Analytic Nonlinear Inverse-Optimal Control for Euler Lagrange System

Transcription

1 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER 847 Analytic Nonlinear Inverse-Optimal Control for Euler Lagrange System Jonghoon Park and Wan Kyun Chung, Member, IEEE Abstract Recent success in nonlinear control design is applied to the control of Euler Lagrange systems. It is known that the eistence of optimal control depends on solvability of the so-called Hamilton Jacobi Isaccs (HJI) partial differential equation. In this article, the associated HJI equation for nonlinear inverse-optimal control problem for Euler Lagrangian system is solved analytically. The resulting control is referred to as the reference error feedback, which takes conventional PID controller form. Consequently, robust motion control can be designed for robot manipulators using -gain attenuation from eogenous disturbance and parametric error. Inde Terms Euler Lagrange system, nonlinear inverse-optimal control, nonlinear -gain attenuation, reference error feedback (REF), reference motion compensation. I. INTRODUCTION Many physical systems can be modeled by Lagrangian equations of motion. Denoting the generalized coordinates of such a system by q =(q 1 ;q ;...;q n ) T < n yields = M (q)q + C(q; _q)_q_q_q + g(q) +d(t) (1) where < n is the input, d(t) is the (nonmeasurable) eogenous disturbance, M (q) < nn is the inertia matri that is symmetric and positive definite for every q, C(q; _q) < nn represents the matri of Coriolis and centripetal forces, and g(q) < n denotes the gravitational forces. Any Euler Lagrangian system has the property that the matrices M (q) and C(q; _q) satisfy _M (q; _q) fc(q; _q) +C T (q; _q)g = () which is a restatement of the well-known skew-symmetry of _ M (q; _q) C(q; _q) [1], where (1) T denotes the transpose of (1). Since tracking of the coordinate q to a given desired trajectory q DES (t) < n bounded functions of time is of concern, let us define e = q DES (t) q: (3) One conventional dynamic controller is based on nonlinear dynamics compensation of (1), e.g., = M (q)q REF + C(q; _q)_q + g(q) (4) Manuscript received April 9, 1998; revised September 17, 1999, February 17,, and September,. This paper was recommended for publication by Associate Editor F. C. Park and Editor A. De Luca upon evaluation of the reviewers comments. This paper was presented in part at the IEEE International Conference on Robotics and Automation (ICRA), Leuven, Belgium, May 16, J. Park is with the Automation Research Center (ARC) and Robotics & Bio- Mechatronics (RNB) Laboratory, Pohang University of Science and Technology (POSTECH), Pohang , Korea. W. K. Chung is with the Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang , Korea. Publisher Item Identifier S 14-96X() where the reference acceleration is defined by q REF =q DES + k v _e + k pe: (5) The closed-loop dynamics is rendered to reference acceleration tracking linear dynamics q =q REF if compensation is perfect and no eogenous disturbances eist. However, in practical robot systems there is uncertainty in identifying real parameters, denoted by < s. The available parameters, denoted by ^, result in ^M (q), ^C(q; _q), and ^g(q), which are different from M (q), C(q; _q), and g(q), respectively. This parameter error also acts as disturbance to the system. Upon these settings, the problem that we address in this paper is the design of a controller that can attenuate the effect of disturbance on control error with control input that is as small as possible. The disturbances consist of eogenous disturbance, d(t), and parametric error, ~ = ^. This problem is referred to as (induced) L -gain attenuation problem, or nonlinear H1 optimal control problem [], [3]. It is well known that the design problem of nonlinear H1 optimal control reduces to solving a partial differential inequality or equation, called Hamilton Jacobi Isaccs (HJI). However, for Euler Lagrange system, the nonlinear H1 optimal control, or the associated HJI partial differential inequality, has long been regarded very difficult to solve analytically. Therefore, much