Automatic Control, IEEE Transactions on

1198

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002

We next show (3). Suppose, on the contrary, that there exist such that

(k1 ; k2 ; k3 ) = 6 (0; 0; 0)

k1 eAt

At

+ k2 e

+ k3 I

b = 0:

Marco A. Arteaga and Yu Tang

Noting that

k1 eAt

At

+ k2 e

2 + k3 I = 1 A + 2 A + 3 I

for some ( 1 ; 2 ; 3 ) 6= (0; 0; 0), we would have 2 [A b

1 Ab b ] 2

3

Adaptive Control of Robots With an Improved Transient Performance

=0

Abstract—By using a robust control technique, this note proposes an adaptive control for rigid robots with the following important features: under a parameter-dependent persistent excitation (PE) condition, it gives a guaranteed transient performance of tracking a smooth desired trajectory while assuring the parameter estimation error to go to a residual set of the origin arbitrarily fast. Simulations are included to support the theoretical results. Index Terms—Adaptive control, robot manipulators, transient performance.

which contradicts the assumption that (A; b) is controllable. I. INTRODUCTION REFERENCES [1] A. T. Fuller, “In-the-large stability of relay and saturating control systems with linear controller,” Int. J. Control, vol. 10, pp. 457–480, 1969. [2] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis and Design. Boston, MA: Birkhäuser, 2001. [3] T. Hu, Z. Lin, and L. Qiu, “Stabilization of exponentially unstable linear systems with saturating actuators,” IEEE Trans. Automat. Contr., vol. 46, pp. 973–979, June 2001. [4] T. Hu, Z. Lin, and Y. Shamash, “Semiglobal stabilization with guaranteed regional performance of linear systems subject to actuator saturation,” Syst. Control Lett., vol. 43, no. 3, pp. 203–210, 2001. [5] Z. Lin, Low Gain Feedback. London, U.K.: Springer-Verlag, 1998, vol. 240, Lecture Notes in Control and Information Sciences. [6] Z. Lin and A. Saberi, “Semiglobal exponential stabilization of linear systems subject to ‘input saturation’ via linear feedbacks,” Syst. Control Lett., vol. 21, pp. 225–239, 1993. BIBO output feedback stabilization with saturated [7] A. Megretski, “ control,” in Proc. 13th IFAC World Congress, vol. D, 1996, pp. 435–440. [8] H. J. Sussmann, E. D. Sontag, and Y. Yang, “A general result on the stabilization of linear systems using bounded controls,” IEEE Trans. Automat. Contr., vol. 39, pp. 2411–2425, Dec. 1994. [9] H. J. Sussmann and Y. Yang, “On the stabilizability of multiple integrators by means of bounded feedback controls,” in Proc. 30th IEEE Conf. Decision and Control, 1991, pp. 70–72. [10] R. Suarez, J. Alvarez-Ramirez, and J. Solis-Daun, “Linear systems with bounded inputs: Global stabilization with eigenvalue placement,” Int. J. Robust Nonlin. Control, vol. 7, pp. 835–845, 1997. [11] A. R. Teel, “Global stabilization and restricted tracking for multiple integrators with bounded controls,” Syst. Control Lett., vol. 18, pp. 165–171, 1992. , “Linear systems with input nonlinearities: Global stabilization by [12] scheduling a family of -type controllers,” Int. J. Robust Nonlin. Control, vol. 5, pp. 399–441, 1995.

