An L Disturbance Attenuation Solution to the Nonlinear Benchmark Problem Panagiotis Tsiotras Department of Mechanical, Aerospace and Nuclear Engineering University of Virginia, Charlottesville, VA 9- Tel: (8) 9- FA: (8) 98- E-mail: tsiotras@virginia.edu Martin Corless School of Aeronautics and Astronautics Purdue University, West Lafayette, IN 9-18 Tel: (5) 9-11 FA: (5) 9- E-mail: corless@ecn.purdue.edu Mario A. Rotea School of Aeronautics and Astronautics Purdue University, West Lafayette, IN 9-18 Tel: (5) 9-1 FA: (5) 9- E-mail: rotea@ecn.purdue.edu 1
Abstract In this paper, we use the theory of L disturbance attenuation for linear (H1) and nonlinear systems to obtain solutions to the Nonlinear Benchmark Problem (NLBP) proposed in the paper by Bupp et. 1 al.. By considering a series expansion solution to the Hamilton-Jacobi-Isaacs Equation associated with the nonlinear disturbance attenuation problem, we obtain a series expansion solution for a nonlinear controller. Numerical simulations compare the performance of the third order approximation of the nonlinear controller with its rst order approximation (which is the same as the linear H1 controller obtained from the linearized problem.) Keywords: Nonlinear Benchmark Problem, Hamilton-Jacobi Equation, Disturbance Attenuation, Series Expansions.
1 Introduction Control of nonlinear systems has received much attention in recent years and many analysis techniques and design methodologies have been developed {11. It is important to determine the advantages and limitations of these dierent nonlinear control design methodologies. The Nonlinear Benchmark Problem (NLBP) proposed by Bupp et. al. 1 is an initial attempt to achieve this objective. The NLBP involves a cart of mass M which is constrained to translate along a straight horizontal line. The cart is connected to an inertially xed point via a linear spring see Figure 1. Mounted on the cart is a \proof body" actuator of mass m and moment of inertia I. Relative to the cart, the proof body rotates about a vertical line passing through the cart mass center. The horizontal external force F acting on the cart is to be regarded as a disturbance force. A motor on the cart can be used to generate a torque N to control the proof mass in such a way that the force F has minimal eect on the cart's position. In other words, it is desirable to attenuate as much as possible the eect of the external (unknown) force F on the cart by appropriate choice of the control input N. The nonlinearity of the problem comes from the interaction between the translational motion of the cart and the rotational motion of the eccentric proof mass. After suitable normalization 1, the equations of motion for this nonlinear system are + = " ( _ sin ; cos )+w (1a) = ;" cos + u (1b) where is the (non-dimensionalized) displacement of the cart and is the angular position of the proof body. For a complete derivation of the equations of motion, see Ref. [1]. In equations (1), w and u are the (non-dimensionalized) disturbance and control inputs, respectively. The coupling between the translational and rotational motions is captured by the parameter " which is dened by " := me p (I + me )(M + m) where e is the eccentricity of the proof body. Clearly, "<1and" = if and only if e = in this case the translational and rotational motions decouple and equations (1) reduce to + = w = u This system is clearly not stabilizable from the control input also, the eect of w is completely decoupled from the eect of u. Letting x := [x 1 x x x ] T := [ _ ] _ T, system (1) can be written compactly in state-space form as _x = x ;x 1+"x sin x 1;" cos x x " cos x (x 1;"x sin x) 1;" cos x 5 + ;" cos x 1;" cos x 1 1;" cos x 5 u + These equations are well dened since 1 ; " cos x = for all x and "<1. A Simple Stabilizing Controller 1 1;" cos x ;" cos x 1;" cos x () 5 w () A minimum requirement for any acceptable controller is that it asymptotically stabilizes the system in the absence of external disturbances. In this section we show that the stabilization problem for the Nonlinear Benchmark Problem has a very simple solution. In particular, we show that a simple linear controller globally asymptotically stabilizes system (). Proposition.1 System () (with w =) is globally asymptotically stabilized using the linear controller where k 1 > and k >. u = ;k 1 ; k _ ()
