An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory

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1 Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July -3, 27 ThA8.4 An analytical aroimation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory Noboru Sakamoto and Arjan J. van der Schaft Abstract In this aer, an analytical aroimation aroach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is roosed. The roosed method gives aroimated flows on the stable manifold of the associated Hamiltonian system and rovides aroimations of the stable Lagrangian submanifold. With this method, the closed loo stability is guaranteed and can be enhanced by taking higher order aroimations. A numerical eamle shows the effectiveness of the method. I. INTRODUCTION When analyzing a control system or designing a feedback control, one often encounters certain tyes of equations that dominate fundamental roerties of the control roblem at hand. It is the Riccati equation for linear systems and the Hamilton-Jacobi equation lays the same role in nonlinear systems. For eamle, an otimal feedback control can be derived from a solution of a Hamilton-Jacobi equation [8] and H feedback controls are obtained by solving one or two Hamilton-Jacobi equations [2], [5], [28], [29]. Closely related to otimal control and H control is the notion of dissiativity, which is also characterized by a Hamilton- Jacobi equation (see, e.g., [3], [32]). Some active areas of research in recent years are the factorization roblem [3], [4] and the balanced realization roblem [] and the solutions of these roblems are again reresented by Hamilton-Jacobi equations (or, inequalities). Contrary to the well-develoed theory and comutational tools for the Riccati equation, which are widely alied, the Hamilton-Jacobi equation is still an imediment to ractical alications of nonlinear control theory. In [9], [], [22], [2] various series eansion techniques are roosed to obtain aroimate solutions of the Hamilton-Jacobi equation. With these methods, one can calculate sub-otimal solutions using a few terms for simle nonlinearities. Although higher order aroimations are ossible to obtain for more comlicated nonlinearities, their comutations are often time-consuming and there is no guarantee that resulting controllers show better erformance. Another aroach is through successive aroimation, where the Hamilton-Jacobi equation is reduced to a sequence of first order linear artial differential equations. The convergence of the algorithm is roven in [7]. In [6] an elicit technique to find aroimate solutions to the N. Sakamoto is with the Deartment of Aerosace Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, , Jaan sakamoto@nuae.nagoya-u.ac.j A. J. van der Schaft is with the Institute for Mathematics and Comuting Science, University of Groningen, 97 AV Groningen, The Netherlands A.J.van.der.Schaft@math.rug.nl sequence of artial differential equation is roosed using the Galerkin sectral method and in [3] the authors roose a modification of the successive aroimation method and aly the conve otimization technique. The advantage of the Galerkin method is that it is alicable to a larger class of systems, while the disadvantages are that it requires the initial iterate to satisfy certain conditions which are difficult to confirm and the multi-dimensional integration which can be significantly time-intensive. The state-deendent Riccati equation aroach is roosed in [4], [2] where a nonlinear function is rewritten in a linear-like reresentation. In this method, feedback control is given in a ower series form and has a similar disadvantage to the series eansion technique in that it is useful only for simle nonlinearities. A technique that emloys oen-loo controls and their interolation is used in [2]. The drawback is that the interolation of oenloo controls for each oint in discretized state sace is time-consuming and the comutational cost grows eonentially with the state sace dimension. A artially related research field to aroimate solutions of the Hamilton- Jacobi equation is the theory of viscosity solutions. It deals with general