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1 This article was downloaded by:[xu, L.] On: 8 November 7 Access Details: [subscription number 78895] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 7954 Registered office: Mortimer House, 7-4 Mortimer Street, London WT JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: Adaptive robust control of mechanical systems with non-linear dynamic friction compensation L. Xu a ; B. Yao bc a Research and Advanced Engineering, Dearborn, MI 484, USA b School of Mechanical Engineering, West Lafayette, IN 4797, USA c State Key Laboratory of Fluid Power Transmission and Control, Hangzhou 7, China First Published on: October 7 To cite this Article: Xu, L. and Yao, B. (7) 'Adaptive robust control of mechanical systems with non-linear dynamic friction compensation', International Journal of Control, 8:, To link to this article: DOI:.8/7779 URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 International Journal of Control Vol. 8, No., February 8, Downloaded By: [Xu, L.] At: : 8 November 7 Adaptive robust control of mechanical systems with non-linear dynamic friction compensation L. XUy and B. YAO*z yresearch and Advanced Engineering, Ford Motor Company, Dearborn, MI 484, USA zschool of Mechanical Engineering, Purdue University, West Lafayette, IN 4797, USA zstate Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 7, China (Received April 7; in final form April 7) In this paper, an adaptive robust control () scheme based friction compensation strategy is presented for a mechanical system in the presence of dynamic friction effects. The system may be subjected to both parametric uncertainties (e.g., unknown parameters in the dynamic friction model) and uncertain non-linearities such as external disturbances and modelling errors. In contrast to existing deterministic robust control (DRC) and robust adaptive control () schemes, the proposed scheme utilizes both the structural information of the dynamic friction model and the a priori information of the system, such as the bounds on the parameters and unmeasured internal friction state. In particular, the bounds on the friction state are fully exploited to construct certain projection-type modifications to the state observers. By doing so, a controlled state estimation process is achieved even in the presence of disturbances. The resulting controller achieves a guaranteed transient performance and final tracking accuracy. Furthermore, in the absence of uncertain non-linearities, asymptotic position tracking is achieved. Comparative simulation results illustrate the effectiveness of the proposed scheme.. Introduction Friction has been shown to have a dynamic non-linear effect which is detrimental to the performance of a number of mechanical systems. Most of the dynamic behaviours of friction, such as the presliding displacement, the friction lag and the Stribeck effect to name a few, all occur in the so-called low-velocity or presliding region. To capture these effects, researchers have proposed a number of dynamic friction models (Olsson et al. 998). For example, Dahl (968) proposed a dynamic model to capture the spring-like behaviour during stiction. Canudas de Wit et al. (995) proposed a dynamic model that is inspired by the bristle interpretation of friction (i.e., surfaces are very irregular at the microscopic level and two surfaces make contact at a number of asperities, which can be thought as elastic bristles), in combination with lubricant effects. This model captures various friction effects such as the Stribeck effect, hysteresis, presliding displacement and *Corresponding author. byao@purdue.edu varying break-away force, and has been used to construct friction state observers and perform friction compensation for position and velocity tracking. The idea behind obtaining these highly sophisticated models is to be able to predict the friction more accurately so that friction compensation can be done more effectively. However, no matter how accurate the mathematical models might be, it is usually difficulty to capture the non-linear features of friction exactly, since every physical system is subjected to certain degrees of model uncertainties. Normally, the causes of model uncertainties can be classified into three categories: (i) repeatable or constant unknown quantities, such as the inertia load and unknown parameters in the friction model; (ii) dynamic uncertainties due to the unmeasured internal state; and (iii) non-repeatable unknown quantities such as external disturbances and imprecise modelling of certain physical terms. To account for these uncertainties, two non-linear control methods, deterministic robust control (DRC) (Corless and Leitmann 98, Utkin 99, Sankaranarayanan and Khorrami 997, 998) and International Journal of Control ISSN 779 print/issn online ß 8 Taylor & Francis DOI:.8/7779

