-modal and 2-dmensonal pecewse affne system that conssts of N regons D ( = N) and N affne systems (2). In ths paper we consder the followng synthess problem on lmt cycles called the lmt cycle synthess problem. 200 P a g e

2 where a skew-symmetrc matrx U f a postve defnte matrx U g and u g such that u g (V ) > 0 V 0 holds. hen for the system (5) wth (6) holds lm V (x(t)) = 0 (7) t Fg. : Example of Polygonal Closed Curve (N = 5) Problem : For the N-modal and 2-dmensonal pecewse affne system (2) desgn a A ( = N) such that a gven polygonal closed curve C () s a unque and stable lmt cycle pf the system. A soluton of Problem has been derved n the author s prevous studes 0 5. However n ths paper we wll derve another soluton whch can be utlzed to consder the lmt cycle control problem shown n Secton III. B. Proposed System and Exstence/Unqueness of Lmt Cycle Next we shall derve a soluton for Problem n ths subsecton. We also consder the exstence and the unqueness of a lmt cycle for the obtaned system. We focus on a behavor of a soluton trajectory of (2) n D. It s easly confrmed that the equaton of C s represented by (p 2 p 2 +)x (p p +)x 2 + p p 2 + p 2 p + = 0. Usng (3) we now defne a lmt cycle functon V as V (x) = (p 2 p 2 +)x (p p +)x 2 + p p 2 + p 2 p +. If V converges to 0 along a soluton trajectory of (2) then the soluton trajectory of (2) also converges to C. Hence a and A should be determned so that V converges to 0 along a soluton trajectory of (2). Now an mportant result on desgn of nonlnear systems s shown as follows 2. heorem 2 : system: (3) (4) Consder the 2-dmensonal nonlnear ẋ = f(x) + g(x) (5) where x R 2 and f g R 2 R 2 are vector felds defned n R 2. In addton consder a radal and unbounded functon defne on R 2 : V : R 2 R such that V (0) = 0 and V (x) 0 x 0 hold. We now defne f and g as f := U f (x) V g := u g (V )U g (x) V (6) x x Applyng heorem we can derve an affne term a and a coeffcent matrx A of the affne system (2) such that V converges to 0 along a soluton trajectory of (2). As a specfc form of (5) and (6) we use ẋ = f + g 0 ω f := V ω 0 x g := V (x) 0 λ x λ 0 V where ω and λ > 0 are desgn parameters. Substtutng (4) nto (8) and comparng t wth (2) we can obtan a and A of (2) as a = λ (p 2 p2 + )(p p2 + p2 p + ) ω (p p + ) λ (p p + )(p p2 + p2 p + ) ω (p 2 p2 + ) A = λ (p 2 p2 + )2 λ (p 2 p2 + )(p p + ) λ (p 2 p2 + )(p p + ) λ (p. p + )2 Compared to the pecewse affne system shown n 0 5 (28) contans a new parameter λ and ths addtonal parameter plays an mportant role n the lmt cycle control problem n Secton III. It s noted that that the system (2) wth (28) satsfes only the convergence property (7) that s ts soluton trajectory converges to C n D. Hence we wll dscuss the exstence of a unque and stable lmt cycle of the system (2) wth (28). o prove ths we frst ndcate three lemmas and then we show the man theorem by usng them. Now we gve the defnton on the clockwse and counterclockwse rotatons of lmt cycle soluton trajectores of the system (2) wth (28) 0 5. Defnton 0 5 : For lmt cycle soluton trajectores of the N-modal and 2-dmensonal pecewse affne system (2) wth (28) one that rotates n the clockwse drecton n R 2 s called a lmt cycle soluton trajectory n the clockwse rotaton. On the contrary one that rotates n the counterclockwse drecton n R 2 s called a lmt cycle soluton trajectory n the counterclockwse rotaton (see Fg. 2). Let us defne a subset n D as (8) (9) M (ε ) := { x D ε V (x) ε + } (0) where ε ε+ R satsfes ε < ε + and we set ε = (ε ε+ ). We also defne a sum of these subsets as M(ε) := M (ε ) () where we use the notatons: ε = (ε ε N ) ε+ = (ε + ε+ N ) ε = (ε ε + ) and the parameters ε and ε + 20 P a g e

