Patchy solution of a Francis Byrnes Isidori partial differential equation

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1 INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust. Nonlinear Control 3; 3:46 6 Published online 4 January in Wiley Online Library (wileyonlinelibrary.com). DOI:./rnc.776 Patchy solution of a Francis Byrnes Isidori partial differential equation Cesar O. Aguilar *, and Arthur J. Krener Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA SUMMARY The solution to the nonlinear output regulation problem requires one to solve a first-order partial differential equation, known as the Francis Byrnes Isidori equations. In this paper, we propose a method to compute approximate solutions to the Francis Byrnes Isidori equations when the zero dynamics of the plant are hyperbolic and the exosystem is two dimensional. With our method, we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two-dimensional analytic center manifolds. Copyright John Wiley & Sons, Ltd. Received 5 August ; Revised 6 November ; Accepted 7 November KEY WORDS: output regulation; zero dynamics; center manifolds; periodic trajectories. INTRODUCTION Consider the control system Px D f.x, u, w/ Pw D s.w/ y D h.x, u, w/, () where x R n is the state variable, u R m is the control variable, w R q is an exogenous variable, and y R p is the output variable. The maps f W R n R m R q! R n, s W R q! R q,and h W R n R m R q! R p are all assumed to be sufficiently smooth and satisfy f.,, / D, s./ D, andh.,, / D. Thevariablew represents a disturbance and/or a reference signal, and its dynamics are commonly referred to as the exosystem. The state feedback regulator problem [] is to find a state feedback control u D.x, w/, with., / D, such that the equilibrium x D of the dynamical system Px D f.x,.x, /, / is exponentially stable and that for each sufficiently small initial condition.x, w /, the solution of () with u D.x, w/ satisfies lim y.t/ D. t! A characterization of the state feedback regulator problem for linear systems was given by Francis [] and later generalized to nonlinear systems by Isidori and Byrnes []. As shown in [], *Correspondence to: Cesar O. Aguilar, Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 9394, USA. coaguilanps.edu Copyright John Wiley & Sons, Ltd.

2 PATCHY SOLUTION TO A FBI PDE 47 the solvability of the regulator problem can be reduced to the solvability of a system of partial differential equations (PDEs), which in the linear case reduce to the Sylvester type equation obtained by Francis. For this reason, we refer to these equations as the Francis Byrnes Isidori (FBI) PDEs. For completeness, we state the main result of [] (for the definition of Poisson stability used, see Remark.). Theorem. ([]) Assume that the equilibrium w D of the exosystem is Lyapunov stable and there is a neighborhood of w D in which every point is Poisson stable. Assume further that the pair f f.,, /,.,, / () x u is stabilizable. Then the state feedback regulator problem is solvable if and only if there exists C k.k > / mappings W! R n, with./ D, and W! R m, with./ D, both defined in a neighborhood R q of w D and satisfying.w/s.w/ D f..w/,.w/, w/ w D h..w/,.w/, w/. Given a solution pair., / to the FBI equations (3), a state feedback solving the regulator problem is given by.x, w/ D.w/ C K.x.w//, where K R mn is any feedback matrix rendering the pair () asymptotically stable. In general, solutions to the FBI equations, being singular quasilinear PDEs with constraints, may not exist. However, for a class of control-affine systems, it is shown in [] that the solvability of the FBI equations is a property of the zero dynamics of (). Roughly speaking, if the zero dynamics of () has a hyperbolic equilibrium at the origin, then a solution to the FBI equations exists by the center manifold theorem [3]. It is known, however, that center manifolds suffer from a number of subtle properties associated with uniqueness and differentiability [4]. Despite these difficulties, a C dynamical system possess a C k center manifold for each k > ; moreover, it is possible to obtain approximate solutions of arbitrarily high-order via Taylor series [3]. In this respect, Huang and Rugh [5] and Krener [6] provide a method to compute approximate solutions to the FBI equations via Taylor polynomials, yielding approximate output regulation. Naturally, if one desires a more accurate output tracking controller, then one can increase the degree of the Taylor approximation; however, there are two main drawbacks in doing so. First, increasing the order increases the accuracy of the approximation but on a possibly smaller domain because of the rapidly growing behavior of high-order polynomials away from the origin. Second, the number of monomials in q variables of degree N is q C N N a number growing rapidly in N. In this paper, we present a method to compute solutions to the FBI equations for the class of real analytic SISO control-affine systems, (3) Px D f.x/cg.x/u Pw D s.w/ y D h.x/ C p.w/, (4) where f W R n! R n, g W R n! R n, s W R q! R q, h W R n! R, andp W R q! R are real analytic about the origin. Furthermore, we will restrict our considerations to two-dimensional Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