The dynamics of rigid robots can be described by a set of nonlinear differential equations. In order to be able to carry out accurate tracking control, the knowledge of the robot model parameters is necessary. However, it is a rather difficult task to calculate the parameter vector accurately. Fortunately, the nonlinear model of rigid robots is linear in its parameters [2], [3]. Thus, adaptive control of robots has received considerable attention during the last two decades (see [4]–[9]). Since the main goal of robot control is to achieve accurate tracking of desired trajectories, many globally stable algorithms have been developed that result in zero tracking error in the steady state (see [10] and [11]). Nevertheless, parameter convergence does not necessarily take place. In fact, even if there is persistent excitation (PE) it may take long time before the estimated parameters tend to the real ones, what decreases the transient performance of the tracking error. Also, without the PE condition being satisfied, in the presence of external perturbations and/or unmodeled dynamics most of the existing adaptation algorithms may present parameter drifting phenomena similar to those observed in adaptive controllers studied in the 1980s. A solution to this consists in modifying the adaptation algorithms. Since robot manipulators constitute a class of passive systems, many authors have exploited this property in order to design and prove the stability of their adaptive control approaches [12]–[14]. In fact, the passivity property can be exploited in a very general framework to design adaptive algorithms, i.e., it can be shown that any adaptive algorithm which is passive can be used to have zero tracking error in the steady state [15]. This fact was used in [16] to slightly relax the general PE condition required for most algorithms in order to guarantee parameter convergence by taking advantage of the transient response of the system. Since a well-known problem in most adaptive controllers is the poor transient response observed when the adaptation is initiated, [17]–[19] present adaptive schemes which give a guaranteed transient performance. In the absence of disturbances, these algorithms are able to guarantee that the tracking errors will tend to zero as well. The transient performance is improved arbitrarily by a proper choice of some control gains. However, tuning these control gains too large to improve the transient performance usually implies that the output torques/forces

Manuscript received March 7, 2000; revised November 21, 2001. Recommended by Associate Editor W. Lin. This work was supported by CONACYTMexico under Grant 27565A and by the DGAPA under Grants IN109799 and IN103798. The authors are with the Sección de Eléctrica, DEPFI, Universidad Nacional Autónoma de México México, D. F. 04510, México (e-mail: [email protected]; [email protected]). Publisher Item Identifier 10.1109/TAC.2002.800672. 0018-9286/02$17.00 © 2002 IEEE


will become larger, what represents a drawback due to possible saturations of the actuators. By using robust control techniques [1], this note proposes an improved adaptive control law for rigid robots based on that given in [16] with the following important features: under a parameter-dependent PE condition, it gives a guaranteed transient performance of tracking a smooth desired trajectory while assuring the parameter estimation error to go to a residual set of the origin arbitrarily fast. The advantage of this parameter-dependent PE condition over the traditional PE condition is that one can choose some parameters in the adaptation algorithm based on the desired trajectory in such a way that this condition is met. In the limit case, this condition can be fulfilled by transient response. It is also guaranteed that the estimated parameters cannot be larger than a predefined bound. Furthermore, on the contrary to the algorithms given in [17]–[19], the improvement in the transient performance is not achieved by making control gains too large, but rather by making the parameter adaptation faster. This approach effectively improves the transient response without increasing input torques and forces. The note is organized as follows. Preliminaries are given in Section II. The control law is proposed in Section III, where the performance of the adaptive law is analyzed. Section IV presents simulation results. The note concludes in Section V. II. PRELIMINARIES The dynamics of a rigid robot arm with described by [11]

n

revolute joints can be

H (q )q + C (q ; q_ ) q_ + Dq_ + g (q ) = :

n is the vector of generalized coordinates. H (q ) 2 n2n is the symmetric positive definite inertia matrix, C (q ; q_ )q_ 2 n is the vector of Coriolis and centrifugal torques, g (q ) 2 n accounts for gravitational torques, D 2 n2n is the positive–semidefinite diagonal matrix of joint viscous friction coefficients and 2 n is the vector of torques acting at the joints. By defining the largest (smallest) eigenvalue of a matrix by max (1)(min (1)), the following properties hold [20]. Property II.1: H (q ) satisfies min (H )kxk2 x T H x 4 max (H )kxk2 8 q ; x 2 n . _ (q ) 0 2C (q ; q_ ) Property II.2: With a proper definition of C (q ; q_ ), H is skew symmetric. 4 Property II.3: With a proper definition of the robot parameters, (1) can be written as

H (q )q + C (q ; q_ ) q_ + Dq_ + g (q ) = = Y (q ; q_ ; q) 2

n p

is the regressor and

2

p

where K p 2 n2n is a diagonal positive–definite matrix. For simplicity, Y a Y (q ; q_ ; q_ r ; qr ) will be used hereafter. The control law (3) can also be written as