Proof. Consider the Lyapunov function candidate V (x) = 1 _ + 1 _ + " cos + 1 + 1 k 1 (5) To demonstrate that V is positive denite, note that V (x) = 1 xt P ()x where P () := 1 1 " cos k 1 " cos 1 Since the eigenvalues of P () are f1 k 1 1 " cos g and j cos j 1, "<1, we have 5 V (x) 1 min(p ()) jjxjj min (P ()) minfk 1 1 ; "g > Hence V is a positive denite function. Dierentiating V along the closed-loop trajectories with controller () we get that dv dt = ;k _ () Thus the closed-loop system is stable about the zero state and all trajectories are bounded. To demonstrate asymptotic stability, consider any solution x() of the closed system for which _ V (x(t)). Then _ (t) this implies that (t) := (), (t), and (using the closed loop system description) + = (a) " cos + k 1 = (b) From equation (b) we have that cos = hence, is constant. Equation (a) then implies that is constant aswell. Since (t) is the only constant solution to this equation, we have (t) (t) _ (t) and, utilizing (b), (t) thus x(t). By LaSalle's results, system (1) with control law () is globally asymptotically stable. We note here that the simple linear controller in Eq. () was simultaneously derived and presented for the rst time by Tsiotras et al.. 1 and Jankovic et al. 1 during the invited session devoted to the NLBP in the 1995 American Control Conference. The derivation of the linear controller in Jankovic et al. 1 was based on passivity arguments, however. Disturbance Attenuation The previous section demonstrated that a linear controller globally asymptotically stabilizes system () when there is no disturbance acting on the system. Ourmainobjective in this paper is to design a controller that will minimize the eect of the disturbance input w on some pre-specied performance output z given by Cx z = (8) u where the matrix C can be regarded as a collection of design parameters. We will suppose that w belongs to the set of functions which are square integrable, that is, we assume that w L [ 1) where L [ 1) denotes the set of square-integrable functions with domain [ 1). We propose the following control design problem to address the qualitative design guidelines given in Bupp et al. 1. Disturbance Attenuation Problem (DAP): Let be a specied positive scalar. Obtain a memoryless state-feedback controller u = k(x) (9) for system () such that the corresponding closed loop system has the following properties.
(a) When w(t) =, the closed loop system is asymptotically stable about the zero state. (b) For zero initial state (x() = ) and for every disturbance input w L [ 1), Z 1 Z 1 jjz(t)jj dt jjw(t)jj dt (1) Note that the second requirement implies that the L -gain of the closed loop system from the disturbance input w to the performance output z is less than or equal to. Since the closed loop system is causal, one can readily show R that the second requirement R above also T implies the following property for any T >. If x() = and jjw(t)jj T dt is nite, then jjz(t)jj dt is nite and satises Z T Z T jjz(t)jj dt jjw(t)jj dt This observation allows us to extend the class of disturbance inputs to those which are square integrable over a nite interval. The Disturbance Attenuation Problem (DAP) has been treated for general classes of systems 15{1. In these references it has been shown that, under mild conditions, the DAP can be solved, provided one has a positive denite solution to the so-called Hamilton-Jacobi-Isaacs Equation. The original idea behind this approach was to formulate the DAP as a dierential game in which u and w are two opposing players. The next section reviews the basic results of Isidori 1 and van der Schaft 1 which are used in this paper. The Hamilton-Jacobi-Isaacs Equation (HJIE) System () along with its performance output is described by _x = F (x)+g 1 (x)u + G (x)w (11a) Cx z = (11b) u where the functions F G 1 G can be obtained from () and F ()=. We assume that the system _x = F (x) z = Cx is observable in the sense that, z(t) =forallt implies x(t) = for all t. One can readily show 1 1 15 that if there is a continuously dierentiable, positive denite, function V which satises the following Hamilton-Jacobi-Isaacs Equation V x (x)f (x) ; 1 V x(x) G 1 (x)g T 1 (x) ; ; G (x)g T (x) V T x (x)+x T C T Cx = (1) where V x is the derivative ofv, i.e, then the feedback controller V x (x) = @V @x 1 (x) @V @x n (x) u = k (x) :=; 1 GT 1 (x)v T x (x) (1) yields a closed loop system with the following property. For every initial condition x() = x and for every disturbance input w L [ 1) onehas Z 1 Also, the \worst case disturbance" is given by Z 1 jjz(t)jj dt jjwjj dt + V (x ) (1) w = l (x) := 1 GT (x)v T x (x) (15) 5
