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Identification of Coulomb Friction-Impeded Systems With a Triple-Relay Feedback Apparatus Si-Lu Chen, Member, IEEE, Kok Kiong Tan, Member, IEEE, and Sunan Huang

Abstract—This paper aims to identify friction-impeded servomechanical systems using a time-domain based approach with a multiple-relay feedback apparatus. By converting the closed-loop system into a triple-relay feedback system, switching conditions of a stable limit cycle are obtained. Following by this, stable oscillations are excited by choosing suitable relay gains. Hence, the level of Coulomb friction and other linear parameters are identified by numerically solving a set of equations with data from just a single limit cycle experiment. This method is also robust to measurement noise. Simulation and real-time experiments on a DC motor show the practical appeal of the proposed method. Index Terms—Friction modeling, hybrid systems, limit cycles, relay feedback, switching systems.

I. INTRODUCTION

T

HE interest in the analysis of limit cycles arising from systems under relay feedback has been generally motivated and reignited following instances of effective applications in the industries in the 1950s. In the early years, relay feedback was mainly used for signal amplification and switching control [30]. In many relay applications, one of the phenomenon which is leveraged on is the resultant limit cycles oscillations and the information they provide. Analysis on relay feedback systems can be carried out using the describing function (DF) approximations [14], which facilitates an approximate analysis of limit cycle behaviors of single-input–single-output (SISO) system. Besides the DF method, phase-plane analysis is another classical technique which is applicable to investigate the existence and stability of limit cycles [16], [30]. Deliberate generated limit cycle oscillations under relay feedback have been successfully used for automatic tuning of controllers since 1980s [2], and many industrial controllers now have built-in relay tuning functions. Linear systems under feedback of single relay with hysteresis has been studied, and the condition of stable limit cycles is given [5]. From the system science point of view, a linear system with relay feedback is a hybrid system and it can be described by Manuscript received December 20, 2009; revised November 07, 2010; accepted February 27, 2011. Manuscript received in final form April 11, 2011. Recommended by Associate Editor G. Guo. S.-L. Chen was with the Department of Electrical and Computer Engineering, National University of Singapore, 117576 Singapore. He is now with Singapore Institute of Manufacturing Technology (SIMTech), Agency for Science, Technology and Research (A*STAR), 638075 Singapore. (Email: [email protected]). K. K. Tan and S. Huang are with the Department of Electrical and Computer Engineering, National University of Singapore, 117576 Singapore (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCST.2011.2143413

a set of differential equations (DEs) with discontinuous parts [12], [27]. These special kinds of DEs may exhibit such phenomena as non-uniqueness of solutions [25], chattering or fast switching [20], sliding modes [7], bifurcations and chaos [8], [18]. The existence of solution has been studied for systems with input-output forms as in [15]. It is known that relay feedback systems often possess limit cycles. Most applications of relay feedback systems (RFSs) are based on stable limit cycle oscillations. The elegant criteria for the local stability of limit cycles is investigated by perturbations of the Poincaré map [5], [24] or harmonic balance condition [1]. If the system starts from an arbitrary set point, the RFS will always be driven to converge to the periodic motions, then the RFS has globally stable limit cycle [17]. However, establishing the exact conditions for existence of limit cycles only limits to certain kinds of RFSs until today [28], and the complete discussion of solutions in hysteresis relay feedback only limits to certain linear systems, such as first-order-plus-dead-time (FOPDT) and second-order type-1 systems [23], [31], [32]. In recent years, the relay feedback approach has also been extended beyond tuning of controllers to the identification of commonly encountered nonlinearities in practical systems, such as friction [4], [6], [9], [10], [29]. One commonly used method to identify friction is based on DF approximation [9], [10], [21], [29]. The harmonics balance methods with DF approximations are constrained by the assumption of sinusoidal oscillations [9], [10], [21], [29], which ceases to be valid when better accuracy is required. In [26], the properties of limit cycle oscillations generated due to friction is further investigated in time-domain. Based on this, a dual-relay configuration [6], where two relays switch asynchronously, is used to generate oscillations. Coulomb friction model is identified recursively by using the corresponding points in two different limit cycles. However, this method has several shortcomings. The most critical weakness is that this identification method is only valid when all the limit cycles start from zero-initial conditions. Second, the linear portion property of the system is totally ignored in this method, which may be critical for controller design. Third, this time-domain-based approach in [6] only uses the data segments before the first switching instants, while the characteristics of relay switchings have not been fully utilized. In this paper, we propose a new method to identify the Coulomb friction-impeded servo-mechanical system using dual-channel relay (DCR) feedback in time-domain. We first investigate the switching condition of stable limit cycle in an equivalent triple-relay feedback system. We further explore the variation of oscillation amplitude and frequency corresponding to different relay gains, hence the suitable oscillations are

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excited for system identification. Later, the Gauss-Newton algorithm is applied to estimate the system parameters including Coulomb friction. By fully utilizing the input and output data, this proposed approach requires only a single closed-loop relay experiment starting from arbitrary initial conditions. More reliable and efficient optimization is achieved if we combine the classical DF-based approach with our new method. The proposed method is also robust to measurement noise by using periodic data within the same limit cycle experiment. Simulation examples and real-time experiments further show the practical appeal of the proposed method.