= H (q )q r + C (q ; q_ )q_ r + Dq_ r + g (q ) + Y a~ 0 K ps: H (q )s_ = 0C (q ; q_ ) s 0 D ps + Y a~

^_ (t) = 0 001 (t) Y Ta s + 1 h + Y Tf " 0 f b Z_ (t) = 0 Z Z (t) + 2Y Tf Y f ; Z (0) = O _ T h(0) = 0: h_ (t) = 0 hh(t) + 2Y f " + Z (t)^(t);

In this section, the tracking control problem of rigid robot arms with model parameter uncertainty is studied. Consider model (1) and define the tracking and parameter errors as q~ q 0q d , ~ ^0 , respectively, where q d is a desired bounded trajectory for q , with bounded first and second derivatives and (^1) denotes the estimate of (1). Before proposing the controller, the following definitions are necessary: q_ r q_ d 0 3q~, s q_ 0 q_ r = q~_ + 3q~, where 3 2 n2n is a diagonal positive–definite matrix. The proposed controller is given by ^ q_ + g =H^ (q )q r + C^ (q ; q_ ) q_ r + D ^(q ) 0 K p s r ^ 0 K ps r ) =Y (q ; q_ ; q_ r ; q

(3)

(6) (7) (8)

In (6), mi î (0) Mi , where mi and Mi are two constants which satisfy: mi i Mi . Note that it is quite reasonable to assume that at least the sign of the parameter i is known (say it is positive). In this case, the lower bound is simply 0. The upper bound can be tuned arbitrarily large. Also in (6)–(8), 2 ; are positive constants. 1 is given by

1 = 1

Y Ta s 11 min (P (t)) + "2 "1

+ 2

(9)

where 2 , "1 and "2 are positive constants and 1 > 2, with P (t) Z (t) + Y Tf (t)Y f (t). The gain matrix 0(t) is given by diag f 1 (t); . . . ; p (t)g

0(t)

0

(10)

with i (t) = ( i (0) 0 di )e + di , where i (0) > di > 0 and ai (t) 0. Note that 0(t) is a positive–definite matrix. Define the columns of Y a and Y f as y ai 2 n and y fi 2 n , respectively, for i = 1; . . . ; p. Since once î < mi or î > Mi the sign of ~i is known, the ith element of f b is defined by

fbi = 0sign ~i

a (t)dt

1 i 1 i y Tais + hi + y Tfi"

^

i =

0

(11)

1

with i = 1; . . . ; p, hi the ith element of h (t), i

is the constant

III. ADAPTIVE CONTROL

(5)

with D p D + K p . Defined the filtered input by f W (s) with W (s) = f =(s + f ) and f > 0. The filtered regressor is given by Y f (q ; q_ ) W (s)Y (q ; q_ ; q). For simplicity, Y f or Y f (1) will be used hereafter as long as there is no confusion. Also, define Y f ^ 0 f = Y f ^ 0 Y f = Y f ~. The the prediction error as " proposed adaptation algorithm is given by

(2)

4

(4)

In view of (4), the tracking error closed-loop dynamics satisfies

(1)

q2

where Y (q ; q_ ; q) 2 vector of parameters.

1199

0 0 0^ 0

if ui

> 1 and

î Ui

if li < î < iu if Li î li

(12)

Li , li , ui and Ui are constants satisfying Li < li < mi and Mi < ui < Ui . Note that fbi is zero if sign(~i ) is unknown. In order to simplify the stability discussion, the state x is introduced T as x q~T q~_ T ~ T . Since ~_ = ^_ , the following theorem establishes the stability of the origin of (5) and (6). Theorem 1: a) Given a bounded continuous desired trajectory q d , with bounded velocity and acceleration, the tracking errors q~ and q~_ in (5) tend asymptotically to zero and the estimated parameters î from (6) are bounded by Li î Ui . b) In addition, if it is satisfied

2

t 0

e0(t0#)Y Tf (#)Y f (#)d# "2I

8tt

1

(13)