3 68 L. Xu and B. Yao Downloaded By: [Xu, L.] At: : 8 November 7 adaptive control () (Canudas de Wit and Lischinsky 997, Tan and Kanellakopoulos 999, Vedagarbha et al. 999), may be applied. In general, the deterministic robust controllers can achieve a guaranteed transient performance and final tracking accuracy in the presence of parametric uncertainties, dynamic uncertainties and uncertain non-linearities. However, since no attempt is made to learn from past behaviour to reduce the effect of parametric and dynamic uncertainties, the design is conservative and may involve either switching or high gain feedback control law (Sankaranarayanan and Khorrami 997, 998) to achieve good tracking accuracy; both means are unattainable in implementation and therefore result in a degraded performance (Yao and Tomizuka 996). In contrast to DRC designs, the adaptive controllers are able to achieve asymptotic tracking in the absence of uncertain non-linearities without resorting to switching or high gain feedback control law. Specifically, Canudas de Wit and Lischinsky (997) illustrated how a previously proposed control structure (Canudas de Wit et al. 995) could be modified to address parameter change in associated with normal force variation or temperature variation. In Vedagarbha et al. (999), an adaptive controller utilizing a non-linear observer/filter structure, was proposed to handle non-uniform variations in the friction force by assuming independent coefficient change as temperature varies. A similar result was reached in Tan and Kanellakopoulos (999), in which a dual-observer structure was utilized to estimate different non-linear effects of the unmeasured friction state. However, these adaptive controllers suffer from two main drawbacks unknown transient performance and possible non-robustness to disturbances. The system may have large tracking errors during the initial transient period or have a sluggish response. In the presence of disturbances, the closed-loop system may lose stability (Reed and Ioannou 989). To overcome such drawbacks of the adaptive control, a robust adaptive controller (Tomei ) was designed to guarantee the transient performance (i.e., arbitrary disturbance attenuation) and preserve asymptotic tracking when disturbances vanish. In this paper, a dynamic friction compensation strategy is proposed by utilizing the idea of adaptive robust control () (Yao and Tomizuka 996, 997, Yao 997). The fundamental problem to be addressed is broader than the one studied in Tomei (), in which the stiffness of bristles is assumed to be known. Under such assumption, the friction arising from the deflection of the bristles is linear with respect to the unmeasured friction state. While in this paper, both the stiffness of bristles and the internal friction state are unknown and have to be estimated. Therefore, the problem is non-linear and more difficult to solve. Aside from this essential difference in problem formulation, Tomei () used over-parameterization to deal with the effect of unknown parameters, an approach which often leads to poor parameter estimation and performance in implementation. The proposed strategy, however, does not suffer from this drawback. Furthermore, as opposed to the mathematically complicated smooth projection (Tomei ), the widely used discontinuous projection method is employed in our design. This results in a much simpler controller architecture while having more transparent performance results, i.e., the transient performance is guaranteed in the sense that the envelope of the tracking error is given and related to certain controller design parameters in a known form.. Problem formulation The mathematical model of a mechanical system in the presence of friction is assumed to be of the form dx dt ¼ x, m dx ¼ uðtþ Fþ, dt ðþ where x ¼ [x, x ] T represents the state vector consisting of position and velocity, m denotes the unknown inertia of the system, F is the friction force, represents the lumped uncertain non-linearity, and u(t) is the control input. Using the dynamic friction model proposed in Canudas de Wit et al. (995), the dynamic friction force F can be described by dz F ¼ z þ dt þ x, ðþ dz dt ¼ x jx gðx Þ z, ð4þ where z represents the unmeasured internal friction state,,, are positive but unknown parameters that can be physically explained as the stiffness of bristles, damping coefficient associated with dz/dt, and viscous coefficient. The function g(x ) is assumed to be known and positive, and can be chosen to describe the Stribeck effect (Canudas de Wit et al. 995) g ðx Þ¼F C þðf S F C Þe ðx =v s Þ, where F C and F S represent the levels of the normalized Coulomb friction and stiction force respectively, and v s is the Stribeck velocity. This model considers the dynamic effects of the friction as arising out of the deflection of bristles used to model the asperities between two contacting surfaces. The friction state ðþ ð5þ

4 Adaptive robust control of mechanical systems 69 Downloaded By: [Xu, L.] At: : 8 November 7 essentially captures the average deflection of these bristles. Understandably, the state z(t) does not proffer itself for direct measurement, which makes the dynamic friction compensation design rather difficult, especially when, and are unknown. It is shown in Canudas de Wit et al. (995) that the model has the following finite bristle deflection property: P: If jzðþj F S, then jzðtþj F S, 8t : This property is physically intuitive and will be used in the subsequent controller design for a guaranteed robust performance. In order to use parameter adaptation to reduce parametric uncertainties, we linearly parameterize the system model with respect to a set of unknown parameters. Substituting equations () and (4) into () gives m dx dt ¼ u jx j z þ gðx Þ z ð þ Þx þ : ð6þ Now, define an unknown parameter set ¼ [,,, 4 ] T, where ¼ m, ¼, ¼ and 4 ¼ þ. Note that,..., 4 are all positive, a fact which will be utilized in the subsequent controller design. Equation (6) can then be rewritten as dx dt ¼ u jx j z þ gðx Þ z 4x þ, ð7þ in which the model uncertainties are assumed to be bounded. Assumption : The extent of parametric uncertainties is known, i.e., ¼ f : < min max g, where min ¼½ min,..., 4 min Š T and max ¼ ½ max,..., 4 max Š T are known. Assumption : The uncertain non-linearity is bounded by a known positive function (x, t) multiplied by an unknown time-varying function d(t), i.e., kðx, z, u, tþk ðx, tþkdðtþk ð9þ where kk stands for the L norm. Let x d (t) be the desired trajectory, which is assumed to be known, bounded with bounded derivatives up to the second order. The control objective is to synthesize a bounded control input u such that the actual position x tracks x d (t) as closely as possible in spite of various model uncertainties. To achieve this objective, existing control tools adaptive control and deterministic robust control will be applied first, from which one can gain insights into the development and the advantages of the proposed adaptive robust control (). ð8þ. Adaptive control Define a sliding surface-like quantity as e ¼ _e þ ke ¼ x x eq, x eq ¼ _x d ke, ðþ where e ¼ x x d is the position tracking error, and k is a positive constant. Then, the problem of tracking x d is equivalent to that of making e as small as possible. The adaptive control law u consists of a model compensation term u a and a robust feedback term u s u ¼ u a þ u s, u a ¼ ^ _x eq þ ^ ^z ^ jx j gðx Þ ^z þ ^ 4 x, u s ¼ k s e, ðþ ðþ ðþ where k s is a positive feedback gain, ^,..., ^ 4 represent the estimates of the unknown parameters, and ^z and ^z represent the estimates of the friction state z. The friction state estimation utilizes a dual-observer design (Tan and Kanellakopoulos 999), which features two nonlinear observers operating in parallel to capture different nonlinear effects associated with z d^z dt ¼ x jx gðx Þ ^z e, ð4þ d^z dt ¼ x jx gðx Þ ^z jx j þ gðx Þ e, ð5þ where and are two positive design parameters. After taking the time derivative of e, substituting equations (7) and () for _x and u, respectively, and then simplifying the resulting expression, one has de dt ¼ k se T jx j ~ þ ~z gðx Þ ~z þ, ð6þ where T ¼½ _x eq, ^z, ðjx j=gðx ÞÞ^z, x Š, ~z ¼ ^z z and ~z ¼ ^z z represent the friction state estimation errors, and ~ ¼ ^ represents the parameter estimation error. The parameter adaptation law is chosen as _^ ¼ e, ð7þ where ^ðþ (i.e., the initial condition) is assigned the nominal value nom, and > is the adaptation rate matrix. Theorem : In the absence of uncertain non-linearities, i.e., ¼, the globally asymptotic tracking of the desired trajectory is achieved by applying the adaptive control law (), parameter adaptation law (7), and friction state observers (4) and (5).

