An L_2 Disturbance Attenuation Solution to the Nonlinear Benchmark Problem (1998)

An L 2 Disturbance Attenuation Solution to the Nonlinear Benchmark Problem Panagiotis Tsiotras Department of Mechanical, Aerospace and Nuclear Engineering University of Virginia, Charlottesville, VA 22903-2442 Tel: (804) 924-6223 FAX: (804) 982-2037 E-mail: tsiotras@virginia.edu Martin Corless School of Aeronautics and Astronautics Purdue University, West Lafayette, IN 47907-1282 Tel: (765) 494-7411 FAX: (765) 494-0307 E-mail: corless@ecn.purdue.edu Mario A. Rotea School of Aeronautics and Astronautics Purdue University, West Lafayette, IN 47907-1282 Tel: (765) 494-6212 FAX: (765) 494-0307 E-mail: rotea@ecn.purdue.edu 1 Abstract In this paper, we use the theory of L 2 disturbance attenuation for linear (H 1 ) and nonlinear systems to obtain solutions to the Nonlinear Benchmark Problem (NLBP) proposed in the paper by Bupp et. al. 1 . 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 H 1 controller obtained from the linearized problem.) Keywords: Nonlinear Benchmark Problem, Hamilton-Jacobi Equation, Disturbance Attenuation, Se- ries Expansions. 2 1 Introduction Control of nonlinear systems has received much attention in recent years and many analysis techniques and design methodologies have been developed 2{11 . It is important to determine the advantages and limitations of these di�erent 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 e�ect on the cart's position. In other words, it is desirable to attenuate as much as possible the e�ect 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  � + � = " ( _ � 2 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. [12]. 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 de�ned by " := me p (I +me 2 )(M +m) (2) where e is the eccentricity of the proof body. Clearly, 0 � " < 1 and " = 0 if and only if e = 0; 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 e�ect of w is completely decoupled from the e�ect of u. Letting x := [x 1 x 2 x 3 x 4 ] T := [� _ � � _ �] T , system (1) can be written compactly in state-space form as _x = 2 6 6 6 4 x 2 �x 1 +"x 2 4 sinx 3 1�" 2 cos 2 x 3 x 4 " cosx 3 (x 1 �"x 2 4 sinx 3 ) 1�" 2 cos 2 x 3 3 7 7 7 5 + 2 6 6 4 0 �" cosx 3 1�" 2 cos 2 x 3 0 1 1�" 2 cos 2 x 3 3 7 7 5 u+ 2 6 6 4 0 1 1�" 2 cos 2 x 3 0 �" cosx 3 1�" 2 cos 2 x 3 3 7 7 5 w (3) These equations are well de�ned since 1� " 2 cos 2 x 3 6= 0 for all x 3 and " < 1. 2 A Simple Stabilizing Controller 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 (3). Proposition 2.1 System (3) (with w = 0) is globally asymptotically stabilized using the linear controller u = �k 1 � � k 2 _ � (4) where k 1 > 0 and k 2 > 0. 3 Proof. Consider the Lyapunov function candidate V (x) = 1 2 _ � 2 + 1 2 _ � 2 + " _ � _ � cos � + 1 2 � 2 + 1 2 k 1 � 2 (5) To demonstrate that V is positive de�nite, note that V (x) = 1 2 x T P (�)x where P (�) := 2 6 6 4 1 0 0 0 0 1 0 " cos � 0 0 k 1 0 0 " cos � 0 1 3 7 7 5 Since the eigenvalues of P (�) are f1; k 1 ; 1� " cos �g and j cos �j � 1, 0 � " < 1, we have V (x) � 1 2 � min (P (�)) jjxjj 2 ; � min (P (�)) � minfk 1 ; 1� "g > 0 Hence V is a positive de�nite function. Di�erentiating V along the closed-loop trajectories with controller (4) we get that dV dt = �k 2 _ � 2 � 0 (6) 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)) � 0. Then _ �(t) � 0; this implies that �(t) � � 0 := �(0),  �(t) � 0, and (using the closed loop system description)  � + � = 0 (7a) "  � cos � 0 + k 1 � 0 = 0 (7b) From equation (7b) we have that cos � 0 6= 0; hence,  � is constant. Equation (7a) then implies that � is constant as well. Since �(t) � 0 is the only constant solution to this equation, we have �(t); _ �(t);  �(t) � 0 and, utilizing (7b), �(t) � 0; thus x(t) � 0. By LaSalle's results 6 , system (1) with control law (4) is globally asymptotically stable. We note here that the simple linear controller in Eq. (4) was simultaneously derived and presented for the �rst time by Tsiotras et al.. 13 and Jankovic et al. 14 during the invited session devoted to the NLBP in the 1995 American Control Conference. The derivation of the linear controller in Jankovic et al. 14 was based on passivity arguments, however. 3 Disturbance Attenuation The previous section demonstrated that a linear controller globally asymptotically stabilizes system (3) when there is no disturbance acting on the system. Our main objective in this paper is to design a controller that will minimize the e�ect of the disturbance input w on some pre-speci�ed performance output z given by z = � Cx u � (8) 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 2 L 2 [0;1) where L 2 [0;1) denotes the set of square-integrable functions with domain [0;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 speci�ed positive scalar. Obtain a memoryless state-feedback controller u = k(x) (9) for system (3) such that the corresponding closed loop system has the following properties. 