ISA Transactions 51 (2012) 801–807

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ISA Transactions
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C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807

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of this uncertainty [15]. This control problem has b...
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C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807

then the state y globally converges to the origin in finit...
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C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807

804

is able to make s being stable and ultimately bounde...
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and second time derivatives, as

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C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807

which coincides with expression in (22). Thus, steps in t...
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Output feedback trajectory stabilization of the uncertainty DC servomechanism system

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This work proposes a solution for the output feedback trajectory-tracking problem in the case of an uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative controller and an uncertain estimator that allows compensating the effects of the unknown bounded perturbation. Because the motor velocity state is not available from measurements, a second-order sliding-mode observer permits the estimation of this variable in finite time. This last feature allows applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov method. Results obtained from numerical simulations and experiments in a laboratory prototype show the performance of the closed loop system.

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Output feedback trajectory stabilization of the uncertainty DC servomechanism system

  1. 1. ISA Transactions 51 (2012) 801–807 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Output feedback trajectory stabilization of the uncertainty DC servomechanism system$ ˜ Carlos Aguilar-Ibanez a,n, Ruben Garrido-Moctezuma b, Jorge Davila c a ´ ´ ´ CIC-IPN, Av. Juan de Dios Batiz s/n Esq. Manuel Othon de M., Unidad Profesional Adolfo Lopez Mateos, Col. Nueva Industrial Vallejo, Del. Gustavo, A. Madero, C.P. 07738 D.F., Mexico b ´ Departamento de Control Automatico, CINVESTAV-IPN, Av. IPN 2508, 07360 D.F., Mexico c National Polytechnic Institute (IPN), School of Mechanical and Electrical Engineering (ESIME-UPT), Section of Graduate Studies and Research, Mexico a r t i c l e i n f o abstract Article history: Received 29 March 2012 Received in revised form 19 June 2012 Accepted 29 June 2012 Available online 9 August 2012 This paper was recommended for publication by Dr. Jeff Pieper This work proposes a solution for the output feedback trajectory-tracking problem in the case of an uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative controller and an uncertain estimator that allows compensating the effects of the unknown bounded perturbation. Because the motor velocity state is not available from measurements, a second-order sliding-mode observer permits the estimation of this variable in finite time. This last feature allows applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov method. Results obtained from numerical simulations and experiments in a laboratory prototype show the performance of the closed loop system. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. Keywords: Servomechanism PD controller Finite time observer Variable structure control 1. Introduction The direct current motor-pendulum system (DCMP) is widely used as a test bed for assessing the effectiveness of several control techniques. This choice is due to the fact that its model captures some of the features found in more complex systems, as in the case of industrial robot manipulators [1,2]. Related to this topic we mention some interesting works; for instance, Ref. [3] applies the well known Generalized Proportional–Integral controller to the tracking control problem for a linear DCMP. In [4] the authors solve the regulation problem using the sliding-mode super-twisting based observer (STBO) method, in conjunction with a twisting controller. An interesting work dealing with the control of the DCMP system using the STBO observer, combined with an identification scheme can be found in [5, Chapter 2]. A close related work is developed by Davila et al. [6]. A closed-loop input error