Fig. 1. Servo-mechanical system with friction under DCR feedback.

II. TRIPLE-RELAY FEEDBACK SYSTEM Consider a DC motor driven by a current amplifier with a [33]. The resulting transfer function between angular gain velocity and input voltage is (1) and are torque constant and combined inertia, rewhere spectively. When the motor velocity is sensed by a tachometer , another velocity feedback loop is formed. Thus with gain (2) Fig. 2. Equivalent standard triple-relay feedback system.

where the velocity time constant transfer function from to position output

. Thus, the is (3)

Set as

, (3) is transformed into the state-space form

first-order portion , in series with two integrators. Three feedback relays RA, RB, and RC are connected to each state of the system. Under triple-relay feedback, the effective control signal is governed by the following equation:

(4) (5) . In addition, where the above servomechanical system experiences the Coulomb during motion. In this paper, we will use friction the dual-channel relay (DCR) feedback to identify the system parameters. The DCR feedback is first proposed in [13], and it is used for the identification of a friction model within a typical servomechanical system configuration [29], as shown in Fig. 1. It consists of a parallel intentional relay construct acting on the linear portion of the dynamic system. The second feedback relay RC, which is cascaded to an integrator, provides a second degree of freedom to adjust the frequency of oscillation and ensure that the phase lag of the oscillation does not exceed . If the above servo-mechanical system (4)–(5) is under the intentional DCR feedback apparatus , where [13], [29], then the actual control signal fed to the linear portion of the system is . Our new method is initiated by transforming the DCR feedback servo-mechanical system to an equivalent triple-relay feedback system, as shown in Fig. 2. The linear portion of the system is essentially a third-order system comprising of a

(6) , the By defining the augmented state vector augmented state-space form of the linear portion of the system can be represented as (7) (8) where (9) (10) Meanwhile, note that . Similarly, . Thus, , and . In our proposed method, we take a closer look at the switching behavior of the stable limit cycles, so that more information will be extracted for system identification. The following subsections give two theorems on the location and stability of limit cycles in the triple-relay feedback system.

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A. Locations of Limit Cycles The study of the limit cycle oscillations arising from the triple-relay feedback system of Fig. 2 will be restricted to the case of simple oscillation, which is defined as one comprising of switching due to a relay is followed by one and only one switching from the other relays before its next switch. This restriction is also usually referred to as the “condition of no additional switching” [30]. Define the switching plane where the outputs of the three re, lays change as and . Since leads by a phase of and leads by the same amount, relays RA, RB, and RC will switch sequentially within half a period of periodic oscillations. Thus, the assumption of simple oscillations is reasonable when the system goes into steady state oscillation. Under this condition, since the control arises from three branches of switching sources, and every branch contributes two possible values, there are six possible states during the steady state oscillation. The following theorem determines the location of limit cycles in the triple-relay feedback system. Theorem 2.1: For the augmented state-space system in the form (7)–(10) under feedback (6), if there exists an odd symmetric and periodic trajectory with period , and the trajectory traverses planes , and at , and accordingly with initial conand , then dition and are given by and , where (11) (12) (13) subject to

being non-singular, and (14) (15) (16)

The control at the three stages within half of the period, are respectively given by (17) (18) (19)

Fig. 3. Sequence of switching arising from the triple-relay feedback.

Fig. 3 illustrates the sequence of switching in the triple-relay feedback systems, while Proof of Theorem 2.1 is given in the Appendix A. This theorem is given to allow numerical compu, and of the three stages of half tation of the duration period of oscillation, so that the locations of the limit cycles can be determined. Compared to DF analysis which can only approximate the period of the resultant limit cycle, Theorem 2.1 is able to provide the exact time duration between two consecutive switchings in the triple-relay feedback system. B. Local Stability of Limit Cycles In practical application, only the stable limit cycles are useful for auto-tuning and system identification since they need to be immune to random noise and perturbation. In Section II-A, the limit cycles have been located for a class of relay feedback system. For further investigating their local stability, the Jacoof the Poincaré map is imported. Physically, it shows bian the variation of states after one period of oscillation w.r.t. the perturbed initial states. For the case of odd symmetric oscillaw.r.t. half of the period suffices for the tion, the Jacobian analysis, while the Jacobian w.r.t. the full period is given by . The following theorem gives the expression for and the condition for local stability of limit cycles. Theorem 2.2: Under condition of no additional switching, the of the Poincaré map is given by Jacobian