3 48 C. O. AGUILAR AND A. J. KRENER exosystems, that is, q D, whose linear part contains non-zero eigenvalues. Our method is based on the existence and uniqueness results for two-dimensional analytic center manifolds in [7] and on a high-order patchy method similar to [8]. A key strength of our approximation method, which relies on the periodic nature of the solution to the FBI equations for (4), is the reduction of the computational effort inherent in a direct Taylor polynomial approximation. The organization of this paper is as follows. In Section, we briefly summarize the key insight provided in [] on how the solvability of the FBI equations can be reduced to the problem of solving a center manifold equation provided that the zero dynamics of (4) are hyperbolic. With this simplification, we show how the standard stability assumptions on the exosystem lead to a direct application of the results in [7] to deduce uniqueness of solutions to the FBI equations of (4). In Section 3, we describe a continuation method to compute a high-order piecewise smooth approximation to the solution of the FBI equations and prove that a sequence of approximations generated by our method converges uniformly to the true solution. Finally, in Section 4, we illustrate our method on examples and then make some concluding remarks. Remark. Henceforth, it will be implicitly assumed that the exosystem has an equilibrium w D that is Lyapunov stable and that there is a neighborhood of w D in which every point is Poisson stable. We will refer to this type of stability as neutral stability. By Poisson stability, we mean the following. An initial condition x of the dynamical system Px D f.x/ is Poisson stable if the flow ˆf t.x / of the vector field f is defined for all t R, and for each neighborhood U of x and each real number T>, there exists a time t >T such that ˆf t.x / U and a time t < T such that ˆf t.x / U.. REAL ANALYTIC AND PERIODIC SOLUTIONS TO THE FRANCIS BYRNES ISIDORI EQUATIONS As shown in [], a key simplification in the problem of solving the FBI equations for a system of the form (4) consists in reducing it to the problem of solving a center manifold equation for the zero dynamics of (4). Following [] and using the now standard notation in [9], assume that the triple ¹f, g, hº from (4) has relative degree 6 r<nat x D and let., / denote the standard normal coordinates, where D.h.x/, L f h.x/, :::, L r h.x// and is such that L f g D. Inthe., / coordinates, (4) takes the form Ṕ D f., / P D, :::, P r D r P r D b., /C a., /u Pw D s.w/ y D C p.w/. The zero dynamics of (4) are given by the dynamical system (5) Ṕ D f., /. (6) Define functions ' i W R q! R by ' i.w/ D L i s p.w/, 6 i 6 r,set'.w/ D.'.w/, :::,' r.w//, and let u e.x, w/ D Lr f h.x/ C Lr s p.w/ L g L r. h.x/ f Then it is straightforward to verify that if satisfies the PDE w.w/s.w/ D f..w/, '.w//, (7) Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

