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Nonlinear Adaptive H∞ Output Feedback Tracking Control for Robotic Systems
Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008
Nonlinear Adaptive H∞ Output Feedback Tracking Control for Robotic Systems I. Levi ∗ N. Berman ∗ A. Ailon ∗∗ ∗
Mechanical Engineering Department, Ben-Gurion University of the Negev, Beer Sheva, Israel. ∗∗ Electrical Engineering Department, Ben-Gurion University of the Negev, Beer Sheva, Israel.
Abstract: This paper presents a solution to the tracking control problem of robotic systems in the presence of exogenous disturbances and model uncertainty with partial state information. The solution yields a Linear Matrix Inequalities (LMIs) based tracking output feedback controller. The main contribution of this paper lies in its particular approach which facilitates an application of the linear H∞ control theory without linearizing the underlying system. This yields a relatively simple and elegant design procedure. In addition, a relatively low gain controller is achieved. Simulation results of application this control algorithm in a two-degree of freedom robot demonstrates the design procedure feasibility. 1. INTRODUCTION This paper introduces a solution to the trajectory tracking control of robotic manipulators which is based on the H∞ control and LMI methods. It is assumed that only a noisy partial state information is available, and that a model uncertainty and exogenous disturbances are present. There are numerous papers which present studies of this subject, see, e.g. [1]-[7] for a state feedback utilization, and [8], [9], [10] for output feedback applications. Studies of this subject which deal with model uncertainty and assume partial information while using adaptive and robust control may be found in [11], [12], [13]. To the best of our knowledge all the studies (excluding those that take the H∞ approach) do not assume the presence of exogenous disturbances, neither a plant noise, nor a measurement noise. The works of Acho et. al ([26]) and Zasadzinski et. al ([10]) which take the H∞ approach, although they assume a presence of exogenous disturbances they do not consider model uncertainty as the theories they develop do not account for it. The novelty of this paper is in its particular approach and in its extent of generality. In particular: 1. the results achieved in this work apply to robotic systems with model uncertainty and with exogenous disturbances that include both, noise associated with the plant and noisy measurements. 2. A particular choice of a storage function which facilitates an application of the linear H∞ control theory and the LMI methods without linearizing the underlying system. In view of the theory of nonlinear H∞ control (see, e.g. [17]-[21]), we formulate the tracking problem as an H∞ control problem, and use the interrelations among the the l2 -gain property, dissipativity and the HamiltonJacobi Inequality (HJI) to derive an output feedback controller, first for the case of absence of uncertainty, and then utilizing these results, we develop a controller that
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accounts for model uncertainty, achieves L2 -gain< γ for a prescribed γ, and a semi-global asymptotic stability. As mentioned above, all this is facilitated by the particular choice of a storage function that takes an advantage of some certain structural properties the underlying system enjoys. This yields sufficient conditions, in terms of certain LMIs for the semi-global asymptotic stability and for the L2 -gain property to hold. The advantage of these sufficient conditions is that they turn to be exactly as the usual ones for an appropriate linear system (see, e.g [14]-[16]). We also introduce an example which demonstrates the algorithm performances by an application in a two-degree of freedom robot where gravity and the model-parameters are only approximately known, while relatively large uncertainties are assumed. 2. PROBLEM FORMULATION: NO MODEL UNCERTAINTY In this section we consider the tracking problem of an nlink robot manipulator with no model uncertainty. In section 2.1 below we introduce a convenient state space representation of the underlying system. The nonlinear H∞ control problem is formulated in section 2.2, while the solution to the nonlinear HJI is introduced in section 2.3. 2.1 The System dynamics The dynamical equations of an n-link robot manipulator with exogenous disturbances is commonly described by the following (see, e.g. Spong and Vidyasagar [1]) M (q)¨ q + (C(q, q) ˙ + H)q˙ + G(q) = τ + ω (1) where q ∈ ℜn is the robot’s joint angular position, M (q) ∈ ℜn×n is the symmetric positive definite inertia matrix, C(q, q) ˙ q˙ is the centripetal and coriolis forces, H q˙ represents the linear frictional forces, G(q) consists of the gravitational forces, τ is the torque applied to the various
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links at the corresponding joints by means of electrical motors and ω represents exogenous R ∞ disturbances, which are assumed to be in L2 , that is 0 ||ω(t)||2 dt < ∞.
The objective is a design of an output feedback which drives the system’s states along a desired trajectory qr (t) starting at a given initial position. For this we define the following error vector: ¸ · ¸ · q − qr e (2) e= 1 = q˙ − q˙r e2
We take: τ =M (qr )¨ qr + (C(qr , q˙r ) + H)q˙r + G(qr ) + u (3) where u is defined in section 2.2 below. Define ∆W =[M (q)−M (qr )]¨ qr +[C(q, q˙r )−C(qr , q˙r )]q˙r+ [G(q)−G(qr )]. (4) Using these in (1), yields the following tracking problem. e˙ = A(q, q, ˙ q˙r )e + B(q)(u − ∆W + ω) (5) where · ¸ 0 In×n A(q, q, ˙ q˙r ) = n×n 0n×n −M −1 (q)(C(q, q)+C(q, ˙ q˙r )+H) ¸ · 0n×n (6) B(q) = M −1 (q) 2.2 Nonlinear H∞ control problem Consider the nonlinear system: ( e˙ = A(q, q, ˙ q˙r )e + B(q)(u − ∆W ) + B(q)ω y = C2 e + D21 ω z = C1 e + D12 u
Where e ∈ ℜ2n , u ∈ ℜm , y ∈ ℜs and ω ∈ ℜd are the state, the control input, the measurement output and disturbances, respectively, while z ∈ ℜh is an objective variable (controlled output). The H∞ output-feedback control objective is a synthesis of an output-feedback that renders the underlying system L2 -gain< γ. In order to achieve this goal the following controller structure ½ is assumed −1 ξ˙ = T (q) [Ak ξ + Bk y] (7) u = Ck ξ + Dk y where ξ ∈ ℜ2n , T (q) is a 2n × 2n matrix, and Ak , Bk , Ck , Dk are constant matrices. Let · ¸ e x= (8) ξ Thus the closed-loop system admits ½ x˙ = Acl (q, q, ˙ q˙r )x − Bcl (q)∆W + B1cl (q)ω z = Ccl x + Dcl ω
Bcl (q) =
We have now the following theorem, the proof of which is omitted for the lack of space. Theorem 1. Given δ > 0. Assume Po (q) has the following structure: Po (q) = Po.c Mo (q) (14) where Mo (q) = blockdiag{Ms (q), T (q)}, (15) Ms (q) given in (13) and Po.c ∈ ℜ4n×4n is a positive symmetric matrix that is to be determined. Then the closedloop system (9) is L2 -gain < γ, and the controller (7) renders the closed-loop system semi-global exponentially stable if the following LMI’s LM I (1) : T ˜ Po.c Acl + ATcl Po.cPo.c B1cl Ccl Po.c Bcl ∆W T ∗ −γ 2 I Dcl 0 0 ∗ ∗ −I 0 0 < 0 1 ∗ ∗ ∗ − δI 0 2 ∗ ∗ ∗ ∗ −δ −1 I (16) LM I (2) : Po.c Mo (q) + (Po.c Mo (q))T > 0 (17) hold for e2 ∈ Br with an arbitrarily fixed r > 0 and for all q, where · ¸ A+BD C BC B(D D +I) k 2 k k 12 Acl B1cl = Bk D21 , Bk C2 Ak Ccl Dcl C1+D12 Dk C2 D12 Ck D12 Dk D21 ˜ = blockdiag{∆W ˜ 1 , 02n×2n }, ∆W ˜ 1 = blockdiag{bmcg In×n , bc˜In×n }, ∆W
and
¸ B(q) . 0 2n×n
Consider the nonlinear system (9) with the following storage function 1 (12) So (x, q) = xT Po (q)x 2 where Po (q) ∈ ℜ4n×4n is a positive C 1 matrix, (note that Po (q) is not necessarily a symmetric matrix). Define, Ms (q) = blockdiag{In×n , M (q)} (13) where M (q) is the inertia matrix and blockdiag{·} denotes a diagonal block matrix. The notation * will be used frequently in the sequel and will denote a symmetric entry of a matrix.