Hamilton-Jacobi equations for which classical (differentiable) solutions do not eist. For introductions to viscosity solutions see, for instance, [5], [8], [9] and for an alication to an H control roblem, see [27]. The finiteelement and finite-difference methods are studied for obtaining viscosity solutions. They, however, require discretization of state sace, which can be a significant disadvantage. Another direction in the research for the Hamilton-Jacobi equation is to study the geometric structure and the roerties of the equation itself and its eact solutions. The aers [28] and [29] give a sufficient condition for the eistence of the stabilizing solution using symlectic geometry. In [25], the geometric structure of the Hamilton-Jacobi equation is studied showing the similarity and difference with the Riccati equation. See also [3] for the treatment of the Hamilton-Jacobi equation as well as recently develoed techniques in nonlinear control theory such as the theory of ort-hamiltonian systems. Recently, the authors roosed a Hamiltonian erturbation aroach to obtain an aroimation of the stabilizing solution when the uncontrolled art of the system is integrable[26]. In this aer, we attemt to develo a method to aroimate the stabilizing solution of the Hamilton-Jacobi equation based on the geometric research in [28], [29] and [25]. The main object of the geometric research on the Hamilton-Jacobi equation is the associated Hamiltonian system. However, most aroimation research aers mentioned above do /7/$ IEEE. 2364

2 ThA8.4 not elicitly consider Hamiltonian systems, although it is well-known that the Hamiltonian matri lays a crucial role in the calculation of the stabilizing solution for the Riccati equation. One of our uroses in this aer is to fill in this ga. The aroach taken in this aer is based on stable manifold theory (see, e.g., [7], [24]). Using the fact that the stable manifold of the associated Hamiltonian system is a Lagrangian submanifold and its generating function corresonds to the stabilizing solution, which is shown in [28], and modifying stable manifold theory, we analytically give the solution sequence that converges to the solution of the Hamiltonian systems on the stable manifold. Thus, each element of the sequence aroimates the Hamiltonian flow on the stable manifold and the feedback control constructed from the each element may serve as an aroimation of the desired feedback. It should be mentioned that comutation methods of stable manifolds in dynamical systems are being develoed and a comrehensive survey of the recent results in this area can be found in [6]. The roosed method in this aer, however, is different from the above numerical methods in that it gives analytical eressions of the aroimated flows on stable manifolds, which may have considerable otential for control system design that often leads to high dimensional Hamiltonian systems. The organization of this aer is as follows. In II, the theory of st-order artial differential equations is reviewed in the framework of symlectic geometry, stressing the one-to-one corresondence between solution and Lagrangian submanifold. In III, a secial tye of solution, called the stabilizing solution, is introduced and the geometric theory for the Riccati equation is also reviewed. In IV, an analytical aroimation algorithm for the stable Lagrangian submanifold is roosed, using a modification of stable manifold theory. In V, we illustrate a numerical eamle showing the effectiveness of the roosed method and discuss some comutational issues. II. REVIEW OF THE THEORY OF ST-ORDER PARTIAL DIFFERENTIAL EQUATIONS In this section we outline, by using the symlectic geometric machinery, the essential arts of the theory of artial differential equations of first order. Let us consider a artial differential equation of the form (PD) F(,, n,,, n )=, where F is a C function of 2n variables,,, n are indeendent variables, z is an unknown function and = z/,, n = z/ n.letm be an n dimensional sace for (,, n ).Weregardthe2ndimensional sace for (, )=(,, n,,, n ) as the cotangent bundle T M of M. T M is a symlectic manifold with