4 PATCHY SOLUTION TO A FBI PDE 49 then.w/ WD..w/, '.w// and.w/ WD u e..w/, w/ constitute a solution pair to the FBI equations of (5). If the origin of (6) is hyperbolic, that is, the matrix f =., / has eigenvalues off the imaginary axis, then (7) is the equation that is satisfied by any center manifold ¹., w/ W D.w/º of the coupled dynamical system Ṕ D f., '.w// Pw D s.w/. (8) Hence, in the hyperbolic case, the problem of solving the FBI equations associated to the original system (4) is reduced to solving the center manifold equation associated to (8). Although this simplification is significant, solutions to center manifolds suffer from subtleties associated with uniqueness and differentiability [4]. For example, it is known that an analytic dynamical system does not generally posses an analytic center manifold, thereby forcing one to seek a center manifold solution that is only C k.k D, 3, :::/ and thus not necessarily unique. As an example, the polynomial dynamical system Ṕ D C w C w Pw D w w.w C w / Pw D w w.w C w /, which is of the form (8), has the property that each center manifold has Taylor series X.i /Š.w C w /i, id which has vanishing radius of convergence. Despite these difficulties, a special case for which sharp uniqueness and differentiability results exist is for two-dimensional center manifolds and is given by the following theorem by Aulbach [7]. Theorem. ([7]) Consider the ordinary differential equation Ṕ D B C Z.w, w, / Pw D w C P.w, w, / Pw D w C Q.w, w, /, (9) where w, w R, R n,andp, Q, andz are real analytic functions about the origin and have Taylor series beginning with quadratic terms. Suppose that the matrix B has no eigenvalues on the imaginary axis. If the local center manifold dynamics of (9) are Lyapunov stable and non-attractive, then (9) has a uniquely determined local center manifold that is analytic and generated by a family of periodic solutions. Aulbach s result has a direct application to the output regulation problem, as given by the following theorem. Theorem. Suppose that in (4) the exosystem is two dimensional and s=w./ has non-zero eigenvalues. Suppose that f, g, h, andp are real analytic mappings about x D and w D, respectively, and that the triple ¹f, g, hº has a well-defined relative degree 6 r<nat x D. If the zero dynamics of (4) are hyperbolic, then there exist unique and real analytic mappings., / solving the associated FBI equations of (4). Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

5 5 C. O. AGUILAR AND A. J. KRENER Proof By assumption and neutral stability of the exosystem, the eigenvalues of the exosystem are nonzero and purely imaginary. Indeed, if the eigenvalues were not purely imaginary, then w D would necessarily be either a repelling or an attractive equilibrium, contradicting the assumption of neutral stability. Now because B WD f =., / contains eigenvalues off the imaginary axis, there exists an analytic coordinate change [] about the origin such that (8) takes the form Ṕ D B C Z.w, w, / Pw D w C P.w, w / Pw D w C Q.w, w /, () where P, Q, andz are analytic at the origin and have Taylor series beginning with quadratic terms. From (), we can observe that the dynamics of any center manifold of () are equivalent to the exosystem dynamics, which by assumption are Lyapunov stable and non-attractive. Aulbach s theorem completes the proof. Remark. Theorem. actually holds for more general MIMO control-affine systems with m D p. In[],it is shown that if the composite control-affine system Px D f.x, w/ C Pw D s.w/ y D h.x, w/ mx g i.x, w/u i has a well-defined relative degree at.x, w/ D., /, then the associated FBI equations are solvable if the zero dynamics of the composite system are hyperbolic. In this case, the FBI equations reduce to a center manifold equation of the form (7) so that Aulbach s theorem can be applied when the exosystem is two dimensional. Example. The dynamics of a cart and inverted pendulum system can be written in the form. id Px D x Px D u Px 3 D x 4 () Px 4 D g` sin.x 3/ ` cos.x 3/u, where x is the position of the cart, x 3 is the angle the pendulum makes with the vertical, g is the acceleration due to gravity, ` is the length of the rod, and u is the control force. With h.x/ D x, the system has relative degree r D at x D and therefore., / D.x/ D.h.x/, L f h.x// D.x, x /. With., / D.x/ D.x 3, x 4 C.x =`/ cos.x 3 //, the zero dynamics are given by Ṕ D Ṕ D sin. /, g` whose linearization at D has eigenvalues pg=`. Hence, with system output y D x C p.w/ (p real analytic) and a two-dimensional real analytic exosystem whose linearization has non-zero eigenvalues, there exists a unique and real analytic solution to the associated FBI equations of the cart and inverted pendulum system (). Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