(9)
where · ¸ Acl (q, q, ˙ q˙r ) B1cl (q) = Ccl Dcl A(q, q, ˙ q˙r )+B(q)Dk C2 B(q)Ck B(q)(Dk D12+I) T (q)−1 Bk C2 T (q)−1 Ak T (q)−1 Bk D21 (10) C1+D12 Dk C2 D12 Ck D12 Dk D21 ·
2.3 Solution To The Nonlinear Hamilton-Jacobi Inequality (HJI)
(11)
bmcg , bc˜ > 0 (18)
and ¸ ¸ · · 0n×n B , B= Bcl = In×n 0 2n×n ¸ · 0 I A = n×n n×n . 0n×n −H
(19)
Remark 1. The L2 -gain< γ means in this theorem that for any initial state x0 there is a neighborhood Bx0 ⊂ L2 ofZ0 such that Z ∞ ∞ ||ω(t)||2 dt}, ∀t ≥ 0. ||z(t)||2 dt < γ 2 {||x0 ||2 + 0
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Theorem 2 below provides sufficient conditions for Theorem 1 to hold. Theorem 2. Fix δ > 0, γ > 0, r > 0. Assume the following LMI’s ˜1 Φ11 Φ12 Φ13 Φ14 XB ∆W ∗ Φ22 Φ23 Φ24 B Y ∆W ˜ 1 ∗ ∗ −γ 2 I 0 0 0 LM I 1 : ∗ ∗ (20) < 0 ∗ −I 0 0 1 ∗ ∗ − δI 0 ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ −δ −1 I ˆk C2+C T B ˆ T , Φ12=Aˆk+AT+C T D ˆ TB T Φ11=AT X+XA+B 2 k 2 k T T ˆT T ˆ Φ13=XB+Bk D21 , Φ14=C1 +C2 Dk D12 ˆ k D21 Φ22=Y AT +AY + B Cˆk+CˆkT B T , Φ23=B+B D T ˆT T Φ24=Y C1 +Ck D12 · ¸ XMs(j) + Ms(j) X Ms(j) LM I 2 : > 0 (21) ∗ Ms(j) Y + Y Ms(j) hold for some symmetric positive definite matrices X, Y ∈ ˆk ∈ ℜ2n×n , Cˆk ∈ ℜ2n×2n and for some Aˆk ∈ ℜ2n×2n , B (j) n×2n ˆ n×n ℜ , Dk ∈ ℜ , where each Ms is the Ms (q) evaluated at the vertex j of the polytope generated by Ms (q). Then the closed-loop system (9), with the controller (3), (7), is L2 -gain< γ, and semi-global exponentially stable, where T (q) is given by T (q) = −M −1 XMs (q)Y N −T (22) If a solution to these LMIs exist, the output feedback gains are given by Aˆk = XAY+M Ak N T+XBCk N T+M Bk C2 Y+XBDk C2 Y ˆk = M Bk +XB2 Dk B Cˆk = Ck N T +Dk C2 Y ˆ k = Dk D
(23)
3. PROBLEM FORMULATION WITH MODEL UNCERTAINTIES This section deals with the tracking problem of an n-link robot manipulator with model uncertainties. In section 3.1 below we introduce the state space equations. The nonlinear H∞ control problem is formulated in section 3.2, while the solution to the nonlinear HJI is introduced in section 3.3. 3.1 The System Dynamics Generally, in addition to external disturbances, there are uncertainties present in the system’s model which must ˆ (q) denote an estimate of M (q), be accounted for. Let M ˆ ˆ an estimate of H and C(q, q) ˙ an estimate of C(q, q), ˙ H ˆ G(q) an estimate of the gravitational forces G(q). The inverse dynamics control law for the nominal system (1) is given by ˆ (qr )¨ ˆ r , q˙r ) + H) ˆ q˙r + G(q ˆ r ) + u. τ =M qr + (C(q (24) Substituting (24) into (1) and subtract M (q)¨qr + (C(q, q) ˙ + H)q˙r from both sides of the equation we obtain ˆ (qr ) − M (q)]¨ M (q)e˙ 2 +(C(q, q) ˙ + H)e2 = u + ω + [M qr + ˆ r , q˙r )−C(q, q)+ ˆ −H]q˙r + [G(q ˆ r ) − G(q)]. [C(q ˙ H (25)
It is easy to show now that (25) may now be expressed as M (q)e˙ 2 + (C(q, q) ˙ + C(q, q˙r ) + H)e2 = u + ω+ ˆ ˆ r , q˙r ) − C(q, q˙r )]q˙r + [M (qr ) − M (q)]¨ qr + [C(q ˆ −H]q˙r + [G(q ˆ r ) − G(q)]. [H (26) from which one obtains M (q)e˙ 2 + (C(q, q) ˙ + C(q, q˙r ) + H)e2 = u + ω+ Y1 (qr , q˙r , q¨r )Po.c − ∆W. (27) where p˜= pˆ−p is the parameter error vector, Y1 (qr , q˙r , q¨r ) is the regressor matrix and ∆W given by (4). Thus, the state space can be written as e˙ = A(q, q, ˙ q˙r )e + B(q)(u + ω + Y1 (qr , q˙r , q¨r )˜ p − ∆W ) (28) where A(q, q, ˙ q˙r ), B(q) are given in (6). 3.2 The Nonlinear H∞ control problem Consider the nonlinear system: ( e˙ = A(q, q, ˙ q˙r )e+B(q)(u+Y1 (qr , q˙r , q¨r )˜ p −∆W )+B(q)ω y = C2 e + D21 ω z = C1 e + D12 u (29) In order to obtain an adaptive H∞ output-feedback control objective our goal is to compute a dynamical outputfeedback controller in the same form as given in (7),i.e ½ −1 ξ˙ = T¯(q) [Ak ξ + Bk y] (30) u = Ck ξ + Dk y where ξ ∈ ℜ2n , T¯(q) is a 2n × 2n matrix (to be determined below) and Ak , Bk , Ck , Dk are constant matrices. In addition, we choose the parameter estimator of the form p˜˙ = Ax (qr , q˙r , q¨r )x (31) where x given by (8) and Ax (qr , q˙r , q¨r ) will be determined later. Let x ˜ be defined by · ¸ x . (32) x ˜= p˜ Then the closed-loop system admits ½ ˜cl (q)(Y1 (qr , q˙r , q¨r )˜ ˜1cl (q)ω x ˜˙ =A˜cl (q, q, ˙ q˙r )˜ x+B p −∆W )+ B ˜ ˜ z=Ccl x ˜ + Dcl ω (33) where ˜1cl (q) A˜cl (q, q, ˙ q˙r ) B Acl (q, q, ˙ q˙r )0r×r B1cl (q) =Ax (qr , q˙r , q¨r )0n×r 0r×d ˜ cl Ccl 0h×r Dcl C˜cl D · ¸ ˜cl (q) = Bcl (q) . B (34) 0r×n
and Acl (q, q, ˙ q˙r ), Bcl (q), Ccl , Dcl , B1cl (q) are given in (10,11) with T¯(q) instead of T (q). 3.3 Solution To The Nonlinear HJI Consider the nonlinear system (33) with the following storage function 1 p} (35) S¯o (x, p˜, q) = {xT P¯o (q)x + p˜T Λ˜ 2 where Λ is a positive definite weighting matrix and P¯o (q) is a positive C 1 matrix, (note that P¯o (q) is not necessary a symmetric matrix).