symlectic form θ = n i= d i d i. Let π : T M M be the natural rojection and V T M be a hyersurface defined by F =.Define a submanifold Λ Z = {(, ) T M i = z/ i (),i=,,n} for a smooth function z(). Then, z() is a solution of (PD) if and only if Λ Z V.Furthermore,π ΛZ :Λ Z M is a diffeomorhism and Λ Z is a Lagrangian submanifold because dim Λ Z = n and θ ΛZ =. Conversely, it is well-known (see, e.g. [], [23]) that for a Lagrangian submanifold Λ assing through q T M on which π Λ :Λ M is a diffeomorhism, there eists a neighborhood U of q and a function z() defined on π(u) such that Λ U = {(, ) U i = z/ i (),i=,,n}. Therefore, finding a solution of (PD) is equivalent to finding a Lagrangian submanifold Λ V on which π Λ :Λ M is a diffeomorhism. Let f = F. To construct such a Lagrangian submanifold assing through q T M, and hence to obtain a solution defined on a neighborhood of π(q), it suffices to find functions f 2,,f n F(T M) with df (q) df n (q) such that {f i,f j } =(i, j =,,n), where{, } is the Poisson bracket, and (f,,f n ) (,, n ) (q). () Using these functions, equations f =, f j = constant, j = 2,...,n define a Lagrangian submanifold Λ V. Note that the condition () imlies, by the imlicit function theorem, that π Λ is a diffeomorhism on some neighborhood of q. Since {F, } is the Hamiltonian vector field X F with Hamiltonian F, the functions f 2,,f n above are first integrals of X F. The ordinary differential equation that gives the integral curve of X F is the Hamilton s canonical equations d i dt = F i d i dt = F i (i =,,n), (2) and therefore, we seek n commuting first integrals of (2) satisfying (). III. THE STABILIZING SOLUTION OF THE HAMILTON-JACOBI EQUATION Let us consider the Hamilton-Jacobi equation often encountered in nonlinear control theory (HJ) H(, )= T f() 2 T R() + q()=, where f : M R n, R : M R n n, q : M R are all C,andR() is a symmetric matri for all M. We also assume that f and q satisfy f() =, q() = and q () =. In what follows, we write f(), q() as f()=a+o( 2 ), q()= 2 T Q+O( 2 ) where A is an n n real matri and Q R n n is a symmetric matri. The stabilizing solution of (HJ) is defined as follows. 2365

3 ThA8.4 Definition : A solution z() of (HJ) is said to be the stabilizing solution if () = and is an asymtotically stable equilibrium of the vector field f() R()(), where ()=( z/ ) T (). It will be imortant to understand the notion of the stabilizing solution in the framework of symlectic geometry described in the revious section. Suose that we have the stabilizing solution z() around the origin. Then, the Lagrangian submanifold corresonding to z() is Λ Z = {(, ) = z/ ()} T M. Λ Z is invariant under the Hamiltonian flow of ẋ = f() R() ṗ = f ()T + (T T R()) q T (3). To see this invariance, one needs to show that the second equation identically holds on Λ Z, which can be done by taking the derivative of (HJ) after relacing with (). Note that the left-hand side in the second equation of (3) restricted to Λ Z is ( / )(f() R()()). The first equation is eactly the vector field in Definition. Therefore, the stabilizing solution is the Lagrangian submanifold on which π is a diffeomorhism and the Hamiltonian flow associated with H(, ) is asymtotically stable. We assume the following throughout this aer. Assumtion : The Riccati equation obtained by linearizing (HJ) PA+ A T P PR()P + Q = has a solution P = Γ such that A R()Γ is stable. IV. ANALYTICAL APPROXIMATION OF THE STABLE LAGRANGIAN SUBMANIFOLD A. Diagonalization of linear Hamiltonian systems Etracting the linear art in (HJ), (3) can be written as (ẋ ) ( )( ) A R() = ṗ Q A T + higher order terms. (4) Using a suitable linear coordinate transformation ( ) ( ) = T (4) can be written as (ẋ ) ṗ = ( A R()Γ (A R()Γ) T B. Aroimation of stable manifolds )( ) (5) + higher order terms. (6) We consider the following system (A in this subsection corresonds to A R()Γ in the revious subsection). { ẋ = A + f(t,, y) ẏ = A T (7) y + g(t,, y) Assumtion 2: A is a stable n n real matri and it holds that e At ae bt, t for some constants a> and b>. Assumtion 3: f,g : R R n R n R n are continuous and satisfy the following. i) For