6 PATCHY SOLUTION TO A FBI PDE 5 3. COMPUTATION OF THE CENTER MANIFOLD In this section, we outline a continuation method to compute the solution to the FBI equations in the case of two-dimensional exosystem and real analytic data. As described in the previous section, for the nonlinear control systems in consideration, the solvability of the FBI equations can be reduced to solving a center manifold equation for a dynamical system of the form Ṕ D B C NZ.w, w, / Pw D w C P.w, w / () Pw D w C Q.w, w /, where w D.w, w / R, R n, NZ, P,andQ are real analytic mappings beginning with quadratic terms, and the eigenvalues of B have non-zero real parts. We will therefore limit our considerations to solving the center manifold equation for (). It will be assumed that the w-dynamics have w D as a Lyapunov stable and non-attractive equilibrium. By Theorem., there exists a unique analytic mapping.w, w /, defined locally about w D, solving the center manifold PDE associated to (). Our method is best described on the representation of () in polar coordinates. Hence, we apply the transformation.w, w, / D.r cos, r sin, / to () yielding a system of the form Pr D r OR., r/ P D C O., r/ (3) Ṕ D B C OZ., r, /, where OR, O, OZ are analytic functions converging for each Œ, and jrj 6 a, k k 6 a, where a>is a positive constant. Define f., O r, / D B C OZ., r, /. The center manifold PDE for (3) is f., O r,., r// D Œ C O., r/ C r r OR., r/ (4) for the unknown analytic mapping., r/ (D.r cos, r sin /). The mapping has a power series representation X., r/ D e i./ r i iš id converging in a cylinder of the form Œ,, jrj 6, and with -periodic coefficients e i./ [7]. By eliminating the time variable t, (3) can be reduced to dr D rr., r/ d (5a) d D B C Z., r, /. d (5b) Define f., r, / D B C Z., r, /. From (4), it follows that f., r,., r// D C dr r d. (6) We now give a brief sketch of our method. Let r./ be a solution to (5a) and define the mapping., /D., r./c / for Œ, and jj small. We note that with r D r./ substituted into the right-hand side of (5b), the curve., / D., r.// is the solution to (5b) with initial condition./ D., r.//.forjj sufficiently small, we have a power series representation X i i., /D., / C., / (7) i iš id Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

7 5 C. O. AGUILAR AND A. J. KRENER converging for all Œ, and having -periodic coefficients i = i., /. In fact, it is easy to see that i i., / D., r.//. (8) i ri By construction, the mapping is a perturbation of., r.// in the radial direction, the amount of perturbation given by the parameter. Our method is based on computing the Taylor series approximation N., /D., / C NX id i i., / i iš and using it to build the center manifold along r./ in the radial direction. Having followed N along a small annular region, say of the form ¹., r/ W 6 6, r./ 6 r<r./c º, we compute a new radial curve 7! Qr./ with initial condition Qr./ D r./c, compute the new corresponding Taylor series approximation Q N for Q., / D., Qr./ C /, and then continue building the center manifold by following Q N along the annular region ¹., r/ W 6 6, Qr./ 6 r<qr./cqº. This process is repeated and the annular regions, along with the corresponding approximations, are patched together to form a piecewise smooth approximation to the true solution. 3.. Computation of the coefficients i i To compute the Taylor series approximation N, it is necessary to compute the -dependent coefficients i = i., / appearing in (7), which can be performed in the following way. From the definition of, a direct computation gives that D., r./c /C., r./c /dr r d, which when combined with (6) yields D f., r./c,., //. (9) Using (9), we can derive a linear inhomogeneous ODE for the coefficient i = i., /. Indeed, differentiating (9) with respect to and interchanging the order of differentiation yields., / D f and therefore., / f., r./c,., //., /C., r./c,., // r D A./ f., / C., r./,., //, r where the matrix A./ D f =., r./,., //. In general, it can be verified by induction that i., / D A./ i i., / C F i i,., /,., /, :::, i., / () i for some mappings F i, i >. In the implementation of our method, it will be convenient to relate the coefficients i = i., / and i Q = i., / obtained from nearby trajectories r./ and Qr./, respectively, of (5a). Suppose Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

