Parameter Update Laws for Adaptive Control with Affine Equality Parameter Constraints

1 P aramet er Update La ws for Adapti v e Control with Af fine Equality Pa rameter Con straints Ashwin P . Dani Abstract —In this paper , constrained p arameter u p date laws fo r adaptive control with conv ex equality constraint on the parameters are dev eloped, one based on a gradient onl y update and the other incorporating concurrent learning (CL) update. The update laws are deriv ed by solving a constrained optimiza- tion problem with affine equality constraints. This constrained problem is ref ormulated as an eq u iv alent un constrained prob- lem in a new v ariable, thereby eliminating the equality con- straints. The resulting update law is integrated with an adap t ive trajectory trackin g controller , enabling online learning of the unknown system parameters. L yapunov stability of the closed- loop system with the equality-constrained param eter up d ate law is established. The effectiv eness of t he p roposed equality- constrained adaptive contr ol law is demonstrated thro ugh simulations, validating its ability to maintain constraints on the p arameter estimates, achieving con ver gence to the true parameters fo r CL-b ased update law , an d achieving asymptotic and exponent i al tracking performance for constrained gradient and constrained CL-based upd ate l aws, re spectively . I . I N T RO D U C T I O N Adaptive control is a con trol m ethod which achie ves control objectiv es in the pr esence o f par ametric m odel u n- certainties. It has a rich history of developments over several decades [ 1 ]–[ 3 ]. Proble m s such as parameter drifts a nd the identification of m odel parameter s wh ile achieving contr o ller tracking/r egulation ob jecti ves have been addr e ssed in sev eral papers. In th is paper, we ad dress the pr oblem of injecting prior knowledge ab out the parameters in the form of con- straints on th e par ameter update laws. Specifically , we d esign parameter update laws that incorpor ate equality constraints on the param eter estimates, given in the f orm of Aθ = d . V ariou s parameter up date laws have been pro posed for adaptive contr ol. Th e grad ient upda te law is one o f the most basic forms studied. T o prevent parameter drift, σ and e - modification algor ithms are developed. Smo oth projectio n algorithm s keep the par ameters b o unded within a prescrib ed conv ex set [ 2 ], [ 4 ]. The L 1 -adaptive contr oller uses a lo w- pass filter in the feedb ack loo p to en f orce d esired transient perfor mance while allowing fast adaptation. Parameter up date laws that use both tracking and prediction errors, referred to as co mposite adap tation [ 5 ] have been developed, and can achieve fast p arameter conv ergence while ach ieving the control objective. The se par ameter update laws typically assume persistency o f excitation (PE) of the regressor to achieve parame te r c o n vergence. Using least squares tech- nique of regression problem, parameter u p date laws ca lled A. P . Dani is with the Departmen t of Electrica l and Computer Engineeri ng at Uni versit y of Connect icut, Storrs, CT 06269. concur r ent learning (CL) and integral concur r ent learning (ICL) are d ev eloped. Th e CL/ICL-based parameter update laws can achieve parame ter conv ergence with finite excitation (FE) c o ndition, which is a wea ker/m ilder cond ition than PE. Dyn amic regressor extension an d mixing (DREM) can achieve param e ter convergence un der milder co nditions than PE [ 6 ]. Using the tech niques in optimiza tio n for pa r ameter learning, momentum - based tools have been used to design parameter up date laws in [ 7 ], [ 8 ], with the g oal of imp roving transient perfo r