Using V as a Lyapunov function one can show that the undisturbed (w = ) closed loop system corresponding to controller (1) is globally asymptotically stable. Hence, a solution to the DAP is given by controller (1). The main stumbling block in using the above result is that only rarely is one able to compute a function V satisfying (1) in closed-form. So, instead of insisting on closed form solutions, we solve (1) in an iterative fashion based on series expansions. This is the methodology proposed in Al'brekht 18, Lukes 19 (see also Yoshida and Loparo ) for the solution of Hamilton-Jacobi equations arising in optimal control problems. We demonstrate here that the same procedure can be applied to nonlinear L disturbance attenuation problems, provided that the linearized version of the problem has a solution. The approach is similar to previous results by van der Schaft 1, Kang et al. 1 and Huang and Lin. An alternative iterative solution to the HJIE is given by Wise and Sedwick. First we rewrite system (11) in the form where Letting _x = F (x)+g(x)v (1a) Cx z = (1b) u G(x) :=[G 1 (x) G (x)] v := Q(x) :=x T C T Cx R := the Hamilton-Jacobi-Isaacs Equation can be rewritten as and letting we have u w 1 ; (1) (18) V x (x)f (x) ; 1 V x(x)g(x)r ;1 G T (x)v T x (x)+q(x) = (19) v (x) := k (x) l (x) v (x) =; 1 R;1 G T (x)v T x (x) () Note that the matrix R in equation (18) is not positive denite. In fact, it is an indenite matrix. 5 A Series Solution Approach to the HJIE The approach we follow in solving HJIE (19) is based on a series expansion of the desired solution V. For problems involving a small parameter (as the parameter " in the NLBP) this methodology typically expands the function V in terms of the parameter. If the zero order problem (setting the parameter to zero) is solvable, then an iterative procedure can be readily devised to generate all the higher order terms in the series expansion. However, according to the discussion at the end of Section 1, the DAP is not solvable for the zero order NLBP hence a perturbation method based on " will not work for the NLBP. Alternatively, one may seek a series expansion of the function V in terms of the state x. Using (1), this will yield a series expansion for the controller k which solves the DAP. This is the approach considered here. Note that this approach pre-supposes that the HJIE has a (real) analytic solution, i.e., a solution with convergent Taylor series expansion. This assumption may be restrictive, in general, since it is well known that solutions to Hamilton-Jacobi type equations may have non-dierentiable (let alone analytic) solutions even if the system dynamics and the cost function are smooth. See Ref. [], Example.1.8.
5.1 Linearized Problem In the next section, it will be shown that the rst term in the series expansion for controller (1) is the solution to the corresponding linearized problem. Thus, we rst consider the linearized DAP. The linearization of system (11) about x = is given by with _x = Ax + B 1 u + B w (1a) Cx z = (1b) u A = F x () B 1 = G 1 () B = G () From standard H 1 theory, the DAP problem for the above linear system is solvable i it is solvable via a linear state feedback controller. In addition, this DAP is equivalent to obtaining a stabilizing controller which, for the closed loop system, achieves an H 1 norm (for the transfer function from w to z) of magnitude less than or equal. Considering a quadratic form V (x) =x T Px () as a candidate solution to the HJIE associated with the linear DAP we obtain x T [PA+ A T P ; PBR ;1 B T P + C T C]x = where B := [B 1 B ]. This is satised for all x i the matrix P solves the following Algebraic Riccati Equation (ARE): PA+ A T P ; PBR ;1 B T P + C T C = () Also, V is positive denite i the matrix P is positive denite. In this case and the controller which solves the linear DAP is given by v (x) =;R ;1 B T Px () k (x) =;B T 1 Px (5) According to standard H 1 theory, ifthepair(c A) is observable and the pair (A B 1 ) is stabilizable, the existence of a positive denite symmetric solution P to the above ARE,with A := A ; BR ;1 B T P () Hurwitz, is a necessary and sucient condition for the linear DAP to have a solution 5. 