Nonlinear H control and the Hamilton-Jacobi-Isaacs equation

Transcription

1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 8 Nonlinear H control and the Hamilton-Jacobi-Isaacs equation Henrique C. Ferreira Paulo H. Rocha Roberto M. Sales University of São Paulo, Department of Telecommunication and Control Engineering, Av. Prof. Luciano Gualberto, trv. 3, n. 58, CEP 558-9, São Paulo (SP), Brazil henrique@lac.usp.br, roberto@lac.usp.br Brazilian Navy Technology Center, Electro-Electronic Project Division, Av. Prof. Lineu Prestes, n. 468, CEP 558-, São Paulo (SP), Brazil Abstract: This paper considers to aspects of the nonlinear H control problem: the use of eighting functions for performance and robustness improvement, as in the linear case, and the development of a Galerkin approximation method for the solution of the Hamilton-Jacobi-Isaacs Equation (HJIE) that arises in the output feedback case. Design of nonlinear H controllers obtained by Taylor approximation and by the proposed Galerkin approximation method applied to a magnetic levitation system are presented. Keyords: H-infinity control; Robust control; Robust estimators; Nonlinear control systems; Partial differential equations.. INTRODUCTION The development of a systematic analysis of the nonlinear equivalent of the H control problem as initiated by the important contributions of Ball and Helton (989), Başar and Bernhard (99), and van der Schaft (99). Although the H norm is defined as a norm on transfer matrices, hen translated into the time domain, it is nothing else than the L -induced norm (from the input time-functions to the output time-functions for initial state zero). van der Schaft (99) shoed that the solutions of the problem in question in the case of state feedback can be determined from the solution of a Hamilton-Jacobi equation, the Hamilton-Jacobi-Isaacs Equation (HJIE) or Inequality, hich is the nonlinear version of the Riccati equation considered in the H control problem for linear systems. The nonlinear H control problem via output feedback as considered, for example, by Isidori and Astolfi (99), Ball et al. (993), and Isidori and Kang (995). The nonlinear output feedback H controllers have separation structures and necessary and sufficient conditions for the H control problem to have solutions involving to HJIE s, associated ith the design of a statefeedback and an output-injection gain, respectively. Although the formulation of the nonlinear theory of H control has been ell developed, solving the HJIE remains a challenge and is the major bottleneck for the practical application of the theory (Beard and McLain, 998; Aliyu, 3; Abu-Khalaf et al., 4). The HJIE is a first-order, nonlinear partial differential equation not solved analytically in the general case and usually very difficult to be solved for specific nonlinear systems. Thus, several numerical methods have been proposed for its solution. Starting ith the ork of Lukes (969), ho proposed a polynomial approximation approach based on Taylor series, many others authors have proposed similar approaches to the solution of the problem. Huang and Lin (995), for example, find a smooth solution to the HJIE by solving for the Taylor series expansion coefficients in a efficient and organized manner. Beard and McLain (998) combine successive approximation and Galerkin approximation methods to derive a novel algorithm that produces stabilizing state feedback control las ith elldefined stability regions. This paper has to purposes. Firstly, it is shon that dynamic eighting functions can be used to improve the performance and robustness of the nonlinear H controller such as in the design of H controllers for linear plants. In the literature only static eighting functions have been explored. Secondly, the Galerkin successive approximation method is used to find approximate solutions for the nonlinear H control problem in the output feedback case. The results are applied to a magnetic levitation system.. NONLINEAR H OUTPUT FEEDBACK CONTROLLER Consider the folloing affine nonlinear state-space system ẋ = f(x)+g (x) + g (x)u, z = h (x)+k + k u, () y = h (x)+k (x), x R n is the state vector, u R m is the control input, R r is the exogenous input hich includes disturbance (to be rejected) and/or references (to be tracked), z R s is the penalty variable and y R p is the measured variables. The mapping f(x), g (x), g (x), h (x), h (x), k (x), k (x) and k (x) are smooth mappings (i.e., mappings /8/$. 8 IFAC 88.38/876-5-KR-.886