research aimed at solving nonlinear H1 optimal control in simpler form was proposed, e.g., Battilotti and Lanari [4] solved disturbance attenuation problem indirectly based on the back-stepping method, and Nakayama and Arimoto [5] proposed a method of tuning passivity based control to endow a sort of H1 optimality. One of the most successful design of nonlinear H1 optimal control is that by Chen et al. [6]. Applying the special coordinate transformation for quadratic optimal control of robot manipulators [7], it was shown that solutions of the associated HJI equation can be found by solving an algebraic matri equation [6]. To the contrary, we aim at solving the HJI equation directly using physical properties of Euler Lagrange system. This paper summarizes the first progress on this direction of research, i.e., directly solving the HJI equation for Euler Lagrange system. In particular, the so-called nonlinear H1 inverse-optimal control is solved analytically for Euler Lagrange system. This paper is organized as follows. First of all, we formulate the problem we are to solve, i.e., nonlinear H1 inverse-optimal control, in Section II. Recognizing that nonlinear H1 optimal control should be defined in terms of disturbance-optimality pair, we provide one favorable pair of disturbance-optimality for Euler Lagrange system, including eogenous disturbance and parametric error disturbance. Then we solve the associated HJI partial differential equation in Section III. First, it is shown that the HJI partial differential equation can be converted to a matri ordinary differential equation if a specific condition is met. Net, the matri differential equation is analytically solved eploiting the Euler Lagrange property (). Surprisingly, the resulting H1 optimal control is defined by simple linear feedback of so-called reference error, where the feedback gain matri is determined by a specified L -gain. It is known that nonlinear H1 optimal control entails closed-loop stability, due to controller properties, against eogenous disturbances which have been used in L -gain attenuation definition [], [3]. However, because disturbances used in current control design include state feedback for parametric error disturbance, hence noneogenous, the resulting control may as well be confirmed of its stability property, which is described in Section IV. It is shown that global 14 96X/$1. 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2 848 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER uniform ultimate boundedness is guaranteed under moderate assumption on norm bounds on system dynamics. with eactly the same disturbance w(; t) formulated in (6). The system can be epressed in state space of = II. FORMULATION OF NONLINEAR H1 INVERSE-OPTIMAL CONTROL PROBLEM To formulate L -gain attenuation problem, one must define a pair of disturbance and cost which is referenced for control design. As mentioned, there are two kinds of disturbances at large, that is eogenous disturbance and parametric error. Eogenous disturbance affects the system directly, whereas parametric error disturbs the controlled system through some motion variables, e.g., velocity, with feedback. Specifically we formulate the disturbance induced by parametric error in the following manner: w PE (q; _q; _q REF ; q REF )=Y (q; _q; _q REF ; q REF ) ~ = ~ M (q)q REF + ~ C(q; _q)_q REF +~g(q) where the reference velocity is _q REF = _q DEF + k ve + k p s e, the reference acceleration is q REF =q DEF + k v _e + k p e, and the matri Y (q; _q; _q REF ; q REF ) < ns is the regressor matri. Then the total disturbance is given by w = ~ M (q)q REF + ~ C(q; _q)_q REF +~g(q) +d(t): (6) Taking into account of the disturbance (6), the Euler Lagrange dynamics (1) becomes = f ^M (q)q REF + ^C(q; _q)_q REF +^g(q)g M (q)e REF C(q; _q)_e REF + w(q; _q; t) (7) where e REF = q REF q and _e REF = _q REF _q. The error _e REF is called the reference error. Since the first three torque components between the curly braces can be determined using only available information, the overall control is defined by = ^M (q)q REF + ^C(q; _q)_q REF +^g(q) u (8) where RMC = ^M (q)q REF + ^C(q; _q)_q REF +^g(q) is called the reference motion compensation (RMC). In the equation, an auiliary control u is inserted to make the system achieve the L -gain attenuation. Substituting the control (8) into the dynamics, (1) or (7), yields the following, called the reference error output system M (q)fe + k v _e + k p eg + C(q; _q) _e + k v e + k p e = u(; t) +w(; t) (9) e T ;e T ; _e T T < 3n as _ = A(; t) + B(; t)u + B(; t)w (1) where the parameters A(; t) and B(; t) are given by (11a) and (11b), shown at the bottom of the page. Here, I and denote an identity and a zero matri, of suitable dimension. Since q = q DES (t) e and _q = _q DES (t) _e, the coefficient matrices A and B can be regarded as function of the state and the time t. In other words, the system is nonlinear and time varying. To define the L -gain attenuation, the cost function must also be defined for optimality evaluation. We define the cost function that will be minimized by the following nonlinear quadratic with a set of possibly state-dependent and time-varying symmetric positive definite weighting matrices Q(; t) to weight the state, and R(; t) to weight the auiliary control u (instead of the overall control ). Then the form of the cost function is given by z T (; u; t)z(; u; t) = T Q(; t) + u T R(; t)u: (1) Using the formulations of disturbance and cost function, the nonlinear H1 inverse-optimal control is stated as: Given a desired L -gain >, we wish to find a state-feedback control u(; t) and a set of possibly time-varying and state-dependent symmetric positive definite matrices Q 3 (; t) and R 3 (; t), such that the following L -gain attenuation problem is satisfied: T f T (t)q 3 (; t)(t) +u T (t)r 3 (; t)u(t)g dt T w T (; t)w(; t) dt (13) for the system (11) for () = and for all T. At this point, we provide the theorem on L -gain attenuation of a general nonlinear time-varying system. Formal definition of nonlinear L -gain attenuation problem is stated as follows. Consider a nonlinear system, with local coordinates =( 1;...; n) T for the state space manifold _ = f (; t) +g(; t)u + p(; t)w (14) where u < m is the control, and w < w is the disturbance. Let performance be evaluated using the cost variable z < z, defined by z(; t) =h(; t) +k(; t)u: (15) A(; t) = B(; t) = I I k p M 1 (q)c(q; _q) k v M 1 (q)c(q; _q)k p I M 1 (q)c(q; _q) k v I M 1 (q) (11a) (11b)

3 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER 849 The H1 control aims at finding a nonlinear state feedback u = (; t);(;t)= (16) such that the closed-loop system from w to z has L -gain from w to z [], [3]. Thus if the following three requirements hold: K 3 should satisfy (R1) K 3 (; t) =K 3 (; t) 1 I > and there eist n n symmetric constant matrices Q 3 1 ; Q3 13, Q 3 3, and a scalar > such that t z T (; t)z(; t) dt t w T (; t)w(; t) dt (17) (R) k p k v K 3 k p K 3 k p K 3 k v K 3 Q3 1 Q 3 13 Q 3 13 Q 3 3 > for all t, for () = 1, and for all w such that the integral in the right-hand side of the equation is well defined. The following assumptions are made. A1) f (; t), g(; t), and p(; t) are sufficiently smooth, and g(; t) and p(; t) have full column rank. A) h T (; t)k(; t) = and k T (; t)k(; t) >. A3) f (;t)= and h(;t)=. Isidori [] and Van der Schaft [3] found the theorem on nonlinear L -gain attenuation problem independently. Nonlinear H1 optimal control is formulated using a solution, if any, of HJI partial differential inequality or equation, as stated in the following theorem [], [3]. The theorem is slightly generalized to include a nonlinear time-varying system such as follows, because the system (11) we are dealing with is typically nonlinear and time varying. Lemma 1: Let >. Suppose there eists a continuously differentiable solution V (; t) with V (;t) = that satisfies HJI (; t; V ) = 1 V t(; t) +V (; t)f (; t) 1 V (; t) 1 g(; t)fk T (; t)k(; t)g 1 g T (; t) 1 p(; t)pt (; t) 1 V T (; t) + 1 ht (; t)h(; t) (18) for the system (14) and the cost function (15). Here, V t = (@V =@t)(; t) <and V =(@V =@ T )(; t) < 1n. The control