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Theorem 3. Given δ > 0. Assume the following structure for P¯o (q): ¯ o (q) P¯o (q) = Po.c M (36) where ¯ o (q) = blockdiag{Ms (q), T¯(q)}, M (37) 4n×4n Ms (q) given in (13) and Po.c ∈ ℜ is a positive symmetric matrix to be determined. Then the closed-loop system (33) is L2 − gain ≤ γ, and the controller (30) renders the closed-loop system semi-global asymptotically stable if the following LMI’s LM I (1) : T ˜ Po.c Acl + ATcl Po.cPo.c B1cl Ccl Po.c Bcl ∆W T ∗ −γ 2 I Dcl 0 0 ∗ ∗ −I 0 0 < 0 1 0 ∗ ∗ ∗ − δI 2 ∗ ∗ ∗ ∗ −δ −1 I (38) LM I (2) : ¯ o (q))T > 0 ¯ o (q) + (Po.c M (39) Po.c M hold for e2 ∈ Br with an arbitrarily fixed r > 0 and for all q, where ¸ · A+BDk C2 BCk B(Dk D12+I) Acl B1cl , = Bk D21 Bk C2 Ak Ccl Dcl C1+D12 Dk C2 D12 Ck D12 Dk D21 ˜ ˜ 1 , 02n×2n }, ∆W = blockdiag{∆W ˜ 1 = blockdiag{bmcg In×n , bc˜0 In×n }, ∆W
bmcg , bc˜0 > 0 (40)
Remark 2. Note that if the uncertainty range of fi shrinks to zero then pav → p. Define, m11 (pav , q) · · · m1n (pav , q) .. .. M av (q) = . ∗ . ∗ ∗ mnn (pav , q)
where M av (q) is the average matrix of the inertia matrix M (q). Obviously, by the above definition of M av (q), this matrix agrees with assumption A1. Finally we define the matrix Msav (q) = blockdiag{I2n×2n , M av (q)} (44) We have now the following result. Theorem 4. Consider the closed-loop system (33) with the storage function (35) where T¯(q) = −M −1 XMsav (q)Y N −1 (45) Given the scalars δ > 0, γ > 0, r > 0, ε > 0, there is an output-feedback controller given by (30) with the parameter update process given by (31), where T ˜T Y1T (qr , q˙r , r¨r )Bcl Po (q) = Ax (qr , q˙r , q¨r ). If the following LMI’s ˜1 Φ11 Φ12 Φ13 Φ14 XB ∆W ∗ Φ22 Φ23 Φ24 B Y ∆W ˜ 1 ∗ ∗ −γ 2 I 0 0 0 LM I 1 : ∗ ∗ ≤0 (46) ∗ −I 0 0 1 ∗ ∗ − δI 0 ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ −δ −1 I ˆk C2+C2T B ˆkT , Φ12=Aˆk+AT+C2T D ˆ kTB T Φ11=AT X+XA+B T T ˆT T ˆ Φ13=XB+Bk D21 , Φ14=C1 +C2 Dk D12 ˆ k D21 Φ22=Y AT +AY + B Cˆk+CˆkT B T , Φ23=B+B D T ˆT T Φ24=Y C1 +Ck D12
and ·
B
¸
, B= 0 2n×n ¸ · 0n×n In×n . A0 = 0n×n 0n×n Bcl =
·
0n×n In×n
¸ (41)
LM I (2) : Ψ11 Ms(j) (q) X(Ms(j)−Msav(j) ) 0 ∗ Ψ22 0 Y >0 ∗ ∗ εI 0 ∗ ∗ ∗ ε−1 I
In what follows we utilize the algorithm introduced in [16] in order to solve the LMI’s of Theorem 3 via LMI’s optimization toolbox in MATLAB. The following notations will be used in the sequel m11 (p, q) · · · m1n (p, q) .. .. M (q) = . ∗ . ∗ ∗ mnn (p, q)
(43)
(47)
Ψ11 = XMs(j) (q)+Ms(j) (q),
where mik (p, q) are bounded, with known bounds. It is well known that the parameters vector p is a function of the physical system’s parameters like: masses, lengths etc.. We take fi to be the i-th physical parameter of the system, therefore if we assume that the system has −l− physical parameters then the vector p may be written as p = F(f1 , f2 , ..., fl ). We denote the upper bound of fi by fi+ (i.e. fi ∈ [fi− , fi+ ]) and the lower bound of fi by fi− and define the following: Let fiav be the average physical parameter of fi , and pav be the average vector parameter of p which are given by: 1 fiav = (fi− + fi+ ) ∀i = 1, ..., l. 2 pav = F(f1av , f2av , ..., flav ). (42)
Ψ22 = Ms(j) (q)Y +Y Ms(j) (q) hold for some symmetric positive definite matrices X, Y ∈ ˆk ∈ ℜ2n×n , Cˆk ∈ ℜ2n×2n and for some Aˆk ∈ ℜ2n×2n , B (j) av(j) n×2n ˆ n×n ℜ , Dk ∈ ℜ , where each Ms , Ms are the av Ms (q), Ms (q) (given by (13),(44) respectively) evaluated at the vertex j of the polytope generated by Ms (q), Msav (q), respectively. If a solution to these LMIs exists, then the closed-loop system (33) has L2 -gain≤ γ (from ω to z) and the tracking error e → 0 as t → ∞ semi-globally. In this case, the output-feedback is given by (24,30,23) with the parameter update process pˆ(t) =ˆ p(0)+Ax (qr (t), q˙r (t), q¨r (t))xI (t)− Z t A˙ x (qr (σ), q˙r (σ), q¨r (σ))xI (σ)dσ (48) 0
where
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a
t
40 q1 q1r 30
(49)
20 R e s p o n s e ( Degrees )
e (τ )dτ 1 Z t 0 e1 (t) x(τ )dτ = xI (t) = Z t 0 ξ(τ )dτ Z
0
(50)
4. EXAMPLE The feasibility of the design of the foregoing sections is demonstrated via simulations of a two-link manipulator. The system is assumed to have known parameters and external disturbances. The H∞ tracking control is then designed according to the proposed procedure. The system’s parameters are: the links’ masses: m1 , m2 (kg), the links’ lengths: l1 , l2 (m), masses’ centers: lc1 , lc2 , the angular positions: q1 , q2 (rad), q1 , q2 (rad), the viscosity coefficient : h(kgm2 ) and the applied torques: τ1 , τ2 (N m). By (1) we have: · ¸ m l l cos(q −q )+I m l2 +m l2+I +I M (q)= 1 c1 2 1 zz1 zz2 2 1 c2 2 1 2 zz2 m2 l1 lc2 cos(q1−q2 )+Izz2 m2 lc2+Izz2 ¸ · 0 q˙2 C(q, q) ˙ = m2 l1 lc2 sin(q1 − q2 ) −q˙1 0 · ¸ −(m1 lc1 + m2 l1 )gsin(q1 ) G(q) = −m2 lc2 gsin(q2 ) · ¸ h 0 H= (51) 0 h
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Fig. 1. Exact model: (a) Positions of link 1. (b) Positions of link 2. (c) Torques. [2]
[3]
[4] [5] [6]
REFERENCES
0
−20
R e s p o n s e (Nm)
where q ∈ ℜ2 and τ ∈ ℜ2 . The nominal parameters of the manipulator are taken to be: m1 = 1(kg), m2 = 5(kg), l1 = l2 = 0.2(m), lc1 = lc2 = 0.1(m), g = 9.8(ms−2 ), h = 5((kgm2 )) where Izz1 = Izz2 = 13 mi l2 (i=1,2) and the initial conditions are q1 (0) = 30◦ , q2 (0) = 100◦ , q˙1 (0) = 0, q˙2 (0) = 0. The desired position is: qr1 = ◦ 30◦ sin(2πt), ·qr2 = ¸ 60 sin(2πt). The exogenous disturω1 bances ω = are chosen to be square wave with ω2 period 2π, that is ½ ½ 1 , 0≤t<π 0 , 0≤t<π ω1 = , ω2 = (52) 0 , π ≤ t < 2π −1 , π ≤ t < 2π For the purpose of simulations, C1 and D12 were chosen as: ¸T · · ¸T 0 0 10 I 0 0 0 C1 = 2 2×2 2×2 2×1 2×1 ,D12 = 0.001 1×2 1×2 01×2 01×2 01 02×2 I2×2 02×1 02×1 and ¸ · 25I2×2 02×2 ˜ , δ = 10−6 ∆W 1 = 02×2 I2×2 In this case:· ¸ 2 m1 lc1 +m2 l12+Izz1+Izz2 m2 l1 lc2 · δj+Izz2 (j) M = 2 m2 l1 lc2 · δj+Izz2 m2 lc2 +Izz2 j=1, 2, δ1=1, δ2 = −1 By applying Theorem 2 we obtain γmin = 4.0467. However, γ = 4.05 was selected to avoid an undesirable highgain controller design corresponding to γ which is close to the optimum. (see Fig.1).
