Connections in control strategy

Proceedings of the Estonian Academy of Sciences, 2014, 63, 1, 26–32 doi: 10.3176/proc.2014.1.05 Available online at www.eap.ee/proceedings Connections in control strategy Maido Rahulaa∗and Petr Vaˇ s´ ıkb aInstitute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia bInstitute of Mathematics, Brno University of Technology, Faculty of Mechanical Engineering, Technick´ a 2, 616 69 Brno, Czech Republic; v[email protected].cz Received 15 August 2013, revised 20 September 2013, accepted 14 October 2013, available online 14 March 2014 Abstract. We present an infinitesimal interpretation of the control theory, particularly of the part concerning dynamic systems. We use the original concept of a bundle connection, which lies in the idea of fibre transportation along a path on the base manifold. The control of a process leads also to the transportation of fibres, and the control strategy, i.e. the choice of a suitable system control in order to optimize the process corresponds to the choice of a path on the base manifold. The triple of crucial terms of control, aim–control–strategy, translates in the terms of connections as fibre–connection–curve. Such a scheme is quite convincing, but it also works well in dynamic systems analysis. Key words: control theory, connection. 1. INTRODUCTION When controlling a system, we not only apply one control model but also try to find a more suitable control model among the possible ones, i.e. we search the control strategy. We distinguish the following stages: controlled process – control correction – strategy choice. Let us describe the mathematical setting. Let X,Y, and Zbe three vector fields and let us denote by at=exptX,b σ =exp σ Y,c τ =exp τ Z the appropriate flows. If we understand the flow as a motion, the vector field can be seen as stopping the motion at the precise moment (stop-scene). Shortly, a vector field is an infinitesimal representation of the flow. The flow c τ of the vector field Zrepresents the controlled process (it is also possible to replace it by a transport of an arbitrary tensor field along the flow c τ ). Furthermore, the flow b σ acts on the vector field Z flow c τ as follows; see [1,3]: c τ Ãb σ c τ b−1 σ ,ZÃZ σ . This corresponds to the change of control. If in addition the flow atacts on b σ , we have a control strategy b σ Ãatb σ a−1 t,YÃYt. ∗Corresponding author, [email protected] M. Rahula and P. Vaˇs´ ık: Connections in control strategy 27 Concerning the strategy {XÃ{YÃZ}}, it is obtained as a composition of the action of b σ on c τ and the action of aton b σ , c τ Ãb σ c τ b−1 σ Ã(atb σ a−1 t)c τ (atb σ a−1 t)−1 . The vector field Yplays the role of the one controlled by the vector field Xand the role of the controlling field over Z. The main goal of the paper is to describe the transformation of the parameters when the process Zis changed according to the strategy appropriate to the vector field Xunder the action of the vector field Y. 2. VECTOR FIELDS Let Mbe a smooth manifold. The derivatives of a function f:M→Ralong the vector fields X,Y, and Z are defined by X f . = (f◦at)0 t=0,Y f . = (f◦b σ )0 σ =0,Z f . = (f◦c τ )0 τ =0. The vector field Yis transported along the flow of X, which can be understood as an infinitesimal interpretation of such transportation (stop-scene) – the bracket of vector fields, i.e. Lie derivative LXY= [X,Y]. Remark 1. One can obtain the bracket of two vector fields [X,Y] = XY −YX by double differentiation of a function falong the vector field flow atb σ a−1 tw.r.t. σ and then w.r.t. t: f◦(atb σ a−1 t)−1(.)0 σ =0 −→ −¡Y(f◦at)¢◦a−1 t (.)0 t=0 −→ (XY −Y X)f. Next, the transport of an arbitrary smooth tensor field along a vector field flow is defined by the Lie– Maclaurin series. For example, the transport of a vector field Zalong the flow b σ is defined by ZÃZ σ =Z+Z0 σ +Z00 σ 2 2+... = ∞ ∑ k=0 Z(k) σ (k) k!, where the coefficients Z(k)=LYZ(k−1) ,k=1,2,..., are Lie derivatives of Zwith respect to Y. In our situation, the vector field Xplays three roles: 1. The vector field Xitself causes the process as a motion in its flow at. 2. The operator LXtransforms the control YÃ[X,Y]. 