From Local to Global Control

Transcription

1 Proceeings of the 47th IEEE Conference on Decision an Control Cancun, Mexico, Dec. 9-, 8 ThB. From Local to Global Control Stephen P. Banks, M. Tomás-Roríguez. Automatic Control Engineering Department, The University of Sheffiel Sheffiel S3 JD Unite Kingom. s.banks@sheffiel.ac.uk Mathematics an Engineering School, City University. Lonon ECV HB. Unite Kingom. Maria.Tomas-Rorigue.@city.ac.uk Abstract In this paper the authors stuy the problem of the existence of multiple local operating points in control systems. In particular, they consier a metho of going from local to global control, i.e. given a number of local, linearize systems, from which global system o they come, an, can global controllers be etermine in this case? I. INTRODUCTION In many practical applications of the control of nonlinear systems, the ynamics are usually linearise about operating points such as trim conitions in aircraft. These local moels are then use to esign local controllers an some gain scheuling proceure is use to switch between the controllers at ifferent operating points. In fact, in many inustrial plants or aerospace systems, only a small number of operating conitions are known an the global nonlinear ynamical system is not known. Gain scheuling methos are wiely use an are object of many research papers, see [], [], [3] an references within as example. In this paper, the problem of going from some known local moels to a global one will be consiere. Furthermore, the obtaine global moel can then be use instea of gain scheuling for control purposes. The contains of this paper are as follows: Section II recalls the basics of nonlinear systems linearisation proceure, operating conitions an trackability properties. Section III introuces an algebraic metho that allows the reconstruction of an unknown nonlinear system taking as starting points its linear representations at some given operating conitions. Section IV presents a global control metho that can be use instea of the well-known of gain scheuling once the nonlinear system is known. Section V tackles the problem of going from the local moels to the topology of the manifol on which the system is efine. Section VI contains a summary of the ieas here presente. II. NONLINEAR SYSTEMS, OPERATING CONDITIONS AND TRACKABILITY Consier the following nonlinear system: ẋ fx,u efine on R n xr m. This is in fact, a kin of local moel for systems on manifols - this point will be aresse later. Intuitively, by an operating point it is unerstoo a pair x t,u t R consisting of an open loop control an a function x t which satisfies equation when the control u t is applie, i.e.: ẋ t fx t,u t On the other han, the esire function x t is trackable if there exists a control u t such that hols. If x t is trackable for, the local variables aroun the operating point yt an vt, are efine as: yt xt x t vt ut u t Then by applying Taylor s theorem: an ẏt ẋt ẋ t fx,u fx t,u t AtytBtvt for small yt an vt. At fx t,u t x Bt fx t,u t If At,Bt is a controllable pair, then the control /8/$5. 8 IEEE 446

2 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 of system aroun the operating conition is achievable by using the control: ut u tvt5 In general, there coul be a number i of these operating points: x i,ui C [, ];R nm, i K 6 associate to a corresponent number i of local moels of the nonlinear system : ẏ it A ity itb itv it, 7 The problem to be consiere now is the inverse one: How to go from a set of i local moels 7 to the global one?. For simplicity, it will be assume that all the operating points are constant, so that the local moels are linear, time-invariant systems of the form: ẏ i A i y itb i v it, i K 8 Also, in this case, becomes an algebraic conition of the form: fx i,ui 9 if the values of x i are constant, i K. fx i,ui B i, 4 for i k, an moreover, f must satisfy: fx i,ui Consier the equations: with an N i N n i n, i k. 5 f p x l,ul 6 M j M m j m a p i...i n, j... j m x l i u l j l k, p n. 7 A l pq N i N q i q N n i n M j M m j m ap i...i n, j... j m i q x l i q u l j 8 B l pq N i N u i n M j M q j q M m j m ap i...i n, j... j m j q x l i u l j q i i,,i n, ThB. 9 Having obtaine a global moel from the i local ones, a global controller which will rive the system from one operating point to another has to be etermine. This will involve an application of an iteration scheme [4] which replaces a nonlinear not necessarily quaratic optimal control problem which is linear time-varying an quaratic, which can be solve by classical methos. Finally, the authors will consier the tracking global problem of etermining to some egree the topology of the manifol on which a nonlinear system is efine from a knowlege of the local representatives assuming that the local systems are complete in the sense that their efining neighborhoos cover the manifol. III. FROM LOCAL TO GLOBAL Suppose the set of k unknown constant operating points: x i,ui, i k at which there exist k linearisations of the form: ẏ it A i y itb i v it, i k of some unknown nonlinear system on R n. If the unknown system is of the form then ẋ fx,u fx i,ui A i, 3 x 447 an j j,, j m q,,, }{{},,, q in the n i N i m j M j variables a p i, j, f is assume to be able to be approximate by a polynomial function. The number of equations is an they can be written in the form: nn nmk LV W L is a linear operator, V is a vector of the unknown parameters a p i, j an W contains the known local representations A i,b i. L maps R α into R β, : an The system is: α ΠN i ΠM j β nn nmk 3 i Overetermine if α < β, ii Determine if α β an L is invertible, iii Uneretermine if α > β. Therefore, accoring to the above classification, there will be a unique solution in case ii, a solution in case i if W RangeL an in case iii if RankL β.