1200


for some t1 > 0 and "2 > 0 given in (9), then the state x of (5) and (6) tends exponentially to zero and is bounded by

kx(t)k Mm kx (t )k e0

0t

(1=2) (t

1

t t1

)

(14)

m ~

where

m

1

min 2 8t0; q 2

with M

min (M )M

M (q ; t) and

1

1

max 2 8t0; q 2

max (M )

(15)

given by

3H (q )3 + 2D p 3 3H (q ) ; 0 M bl.diag (16) H (q )3 H (q ) 1 n where n min (min (3Dp 3) ; min (D p ) ; 2 "2 ) : M

(17)

Also, ~ ! S = f~ : k~k (2"1 =1 ) M =m g, in a time ts given by 2"1 ts t1 0 2 M In 2 "2 1 ~ (t1 )

(18)

where

m

i (t) M 8t0min ; 1ip

i (t): 8tmax 0; 1ip

(19)

Proof: a) To prove that

Li î Ui 8 t 0, define the function Vf (t) = 1=2 i ~i2 , whose derivative satisfies V_ f (t) ~i y Tais + 1 hi + y Tfi"

+ ~i fbi

since _ i (t) 0. For the worst case î = Li or î = Ui , one has V_ f (t) 0(i 0 1)j~i jjy Tai s + 1 (hi + y Tfi ")j 0; so that T Li î Ui holds. Note that ~ f b = f Tb ~ 0 8 t 0. To prove that q~ and q~_ tend to zero, consider the Lyapunov function V (t) = 1=2xT M (q ; t)x, whose derivative satisfies

V_ (t) 0q~T 3D p 3q~ 0 q~_ T D pq~_ 0 1 hT

~T Y T Y f + f

~

(20)

along (5) and (6). Since from (7), Z (t) satisfies

Z (t) = 2

t 0

e0(t0#)Y Tf (#)Y f (#)d# = Z T (t)

(22)

where N block diagf N q ; 1 P g with N q block diagf 3D p 3; D p g. Recall that P (t) = Z (t) + Y Tf (t)Y f (t). Clearly, N is at least positive semidefinite. Thus, q~ and q~_ tend asymptotically to zero according to Barbalat’s Lemma [12], [21]. b) If (13) holds, then Z (t) "2I 8 t t1 , N becomes positive definite and x tends to zero exponentially. From (17) and (22) V_ (t) 0n kxk2 . Since from (15) one gets

m kxk2 V (t) M kxk2

2

V (t) M ~

2

(24)

where m and M are given in (19). From (9) and the fact that min (P (t)) "2 , it can be shown that

V_ (t) 0 ~ Y Ta s

1 2

~

1 " 0 1 0 " ~ : 2

2 2

1

(25)

Since 1 > 2, if k~k "1 then V_ (t) 02 "2 k~k2 . If k~k < "1 , V_ (t) will be negative if k~k 2"1 =1 . Thus, ~ will tend to S . From (24), if V_ (t) < 0 from an initial time ti , then k~k k~(ti )k M =m 8 t ti . Therefore, if k~(t1 )k 2"1 =1 , then ts = t1 . If k~(t1 )k > 2"1 =1 , then V_ (t) 02 "2 k~k2 for t1 t < ts and

V_ (t) 0 2 "2 V (t) M

0 V (t): 2

(26)

This implies that k~(t)k M =m k~(t1 )ke01=2 (t0t ) . At t = ts > t1 one has (18) and the proof is concluded. Remark III.1: The algorithm (6)–(8) makes use of robust control techniques to achieve a faster parameter estimation (see (9)). f b is aimed at avoiding î from deriving too far away from the known bounds 4 mi and Mi . Remark III.2: The convergence to zero of ~(t) is guaranteed under condition (13), which is a parameter-dependent PE condition (see [16] for details). Recall that a (bounded) regressor Y f (t) is said to be of t+t Y T ( )Y f ( )d 0I for some persistent excitation (PE), if t 0t , 0 > 0 and for all t 0 and fof sufficient excitation (SE), if t Y Tf ( )Y f ( )d 0 I , for some t0 , 0 > 0. The PE condition 0 on the regressor is required for most existing algorithms for their convergence. The SE condition is strictly weaker than PE and it may be satisfied with transient responses. For > 0, condition (13) is equivt+t Y Tf ( )Y f ( )d 0 I . For = 0, condition (13) is alent to t SE. Since Z (t) is continuous on , condition (13) tends to be SE for small . Therefore, one can choose based on the desired trajectory such that condition (13) is satisfied. 4 Remark III.3: Once (13) is satisfied, the time ts given in (18) can be done arbitrarily close to t1 , while the ball S can be done arbitrarily small by a proper choice of "1 , 1 and 2 . 4 IV. SIMULATION RESULTS