5. Nonlinear Problem We seek to obtain a solution V to the HJIE by considering a series expansion of the form V (x) =V [] (x)+v [] (x)+ () where V [k] is a homogeneous function of order k. A homogeneous function of order k in n scalar variables x 1 x ::: x n is a linear combination of N n k := n + k ; 1 k terms of the form x i1 1 xi :::xin n, where i j is a nonnegative integer for j =1 ::: n and i 1 + i + + i n = k. The vector whose components consist of these terms is denoted by x [k] for example, with two scalar variables
one has x [1] = x1 x x [] = " x1 x 1x x # x [] = x 1 x 1x x 1x x 1x x 1x x x 1x x x x x x x Therefore, a homogeneous function [k] of order k can be written as [k] (x) = x [k] where IR 1Nn k. We assume that F (x) andg(x) have series expansions of the form F (x) = F [1] (x)+f [] (x)+ (8a) G(x) = G [] (x)+g [1] (x)+ (8b) where each componentoff [k] and G [k] are homogeneous functions of order k. Note that F [1] (x) =Ax G [] (x) =B Substituting () into () one obtains a series expansion for v of the form where v [k] is the homogeneous function of order k given by v [k] v = v [1] + v [] + (9) = ; 1 R;1 k;1 j= 5 G [j] T V [k+1;j] T x () and where explicit dependence on x has been dropped for notational simplicity. Also, one can obtain a series expansion for the desired controller k of the form k = k [1] + k [] + (1) where k [k] is a homogeneous function of order k consisting of the rst p components of v [k] and u IR p. To compute the terms in the series expansion for V, rst note that HJIE (19) can be written as V x (x)f (x) ; v T (x)r(x)v (x)+q(x) = (a) v (x)+ 1 R;1 (x)g T (x)v T x (x) = (b) Substitution of the expansions in ()-(9) into (a) and equating terms of order m to zero yields m; k= For m = equation () simplies to m;1 V x [m;k] F [k+1] ; k=1 v [m;k] T Rv [k] + Q [m] = () Since F [1] (x) =Ax and V [] x F [1] ; v [1] Rv [1] + Q [] = v [1] (x) =; 1 R;1 B T V [] T x (x) Q [] (x) =x T C T Cx we obtain V x [] (x)ax ; 1 V x [] T (x)br ;1 B T V x [] (x)+x T C T Cx = 8
which is the HJIE for the linearized problem. Hence V [] (x) =x T Px where P T = P > solves the ARE with A := A ; BR ;1 B T P Hurwitz also, and Consider now any m and rewrite () as m; k= v [1] (x) =;R ;1 B T Px () k [1] (x) =;B T 1 Px (5) m; V x [m;k] F [k+1] ; v [m;1] T Rv [1] ; k= v [m;k] T Rv [k] = Note that the last term in the above expression does not depend on V [m]. Using and dening the rst two terms can be written as where m; k= m; V x [m;k] F [k+1] + v [m;1] T = ; 1 m; V x [m;k] G [k] R ;1 k= k= f(x) :=F (x)+g(x)v [1] (x) () V [m;k] x G [k] v [1] = m; and A is given by (). For m, equation () can now be written as m; V x [m] f [1] = ; k=1 k= V x [m;k] f [k+1] m; = V x [m] f [1] + V x [m;k] f [k+1] k=1 f [1] (x) =A x () m; V x [m;k] f [k+1] + k= v [m;k] T Rv [k] (8) Equation (8) can be solved for V [m] as follows. Consider an expression for V [m] (x) of the form V [m] (x) = V m x [m], with V m IR 1Nn m. Substitute this expression for V [m] (x) into (8) and solve the resulting linear system of Nm n equations for the unknown Nm n elements of the coecient vector V m. It can be shown 1, that if the eigenvalues of the matrix A in () are nonresonant y then this linear equation has a unique solution for all m. In other words, the HJIE can be solved to any order. Moreover, since the eigenvalues of A are in the left-half of the complex plane, the solution V of the HJIE is analytic and the series () converges. Thus, starting with V [] (x) =x T Px and v [1] (x) =;R ;1 B T Px one can use equations (8) and () to compute consecutively the sequence of terms V [] (x) v [] (x) V [] (x) v [] (x) ::: (9) and construct iteratively the solution V of HJIE and the associated v. Notice that this procedure generates not only the feedback controller k (x) dened in (1) for disturbance attenuation, but also the worst case disturbance strategy l (x) given in (15). y A set of eigenvalues f1 ::: ng is called resonant if P n j=1 i j j = for some nonnegative integers i 1 i ::: in such that P n j=1 i j >. Otherwise, it is called nonresonant 1. 9