8 PATCHY SOLUTION TO A FBI PDE 53 then that r./ and Qr./ are two solutions of (5a), and let./ D Qr./ r./. As before, let., /D., r./c / and Q., /D., Qr./C /.Then Q., / D., Qr.// D r r., r./c.// D X j D Cj., r.//./j, rcj jš where the last series converges for all Œ, provided./ is sufficiently small. Using (8), we can write that Q., / D X j D Cj., /./j. Cj jš Therefore, if =., /, :::, N = N., / are known, we can approximate Q =., / as Q., /., / C., /./ CC N., /./N N NŠ A similar argument shows that for i ¹,, :::º i Q., / D X i j D icj., /./j icj jš and thus if i = i., /, :::, N = N., / are known we can approximate i Q = i., / as i N Q., / X i i j D icj., /./j icj jš () provided i 6 N. 3.. Continuation algorithm With the previous constructions in mind, we are now ready to describe our continuation algorithm for computing the solution to the center manifold (6). () Let N > be a fixed positive integer and let N N., r/ D X e i./ r i iš, that is, N is the N th-order Taylor approximation of in r. To compute N, one can use the method in [5] to generate a N th-order Taylor polynomial approximation of.w, w /, say N.w, w /,andthen N., r/ D N.r cos, r sin /. For notational consistency, set D and set r./ D for R. The initial approximation N will be accepted in an annular region of the form id ¹., r/ W 6 6, 6 r<r./º, where r./ is the solution to (5a) with some prescribed initial condition r./ D >. To compute accurate numerical solutions to r, we solve a BVP using (5a) with boundary conditions r./ D r./ D and constant initial guess on Œ,. () Define., / D., r./ C /. From (7), can be approximated by the truncated series N., /D., / C NX id i., / i i iš Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

9 54 C. O. AGUILAR AND A. J. KRENER for jj small. To compute., / D., r.//, we solve a BVP using (5b) with boundary conditions./ D./ and initial guess., / N., r.//. Similarly, to compute the coefficient i = i., /, 6 i 6 N, we solve a BVP using the ODE () with boundary conditions i = i., / D i = i., / and initial guess i., / i N., r i r i.//. Having computed., /, =., /, :::, N = N., /, we obtain an approximation., r/ to., r/ defined by which is accepted in the region., r/ D N., r r.//, ¹., r/ W 6 6, r./ 6 r<r./ C º () for some desired >. In this way, we have extended our original approximation of to the domain (). Our running approximation of is given by 8 <., r/, 6 r<r./,., r/ :., r/, r./ 6 r 6 r./ C for Œ,. (3) We now proceed to augment to our running approximation a mapping that will be defined on an annular region surrounding the domain of in the following way. We first compute the solution r./ to (5a) with initial condition r./ D r./ C. This is performed by solving a BVP using (5a) with boundary conditions r./ D r./ D r./ C and taking the curve r./ C as an initial guess to r. Here, we note that to avoid overlapping domains of definition between and, the domain () of is redefined to be ¹., r/ W 6 6, r./ 6 r<r./º. (4) We now repeat Step with r./ and build an approximation to., /D., r./ C / of the form N., /D., / C NX id i i., / i iš, for 6 6 and >small. To compute., / D., r.//, we solve a BVP using (5b) with boundary conditions./ D./ and initial guess., /., r.//. Similarly, the coefficients i = i., /, 6 i 6 N, are computed by solving a BVP using the ODE () with boundary conditions i = i., / D i = i., / and initial guess (see ()) i N X i., / i j D icj., /.r./ r.// j. icj jš Having computed., /, =., /, :::, N = N., /, we obtain the approximation., r/ D N., r r.//, Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