mance. For deep neur al netw ork, parameter update laws are designed in [ 9 ]. In certain applications, addition a l constraints on the pa- rameters, such as sym metry with eigenv alue bound s or positive definiten ess of p arameter matrices, are importan t for ensuring physically con sistent para m eter estimates and improving controller ro bustness. Th e metho ds in [ 10 ], [ 1 1 ] develop parameter u pdate laws based on Bregman diver gence metrics to enfor ce positive definiten e ss of param eter matrices. The method in [ 12 ] incorporates symmetric matrix con- straints with eigenv alue b ounds into the p arameter estima tio n law . These a p proach es enable the incor poration of specific structural co nstraints present in the system dynamics. In [ 13 ], an adaptive c o ntroller that incorpor a tes knowledge o f dynamics is developed using L y apunov-b ased co n trol and parameter update law desig n . I n so me application s, it is critical to strictly en force up per an d lower boun d co nstraints on the parameters. For example, this ar ises when estimating control effecti veness m atrix g ( x ) in the con tr ol-affine system ˙ x = f ( x ) + g ( x ) u [ 14 ], or wh en parameter constra in ts require maintainin g the norm of th e p arameter vector within prescribed bo unds. Constrained p arameter update laws usin g Barrier metho ds, as stud ied in optim iz a tio n, are developed in [ 14 ], [ 15 ]. In this paper, a n ew par ameter update law that incorp orates affine equality constraints of the form Aθ = d on the parameter estimates is developed. Th e op tim ization viewpoint of developing par ameter upd ate law shown in [ 2 ], [ 10 ], [ 16 ] is used to inco rporate suc h constraints. As shown in [ 1 0 ], [ 16 ], the ob jecti ve fu nction used to derive gradien t update law is conv ex. When affine eq uality constraints on the p arameters (which are con vex con straints) are added, the resulting pa- rameter estimation p roblem b ecomes a co n vex program . T w o new equality-c o nstrained p arameter u pdate laws ar e desig n ed - on e based o n a gradient upd ate law and the other ba sed on a CL-based upd ate law . The objective fun ction is con vex for the gradient-b ased update law , and strictly conv ex for the 2 CL-based update law . T o d esign p arameter up date laws with equality co nstraints, an equ iv a lent u n constrained optimizatio n is formu late d by p rojecting th e p arameter estimates o nto th e constrained set [ 17 ]. This y ields an op timization problem in a new r educed-d imensional variable z ( t ) , wher e th e par a meter estimate ˆ θ ( t ) is a f unction o f z ( t ) . Explicit formu las f or ˆ θ ( t ) in terms of z ( t ) ar e derived, where z ( t ) is updated using a differential equation, resulting in a red u ced-dime nsional parameter up date law . Stability of the p a r ameter update law , together with the tracking error d y namics induced by a tracking controller, is established via L yapunov analysis. Using this co nstraint-elimin ation formu lation of the con vex progr am, ther e is no need to introdu ce Lagrang e mu ltipliers to en force the p arameter constra ints, w h ich simplifies the sta- bility analy sis and yields a reduced - dimensiona l upd ate law . Both equality-co n strained gradient and CL-based parameter update laws are tested in simulation s using constrain ts of the form θ i − θ j = 0 . Th e simulations validate th at the pr oposed update laws maintain the param eter con straints f or all time while achie ving the control objec ti ves. In the simulations, the parameters may not con verge to true values for the equality-co nstrained grad ient-based update law , whereas the parameters co nvergence is obtained for equ ality-constrain ed CL-based