The Nonlinear Benchmark Problem According to the specications set forth by Buppet al. 1 it is desired that the (non-dimensionalized) variables satisfy jj 1:8 and juj 1:11 () These performance specications suggest that the (1 1) element of the matrix C T C be equal to 1:8 =1:11 :8). For simplicity, we choose the (1 1) element to be unity. No specic requirements are provided for the other state variables, so the elements penalizing these variables are chosen to be one order of magnitude smaller. Thus, the following matrix is chosen for the performance output z C = diag(1 p :1 p :1 p :1) (1) The eccentricity parameter is given as " = :. In order to apply the proposed methodology to the NLBP, we rst expand the F and G vector elds in the right hand side of () in a series expansion. Noting that 1 ; " cos x =, these expansions can be readily computed as F (x) = x ; 5 x 1 + 5 x x + 5 5 x 1x + x 5 x 1 ; 1 x x ; 5 x 5 1x +.1 Linear problem The linearized system is described by (1) with A = 1 ; 5 1 5 5 G(x) = 5 B 1 = ; 5 5 ; 5 + 5 5 x + 5 ; 5 5 x + 5 ; 5 5 x + ; 5 + 5 5 x + 5 B = 5 ; 5 5 5 () Using this data one can show that the linearized DAP has a solution if and only if > 5:5. From the results of van der Schaft, this value provides a lower bound of the achievable L -gain of the nonlinear system. Choosing = one can then solve ARE () for P > to obtain The matrix A in () is P = A = 19:8 ;:88 ;:8 ;:88 ;:88 15:59 :89 1:9915 ;:8 :89 : :9 ;:88 1:9915 :9 1:15 1: ;1:1 :1 :9 : 1: :8 1:1 ;: ;1:1 and has eigenvalues f;:15 i 1:15 ;: i :8g. The linear term of v (x) can be computed from () to obtain v [1] :18 x (x) = 1 +1:15 x ; :1 x ; 1:8 x ;:5 x 1 +:8 x +:15 x +:9 x 5 () 5 () where the rst row is the controller k [1] (x) and the second row the disturbance strategy l [1] (x). 1
. Higher order terms Since A is Hurwitz, its set of eigenvalues is nonresonant. Therefore, according to the discussion in Section 5.1 the series solution to the HJIE can be computed to any order and this series converges 1. The calculations are simplied for the NLBP because, as it is evident from the expressions for F (x) and G(x), F [k] (x) = G [k;1] (x) = k =1 ::: (5) As a result, V [] (x) =andv [] (x) =. The rst nonzero higher order term for the controller is third order and can be computed from v [] (x)=; 1 R;1 B T V [] T V [] The solution of these equations yields x x (x)a x=;v x [] (x) (x)+g []T (x)v x [] T (x) F [] (x)+g [] (x)v [1] (x) V [] (x) = 1:111 x 1 +91:15 x ; :9 x x x 1 ; :1 x x x ; :1911 x x +59:9 x x 1 ; :818 x x x 1 +:1915 x x ; 8:9 x x x 1 +:89 x ; 151:885 x x 1 +5:8 x x x 1 ; 19:918 x x x 1 +9:5 x x +:5 x x x 1 ; 5:9 x x x x 1 ; 58:9 x x 1 +:1 x x 1 ; 18:18 x x 1 ; 9:85 x x 1 + 9:891 x x x 1 ; : x x x 1 ; :8 x x +:1118 x x x + 9:158 x x x ; :8 x x ; 9: x x 1 +8:58 x x + :88 x x +:15 x x +9:9 x x 1 ; :15 x x x 1 and v [] (x) = + 8:5 x x ; :95 x x 1 ; :15 x ;:185 x x +:99 x x +8:19 x x 1 ; :89 x x ;:51 x ; 1:8 x x x ; :911 x x +:18 x x ;:11 x x 1 +8:55 x x x 1 ;1:8 x x +:5 x x x 1 +9:1991 x x x 1 +: x ;5:555 x x 1 +:8 x x 1 ; 1:99 x x 1 +5:9 x 1 ;1:158 x +:81 x 1 x :11 x x ; : x x ; :8 x x 1 +1:159 x x +:189 x +1:8 x x x +1:895 x x +: x x +:55 x x 1 ; 5:8 x x x 1 +:95x x ; :18 x x x 1 ;1:919 x x x 1 ; :18 x +8:888 x x 1 ; :51 x x1 +1:19 x x 1 ; :95 x 1 +:995 x ; :98 x 1x 5 () Specically, the rst row in equation () yields the controller term k [] and the second row yields the disturbance strategy term l []. In fact, because of (5), one can show that all the even terms of the series expansion for v are zero, i.e., v [k] = for k =1 :::. Discussion As mentioned at the end of Section the HJIE may, in general, fail to have (real) analytic solutions. For the NLBP, nevertheless, the existence of a (convergent) analytic solution is insured by the analyticity of the Hamiltonian of this problem and the fact that the matrix A in equation () has all its eigenvalues in the left-half of the complex plane (cf. Corollary. in Kang et al. 1 ). The proposed methodology has certainly some limitations. First, in contrast to linear problems, one cannot guarantee that (even global) asymptotic stability of the closed-loop system together with w L [ 1) 11