Furthermore, the solution must satisfy the following con: ditions within the time interval

(23) where

(20) (21) (22)

; and , and , with various notations in Theorem 2.1 are inherited. The limit cycle are inside the unit is locally stable iff all the eigenvalues of circle. The proof of Theorem 2.2 is given in Appendix B. Notice that the eigenvalues of may not all fall inside the unit circle, since

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the triple-relay feedback system can be viewed as a switching system. Thus, the overall stability of the trajectory does not necessarily imply the stability of every individual segment.

the set of equations (11)–(13). For convenience, (11)–(13) can be rewritten as

(24) (25) (26)

III. IDENTIFICATION PROCEDURES

A. Excitation of Suitable Limit Cycles Our motivation is to use the closed-loop relay test to excite the stable oscillation via choosing appropriate gain, and use the relay and position signals for system identification. Although Theorem 2.1 provides numerical method to solve the limit cycle and location, the relationship between limit cycle amplitude w.r.t. relay gains cannot be explicitly given since frequency (11)–(13) are not analytical. In this case, the DF-based method become a good supplement to give a approximate explicit expression to guide our tuning of relay gains to achieve appropriate limit cycles. Similar to [9], by grouping the linear and nonlinear portion of whole relay feedback systems as shown in Fig. 2, two properties are summarized as following. Property 3.1: The limit cycle is excitable when under describing function approximation. increases with increasing and decreasing Property 3.2: , while increases with decreasing and increasing . The proofs of the two properties are given in Appendix C. A large amplitude of oscillation is not desired since it may exceed the travel limits or damage the mechanical components. Similarly, high frequency signals are also not suitable for system identification too since it will excite high frequency resonance. The above undesired oscillations can be avoided if we fixed a and choosing a small , the oscillation may not be certain now by Property 3.1. However, as we excited since gradually increase , the oscillation is eventually observed. Then, we can further fine tune and to achieve a satisfied and , according to Property 3.2. Then we use the relay and position signal of proper oscillations for signal identifying parameters, as shown in the following parts.

B. Data Extraction In the position feedback system under study, the position and the input signal from the DCR are measurable. signal Under the condition of simple oscillation, without loss of generality, select the instant when the DCR switches to the maximum value as the starting time . Inferring from the (position) reaches the maximum nature of integration, when value, the time instant then can be denoted as for relay RA (velocity) as input to switch from a positive to negative with state. Furthermore, the switching instants and of RB and RC are directly observed from . In this way, the durations between two consecutive switchings and in the limit cycle are known. By Theorem 2.1, with just one relay experiment, it is possible to estimate and by numerically solving the model parameters, i.e.,

where are the state vectors when the trajectory traverses is the switching planes as defined in (40)–(42), is the identifying parameter vector, and the experiment data vector. However, due to the nonlinear nature of the equation set, it is useful to use additional characteristics of the oscillations to increase the robustness of the parameter estimation. To this end, it may be noted that when the trajectory traverses the switching at time , not only the switching condition plane is satisfied, but also the position is measurable de. Similarly, when the trajectory traverses noted as at time , an additional condition holds. Hence, we have another two equations (27) (28)

C. Recursive Approach Now, we are able to numerically solve from (24)–(28) using the Gauss-Newton iterative method [19], with an initial guess of . parameters , the Jacobian For the vector function of at at the th iteration, is defined as .. .

The incremental

.. .

.. .

at th iteration, is solvable from

where is the value of function vector at the th iterative based on current guess of parameters’ value. The parameter vector is updated as (29) The iterative search for the parameters terminates when , where is a small positive value. To avoid the situation of the gradient-based optimization indulging in a local minimum, different sets of initial guess values may be used. D. Summary and Remarks This approach is a closed-loop approach based on position feedback information only. Thus, it has the advantage of low cost and high noise immunity in practical applications. In addition, the new time-domain-based method releases the constraint

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to have sinusoidal-like output waveform in DF-based method [9], [29]. Instead, the approach only requires the switching information from stable limit cycles with “simple oscillation” behavior. Thus, compared to the method proposed in [6], it is able to identify all the linear and nonlinear parameters by a single relay experiment, starting from any non-zero initial condition. Remark 3.1: In general, the position signal is relatively smooth due to the low-pass property of the integrator in the pois corrupted with white sition loop. However, if the output , artificial filtering may not be applicable since it will noise impose additional time delay and increase system complexity. Instead, by observing the periodic property of the limit cycle, additional steps may be taken to remove the noise without requiring additional set of experiment data. Denote the measured corrupted signal as