2 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 of class C k for some sufficiently large k) defined in a neighborhood of the origin in R n. It is also assumed that f() =, h () =, andh () =. The control action to () is provided by a controller, hich processes the measured variable y and generates the appropriate control input u by ξ = η(ξ,y), () u = θ(ξ), ξ is defined on a neighborhood Ξ of the origin in R ν and η :Ξ R p R ν, θ :Ξ R m are C k functions (for some k ), satisfying η(, ) = and θ() =. The purposes of the controller are: to achieve closed-loop stability and to attenuate the influence of the exogenous input on the penalty variable z. A controller hich locally asymptotically stabilizes the equilibrium (x, ξ) = (, ) of the closed-loop system is said to be an admissible controller. The requirement of disturbance attenuation is characterized in the folloing manner: given a real number γ >, it is said that the exogenous signals are locally attenuated by γ if there exists a neighborhood U of the point (x, ξ) =(, ) so that for every T>and for every pieceise continuous function :[,T R r for hich the state trajectory of the closed-loop system (), () starting from the initial state (x(),ξ()) = (, ) remains in U for all t [,T, the response z :[,T R s of (), () satisfies T T z T (s)z(s)ds γ T (s)(s)ds. The nonlinear H control problem using output feedback consists in finding an admissible controller yielding local attenuation of the exogenous input. In order to describe the solution of the nonlinear H control problem using output feedback, a notion of detectability is necessary. Definition. Suppose f() = and h() =. The pair {f,h} is said to be locally detectable if there exists a neighborhood U of the point x =so that, if x(t) is any integral curve of ẋ = f(x) satisfying x() U,thenh(x(t)) is defined for all t and h(x(t)) = for all t implies lim t x(t) =. Theorem. (Isidori and Astolfi (99)). Consider system () and suppose the folloing H) The pair (f,h ) is locally detectable. H) There exists a smooth positive definite function V (x), locally defined in a neighborhood of the origin in R n, hich satisfies the HJIE V x f + h T h + γ α T α α T α =, (3) α = γ gt Vx T, α = gt Vx T. (4) H3) There exists an n p matrix G, so that the equilibrium ξ =of the system ξ = f(ξ)+g (ξ)α (ξ) Gh (ξ) (5) is locally asymptotically stable. H4) There exists a smooth positive semidefinite function W (x, ξ), locally defined in a neighborhood of the origin R n R n and such that W (,ξ) > for each ξ,hichsolvesthehjie ( W x W ξ )f e + h T e h e + γ Φ T Φ=, (6) f(x)+g (x)α (x)+g (x)α (x) f e = f(ξ)+g (ξ)α (ξ)+g (ξ)α (ξ), +G(h (x) h (ξ)) (7) h e = α (ξ) α (x), Φ= γ (W xg (x)+w ξ Gk (x)) T. Then, the problem of nonlinear H control is solved by the output feedback ξ = f(ξ)+g (ξ)α (ξ)+g (ξ)α (ξ)+g(y h (ξ)), (8) u = α (ξ). Remark 3. In order to simplify the analysis and to provide a reasonable expression of the controller, it is assumed that the coefficient matrices hich characterize the plant () satisfy the folloing assumptions, hich are the nonlinear versions of the standing assumptions considered, for example, by Doyle et al. (989) k (x) =, h T (x)k (x) =, k(x)k T (x) =I, k (x)g T (x) =,k(x)k T (9) (x) =I. Existence of a solution to (3) and (6) is shon to exist in the conditions of the folloing proposition. Proposition 4. (Isidori and Astolfi (99)). Let system () linearized around the origin, given by ẋ = Ax + B + B u, z = C x + D + D u, () y = C x + D. Assume that the linear system () satisfies L) The pair (A, B ) is stabilizable. L) The pair (A, C ) is detectable. L3) There exists a positive definite symmetric solution X of the Riccati equation A T X + XA + C T C XB B T X + γ XB B T X =.() L4) There exists a positive definite symmetric solution Y of the Riccati equation YA T + AY + B B T YCT C Y + γ YCT C Y =. () L5) ρ(xy ) <γ. (that is, there exists a solution for the linear H control problem via output feedback). Then hypotheses H) H4) of theorem hold and the nonlinear H control problem via output feedback is solvable ith G = ZC T, V(x) =xt Xx, W (x, ξ) =γ (x ξ) T Z (x ξ), ( Z = Y I ) γ XY. 3. THE WEIGHTED NONLINEAR H CONTROL PROBLEM The penalty variable z comprises any output hose L norm is desired to be minimized. In the linear H con- 89