implying that z L [ 1) andul [ 1). This technical diculty limits the attenuation properties of the controller to those disturbance inputs w L [ 1) which do not drive the state of the closed system outside some neighborhood of the origin van der Schaft, for example, considers only w with compact support in order to ensure that x(t)! ast!1. Another potential problem associated with the series expansion method for solving the HJIE is the lack of ecient algorithms for checking positive deniteness of the solution () in the large. By virtue of the positive deniteness of the matrix P in equation (), we know that V is at least locally positive denite. This ensures local asymptotic stability with the (truncated) optimal controller. Nevertheless, global results are not available, at least as far as the authors know. We mention, however, that a conservative approach for ensuring positive deniteness of the solution of the HJIE has been recently reported by Shue et al.. In light of the previous discussion it should be apparent that the truncated controller in equation (9) can only guarantee local asymptotic stability and disturbance attenuation level less than or equal to for disturbances which do not \push" the trajectories \too far away" from the origin 1,. In that respect, the truncated controller can be naturally viewed as a higher order correction to the linear controller. Both the linear and the nonlinear H 1 controllers locally asymptotically stabilize the system. In general, one expects that the region of attraction of the nonlinear controller is larger than the region of attraction of the linear controller. For the NLBP this was checked via numerical simulations. One should be cautioned, however, to the fact that due to the higher order polynomial terms the nonlinear controller may exhibit a more dramatic onset of instability thanthelinearh 1 controller. For the NLBP this could be remedied, for example, by using the simple linear controller of equation () far away from the origin and switching to the nonlinear (or even the linear) H 1 controller once close to the origin. This, we believe, will oer the most practical solution to the NLBP. 8 Numerical Simulations Here we present some numerical results for the NLBP. Symbolic calculations for controller design were performed using Maple and numerical simulations were carried using the ode5 command of Matlab. All the plots correspond to non-dimensionalized variables. We rst demonstrate the global stabilizing properties of the simple linear controller. Simulations were carried out for dierent initial conditions and asymptotic stability was veried in all cases, as predicted by Proposition.1. The results of one of these simulations are illustrated in Figure. The initial condition for these simulations is x()=[11] T and the controller parameters were chosen as k 1 =1andk =1. Although it has been shown that the simple linear controller in equation () is globally asymptotically stabilizing for the nonlinear system, its has no obvious disturbance attenuation properties. On the other hand, the linear H 1 controller k [1] is only locally asymptotically stabilizing. It is expected, however, that the controller k [1] will have superior disturbance attenuation properties than the simple. In order to verify this, we numerically simulated the system in Eqs. () both with the simple linear controller and the linear H 1 controller k [1] subject to zero initial conditions. The disturbance was chosen as w = :1 sin(t). This is a sinusoidal disturbance with frequency which corresponds to the peak of the singular value plot (magnitude of transfer function from w to z versus frequency) of the closed-loop linearized systems see Figure. The gains k 1 and k in equation () were chosen to be approximately equal to the corresponding