5 7 L. Xu and B. Yao Downloaded By: [Xu, L.] At: : 8 November 7 Proof: The proof is similar to that of Theorem in Tan and Kanellakopoulos (999). Consider the Lyapunov function candidate V a ¼ e þ ~z þ ~z þ ~ T ~: ð8þ In the absence of uncertain non-linearities, i.e., ¼, the derivative of V a along the solutions of (6), (4), (5) and (7), is _V a ¼ k s e jx j gðx Þ ~z jx j gðx Þ ~z k se : ð9þ T This shows that e L L. From (6), it is easy to verify that _e L as well. Using Barbalat s lemma, we conclude that lim t! e ðtþ ¼, which further implies that lim t! e ðtþ ¼ in view of equation (). Remark : In Tan and Kanellakopoulos (999), the inertia m is assumed to be known, and the two design parameters and are set to be. Therefore, it can be viewed as a special case of the above design. Also note that Assumption is not required in this section. Remark : Traditional adaptive controllers have two main drawbacks unknown transient performance and possible non-robustness to disturbances. The closed-loop system may have large tracking errors during the transient or have a sluggish response. In the presence of uncertain non-linearity, instability may occur (Reed and Ioannou 989), since parameter adaptation may learn in a wrong way or the state estimation may have large errors. Improvement of transient performance and robustness is a topic of intensive current studies in adaptive control (Krstic et al. 995, Yao and Tomizuka 995, Zhang and Ioannou 998). 4. Deterministic robust control Instead of using parameter adaptation and observers for the unmeasured friction state, deterministic robust control (DRC) only uses robust feedback to attenuate the effect of various model uncertainties for a guaranteed performance. Consider the following DRC law: u ¼ u a þ u s þ u s, ðþ where u a has the same form as that in equation () but with ^ nom, and the state estimates equal to a nominal value z nom (i.e., ^z ¼ ^z z nom ); u s is given by () and u s is a robust control term to be synthesized later. Substituting the control law () into (7) results in the following error dynamics: de dt ¼ k se T jx j ~ þ ~z gðx Þ ~z þ þ u s : ðþ Note that ~z ¼ ~z ¼ z nom z and ~ ¼ nom. The essence of DRC lies in the added robust control term u s which is synthesized to attenuate the effect of the model uncertainties. Specifically, noting Assumption, Assumption and property P, a robust control term u s can be synthesized such that the following condition is satisfied: e T jx j ~ þ ~z gðx Þ ~z þ þ u s " þ " kdk, ðþ where " and " are two design parameters which may be arbitrarily small. Consider a different Lyapunov function candidate V s ¼ e : ðþ With the help of (), the derivative of V s along the trajectory of () satisfies _V s ¼ k s e þ e T jx j ~ þ ~z gðx Þ ~z þ þ u s l V V s þ " þ " kdk, where l V ¼ k s / max. From (4), it follows that ð4þ V s ðtþ expð l V tþv s ðþþ " þ " kdk ½ expð l V tþš: l V ð5þ Thus V s (t) exponentially decays and is ultimately bounded. The decay rate l V and the final tracking accuracy index ðjv s ðþj ð" þ " kdk Þ=l VÞ can be adjusted by the design parameters " and ", as well as the feedback gain k s. In other words, both the transient performance and final tracking accuracy are guaranteed in the presence of. Remark : One example of a smooth u s satisfying () can be formulated as follows. Let h, h and h be any smooth bounding functions satisfying the following conditions: h k k k max min k, h max ðz max z min Þ, h max jx j gðx Þ ðz max z min Þ, ð6þ ð7þ ð8þ