u = fk T (; t)k(; t)g 1 g T (; t)v T (; t) (19) satisfies the L -gain attenuation requirement (17) for the L -gain >. III. NONLINEAR H1 INVERSE-OPTIMAL CONTROL VIA REFERENCE ERROR FEEDBACK The main result on the nonlinear H1 inverse-optimal control for the system (9) or (11) is summarized by the following theorem. It states that L -gain attenuation (13) by an arbitrarily specified > is possible by a simple reference error feedback (REF) for a set of special weight matrices Q 3 (; t) and R 3 (; t). Theorem 1: Let > and a set of (k p ;k v ) be given. For the Euler Lagrangian reference error output system (9), or its state-space representation (11), the following REF is the solution to the nonlinear H1 inverse-optimal control problem stated in the previous section u 3 = K 3 (; t) _e + k v e + k p e () 1 The case of nonzero initial condition can easily be solved if this zero initial condition problem is solved. (R3) k pk 3 Q 3 1 Q 3 13 Q 3 1 (k v k p)k 3 +Q 3 13 Q 3 3 Q 3 13 Q 3 3 K 3 > where K 3 = K 3 (1= )I with K 3 being the largest constant portion of K 3 (; t) such that ~ K 3 (; t) =K 3 (; t) K 3. Then the cost attenuated in the L -gain sense is defined by (13) using and kpk 3 Q 3 Q 3 1 Q 3 13 (; t) = Q 3 1 (k v k p )K 3 +Q 3 13 Q 3 3 Q 3 13 Q 3 3 K 3 k p K ~ 3 k p k v K ~ 3 k p K ~ 3 + k p k ~ v K 3 k ~ v K 3 k ~ v K 3 (1) k ~ p K 3 k ~ v K 3 K ~ 3 R 3 (; t) =K 31 (; t): () The proof of this theorem consists of two steps as follows. 1) Conversion of the HJI partial differential inequality (18) to a nonlinear matri ordinary differential Riccati inequality defined by _P (; t) +A T (; t)p (; t) +P (; t)a(; t) P (; t)b(; t) 1 R 31 (; t) 1 I BT (; t)p (; t) +Q 3 (; t) under the assumed solution (3) V (; t) = 1 T P (; t) (4) where _ P (; t) is the total derivative of P (; t) with respect to time t, i.e., dp (; t) (; t) _P (; t) = = + i (; t) i _ i: ) Analytic solution to the differential Riccati equation _P (; t) +A T (; t)p (; t) +P (; t)a(; t) P (; t)b(; t) 1 R 31 (; t) 1 I BT (; t)p (; t) +Q 3 (; t) = using the property () of Euler Lagrangian system. (5) A. Differential Riccati Inequality for Euler Lagrangian System The assumed solution (4) is in quadratic form with a nonlinear timevarying and state-dependent matri P (; t) = P ( 1 ; ; 3 ;t). The following lemma provides a general case where the HJI inequality (18)

4 85 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER reduces to a nonlinear matri differential Riccati inequality for a class of the system for which (14) can be represented by _ = A(; t) + B(; t)u + D(; t)w I = I a 1 (; t) a (; t) 111 a r (; t) u + w (6) b(; t) c(; t) for the state = ( T 1 ; T ;...; T r ) T. Note, in particular, that the number r is 3 for the state of the Euler Lagrangian system (11). Lemma : For the system represented by (6), assume that h(; t) and k(; t) in (15) satisfy h T (; t)h(; t) = T Q(; t) and k T (; t)k(; t) = R(; t) for two symmetric matrices Q(; t) and R(; t). IfP (; t) in (4) is not an eplicit function of r that is (=@ T r ) =, then the HJI inequality (18) for V (; t) of (4) reduces to the differential Riccati inequality _P (; t) +A T (; t)p (; t) +P (; t)a(; t) P (; t) 1 B(; t)r 1 (; t)b T (; t) 1 D(; (; t)dt t) 1 P (; t) +Q(; t) : (7) If there eist a solution matri P (; t) >, then H1 optimal control defined by (19) also reduces to u 3 = R 1 (; t)b T (; t)p (; t): (8) Proof: See Appendi A. In particular, for the Euler Lagrangian reference error output system (11), the differential Riccati inequality (7) is further reduced to (3) since B(; t) =D(; t) in (6). The above lemma can be applied to design of nonlinear H1 inverse optimal, or optimal, control for any nonlinear system of the form (6) and for any nonlinear quadratic cost function z T (; t)z(; t) = T Q(; t) +u T R(; t)u. In many cases, the corresponding differential Riccati inequality is easier to analyze and solve than the HJI inequality. B. Analytic Solution to Differential Riccati Equation In this section, analytic solution to the differential Riccati equation (5), a special case of the inequality (3), is found by eploiting the