approach for on-line estimation of a continuous-time model of the DCMP $ ´ ´ This research was supported by the Centro de Investigacion en Computacion ´ of the Instituto Politecnico Nacional (CIC-IPN), and by the Secretarıa de Investiga´ cion y Posgrado of the Instituto Politecnico Nacional (SIP-IPN), under Research Grant 20121712. n ´ Corresponding author at: CIC-IPN, Av. Juan de Dios Batiz s/n Esq. Manuel ´ ´ Othon de M., Unidad Profesional Adolfo Lopez Mateos, Col. Nueva Industrial Vallejo, Del. Gustavo, A. Madero, C.P. 07738 D.F., Mexico. ˜ E-mail address: caguilar@cic.ipn.mx (C. Aguilar-Ibanez). was developed in [7]; while in [8] a parameter identification methodology based on the discrete-time Least Squares algorithm and a parameterization using the Operational Calculus is proposed. In [9] an adaptive neural output feedback controller design is used to solve the tracking problem of the system studied here, having the advantage of including the model of the actuator. Ref. [10] employs H1 techniques to deal with uncertainties; performance of the proposed approach is evaluated through numerical simulations. Another interesting work [11] reports the application of second order sliding mode control applied to an uncertain DC motor; the motor under control receives disturbance torques produced by another motor directly coupled to its shaft of the first motor. A smooth hyperbolic switching function eliminates chattering phenomena. The Hoekens mechanical system is the subject of research in [12]; here the authors apply a sliding mode control technique with uncertainty estimation combined with a learning technique. A key feature of this approach is the fact that it applies the switching to the plant indirectly through the learning process and the disturbance estimator thus reducing the occurrence of chattering; experiments validate the findings. In this context, perhaps one of the most challenging control problems consists in designing a smooth output-feedback stabilization algorithm for an uncertain and perturbed DCMP [13,14]. Generally speaking, this problem is by no means easy because it is not possible to fully compensate the effects of the system uncertainty without having information about the time derivative 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.06.015
  2. 2. ˜ C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807 802 of this uncertainty [15]. This control problem has been solved using a combination of neuronal networks and adaptive control theory, as in [9,16]. On the other hand, the problem is easily solved using a sliding mode controller (SMC), a methodology ensuring robustness against disturbances and parameter variations. However, its main drawback is that it may generate high frequency violent control signals, a behavior known as chattering. Moreover, the presence of chattering may excite unmodeled high-frequency dynamics, resulting in unforeseen instability and damage to the actuators. [17–19]. There exists essentially three main approaches to eliminate the chattering effects. The first approach uses continuous approximations, as the saturation function, of the sign function appearing in the sliding mode controllers [1,20,21]. This approach trades the robustness on the sliding surface and the system convergences to a small domain [22]. Observed-based approaches are another way of overcoming chattering; it consists in bypassing the plant dynamics by a chattering loop, then reducing the robust control problem to an exact robust estimation problem. However, this action can deteriorate the robustness with respect to the plant uncertainties and disturbances [19,23,24]. The last approach, based on the high-order sliding-mode method guaranties convergence to the origin of the sliding variable and its corresponding derivatives; the high-order sliding-mode algorithms translate the discontinuity produced by the sign function to the higher order derivatives, producing continuous control signals; however, these algorithms require a great computing effort [6,25–28]. In this work we introduce a smooth controller for output feedback trajectory tracking in an uncertain DCMP. The solution consists of a PD controller and a robust uncertain estimator. A super-twisting second-order sliding-mode observer estimates motor velocity. The observer finite time of convergence ensures that the estimation error will vanish after a finite time transient, the allowing the use of the Separation Principle. The corresponding convergence analysis is carried out using