(21)

h(t) is given by h(t) = Z (t)~(t). This means that V_ (t) 0xT N x

then V (t) V (t1 )e0 (t0t ) 8 t t1 . Equation (14) is obtained by taking (23) into account. To prove that ~ will reach S T in a time ts given by (18), consider V (t) = 1=2~ 0(t)~, which satisfies

(23)

To test the adaptive algorithm given by (6)–(8), a simulation has been carried out. To have a better insight of the performance of the proposed scheme, the robot model of the two-link manipulator given in [22] has been used, since it provides a good matching with the dynamics of the physical system. The model can be written as a function of nine parameters and, unlike (1), it includes a Coulomb friction term. This is a discontinuous term which affects the stability analysis of the previous section and that will be introduced in model (1) as f c (q_ ). The controller gains were chosen as K p = block diag f150 50g , 3 = block diag f13:34 66:66g . The parameter for the adaptive law are f = 1, "1 = 0:01, "2 = 0:01, 1 = 2:5, 2 = 2:5 and 2 = 2:5. It is assumed that the bounds mi are zero and it is chosen li = 00:001 and Li = 01:0. The upper known bounds are assumed to be 10 times the nominal parameters, i.e., Mi = 10i (M 8 = 76:095 Nm), with ui = Mi + 0:001 and U 1 = Mi + 1:0. One has = 0:005, i (0) = di = 0:1, ai (t) = 0 and i = 1:001. In


Fig. 1. (a)

~

. (b)

_ ~

. (c)

. (d)

.(

1201

(a)

(b)

(c)

(d)

), (

), (

) represent the results obtained with the control laws (3), (28), and (30), respectively.

all cases i = 1; . . . ; 9. The adaptive algorithms given in [18] and [19] will be used as well, since they are also designed to improve the transient performance of a robotic system. Using the same notation as in this note and defining

Y a^ Y (q ; q_ ; q_ r ; qr ) ^ ^ q_ + f^ c (_q ) + g^(q ) ^ (q ) =H q r + C^ (q ; q_ ) q_ r + D

(27)

the control law in [18, eq. (28)] is given by

= Y a^ 0 K ps + a :

(28)

In this case, q_ r is computed through q_ r q_ d 0 y s and y s = C sz + D sq~, z_ = Asz + B sq~, with As = O , B s = 1=432 , C s = I and D s = 3. K p and 3 are the same as for control law (3). a 0(1 + 4 kY a kM )kY a kM s=(ksk + "). 4 = 0:01 and " = 250. M 1=2 (M 1 0 m1 )2 + 1 1 1 + (Mp 0 mp )2 , where the constants mi and Mi for i = 1; . . . ; p have been defined before. By defining i T (001Y a s)i , the corresponding adaptive algorithm is given by [18, eq. (31)] as

0

^_ i =

0 i 0

if î = Mi and i < 0 mi < î < Mi if î = Mi and i 0 î = mi and i 0 if î = mi and i > 0

(29)

The control law in (10) of [19] is given by1

= Y a ( 0 + ^) 0 K ps 0 k s 0 k Y aY Ta s 0 q~ 4

4

(30)

where K p is the same as for (3), k = 15 and 0 = 0. The corresponding adaptive algorithm is given by [19, eq. (14)] as