10 PATCHY SOLUTION TO A FBI PDE 55 which is accepted in the region ¹., r/ W 6 6, r./ 6 r<r./ C º. (3) In this way, we extend our approximation of the true solution to the annulus (3), and our running approximation is 8., r/, 6 r<r./, ˆ<., r/., r/, r./ 6 r<r./, ˆ:., r/, r./ 6 r 6 r./ C for Œ,. (5) Steps 3 4 can now be iterated. Indeed, suppose we have computed an approximation k., r/ to., r/ of the form k., r/ D N k., r r k.// and defined in the region where k., /D., r k./ C /, ¹., r/ W 6 6, r k./ 6 r<r k./ C k º, (4) N k., /D k., / C NX id i k., / i i iš and r k./ is the solution to (5a) with initial condition r k./ D k id i. To extend our current approximation of beyond the domain of k, we begin by computing the solution r k./ to (5a) with initial condition r k./ D r k./ C k. This is performed by solving a BVP using (5a) with boundary conditions r./ D r./ D r k./ C k and initial guess r k./ C k.letnow kc., /D., r k./ C / and form N kc., /D kc., / C NX id i kc., / i i iš. To compute kc., /, we solve a BVP using (5b) with boundary conditions./ D./ and initial guess kc., / k., r k.//. Similarly, the coefficients i kc = i., / are computed by solving a BVP using the ODE () with boundary conditions i kc = i., / D i kc = i., / and initial guess i N kc X i., / i j D icj k., /.r k./ r k.// j. icj jš Having computed kc., /, kc =., /, :::, N kc = N., /, we obtain the approximation kc., r/ D N kc., r r k.//, which is accepted in the region ¹., r/ W 6 6, r k./ 6 r<r k./ C kc º. (5) To avoid overlapping the domains of k and kc, the domain of definition (4) of k is redefined to be ¹., r/ W 6 6, r k./ 6 r 6 r k./º. Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

11 56 C. O. AGUILAR AND A. J. KRENER After iterating Steps 3 4 a k > number of times, we obtain the following piecewise smooth approximation to., r/: 8., r/, 6 r<r./, ˆ<., r/, r 6 r<r./,., r/ Q k., r/ WD.. ˆ: k., r/, r k 6 r 6 r k./ for Œ,. Let us make a few remarks about our algorithm. (6) Remark 3. () The coefficients i = i., /, i D,, :::, N, are computed by solving boundary value problems for mainly two reasons: () they are known apriorito be periodic, and () we have good approximations of them from the previously computed coefficients (). () The main computational effort of our method is in computing the coefficients i = i., /. From (), we see that the ODE for i = i., / is linear in i = i., / and is polynomial in the previously computed., /, :::, i = i., /. Hence, a way to speed up the computation is to solve for the coefficients i = i., / order by order. This can result in computational savings when the right-hand side of () is complicated to evaluate or when n N is large. (3) When the w dynamics are given by the harmonic oscillator Pw D w Pw D w, the computation of the radial curves r./ is trivial and is given by the constant curves r./ D r./. In this case, there is no need to redefine the outer boundary of the successive approximations i when going from one annulus to another. (4) For the algorithm to produce a meaningful approximation to, the domain on which the approximation (6) is defined, namely ¹., r/ W 6 6, 6 r 6 r k./º, mustbe contained in the cylinder Œ,, jrj 6 on which is defined. Because is not known apriori, the algorithm must proceed from r k to r k by taking small increments r k./ r k./ D k and choosing r./ sufficiently small. To end this section, we prove that the sequence of approximations ¹ Q k º obtained from (6) kd converges uniformly to. Theorem 3. Suppose that is defined on the cylinder D¹., r/ W 6 6, 6 r 6 º and let Q <be chosen so that if r W Œ,! R is a trajectory of (5a) with r./ 6 Q, thenr./ <.Let Q k be defined as in (6), with radial step-size r j./ r j./ D j WD.=.k C //Q, forj D,, :::, k. Then Q k! uniformly in Q D¹., r/ W 6 6, 6 r 6 r k./º. Proof We first note that the existence of Q follows by Lyapunov stability of the exosystem. By construction, r j./ D..j C /=.k C //Q for j D,, :::, k, and in particular r k./ DQ, thereby rendering the domain Q independent of k. Also, we note that by shrinking Q if necessary, by Gronwall s lemma it follows that jqr./ Nr./j 6 jqr./ Nr./je K, (7) for all trajectories 7! Qr./ and 7! Nr./ of (5a) such that 6 Qr./, Nr./ 6 Q, wherek is a Lipschitz constant independent of. Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