,

The augmented system can be represented as , where and

Numerical calculations with Theorem 2.1 gives only one set of , and positive solutions for (11)–(13). Thus, the limit cycle has period . Furthermore

and where is additive white gaussian noise with zero mean and variance . Since the data collected from the relay test is generally periodic with period , starting from arbitrary data point of stable limit cycle, we denote the th period segment of as , while . Since

The noise can be ideally removed by averaging infinite number of periodic signal together. In practice, the noise will be greatly 5–20. Notice that this special method reduced by letting is based on the averaging of corresponding signal points in different periods, while the classical filtering method is based on the averaging of signal points at their neighborhood. is noisy, small chartering Remark 3.2: If output signal during state transiwill be also experienced in relay signal . When applying the noise reduction from State to State tion method in Remark 3.1, the corresponding relay mean signal becomes stair-shape. Thus, we estimate the switching instant as

where is the relay chattering duration, and is the value of relay signal before and after state transition. Remark 3.3: The efficiency of optimization can be greatly improved if we combine the newly developed time-domain-based approach with the classical DF based relay feedback approach. First, the DF-based approach is applied to get a reasonable estimation of the parameters. Later, this rough estimation is used as a wise initial guess for our time-domain-based method to speed up the parameter convergence and reduce the chances of falling into local minima. IV. SIMULATION AND DISCUSSIONS Example 4.1: To verify Theorem 2.1 and 2.2, consider a , under the triple-relay feedback as in system , and . Fig. 2, where

The Jacobian of the Poincaré map can be computed from Theorem 2.2 as

The eigenvalues of are , and 0. It can be concluded that the limit cycle is locally stable according to Theorem has one eigenvalue 1.4823 2.2. In addition, we notice that and . However, all the eigenoutside the unit circle, so do are still within the unit circle, invariant to stability values of of individual segments. Example 4.2 (Reverse of Example 4.1): To compare the effectiveness of different identification methods, consider the , under the second-order positioning system effect of friction simulated via the Coulomb friction model denote the Coulomb friction model. The , parameters used in simulation are: same as Example 4.1. The sampling period is set as s. For examining the robustness of our estimation method, a with zero mean and variance of is white noise signal also added in the position feedback loop. a) Linear Least-Square Approach: In this part, we will follow the method proposed in [6] to use a single hysteretic relay feedback, to identify the Coulomb friction level, as shown in Fig. 4. To satisfied the operation condition of this method, the initial positions of two experiment have to be both set as zero. Select the levels of hysteresis and gain in two relay experiments and accordingly, as the relay and output signals are shown in Fig. 5. By taking the segments before the first switching instants of both experiment, the Coulomb friction parameter can be identified using the following linear regression model:

where

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Fig. 4. Servo-mechanical system with friction under single relay feedback.

Fig. 6. Friction identification by linear least square with x .

(0) = x (0) =

Fig. 7. Friction identification by linear least square with x

(0) = x (0) =

0

Fig. 5. Input and output waveform by linear least square estimation with zero initial condition. Solid line: Output position waveform x . Dashed line: Input : ;d : . (b)  : ;d . relay waveform v . (a) 

=05

=08

=03

=1

and are in first and second relay experiments, respectively. The following recursive linear least square method is applied to identify the Coulomb friction parameter [3]

Fig. 6 shows the identification result corresponding to number of samples. Notice that the noise reduction technique in Remark 3.1 does not apply here since the oscillation has not reach , there is a steady-state. Depending on different choice of tradeoff between estimation consistency and convergent speed, and the estimation of varies from 0.4921 to 0.4988 after 1000 data samples, which is close to the true value. However, by using this method, the linear portion of parameters cannot be obtained. Moreover, this method incurs an estimation error if the initial conditions of relay experiments are non-zero. When other

3.

simulation parameters remain unchanged, Fig. 7 shows the esin timation results when initial position the two relay experiment, while Fig. 8 shows the estimation reand in the first and second sults when experiments accordingly. From these two figures, we conclude varies, both of estimations that no matter how the value of fail to converge to the true value of . b) DF-Based Approach: Through this simulation, we will examine the efficiency and accuracy of estimating the Coulomb (i.e., ) as well as linear system pafriction coefficient rameters and using DCR apparatus as shown in Fig. 1 by and as the gain DF-based approach. In this part, denote and of relays in th experiment accordingly. The gains of . From Remark 3.1, DCR are selected as we know that by averaging the corresponding data points in several consecutive periods of waveforms, the noise can be greatly reduced. This has been verified by Fig. 9, which shows the noise

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Fig. 8. Friction identification by linear least square with x x (0) = 3.

