Hopf Bifurcation Control for Affine Systems

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1 opf Bifurcation Control for Affine Systems Ferno Verduzco Joaquin Alvarez Abstract In this paper we establish conditions to control the opf bifurcation of nonlinear systems with two uncontrollable modes on the imaginary ais We use the center manifold to reduce the system dynamics to dimension two, find epressions in terms of the original vector fields I INTRODUCTION There eists a great interest to analyze control systems that can ehibit comple dynamics An emerging research field that has become very stimulating is the bifurcation control which, for eample, tries to modify the dynamical behavior of a system around bifurcation points, generate a new bifurcation in a desirable parameter value [3], delay the onset of an inherent bifurcation [], or stabilize a bifurcated solution [], [] In [6] an overview of this field is included There are many works that study the bifurcation control problem In [], [], [] this problem is analyzed using state feedback control In [9], [5], [8] the problem is investigated using normal forms invariant In this paper, we analyze control systems with two uncontrollable modes on the imaginary aes We propose a state feedback control u uz; µ, β,β such that µ causes the opf bifurcation, β determines the stability of the equilibrium point, β establishes the orientation stability of the periodic orbit This analysis is based on the opf bifurcation center manifold theorems [7], [4] II STATEMENT OF TE PROBLEM Consider the nonlinear system ξ F ξ+gξu, ξ R n is the state u R is the control input The vector fields F ξ Gξ are assumed to be sufficiently smooth, with F Assume that J J DF J S ω with J, J ω S R n n a urwitz matri Suppose that F ξ G ξ z, ξ G ξ w F ξ F ξ, Gξ, with z R, w R n, F Verduzco is with the Department of Mathematics, University of Sonora, 83 ermosillo, Sonora, Meico verduzco@gaussmatusonm J Alvarez is with the Scientific Research Advanced Studies Center of Ensenada CICESE, 86 Ensenada, BC, Meico jqalvar@cicesem F,G : R R n R, F,G : R R n R n Then, eping system around ξ yields ż J z + F z, w+f 3 z, w+ +b + M z + M w + G z, w+ u, ẇ J S w + F z, w+f 3 z, w+ +b + M 3 z + M 4 w + G z, w+ u, b G b, DG b with b R, b R n, M M, M 3 M 4 F j z, w zt F j, z + zt F j, w w + wt F j, w, w G j z, w zt G j, z + zt G j, w w + wt G j, w, w F 3j z, w 3 F j, z, z, z+, 6 3 for j, We wish to design a control law u uz, µ, with µ a real parameter, such that the original system undergoes a opf bifurcation at ξ µ, that we could control it, ie, that we could decide the stability direction of the emerging periodic solution We suppose that rank bjb J n b n There are many ways to satisfy the condition ; in this paper we analyze the case b b j for j,,,n, b b,b,,b,n T This corresponds to the case the linear approimation of has two uncontrollable modes, ±iω,atξ Consider the control law uz, µ β µ + β z + z β µ + β z T z, 3 β,β R Now, using the control law 3 in system we obtain the closed-loop system ż J z + F z, w, µ, 4 ẇ β b µ + J S w + F z, w, µ,

2 F z, w, µ β µm z + β µm w + F z, w +β µg z, w+β z T z M z + M w +F 3 z, w+, F z, w, µ β µm 3 z + β µm 4 w + F z, w +β z T z b + M 3 z + M 4 w +β µg z, w+f 3 z, w+ Then, our goal is to find β β such that system 4 undergoes a opf bifurcation can be controllable For this, we use the center manifold theory III CENTER MANIFOLD A Quadratic terms Equation 4 represents a µ-parameterized family of systems, which we can write as an etended system ż µ ẇ J z µ β b J S w + F z, w, µ F z, w, µ In this form, the system has a three-dimensional center manifold through the origin To find this manifold, we need to change coordinates to put the linear part in diagonal form Then, using the transformation matri P P z µ w P µ y J β J S b J S J β J S b J S we can put 4 into stard form ẋ µ J µ ẏ J S y or + f, µ, y g, µ, y ẋ J + f, µ, y, µ, 5 ẏ J S y + g, µ, y, f, µ, y J F J, µ, β J S b µ + J S y, 6 g, µ, y J S F J, µ, β J S b µ + J S y 7 We seek a center manifold y h, µ T + T µ + 3µ + 8 such that h,, Dh, h i, µ T i + T i µ + 3iµ + for i,,,n Substituting 8 into 5 using the chain rule, we obtain h, µ [J + f, µ, h, µ] J S h, µ g, µ, h, µ 9 This partial differential equation for h will be solved in the simplest case, that is, when J S is diagonal, ie, J S λ λ n with λ j < for each j Besides, we are just interested to calculate because we will make µ when we calculate the first Lyapunov coefficient a Now,if g, µ T N + T N µ + N 3µ, with g i, µ T N i + T N i µ + N 3iµ, for i,,n, represents the quadratic terms of g, µ, h, µ, then from 9 we obtain, h i,µ J λ i h i, µ g i, µ+hot T i + i T µ J λ i T i + T i µ + 3iµ T N i + T N i µ + N 3iµ + hot T i J λ i i N i + T J T i λ i i N i µ λ i 3i + N 3i µ + hot, for i,,n, we consider only the quadratic terms Then i N i J λ ii N i R i I R i J λ ii λi ω λ i +4ω ω λ i Now we are going to calculate N Observe that, from 7, g, µ, h, µ JS F J, µ, β JS b µ + J S h, µ J S T JF T zz, J +β ω T J S b +,