3 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 u Plant + y W W (s) 3 W (s) z z 3 z Then, under the conditions of proposition 4, there ill be a nonlinear H controller for the system (3). In other ords, if the Riccati equations () and () are solvable for the linearized system, then the HJIEs (3) and (6) are solvable in a neighborhood of the origin in R n. 4. GALERKIN SUCCESSIVE APPROXIMATION FOR NONLINEAR H CONTROL PROBLEM VIA OUTPUT FEEDBACK Controller Fig.. Closed loop system including eighting functions troller design, the penalty variable can be eighted as required, either through static eighting, frequency (linear) eighting, nonlinear eighting function or any combination of these. The penalty variable can include tracking error, actuator effort, sensitivity specifications via norm bounds, robustness specifications via norm bounds and frequency response criteria. The selection of eighting functions for linear design problem is detailed in (Zhou and Doyle, 998). On the other hand, in the nonlinear H controller design, only static eighting functions have been explored (e.g., Sinha and Pechev, 4). In the present paper it is shon that dynamic eighting functions can also be used in the nonlinear H controller design, ith similar effects to the linear case. Consider a nonlinear system modeled by equations of the form ẋ p = f p (x p )+g p (x p ) + g p (x p )u, (3) y = h p (x p )+k p (x p ). Consider that linear dynamic eighting functions W (s) and W 3 (s) ith state space description ẋ = A x + B (h p (x p )+k p (x p )), z = C x, ẋ 3 = A 3 x 3 + B 3 h p (x p ), z 3 = C 3 x 3 + D 3 h p (x p ), (4) are included in the design as in Fig.. In addition, consider that z = W u (5) eights the control effort through the constant gain W, for simplicity. Combining (3), (4) and (5), the system canbeexpressedas()ith [ [ xp f p (x p ) x = x, f(x) = A x + B h p (x p ), x 3 A 3 x 3 + D 3 h p (x p ) [ [ g p (x p ) gp (x p ) g (x) = B k p (x p ), g (x) =, [ C x h (x) =, k (x) =[, C 3 x 3 + D 3 h p (x p ) k (x) =[ W, h (x) =h p (x p ), k (x) =k p (x p ). Suppose that there is a linear H controller to the linear system obtained from the linearization of system (3). 4. Successive approximation The HJIE (6) can be reritten as K(x, Wx T,,y ) H(x, Vx T,,u )=, (6) K(x, W T x,,y )=W x f + W x g y T h y T k +h T h γ T = γ gt W x k T h and H(x, Vx T,,u ) is the left-hand side of (3). The basic idea of successive approximation is to compute W and iteratively, as shon belo. Algorithm. Let be an exogenous input ith stability region. For i =to Solve for (i) from W x f + W x g (i) y T h y T k (i) + h T h γ (i) H Update the exogenous input (i+) = γ g T W x kh T End. The essence of algorithm is to reduce the HJIE to an infinite sequence of linear partial differential equations. Since these equations are difficult to be solved analytically, in the next section a Galerkin approximation method ill be used to construct an approximate solution to the HJIE. 4. Galerkin approximation method LetapartialdifferentialequationA(V )=ith boundary conditions V () =. Galerkin method assumes a complete set of basis functions {φ j } j=, so that φ j() =, j and V (x) = j= c jφ j (x), the sum is assumed to converge pointise in some set. An approximation to V is formed by truncating the series to V N (x) = N j= c jφ j (x). The coefficients c j are obtained by solving the algebraic equations A(V N (x))φ l (x)dx =, l =,...,N. (7) For a more rigorous and complete treatment see, for example, (Fletcher, 984). 4.3 Output feedback In this section an algorithm for output feedback control design is proposed. In fact, the proposed algorithm is a 9

4 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 dual version of the state feedback algorithm originally proposed in (Beard and McLain, 998). Assume that : R s is the exogenous input for the system () on a compact set. Also assume that the set {φ (j) } j= is a complete basis set for the domain of the HJIE (6). Then, according to (7), an approximate solution to (6) is given by W N (x) = N j= c(j) φ (j) (x), thecoefficients satisfy the equation [K(x, W T x,,y ) H(x, V T x,,u )φ l dx =. (8) Substituting W (x) by W N (x) in (8) and defining c = [ c ()... c(n) T, φ (x) =[φ ()... φ(n) and φ (x) = [ φ () / x... φ(n) / x, (8), after some algebraic manipulations, can be reritten as ( ) A + A (c (n) ) γ c (n+) = γ b + b + A (c (n) )c (n) 4γ + b 3 4γ + b 4 4, N A = φ f T φ T dx, A (c )= c (j) X(j), j= X (j) φ (j) T = φ g g T φ T x dx, b = φ h T h dx, b = φ f T φ T v c vdx, b 3 = φ c T v φ v g g T φ T v c v dx, b 4 = φ c T v φ v g g T φ T v c v dx. The coefficients c pull outside the integrals, and A (), b () and b 3 (u) can be computed iteratively once the matrices {X (j) } N j= have been calculated. Algorithm. Let W () be an initial stabilizing solution to (). Let c v be the coefficients obtained by algorithm to approximate (3). Pre compute the integrals A, A (), b, b, b 3, b 4 and {X (j) } N j=. For i =to If i = A (i) = A + γ A (W x () ) b (i) = γ b + 4γ A (W x () )c () + b + 4γ b 3 4 b 4 Else A (i) = A + (j)x (j) b (i) = γ b + + 4γ b 3 4 b 4 End c (i) N γ j= c(i ) N 4γ j= c(i ) (j)x (j) c (i ) + b = [ A (i) b (i) End Extract R (x) from x T R (x) =c T φ c T v φ v Extract L(x) from x T L(x) =γ h Compute the output feedback ith G(x) =R (x)l(x). F(t) i(t) mg Fig.. Magnetic levitation system x x (t) 5. NONLINEAR H CONTROL DESIGN FOR A MAGNETIC LEVITATION SYSTEM In this section, nonlinear H controllers are designed and applied to a magnetic levitation system. In section 5. a nonlinear controller based on the Taylor approximation method, as proposed in (van der Schaft, 99), is designed to illustrate the benefits produced by dynamic eighting functions. A similar design using static eighting functions is presented in (Sinha and Pechev, 4). In section 5.3 a controller based on the proposed Galerkin approximation method is designed and compared to the Taylor approximation controller. In order to focus on the controllers, eighting functions are not employed here. 5. Nonlinear model for the magnetic levitation system The magnetic levitation system considered is schematically represented in Fig., i(t) indicates the current through the magnetic bearing, that produces the attraction force F (t), m is the mass to be levitated and x (t) represents the deviation from the desired levitation gap x. Defining u(t) =i (t) as the control action and considering and as disturbances and g the gravity force, the system is described by the folloing equations ẋ = x, x ẋ = g g ( x + x ) k u m ( x + x ) + m, y = x +, hich may be put in the affine form (3), x f p (x)= x, g g g p (x) = ( x + x ) g p (x)= k, h p (x) = m ( x + x ) k p (x)=, k p (x) =[ T, h p (x)=x, k p =[. [ m [ x The adopted parameter values for the system are.6 4 N/m, m =.4kg, x =5mm, g =9.8m/s. x, (9), 9

5 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, x (mm).5 ithout eighting filters ith eighting filters x ( mm) (a) Disturbance rejection - ithout eighting filters ith eighting filters (b) Noise atenuation Fig. 3. Performance and robustness simulation results 5. Taylor approximation design including dynamic eighting functions The eighting function W is specified to produce an integral effect at lo frequencies and W 3 aims at noise attenuation. W (s) = s +.,W (s) =., W 3 (s) = s +. s +. These eighting functions are included as shon in Fig., and the global system is ritten in the affine form as in section 3. Thus, the first order Taylor approximation to the corresponding HJIE (3) produces the folloing state feedback la u = 5.47x x x x 4. () In order to get a simpler observer, W and W 3 ere implemented and the corresponding states, x 3 and x 4, directly measured. Thus, the first order Taylor approximation to the corresponding HJIE (6) produces the folloing output injection [ 6.9 G =. 96. The loest value of γ for hich positive definite solutions ere found to () and (), hich correspond to the first order Taylor approximations to (3) and (6), is γ =. Fig. 3(a) shos the integral effect produced by W ; a unit step disturbance, applied at t =.s, produces a steady state error hen W =and zero steady state error for W as specified. Fig. 3(b) shos the noise attenuation effect produced by W 3 ; for a hite noise ith standard deviation. N, added to, the output variance for W 3 =is mm and 4. 5 mm for W 3 as specified. 5.3 Galerkin Successive Approximation design In this section nonlinear output feedback controller designs based on Taylor approximation and on the proposed Gallerkin approximation are presented. Weighting functions are not included here. In this case, the Taylor approximation design is given by u = [ 476.x + 5.9x, 6.9 G =. 