12 PATCHY SOLUTION TO A FBI PDE 57 Now by definition, we have that j., /D., r j./ C / and therefore N C j., /D N C N C r., r N C j./ C /. Hence, by Taylor s theorem with integral remainder, j., /D N j., /C NŠ Z Z. / N N C j., /d N C C N N. / r., r N C j./ C /d. D j N., /C NŠ Let C D max.,r/ k N C =r N C., r/k, and we note that C N C =r N C on the compact set. We therefore have that exists by continuity of k j., / j N C jjn., /k 6 C.N C /Š provided 6 6 r j./ r j./ for Œ,, forallj D,, :::, k. Now because r j./ 6 Q for j D,, :::, k, it follows by (7) that jr j./ r j./j 6 jr j./ r j./je K 6 k C QeK. Therefore, given., r/ Q, say that r j./ 6 r<r j./ for some j ¹,, :::, kº, it follows that k., r/ Q k., r/kdk j., r r j.// j., r r j.//k Dk j., r r j.// j N., r r j.//k 6 6 C.N C /Š jr r j./j N C C N C.N C /Š k C QeK. Hence, k., r/ Q k., r/k! as k!uniformly in Q. This completes the proof. 4. EXAMPLES In this section, we present examples illustrating our method. Example 4. In this example, we take a linear dynamical system of the form (), whose center manifold is easily computed, and perform a nonlinear change of coordinates and obtain a nonlinear system on which we apply our method. The true solution to the nonlinear system is then readily available, and we can compare the approximations produced by our method with the true solution. Consider then the linear dynamical system Px D x C w C w Px D x 3 C 3 w C 3 w Px 3 D x w C w Pw D w Pw D w. Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

13 58 C. O. AGUILAR AND A. J. KRENER The center manifold equation for this system in the unknown mapping x D.w/ is.w/sw D C.w/C Dw, w where x D.x, x, x 3 /, w D.w, w /,and S D, C D A, D D A. 3 3 It is straightforward to verify that.w/ D. =3w, =w =6w, =w =6w / is the unique solution to the center manifold equation for this system. Consider the coordinate change D Z.x/ D. 3x, 9x 6x, x C x 3 C. 3x, 9x 6x //, where W R! R is a smooth function. The system in., w/ coordinates takes the form of () with matrix B having eigenvalues, = p3i. By direct substitution, the solution to the center manifold equation of the system in., w/ coordinates is D Z..w// D.w, w,.w, w //, thatis, the graph of the function. For purposes of illustration, we take the egg carton shaped function.w, w / D sin.w / sin.w /, whose graph is shown in Figure (left). The patchy approximation Q k.w, w / D.w, w, Q.w, w // computed with our method with k D annular regions of thickness D.5 and of order N D is shown in Figure (center). The error between the patchy approximation and the true solution is shown in Figure (right). Example 4. The following is an example of the rapidly growing behavior of polynomial approximations that can lead to destabilizing effects in an output tracking controller. We proceed as in Example 4. but instead use the volcano type function.w, w / D sin.w C w /e w w, whose graph is shown in Figure (left). The patchy approximation and the error in using the patchy approximation are shown in Figure (center and right). The patchy approximation is constructed with k D 6 annular regions of thickness D.5 and of order N D ; that is, we only use a first-order Taylor series in the radial direction. For this example, polynomial approximations of orders up to 3 where tested on the domain in consideration, and it was verified that as one increases the order of the polynomial approximation the error in fact increases. Example 4.3 Consider the inverted pendulum cart system from Example. with two-dimensional exosystem given by Pw D s.w/ D w w aw 3, (8) w w w w 5 w 5 5 w 5 Figure. True solution.w, w / D sin.w / sin.w / (left); patchy solution Q.w, w / with degree N D, k D annular regions, and radial step-size D.5 (center); error.w, w / Q.w, w / (right). Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