terms in the linear H 1 gain matrix (k 1 =:5 k = 1). The results of these simulations are shown in Figure. The linear controller k [1] (although only locally asymptotically stabilizing) has much better disturbance attenuation properties than the linear controller in Eq. (). We next compare the linear controller k [1] and the nonlinear controller k [1] + k [] with respect to the region of attraction they achieve for the closed loop system. This is illustrated in Figures 5- which contain the \phase portraits" of the variables and. For initial condition x() = [;1 1 ; 1 1] T, the dashed lines denote the response due to the linear controller and the solid lines denote the response due to the nonlinear controller. From this simulation, it seems that the region of attraction due to the nonlinear controller is larger than that due to the linear controller: for the chosen initial state, the state trajectory resulting from the nonlinear controller tends asymptotically to the origin, whereas the trajectory resulting from the linear controller tends to a limit cycle. The corresponding histories of the states and are shown in Figure. Figure 8 depicts the control history. 1
Finally, we compare the disturbance attenuation properties of the two controllers k [1] and k [1] + k []. For sinusoidal disturbances of small magnitude the two controllers had almost identical performance. This is to be expected, since in these cases the term k [] is negligible. For larger magnitude of the disturbance the nonlinear controller performed better. Figure 9, for example, shows the case when w = :sin(t) (zero initial conditions). The linear controller in this case is unable to keep the motion of the system bounded, whereas the nonlinear controller results in a bounded motion. For much larger values of the disturbance, both controllers were unable to keep the motions bounded. 9 Conclusion We have applied the theory of L disturbance attenuation for nonlinear systems to the recently proposed nonlinear benchmark problem. A nonlinear state-feedback controller is computed recursively by considering a series expansion solution to the associated Hamilton-Jacobi-Isaacs Equation. The procedure is straightforward and can be readily automated in a computer. Numerical simulations indicate that the performance of the third order approximation of the nonlinear controller provides some improvement over its rst order approximation (which is the same as a linear H 1 controller obtained from the linearized problem). This improvement is however not very signicant, thus indicating that higher order terms may be necessary to extend the region of attraction of the closed-loop system, or to enhance the performance of the controller. Future issues should include the choice of weighting lters for shaping the system response, a rather dicult task for nonlinear systems. Also, alternative approaches to the solution of Hamilton-Jacobi type equations, as well as non-conservative algorithms for investigating positive deniteness of the solutions to these equations are highly desirable. Acknowledgments This research was supported in part by the National Science Foundation under Grants CMS-9-188, MSS- 9-59 and ECS-9-5888, and in part by the Boeing Company. References 1. Bupp, R. T., Bernstein, D. S., and Coppola, V. T., \Benchmark Problem for Nonlinear Control Design,", Proceedings of the American Control Conference (1995), {, Seattle, WA.. Corless, M., \Control of Uncertain Nonlinear Systems,", ASME Journal of Dynamic Systems, Measurement, and Control, 115, { (199).. Desoer, C. A., and Vidyasagar, M., Feedback Systems: Input-Output Properties, Academic Press, New York, 195.. Hedrick, K. H., \Analysis and Control of Nonlinear Systems,", ASME Journal of Dynamic Systems, Measurement, and Control, 115, 51{1 (199). 5. Isidori, A., Nonlinear Control Systems: An Introduction, Springer-Verlag, New York, 1989.. Khalil, H. K., Nonlinear Systems, nd ed., Prentice Hall, New Jersey, 199.. Krstic, M., Kanellakopoulos, I., and Kokotovic, P., Nonlinear and Adaptive Control Design, Wiley and Sons, New York, 1995. 8. Leitmann, G., \On One Approach to the Control of Nonlinear Systems,", ASME Journal of Dynamic Systems, Measurement, and Control, 115, {8 (199). 9. Nijmeijer, H., and van der Schaft, A. J., Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 199. 1