3 are determned such that M (ε ) = M + (ε + ) = { x D V (x) = ε } { x D V (x) = ε + } (2) form closed polygons. If M (ε ) and M + (ε + ) are closed polygons that s M(ε) s a bounded and closed set then ε s sad to be admssble (see Fg. 3). We can derve the followng proposton on M(ε). soluton trajectory of the system (2) wth (28) we have d 2 dt (V 2 V ) = V V = V x ẋ = V V x (f + g ) V 0 ω = V V x ω 0 x V 2 V x 0 λ x λ 0 V = λ 2 V 2 V V < 0. x x Hence t turns out that the velocty vector feld of the system (2) wth (28) ponts to the drecton of the nner sde of the bounded and closed set M at any ponts on the boundary M (ε ) M + (ε + ) of M. Consequently M(ε) s a postvely nvarant bounded and closed set. Next we consder equlbrum ponts of the system (2) wth (28). he followng lemma on equlbrum ponts can be obtaned. Lemma 2 : Assume ω 0 ( = N). hen the N-modal and 2-dmensonal pecewse affne system (2) wth (28) does not have any equlbrum ponts n M(ε) for any admssble ε. a Crockwse Rotaton b Countercrockwse Rotaton Fg. 2 : Clockwse and Counterclockwse Rotatons of Lmt Cycle Soluton rajectores (Proof) he unt vector whch s on a parallel wth C n D and ponts to the counterclockwse rotaton s gven by (p p + )/ p p +. By consderng the nner product of ths unt vector and the velocty vector feld of the system (2) wth (28) we have the magntude of the velocty component to the drecton of p p + for a soluton trajectory v of the system (2) wth (28) n D as p p + v = (a + A x) p p +. (3) Now we denote a pont n D by x = α p +β p + α β 0. Hence we can calculate (3) as p p + v = {a + A (α p + β p + )} p p + (4) = ω (p p + )2 + (p 2 p2 + )2. Note that λ does not appear n (4). From (4) we can see that the parameters α and β vansh and hence v s constant at any pont x D. Snce v does not vansh at any pont x D the system (2) wth (28) does not have any equlbrum ponts n M(ε). Now a defnton on the concept traversal for the system (2) wth (28) s gven as follows. Fg. 3 : Example of M (ε ) M + (ε + ) and M(ε) Lemma : For the N-modal and 2-dmensonal pecewse affne system (2) wth (28) M(ε) s a postvely nvarant bounded and closed set for any admssble ε. (Proof) Calculatng the tme dfferental of /2V 2 along a Defnton 2 : Let Σ be a lne segment n the postvely nvarant bounded and closed set M(ε). If the value of an nner product of the unt normal vector to Σ: e Σ and the velocty vector of the N-modal and 2-dmensonal pecewse affne system (2) wth (28) s not equal to 0 and ts sgn does not change at any pont n Σ then Σ s sad to be traversal wth respect to the system (2) wth (28). In addton to Lemma 2 under the condton of ω > 0( = N) a soluton trajectory vector of the system (2) wth (28) always has a velocty component n the counterclockwse 202 P a g e

4 rotaton. On the other hand under the condton of ω < 0( = N) a soluton trajectory vector of (2) wth (28) always has a velocty component n the clockwse rotaton. From ths fact we can derve the followng lemma. Lemma 3 : For the N-modal and 2-dmensonal pecewse affne system (2) wth (28) assume that ω > 0( = N) or ω < 0 ( = N) holds. hen there exsts a traversal lne segment Σ at any pont n x M(ε) and t s satsfed that x Σ and Σ nfntely ntersects wth soluton trajectores of the system (2) wth (28). (Proof) We assume that ω > 0 ( = N) or ω < 0 ( = N) holds. hen a soluton trajectory of the system (2) wth (28) always crcles to the counterclockwse rotaton or to the clockwse rotaton. Now for a pont x M we consder a half lne whose orgn s O and that passes through x and defne a subset Σ M as the ntersecton of the half lne and M. Snce the velocty vector feld of the system (2) wth (28) always has the velocty component of the counterclockwse rotaton or the