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[1] M. W. Spong and M. Vidyasagar , Robot Dynamics and Control, New york: Wiley, 1989,
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N. Sadegh, and R. Horowitz, ”Stability and Robustness Analysis of a Class of Adaptive Controllers for Robotic Mnipulators”, Int. J. Robotics Research, Jun 1990, Vol. 9, No. 3, p: 74-92, B. S. Chen and Y. C. Chang, ”Nonlinear Mixed H2 /H∞ control for robust tracking design of robotic systems”, Int. J. Control, 1997, Vol. 67, No. 6, p: 837857. H. L. Trentelman, J. C. Willems,, ”Nonlinear H∞ Control of Robotic Manipulator”, 1999, T-Y. Kuc, W-G. Han, ”An adaptive PID learning control of robot manipulatorsq”, Automatica, Sep 2000, Vol. 36, p: 717-725, J. Alvarez-Ramirez, R. Kelly and I. Cervantes, ”Semiglobal stability of saturated linear PID control for robot manipulators”, Automatica, Jun 2003, Vol.
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
39, Issue 6, p: 989-995, [7] T. Ravichandran, D. W. L. Wang and G. R. Heppler, ”Stability and Robustness of a Class of Nonlinear Controllers for Robot Manipulators”, Automatica, July 2004, Vol. 2, Issue 6, p: 5262-5267, [8] E.S. Shin and K.W.Lee, ”Robust Output Feedback Control of Robot Manipulators Using High-Gain Observer”, Proceedings of the 1999 IEEE Int. Conf. on Control Applicatlons, July 1999, p: 881-886, [9] K.W. Lee and H.K. Khalil, ”Adaptive output feedback control of robot manipulators using high-gain observer”, Int. J. Control , July 1997, Vol. 67, No. 6, p: 869-886, [10] M. Zasadzinski, E. Richard, M. F. Khelfi and M. Darouach, ”Disturbance Attenuation and Trajectory Tracking via a Reduced-order Output Feedback Controller for Robot Manipulators”, Automatica, 1998, Vol. 33, No. 12, p: 1539-1546, [11] P. R. Pagilla, M. Tomizuka, ”An adaptive output feedback controller for robot arms: stability and experiments”, Automatica, 2001, Vol. 37, p: 983-995, [12] M. A. Arteaga, ”Robot control and parameter estimation with only joint position measurements”, Automatica, January 2003, Vol. 39, Issue. 1, p: 67-73, [13] E. Kim, ”Output Feedback Tracking Control of Robot Manipulators With Model Uncertainty via Adaptive Fuzzy Logic”, IEEE Tran. on fuzzy systems,, June 2004, p: 368-378,An LPD Approach to Robust and Static Output-Feedback Design [14] P. L. D. Peres, J. C. Geromel, and S. R. Souza, ” H∞ control design by static output-feedback”, Proc. IFAC Symp. Robust Control Design, Rio de Janeiro, Brazil, 1994, p: 243248, [15] U. Shaked, ”An LPD Approach to Robust H2 and H∞ Static Output-Feedback Design”, IEEE Tran. on automatic control , May 2003, Vol. 48, No. 5, p: 866872, [16] C. Scherer, P. Gahinet and M. Chilali, ”Multiobjective Output-Feedback Control via LMI Optimization”, IEEE Tran. on automatic control , July 1997, No. 7, p: 866-872, [17] A. J. Van der Shcaft , On a state space approach to nonlinear H∞ control., Systems and Control Latters, 1991, Vol. 16, p: 1-8. [18] A. Isidori and A. Astolfi , Nonlinear H∞ control via measurement feedback, Journal of Mathematical Systems, Estimation, and control, 1992, Vol. 2, p: 3144. [19] A. J. Krener , Necessary and sufficient conditions for nonlinear worst case (H∞ ) control and estimation, Journal of Mathematical Systems, Estimation, and control, 1994, Vol. 4, p: 1-25. [20] A. J. Van der Shcaft , L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control., IEEE Tran. on automatic control, 1992, Vol. 37, p: 770-784 [21] A. J. Van der Shcaft , L2 -gain and Passivity Techniques in nonlinear control., 2002, Springer, [22] R. Ortega, A. Lori’a, P. J. Nicklasson and H. SiraRami’rez, ”Passivity-based Control of Euler-Lagrange Systems: Mechanical, electrical and electromechanical applications”, Springer-Verlag London, 1998. [23] A. Zemouche, M. Boutayeb and G. I. Bara, Observer
Design for Nonlinear Systems: An Approach Based on the Differential Mean Value Theorem., Proceedings of the 44th IEEE Conf. on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005, [24] Slotine, J.-J. E., and Li, W., ”On the adaptive control of robot manipulators”, Int. J Control , 1987, Vol. 6, No. 3, p: 49-59, [25] Hassan K. Khalil, ”Nonlinear Systems”, Second edition, Jun 1996, [26] L. Acho, Y. Orlov and V. Solis ”Nonlinear measurement feedback H∞ -control of time-periodic systems with application to tracking of robot manipulators”, Int. J Control 2001 Vol. 74 No. 2 pp. 190-198. Appendix A. PROPERTIES AND ASSUMPTIONS A 1. The matrix M (q) is symmetric positive definite and there exists some positive number σm and σM such that σm I ≤ M (q) ≤ σM I (A.1) A 2. There exist some positive constants σ1 and σ2 such that σ1 ≥ sup kg(q)k (A.2) q∈ℜn ° ° ° ∂g(q) ° ° (A.3) σ2 ≥ sup ° ° ∂q ° q∈ℜn
P 1. The representation of the matrix C(q, q) ˙ is unique and it can be obtained by the entries of the inertia matrix M (q). Let the ij-th element of the inertia matrix M (q) be denoted by mij , and let the ik-th element of the matrix C(q, q) ˙ be given by n X Cik (q, q) ˙ = cijk (q) (A.4) j
where
¶ µ 1 ∂mik (q) ∂mjk (q) ∂mij (q) (A.5) + − 2 ∂qj ∂qi ∂qk are the Christoffel symbols of the first kind. Then the property M˙ (q) = C(q, q) ˙ + C T (q, q), ˙ ∀q, q˙ (A.6) cijk (q) ≡
holds. ( The proof can be found in [1]). P 2. The matrix C(v1 , v2 ) is bounded in v1 and linear in v2 , then C(v1 , v2 )v3 = C(v1 , v3 )v2 , ∀v1 , v2 , v3 ∈ ℜn (A.7) kC(v1 , v2 )k ≤ σ4 kv2 k, for some σ4 > 0, ∀v1 , v2 (A.8) P 3. The system can be parameterized as follows M (q)¨ q + C(q, q) ˙ + H q˙ + G(q) = Y1 (q, q, ˙ q¨)T p (A.9) p where p ∈ ℜ is a vector of constant parameters and Y1 (q, q, ˙ q¨) ∈ ℜp×n is called regressor matrix (see [1]). P 4. Define ∆W=[M (q)−M (qr )]¨ qr+[C(q, q˙r )−C(qr , q˙r )]q˙r+ [G(q)−G(qr )], e1 = q − qr (A.10) where (qr , q˙r , q¨r ) is the desired trajectory, which is bounded. Therefore, there is a positive number bmcg such that k∆W k ≤ bmcg ke1 k, ∀e1 , (see [2]).