3 but J S T JF T zz, J T A A Aω,λ i, F,, β ω T J S b β ω T β ω β ω β ω T B B i bi λ i I Therefore, b λ b,n λ n b λ T b,n λ n T T b λ I T b,n λ n I g, µ, h, µ T A + β ω T B + T A +β ω B +, then, from, N A +β ωb Now then, from, we obtain i N i R i A i +β ω B i Ri Ā i +β ω B i, b i λi ω B i λ i λ i +4ω ω λ i Finally, from 8, h, µ T +, Ā +β ω B B Dynamics on the center manifold On the center manifold the dynamics is given by ẋ J + f, µ, h, µ, 3, from 6, f, µ, h, µj F J, µ, β J S b µ + J S h, µ β µj M b T JS T F wz, J + J T J T F zz, J T JF T zw, J S h, µ +J +β J T JJ T M J + 6 J T JF T zzz, J J +, but, we define β µm β µj M J M b T M b T JS T F wz, J J T S Fwz, J ; 4 T Q J T JF T zz, J Q Qω, F,, C,, J T J T 6 F zw, J S h, µ +β J T J T J M J + 6 J T J T F zzz, J J 5 Observe that 6 J T JF T zzz, J J 6 C,, with C C ω, 3 F, ; 3 β J T JJ T M J β ω T J M J β ωc M,,, m m with M m m m I C M m I, 6 m I m I J T J T F zw, J S h, µ J T J T F zw, J S T Ā +β ω B J T J T F zw, J S T Ā +β ω J T J T F zw, J S T B CĀ,, +β ω C B,, with C Ā C Ā ω,λ i, F,, F w C B,, J T J T F zw, J S T B We are going to calculate C B Observe that F F zw, zw, Fzw,, 7 F j F j zw, z w, F j z w, F j, w,, F j, w n F j, w,, J S T B T λ B T λ n Bn F j, w n

4 then F zw, J S T B F zw, J S T B F zw, J S T B, n F j zw, J S T B T F j, k w k Bk n T F j, k w k Bk T S j, with S j i n k n F j, Bk w k i F j, b k w k i λ k +4ω k for i, j, Now then, T J T ω,, then, T J T F zw, J S T B λk ω ω, ω T S, T S ω T S + T S T S + T S, therefore, T J T F zw, J S T B J T S + T S T S + T S T S + T S T S T S S S S S then,,, C B S S S S Now, we re-write 5 6 C 6 C + β ωc M + CĀ + β ωc B 6 C +3C Ā + 6β ω C M + C B C + C, 9 C C ω,λ i, 3 F, 3, F,, F, w C 6β ω m I + S m I S m I S m I + S Finally, the dynamics on the center manifold is given by ẋ J µ + Q, + C,, +, 6 J µ J + β µm, M C are given by 4,9, Remark Remember that we just consider those quadratic cubic terms in that do not depend on µ because we put µ to find the first Lyapunov coefficient At the same time, we have just found epressions for those terms that depend on β that are needed to find the mentioned coefficient IV CONTROL OF TE OPF BIFURCATION In this section we will find conditions to ensure that system undergoes a opf bifurcation that can be controlled A opf bifurcation theorem Theorem : [7] Suppose that the system ẋ f, µ with R n, µ R has an equilibrium,µ at which the following properties are satisfied: A D f,µ has a simple pair of pure imaginary eigenvalues no other eigenvalues with zero real parts A Let λµ, λµ be the eigenvalues of D f,µ which are imaginary at µ µ, such that d dµ Reλµ µµ d Then there is a unique three-dimensional center manifold passing through,µ R n R a smooth system of coordinates for which the Taylor epansion of degree three on the center manifold, in polar coordinates, is given by ṙ dµ + ar r, θ ω + cµ + br If a, there is a surface of periodic solutions in the center manifold which has quadratic tangency with the eigenspace of λµ, λµ agreeing to second order with the paraboloid µ a d r Ifa<, then these periodic solutions are stable limit cycles, while if a >, are repelling For bidimensional systems, there eists an epression to find the called first Lyapunov coefficient a Consider the system ẋ J + F, ω J ω DF Then, F F F, F a 6ω R + ωr, 3 R F F + F F F + F F F + F F R F + F + F + F