the Lyapunov method. This work continues with Section 2, where the model of the DCMP system and the problem statement are presented. In Section 3 the control strategy and the corresponding convergence analysis are developed. Sections 4 and 5 are devoted respectively to the numerical and experimental results and the conclusions. perturbation, defined as wðx,tÞ ¼ 1 _ ðÀf c sign½xŠ þ ZÀgmL sin xÞ, J ð2Þ which is a bounded function. In order to simplify the forthcoming developments, the system (1) is re-written in its state space form as _ x 1 ¼ x2 , _ x 2 ¼ Àf 0 x2 þ wþ u, ð3Þ _ where x1 ¼ x and x2 ¼ x, and f0 ¼ fd , J u¼ ku t: J ð4Þ Problem statement. Consider the uncertain nonlinear system (3) and the corresponding state x1 regarded as the measured system output, where the perturbation is uniformly bounded by 9w9 r 1 ðZ þgmL þ f c Þ r c: J ð5Þ Then, the control goal is to design a controller such that the angular pendular position tracks a given continuous reference trajectory xr(t), with their first and second time derivatives being also continuous. In other words, we want to control the pendular angular position x towards a pre-specified desired trajectory xr ðtÞ, in spite of the perturbation w. For simplicity, we use the symbol q to denote the upper bound of the term q 4 0, i.e. 0 o q rq. It is important to notice that the system (3) can be seen as a general electro-mechanical system, because a wide range of robots admit this configuration. Consequently, the solution that we propose in this work can be straightforwardly applied to more complex configurations, as for example a robot manipulator. On the other hand, it is important to remark that reference xr has to be continuous, and the corresponding first and second time derivative being at least piece-wise continuous ([29]. 3. The control strategy 2. DCMP dynamic model Consider the actuated second order DCMP system composed of a servomotor driving a pendulum, a servo-amplifier and a position sensor. The corresponding model of this system has the following form: € x¼ 1 _ _ ðÀf d xÀf c sign½xŠ þ ZÀgmL sin x þ ku tÞ, J ð1Þ _ Variables x and x are, correspondingly, the pendular angle position and the pendular angle velocity; t is the control input voltage; parameters m, L and g are respectively the pendulum mass, the pendulum arm length and the gravity constant; the terms fd and fc are in that order the pendulum viscous friction and Coulomb friction coefficients. The parameter J stands for the pendulum and the rotor inertias; the parameter ku is related to the amplifier gain and to the constant motor torque. Finally, Z is the unknown time varying disturbance. We underscore that in many cases found in practice the disturbance Z, and the gravitational and Coulomb friction torques are unknown. On the other hand, most of the amplifiers used in practice only accept smooth control voltages and has a relatively low bandwidth; thus, they could not withstand the switching signals produced by a classic sliding mode controller. In this context, we solve the trajectory tracking control problem, assuming that we do not know, both the velocity position y and the In this section we first develop a robust control scheme based on the approximation of the sign function. Then, we introduce a robust observer able to estimate, in finite-time, the motor velocity. Finally, we propose an output-feedback controller that solves the trajectory tracking control problem for the DCMP. Finally, the convergence analysis of the closed-loop system is carried out using the Lyapunov method. 3.1. Robust control scheme In this section we propose a simple control scheme, which emulates the robust behavior of a first order sliding mode controller. To this end, we approximate the sign function by means of a saturation function in conjunction with a first order filter. The saturation function is a smooth approximate to the sign function; the filter attenuates the high frequencies produced by the saturation function. Consider the following system: _ y ¼ ÀayÀv þ rðy,tÞ, ð6Þ where a 4 0 is a constant, y A R and v A R are the state and the input. Function rðnÞ A R is unknown and continuous satisfying 9rðy,tÞ9 r r, for a known constant r and for all, t Z 0. It is well known that if we select v as v ¼ k sign½yŠ, k 4r, ð7Þ