6 Adaptive robust control of mechanical systems 7 Downloaded By: [Xu, L.] At: : 8 November 7 where kk stands for the Euclidean norm, and z max ¼ F S and z min ¼ F S in view of property P. Note that both h and h can be calculated in real time, since and jx j/g(x ) only depend on signals which are available for measurement. Then, u s can be chosen as u s ¼ h 4 " þ h " þ h " þ e : ð9þ " Using the same technique as in Yao (997), it can be verified that e T jx j ~ þ ~z gðx Þ ~z þ þ u s je jh þje jh þje jh þje jkdk þ e u s h pffiffi rffiffiffiffi p ffiffiffiffi " je j " h pffiffi rffiffiffiffi p ffiffiffiffi " je j " h pffiffi rffiffiffiffi p ffiffiffiffi " je j " þ " p ffiffiffiffi je j ffiffiffiffi p " kdk " þ " kdk " þ " kdk : ðþ Thus () is satisfied. Other smooth examples of u s can be worked out in the same way as in Yao (997), Yao and Tomizuka (997). 5. Adaptive robust control It seems that the DRC presented above has solved the robust control problem since theoretically both transient error and final tracking error can be made arbitrarily small by increasing k s and/or decreasing " and ". However this means the increase of the feedback gains, which in turn raises the bandwidth of the closed-loop system. Due to the neglected factors such as high-frequency dynamics and sampling rate, every physical system has a finite bandwidth. The accuracy that DRC can achieve in practice is thus limited. To further improve performance, it is necessary to introduce dynamic friction state observers and parameter adaptation as in adaptive control to reduce model uncertainties. Yao (997) has shown that the discontinuous projection based adaptive robust control () can be used to achieve this goal for a class non-linear systems transformable to the semi-strict feedback form. However, the effect of dynamic uncertainties (e.g., the unmeasured friction state) is not considered. In the following, we will extend the discontinuous projection based design and propose a dynamic friction compensation scheme for a mechanical system where the bounds of parameter variations and internal friction state are known, as outlined in x. 5. Discontinuous projection mapping By exploiting practically reasonable a priori information on the physical systems such as the bounds of parameter variations and internal friction state, a discontinuous projection based design is constructed to solve the robust tracking control problem. Specifically, the parameter estimate ^ is updated through a parameter adaptation law having the form of _^ ¼ Proj ð e Þ, ðþ where Proj ð e Þ¼½Proj ðð e Þ Þ,...,Proj 4 ðð e Þ 4 ÞŠ T, and ( e ) i represents the ith component of e. The unmeasured friction state z is estimated by the following robust observers with projection-type modifications d^z dt ¼ Proj z x jx gðx Þ ^z e, ðþ : ðþ d^z dt ¼ Proj z x jx gðx Þ ^z jx j þ gðx Þ e In () (), the projection mapping (Sastry and Bodson 989) 8 if ^ ¼ max and > >< Proj ðþ ¼ if ^ ¼ min and < ð4þ >: otherwise, where is a symbol that can be replaced by,..., 4, z or z. Taking equation () as an example, it is interpreted as 8 if ^z ¼ z max and x jx gðx Þ ^z e > d^z dt ¼ >< >: x jxj gðx Þ ^z e if ^z ¼ z min and x jx gðx Þ ^z e < otherwise, ð5þ where z max ¼ F S and z min ¼ F S. Using similar arguments as in Goodwin and Mayne (989), Sastry and Bodson (989), one can show that the

7 7 L. Xu and B. Yao Downloaded By: [Xu, L.] At: : 8 November 7 above projection mappings have the following nice properties P: min ^ max ð6þ P: ~ T ð Proj ð e Þ e Þ ð7þ P4: ~z Proj z x jx gðx Þ ^z e x jx gðx Þ ^z e ð8þ P5: ~z Proj z x jx gðx Þ ^z jx j þ gðx Þ e x jx gðx Þ ^z jx j þ gðx Þ e : ð9þ 5. controller design The proposed law has a similar structure as the DRC law () u ¼ u a þ u s þ u s, ð4þ where u a is given by () but with ^ updated by (), and ^z and ^z updated by () and (), respectively. The robust feedback term u s is the same as (), and u s is synthesized in a similar way as in the DRC design. Substituting the law (4) into (7) results in the following error dynamics de dt ¼ k se T jx j ~ þ ~z gðx Þ ~z þ þ u s, ð4þ where the definitions of, ~, ~z and ~z are given in x. As in (), u s is synthesized to satisfy the following constraint: e T jx j ~ þ ~z gðx Þ ~z þ þ u s " þ " kdk : ð4þ Such a robust control term u s always exists since both parameter and state estimates belong to some known bounded regions according to property P. To ensure that the robust control term u s does not interfere with parameter adaptation and