5 There eists another way to epress R If Q R, then F Q, + C,, + 6 Q Q, C with tr trace B Control law design C C with Q C C i,c ij R trc + C, 4 In this section we are going to prove, using the theorem, that system undergoes the opf bifurcation at µ First, we are going to prove that the eigenvalues of J µ cross the imaginary aes when µ, second, we will show that the first Lyapunov coefficient a is different of zero Eigenvalues of J µ : The characteristic equation of the linear part of is given by λ trj µ λ + detj µ trj µ β µtrm detj µ ω + β µω M M +β µ detm, with M M ij Then, for µ sufficiently small, the eigenvalues are given by λµ trj µ ± i Then, λ ±iω detj µ trj µ Reλµ trj µ β µtrm but, from 4, trm trm b T J T S Fwz, trm trb T J T S Fwz, trm trb T J T S Fwz, T trm trfzw, T J S b, from 7, Fzw T, JS b Fzw J S b Fzw J S b F j zw, JS b F j z w, JS b, J F j z w S b n F j, b k k w k n F j, b k k w k, then trfzw, T J S b n therefore, k n k n k n k b k F n, + w k b k b k b k n K trm from, k b k F, w k F, + F, w k w k w k F, + F, w k div z F,, Reλµ β µ K, k b k w k div z F,, 5 d d dµ Reλµ µ β K 6 First Lyapunov coefficient: From 3-4, a R + ω R 6ω, for our system, R trc + C trc +trc δ + δ δ δ ω,λ i,f zz,,f zzz, trc δ trc tr 6β ω m I + S m I S m I S m I + S β ω K, n b k K trm λ div z F, 7 k k +4ω w k Then, a 3 4 β ω K + δ, 8 We have then proved the net result

6 Theorem : Consider the system ξ F ξ+gξu, J with F DF J, with J S ω J J ω S diag{λ,,λ n } urwitz If K K, given by 5 7, respectively, b are different of zero, G, with b b, rank bjb J n b n, then there eists β, β such that, with the control law u β µ + β z + z, the system undergoes the opf bifurcation at µ Moreover, it is possible to control the stability direction of the emerging periodic solution near the origin, by selecting the signs of d a in 6 8, respectively For the case n, K K trm This case was reported in [] [9] W Kang, Bifurcation normal form of nonlinear control systems, Part I Part II, SIAM J Control Optimization, 36-, 988, pp 93-, pp 3-3 [] A Tesi, E Abed, R Genesio O Wang, armonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics, Automatica 3, 996, pp 55-7 [] F Verduzco Leyva, Control of codimension one bifurcations in two dimensions In Spanish, Proc AMCA, Ensenada, BC, Meico 3 V CONCLUSIONS In this paper we have derived sufficient conditions to ensure the control of the opf bifurcation in nonlinear systems with two uncontrollable modes in the imaginary aes We have used the center manifold theorem to reduce the analysis to dimension two; nevertheless, we have obtained epressions in terms of the original vector fields The control law designed has a constant term, which establish the stability of the equilibrium point, a quadratic term, which determines the orientation stability of the periodic solution near the origin REFERENCES [] E Abed J Fu, Local feedback stabilization bifurcation control, I opf bifurcation, Systems & Control Letters, 7, 986, pp -7 [] E Abed J Fu, Local feedback stabilization bifurcation control, II Stationary bifurcation, Systems & Control Letters, 8, 987, pp [3] E Abed, O Wang A Tesi, Control of bifurcation chaos, in The Control book, WS Levine, Ed Boca Raton, FL CRC Press, 995, pp [4] J Carr, Application of Center Manifold Theory, Springer 98 [5] DE Chang, W Kang AJ Krener, Normal forms bifurcations of control systems, Proc 39th IEEE CDC, Sydney, Australia, [6] G Chen, JL Moiola O Wang, Bifurcation control: theories, methods, applications, Int J Bif Chaos, vol, No 3,, pp [7] J Guckenheimer P olmes, Nonlinear oscillations, Dynamical systems, bifurcations of vector fields, Springer-Verlag 993 [8] B amzi, W Kang JP Barbot, On the control of opf bifurcations, Proc 39th IEEE CDC, Sydney, Australia

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