4 (a) When w(t) = 0, the closed loop system is asymptotically stable about the zero state. (b) For zero initial state (x(0) = 0) and for every disturbance input w 2 L 2 [0;1), Z 1 0 jjz(t)jj 2 dt � 2 Z 1 0 jjw(t)jj 2 dt (10) Note that the second requirement implies that the L 2 -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 that the second requirement above also implies the following property for any T > 0. If x(0) = 0 and R T 0 jjw(t)jj 2 dt is �nite, then R T 0 jjz(t)jj 2 dt is �nite and satis�es Z T 0 jjz(t)jj 2 dt � 2 Z T 0 jjw(t)jj 2 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{17 . In these references it has been shown that, under mild conditions, the DAP can be solved, provided one has a positive de�nite solution to the so-called Hamilton-Jacobi-Isaacs Equation. The original idea behind this approach was to formulate the DAP as a di�erential game in which u and w are two opposing players. The next section reviews the basic results of Isidori 16 and van der Schaft 17 which are used in this paper. 4 The Hamilton-Jacobi-Isaacs Equation (HJIE) System (3) along with its performance output is described by _x = F (x) +G 1 (x)u+G 2 (x)w (11a) z = � Cx u � (11b) where the functions F;G 1 ; G 2 can be obtained from (3) and F (0) = 0. We assume that the system _x = F (x) z = Cx is observable in the sense that, z(t) = 0 for all t � 0 implies x(t) = 0 for all t � 0. One can readily show 10;14;15 that if there is a continuously di�erentiable, positive de�nite, function V which satis�es the following Hamilton-Jacobi-Isaacs Equation V x (x)F (x) � 1 4 V x (x) � G 1 (x)G T 1 (x)� �2 G 2 (x)G T 2 (x) � V T x (x) + x T C T Cx = 0 (12) where V x is the derivative of V , i.e, V x (x) = � @V @x 1 (x) � � � @V @x n (x) � then the feedback controller u = k � (x) := � 1 2 G T 1 (x)V T x (x) (13) yields a closed loop system with the following property. For every initial condition x(0) = x 0 and for every disturbance input w 2 L 2 [0;1) one has Z 1 0 jjz(t)jj 2 dt � 2 Z 1 0 jjwjj 2 dt+ V (x 0 ) (14) Also, the \worst case disturbance" is given by w = l � (x) := 1 2 2 G T 2 (x)V T x (x) (15) 5 Using V as a Lyapunov function one can show that the undisturbed (w = 0) closed loop system corresponding to controller (13) is globally asymptotically stable. Hence, a solution to the DAP is given by controller (13). The main stumbling block in using the above result is that only rarely is one able to compute a function V satisfying (12) in closed-form. So, instead of insisting on closed form solutions, we solve (12) in an iterative fashion based on series expansions. This is the methodology proposed in Al'brekht 18 , Lukes 19 (see also Yoshida and Loparo 20 ) 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 2 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 17 , Kang et al. 21 and Huang and Lin 22 . An alternative iterative solution to the HJIE is given by Wise and Sedwick 23 . First we rewrite system (11) in the form _x = F (x) +G(x)v (16a) z = � Cx u � (16b) where G(x) := [G 1 (x) G 2 (x)] ; v := � u w � (17) Letting Q(x) := x T C T Cx; R := � 1 0 0 � 2 � (18) the Hamilton-Jacobi-Isaacs Equation can be rewritten as V x (x)F (x) � 1 4 V x (x)G(x)R �1 G T (x)V T x (x) +Q(x) = 0 (19) and letting v � (x) := � k � (x) l � (x) � we have v � (x) = � 1 2 R �1 G T (x)V T x (x) (20) Note that the matrix R in equation (18) is not positive de�nite. In fact, it is an inde�nite 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 (13), 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-di�erentiable (let alone analytic) solutions even if the system dynamics and the cost function are smooth � . � See Ref. [24], Example 6.1.8. 6 5.1 Linearized Problem In the next section, it will be shown that the �rst term in the series expansion for controller (13) is the solution to the corresponding linearized problem. Thus, we �rst consider the linearized DAP. The linearization of system (11) about x = 0 is given by _x = Ax+B 1 u+B 2 w (21a) z = � Cx u � (21b) with A = F x (0); B 1 = G 1 (0); B 2 = G 2 (0) 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 (22) 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 = 0 where B := [B 1 B 2 ]. This is satis�ed 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 = 0 (23) Also, V is positive de�nite i� the matrix P is positive de�nite. In this case v � (x) = �R �1 B T Px (24) and the controller which solves the linear DAP is given by k � (x) = �B T 1 Px (25) According to standard H 1 theory, if the pair (C; A) is observable and the pair (A;B 1 ) is stabilizable, the existence of a positive de�nite symmetric solution P to the above ARE, with A � := A�BR �1 B T P (26) Hurwitz, is a necessary and su�cient condition for the linear DAP to have a solution 25 . 