(0)

=

7

0;

Fig. 10. Limit cycles with Coulomb friction model. (a) h = 0:8; h = 1. (b) h = 0:5; h = 1:2. Solid line: Average dual-relay signal v(t). Dashed-dotted line: Average output position signal x  (t). Dashed line: Average actual input signal u (t) fed to linear portion (not measurable in practice).

Fig. 9. Noise reduction by taking average of corresponding points in ten consecutive periods of position waveform. Left: A segment of the average position signal x  (t). Right: Corresponding original noisy position signal x (t).

effect before and after taking average of ten periods of superimposed waveform. As illustrated in [9], [29] and Appendix C, by setting up the harmonic balance condition, the frequency and of the limit cycle are determined by amplitude (30) (31) (32) and are limit cycle amplitude and frequency in where and the th experiment accordingly. By choosing , amplitude and frequency of the limit cycle are obtained as and , as in Fig. 10(a). However, there are three unknowns but only two equations available. Thus, for the DF-based approach, we require an additional data set to solve parameter values. The simulation is rerun by setting and , another stable limit cycle is obtained and , as shown in Fig. 10(b). with Thus, the system parameters are solved as , and (or ) .

c) New Approach: For a comparison study, we would like to examine the identification results via our time-domain ap. The proach, with gains of DCR selected as average waveforms of various input and output signals during steady-state oscillation are shown in Fig. 10(a). By Theorem 2.1, the starting time corresponds to the instant of relay control signal switching to its maximum amplitude. Although the velocity is not assumed to be measurable, the RA switching incan still be tracked from the time the position signal stant reaches its minimum. The switching instants of RB and RC are detectable from the relay control signal . Thus, the respective time durations required for modeling are obtained as . The Jacobian of is computed by MATLAB numerically [34]. By the function , the (24)–(29), starting from initial guess of parameters are identified as , and after 5 iterations, which are very close to their true values, as shown in Fig. 11. The comparison of results between these two approaches is shown in Table I. From this table, it is concluded that the new method differs a superior performance in both identification accuracy and data utilization. d) Two-Stage Approach: In this part, we use the identification results by DF based method in Part b) as the initial guess, to start the identification use the new time-domain based method as done in Part c). Notice that one of the two sets of input-output data used for DF-based method is reused in the second stage

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Fig. 11. Convergence of parameter estimations with Coulomb friction model. Solid line: Without assistance of DF-based method. Dashed line: Using results from DF-based method as initial guess.

Fig. 13. Convergence of parameter estimation with Tustin friction model.

TABLE I COMPARISON OF IDENTIFICATION RESULTS

=08 =1 ( )

Fig. 12. Limit cycle with Tustin friction model and h : ;h . Solid line: Average dual-relay signal v t . Dashed-dotted line: Average output position signal x t . Dashed line: Average actual input signal u t fed to linear portion (not measurable in practice).

()

( )

time-domain based approach. Thus, no additional experiment is required. After applying Gauss-Newton algorithm, we find that we only need two iterations to get the same estimation result instead of five iterations. This verifies that the optimization process has become more efficient by using the two-stage approach, as stated in Remark 3.3. Example 4.3 (Model Uncertainty): Consider the secondorder positioning system , under the effect of friction simulated via Tustin friction model

period is set as s and variance of white noise are too. The additive noise is greatly reduced in the same way by Reand , and mark 3.1. By choosing are now observed as 0.0080, 0.1400, and 0.1566. The additional for modeling are obtained as position information and 0.3871, respectively. By using the Gauss-Newton iterative method implemented by MATLAB, starting from ini, after four iterations, the patial guess of rameters are identified as , and . and are close to the actual values. The apparent deviation from its true value of is not due to inaccuracy arising from the identification process. The model structure used in this paper does not include a viscous friction component. simulated is Thus, the equivalent effect of viscous friction absorbed into the time constant of the dynamics of the servo system, so that the model is able to account for the viscous friction dynamics via a modified linear portion. We may verify that this is indeed true by checking if

Indeed, where , and denote the static, Coulomb, and viscous denotes the Stricbeck vefriction coefficient accordingly, locity. Similar to Example 4.1, the linear portion parameters are . The parameters used in this friction set as , and they model are: will test the robustness of our estimation method. The sampling

is close to

consistent with the simulation results.