u p date law . Notations: For any matr ix A ∈ R n × m , N ( A ) denotes the null space, R ( A ) den otes the r a nge space. For a square matrix A , A ≻ 0 d enotes positive definite matrix A I I . S Y S T E M M O D E L A N D C O N T RO L O B J E C T I V E A. System Dynamics Consider the following sy stem ˙ x = f ( x ) + u (1) where x ( t ) ∈ R n is the system state, u ∈ R n is the co ntrol input, f : R n → R n is a locally Lipschitz continuous function wh ich is line arly para m etrizable, defined as f ( x ) = Y ( x ) θ (2) where Y : R n → R n × p is the regressor matrix and θ ∈ R p is an unknown parameter . The par ameters ar e suc h that they satisfy fo llowing affine equality co n straints Aθ = d (3) where d ∈ R m and A ∈ R m × p are constants. An example of such a constraint would b e ∃ i, j : θ i ( t ) − θ j ( t ) = 0 , i, j ∈ { 1 , .., p } , i 6 = j (4) Remark 1. The equality con straints are co n vex since the constraints ar e affine in θ . Remark 2. S uch parameter constraints imply tha t there is some prio r knowledge a bout parameters, e.g., r o bot lin k masses are equal to each other fo r r o bot dyn amics. Assumption 1 . The matrix A is full r ow rank and ∃ κ 1 , κ 2 > 0 such th at κ 1 I ≤ AA T ≤ κ 2 I . B. Contr oller Objective The control o b jectiv e is to track a desired trajectory x d ( t ) ∈ R n and com pute p a r ameter vector estimate ˆ θ ( t ) ∈ R p while maintaining p rescribed equ ality con stra ints of the form ( 3 ) on the parameter estimates. For th e contro l design , let’ s define trac k ing and parameter estimatio n erro rs as fol- lows e ( t ) = x ( t ) − x d ( t ) , ˜ θ ( t ) = θ − ˆ θ ( t ) (5) where ˜ θ ( t ) ∈ R p is th e parame ter estimation erro r . C. Adaptive Contr oller Design T o achieve the contro l objective, an adap ti ve trajectory tracking con troller is de sig n ed as u = ˙ x d − Y ˆ θ − k e (6) where ˙ x d ( t ) is the deriv ativ e of the desired tr ajectory , and k > 0 is the con trol gain. T aking the time deriv ati ve of ( 5 ) and substituting th e c o ntroller ( 6 ), the following clo sed -loop error system can be obtain e d ˙ e = Y ˜ θ − k e (7) The par ameter estimation er ror dynam ics is written as ˙ ˜ θ = − ˙ ˆ θ (8) In next sectio ns, gradient and CL-b ased parameter update laws ar e p resented that satisfy the affine equ ality con straints of the form ( 3 ). I I I . E Q UA L I T Y - C O N S T R A I N E D G R A D I E N T U P DAT E L AW In th is section , an equality-co nstrained g radient par ameter update law is d esigned. Th e affine equality constraint o n the parameter estimates can be written u sing ( 3 ) as follows A ˆ θ = d. (9) T o design th e constrained p a r ameter update law , consider th e following constrained minimization p roblem ˆ θ ∗ = min ˆ θ γ e T Y ˜ θ s . t . c ( ˆ θ ) = A ˆ θ − d = 0 , (10) where γ > 0 is a constant gain. An equiv alent constraint free for mulation of the c on vex op timization can be written by formulatin g a new variable z ∈ R p − m (see [ 17 ] Chapter 4, Elimina ting Equality Co n straints) ˆ θ 0 = A T ( AA T ) − 1 d ˆ θ = ˆ θ 0 + F z (11) where ˆ θ 0 ∈ R p represents a solution to e q uality constraint c ( ˆ θ ) = 0 , F ∈ R p × ( p − m ) is selected such that R ( F ) = N ( A ) . Remark 3. One way to choose F is to perform sin g ular va lue decompo sition (SVD) o f A = U Σ V T and choose last p − m 3 columns of V corr espon ding to N ( A ) . Th is will ensure that F is full colum n rank, i.e., rank p − m and R ( F ) = N ( A ) . The optimization p roblem can be refo rmulated in new variable z and the equ ality constraint can be eliminated as follows z ∗ = min z f ( z ) = min z γ e T Y ( θ − ˆ θ 0 − F z ) (12) The co n vex optimizatio n prob lems ( 10 ) and ( 12 ) are eq u iv a- lent ( see [ 17 ]) an d the optimal solu tion satisfies ˆ θ ∗ = ˆ θ 0 + F z ∗ (13) The gr a dient dynam ics of z can be comp uted u sing ( 12 ) as ˙ z = − ∇ z f ( z ) = γ F T Y T e (14) The par ameter u pdate law in redu ced dim ension is g iven by ˆ θ = ˆ θ 0 + F z ˙ z = γ F T Y T e (15) A. Err o r dyna mics The track ing error dynamics from ( 7 ) is expressed as ˙ e = − k e + Y ˜ θ (16) Using ( 15 ) the par ameter estimation error dynam ics is g i ven by ˙ ˜ θ = − ˙ ˆ θ = − ˙ ˆ θ 0 − F ˙ z ˙ ˜ θ = − γ F F T Y T e (17) B. Stability Analysis In th is subsectio n, stability of the tracking error d y namics ( 7 ) a n d eq uality-con strained g r adient parameter update law ( 15 ) is established. Theorem 1. If Assumptio n 1 is satisfied, for th e system shown in ( 1 ), the eq uality-con strained pa rameter upda te law ( 15 ) and the ad a ptive con tr oller ( 6 ) ensure global a symptotic trac king, i.e., k e ( t ) k → 0 as t → ∞ (18) and bounded parameter estimatio n err or with constrained satisfaction on the parameter estimates. Pr oo f. Let y = [ e T , ˜ θ T ] T ∈ R n + p be an au xiliary vector . Consider a L yap unov f unction V ( y ) = 1 2 e T e + 1 2 γ ˜ θ T ˜ θ (19) The bo unds on V ( y ) ar e λ 1 k y k 2 ≤ V ( y ) ≤ λ 2 k y k 2 (20) where λ 1 = min  1 2 , 1 2 γ  , λ 2 = max  1 2 , 1 2 γ  . T ime d eriv a- ti ve of V ( y ) y ields ˙ V = e T ( − k e + Y ˜ θ ) + ˜ θ T ( − F F T Y T e ) ˙ V = e T ( − k e ) + ˜ θ T ( Y T e − F F T Y T e ) ˙ V = − k e T e + ˜ θ T ( I − F F T ) Y T e (21) Since θ satisfies Aθ = d and ˆ θ is designe d to satisfy A ˆ θ = d , we have A ( θ − ˆ θ ) = 0 , = ⇒ A ˜ θ = 0 (22) This imp lies that ˜ θ ∈ N ( A ) . Since F is co nstructed using the orth onorma l column s of the N ( A ) (Remark 3 ), F F T is an o r thonor m al projecto r on to N ( A ) . So, for any ˜ θ ∈ N ( A ) F F T ˜ θ = ˜ θ = ⇒ ( I − F F T ) ˜ θ = 0 (23) Using ( 23 ) the deriv ati ve o f L yapun ov fu nction can be written as ˙ V = − k e T e ≤ 0 (24) Since V ( y ) is po siti ve definite ( PD) and ˙ V is negativ e sem i- definite (NSD), the system ( 7 )-( 17 ) is stable in the sense of L yapun ov , which means y ( t ) ∈ L ∞ . Computing the deriv ativ e of ˙ V y ields ¨ V = − k e T ( − k e + Y ˜ θ ) (25) Since e ( t ) , ˜ θ ( t ) ∈ L ∞ , ¨ V ∈ L ∞ , which mean s ˙ V is unifor m ly continuou s [ 5 ]. Using Bar b alat’ s lemma ˙ V → 0 as t → ∞ , which im plies k e k → 0 as t → ∞ . I V . E Q UA L I T Y - C O N S T R A I N E D C O N C U R R E N T L E A R N I N G U P DAT E L A W In this section, CL-ba sed parameter up date law with affine equality co nstraints is de r i ved. For the CL- b ased equality - constrained par ameter update law , pr ior data of state and control inp u t called as a history stack H = { x k , ˆ u k , ˙ ˆ x k } k = h k =1 is collected . Before d iscussing the parameter up date law , finite excitation (FE ) assumption is stated. Assumption 2. F or the history stack H = { x k , ˆ u k , ˙ ˆ x k } k = m k =1 the following condition is satisfied σ 2 I ≥ Y R ≥ σ 1 I (26 ) wher e Y R = P h k =1 Y T k Y k , σ 1 , σ 2 ∈ R + . The numerically computed derivatives of x ( t ) , ˙ ˆ x k computed at k th data point satisfies k ˙ ˆ x k − ˙ x k k ≤ ǫ fo r a small p ositive ǫ ∈ R + . Remark 4. Assump tio n 2 is a finite excitation conditio n that can be verified in r eal-time [ 18 ]. T o design the eq uality-con strained CL parameter up date law , consider a conve x con strained minimizatio n problem ˆ θ ∗ = min ˆ θ γ e T Y ˜ θ + k cl 2 ˜ θ T Y R ˜ θ s . t . c ( ˆ θ ) = A ˆ θ − d = 0 , (27) 4 where k cl is a positive co nstant. Similar to previous section, an equiv alent equ ality constraint free form u lation of the conv ex optimization can be written b y form ulating a new variable ˆ θ 0 = A T ( AA T ) − 1 d ˆ θ = ˆ θ 0 + F z (28) where ˆ θ 0 ∈ R p , F ∈ R p × ( p − m ) and z ∈ R p − m are defin ed in ( 11 ). The optimizatio n p roblem c an be written in new variable z as an uncon strained co n vex optimizatio n problem z ∗ =min z f C L ( z ) z ∗ =min z γ e T Y ( θ − ˆ θ 0 − F z ) + k cl 2 ( θ − ˆ θ 0 − F z ) T Y R ( θ − ˆ θ 0 − F z ) ( 29) The conve x o ptimization prob lems ( 27 ) and ( 29 ) are equ i v- alent (see Section 4.1.3 of [ 17 ]), which mean s the optimal solution satisfies ˆ θ ∗ = ˆ θ 0 + F z ∗ (30) The gr a dient dynam ics of z can be comp uted b y ˙ z = − ∇ z f C L ( z ) = γ F T Y T e + k cl F T Y R ˜ θ (31) The par ameter update law is given b y ˆ θ = ˆ θ 0 + F z ˙ z = γ F T Y T e + k cl F T Y R ˜ θ (32) Since ˜ θ ( t ) is not measu r able, the up date law in ( 32 ) is not implementab le, b ut can be used for analysis. In an imple - mentable f orm the eq uality-con strained CL-based parameter update law is given by ˆ θ = ˆ θ 0 + F z ˙ z = γ F T Y T e + k cl F T h X k =1 Y T k ( ˙ ˆ x k − u k − Y k ˆ θ ) (33) where ( 1 ) and ( 2 ) ar e used. A. Err o r dyna mics The tr acking error dynam ics using ( 7 ) c a n be expressed as ˙ e = − k e + Y ˜ θ (34) Using ( 32 ) th e pa rameter estimation error dyn a mics is giv en by ˙ ˜ θ = − ˙ ˆ θ = − γ F F T Y T e − k cl F F T Y R ˜ θ (35) B. Stability analysis In th is subsectio n, stability of the tracking err or d ynamics ( 7 ) and equality- constrained CL-based par a meter upd ate law ( 33 ) is established u sing L y apunov ana lysis. Theorem 2. If Assumptions 1 - 2 ar e satisfied, for the system shown in ( 1 ), the equality-co n strained parameter update law ( 3 3 ) an d the ada ptive con tr oller ( 6 ) ensur e g lobally exponentially stable tr acking performance with constrained satisfaction on the pa rameter estimates an d bo unded closed - loop signals. Pr oo f. Let y = [ e T , ˜ θ T ] T ∈ R n + p be an au xiliary vector . Consider a L yap unov fun ction V ( y ) = 1 2 e T e + 1 2 γ ˜ θ T ˜ θ (36) The bo unds on V ( y ) ar e λ 1 k y k 2 ≤ V ( y ) ≤ λ 2 k y k 2 (37) where λ 1 = min  1 2 , 1 2 γ  , λ 2 = ma x  1 2 , 1 2 γ  . Using ( 7 ) and ( 35 ), the tim e derivati ve of V ( y ) is com puted as ˙ V = e T ( − k e + Y ˜ θ ) + ˜ θ T ( − F F T Y T e − k cl γ F F T Y R ˜ θ ) (38) Using F F T ˜ θ = ˜ θ f r om ( 23 ) yields ˙ V = − k e T e − k cl γ ˜ θ T Y R ˜ θ (39) Using Assumptio n 2 Y R ≻ 0 , wh ich leads to ˙ V = − k e T e − k cl σ 1 γ ˜ θ T ˜ θ = − min (2 k , 2 k cl σ 1 ) V . (40) The following b ound can be developed o n y using ( 40 ) an d ( 37 ) k y ( t ) k ≤ µ o k y ( t 0 ) k e − µ 1 t (41) where µ 0 = q λ 2 λ 1 and µ 1 = min (2 k , 2 k cl σ 1 ) . Since k y k ∈ L ∞ , k ˜ θ k ∈ L ∞ , which implies ˆ θ ∈ L ∞ . Using ( 32 ), since ˆ θ 0 and F are constants, z ∈ L ∞ . Remark 5. F or the time period when the history sta c k data is co llec te d , λ min ( P m k =1 Y T k Y k ) is n ot full rank, thus, Assumption 2 is not satisfied. F or th is time period, the equality-co nstrained pa rameter upda te law ( 1 5 ) can be used, which ensur es tha t all th e closed-lo op sign als are b ounded . Remark 6. The da ta po ints H collected in history stack P m k =1 Y T k Y k can be replaced using sing ular value maximiza- tion algo rithm in [ 18 ], wh ich en sur es P m k =1 Y T k Y k is always incr easing, thus, Lyapunov functio n ( 36 ) serves as a commo n L yapu nov functio n [ 19 ]. 