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Nonlinear Adaptive H∞ Output Feedback Tracking Control for Robotic Systems I. Levi ∗ N. Berman ∗ A. Ailon ∗∗ ∗
Mechanical Engineering Department, Ben-Gurion University of the Negev, Beer Sheva, Israel. ∗∗ Electrical Engineering Department, Ben-Gurion University of the Negev, Beer Sheva, Israel.
Abstract: This paper presents a solution to the tracking control problem of robotic systems in the presence of exogenous disturbances and model uncertainty with partial state information. The solution yields a Linear Matrix Inequalities (LMIs) based tracking output feedback controller. The main contribution of this paper lies in its particular approach which facilitates an application of the linear H∞ control theory without linearizing the underlying system. This yields a relatively simple and elegant design procedure. In addition, a relatively low gain controller is achieved. Simulation results of application this control algorithm in a two-degree of freedom robot demonstrates the design procedure feasibility. 1. INTRODUCTION This paper introduces a solution to the trajectory tracking control of robotic manipulators which is based on the H∞ control and LMI methods. It is assumed that only a noisy partial state information is available, and that a model uncertainty and exogenous disturbances are present. There are numerous papers which present studies of this subject, see, e.g. [1]-[7] for a state feedback utilization, and [8], [9], [10] for output feedback applications. Studies of this subject which deal with model uncertainty and assume partial information while using adaptive and robust control may be found in [11], [12], [13]. To the best of our knowledge all the studies (excluding those that take the H∞ approach) do not assume the presence of exogenous disturbances, neither a plant noise, nor a measurement noise. The works of Acho et. al ([26]) and Zasadzinski et. al ([10]) which take the H∞ approach, although they assume a presence of exogenous disturbances they do not consider model uncertainty as the theories they develop do not account for it. The novelty of this paper is in its particular approach and in its extent of generality. In particular: 1. the results achieved in this work apply to robotic systems with model uncertainty and with exogenous disturbances that include both, noise associated with the plant and noisy measurements. 2. A particular choice of a storage function which facilitates an application of the linear H∞ control theory and the LMI methods without linearizing the underlying system. In view of the theory of nonlinear H∞ control (see, e.g. [17]-[21]), we formulate the tracking problem as an H∞ control problem, and use the interrelations among the the l2 -gain property, dissipativity and the HamiltonJacobi Inequality (HJI) to derive an output feedback controller, first for the case of absence of uncertainty, and then utilizing these results, we develop a controller that
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accounts for model uncertainty, achieves L2 -gain< γ for a prescribed γ, and a semi-global asymptotic stability. As mentioned above, all this is facilitated by the particular choice of a storage function that takes an advantage of some certain structural properties the underlying system enjoys. This yields sufficient conditions, in terms of certain LMIs for the semi-global asymptotic stability and for the L2 -gain property to hold. The advantage of these sufficient conditions is that they turn to be exactly as the usual ones for an appropriate linear system (see, e.g [14]-[16]). We also introduce an example which demonstrates the algorithm performances by an application in a two-degree of freedom robot where gravity and the model-parameters are only approximately known, while relatively large uncertainties are assumed. 2. PROBLEM FORMULATION: NO MODEL UNCERTAINTY In this section we consider the tracking problem of an nlink robot manipulator with no model uncertainty. In section 2.1 below we introduce a convenient state space representation of the underlying system. The nonlinear H∞ control problem is formulated in section 2.2, while the solution to the nonlinear HJI is introduced in section 2.3. 2.1 The System dynamics The dynamical equations of an n-link robot manipulator with exogenous disturbances is commonly described by the following (see, e.g. Spong and Vidyasagar [1]) M (q)¨ q + (C(q, q) ˙ + H)q˙ + G(q) = τ + ω (1) where q ∈ ℜn is the robot’s joint angular position, M (q) ∈ ℜn×n is the symmetric positive definite inertia matrix, C(q, q) ˙ q˙ is the centripetal and coriolis forces, H q˙ represents the linear frictional forces, G(q) consists of the gravitational forces, τ is the torque applied to the various
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links at the corresponding joints by means of electrical motors and ω represents exogenous R ∞ disturbances, which are assumed to be in L2 , that is 0 ||ω(t)||2 dt < ∞.
The objective is a design of an output feedback which drives the system’s states along a desired trajectory qr (t) starting at a given initial position. For this we define the following error vector: ¸ · ¸ · q − qr e (2) e= 1 = q˙ − q˙r e2
We take: τ =M (qr )¨ qr + (C(qr , q˙r ) + H)q˙r + G(qr ) + u (3) where u is defined in section 2.2 below. Define ∆W =[M (q)−M (qr )]¨ qr +[C(q, q˙r )−C(qr , q˙r )]q˙r+ [G(q)−G(qr )]. (4) Using these in (1), yields the following tracking problem. e˙ = A(q, q, ˙ q˙r )e + B(q)(u − ∆W + ω) (5) where · ¸ 0 In×n A(q, q, ˙ q˙r ) = n×n 0n×n −M −1 (q)(C(q, q)+C(q, ˙ q˙r )+H) ¸ · 0n×n (6) B(q) = M −1 (q) 2.2 Nonlinear H∞ control problem Consider the nonlinear system: ( e˙ = A(q, q, ˙ q˙r )e + B(q)(u − ∆W ) + B(q)ω y = C2 e + D21 ω z = C1 e + D12 u
Where e ∈ ℜ2n , u ∈ ℜm , y ∈ ℜs and ω ∈ ℜd are the state, the control input, the measurement output and disturbances, respectively, while z ∈ ℜh is an objective variable (controlled output). The H∞ output-feedback control objective is a synthesis of an output-feedback that renders the underlying system L2 -gain< γ. In order to achieve this goal the following controller structure ½ is assumed −1 ξ˙ = T (q) [Ak ξ + Bk y] (7) u = Ck ξ + Dk y where ξ ∈ ℜ2n , T (q) is a 2n × 2n matrix, and Ak , Bk , Ck , Dk are constant matrices. Let · ¸ e x= (8) ξ Thus the closed-loop system admits ½ x˙ = Acl (q, q, ˙ q˙r )x − Bcl (q)∆W + B1cl (q)ω z = Ccl x + Dcl ω
Bcl (q) =
We have now the following theorem, the proof of which is omitted for the lack of space. Theorem 1. Given δ > 0. Assume Po (q) has the following structure: Po (q) = Po.c Mo (q) (14) where Mo (q) = blockdiag{Ms (q), T (q)}, (15) Ms (q) given in (13) and Po.c ∈ ℜ4n×4n is a positive symmetric matrix that is to be determined. Then the closedloop system (9) is L2 -gain < γ, and the controller (7) renders the closed-loop system semi-global exponentially stable if the following LMI’s LM I (1) : T ˜ Po.c Acl + ATcl Po.cPo.c B1cl Ccl Po.c Bcl ∆W T ∗ −γ 2 I Dcl 0 0 ∗ ∗ −I 0 0 < 0 1 ∗ ∗ ∗ − δI 0 2 ∗ ∗ ∗ ∗ −δ −1 I (16) LM I (2) : Po.c Mo (q) + (Po.c Mo (q))T > 0 (17) hold for e2 ∈ Br with an arbitrarily fixed r > 0 and for all q, where · ¸ A+BD C BC B(D D +I) k 2 k k 12 Acl B1cl = Bk D21 , Bk C2 Ak Ccl Dcl C1+D12 Dk C2 D12 Ck D12 Dk D21 ˜ = blockdiag{∆W ˜ 1 , 02n×2n }, ∆W ˜ 1 = blockdiag{bmcg In×n , bc˜In×n }, ∆W
and
¸ B(q) . 0 2n×n
Consider the nonlinear system (9) with the following storage function 1 (12) So (x, q) = xT Po (q)x 2 where Po (q) ∈ ℜ4n×4n is a positive C 1 matrix, (note that Po (q) is not necessarily a symmetric matrix). Define, Ms (q) = blockdiag{In×n , M (q)} (13) where M (q) is the inertia matrix and blockdiag{·} denotes a diagonal block matrix. The notation * will be used frequently in the sequel and will denote a symmetric entry of a matrix.