V. REAL-TIME EXPERIMENT ON A DC MOTOR To illustrate the effectiveness of proposed method, real-time experiments are carried out on a LJ Electronic MS15 DC Motor platform, as shown in Fig. 14(a). The DC Motor accepts the analog input voltage to generate different rotation speeds. In our

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Fig. 14. DC motor experiment. (a) Setup. (b) Block diagram.

proposed method, only the position signal is required, which is provided by the on-shaft potentiometer. The PC-based National Instrument (NI) Data Acquisition (DAQ) Card, together with LabVIEW virtual instrument platform, integrate the whole development cycle to a single user interface. MATLAB optimization toolbox is also used for data analysis [34]. Fig. 14(b) further illustrates the block diagram of the experiment setup. In the experiment, a Virtual Instrument (VI) is built using LabVIEW based on the DCR feedback configuration of Fig. 1. Note that the potentiometer in this DC motor is attached to the slave shaft, rather than the master one, and the ratio of angular . The gains of the DCR are sevelocity between them is lected according to Property 3.1 and 3.2, such that the angular , for cordisplacement of the slave shaft does not exceed rectly detecting the position signal.

Fig. 15. Limit cycle in the DC motor experiment by the DCR feedback (a) h ;h : ; (b) h : ;h : . Solid line: Average waveform of DC motor position. Dotted line: Average waveform of relay signal.

=1

= 1 25

= 12

= 1 35

A. Parameter Estimation and ; the experimental reBy selecting sults of limit cycles of DC motor position signal under DCR feedback is shown in Fig. 15(a), with a sampling period of 10 ms. To eliminate the noise, we take the average of 10 periods of input and output signals. The various data required for modeling are extracted from the oscillation as , and . Starting from , by Gauss-Newton method, the system parameters are identified as after just four iterations, as illustrated in Fig. 16. For comparison, we estimate the parameters using a DF-based method as shown in [9], for experiment with and , we obtain , and . In addition, by selecting , , as shown in Fig. 15(b). we get By (30)–(32), the parameters are identified as , and .

Fig. 16. Convergence of parameter estimation and residue of cost function in DC motor experiment.

B. Model Verification via Feedback Compensation With the model identified, a full-state feedback linearization controller is designed to verify the adequacy of model parameters thus obtained. Due to the placement of velocity and position sensors, a gain of 9 exists between velocity and position signals

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triple-relay feedback system, the system parameters including Coulomb friction is simply identified using Gauss-Newton approach to solve a set of equations from a single, proper relay experiment under position feedback. The measurement noise is greatly removed by superposition of corresponding points in the periodic waveform. Simulation and real-time experiment have verified the superior performance of the new method over existing time-domain and DF-based methods.

APPENDIX A PROOF OF THEOREM 2.1

Fig. 17. Design of feedback controller with compensation.

at Note that the trajectory will traverse again by symmetry. Define as the value of at the th switching instant of the th period of oscillation. Then, under the condition of no additional switching, the sequential chart of switching instants and variation of relative parameters is de, picted in Fig. 3. In this figure, for simplicity, are simply abbreviated as . In this way, the and state variable for different switching states is listed for ease of in system (7)–(10) are reanalysis. The traversing points by the same function , w.r.t. the same lated to Fig. 18. Tracking error with model-based feedback controller. Solid line: with parameters from our new approach. Dotted line: with parameters from DF-based approach.

obtained, as shown in Fig. 17. Thus, in the velocity loop model, , where , and is the Coulomb friction force. (i.e., ), in order to With the identified model , we set achieved the tracking of the trajectory profile , or (33) so that , so that

. Define the tracking error as and . If

(36) In fact, is the Poincaré Map [16] on the switching plane (or or ). In the case of periodic oscillations, the traversing of the trajectory with is fixed. Furthermore, if the point periodic oscillation is odd symmetric, . Similar properties exist for other switching planes and . for with control and initial The solution condition is given by . , and . For simplicity, set Thus, at which can be simply written as

is defined as (34)

Then the following closed-loop error dynamics is achieved (35) In our experiment, the desired time-varying trajectory is de. We compare the control perfined as formance by applying the control scheme in (33) and (34), with the parameters obtained from our new approach and DF-based approach. The comparison of tracking error is shown in Fig. 18. From these two figures, we are able to observe that, with our new approach, the root-mean-square (RMS) of the tracking error is only 0.0602, compared to 0.1027 in DF based approach. This verifies the effectiveness of our new identification method VI. CONCLUSION This paper proposed a new method to identify Coulomb friction impeded systems using DCR feedback. By analyzing the location of a stable limit cycle induced in the equivalent

(37) Inferring from the symmetric property of oscillation, . Then, considering the time intervals , and , it follows that (38) (39) are defined in (14)–(19). Note where and commutes. (17)–(19) can be obtained from that Fig. 3 under the simple oscillation condition. Solving the equation set (37)–(39) yields (40) (41) (42) and will Conditions give rise to the (11)–(13). Equations (20)–(22) can be obtained from Fig. 3, which shows the change of signs of key state variables at the switching instants.