5.2 Nonlinear Problem We seek to obtain a solution V to the HJIE by considering a series expansion of the form V (x) = V [2] (x) + V [3] (x) + � � � (27) where V [k] is a homogeneous function of order k. A homogeneous function of order k in n scalar variables x 1 ; x 2 ; : : : ; x n is a linear combination of N n k := � n+ k � 1 k � terms of the form x i 1 1 x i 2 2 : : : x i n n , where i j is a nonnegative integer for j = 1; :::; n and i 1 + i 2 + � � �+ i n = k. The vector whose components consist of these terms is denoted by x [k] ; for example, with two scalar variables 7 one has x [1] = � x 1 x 2 � ; x [2] = " x 1 2 x 1 x 2 x 2 2 # ; x [3] = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 x 3 1 x 2 1 x 2 x 2 1 x 3 x 1 x 2 2 x 1 x 2 x 3 x 1 x 2 3 x 3 2 x 2 2 x 3 x 2 x 2 3 x 3 3 3 7 7 7 7 7 7 7 7 7 7 7 7 5 Therefore, a homogeneous function [k] of order k can be written as [k] (x) = x [k] where 2 IR 1�N n k . We assume that F (x) and G(x) have series expansions of the form F (x) = F [1] (x) + F [2] (x) + � � � (28a) G(x) = G [0] (x) +G [1] (x) + � � � (28b) where each component of F [k] and G [k] are homogeneous functions of order k. Note that F [1] (x) = Ax; G [0] (x) = B Substituting (27) into (20) one obtains a series expansion for v � of the form v � = v [1] � + v [2] � + � � � (29) where v [k] � is the homogeneous function of order k given by v [k] � = � 1 2 R �1 k�1 X j=0 G [j] T V [k+1�j] T x (30) 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 [2] � + � � � (31) where k [k] � is a homogeneous function of order k consisting of the �rst p components of v [k] � and u 2 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) = 0 (32a) v � (x) + 1 2 R �1 (x)G T (x)V T x (x) = 0 (32b) Substitution of the expansions in (27)-(29) into (32a) and equating terms of order m � 2 to zero yields m�2 X k=0 V [m�k] x F [k+1] � m�1 X k=1 v [m�k]T � Rv [k] � +Q [m] = 0 (33) For m = 2 equation (33) simpli�es to V [2] x F [1] � v [1] � Rv [1] � +Q [2] = 0 Since F [1] (x) = Ax and v [1] � (x) = � 1 2 R �1 B T V [2]T x (x) ; Q [2] (x) = x T C T Cx ; we obtain V [2] x (x)Ax � 1 4 V [2]T x (x)BR �1 B T V [2] x (x) + x T C T Cx = 0 8 which is the HJIE for the linearized problem. Hence V [2] (x) = x T Px where P T = P > 0 solves the ARE with A � := A�BR �1 B T P Hurwitz; also, v [1] � (x) = �R �1 B T Px (34) and k [1] � (x) = �B T 1 Px (35) Consider now any m � 3 and rewrite (33) as m�2 X k=0 V [m�k] x F [k+1] � 2v [m�1]T � Rv [1] � � m�2 X k=2 v [m�k] T � Rv [k] � = 0 Note that the last term in the above expression does not depend on V [m] . Using v [m�1]T � = � 1 2 m�2 X k=0 V [m�k] x G [k] R �1 and de�ning f(x) := F (x) +G(x)v [1] � (x) (36) the �rst two terms can be written as m�2 X k=0 V [m�k] x F [k+1] + m�2 X k=0 V [m�k] x G [k] v [1] � = m�2 X k=0 V [m�k] x f [k+1] = V [m] x f [1] + m�2 X k=1 V [m�k] x f [k+1] where f [1] (x) = A � x (37) and A � is given by (26). For m � 3, equation (33) can now be written as V [m] x f [1] = � m�2 X k=1 V [m�k] x f [k+1] + m�2 X k=2 v [m�k] T � Rv [k] � (38) Equation (38) 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 2 IR 1�N n m . Substitute this expression for V [m] (x) into (38) and solve the resulting linear system of N n m equations for the unknown N n m elements of the coe�cient vector V m . It can be shown 21,22 that if the eigenvalues of the matrix A � in (26) are nonresonant y then this linear equation has a unique solution for all m � 3. 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 (27) converges. Thus, starting with V [2] (x) = x T Px and v [1] � (x) = �R �1 B T Px one can use equations (38) and (30) to compute consecutively the sequence of terms V [3] (x); v [2] � (x); V [4] (x); v [3] � (x); : : : (39) and construct iteratively the solution V of HJIE and the associated v � . Notice that this procedure generates not only the feedback controller k � (x) de�ned in (13) for disturbance attenuation, but also the worst case disturbance strategy l � (x) given in (15). y A set of eigenvalues f� 1 ; � 2 ; : : : ; � n g is called resonant if P n j=1 i j