(9)
where · ¸ Acl (q, q, ˙ q˙r ) B1cl (q) = Ccl Dcl A(q, q, ˙ q˙r )+B(q)Dk C2 B(q)Ck B(q)(Dk D12+I) T (q)−1 Bk C2 T (q)−1 Ak T (q)−1 Bk D21 (10) C1+D12 Dk C2 D12 Ck D12 Dk D21 ·
2.3 Solution To The Nonlinear Hamilton-Jacobi Inequality (HJI)
(11)
bmcg , bc˜ > 0 (18)
and ¸ ¸ · · 0n×n B , B= Bcl = In×n 0 2n×n ¸ · 0 I A = n×n n×n . 0n×n −H
(19)
Remark 1. The L2 -gain< γ means in this theorem that for any initial state x0 there is a neighborhood Bx0 ⊂ L2 ofZ0 such that Z ∞ ∞ ||ω(t)||2 dt}, ∀t ≥ 0. ||z(t)||2 dt < γ 2 {||x0 ||2 + 0
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Theorem 2 below provides sufficient conditions for Theorem 1 to hold. Theorem 2. Fix δ > 0, γ > 0, r > 0. Assume the following LMI’s ˜1 Φ11 Φ12 Φ13 Φ14 XB ∆W ∗ Φ22 Φ23 Φ24 B Y ∆W ˜ 1 ∗ ∗ −γ 2 I 0 0 0 LM I 1 : ∗ ∗ (20) < 0 ∗ −I 0 0 1 ∗ ∗ − δI 0 ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ −δ −1 I ˆk C2+C T B ˆ T , Φ12=Aˆk+AT+C T D ˆ TB T Φ11=AT X+XA+B 2 k 2 k T T ˆT T ˆ Φ13=XB+Bk D21 , Φ14=C1 +C2 Dk D12 ˆ k D21 Φ22=Y AT +AY + B Cˆk+CˆkT B T , Φ23=B+B D T ˆT T Φ24=Y C1 +Ck D12 · ¸ XMs(j) + Ms(j) X Ms(j) LM I 2 : > 0 (21) ∗ Ms(j) Y + Y Ms(j) hold for some symmetric positive definite matrices X, Y ∈ ˆk ∈ ℜ2n×n , Cˆk ∈ ℜ2n×2n and for some Aˆk ∈ ℜ2n×2n , B (j) n×2n ˆ n×n ℜ , Dk ∈ ℜ , where each Ms is the Ms (q) evaluated at the vertex j of the polytope generated by Ms (q). Then the closed-loop system (9), with the controller (3), (7), is L2 -gain< γ, and semi-global exponentially stable, where T (q) is given by T (q) = −M −1 XMs (q)Y N −T (22) If a solution to these LMIs exist, the output feedback gains are given by Aˆk = XAY+M Ak N T+XBCk N T+M Bk C2 Y+XBDk C2 Y ˆk = M Bk +XB2 Dk B Cˆk = Ck N T +Dk C2 Y ˆ k = Dk D
(23)
3. PROBLEM FORMULATION WITH MODEL UNCERTAINTIES This section deals with the tracking problem of an n-link robot manipulator with model uncertainties. In section 3.1 below we introduce the state space equations. The nonlinear H∞ control problem is formulated in section 3.2, while the solution to the nonlinear HJI is introduced in section 3.3. 3.1 The System Dynamics Generally, in addition to external disturbances, there are uncertainties present in the system’s model which must ˆ (q) denote an estimate of M (q), be accounted for. Let M ˆ ˆ an estimate of H and C(q, q) ˙ an estimate of C(q, q), ˙ H ˆ G(q) an estimate of the gravitational forces G(q). The inverse dynamics control law for the nominal system (1) is given by ˆ (qr )¨ ˆ r , q˙r ) + H) ˆ q˙r + G(q ˆ r ) + u. τ =M qr + (C(q (24) Substituting (24) into (1) and subtract M (q)¨qr + (C(q, q) ˙ + H)q˙r from both sides of the equation we obtain ˆ (qr ) − M (q)]¨ M (q)e˙ 2 +(C(q, q) ˙ + H)e2 = u + ω + [M qr + ˆ r , q˙r )−C(q, q)+ ˆ −H]q˙r + [G(q ˆ r ) − G(q)]. [C(q ˙ H (25)
It is easy to show now that (25) may now be expressed as M (q)e˙ 2 + (C(q, q) ˙ + C(q, q˙r ) + H)e2 = u + ω+ ˆ ˆ r , q˙r ) − C(q, q˙r )]q˙r + [M (qr ) − M (q)]¨ qr + [C(q ˆ −H]q˙r + [G(q ˆ r ) − G(q)]. [H (26) from which one obtains M (q)e˙ 2 + (C(q, q) ˙ + C(q, q˙r ) + H)e2 = u + ω+ Y1 (qr , q˙r , q¨r )Po.c − ∆W. (27) where p˜= pˆ−p is the parameter error vector, Y1 (qr , q˙r , q¨r ) is the regressor matrix and ∆W given by (4). Thus, the state space can be written as e˙ = A(q, q, ˙ q˙r )e + B(q)(u + ω + Y1 (qr , q˙r , q¨r )˜ p − ∆W ) (28) where A(q, q, ˙ q˙r ), B(q) are given in (6). 3.2 The Nonlinear H∞ control problem Consider the nonlinear system: ( e˙ = A(q, q, ˙ q˙r )e+B(q)(u+Y1 (qr , q˙r , q¨r )˜ p −∆W )+B(q)ω y = C2 e + D21 ω z = C1 e + D12 u (29) In order to obtain an adaptive H∞ output-feedback control objective our goal is to compute a dynamical outputfeedback controller in the same form as given in (7),i.e ½ −1 ξ˙ = T¯(q) [Ak ξ + Bk y] (30) u = Ck ξ + Dk y where ξ ∈ ℜ2n , T¯(q) is a 2n × 2n matrix (to be determined below) and Ak , Bk , Ck , Dk are constant matrices. In addition, we choose the parameter estimator of the form p˜˙ = Ax (qr , q˙r , q¨r )x (31) where x given by (8) and Ax (qr , q˙r , q¨r ) will be determined later. Let x ˜ be defined by · ¸ x . (32) x ˜= p˜ Then the closed-loop system admits ½ ˜cl (q)(Y1 (qr , q˙r , q¨r )˜ ˜1cl (q)ω x ˜˙ =A˜cl (q, q, ˙ q˙r )˜ x+B p −∆W )+ B ˜ ˜ z=Ccl x ˜ + Dcl ω (33) where ˜1cl (q) A˜cl (q, q, ˙ q˙r ) B Acl (q, q, ˙ q˙r )0r×r B1cl (q) =Ax (qr , q˙r , q¨r )0n×r 0r×d ˜ cl Ccl 0h×r Dcl C˜cl D · ¸ ˜cl (q) = Bcl (q) . B (34) 0r×n
and Acl (q, q, ˙ q˙r ), Bcl (q), Ccl , Dcl , B1cl (q) are given in (10,11) with T¯(q) instead of T (q). 3.3 Solution To The Nonlinear HJI Consider the nonlinear system (33) with the following storage function 1 p} (35) S¯o (x, p˜, q) = {xT P¯o (q)x + p˜T Λ˜ 2 where Λ is a positive definite weighting matrix and P¯o (q) is a positive C 1 matrix, (note that P¯o (q) is not necessary a symmetric matrix).