96. () In the Galerkin approximation design the basis functions [ ( x + x ) x x [ x 3 φ v = ( x + x ) x, φ = + 3 x x +3 x 3, are proposed to the state feedback HJIE (3) and to the output feedback HJIE (6), respectively. The state feedback basis is chosen so that the control la given by (4) is linear. Using Galekin successive approximation algorithm for state feedback proposed by Beard and McLain (998), the folloing control la is obtained u = x + 6.7x. The basis for the output feedback design is chosen so that G(x) is different from zero hen the observer error is zero. Thus, algorithm, initialized ith the control la obtained by Taylor approximation (), produces 8, 4 (, 5 3, 95 x +7, 8x +7, 8 x )/(, 5 7 x 5, 76 4 x x +4, 87 6 x +4, 44 7 x 3, 64 7, 4 5 x +3, 48 5 x x +, 65 9 x 4 +, 3 7 x 3 G = x ) 8, 4 (, 49 +5, 95. x +8, 9 4 x +3, 9x +7, 8 x x )/ (, 5 7 x 5, 76 4 x x +4, 87 6 x +4, 44 7 x 3, 64 7, 4 5 x +3, 48 5 x x +, 65 9 x 4 +, 3 7 x 3 x ) The loest value for γ, for hich algorithm converges is γ =, indicating the possibility of a higher noise attenuation. Notithstanding, for comparison purposes ith the Taylor approximation design, γ = as used. The adopted stability domain is x <. 3 and x <.3. In this domain the control la used for initializing algorithm stabilizes the closed loop system. It is orth observing that, hile the Galerkin method does not decrease the stability domain, in the Taylor 9

6 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8..8 Taylor Galerkin x (mm).4 x (m/s) Taylor Galerkin (a) Position (b) velocity Fig. 4. Position and velocity for output feedback for Taylor and Galerkin approximation controllers x (mm) Taylor Galerkin Fig. 5. Noise attenuation approximation method the stability domain is not knon a priori, either. Fig. 4 shos the position x and the velocity x in the levitation transient for the Taylor and Galerkin controllers. As observed, Galerkin controller produces a smoother transient. Fig. 5 shos the higher noise attenuation produced by Galerkin controller. 6. CONCLUSION In this paper, benefits of dynamic eighting functions are explored in nonlinear H control design, similarly to the linear case. An output feedback method based on Galerkin approximation is also proposed and shon to be more efficient than the already established Taylor approximation method in the case of a magnetic levitation system. REFERENCES M. Abu-Khalaf, F. L. Leis, and J. Huang. Hamilton- Jacobi-Isaacs formulation for constrained input nonlinear systems. In 43rd IEEE Conference on Decision and Control, volume 5, pages , 4. M. D. S. Aliyu. An approach for solving the Hamilton- Jacobi-Isaacs equation (HJIE) in nonlinear H control. Automatica, 39(5): , 3. J. A. Ball, J. W. Helton, and M. L. Walker. H control for nonlinear systems ith output feedback. IEEE Transactions on Automatic Control, 38(4): , April 993. J.A. Ball and J.W. Helton. H control for nonlinear plants: connections ith differential games. In Proceedings of the 8th IEEE Conference on Decision and Control, volume, pages , 989. T. Başar and P. Bernhard. H optimal control and related minimax design problems. Birkhäuser, Boston, 99. R. W. Beard and T. W. McLain. Successive Galerkin approximation algorithms for nonlinear optimal and robust control. International Journal of Control, 7(5): , 998. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis. State-space solutions to standard H and H control problems. IEEE Transactions on Automatic Control, 34(8):83 847, Aug C. A. J. Fletcher. Computational Galerkin Methods. Springer Series in Computational Physics. Spring, Ne York, 984. J. Huang and C.-F. Lin. Numerical approach to computing nonlinear H control las. Journal of Guidance, Control and Dynamics, 8(5): , Sept.-Oct A. Isidori and A. Astolfi. Distubance attenuation and H - control via measurement feedback in nonlinear systems. IEEE Transactions on Automatic Control, 37(9):83 93, Sept. 99. A. Isidori and W. Kang. H control via measurement feedback for general nonlinear systems. IEEE Transactions on Automatic Control, 4(3):466 47, March 995. D. L. Lukes. Optimal regulation of nonlinear dynamical systems. SIAM Journal of Control, (7), 969. P. K. Sinha and A. N. Pechev. Nonlinear H controllers for electromagnetic suspension systems. IEEE Transactions on Automatic Control, 49(4): , April 4. A. J. van der Schaft. On a state space approach to nonlinear H control. Systems & Control Letters, 6 (): 8, Jan. 99. A. J. van der Schaft. L -gain analysis of nonlinear systems and nonlinear state feedback H control. IEEE Transactions on Automatic Control, 37(6):77 784, June 99. K. Zhou and J. C. Doyle. Essentials of robust control. Prentice Hall, Upper Saddle River,