  3. 3. ˜ C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807 then the state y globally converges to the origin in finite-time. Unfortunately, in practice using this scheme would produce chattering together with its undesirable effects. A way to emulate the robust property of the sliding-mode method consists of using a nonlinear smooth saturation together with a first order filter. Let define the following auxiliary signal: s ¼ vÀk tanhðLyÞ, ð8Þ 1 where L 4 1 is a constant large enough. Intuitively, if s % 0, then v % k sign½yŠ implying that all the solutions of the system (6) will be confined to moved close enough to the origin. In order to design a filter that emulates the behavior of the sign function, we compute the time derivative of s which produces 2 _ s ¼ ur þ kL sech ðLyÞðay þvÞ þ Rðy,tÞ: ð9Þ _ In this last dynamics, ur ¼ v plays the role of a control signal and 2 Rðy,tÞ ¼ kL sech ðLyÞrðy,tÞ is the uncertainty with 9Rðy,tÞ9 rkLr. Hence, to make the variable s to move inside a small neighborhood of the origin, we propose ur as 2 ur ¼ ÀK M sÀkL sech ðLyÞðay þvÞ, ð10Þ where K M 4 0. Consequently, substituting (10) into (9), we have the following stable dynamics: _ s ¼ ÀK M s þ Rðt,yÞ: 803 where k ¼ br, K M 4 bLr 2 and b,L 41. Then, y(t) is uniformly ultimately bounded with the estimation bound given by (11). & Comment 1. If we directly introduce in Eq. (6) v ¼ k tanhðLyÞ with k4 r and L 40 large enough, evidently the closed-loop solutions are confined to move inside of a small origin vicinity, but having the presence of chattering phenomena. We underscore that the proposed control scheme can be seen as a nonlinear version of a PI controller, because the auxiliary variable v adds an integral term. Remark 1. As mentioned in the Introduction, there are several SMO based control methods with application results especially in motor control. However, the majority of these works are based on the usage of function sign, which involves some difficulties such as the sign of zero is not well defined or high frequencies are present in the control. These inconveniences make very difficult to build an actual implementation of these methods. In our control scheme we approximated the sign function by means of continuous and derivable functions (see the auxiliary signal ‘‘s’’ defined in (8)). Effectively, this solution brought a small stationary error, however it can be as small as desired. On the other hand, the high frequency problem can be overcome by introducing auxiliary high order time derivatives of s. For instance, the performance of our control strategy can be improved if we assure _ € that the auxiliary signals s, s and s are almost equal to zero. The above implies the following inequality when t-1: 9sðtÞ9 r eÀK M T 9sð0Þ9 þ 3.2. State observer kLr kLr ð1ÀeÀK M T Þ r K c 9sð0Þ9 þ ¼ : KM KM If the initial conditions s(0) are set to zero, then the following inequality holds 9sðtÞ9 r r M ¼ kLr : KM Now, to compute the final value of y, in (6) we use the following positive function: V ¼ 1y2 , 2 whose time derivative evaluated along the solutions of (6) leads to _ V ¼ yðÀayÀk tanhðLyÞÀs þ rðy,tÞÞ, _ where s is defined in (8). Hence, V can be upper bounded by: _ V r 9y9ðÀa9y9Àk9 tanhðLyÞ9 þr þ 9s9Þ: Since the values of 9s9 continuously decrease until 9sðtÞ9 rr M for some t 4 T, then we have kLr _ , t 4T, V ðtÞ r9y9 Àa9y9Àk9 tanhðLyÞ9 þ r þ KM _ which implies that V is negative outside of BM ¼ fyA R : 9y9 rMg, where M is the single root of kLr pðyÞ c a9y9 þ ktanhðL9y9ÞÀrÀ ¼ , KM ð11Þ that is pðMÞ ¼ 0. Therefore, y is bounded and converges to the smallest level set of V that includes BM. In other words, there is a finite time T after which y(t) is confined to move inside the compact set BM. The following lemma summarizes the stability result previously presented: Lemma 1. Consider the scalar system (6) with 9rðy,tÞ9 r r in closed loop with the nonlinear controller (10) 2 _ v ¼ ÀK M ðvÀk tanhðLyÞÞÀkL sech ðLyÞðay þvÞ, 1 For simplicity, we use the tanh(s) as a smooth nonlinear saturation function; however, we can use any kind of smooth saturation function. In order to estimate the velocity x2 in (3) in a short period, we use the super-twisting based observer (STBO) [4], defined by the following second order system: _ b 1 ¼ Àa1 9e1 91=2 sign½e1 Š þ b2 , x x _ b 2 ¼ Àa2 sign½e1 Š þf 0 b2 þ u, x x x e1 ¼ b1 Àx1 : ð12Þ x Variables b1 and b2 are, respectively, the estimates of the state x variables x1 and x2; a1 and a2 are strictly positive constants. Defining the observation errors as e1 ¼ b1 Àx1 and e2 ¼ b2 Àx2 , the x x error equation takes the following form: 1=2 _ sign½e1 Š þ e2 , e 1 ¼ Àa1 9e1 9 _ e 2 ¼ Àa2 sign½e1 ŠÀf 0 e2 þw, ð13Þ where wðÁÞ can be considered as the system perturbation and is defined in (2). From the definition of (2), we have that the above term can be bounded by: 9w9 o d0 c ¼ 1 ðZ þgmL þf c Þ: J ð14Þ The following theorem gives a tuning rule for the observer gains. Lemma 2. Under condition (14) and assuming that a1 and a2 satisfy the following inequalities: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða1 þ d0 Þð1 þ pÞ , a1 4 d0 , a2 4 ð1ÀpÞ a1 Àd0 with 0 o p o 1, then the observer (12) assures finite time convergence of the estimated states ðb1 , b2 Þ to the actual states ðx1 ,x2 Þ. x x The corresponding proof can be found in [4,6,30,26]. 