3. The operator LLXdefines the control strategy – control of control LYÃ[LX,LY] = L[X,Y]. Note that the above equality can be obtained from the Jacobi identity: £[X,Y],Z¤+£[Y,Z],X¤+£[Z,X],Y¤=0 or equivalently £[X,Y],Z¤=£X,[Y,Z]¤−£Y,[X,Z]¤. Note that the process can be influenced only by some outer process, not by it itself. Indeed, if we admit that the operator LXacts on the control by the process X, we obtain: LXX= [X,X] = 0. We assign the following operators to the vector fields X,Y, and Z: 1. the operator Zimplementing the process (motion in the flow c τ ); 2. the operator LYimplementing the control of the process ZÃ[Y,Z](motion of the flow c τ in the flow b σ ); 3. the operator LLXdefining the control strategy LY(motion in the flow atof a motion of the flow c τ in the flow b σ ). 28 Proceedings of the Estonian Academy of Sciences, 2014, 63, 1, 26–32 3. CONTROL AND CONNECTION We will follow the notations appropriate to the theory of connections on fibred manifolds; see, e.g., [1,4]. Let us consider a vector bundle π :M1→Mwith n-dimensional base manifold Mand r-dimensional fibres. The standard fibre is isomorphic to Rr. On a neighbourhood U⊂M1we have local coordinates (ui ,u α ), where (ui)denotes the base coordinates and (u α )the fibre coordinates. Precisely, ui=¯ui◦ π , where ¯uidenotes the local coordinates on the neighbourhood π (U)⊂M. The coordinates (u α )are the coordinates of Rr .Latin indices i,j, . . . range from 1 to n, Greek indices α , β , . . . range from n+1 to n+r. We define two vector fields: Y=y α∂α and Z=z α∂α . Here z α are the functions depending on the fibre coordinates u α only, while y α are the functions of all coordinates (ui ,u α ). The flow b σ =exp σ Yis defined on the neighbourhood Uby a system of ODEs du α d σ =y α (ui ,u β ).(1) Indeed, now we can see the connection between dynamic systems, see [2], and the controlling parameters (ui). As mentioned above, these parameters are lifted from the base π (U)⊂Mto the neighbourhood U⊂M1, i.e. ui=¯ui◦ π . On every fibre, the vector field Yinduces a family of trajectories – phase portrait. When the fibre is changed, the vector field Ychanges too and so does the phase portrait, i.e. the control {YÃZ}. A question arises: how do the parameters (ui)affect the controlling process? Let us consider the coordinate map Φ:(ui ,u α )Ã(ui ,s,I κ ),k=n+2,...,n+r, where sis a canonical parameter, i.e. LYs=0, and I κ is a system of r−1 independent invariants of the vector field Y. The coordinates (ui ,I κ )form a complete system of local invariants of Yon the manifold M1. Now we can define the submersion of the manifold M1onto the fibre Rr , ϕ :M1→Rr:(ui ,u α )Ã(s,I κ ). A fibre of the submersion ϕ has the dimension nand forms the family of the integral surfaces which define a horizontal distribution 4h.Thus on the fibration π , a zero torsion connection structure 4h⊕4vis defined. Let us consider the adapted basis (XiX α ) = µ ∂ ∂ uj ∂ ∂ u β ¶·Ã δ j i0 Γ β i δβ α !,µ ω i ωα ¶=µ δ i j0 −Γ α j δα β ¶.µduj du β ¶, where the vector fields Xi= ∂ ∂ ui+Γ α i ∂ ∂ u α form a base of the distribution ∆hand the forms ωα =du α −Γ α idui vanish on the distribution ∆h. The number of parameters Γ α iequals nr and they define the distribution ∆huniquely. On the other hand, the parameters Γ α iare determined by setting the functions ϕα equal to a constant on the fibres of the submersion ϕ , more precisely by their differentials: d ϕα = ϕα idui+ ϕα β du β = ϕα β (du β +¯ ϕβ γϕγ idui) = ϕα βωβ =⇒Γ α i=¯ ϕα γϕγ i, where the coefficients of d ϕα are the partial derivatives of ϕα .The matrix ( ϕα β )is the integrating matrix with respect to the