clockwse rotaton the nner product of a normal vector of Σ and the vector feld of the system (2) wth (28) at any pont n Σ s not equal to 0 and ts sgn does not change that s Σ s traversal. Moreover n each M ( = N) snce the velocty vector feld of the system (2) wth (28) always has the velocty component of the clockwse rotaton or the counterclockwse rotaton a soluton trajectory of the system (2) wth (28) x(t) that ntersected Σ ntersects Σ n a fnte tme agan. Consequently t turns out the soluton trajectory ntersects Σ nfntely. We have to note that the result n Lemma 3 does not depend on λ ( = N). Usng Lemmas 3 we can derve the man theorem on the exstence of the lmt cycle of the system (2) wth (28). heorem 2 : For the N-modal and 2-dmensonal pecewse affne system (2) wth (28) assume that ω > 0( = N) or ω < 0 ( = N) holds. hen the unque and stable lmt cycle of the system (2) wth (28) s equvalent to C. (Proof) By the result on the hybrd Poncare-Bendxson theorem derved n 3 t turns out that suffcent condtons for the exstence of stable lmt cycles of the system (2) wth (28) n M(ε) are the followng three: () M(ε) s a postvely nvarant bounded and closed set () there do not exst any equlbrum ponts at the boundary and n the nteror of M(ε) () there exsts a traversal lne segment Σ M(ε) such that x Σ and Σ nfntely ntersects wth soluton trajectores of the system (2) wth (28). Snce we have confrmed these three condtons n Lemma 2 and 3 we can see that there exsts a stable lmt cycle n M(ε) for the system (2) wth (28) for any admssble ε. Moreover snce M(ε) converges to C as the values of ε goes to 0 t can be confrmed that C s a unque and stable lmt cycle. Hence the proof s completed. In ths paper we consder an addtonal parameter λ n the system (2) (28). However from the results obtaned n ths subsecton we can see that the exstence and the unqueness of a lmt cycle of the system (2) (28) are ndependent of λ. hs fact s qute mportant n the next secton. C. heoretcal Analyss Fnally ths subsecton gves theoretcal analyss on rotatonal drectons and perods of lmt cycle soluton trajectores of the system (2) wth (28). Frst we consder the relatonshp between rotatonal drectons of lmt cycles and the parameters n (28). he followng proposton can be derved. Proposton : For the N-modal and 2-dmensonal pecewse affne system (2) wth (28) ts lmt cycle soluton trajectory moves n the counterclockwse rotaton for ω > 0 ( = N) and conversely t moves n the clockwse rotaton for ω < 0 ( = N). (Proof) he proof of ths proposton s trval from the dscusson n the prevous secton. From Proposton t s confrmed that the rotatonal drectons of lmt cycles do not depend on λ. Next we analyze perods of lmt cycles of the system (2) wth (28). It can be expected that after a soluton trajectory of the system (2) wth (28) converges to C t behaves as a perodc trajectory. By calculatng the velocty component of the vector of the system along C we can derve the next proposton. Proposton 2 : When a lmt cycle soluton trajectory of the N-modal and 2-dmensonal pecewse affne system (2) wth (28) s suffcently close to C the perod wth whch t rotates around C s gven by N ω. (5) (Proof) he velocty component of a soluton trajectory v of (2) wth (28) n D to the drecton of p p + s gven by (3). he length of C : L can be calculated as L = (p p + )2 + (p 2 p2 + )2. (6) herefore we can obtan the perod as N L N v = ω. (7) hs completes the proof of ths proposton. From Proposton 2 we can also see that the perod of a lmt cycle soluton trajectory of the system (2) (28) s not ndependent of λ. So we can freely choose the value of λ. III. LIMI CYCLE CONROL FOR PIECEWISE AFFINE SYSEMS A. Formulaton of Lmt Cycle Control In ths secton we consder a controller desgn problem on generaton of lmt cycles for gven pecewse affne control systems. Frst ths sub-secton gves the problem formulaton. Consder the next pecewse affne control system defned n D : ẋ = a + A x + b u x D (8) 203 P a g e