3.3. Perturbation estimation and robust controller In this section we focus our attention on a robust control strategy to compensate the effects of the perturbation w appearing in (3). Lemma 2 assures that, after a finite time, the estimated state variables ðb1 , b2 Þ converge to the corresponding actual values x x
  4. 4. ˜ C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807 804 is able to make s being stable and ultimately bounded in a finite time, T 4 0. Besides, variable s remains inside the compact set: Bm ¼ fs A R : 9s9 r mg, where m is the single root of: ðx1 ,x2 Þ. Hence, the separation principle holds and the controller can be designed without taking into account the observers dynamics. Therefore, the stability analysis of the system (3) together with the observer (12) is equivalent to examine the stability of the system (3), for the case where b2 ¼ x2 . The formal x and detailed stability analysis of the closed-loop system can be found in the Appendix. Let us consider the smooth reference signal xr and its time _ _ derivatives x r and x r . We propose the following tracking control law: We must underscore that b is required to be slightly larger than one, while L being much more greater than one. We end this section introducing some assumptions for establishing Proposition 1, which resumes the main result of this work. b _ € u ¼ ÀwÀkp ðx1 Àxr ÞÀkd ðx2 Àx r Þ þ f 0 x2 þ x r , Assumptions. ð15Þ pðxÞ ¼ ki 9x9 þ br tanhðL9x9ÞÀrÀ Lbr 2 : KM ð25Þ b where kp 4 0 and kd 4 0 are the positive constants, w is an auxiliary signal that allows us to compensate for the effects of the disturbance w. Therefore, substituting (15) into (3), we have the following closed-loop system: (A1) b2 is obtained by using the STBO (12). x (A2) The set of positive control parameters fL,K M ,ki , b, lg is selected according to the previous discussion. _ x 2 ¼ x2 , _ 2 ¼ ue ðx1 ,x2 , b2 , wÞ þw, x b x Proposition 1. Consider the uncertain DCMP system (3), under the above assumptions, and in closed-loop with ð16Þ where € b ue ðx1 ,x2 , b2 , wÞ ¼ Àkp ðx1 Àxr ÞÀkd ðb2 Àx r Þ þ x r Àf 0 ðx2 Àb2 ÞÀw: x b x _ x We must clarify that the solutions of the system (3) under the control law (15) are understood in Filippov’s sense [31], because the discontinuities appearing in the closed loop system. _ Define the tracking errors x1r ¼ x1 Àxr and x2r ¼ x2 Àx r . Using the above definitions, and substituting (15) into (3), it is not difficult to show that the following equations describe the tracking errors dynamics: _ x 1r ¼ x2r , b _ x 2r ¼ Àkp x1r Àðkd þ f 0 Þx2r Àw þ w, ð17Þ Let us define the following auxiliary variable: s ¼ x2r þ lx1r , ð18Þ where l 4 0. Evidently, if we assure that s converges to zero, or at least remains close enough to zero, then the tracking errors stay very close to zero too. Computing the time derivative of s along the trajectories of (17), and using (17) we have, after some simple algebra, the following: k b _ s ¼ Àki x2r þ p x1r Àw þw, ð19Þ ki where: ki ¼ kd þ f 0 Àl 4 0: ð20Þ Hence, fixing l ¼ kp =ki it is not difficult to show that the parameter l is given by l¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kd þ f 0 À ðkd þ f 0 Þ2 À4kp 2 ð21Þ ð23Þ Notice that control parameters kp and kd have to be selected taking into account ðkd þ f 0 Þ2 4 4kp to assure the feasibility of the parameters ki and l, respectively. Therefore, applying Lemma 1 to (22), we can assure that the following controller: 2 _ b b b w ¼ ÀK M ðwÀk tanhðLsÞÞ þkL sech ðLsÞðki s þ wÞ, where r ¼ d0 , Remark 2. Our control strategy is based on the assumption that the parameter f0 is known. However, if this parameter was unknown, then it can be easily estimated online or identified by some identification method, like the one found in [4,6]. In general, the parameter value is very small in comparison with the actual kd, and can be neglected. We want to point out that our control strategy was developed ad hoc to be applied to an uncertain second order nonlinear system, based on using the traditional PI controller with a correction or a compensation term, which emulates a smooth sliding mode controller. But we should not forget that there are more general strategies which have been successfully designed for systems with more than two state variables. 