^_ =

0001Y Ta s 0001Y Ta s 0001GY Ta s

0 if p 0 and 0p^ ^ Y Ta s 0 T if p ^ > 0 and 0p^ ^ Y a s > 0 if p ^

^

(31)

where G (I 0 p(^)pT^ (^)p^ (^)=kp^ (^)k2 ), p(^) T 2 2 (^ ^ 0 M )=(" + 2"M ) and p^ (^) dp(^)=(d^). " = 0:01 and M is chosen as before. Remark IV.1: To improve transient performance, control laws (28) T and (30) increase gains. Control (30) does this through k and Y a Y a , while control (28) carries it out through a by 4 and kY a kM . Also, in this last equation " can be tuned arbitrarily small. The disadvantage in doing so is that the output controller may become saturated and/or show many peaks or oscillations. On the contrary, the approach proposed in this note consists in carrying out a fast parameter adaptation. Thus, no extra term is added as proportional gain in the control law (3), which results in a smoother output. 4 The desired trajectories are chosen to be qdi = (ki1 + ki2 sin(!i t))(1 0 exp(0kei t3 )), with k11 = 45 , k12 = 10 , ke1 = 2, k21 = 60 , k22 = 125 and ke2 = 1:8 [22]. In Fig. 1, it can be seen that the results are better for the new adaptive scheme and that 1This is actually a special case of [19, eq. (10)], since the original control law has been set to takes into account time varying parameters. Thus, the term zero.

1202


the control law (3) does not only improves the transient performance, but also provides smoother outputs. It is important to stress that a better performance can still be achieved with (28) and (30). However, as discussed in Remark IV.1, increasing gains would not only cause more peaks in the outputs, but it might also yield saturation, especially for 2 .

[17] A. Yao and M. Tomizuka, “Robust desired compensation adaptive control of robot manipulators with guaranteed transient performance,” in Proc. IEEE Int. Conf. Robotics Automation, vol. 3, 1994, pp. 1830–1836. , “Smooth robust adaptive sliding mode control of manipulators [18] with guaranteed transient performance,” in Proc. Amer. Control Conf., Baltimore, MD, June 1994, pp. 1176–1180. [19] P. Tomei, “Robust adaptive control of robots with arbitrary transient performance and disturbance attenuation,” IEEE Trans. Automat. Contr., vol. 44, pp. 654–658, MAr. 1999. [20] M. A. Arteaga Pérez, “On the properties of a dynamic model of flexible robot manipulators,” ASME J. Dyna. Syst., Meas., Control, vol. 120, pp. 8–14, 1998. [21] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Upper Saddle River, NJ: Prentice-Hall, 1991. [22] F. Reyes and R. Kelly, “Experimental evaluation of model-based controllers on a direct-drive robot arm,” Mechatronics, vol. 11, pp. 267–282, 2001.

V. CONCLUSION The tracking control problem for rigid robots with model parameter uncertainty has been studied in this note. In order to improve the parameter error convergence to zero, robust control techniques have been used. It was shown that with only parameter-dependent persistent excitation, the transient response of the parameter and tracking errors can be improved notably. Unlike other existing algorithms, the improvement of the transient performance is not achieved by increasing control gains, but by achieving a fast parameter adaptation, what results in smoother control outputs. Under the assumption of known bounds for the real parameters, it is also guaranteed that even in the absence of excitation the estimated parameter will remain bounded. By using the model of a two-link robot available in the literature, the proposed algorithm was tested in simulation. It was shown that the transient performance of this new adaptive algorithm is better in comparison with other well-known algorithms in the literature.