all t R, + y <land + y <l, f(t,, y) f(t,,y ) δ (l)( + y y ). ii) For all t R, + y <land + y <l, g(t,, y) g(t,,y ) δ 2 (l)( + y y ), where δ j :[, ) [, ), j =, 2 are continuous and monotonically increasing on [,L j ] for some constants L,L 2 >. Furthermore, there eist constants M, M 2 > such that δ j (l) M j l holds on [,L j ] for j =, 2. Let us define the sequences { n (t, ξ)} and {y n (t, ξ)} by t n+ = e At ξ + e A(t s) f(s, n (s),y n (s))ds (8) y n+ = e AT (t s) g(s, n (s),y n (s))ds t for n =,, 2,..., and { = e At ξ (9) y = with arbitrary ξ R n. The following theorem states that the sequences { n (t, ξ)}, {y n (t, ξ)} are the aroimating solutions to the eact solution of (7) on the stable manifold with the roerty that each element of the sequences is convergent to the origin. Due to sace limitation, we did not include the roof. Theorem 2: Under Assumtions 2 and 3, n (t, ξ) and y n (t, ξ) are convergent to zero for sufficiently small ξ, that is, n (t, ξ), y n (t, ξ) as t for all n =,, 2,... Furthermore, n (t, ξ) and y n (t, ξ) are uniformly convergent to a solution of (7) on [, ) as n.let(t, ξ) and y(t, ξ) be the solution obtained as the limits of n (t, ξ) and y n (t, ξ), resectively. Then, it holds that (t, ξ), y(t, ξ) as t. C. The aroimation algorithm For (6), Assumtion imlies Assumtion 2 and Assumtion 3 is satisfied if f, R and q in (HJ) are sufficiently smooth. Thus, we roose the following rocedure for aroimation of V/. Algorithm: (i) Construct the sequences (8) for (6) and obtain the sequences { n (t, ξ)}, { n (t, ξ)} in the original coordinates using (5). (ii) Form an n dimensional manifold in ξ-sace, say, (ξ (η,...,η n ),...,ξ n (η,...,η n )). Eliminate t and n variables η,...,η n from = n (t, ξ(η,...,η n )), = n (t, ξ(η,...,η n )) to get = π n (). 2366

4 ThA8.4 (iii) The function π n serves as an aroimation of V/. Remark 4.: (i) Tyically, one can choose an n - shere as the n dimensional initial manifold. (ii) Elimination of variables is an algebraic oeration but not necessarily easy to carry out in ractice. An effective use of software is required for this urose. In V, we interolate the values of n for samle oints of n to get the function π n (). Furthermore,π n () deends on the initial manifold in ξ-sace. The deendence, however, can be smaller by taking larger n. (iii) What one obtains from the algorithm is equivalent to aroimations of the stable Lagrangian submanifold. They, in general, do not satisfy the integrability condition for finite n. Therefore, it is difficult to get an aroimation of the generating function for the Lagrangian submanifold in a geometric manner. However, since we have the analytical eression of the aroimations, we can write down aroimations of generating function as described below. Let us consider the following otimal control roblem. J = ẋ = f()+g()u L((t),u(t))dt, where L takes the form of, for eamle, L = (h() T h()+u T u)/2. The otimal feedback control is given by u = T V g()t 2 (), where V () is the stabilizing solution of the corresonding Hamilton-Jacobi equation. By the algorithm, the n-th aroimation of the Lagrangian submanifold is obtained from { = n (t, ξ) = n (t, ξ) and the n-th aroimation of the otimal feedback can be described with t and ξ as u n (t, ξ)= 2 g( n(t, ξ)) T n (t, ξ). Since the generating function is the minimum value of J for each ξ, its aroimation can be written as V n (ξ)= L( n (t, ξ),u n (t, ξ))dt. Similar eressions of the generating function for the H roblem may be ossible using dissiative system theory[32]. V. NUMERICAL EXAMPLE Let us consider the -dimensional nonlinear otimal control roblem; ẋ = 3 + u q J = r 2 u2 dt. The Hamilton-Jacobi equation for this roblem is H = ( 3 ) 2r 2 + q 2 2 = and the Hamilton s canonical equations are ẋ = 3 r ṗ = ( 3 2 ) q. A. The stable manifold aroimation method () The coordinate transformation that diagonalizes the linear art of () is ( ( ) = T ), ( ( + ) +q/r) T = r + r 2. + qr q The equations in the new coordinates are where (ẋ ṗ ) = ( ) ( +q/r +q/r + (, ) 3 3(, ) 2 (, ) (, )= ( + +q/r), (, )=(r + r 2 + qr) + q. We construct the sequences (8) with f(, )= (, ) 3, g(, )=3(, ) 