4. Numerical and experimental results This section shows the effectiveness of the proposed controller applied to the output-feedback trajectory tracking problem of the DCMP. To this end we carried out a numerical simulation and an experiment using a laboratory prototype. 4.1. Numerical simulation f 0 ¼ 0:1, where k ¼ br, where ðkd þf 0 Þ 44kp . Then, the closed-loop system is uniformly ultimately bounded, with the estimation bound given by (25). ð22Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ki ¼ 1 ðkd þ f 0 Þ þ 1 ðkd þf 0 Þ2 À4kp : 2 2 2 b,L 4 1, K M 4Lbr , a ¼ ki ð26Þ 2 The physical parameters of the DCMP are set to the following values: with ðkd þf 0 Þ2 44kp . So, system (19) can be written as b _ s ¼ Àki sÀw þw, b b € b _ u ¼ Àkp ðx1 Àxr ÞÀkd ðx 2 Àx r Þ þ f 0 x 2 Àw þ x r , d 2 b b b w ¼ ÀK M ðwÀk tanhðLsÞÞ þ kL sech ðLsÞðki s þ wÞ, dt s ¼ ðb2 Àx r Þ þ lðx1 Àxr Þ, x _ ð24Þ fc ¼ 1:3, J Z J ¼ 0:84, gmL ¼ 14:03, J ku ¼ 5:4: J These values are close to the one proposed in [7]. In order to make the experiment more interesting, we added an external perturbation to the parameter Z=J, indeed Z=J ¼ 0:84 þ 0:2 sinðt=5Þ. The objective is to track the reference xr ¼ sinðt=2Þ; for this system, the bound in the disturbance is fixed to r ¼ 15:32. Hence, the control parameters are set as b ¼ 4, kp ¼ 16, kd ¼ 8:4, L ¼ 40, K M ¼ 12,000, l ¼ 2:81: According to (25), it is quite easy to show numerically that the manifold s satisfies 9s ¼ x2r þ lx1r 9 o 8 Â 10À3 , which implies that the steady state error is given by 9x1r 9 r 2:3 Â 10À3 rad. Notice
  5. 5. Velocity [rad/s] Position [rad] ˜ C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807 2 -2 x 0.2 0 2 4 6 8 10 12 14 16 18 6 4 2 0 -2 0.24 20 x2 vr 0.1 xr 0.238 0 0 Input [Nw m] Reference output x1 xr 0 805 2 4 6 8 10 12 14 16 18 20 10 -0.1 τ -0.2 0 -10 0 2 4 6 8 10 12 Time [sec] 14 16 18 20 x 10-3 Output tracking error Fig. 1. Closed-loop responses of the uncertain DC servo pendulum. [rad] 2 x 10 2 Tracking position error -3 0 -2 [rad/s] 2 6 8 -4 5 x 10 10 12 14 Tracking velocity error 16 18 20 0 -2 0 -5 2 [rad/s] 4 4 6 8 10 12 14 Observed velocity error 16 18 20 -4 2 x 10 Control action 1 0 -2 2 4 6 8 10 12 Time [sec] 14 16 18 20 0.5 Fig. 2. State errors of position, velocity and observed velocity, from 2 to 20.5 s. 0 -0.5 -1 0 5 10 15 Time [sec] Fig. 4. Closed-loop response to the proposed control law (26), when applied to the laboratory prototype. of the reference. In Fig. 2, the tracking position, velocity position and observed velocity errors are illustrated in steady state. For this case, we only show the simulations from 2 s to 20.5 s. 4.2. Experimental result Fig. 3. DC servo pendulum prototype used to carried out the actual experiments. that for this case d0 ¼ 17:4. On the other hand, according to Lemma 2, the STBO parameters were fixed as a1 ¼ 3:7, a2 ¼ 41:6, p ¼ 0:5: The DCMP initial conditions are chosen as x1 ¼ À1:57 rad and x2 ¼ 0 rad=s, the corresponding state variables of the STBO are set at the origin. The simulation was carried out by using the Runge– Kutta algorithm with a sampling integration interval of, 1 Â 10 À 4 s. Fig. 1 shows the closed-loop response of the whole state. From this figure, we can see that the proposed controller effectively makes the pendulum to follow the reference signal xr ¼ sinðt=2Þ after one second. Moreover, the state x2 tracks the time derivative In order to carry out this experiment we used a laboratory prototype, which consists of a DC servomotor from Moog, model C34-L80-W40 (Fig. 3) driven by a Copley Controls power servoamplifier model 423 configured in current mode. A BEI optical encoder model L15 with 2500 pulses per revolution allows measuring the servomotor position. The algorithms are coded using the MatLab/ Simulink software platform under the program Wincon from Quanser Consulting, and a Quanser Consulting Q8 board performs data acquisition. The data card electronics increases four times the optical encoder resolution up to 2500 Â 4¼10,000 pulses per revolution. The control signal produced by the Q8 board passes through a galvanic isolation box. The Q8 board is allocated in a PCI slot inside a personal computer, which runs the control software.