property () of an Euler Lagrangian system. By partitioning the 3n 3n P (; t) matri as P (; t) = P 11 (; t) P 1 (; t) P 13 (; t) P 1 (; t) P (; t) P 3 (; t) P 13(; t) P 3(; t) P 33(; t) and Q(; t) similarly, (5) reduces to the following set of si simultaneous matri differential equations, each of dimension n n: = P _ 11 k p (P 13 M 1 C + C T M 1 P 13 ) P 13M 1 R 31 M 1 P 13 + Q 3 11 = P _ +P 1 k v (P 3 M 1 C + C T M 1 P 3 ) k p P 3 P 3 M 1 R 31 M 1 P 3 + Q 3 = P _ 33 +P 3 (P 33M 1 C + C T M 1 P 33) k v P 33 P 33 M 1 R 31 M 1 P 33 + Q 3 33 = P _ 1 + P 11 k pp 13 k vp 13M 1 C k pc T M 1 P 3 P 13 M 1 R 31 M 1 P 3 + Q 3 1 = P _ 13 + P 1 k v P 13 P 13 M 1 C k p C T M 1 P 33 P 13M 1 R 31 M 1 P 33 + Q 3 13 = P _ 3 + P k v P 3 + P 13 k p P 33 P 3 M 1 C k vc T M 1 P 33 P 3M 1 R 31 M 1 P 33 + Q 3 3 : (9) These differential equations must be solved analytically. The following lemma proposes one class of analytic solutions: Lemma 3: The differential Riccati equation (5) for the Euler Lagrangian reference error output system, i.e., (1) with (11), has solution P (; t) of the form (3), shown at the bottom of the page, if the state weighting matri Q 3 (; t) and the control weighting R 3 (; t) are given by (1) and (). Here, K 3 = K 3 (1= )I for K 3 being the largest constant portion of K 3 (; t) such that K ~ 3 (; t) =K 3 (; t) K 3, and constant symmetric matrices Q1, 3 Q13, 3 and Q3. 3 Further, the solution satisfies (=@_e T )=. The control (8) reduces to the REF given in (). Proof: See Appendi B. It should be noted that the nonlinear H1 inverse-optimal control () does not depend on any dynamic parameters of the system (1), although it was formulated using actual dynamic parameters, see (11). Due to this property, the REF H1 control can be applied generally without any knowledge of a system s dynamics, provided it is an Euler Lagrangian system. Positive definiteness of P, Q 3, and R 3 is ensured using the following lemmas. Lemma 4: The matri K 3 (; t) satisfies (R1) if min (K 3 ) > 1 and ~ K 3 (; t) (31) where min(a) denotes the minimum eigenvalue of A. If (31) holds, then R 3 (; t) =K 31 (; t) >. Lemma 5: For the 3n 3nP (; t) matri given in (3) to be positive definite, it is necessary and sufficient that the n n constant portion matri given in (R) is positive definite. Lemma 6: Similarly, the matri Q 3 (; t) is positive-definite if the constant portion Q 3 given in (R3) is positive-definite for ~ K 3. P (; t) = kpm (q) + k p k v K 3 Q 3 1 k p k v M (q) + k p K 3 Q 3 13 k p M (q) k p k v M (q) + k p K 3 Q 3 13 kvm (q) + k v K 3 Q 3 3 k v M (q) k pm (q) k vm (q) M (q) (3)

5 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER 851 Remark 1: The proposed solution V (; t) using P (; t) can be interpreted elegantly as the kinetic energy plus potential energy since T P (; t) = _e REF M (q)_e REF + 1 k pk vk Q 1 k pk Q 13 k pk Q 13 k vk Q 3 e e As the REF () involves a constant and three constant matrices Q 3 1, Q13, 3 and Q 3 3 as free parameters, they should be chosen to verify the requirements stated in Theorem 1, even if and (k p ;k v ) are fied. In general, for a prespecified, R 3 should be selected to satisfy (31), or (R1). Net, the free parameter can be determined so that (R) and (R3) can hold for a set of constant matrices Q 3 ij s. In the meantime, the requirements can be relaed by choosing a constant matri R 3 and by setting all Qij s 3 to zero. Then (R1) is easily satisfied if one chooses R 3 > such that ma (R 3 ) <, where ma (A) denotes the maimum eigenvalue of A. (R) and (R3) reduce to the simple gain inequality of k p and k v regardless of, since (R) is simplified to k v k p K 3 (k p K 3 1 ) R 3 (k p K 3 ) k v = k vk p k p K 3 k v = k p k v (k v k p )K 3 > and (R3) to k v > k p. Hence, just the gain inequality k v > k p suffices for both (R) and (R3) to hold. The following corollary is a simplified version of Theorem 1 employing constant Q 3 and R 3. Corollary 1: Let the reference acceleration generation gain k v and k p satisfy Then for a given >, the REF e e T k v > k p : (3) u = K 3 _e + k ve + k p e (33) satisfies the L -gain attenuation requirement