REFERENCES [1] M. Corless, “Tracking controllers for uncertain systems: Application to a manutec r3 robot,” ASME J. Dyna. Syst., Meas., Control, vol. 111, pp. 609–618, 1989. [2] C. H. An, C. G. Atkeson, and J. M. Hollerbach, “Estimation of inertial parameters of rigid body links of manipulators,” in Proc. 24th IEEE Conf. Decision Control, Ft. Lauderdale, FL, Dec. 1985, pp. 990–995. [3] P. K. Khosla and T. Kanade, “Parameter identification of robot dynamics,” in Proc. 24th IEEE Conf. Decision and Control, IEEE, Ft. Lauderdale, FL, Dec. 1985, pp. 1754–1760. [4] R. H. Middleton and G. C. Goodwin, “Adaptive computed torque control for rigid link manipulators,” Syst. Control Lett., vol. 10, pp. 9–16, 1988. [5] J. J. E. Slotine and W. Li, “Adaptive manipulator control: A case study,” IEEE Trans. Automat. Contr., vol. 33, pp. 995–1003, Nov. 1988. [6] Z. Qu, D. M. Dawson, and J. F. Dorsey, “Exponentially stable trajectory following of robotic manipulators under a class of adaptive controls,” Automatica, vol. 28, no. 3, pp. 579–586, 1992. [7] S. Nader and R. Horowitz, “Stability and robustness analysis of a class of adaptive controllers for robotic manipulators,” Int. J. Robot. Res., vol. 9, no. 3, pp. 74–92, 1990. , “An exponentially stable adaptive control law for robotic manip[8] ulators,” IEEE Trans. Robot. Automat., vol. 6, pp. 491–496, Apr. 1990. [9] M. S. de Queiroz, D. M. Dawson, and M. Agarwal, “Adaptive control of robot manipulators with controller/update law modularity,” Automatica, vol. 35, pp. 1379–1390, 1999. [10] R. Ortega and M. W. Spong, “Adaptive motion control of rigid robots: A tutorial,” Automatica, vol. 25, no. 6, pp. 877–888, 1989. [11] L. Sciavicco and B. Siciliano, Modeling and Control of Robot Manipulators, 2nd ed. London, U.K.: Springer-Verlag, 2000. [12] J. J. E. Slotine and W. Li, “On the adaptive control of robot manipulators,” Int. J. Robot. Res., vol. 6, no. 3, pp. 49–59, 1987. [13] , “Composite adaptive control of robot manipulators,” Automatica, vol. 25, no. 4, pp. 509–519, 1989. [14] R. Lozano and C. C. de Wit, “Passivity based adaptive control for mechanical manipulators using ls-type estimation,” IEEE Trans. Automat. Contr., vol. 35, pp. 1363–1365, Dec. 1990. [15] B. Brogliato, I. D. Landau, and R. L. Real, “Adaptive motion control of robot manipulators: A unified approach based on passivity,” Int. J. Robust Nonlinear Control, vol. 1, pp. 187–202, 1991. [16] Y. Tang and M. A. Arteaga Pérez, “Adaptive control of robot manipulators based on passivity,” IEEE Trans. Automat. Contr., vol. 39, pp. 1871–1875, Sept. 1994.

A Continuous-Time Observer Which Converges in Finite Time Robert Engel and Gerhard Kreisselmeier Abstract—It is shown that a continuous-time observer, which comprises two standard th order observers and a delay , can observe the state of an th order linear system in finite time exactly. In particular, (almost) any convergence time can be assigned, independent of the observer eigenvalues. Index Terms—Convergence, delay time, linear systems, observer.

I. INTRODUCTION Consider an observable linear system in continuous time x_ =Ax + Bu y =Cx

x (t0 ) = x0 ;

t

t0

(1a) (1b)

with state x 2 n , input u 2 m and output y 2 p . The theory of observers for such systems, which reconstruct the state x from measurements of the input and output, is well established, see, e.g., [2], [6], and [7]. In a continuous-time setting, the convergence of the state observation to zero is always asymptotic with time. The convergence rate is exponential and can be assigned by suitably choosing the observer eigenvalues [6]. In contrast, the observation problem in a discrete-time setting allows the choice of zero eigenvalues and thereby a dead-beat response, i.e., a transient evolution which converges in finite time. The guaranteed convergence time is then n times the sampling time and can be assigned by choosing the latter [1], [3], [5]. Convergence in finite time is an attractive feature and, as this note shows, not restricted to the use of sampled-data or discrete-time techniques. This note presents a purely continuous-time observer which Manuscript received September 4, 2001; revised December 5, 2001. Recommended by Associate Editor M. E. Valcher. The authors are with the Department of Electrical Engineering, University of Kassel, D-34109 Kassel, Germany (e-mail: [email protected] [email protected]). Publisher Item Identifier 10.1109/TAC.2002.800673.

0018-9286/02$17.00 © 2002 IEEE