5 where u R s the control nput and b R 2 s the coeffcent vector for the control nput. We next consder the state feedback law: u = k x + l x D (9) where k R 2 and l R. We assume that p p + a A are gven parameters. Now we formulate a problem on generatng a desred lmt cycle for the pecewse affne control system (2) and the state feedback law (9) as follows. Problem 2 : For the N-modal and 2-dmensonal pecewse affne control system (8) wth the state feedback law (9) desgn b k l ω λ ( = N) such that a gven polygonal closed curve C () s a unque and stable lmt cycle of the closed-loop system. hroughout ths paper we call Problem 2 a lmt cycle control problem for pecewse affne control system. B. Matchng Condtons for Lmt Cycle Control Problem he purpose of ths subsecton s to derve a soluton method of Problem 2 for the pecewse affne control system (8) wth the state feedback law (9). o fulfll ths we shall utlze the lmt cycle synthess method obtaned n Secton 2. he results n Secton II show that the unque and stable lmt cycle of the system (2) (28) concdes wth C. Hence by tunng desgn parameters b k l ω λ ( = N) we conform the closed-loop system (8) (9) to the system (2) (28). We here call the system (2) (28) the reference system. Use the followng notatons for the system (2) (28): a a = a 2 b b = b 2 A A = k = k k 2 A 2 A 2 A 22. (20) system: ẋ = a + A x + b (k x + l ) = a + b l + (A + b k )x a = + b l A a 2 + b2 l + + b k A 2 + b k2 A 2 + b 2 k A 22 + b 2 k2 x. (27) Comparng the components of the reference system (2) (28) to the closed-loop system (27) we can obtan the matchng condtons (2) (26). he matchng condtons (2) (26) conssts of 6 algebrac equatons and 7 unknown varables: b b2 k k2 l ω λ. Hence by solvng them under the condton λ > 0 we can obtan these unknown varables that s a soluton of Problem 2. IV. SIMULAIONS hs secton presents a numercal example n order to confrm the effectveness of the results derved n the prevous sectons. We now gve data of a polygon wth N = 4 as P = ( 0) P 2 = (0 ) P 3 = ( 0) P 4 = (0 ).0 he polygon s shown n Fg. 4. Condtons such that the closed-loop system (8) (9) s consstent wth the reference system (2) (28) can be obtaned by the followng theorem. heorem 3 : he N-modal and 2-dmensonal pecewse affne control system (8) wth the state feedback control law (9) s equvalent to the reference system (2) (28) f and only f the matchng condtons: hold. (Proof) a + b l = λ (p 2 p 2 +)(p p 2 + p 2 p +) a 2 + b 2 l = λ (p p +)(p p 2 + p 2 p +) ω (p p +) (2) ω (p 2 p 2 +) (22) A + b k = λ (p 2 p 2 +) 2 (23) A 2 + b k 2 = λ (p 2 p 2 +)(p p +) (24) A 2 + b 2 k = λ (p 2 p 2 +)(p p +) (25) A 22 + b 2 k 2 = λ (p p +) 2 (26) Substtutng (9) nto (8) we get the closed-loop Fg. 4 : Polygonal Closed Curve of Example he coeffcents of the pecewse affne system are gven by 3 0 a = A 2 = 3 0 a 2 = A 3 2 = 0 2 (28) 2 a 3 = A 3 = a 4 = A 4 =. 0 8 By solvng the matchng condtons for Problem 2 (2) (26) n terms of the problem formulaton we can calculate 204 P a g e