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APPENDIX B PROOF OF THEOREM 2.2

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of nonlinear portion, inThe describing function cluding three relay branches, is given as

. If the Consider a trajectory with initial condition initial value varies within the switching plane by , i.e., . If reaches the switching plane at time and the control signal for , it yields

(50) while for linear portion, the frequency response by

is given

(43) (51) reaches the switching plane at time , i.e., If some perturbations exist due to the variation of initial conditions

Substitute (50) and (51) into the harmonic balance condition , we have (52) From (52), since and , the solution exists only when , this proves Property 3.1. Solving (52) also yields (53) (54)

(44) , which is the velocity of the Set is on the switching plane trajectory at time . Since . By ignoring the higher order terms , yielding . Note that the conmust hold to satisfy the switching direction dition condition if there is a symmetric limit cycle. Hence (45) Substituting (45) into (44), it follows that (46) Similarly, consider the time interval , we have

and

(47) (48) and with the switching conditions ingly. From (47) and (48), similarly we have

accord-

(49) where the Jacobian by

w.r.t. half period of oscillation is given

Similar to the analysis in [5], the limit cycle is stable if and only if .

APPENDIX C PROOF OF PROPERTY 3.1 AND 3.2

Thus, Property 3.2 is proven too. REFERENCES [1] L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, “Generating self-excited oscillations via two-relay controller,” IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 416–420, Feb. 2009. [2] K. J. Åström and T. Hägglund, “Automatic tuning of PID controllers,” ISA, Research Triangle Park, NC, 1988. [3] K. J. Åström and B. Wittenmark, Adaptive Control, 2nd ed. Boston, MA: Addison-Wesley, 1994. [4] B. Armstrong-Helouvry, P. Dupont, and C. C. de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138, 1994. [5] K. J. Åström, , K. J. Åström, G. C. Goodwin, and P. R. Kumar, Eds. , “Oscillations in systems with relay feedback,” in Adaptive Control, Filtering and Signal Processing. New York: Springer-Verlag, 1995. [6] A. Besançon-Voda and G. Besançon, “Analysis of a two-relay system configuration with application to Coulomb friction identification,” Automatica, vol. 35, pp. 1391–1399, 1999. [7] M. di Bernardo, K. H. Johansson, and F. Vasca, “Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations,” Int. J. Bifurcation Chaos, vol. 11, no. 4, pp. 1121–1140, 2001. [8] S. L. Chen, K. K. Tan, and S. Huang, “Two-layer binary tree datadriven model for valve stiction,” Ind. Eng. Chem. Res., vol. 47, no. 8, pp. 4549–4560, 2008. [9] S. L. Chen, K. K. Tan, and S. Huang, “Friction modeling and compensation of servo-mechanical system using a dual-relay feedback approach,” IEEE Trans. Control Syst. Technol., vol. 17, no. 6, pp. 1295–1305, Nov. 2009. [10] S. L. Chen, K. K. Tan, and S. Huang, “Modeling and compensation of ripples and friction in permanent magnet linear motor using a hysteretic relay,” IEEE/ASME Trans. Mechatron., vol. 15, no. 4, pp. 586–594, Aug. 2010. [11] D. Dai, R. G. Sanfelice, T. Hu, and A. R. Teel, “Analysis of hybrid systems resulting from relay-type hysteresis and saturation: A Lyapunov approach,” in Proc. 47th IEEE Conf. Decision Control, 2008, pp. 2764–2769. [12] A. F. Filippov, Differential Equations With Discontinous Righthand Sides. Norwell, MA: Kluwer, 1988. [13] M. Friman and K. V. Waller, “A two-channel relay for autotunning,” Ind. Eng. Chem. Res., vol. 36, pp. 2662–2671, 1997. [14] A. Gelb and W. E. V. Velde, Multiple-Input Describing Functions and Nonlinear System Design. New York: McGraw-Hill, 1968. [15] T. T. Georgiou and M. C. Smith, “Robustness of a relaxation oscillator,” Int. J. Robust Nonlinear Control, vol. 10, pp. 1005–1024, 2000.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