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Theorem 3. Given δ > 0. Assume the following structure for P¯o (q): ¯ o (q) P¯o (q) = Po.c M (36) where ¯ o (q) = blockdiag{Ms (q), T¯(q)}, M (37) 4n×4n Ms (q) given in (13) and Po.c ∈ ℜ is a positive symmetric matrix to be determined. Then the closed-loop system (33) is L2 − gain ≤ γ, and the controller (30) renders the closed-loop system semi-global asymptotically stable if the following LMI’s LM I (1) : T ˜ Po.c Acl + ATcl Po.cPo.c B1cl Ccl Po.c Bcl ∆W T ∗ −γ 2 I Dcl 0 0 ∗ ∗ −I 0 0 < 0 1 0 ∗ ∗ ∗ − δI 2 ∗ ∗ ∗ ∗ −δ −1 I (38) LM I (2) : ¯ o (q))T > 0 ¯ o (q) + (Po.c M (39) Po.c M hold for e2 ∈ Br with an arbitrarily fixed r > 0 and for all q, where ¸ · A+BDk C2 BCk B(Dk D12+I) Acl B1cl , = Bk D21 Bk C2 Ak Ccl Dcl C1+D12 Dk C2 D12 Ck D12 Dk D21 ˜ ˜ 1 , 02n×2n }, ∆W = blockdiag{∆W ˜ 1 = blockdiag{bmcg In×n , bc˜0 In×n }, ∆W
bmcg , bc˜0 > 0 (40)
Remark 2. Note that if the uncertainty range of fi shrinks to zero then pav → p. Define, m11 (pav , q) · · · m1n (pav , q) .. .. M av (q) = . ∗ . ∗ ∗ mnn (pav , q)
where M av (q) is the average matrix of the inertia matrix M (q). Obviously, by the above definition of M av (q), this matrix agrees with assumption A1. Finally we define the matrix Msav (q) = blockdiag{I2n×2n , M av (q)} (44) We have now the following result. Theorem 4. Consider the closed-loop system (33) with the storage function (35) where T¯(q) = −M −1 XMsav (q)Y N −1 (45) Given the scalars δ > 0, γ > 0, r > 0, ε > 0, there is an output-feedback controller given by (30) with the parameter update process given by (31), where T ˜T Y1T (qr , q˙r , r¨r )Bcl Po (q) = Ax (qr , q˙r , q¨r ). If the following LMI’s ˜1 Φ11 Φ12 Φ13 Φ14 XB ∆W ∗ Φ22 Φ23 Φ24 B Y ∆W ˜ 1 ∗ ∗ −γ 2 I 0 0 0 LM I 1 : ∗ ∗ ≤0 (46) ∗ −I 0 0 1 ∗ ∗ − δI 0 ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ −δ −1 I ˆk C2+C2T B ˆkT , Φ12=Aˆk+AT+C2T D ˆ kTB T Φ11=AT X+XA+B T T ˆT T ˆ Φ13=XB+Bk D21 , Φ14=C1 +C2 Dk D12 ˆ k D21 Φ22=Y AT +AY + B Cˆk+CˆkT B T , Φ23=B+B D T ˆT T Φ24=Y C1 +Ck D12
and ·
B
¸
, B= 0 2n×n ¸ · 0n×n In×n . A0 = 0n×n 0n×n Bcl =
·
0n×n In×n
¸ (41)
LM I (2) : Ψ11 Ms(j) (q) X(Ms(j)−Msav(j) ) 0 ∗ Ψ22 0 Y >0 ∗ ∗ εI 0 ∗ ∗ ∗ ε−1 I
In what follows we utilize the algorithm introduced in [16] in order to solve the LMI’s of Theorem 3 via LMI’s optimization toolbox in MATLAB. The following notations will be used in the sequel m11 (p, q) · · · m1n (p, q) .. .. M (q) = . ∗ . ∗ ∗ mnn (p, q)
(43)
(47)
Ψ11 = XMs(j) (q)+Ms(j) (q),
where mik (p, q) are bounded, with known bounds. It is well known that the parameters vector p is a function of the physical system’s parameters like: masses, lengths etc.. We take fi to be the i-th physical parameter of the system, therefore if we assume that the system has −l− physical parameters then the vector p may be written as p = F(f1 , f2 , ..., fl ). We denote the upper bound of fi by fi+ (i.e. fi ∈ [fi− , fi+ ]) and the lower bound of fi by fi− and define the following: Let fiav be the average physical parameter of fi , and pav be the average vector parameter of p which are given by: 1 fiav = (fi− + fi+ ) ∀i = 1, ..., l. 2 pav = F(f1av , f2av , ..., flav ). (42)
Ψ22 = Ms(j) (q)Y +Y Ms(j) (q) hold for some symmetric positive definite matrices X, Y ∈ ˆk ∈ ℜ2n×n , Cˆk ∈ ℜ2n×2n and for some Aˆk ∈ ℜ2n×2n , B (j) av(j) n×2n ˆ n×n ℜ , Dk ∈ ℜ , where each Ms , Ms are the av Ms (q), Ms (q) (given by (13),(44) respectively) evaluated at the vertex j of the polytope generated by Ms (q), Msav (q), respectively. If a solution to these LMIs exists, then the closed-loop system (33) has L2 -gain≤ γ (from ω to z) and the tracking error e → 0 as t → ∞ semi-globally. In this case, the output-feedback is given by (24,30,23) with the parameter update process pˆ(t) =ˆ p(0)+Ax (qr (t), q˙r (t), q¨r (t))xI (t)− Z t A˙ x (qr (σ), q˙r (σ), q¨r (σ))xI (σ)dσ (48) 0
where
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17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
a
t
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e (τ )dτ 1 Z t 0 e1 (t) x(τ )dτ = xI (t) = Z t 0 ξ(τ )dτ Z
0
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4. EXAMPLE The feasibility of the design of the foregoing sections is demonstrated via simulations of a two-link manipulator. The system is assumed to have known parameters and external disturbances. The H∞ tracking control is then designed according to the proposed procedure. The system’s parameters are: the links’ masses: m1 , m2 (kg), the links’ lengths: l1 , l2 (m), masses’ centers: lc1 , lc2 , the angular positions: q1 , q2 (rad), q1 , q2 (rad), the viscosity coefficient : h(kgm2 ) and the applied torques: τ1 , τ2 (N m). By (1) we have: · ¸ m l l cos(q −q )+I m l2 +m l2+I +I M (q)= 1 c1 2 1 zz1 zz2 2 1 c2 2 1 2 zz2 m2 l1 lc2 cos(q1−q2 )+Izz2 m2 lc2+Izz2 ¸ · 0 q˙2 C(q, q) ˙ = m2 l1 lc2 sin(q1 − q2 ) −q˙1 0 · ¸ −(m1 lc1 + m2 l1 )gsin(q1 ) G(q) = −m2 lc2 gsin(q2 ) · ¸ h 0 H= (51) 0 h
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Fig. 1. Exact model: (a) Positions of link 1. (b) Positions of link 2. (c) Torques. [2]
[3]
[4] [5] [6]
REFERENCES
0
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where q ∈ ℜ2 and τ ∈ ℜ2 . The nominal parameters of the manipulator are taken to be: m1 = 1(kg), m2 = 5(kg), l1 = l2 = 0.2(m), lc1 = lc2 = 0.1(m), g = 9.8(ms−2 ), h = 5((kgm2 )) where Izz1 = Izz2 = 13 mi l2 (i=1,2) and the initial conditions are q1 (0) = 30◦ , q2 (0) = 100◦ , q˙1 (0) = 0, q˙2 (0) = 0. The desired position is: qr1 = ◦ 30◦ sin(2πt), ·qr2 = ¸ 60 sin(2πt). The exogenous disturω1 bances ω = are chosen to be square wave with ω2 period 2π, that is ½ ½ 1 , 0≤t<π 0 , 0≤t<π ω1 = , ω2 = (52) 0 , π ≤ t < 2π −1 , π ≤ t < 2π For the purpose of simulations, C1 and D12 were chosen as: ¸T · · ¸T 0 0 10 I 0 0 0 C1 = 2 2×2 2×2 2×1 2×1 ,D12 = 0.001 1×2 1×2 01×2 01×2 01 02×2 I2×2 02×1 02×1 and ¸ · 25I2×2 02×2 ˜ , δ = 10−6 ∆W 1 = 02×2 I2×2 In this case:· ¸ 2 m1 lc1 +m2 l12+Izz1+Izz2 m2 l1 lc2 · δj+Izz2 (j) M = 2 m2 l1 lc2 · δj+Izz2 m2 lc2 +Izz2 j=1, 2, δ1=1, δ2 = −1 By applying Theorem 2 we obtain γmin = 4.0467. However, γ = 4.05 was selected to avoid an undesirable highgain controller design corresponding to γ which is close to the optimum. (see Fig.1).