for if Q 3 = kpk 3 (k v k p )K 3 K 3 R 3 = K 31 : (34) K 3 = K 3 1 I > : (35) Remark : We would like to remark again that we solved the nonlinear H1 inverse optimal control problem, which means that the cost function represented by the state weight Q 3 (; t) and R 3 (; t) is given at the same time as well as the optimal control u(; t). That is, for a given, there eists a set of solutions fr 3 ;Q 3 ;u 3 g. Second point we would like to comment that the L -gain attenuation should be evaluated for a specific disturbance (input)-cost(output) pair. Note that the disturbance formulation is independent of the cost function or L -gain. Remark 3: Understanding of rough behavior of the epected inverse optimal solution, as the L -gain changes, will be of help. Given an L -gain 1, we can find a set of solutions fr 1;Q 1 ;u 1g. Meanwhile if another L -gain = 1 ( < < 1) is desired, we can find another set of solutions fr ;Q ;u g. To make discussion simple, just consider the constant diagonal weight matrices. First note that the control is given directly by the control weight matri R, that is the inverse optimal control for both cases are given by u 1 = R 1 1 _eref u = R 1 _eref : As the control weight R should be determined to satisfy R < I: Therefore, the following should be met: R 1 < 1 I R < I = 1 I: Rough estimation being applied, we have Then, the control becomes R R 1: u 1 u 1 : Furthermore, in general one can set K 1 = R I K = R 1 1 I: (36) In this case, the state weight matrices for two cases become almost equal to each other, i.e Q 1 Q : (37) That is, as a smaller L -gain is required, the control weight becomes smaller, whereas the state weight does not change. Conclusively, the RMC with REF is computed by = ^M (q)_q REF + ^C(q; _q)_q REF +^g(q) +K 3 (q)_e REF (38) where K 3 (q) < nn is the REF gain matri, possibly state-dependent. IV. STABILITY ANALYSIS OF REFERENCE MOTION COMPENSATION WITH REFERENCE ERROR FEEDBACK The property of L -gain attenuation from disturbance to cost function does not directly entail the stability, or boundedness, of the state (t) because the notion of stability concerns the norm of (t), not to s T dt [1]. Fortunately the nonlinear H1 optimal control satisfies closed-loop stability when w =. The general stability properties of the solution to the HJI inequality or equation must be considered. Choosing the solution V (; t) > to the HJI equation (18) as the Lyapunov function for the system (14) with the nonlinear H1 optimal control (19) for the cost function (15) yields the following derivative along the system s trajectory: dt = Vt + V f V gfkt kg 1 g T V T + V pw:

6 85 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER As V (; t) is a solution of the HJI equation (18), the above derivative can be written as dt = 1 ht h 1 V gfk T kg 1 g T V T 1 V pp T V T + V pw: Then the asymptotic stability of the closed-loop system when w = directly follows. However, stability when w = does not always imply that the total system remains still stable, or bounded, especially when there eists disturbance involving state feedback. An alternative to assess stability of such disturbed system is to define a specific Lyapunov function and check stability of the closed-loop system. This analysis can produce more direct estimate of error norm if more specific information on the system s dynamic parameters is available. In this section, we use the solution to the HJI equation V (; t) of the form (4) with (3) to analyze the stability of Euler Lagrangian system under the control of RMC with REF. A. Closed-Loop Stability Without the Disturbance First assess the global eponential stability of the Euler Lagrange system under the proposed RMC with REF control, if w =. Closing the control (38) around the Euler Lagrangian system (1) yields the following error dynamics: w = M (q)(e + k v _e + k p e) + fc(q; _q) +K 3 (; t)g _e + k v e + k p e (39) where w is defined by (6). Applying the solution V (; t) > of the form (4) with (3) as the Lyapunov function, similar analysis leads to the following derivative: since dt = 1 T Q 3 _eref K I _eref + _e REF w (4) h T h = T Q 3 > p T V T = g T V T = [k pi k vi I] = _e REF : Then it can be shown that the closed-loop system is in fact globally eponentially stable when w =. B. Stability Analysis Under Disturbance If the disturbance eists, one can see that the term detrimental to stability is the last term _e REF w in (4). By the manipulation of ( = )_e REF _e REF + _e REF w using (4) is simplified to _eref _e REF + _e REF w = _e REF w + wt w dt = 1 T Q 3 (; t) _eref K 3 (; t)_e REF _e REF w + wt w: (41) Now it is revealed that the term detrimental to closed-loop stability is the disturbance norm itself, w T w, premultiplied by the square of the L -gain. One can see the direct effect of the L -gain on closed-loop stability in the presence of disturbances. Note that as the L -gain gets smaller, the value of K 3 becomes larger. Consequently, the detrimental term ( =)w T w becomes less effective because the disturbance norm is reduced by = and the term ( =)_e REF K 3 _e REF becomes more positive. A more specific stability result, e.g., the global uniform ultimate boundedness [8], can be stated. Only a basic procedure is described herein. Assume that the disturbance is bounded by kw(; t)k T (; t) + 1(; t)kk + (; t) (4) where (; t) is a positive semidefinite matri function, 1(; t) and (; t) are positive semidefinite scalar functions of and t. The Lyapunov analysis is more specialized under this assumption, as dt 1 T fq 3 g + 1kk + _eref K 3 _e REF _e REF w From the equation, global uniform ultimate boundedness can easily be derived by applying the relevant theorem. V. CONCLUSION We have presented the analytic solution to the nonlinear H1 inverse-optimal control problem for Euler Lagrangian system. The control based on this analytic solution consists of RMC with REF. Robustness against eogenous disturbance and parametric error was shown eplicitly using Lyapunov analysis. Although the proposed control is valid as the nonlinear H1 inverse-optimal control, the solution is very general, for eample only the skew-symmetry of the Euler Lagrangian system was used to prove this result. Further, the technique of solving the HJI partial differential equation for Euler Lagrange system can be applied for more general controller design. One notable thing is that the REF which takes PID form is a class of solution to nonlinear H1 optimal control problem. APPENDIX A PROOF OF LEMMA Assume that the solution V (; t) = 1 P (; t) given in (4) satisfies (=@ T r ) for the system (1). In the following, the ar- T gument and t will not be eplicitly written, when it is obvious by contetual flow. First note that where it is easy to see that V = 1 T = T P + 1 = T T 1 T T T r : Computing first two terms of the HJI epression in (18) yields V t + V f = V t + V A = 1 + PA + AT P + 1 T A: :

7 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER 853 By rearranging the equation, there follows V t + V f = 1 T fpa + A T P g for which we are to show that In the equation, T A = T 1 r1 T A = _ T using the fact that (=@ T r ), T k T r1 = k;1 111 = n T A r k;j _ k;j k;n where k;j is the jth element of k, there follows T A = V t + V A = 1 T r n k=1 k;j _ k;j = + i=1 rn i=1.. r T k _ k i _ i i _ i + PA + A T P = 1 T f _ P + PA + A T P g: (43) For the third term of the HJI epression given by V g(k T k) 1 g T 1 ppt V T = V BR 1 B T V T 1 V DDT V T note that V B and V D are simplified to T B = V B = T PB + 1 T Similarly there T 1 T B = T T r1 V D = T PD:... b(; t) = : Therefore, the third term is simply written as V g(k T k) 1 g T 1 ppt V T = T P BR 1 B T 1 DDT P: (44) Combining (43) and (44) with h T h = (1=) T Q, the original HJI partial differential epression reduces to the following ordinary differential Riccati epression: where HJI = 1 T DR DR = P _ + PA + A T P P BR 1 B T 1 DDT P + Q: Because the HJI inequality should hold for all, the differential Riccati inequality (7) follows. The original nonlinear H1 optimal control given by (19) reduces to the control given in (8), since u = (k T k) 1 g T V T = KB T V T = KB T P: APPENDIX B PROOF OF LEMMA 3 In the following derivation, all the