friction state estimation, the following passivity-like requirement is also imposed: e u s : ð4þ One example of u s satisfying (4) and (4) is given by (9). Theorem : Consider the plant () (5) subject to Assumptions and, and the adaptive robust controller which consists of the control law (4), the parameter adaptation law (), and the friction state observers () (). We have (a) all the signals are bounded. Furthermore, there exist positive constants ", " and k s such that V s ðtþ expð l V tþv s ðþþ " þ " kdk ½ expð l V tþš, l V ð44þ where V s is the Lyapunov function candidate defined in (), and l V ¼ k s / max. (b) If after a finite time t f, ¼, i.e., in the presence of parametric uncertainties and dynamic friction only, then, in addition to result in A, asymptotic position tracking is also achieved. Proof: Consider the same Lyapunov function candidate V s as that in DRC design (). With the help of (4), its derivative along the trajectory of (4) satisfies _V s ¼ k s e þ e T jx j ~ þ ~z gðx Þ ~z þ þ u s l V V s þ " þ " kdk, ð45þ which lead to (44). Thus e (t) is bounded. Since e (t) is related to e (t) via an exponentially stable transfer function G(s) ¼ (/(s þ k)), e (t) is also bounded. Since x d (t) is assumed to be a bounded signal with bounded derivatives up to second order, from equation (), it follows that x eq is bounded. Since e ¼ x x d and e ¼ x x eq, therefore, the state vector x is also bounded. From property P, the boundedness of ^, ^z and ^z is apparent. The control input u is thus bounded. This proves (a) of the theorem. Now consider the situation in (b) of the theorem, i.e., ¼. Choose the same Lyapunov function candidate V a as that in design (8). Its derivative along the trajectory of (4) is _V a ¼ k s e þ e T jx j ~ þ ~z gðx Þ ~z þ u s þ ~z ð _^z _zþþ ~z ð _^z _zþþ ~ T _^, ð46þ which, recalling the constraint placed on u s (4), implies _V a k s e þ ~z _^z þ e! ~z _z þ ~ T ð _^ e Þ: ~z _z þ ~z! _^z jx gðx Þ e ð47þ

8 Adaptive robust control of mechanical systems 7 Downloaded By: [Xu, L.] At: : 8 November 7 Substituting (), () and () into (47), and rearranging the results shows that _V a k s e þ ~z _^z x jx gðx Þ ^z e ~z _z x þ jx gðx Þ ^z þ ~z _^z x jx gðx Þ ^z jx j þ gðx Þ e ~z _z x þ jx gðx Þ ^z þ ~ T ð _^ e Þ, ð48þ which can be further simplified, in view of (7) (9), as _V a k s e ~z _z x þ jx gðx Þ ^z ~z _z x þ jx gðx Þ ^z : ð49þ Then by replacing _z with equation (4), it follows that _V a k s e jx j gðx Þ ~z jx j gðx Þ ~z k se : ð5þ T Thus, e L L. It is clear that _e L based on (4). So, e! ast! by the Barbalat s lemma. Since e (t) is related to e (t) via an exponentially stable transfer function, we conclude that lim t! e ðtþ ¼, which leads to (b) of the theorem. Remark 4: Theorem (a) indicates that the proposed controller has an guaranteed transient performance and final tracking accuracy. Theoretically, this result is what a well-designed robust controller can achieve. In fact, when the parameter adaptation law and state observers are switched off, the proposed law becomes a deterministic robust control law and (a) of the theorem remains valid (Yao and Tomizuka 996, 997). Theorem (b) implies that without using switching or high gain feedback control law, the controller can achieve a very small tracking error due to the reduced parametric uncertainties and dynamics uncertainties. Theoretically, (b) of the theorem is what a well-designed adaptive controller can achieve. Remark 5: It is seen from (44) that the tracking error during transient is affected by the initial value V s (). To further reduce the error, the idea of trajectory initialization (Yao and Tomizuka 996) can be used to render V s () ¼. Remark 6: In the above design, we assume that the variation of friction is caused by temperature changes or material wear (Canudas de Wit and Lischinsky 997). It is noted that the above design procedure can be adopted for a different task where variation of friction is caused by the change of normal force (Canudas de Wit and Lischinsky 997). The details are omitted. 