3 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 ThB. IV. GLOBAL CONTROL In the previous section, a way to fin a global moel from the local moels aroun some operating conitions has been presente. Now, a global controller which rives the system from one operating conition to another is foun. Suppose the existence of two istinct operating points x,u an,u, so that in the obtaine global moel of the form: ẋ fx,u4 this will be, fx i,u i, i,. 5 Now the control objective is to rive the global nonlinear system 4 from x to, this is, to seek a esire trajectory x t such that: x x, x t f Therefore, the question is: Does there exist a control u t so that: ẋ t fx t,u t6 In orer to answer this question, the notion of projection fiel shoul be introuce: This is efine as a section of the bunle of projection operators on R n of rank m. Thus, a projection fiel on R n associates a projection operator P x : R n R m to each point x R n such that the function x P x is smooth. The main result can be summarize as: Theorem: Given a esire trajectory x : R R n, there is a control u : R R m satisfying 6, if there exist a projection fiel x P x such that the function gx,u P x fx,u satisfies: gx t,u t for each t R an ẋ t RP x t for all t R, RP is the range of the projection P. Proof: Since ẋ RP x, then: Hence, P x tẋ P x t fx t,u t gx t,u t ẋ gx t,u t an since gx t,u t the results follow from the implicit function theorem. Suppose there exists a control u t satisfying the above theorem, then from 4 an 6, it can be written that: ẋ ẋ fx,u fx,u fx x x,u u u fx,u gx x,u u,t g,,t, t. Hence, if y x x an v u u, then the system 7 can be written as: ẏ gy,v,t7 therefore, the regulator problem for v can be solve, an then write u vu. To solve this problem, a well establishe technique of linear, time-varying approximations to the problem see [4] is applie: It is assume that the system 7 can be written on the form: ẏt Ay,tytBy,tvt, 8 In the case the system is not affine in the control, nonlinear control terms can be inclue in Ay,t an By,t. Now, equation 8 is replace by the following sequence of LTV systems: ẏ [i] t Ay [i ] ty [i] tby [i ] tv [i] t9 an to each of the equations 9, the following quaratic cost functional is applie: J x[i]t t f Fx [i] t f t f [ ] x [i]t tqx [i] tu [i]t tru [i] t t 3 The problems 9 an 3 can then be solve by stanar methos see [5] or [7]. Example Let ξ,ν, ξ,ν R be two points which satisfy: ν ξ ξ ξ 3 ν ξ ξ 3, an suppose the two local systems: ẋ ξ i x These systems are local versions of: ẋ ξ an if: Then the control: x x t ξ tξ t t ẋ x x 3 u x ẋ 3 ẋ x will erive the system between the two points. u, i,. 3 u 3 448