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[16] J. Guckenheimer and P. Holomes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983. [17] J. M. Goncalves, A. Megretski, and M. A. Daleh, “Global stability of rela feedback systems,” IEEE Trans. Autom. Control, vol. 46, no. 4, pp. 550–562, Apr. 2001. [18] Y. Hong and B. Yao, “A globally stable high-performance adaptive robust control algorithm with input saturation for precision motion control of linear motor drive systems,” IEEE/ASME Trans. Mechatron., vol. 12, no. 2, pp. 198–207, Apr. 2007. [19] S. L. S. Jacoby, J. S. Kowalik, and J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems. Englewood Cliffs, NJ: PrenticeHall, 1972. [20] K. H. Johansson, A. Rantzer, and K. J. Astrom, “Fast switching in relay feedback systems,” Automatica, vol. 35, pp. 539–552, 1999. [21] M.-S. Kim and S. C. Chung, “Friction identification of ball-screw driven servomechanisms through the limit cycle analysis,” Mechantron., vol. 16, pp. 131–140, 2006. [22] L. H. Lim, A. P. Loh, and W. W. Tan, “On forced and subharmonic oscillations under relay feedback,” IET Control Theory Appl., vol. 2, no. 9, pp. 829–840, 2008. [23] C. Lin, Q.-G. Wang, and T. H. Lee, “Relay feedback: A complete analysis for first-order systems,” Ind. Eng. Chem. Res., vol. 43, no. 26, pp. 8400–8402, 2004. [24] C. Lin, Q.-G. Wang, T. H. Lee, A. P. Loh, and K. H. Kwek, “Stability criteria and bounds for limit cycles of relay feedback systems,” Dyn. Syst., vol. 19, no. 2, pp. 161–170, 2004. [25] Y. J. Lootsma, A. J. van der Schaft, and M. K. Camlibel, “Uniqueness of solutions of linear relay systems,” Automatica, vol. 35, pp. 467–478, 1999. [26] H. Olsson and K. J. Åström, “Friction generated limit cycles,” IEEE Trans. Control Syst. Technol., vol. 9, no. 4, pp. 629–636, Jul. 2001. [27] A. Yu, W. P. Pogromskya, M. H. Heemelsb, and H. Nijmeijera, “On solution concepts and well-posedness of linear relay systems,” Automatica, vol. 39, pp. 2139–2147, 2003. [28] A. Pogromsky, H. Nijmeijer, and J. Rooda, “A negative Bendixson-like criterion for a class of hybrid systems,” IEEE Trans. Autom. Control, vol. 52, no. 4, pp. 586–595, Apr. 2007. [29] K. K. Tan, T. H. Lee, S. N. Huang, and X. Jiang, “Friction modeling and adaptive compensation using a relay feedback approach,” IEEE Trans. Ind. Electron., vol. 48, no. 1, pp. 169–174, Jan./Feb. 2001. [30] Y. Z. Tsypkin, Relay Control Systems. Cambridge, U.K.: Cambridge Univ. Press, 1984. [31] Q.-G. Wang, T. H. Lee, and C. Lin, Relay Feedback: Analaysis, Identification and Control. London, U.K.: Springer-Verlag, 2002. [32] Z. Ye, Q.-G. Wang, C. Lin, C. C. Hang, and A. E. Barabanov, “Relay feedback analysis for a class of servo plants,” J. Math. Analysis Appl., vol. 334, pp. 28–42, 2007.

[33] Galil Motion Control Inc., Rocklin, CA, “DMC18X6 User Manual, Rev. 1.0e,” , 2009. [34] The MathWorks, Inc., Natick, MA, “Optimization Toolbox User’s Guide,” 2000.

Si-Lu Chen (S’07–M’11) received the B.Eng. and Ph.D. degrees from the National University of Singapore (NUS), Singapore, in 2005 and 2010, respectively. He was a Senior Engineer in Manufacturing Integration Technology Ltd from April 2010 to March 2011, working on motion control design of semiconductor machine. Since April 2011, he has been with the Mechatronics Group, SIMTech, Singapore, as a Scientist. His research interests include system identification, motion control and industrial drive tuning.

Kok Kiong Tan (S’94–M’99) received the Ph.D. degree from the National University of Singapore (NUS), Singapore, in 1995. Prior to joining the university, he was a research fellow with SIMTech, Singapore, a national R&D institute spearheading the promotion of R&D in local manufacturing industries, where he has been involved in managing industrial projects. He is currently an Associate Professor with NUS. His current research interests include precision motion control and instrumentation, advanced process control and autotuning, and general industrial automation.

Sunan Huang received the Ph.D. degree from Shanghai Jiao Tong University, Shanghai, China, in 1994. He is a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests include adaptive control, neural network control, and automated vehicle control.

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