6. Comparative simulation results In order to illustrate the proposed algorithm and compare it with other friction compensation designs, comparative simulation results are obtained for the mechanical system described by equations () (5), in which the numerical values of the parameters are (Vedagarbha et al. 999) m ¼ :5kgm, ¼ Nm=rad, ¼ :Nms=rad, ¼ :Nms=rad, F c ¼ :4, F s ¼ 8:45, v s ¼ :: For comparison purpose, the inertia m is assumed to be known as in Tan and Kanellakopoulos (999). Thus there are only three parameters to be adapted. The nominal values of the parameters are chosen as nom ¼ [.5, 6,.5,.7] T, while their lower and upper bounds are given by min ¼ [.5,,, ] T and max ¼ [.5,,., ] T, respectively. From the finite bristle deflection property, the bounds of the friction state z are z min ¼ 8.45 and z max ¼ The positive function (x, t) in (9) is set to. In the simulation, the controllers synthesized in previous sections are compared. () The adaptive robust () controller proposed in x 5. The control law is summarized as follows: u ¼ u a þ u s þ u s : ð5þ u s ¼ 4 u a ¼ T ^, ð5þ jx j T ¼ _x eq ^z gðx Þ ^z x, ð5þ u s ¼ k s e, h " þ h " þ h " þ e, " h ¼k k k max min k, ð54þ ð55þ ð56þ h ¼ max ðz max z min Þ, ð57þ jx j h ¼ max gðx Þ ðz max z min Þ: ð58þ The parameter adaptation law and state observers are given by equations () (). () The adaptive controller () presented in x, which can be summarized by equations (5) (54), with u s ¼. The state observers and parameter adaptation law are given by equation (4), (5),

9 74 L. Xu and B. Yao Downloaded By: [Xu, L.] At: : 8 November 7 and (7). For comparison purpose, initial conditions and common design parameters of and are kept the same, 9 zðþ ¼^z ðþ ¼^z ðþ ¼x ðþ ¼x ðþ ¼, ^ðþ ¼ nom, k ¼ 5, k s ¼ 5, ¼ diag½,,, 5Š, ¼ ¼, " ¼ " ¼ 4 : >= >; ð59þ () The deterministic robust controller (DRC) presented in x 4, which can also be represented by (5) (58), but with ^ nom and ^z ¼ ^z. Parameter adaptation law and state observers are not used. Initial conditions and design parameters remain the same except k ¼ k s ¼ 5. The desired position trajectory (see figure ) is selected to be (Vedagarbha et al. 999) x d ¼ : tan ð4 sinð:5tþð expð :t ÞÞ rads: ð6þ The following two typical cases are considered. No disturbance: To test the nominal performance of all three controllers, simulation is first run in the absence of uncertain non-linearities, i.e., ¼. The tracking errors of all three controllers are shown in figure. It is seen that the proposed scheme has the advantages of both DRC and : the transient period of is comparable to that of DRC while asymptotic tracking is achieved as in, which agrees with the theoretical results in Theorem. Overall, has the worst transient performance, while the tracking error of DRC does no reduce with time. The control inputs of and are given in figure for reference. Figures 4 and 5 show that has a more stable parameter estimation process and state estimation process than that of. With disturbance: To test the performance robustness of the controllers, a random signal with an amplitude of 6.5 (Nm) is injected to the input channel of the system between t ¼ s and t ¼ s to simulate external disturbance (t). The tracking errors of and are shown in figure 6. It is seen that achieves a much smaller tracking error when the disturbance occurs. In addition, recovers to its nominal performance much faster after the disturbance is removed. One reason is that the disturbance is dealt with via the robust feedback term u s. The other equally important reason is that due to the use of projection mappings, has a more stable parameter estimation process and state estimation process, which are shown in figures 7 and 8. Desired trajectory (deg) Tracking errors (deg) Control inputs (Nm) Figure. Desired position trajectory Figure Figure. DRC Tracking errors in the absence of disturbances. Control inputs in the absence of disturbances.