forms ωα and its inverse is (¯ ϕβ α ). M. Rahula and P. Vaˇs´ ık: Connections in control strategy 29 Theorem 1. The vector field Y is projected by the submersion ϕ :M1→Rronto the vector field T ϕ Y on the standard fibre Rr. In the coordinates (s,I),where s denotes the canonical parameter and I is a system of the base invariants,the vector field T ϕ Y represents the operator ∂ s . = ∂ ∂ s. The vector field Z is expressed uniquely in the basis ( ∂ s, ∂ I)and the process controlled by Z is,in the coordinate system (s,I),described by the functions s◦ ϕ and I ◦ ϕ . These functions depend on the parameters u α and the controlling parameters ui. Proof. A family of the fibres corresponding to the submersion ϕ is defined by the solution of the system of differential equations (u α ) σ = ϕα ( σ ,ui ,u β ); see system (1). Furthermore, an arbitrary section of the fibration π can be extended into the system of imprimitivity appropriate to the flow bs, i.e. the family of the fibres corresponding to the submersion ϕ . The vector field Yis ϕ -projected on the fibre Rr. An integrable distribution ∆h=KerT ϕ in the fibration π defines a zero curvature connection and thus on the neighbourhood Uthe basis and the co-basis of the distribution ∆his defined as follows: Xi= ∂ i+Γ α i ∂α , ωα =du α −Γ α idui . Let us recall that an arbitrary vector field ¯ Xon the base manifold Mcan be lifted from Mto the horizontal distribution ∆h: ¯ X=¯xi¯ ∂ iÃX=xiXi,where xi=¯xi◦ ϕ . In our notations, the basis Xirepresents the operators ¯ ∂ ifrom the neighbourhood π (U)lifted to the distribution ∆h. It is now clear that the vector field Xbehaves with respect to the vector field Yas an infinitesimal symmetry, i.e. [X,Y] = 0, and thus the impact on the vector field Yvanishes. In other words, the process appropriate to the vector field Zis defined on the fibre in the coordinates (s,I), where the functions s◦ ϕ and I◦ ϕ depend on the parameters u α and the controlling parameters ui. The vector field Xaffects the vector field Zindirectly by means of the invariants of the vector field Y. Remark 2. The components y α of the vector field Ydepend linearly and homogeneously on the fibre coordinates. Thus the defining system is described by the system of linear differential equations du α d σ =y α β (ui)u β . 4. APPLICATION On the bundle1 π :R3→R:(u,x,y)Ã(u) with the fibre coordinates (x,y)and the controlling parameter (or base coordinate) (u)we have the vector field Y= ∂ ∂ x+ux ∂ ∂ y. We define its flow bs=expsY , the canonical parameter s, and the invariant Iof Yas follows: ½˙x=1 ˙y=ux ⇒½xs=x+s ys=y+u(xs +s2 2),½s=x I=y−ux2 2. We check that LYs=1,LYI=0. The trajectories on the fibres are parabolas depending on the parameter u. 1Here, for the sake of simplicity, we denote the local coordinates by (u,x,y)instead of (u1 ,u2 ,u3)but note that the fibre coordinates (x,y)are in no way related to the components (xi ,y α )of the vector fields Xand Y. 30 Proceedings of the Estonian Academy of Sciences, 2014, 63, 1, 26–32 The submersion ϕ :R3→R:(u,x,y)Ã(s,I)projects the space R3onto the plane sI. The tangent mapping T ϕ is defined by the following differentials and by the Jacobi matrix: ½ds =dx dI =−x2 2du −uxdx +dy,µ0 1 0 −x2 2−ux 1¶. The vector field Ywith the components (0,1,ux)is projected to the plane sI in which it forms the operator T ϕ Y= ∂ s(see Fig. 1). Thus on the bundle π a horizontal distribution 4h=KerT ϕ is defined. The co-basis on 4his of the form ½ ω 2=ds =dx = (dx −Γ2 1du) ω 3=uxds +dI =dy −x2 2du = (dy −Γ3 1du), and the connection coefficients are µΓ2 1 Γ3 1¶=µ0 x2 2¶. The adapted basis of the distribution 4his characterized by the following: X1= ∂ u+x2 2 ∂ y,µ ω 2 ω 3¶=µdx dy ¶−µ0 x2 2¶·(du). The operator X1commutes with the vector field Y, i.e. [X1,Y] = 0,and vanishes under the projection T ϕ , i.e. T ϕ X1=0. The co-basis admits an integrating matrix as follows: µ ω 2 ω 3¶=µ1 0 us 