6 desgn parameters as follows: b = k 2 = 2 l = ω = 3 λ = b 2 = k 2 2 = 2 l 2 = 2 ω 2 = λ 2 = 2 b 3 = k 2 3 = 2 l 3 = 2 ω 3 = 2 λ 3 = 3 b 4 = k 4 = 4 4 l 4 = 4 ω 4 = λ 4 = 4. (29) Note that λ > 0 ( = 2 3 4) holds n (29). It can be confrmed that from Proposton a lmt cycle soluton trajectory moves n the counterclockwse rotaton snce ω > 0 ( = 2 3 4) holds. In addton from Proposton 2 we can estmate the perod of a lmt cycle soluton trajectory as 4 ω = 7 6. (30) We set the ntal state as x 0 = for the numercal smulaton. he smulaton results are llustrated n Fgs Fg. 5 shows the soluton trajectory on the x x 2 -plane. In Fgs. 6 and 7 the tme seres of x and x 2 are shown respectvely. From these smulaton results we can see that the soluton trajectory that starts from x 0 behaves as a lmt cycle for the desred polygonal closed curve C and hence heorem holds. As we expected above the soluton trajectory moves n the counterclockwse rotaton and ths result s concdent wth Proposton. Moreover the estmated perod 7/6 s mostly agree about the smulaton result from Fgs. 6 and 7..5 x x t Fg. 6 : me Seres of x t Fg. 7 : me Seres of x 2 V. CONCLUSION In ths paper we have consdered a lmt cycle control problem for a mult-modal and 2-dmensonal pecewse control affne system. We have derve the matchng condtons such that the pecewse control affne system wth the state feedback law corresponds wth the reference system whch generates a unque and stable lmt cycle. It has been confrmed by solvng the matchng condtons we can obtan the values of desgn parameters. A numercal smulatons show the avalablty and the applcaton potentalty of the proposed method. Our future work ncludes applcatons of the proposed control method to real systems and extensons to multdmensonal pecewse affne systems. x Fg. 5 : Soluton rajectory on x x 2 -Plane x REFERENCES J. Buchl L. Rghett and A. Ijspeert Engneerng Entranment and Adaptaton n Lmt Cycle Systems Bologcal Cybernetcs vol.95 pp D. N. Green Synthess of Systems wth Perodc Solutons Satsfyng V(x) = 0 IEEE rans. Crcuts and Systems vol.3 no.4 pp S. N. Smc K. H. Johansson J. Lygeros and S. Sastry Hybrd Lmt Cycles and Hybrd Poncare-Bendxson n Proc. of IFAC World Congress Barcelona Span pp A. Grard Computaton and Stablty Analyss of Lmt Cycles n Pecewse Lnear Hybrd Systems n Proc. of st IFAC Conference on Analyss and Desgn of Hybrd Systems Sant-Malo France pp M. Adach and. Usho Synthess of Hybrd Systems wth Lmt Cycles Satsfyng Pecewse Smooth Constant Equatons IEICE rans. Fundamentals vol.e87-a No.4 pp F. Gómez-Estern J. Aracl F. Gordllo and A. Barrero Generaton of Autonomous Oscllatons va Output Feedback n Proc. of IEEE CDC 2005 Sevlle Span pp P a g e

7 7 A. Ohno. Usho and M. Adach Synthess of Nonautonomous Systems wth Specfed Lmt Cycles IEICE rans. Fundamentals vol.e89-a No.0 pp F. Gómez-Estern A. Barrero J. Aracl and F. Gordllo Robust Generaton of Almost-perodc Oscllatons n a Class of Nonlnear systems Int. J. Robust Nonlnear Control vol.6 no.8 pp D. Fleller P. Rednger and J. Lous Computaton and Stablty of Lmt Cycles n Hybrd Systems Nonlnear Analyss Vol. 64 pp Ka and R. Masuda Lmt Cycle Synthess of Mult-Modal and 2- Dmensonal Pecewse Affne Systems Mathematcal and Computer Modellng Vol. 55 pp M. Suenaga and. Hayakawa Exstence Condton of Perodc Orbts for Pecewse Affne Planar Systems n Proc. of SICE 8th Annual Conference on Control Systems Kyoto Japan Ka and R. Masuda Controller Desgn for 2-Dmensonal Nonlnear Control Systems Generatng Lmt Cycles and Its Applcaton to Spacerobots n Proc. of NOLA 2008 Budapest Hungary pp Ka and M. Katsuta Lmt Cycle Control for 2-Dmensonal Dscrete-tme Nonlnear Control Systems and Its Applcaton to Chaos Systems n Proc. of NOLA 2009 Sapporo Japan pp A. Schld Xu Chu Dng M. Egerstedt J. Lunze Desgn of optmal swtchng surfaces for swtched autonomous systems n Proc. of IEEE Conference on Decson and Control & Chnese Control Conference 2009 Shangha Chna pp Ka and R. Masuda A Lmt Cycle Synthess Method of Mult- Modal and 2-Dmensonal Pecewse Affne Systems n Proc. of 50th IEEE Conference on Decson and Control and European Control Conference Orlando USA pp P a g e