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[1] M. W. Spong and M. Vidyasagar , Robot Dynamics and Control, New york: Wiley, 1989,
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N. Sadegh, and R. Horowitz, ”Stability and Robustness Analysis of a Class of Adaptive Controllers for Robotic Mnipulators”, Int. J. Robotics Research, Jun 1990, Vol. 9, No. 3, p: 74-92, B. S. Chen and Y. C. Chang, ”Nonlinear Mixed H2 /H∞ control for robust tracking design of robotic systems”, Int. J. Control, 1997, Vol. 67, No. 6, p: 837857. H. L. Trentelman, J. C. Willems,, ”Nonlinear H∞ Control of Robotic Manipulator”, 1999, T-Y. Kuc, W-G. Han, ”An adaptive PID learning control of robot manipulatorsq”, Automatica, Sep 2000, Vol. 36, p: 717-725, J. Alvarez-Ramirez, R. Kelly and I. Cervantes, ”Semiglobal stability of saturated linear PID control for robot manipulators”, Automatica, Jun 2003, Vol.
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
39, Issue 6, p: 989-995, [7] T. Ravichandran, D. W. L. Wang and G. R. Heppler, ”Stability and Robustness of a Class of Nonlinear Controllers for Robot Manipulators”, Automatica, July 2004, Vol. 2, Issue 6, p: 5262-5267, [8] E.S. Shin and K.W.Lee, ”Robust Output Feedback Control of Robot Manipulators Using High-Gain Observer”, Proceedings of the 1999 IEEE Int. Conf. on Control Applicatlons, July 1999, p: 881-886, [9] K.W. Lee and H.K. Khalil, ”Adaptive output feedback control of robot manipulators using high-gain observer”, Int. J. Control , July 1997, Vol. 67, No. 6, p: 869-886, [10] M. Zasadzinski, E. Richard, M. F. Khelfi and M. Darouach, ”Disturbance Attenuation and Trajectory Tracking via a Reduced-order Output Feedback Controller for Robot Manipulators”, Automatica, 1998, Vol. 33, No. 12, p: 1539-1546, [11] P. R. Pagilla, M. Tomizuka, ”An adaptive output feedback controller for robot arms: stability and experiments”, Automatica, 2001, Vol. 37, p: 983-995, [12] M. A. Arteaga, ”Robot control and parameter estimation with only joint position measurements”, Automatica, January 2003, Vol. 39, Issue. 1, p: 67-73, [13] E. Kim, ”Output Feedback Tracking Control of Robot Manipulators With Model Uncertainty via Adaptive Fuzzy Logic”, IEEE Tran. on fuzzy systems,, June 2004, p: 368-378,An LPD Approach to Robust and Static Output-Feedback Design [14] P. L. D. Peres, J. C. Geromel, and S. R. Souza, ” H∞ control design by static output-feedback”, Proc. IFAC Symp. Robust Control Design, Rio de Janeiro, Brazil, 1994, p: 243248, [15] U. Shaked, ”An LPD Approach to Robust H2 and H∞ Static Output-Feedback Design”, IEEE Tran. on automatic control , May 2003, Vol. 48, No. 5, p: 866872, [16] C. Scherer, P. Gahinet and M. Chilali, ”Multiobjective Output-Feedback Control via LMI Optimization”, IEEE Tran. on automatic control , July 1997, No. 7, p: 866-872, [17] A. J. Van der Shcaft , On a state space approach to nonlinear H∞ control., Systems and Control Latters, 1991, Vol. 16, p: 1-8. [18] A. Isidori and A. Astolfi , Nonlinear H∞ control via measurement feedback, Journal of Mathematical Systems, Estimation, and control, 1992, Vol. 2, p: 3144. [19] A. J. Krener , Necessary and sufficient conditions for nonlinear worst case (H∞ ) control and estimation, Journal of Mathematical Systems, Estimation, and control, 1994, Vol. 4, p: 1-25. [20] A. J. Van der Shcaft , L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control., IEEE Tran. on automatic control, 1992, Vol. 37, p: 770-784 [21] A. J. Van der Shcaft , L2 -gain and Passivity Techniques in nonlinear control., 2002, Springer, [22] R. Ortega, A. Lori’a, P. J. Nicklasson and H. SiraRami’rez, ”Passivity-based Control of Euler-Lagrange Systems: Mechanical, electrical and electromechanical applications”, Springer-Verlag London, 1998. [23] A. Zemouche, M. Boutayeb and G. I. Bara, Observer
Design for Nonlinear Systems: An Approach Based on the Differential Mean Value Theorem., Proceedings of the 44th IEEE Conf. on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005, [24] Slotine, J.-J. E., and Li, W., ”On the adaptive control of robot manipulators”, Int. J Control , 1987, Vol. 6, No. 3, p: 49-59, [25] Hassan K. Khalil, ”Nonlinear Systems”, Second edition, Jun 1996, [26] L. Acho, Y. Orlov and V. Solis ”Nonlinear measurement feedback H∞ -control of time-periodic systems with application to tracking of robot manipulators”, Int. J Control 2001 Vol. 74 No. 2 pp. 190-198. Appendix A. PROPERTIES AND ASSUMPTIONS A 1. The matrix M (q) is symmetric positive definite and there exists some positive number σm and σM such that σm I ≤ M (q) ≤ σM I (A.1) A 2. There exist some positive constants σ1 and σ2 such that σ1 ≥ sup kg(q)k (A.2) q∈ℜn ° ° ° ∂g(q) ° ° (A.3) σ2 ≥ sup ° ° ∂q ° q∈ℜn
P 1. The representation of the matrix C(q, q) ˙ is unique and it can be obtained by the entries of the inertia matrix M (q). Let the ij-th element of the inertia matrix M (q) be denoted by mij , and let the ik-th element of the matrix C(q, q) ˙ be given by n X Cik (q, q) ˙ = cijk (q) (A.4) j
where
¶ µ 1 ∂mik (q) ∂mjk (q) ∂mij (q) (A.5) + − 2 ∂qj ∂qi ∂qk are the Christoffel symbols of the first kind. Then the property M˙ (q) = C(q, q) ˙ + C T (q, q), ˙ ∀q, q˙ (A.6) cijk (q) ≡
holds. ( The proof can be found in [1]). P 2. The matrix C(v1 , v2 ) is bounded in v1 and linear in v2 , then C(v1 , v2 )v3 = C(v1 , v3 )v2 , ∀v1 , v2 , v3 ∈ ℜn (A.7) kC(v1 , v2 )k ≤ σ4 kv2 k, for some σ4 > 0, ∀v1 , v2 (A.8) P 3. The system can be parameterized as follows M (q)¨ q + C(q, q) ˙ + H q˙ + G(q) = Y1 (q, q, ˙ q¨)T p (A.9) p where p ∈ ℜ is a vector of constant parameters and Y1 (q, q, ˙ q¨) ∈ ℜp×n is called regressor matrix (see [1]). P 4. Define ∆W=[M (q)−M (qr )]¨ qr+[C(q, q˙r )−C(qr , q˙r )]q˙r+ [G(q)−G(qr )], e1 = q − qr (A.10) where (qr , q˙r , q¨r ) is the desired trajectory, which is bounded. Therefore, there is a positive number bmcg such that k∆W k ≤ bmcg ke1 k, ∀e1 , (see [2]).
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