superscripts 3 are suppressed, which indicates that two matrices Q and R are solutions of the nonlinear H1 inverse-optimal control. After some manipulation of the first three equations of (9), epressing the diagonal blocks of the original DRE (5), using the property (EL) or (), they can be reduced to = 13R 1 + Q11 =P1 k p 3M 3R 1 + Q =3M k v 33M 33R 1 + Q33 if the following are substituted: P 13M 1 := 13I ) P 13 = 13M P 11 := k p13m + P 11 P 3M 1 P 3 = 3M := 3I ) P := k v3m + P P 33M 1 := 33I ) P 33 = 33M: For eample, consider the first equation of (9). As one can see, the first three elements _ P 11 k p(p 13M 1 C + C T M 1 P 13) can be eliminated by the property (EL), if P 13M 1 = 13 and P 11 = k p 13M + constant portion, which is the first assignment of the above equation. Then the first three elements reduce to k p13( _ M C C T ), which is zero by the property (EL). The other two assignments are derived similarly. In the above, the term denoted by (1) denotes the constant portion of (1). Substituting the above results to the remaining three equations of (9) and applying similar processes to them, one can see that the assignments on the left-hand side of the following equations make the remaining three equations hold: k p3 := k v13 ) P 1 := k p 3M + P 1 P R 1 + Q1 = k p33 := 13 ) P R 1 + Q13 = k v 33 := 3 ) P 333R 1 + Q3 = :

8 854 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER Arranging the left-hand side assignments between scalars in matri form yields and k v k p 1 k p 1 k v 13 3 = : 33 P 11 = k pm + k pk vk Q 1 P = k vm + k v K Q 3 P 33 = M In order not to result in only trivial solutions for s the determinant of the coefficient matri should be zero, which is indeed the case. In fact, the matri has one-dimensional null space, and any elements belonging to the null space can be epressed as 33 := ) 13 = k p 3 = k v using a single free parameter. Then summarizing the above assignments yields the following P matri: and P 11 = k p k v R 1 Q 1 P = k vr 1 Q 3 P 1 = k p R 1 Q 13 (45) P 11 = k pm + k p k v R 1 Q 1 P = k vm + k vr 1 Q 3 P 33 = M P 1 = k pk vm + k pr 1 Q 13 P 13 = k p M P 3 = k vm: (46) From the constraint that P s of (45) are constant, one can conclude that, by decomposing K (; t) as in Lemma 3, the off-diagonal blocks of Q should be of the following form: P 1 = k p k v M + k p K Q 13 P 13 = k pm P 3 = k v M: Using the obtained solution P (; t), the H1 control (8) reduces to () since B T P = [k pi k vi I]: REFERENCES [1] F. L. Lewis, C. T. Abdallah, and D. M. Dawson, Control of Robust Manipulators. New York: Macmillan, [] A. Isidori, Feedback control of nonlinear systems, Int. Jr. Robust Nonlinear Contr., vol., pp , 199. [3] A. J. Van Der Schaft, L -gain analysis of nonlinear systems and nonlinear state feedback H control, IEEE Trans. Automat. Contr., vol. 37, no. 6, pp , 199. [4] S. Battilotti and L. Lanari, Tracking with disturbance attenuation for rigid robots, in Proc IEEE Int. Conf. Robotics and Automation, 1996, pp [5] T. Nakayama and S. Arimoto, H control for robotic systems using the passivity concept, in Proc IEEE Int. Conf. Robotics and Automation, 1996, pp [6] B. S. Chen, T.-S. Lee, and J.-H. Feng, A nonlinear H control design in robotics systems under parametric perturbation and eternal disturbance, Int. J. Contr., vol. 59, no. 1, pp , [7] R. Johansson, Quadratic optimization of motion coordination and control, IEEE Trans. Automat. Contr., pp , 199. [8] Z. Qu and J. Dorsey, Robust tracking control of robots by a linear feedback law, IEEE Trans. Automat. Contr., pp , Q 1 (; t) =Q 1 + k p k v ~ K (; t) Q 3 (; t) =Q 3 + k v ~ K (; t) Q 13 (; t) =Q 13 + k p ~ K (; t): The same decomposition applies to the diagonal blocks of Q matri, i.e., Q 11 (; t) =Q 11 + k ~ p K (; t) Q (; t) =Q + k ~ v K (; t) Q 33 (; t) =Q 33 + K ~ (; t): The constant portions should satisfy Q 11 = k pk Q = (k v k p )K +Q 13 Q 33 = K : Then, the following assignment of P matri will suffice to solve all the equations in (45) and (46), and consequently the differential Riccati equation (5) P 11 = k p k v K Q 1 P = k v K Q 3 P 1 = k p K Q 13