  6. 6. ˜ C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807 806 and second time derivatives, as Reference output 0.24 0.2 € _ xð3Þ þ 3a x r þ 3a 2 x r þ a 3 xr ¼ a 3 r, r xr x 0.22 0.2 0.1 0.18 2.5 0 3 3.5 4 -0.1 where, a ¼ 20, and, r ¼ 0:25 sinðptÞ. Fig. 4 shows the reference and the DCMP output, the tracking position error when the proposed control law (26) was used. Fig. 5 b shows the same results but in this case the compensator w is set to zero. From these figures it is clear that the proposed control law effectively compensates the uncertain term, w, and tracks the reference signal with an error position ranging in, 10 À 3 rad. -0.2 5. Conclusions This work proposes a new output feedback control scheme for solving the stabilization and trajectory tracking problem in an uncertain DC servomechanism system. A Proportional Derivative algorithm plus a robust uncertain compensator composes the control law. A super-twisting second-order sliding-mode observer recovers the pendulum velocity. Owing to the finite time convergence of this observer, the Separation Principle holds thus making possible to develop the control law without taking into account the observer dynamics. The effectiveness of the controller is tested by performing a numerical simulation, and an experiment using a laboratory prototype. These results show that the proposed algorithm is able to compensate for the disturbances affecting the DC servomechanism system without generating chattering in the control signal. As a future extension of the obtained control scheme, it can be generalized to a higher order system. Output tracking error 0.01 0 Control Action 1 ^ Appendix A. Finite time convergence of x 2 to x2 0.5 Proof. First of all notice that the output-feedback controller (15) with (3) is given by _ x 1 ¼ x2 , _ x 2 ¼ Àf 0 x2 þ wþ ua , 0 ð27Þ where ua is in fact the actual controller: b € ua ¼ Àwðx1 , b2 ÞÀkp ðx1 Àxr ÞÀkd ðb2 Àx r Þ þ f 0 b2 þ x r : x x _ x -1 0 5 b Remember that w, kp and kd are control parameters, previously defined. Now, by simple algebra, from (27), (28) and (12) we have the following dynamic error: 10 Time [sec] Fig. 5. Closed-loop response to the classical PD controller, when applied to the laboratory prototype. The physical parameters are fixed as f 0 ¼ 0:13, fc ¼ 1:27, J Z J ¼ À0:17, gmL ¼ 14:21, J ku ¼ 18:56: J These parameters were previously identified in Ref. [7]. For this case, the uncertainty in the pendulum model is given by 9wðx,tÞ9 rr ¼ 16. The controller parameters used during the experiments are k ¼ 20, kp ¼ 1600, kd ¼ 80, L ¼ 15, K M ¼ 6000, ð28Þ l ¼ 38:04: From (25), we have that s satisfies 9s ¼ x2r þ lx1r 9 o 0:06, implying that the steady state error is given by 9x1r 9 r 1:5  10À3 rad. The STBO parameters were fixed as, a1 ¼ 25, and, a2 ¼ 47. The reference applied to the closed loop system consists of a filtered sinusoid. The filter allows generating the reference and its first _ b 1 ¼ Àa1 9e1 91=2 sign½e1 Š þ b2 , e e _ b 2 ¼ Àa2 sign½e1 Š þ f 0 b2 þ w, e e where 9w9 o d0 . So, selecting a1 and a2 according to Lemma 2, we can always assure that e1 and e2 asymptotically converge to zero in a finite time. In other words, b2 ¼ x2 þ fðtÞ, with fðtÞ being a x continuous functions, and provided that limt-tf fðtÞ ¼ 0, with tf as short as needed. Therefore, the actual controller can be read as b € ua ¼ Àwðx1 , b2 ÞÀkp x1r Àðkd þf 0 Þx2r þ x r þ ðkd þ f 0 ÞfðtÞ: x Now, we must remember that ua is proposed such that the values _ of s and s are always almost zero. Therefore, computing the time derivative of s along the trajectories of (27) and (28), we have b _ s ¼ Àki sÀw þ w þðkd þ f 0 ÞfðtÞ, where ki is defined in (20) and fðtÞ converges to zero in a short period of time. This discussion justify that the filter variable b2 is x not needed to carry out the stability analysis. It means that b2 is x equivalent to x2. Formally, the last equation converts to b _ s ¼ Àki sÀw þ w,
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