2 (, ), ), and q =, r =.From n (t, ξ) and n (t, ξ), the relation of n and n is obtained by eliminating t, which will be denoted as = π n (). We note that π n () deends on ξ.the aroimated feedback functions are u = (/r)π n () = π n (). Figures -3 show the results of calculation for π n (). To guarantee the convergence of the solution sequence (8), ξ has to be small enough. If ξ is too large, the sequence is not convergent (see, Fig. and Fig. 3). If ξ is small and only ositive t is used in n (t, ξ) and n (t, ξ), then the resulting trace in the lane is short, hence, the function π n () is defined in a small set around the origin. Therefore, we substitute negative values in t to etend the trace toward the oosite direction. This, however, creates a divergent effect on the sequence (see, Fig. and Fig. 2). And this effect becomes smaller as n increases. In Fig. 4, the calculation result by the Hamiltonian erturbation aroach in [26] is shown. Since the integrable nonlinearity is fully taken into account in this aroach, the feedback function is better aroimated in the region further from the origin. Also, we showed the result by the Taylor series eansion of order n =6in Fig

5 ThA n=2 n=3, n=3.8.6 n= otimal otimal n=4.4.2 n=2.2.2 n= Fig.. ξ =.42 and etended to the negative time Fig. 3. ξ =.6 and etended to the negative time.5.4 n=2.5 linearization (Γ) n=3 n=4 otimal.5 Taylor(n=6) erturbation otimal.2 n= Fig. 2. ξ =.42 and etended to the negative time.8 Fig. 4. Perturbation and Taylor eansion solutions VI. CONCLUDING REMARKS In this aer, we roosed an analytical aroimation aroach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory. The roosed method gives aroimated flows on the stable manifold of the associated Hamiltonian system as functions of time and initial states. Feedback controls are calculated by elimination of variables. Since this method focuses on the stable manifold of the Hamiltonian system, the closed loo system stability is guaranteed and can be enhanced by taking higher order aroimation. The elimination of variables is not necessarily easy in ractice, although it is an algebraic oeration. Effective software use should be discussed in the future research. Acknowledgement. The first author was suorted by the Scientist Echange Program between the Jaan Society for the Promotion of Science (JSPS) and the Netherlands Organisation for Scientific Research (NWO). REFERENCES [] R. Abraham and J. E. Marsden. Foundations of Mechanics. Addison- Wesley, New York, 2nd edition, 979. [2] J. A. Ball, J. W. Helton, and M. L. Walker. H control for nonlinear systems with outut feedback. IEEE Trans. Automat. Control, 38(4): , 993. [3] J. A. Ball, M. A. Petersen, anda. van der Schaft. Inner-outer factorization for nonlinear noninvertible systems. IEEE Trans. Automat. Control, 49(4): , 24. [4] J. A. Ball and A. J. van der Schaft. J-inner-outer factorization, J- sectral factorization, and robust control for nonlinear systems. IEEE Trans. Automat. Control, 4(3): , 996. [5] M. Bardi and I. Cauzzo-Dolcetta. Otimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, 997. [6] R. W. Beard, G. N. Sardis, and J. T. Wen. Galerkin aroimations of the generalized Hamilton-Jacobi-Bellman equation. Automatica, 33(2): , 997. [7] S.-N. Chow and J. K. Hale. Methods of Bifurcation Theory. Sringer- Verlag, 982. [8] M. G. Crandall and P. L. Lions. Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS, 277: 43, 983. [9] W. H. Fleming and H. M. Soner. Controlled Markov Processes and Viscosity Solutions. Sringer, 2nd edition, 25. [] K. Fujimoto and J. M. A. Scheren. Nonlinear inut-normal realizations based on the differential eigenstructure of Hankel oerators. IEEE Trans. Automat. Control, 5():2 8, 25. [] W. L. Garrard. Additional results on sub-otimal feedback control of non-linear systems. Int. J. Control, (6): , 969. [2] W. L. Garrard and J. M. Jordan. Design of nonlinear automatic flight control systems. Automatica, 3(5):497 55, 977. [3] D. Hill and P. Moylan. The stability of nonlinear dissiative systems. IEEE Trans. Automat. Control, 2():78 7,

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