1¶·µds dI ¶⇒µ1 0 −ux 1¶·µ ω 2 ω 3¶=µds dI ¶. H H H H H H H H H H H H C C C C C C C C C CO r + = r - u r r r 6 ¤¤¤¤¤¤¤¤¤¤ ¤º ¢¢¢¢¢¢¢¢¢ ¢¸ rr r ¾ T ϕ ∂ s sI Y Fig. 1. Mapping T ϕ :Y→ ∂ s. M. Rahula and P. Vaˇs´ ık: Connections in control strategy 31 The direct impact of the parameter (u)on the operator Yis eliminated. Indeed, because the projection ϕ targets on the fibre xy, it is possible to change the coordinates under the condition u=const from (x,y) to (s,I), ½x=s y=us2 2+I,µ1 0 us 1¶. Using the Jacobi matrix (the right-hand side), we can change the basis to the new natural one and we obtain the following frames and co-frames: ³ ∂ ∂ x ∂ ∂ y´=³ ∂ ∂ s ∂ ∂ I´·µ1 0 −us 1¶,µdx dy ¶=µ1 0 us 1¶·µds dI ¶. Let us focus on the fibre. Note that the action of the vertical vector field Zcan be understood as an action on a tensor field. Concerning the action of the operator Yon the vector field Zin the form YÃZ= µ∂ ∂ x+ ν∂ ∂ y, with the components ( µ , ν ), we can see that in new coordinates it reduces to the action of the operator ∂ son the vector field ˜ Zdepending on the parameter uonly: ∂ sØ Z= µ∂ s+ ν∂ I−u µ s ∂ I. Note that Zand ˜ Zare the same vector field, only expressed in the coordinates (x,y)and (s,I),respectively. The operators T ϕ Yand ∂ sare the same operators expressed in different coordinate systems. Thus we change the control: {YÃZ}Ã{ ∂ sØ Z}. Remark 3. As an example, let us consider the operator of rotation Z=−y ∂ ∂ x+x ∂ ∂ y. In coordinates (s,I),it can be written in the form ˜ Z=−I ∂ s+s ∂ I+u(...), i.e. in such a form that some new operator with coefficient uis added. Such a property holds for an arbitrary linear dynamic system. The control {YÃZ}is described in the coordinates (x,y), while the control { ∂ sØ Z}is expressed in the coordinates (s,I). The parameter uaffects the controlled field ˜ Zdirectly. 5. CONCLUSION The control of a dynamic system is viewed by means of differential geometry as the vector field Yon the bundle π :M1→Mwith the standard fibre Rrand the base manifold M=Rn .The submersion ϕ is defined in such a way that the vector field Yis projected to the fibre Rr .The distribution 4h=KerT ϕ gives rise to the possibility of eliminating the dependence of the vector field Yon the controlling parameter u. The change of variables to (s,I), where sis the canonical parameter and Iis the invariant of the field Y, changes the control (Y→Z)to the control { ∂ sØ Z}, where the field ∂ sno longer depends on the parameter uwhile the controlled field ˜ Zdoes so. 32 Proceedings of the Estonian Academy of Sciences, 2014, 63, 1, 26–32 ACKNOWLEDGEMENTS The first author was supported by Estonian Targeted Financing Project SF0180039s08. The second author was supported by the project NETME CENTRE PLUS (LO1202). The results of the project NETME CENTRE PLUS (LO1202) were co-funded by the Ministry of Education, Youth and Sports within the support programme “National Sustainability Programme I”. REFERENCES 1. Atanasiu, Gh., Balan, V., Brˆ ınzei, N., and Rahula, M. 2009. Differential-Geometric Structures. Tangent Bundles, Connections in Fiber Bundles, Exponential Law and Jet Spaces. Librokom, Moscow (in Russian). 2. Perko, L. Differential Equations and Dynamic Systems. Springer, 1991. 3. Rahula, M. New Problems in Differential Geometry. World Scientific Publishing Co. Pte. Ltd., 1993. 4. Rahula, M. and Vaˇ s´ ık, P. A note on jet and geometric approach to higher order connections. Springer Proc. Math. Statistics (to appear). Seostused juhtimise teoorias Maido Rahula ja Petr Vaˇ s´ ık Juhtimisel on kolm aspekti: juhitav protsess, juhtiv protsess ja juhtimise valik/strateegia. Matemaatiliseks mudeliks on kihtkond, kus seostus m˜ ojutab toimuvat kihil, ja baasiparameetrid, millest s˜ oltub seostus. Need m¨ a¨ aravad juhtimise strateegia. Osutub, et baasiparameetreid v˜ oib seostuse abil otsekohe kihile suunata, st juhtivast protsessist juhitavasse.