5 0 10 20 30 40 -10 0 10 x 1 0 10 20 30 40 Time (s) -5 0 5 x 2 Figure 1. T rajecto ry tracking with equality-co nstrained gradient parameter update law . V . S I M U L AT I O N S Simulations are carried out to demo nstrate the ap plicability of th e equa lity -constrained parame te r up date law in adaptive control setting. T h e following system dynamics similar to [ 19 ] is considered ˙ x =  x 2 1 sin ( x 2 ) 0 0 0 x 2 sin ( x 1 ) x 1 x 1 x 2  θ + u (42) where x = [ x 1 x 2 ] T ∈ R 2 is th e state, u ∈ R 2 is th e contro l input, θ ∈ R 4 are the true par ameters whose values are given by θ = [5 5 10 20] T . The true parameters satis fy equality constraint θ 1 = θ 2 , which can b e written as ( 3 ) using A =  1 − 1 0 0  , d = 0 . (43) The desired trajectory is given by x d ( t ) = 10(1 − e − 0 . 1 t )  sin(2 t ) 0 . 4cos(3 t )  (44) A. Simulatio n for Equ ality-Constrained Gradient Law The contro ller in ( 6 ) is impleme n ted along with the equality-co nstrained par a m eter upd ate law g i ven in ( 15 ). Th e system state is initialized to x ( t 0 ) = [10 5] T , the parameters are initialized to ˆ θ ( t 0 ) = [4 . 5 4 . 5 4 . 5 1 5] T . This satisfies the parameter con straint a t the initial time. The contro l g ain is selected as k = 2 dia g { 10 , 50 } and γ = 0 . 4 , F is co mputed using N ( A ) giv en by F = h 0 . 7071 0 . 7071 0 0 0 0 1 0 0 0 0 1 i T (45) and ˆ θ 0 = 0 . The perf ormance of the equality -constrained ad aptive con- troller is sh own in Figs. 1 and 2 . It is o bserved fro m Fig. 2 that the p arameters estimated by the p roposed equality- constrained param eter up date law satisfy the parameter con - straints, i.e., ˆ θ 1 ( t ) = ˆ θ 2 ( t ) and the trackin g errors converge to zer o. 0 10 20 30 40 Time (s) 0 5 10 15 20 Figure 2. Paramet er estimates using equality-con strained gradient parameter update law . B. Simulatio n for Equ ality-Constrained CL Update Law In this simulation, equality-co nstrained CL- b ased update law is tested. The controller is implemen ted using con straints on ˆ θ for the same dyn amics and desired trajectories as th at of previous simulation use case but w ith different parameter set θ = [5 20 10 20 ] T . The parameter estimates are initialized to ˆ θ ( t 0 ) = [3 10 5 10] T such that the parameter constraints are satisfied. T h e contr ol gain is selected as k = 2diag { 10 . 50 } and parameter adaptation law gains are γ = 0 . 05 and k cl = 0 . 0008 . In th is ca se, A = [0 − 1 0 1] and d = 0 , for which F is compu ted u sing N ( A ) g i ven by F = h 0 . 7071 0 . 5 0 0 . 5 0 0 1 0 − 0 . 7071 0 . 5 0 0 . 5 i T (46) and ˆ θ 0 = 0 . From the simulation r esults sho wn in Figs. 3 - 4 , it is observed th a t the parame ters are e stimated wh ile the eq uality constraints on the p arameters are maintain ed f or all time, i.e., ˆ θ 2 ( t ) = ˆ θ 4 ( t ) and the tracking error conver ges to zero. The parameter estimates using gr adient u pdate law are shown in Fig. 5 u sing th e same gains used for the equality- constrained CL upd a te law . The paramete r s may n ot be identified corr e c tly by the gr adient u pdate law if the regressor is not PE. V I . C O N C L U S I O N In this paper, an eq uality-con strained parameter upda te law is d e veloped f or adap tive tra c king con trol u sing a conve x optimization formulatio n . Th e appr oach of d eriving a d aptive parameter update law using a minimization problem of an objective func tio n sub ject to affine equality con straints on the parameter estimates yie ld s co ntinuou s u pdate law solutions in a mo dified fo rm for bo th grad ie n t and CL-based update laws. L yapu nov stability analysis of the tracking error, parameter estimation erro r dy namics is con ducted which shows that the tracking error conver ge asymptotically and parameter estimation erro r rema ins boun ded fo r gradien t- based up date 6 0 10 20 30 40 -10 0 10 0 10 20 30 40 Time (s) -5 0 5 Figure 3. Traj ectory tracking with equalit y-constrain ed CL parameter update law . 0 10 20 30 40 Time (s) 0 5 10 15 20 Figure 4. 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