1. van der Schaft, A. J., \Nonlinear State Space H 1 Control Theory,", In Essays on Control, H. L. Trentelman and J. C. Willems, Eds. Birkhauser, Boston, 199, 15{19. 11. Utkin, V. I., \Variable Structure Systems with Sliding Modes,", IEEE Transactions on Automatic Control,, 1{ (19). 1. Bupp, R., Coppola, V. T., and Bernstein, D. S., \Vibration Suppression of Multi-Modal Translational Motion Using a Rotational Actuator,", Proceedings of the rd Conference on Decision and Control (199), {, Lake Buena Vista, FL. 1. Tsiotras, P., Corless, M., and Rotea, M., \An L Disturbance Attenuation Approach to the Nonlinear Benchmark Problem,", Proceedings of the American Control Conference (1995), 5{5, Seattle, WA. 1. Jankovic, M., Fontaine, D., and Kokotovic, P. V., \TORA Example: Cascade and Passivity Control Designs,", Proceedings of the American Control Conference (1995), {51, Seattle, WA. 15. Ball, J. A., and Helton, W., \H 1 Control for Nonlinear Plants: Connections with Dierential Games,", Proceedings of the 8th IEEE Conference on Decision and Control (1989), 95{9, Tampa, FL. 1. Isidori, A., \Feedback Control of Nonlinear Systems,", International Journal on Robust and Nonlinear Control,, 91{11 (199). 1. van der Schaft, A. J., \L -Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H 1 Control,", IEEE Transactions on Automatic Control,,, {8 (199). 18. Al'brekht, E. G., \On the Optimal Stabilization of Nonlinear Systems,", Journal of Applied Mathematics and Mechanics, 5, 15{1 (19). 19. Lukes, D. L., \Optimal Regulation of Nonlinear Dynamical Systems,", SIAM Journal on Control and Optimization,, 1, 5{1 (199).. Yoshida, T., and Loparo, K. A., \Quadratic Regulatory Theory for Analytic Non-linear Systems with Additive Controls,", Automatica, 5,, 51{5 (1989). 1. Kang, W., De, P. K., and Ididori, A., \Flight Control in a Windshear via Nonlinear H 1 Methods,", Proceedings of the 1st Conference on Decision and Control (199), 115{11, Tucson, AZ.. Huang, J., and Lin, C., \Numerical Approach to Computing Nonlinear H 1 Control Laws,", Journal of Guidance, Control, and Dynamics, 18, 5, 989{99 (1995).. Wise, K. A., and Sedwick, J. L., \Nonlinear H 1 Optimal Control for Agile Missiles,", Journal of Guidance, Control, and Dynamics, 19, 1, 15{15 (199).. van der Schaft, A. J., \L -Gain and Passivity Techniques in Nonlinear Control,", In Lecture Notes in Control and Information Science, vol. 18. Springer-Verlag, London, 199. 5. Green, M., and Limebeer, D., Linear Robust Control, Prentice Hall, New Jersey, 1995.. van der Schaft, A. J., \On a State Space Approach to Nonlinear H 1 Control,", Systems and Control Letters, 1, 1{8 (1991).. Shue, S.-P., Sawan, M. E., and Rokhsaz, K., \Optimal Feedback Control of a Nonlinear System: Wing Rock Example,", Journal of Guidance, Control, and Dynamics, 19, 1, 1{11 (199). 1
k M N F θ e m Figure 1: Nonlinear Benchmark Problem. 15
1.5 LINEAR GLOBALLY ASYMPTOTICALLY STABILIZING CONTROLLER ξ ξ States.5 1 5 1 15 5 5 1 Time States.5 θ θ.5 5 1 15 5 5 Time Figure : State histories due to the simple linear controller. 1
Linear Hoo Controller Singular Values db 1 1 Simple Linear Controller 1 1 1 1 1 Frequency Figure : Singular value plots of linear controllers. 1
1 ξ 1 Linear Hoo Controller Simple Linear Controller 1 5 8 9 1 Time Figure : Comparison of disturbance attenuation properties of the linear controllers. 18
Dashed: Linear Solid: Nonlinear 1.5 1.5 ξ.5 1 1.5.5 1.5 1.5.5 1 1.5 ξ Figure 5: Phase portrait of for linear and nonlinear H 1 controllers. 19
Dashed: Linear Solid: Nonlinear 1 θ 1 5 1 1 5 θ Figure : Phase portrait of for linear and nonlinear H 1 controllers.
Dashed: Linear Solid: Nonlinear ξ 1 1 5 1 15 5 Time θ 5 1 15 5 Time Figure : Time history of and for linear and nonlinear H 1 controllers. 1
Dashed: Linear Solid: Nonlinear u 5 1 15 5 Time Figure 8: Control history for linear and nonlinear H 1 controllers.
Dashed: Linear Solid: Nonlinear 15 1 5 ξ 5 1 15 Linear Hoo Controller Nonlinear Hoo Controller 5 1 15 5 5 Time Figure 9: Disturbance attenuation for linear and nonlinear H 1 controllers.