10 Adaptive robust control of mechanical systems 75 Downloaded By: [Xu, L.] At: : 8 November 7 Estimate for θ Estimate for θ Estimate for θ Estimation errors of the first observer Estimation errors of the second observer Time (sec) Figure 5. Friction state estimation errors in the absence of disturbances. Tracking errors (deg) Figure Parameter estimates in the absence of disturbances Figure 6. Tracking errors of and (with disturbance). Estimate for θ Estimate for θ Estimate for θ Conclusion Estimation errors of the first observer Estimation errors of the second observer Figure 7. Parameter estimates of and (with disturbance). Figure 8. Friction state estimation errors of and (with disturbance). In this paper, an adaptive robust controller with dynamic friction compensation is presented. The design utilizes the structural information of the friction dynamics to construct non-linear observers to recover the unmeasured friction state. In addition to that, by utilizing a priori information of the system such as the bounds on parameter variations and unmeasured friction state, the proposed controller uses the discontinuous projection mapping to improve the stability of parameter adaptation and friction state estimation. The estimation errors as well as uncertain non-linearities are attenuated by certain robust feedback

11 76 L. Xu and B. Yao Downloaded By: [Xu, L.] At: : 8 November 7 terms to achieve a guaranteed robust performance. In the absence of uncertain non-linearities, the effect of dynamic uncertainties is eliminated and an improved tracking performance is achieved without resorting to discontinuous or high gain feedback control laws. Comparative simulation results are presented to illustrate the effectiveness of the proposed scheme. Acknowledgement This work is supported in part by the US National Science Foundation grant CMS-656 and in part by the National Natural Science Foundation of China (NSFC) under the Joint Research Fund (grant 55855) for Overseas Chinese Young Scholar. References C. Canudas de Wit and P. Lischinsky, Adaptive friction compensation with partially known dynamic friction model, Inter. J. Adaptive Cont. and Signal Proc.,, pp. 65 8, 997. C. Canudas de Wit, H. Olsson, K.J. Åstro m and P. Lischinsky, A new model for control of systems with friction, IEEE Trans. on Auto. Cont., 4, pp , 995. M.J. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. on Auto. Cont., 6, pp. 9 44, 98. P. Dahl, A solid friction model. Technical report TOR- 58(7-8), The Aerospace Corporation, El Segundo, CA (968). G.C. Goodwin and D.Q. Mayne, A parameter estimation perspective of continuous time model reference adaptive control, Automatica,, pp. 57 7, 989. M. Krstic, I. Kanellakopoulos and P.V. Kokotovic, Nonlinear and Adaptive Cont. Des., New York: Wiley, 995. H. Olsson, K.J. Åstro m, C. Canudas de Wit, M. Ga fvert and P. Lischinsky, Friction models and friction compensation, Euro. J. Cont., 4, pp , 998. J.S. Reed and P.A. Ioannou, Instability analysis and robust adaptive control of robotic manipulators, IEEE Trans. on Rob. and Auto., 5, pp. 8 86, 989. S. Sankaranarayanan and F. Khorrami, Adaptive variable structure control and application to friction compensation, in Proc. IEEE Conf. on Decision and Control, pp , California, 997. S. Sankaranarayanan and F. Khorrami, Model independent friction compensation, in Proc. of the American Control Conference, pp , Philadelphia, 998. S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Englewood Cliffs, NJ 76, USA: Prentice Hall, Inc., 989. P. Tomei, Robust adaptive friction compensation for tracking control of robot manipulators, IEEE Trans. on Auto. Cont., 45, pp ,. Y. Tan and I. Kanellakopoulos, Adaptive non-linear friction compensation with parametric uncertainties, in Proc. of the American Control Conference, pp. 5 55, San Diego, 999. V.I. Utkin, Sliding Modes in Control Optimization, New York: Springer Verlag, 99. P. Vedagarbha, J.M. Dawson and M. Feemster, Tracking control of mechanical systems in the presence of non-linear dynamic friction effects, IEEE Trans. on Cont. Syst. Tech., 7, pp , 999. B. Yao, High performance adaptive robust control of non-linear systems: a general framework and new schemes, in Proc. of IEEE Conference on Decision and Control, San Diego, pp , 997. B. Yao and M. Tomizuka, Robust adaptive non-linear control with guaranteed transient performance, in Proc. of American Control Conference, Seattle, pp. 5 55, 995. B. Yao and M. Tomizuka, Smooth robust adaptive sliding mode control of manipulators with guaranteed transient performance, ASME J. Dyn. Syst., Measurement and Cont., 8, pp , 996. B. Yao and M. Tomizuka, Adaptive robust control of SISO non-linear systems in a semi-strict feedback form, Automatica,, pp. 89 9, 997. Y. Zhang and P.A. Ioannou, Robustness and performance of a modified adaptive backstepping controller, Inter. J. Adap. Cont. and Signal Proc.,, pp , 998.