4 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 ThB. The system 7 becomes: ẏ y x y x x ẏ y v y 33 an this generates a close-loop control so that the following expression of ut, u vu, 34 will regulate the system between the two given operating values. V. GLOBAL SYSTEMS ON MANIFOLDS In this section, a more general problem is consiere; that of going from local moels to the topology of the ambient manifol on which the system is efine. Hence, consier a compact n-imensional ifferentiable manifol M an let Xu be a parameterize vector fiel on M, u R m. Let φ i,u i, i K be a finite covering of M by local parameters, each φ i : U i R n is a homeomorphism. In each coorinate neighborhoo, the local form of the ynamical system corresponing to the vector fiel Xu ui can be written as: ẋ i f ix i,u i. 35 It will be assume that, in each neighborhoo U i there is an operating point x i φ iu i. This, as before, means that there is a constant open loop control u i such that f ix i,ui 36 Then the new local coorinates are efine, an the new control so the local moel is of the form: y i x i x i 37 v i u i u i y 38 y i g iy i,v i 39 g iy i,v i f iy i x i,vi u i Note that g,. The question is now: Knowing the local moels 39 an the operating points x i,ui, what can be sai about the topology of M?. It is assume that complete information is known, in the sense that the coorinate patches on which the local systems are efine cover the unknown ambient manifol. So the problem is, how o they fit together?. Taking zero controls from 39: ẏ i g iy i,, i K each of these systems is efine on some region V i, say of R n. Now, it is sai that two local systems ẏ i g iy i,, ẏ j g jy j,4 are compatible if there exist a iffeomorphism φ i j : ˆV i V j from some nonempty subset ˆV i of V i onto some subset ˆV j of V j such that the system is a topologically conjugate on ˆV j, i.e. φ i j maps trajectories of systems i on those of system j. Note that, if these two systems are compatible, then y it φ i j y it an so i.e., ẏ j φ i j ẏi y i g jy i, φ i j y i giy i,. Hence the matrices φ i j y i gi from the transmition matrices for this tangent bunle of a manifol. The transition matrices are enote by γ i j φ i j y iyi Note that they obey the stanar cocycle conitions: γ i j γ jk γ ik on U i U j U k, γ ii I, γ i j γ ji I on U i U j. For example, for a sphere S, regare as S C { }: g : U U GLI,C g z z n for each integer n an U, U are neighborhoos of an respectively. This efines a complex line bunle on S, usually enote by H n. A connection on the vector bunle efine by transition functions γ i j is a collection of ifferential operators w i efine on U i such that w i γ i j γi j γ i j w j γi j on U i U j is the exterior erivative. These glue together to give a global map: for a bunle E. A : Ω E Ω E4 The curvature of A is A an is represente locally by a matrix K i of two-forms for which K i γ i j K j γ i j 4 The K th characteristic class of the bunle is efine by τ k E [τ k A] H k M;R43 449

5 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 ThB. τa trace [ π i K j k]. The first Chern class c E τ E. Then, < c τm,[m] > π M KA XM by the Gauss-Bonnet theorem, so the Chern class states if the system is trivial or not. Example The systems: ẋ x u 44 ẏ ẏ y y u 45 Generate a system in S, since the transition functions look like g z z in local coorinates. It can be seen that < c,τm,[m] > XM VI. CONCLUSIONS In this paper the authors have consiere the problem of piecing together a set of given local systems to form a global one on some manifol. If the information is incomplete as in most practical cases interpolation methos can be use an if it is complete, then topological manifol theory will be use to escribe the ambient manifol. Using an approximation metho for the obtaine nonlinear system, global controllers can be now esigne in orer to rive the system from one local operating point to another. REFERENCES [] Hunt, K.J., Johasen,T.A. Design an analysis of gain-scheule control using local controller networks. International Journal of Control, 66, [] Hye, R.A., Glover, K.A. Comparison of Different Scheuling Techniques for H Controllers. Proceeings of the Institute of Measurement an Control Symposium on Robust Control, Cambrige. 99. [3] Hye, R.A., Glover, K.A. The Application of Scheule H Controllers to a VSTOL Aircraft. IEEE Transactions on Automatic Control, 38, [4] M. Tomas-Roriguez an S.P. Banks, Linear Approximations to Nonlinear Dynamical Systems with Applications to Stability an Spectral Theory. IMA J. Math. Cont Inf.,, 3, pp [5] M. Tomas-Roriguez, M., Navarro Hernanez, C. an Banks, S.P., Parametric Approach to Optimal Nonlinear Control Problem using Orthogonal Expansions, In proceeings of 6th IFAC Worl Congress, Prague, July 5, Cz. [6] Navarro Hernanez, C, Banks, S P an Aleen. Observer Design for Nonlinear Systems using Linear Approximations. IMA J. Math. Cont Inf, [7] Tayfun Cimen an Stephen P. Banks. Nonlinear optimal tracking control with application to super-tankers for autopilot esign. Automatica, Volume 4, Issue, November 4, Pages