14 59 PATCHY SOLUTION TO A FBI PDE w w w w w w Figure. True solution.w, w / D sin.w Cw /e w w (left); patchy solution Q.w, w / with degree N D, k D 6 annular regions, and radial step-size D.5 (center); error.w, w / Q.w, w / (right) w 4 3 w w w Figure 3. Patchy approximation to.w, w / (left) and.w, w / (right) with degree N D, k D 4 annular regions, and radial step-size D.5. where a >. System (8) is a special case of the unforced Duffing s oscillator with no damping []. The equilibrium w D of (8) is a center. Let p.w/ D w. As in Example., let., / denote the standard normal coordinates, where D.x3, x4 C.x =`/ cos.x3 // and D.h.x/, Lf h.x// D.x, x /. Following the notation at the beginning of Section, let '.w/ D p.w/ D w and '.w/ D Ls p.w/ D w. Then the differential equation (8) for this system is given by P D w cos. / ` g P D sin. / w sin. / C w sin. / cos. / ` ` ` wp D w (9) wp D w aw3. A patchy approximation to the solution.w, w / D..w, w /,.w, w // of the center manifold equation of (9) was computed and is illustrated in Figure 3. The data g D, ` D =3, and a D =4 were used. The solution was computed with k D 4 annular regions, and the radial Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

15 6 C. O. AGUILAR AND A. J. KRENER y ref (t) y(t) 3 4 time [sec] time [sec] Figure 4. Output y.t/ D x.t/ and reference y ref.t/ D w.t/ (left), and tracking error e.t/ D y.t/ y ref.t/ (right). step at the initial angle D was taken to be i D.5, i D,, :::, 4. The order of the radial Taylor polynomials was chosen as N D. The computed patchy approximation to the center manifold PDE for (9) is used in an output tracking controller of the form., w/ D.w/ C K.., /.w//, where.w/ D..w/, '.w// and.w/ D u e..w/, w/, and where the gain matrix K is chosen as the solution to a linear-quadratic regulator problem for the linearization of the inverted pendulum. In the linear-quadratic regulator problem, the matrices Q D diag.4, 4, 4, 4/ and R D were chosen. A simulation is performed in which the pendulum is initialized at an angle of 5 from the vertical, and the cart is initialized at.5 from the origin. The reference trajectory, y ref.t/ D w.t/, is chosen with initial condition y ref./ D.. The results of the simulation are shown in Figure CONCLUSION We have presented a method to compute solutions to the FBI equations of real analytic controlaffine systems with two-dimensional exosystems. Our technique is based on the periodic nature of two-dimensional analytic center manifolds. In comparison with direct Taylor polynomial approximations [5, 6], our method lessens the computational effort needed to produce approximate solutions by taking into account the periodicity of the center manifold. We proved that our method generates a sequence of approximations converging uniformly to the true solution. ACKNOWLEDGEMENTS The authors thank the anonymous referees for their valuable suggestions. The research in this paper was performed while the first author held a National Research Council Postdoctoral Fellowship. The research was supported in part by AFOSR and NSF. REFERENCES. Isidori A, Byrnes CI. Output regulation of nonlinear systems. IEEE Transactions on Automatic Control 99; 35:3 4.. Francis BA. The linear multivariable regulator problem. SIAM Journal on Control and Optimization 977; 5: Carr J. Applications of Centre Manifolds Theory. Springer-Verlag: New York, NY, USA, Sijbrand J. Properties of center manifolds. Transactions of the American Mathematical Society 985; 89(): Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc

16 PATCHY SOLUTION TO A FBI PDE 6 5. Huang J, Rugh WJ. An approximation method for the nonlinear servomechanism problem. IEEE Transactions on Automatic Control 99; 37: Krener AJ. The construction of optimal linear and nonlinear regulators. In Systems, Models and Feedback: Theory and Applications, Isidori A, Tarn TJ (eds). Birkhäuser: Boston, 99; Aulbach B. A classical approach to the analyticity problem of center manifolds. Journal of Applied Mathematics and Physics 985; 36(): Navasca C, Krener AJ. Patchy solution of the HJB PDE. In Modeling, Estimation and Control, Lecture Notes in Control and Information Sciences, Vol. 364, Chiuso A, Ferrante A, Pinzoni S (eds): Berlin, Germany, 7; Isidori A. Nonlinear Control Systems, (3rd edn). Springer: London, Great Britain, Perko L. Differential Equations and Dynamical Systems. Springer-Verlag: New York, NY, USA, 99.. Huang J. On the solvability of the regulator equations for a class of nonlinear systems. IEEE Transactions on Automatic Control 3; 48(5): Guckenheimer J, Holmes P. Nonlinear oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag: New York, NY, USA, 983. Copyright John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 3; 3:46 6 DOI:./rnc