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Observadores GPI DOB PID

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Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 1 Center for Research and Advanced Studies- National Polytechnic Institute (CINVESTAV-IPN) Position control of a linear ultrasonic motor: An active disturbance rejection approach. Jose Luis Luna CINVESTAV-Departamento de Control Automático, México México City, México November 22, 2022
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 2 JOSE LUIS LUNA PINEDA • I received the B.Sc. (2013) in Electrical Engineering from the Instituto Tecnologico de Pachuca, Hidalgo, Mexico. • The M.Sc. and the Dr. Sc (2017 and 2021) in Automatic Control from the Center for Research and Advanced Studies (CINVESTAV), Mexico City, Mexico. • Professor, Anahuác University, State of Mexico (Mexico). Teaching classes of Industrial Robotics with UR5, Fanuc and Mitsubishi Robots, Classical Control Theory, Basic Mathematics, Advanced Linear Algebra, Discrete Mathematics, Numerical Methods. • Professor at National Polytechnic Institute, Mexico City, Mexico. Teaching classes of Physics, Arithmetic, Algebra, Statistics, Probability, Geometry, Trigonometry, Differential and Integral Calculus.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 3 Research interest: • Design of Active Disturbance Rejection Control. • Proportional Integral Derivative controllers. • Parameter identification and Time Delay Based-Control applied to servodrives, Linear Piezoelectric Ultrasonic Motors and robots. • Adaptive and Chaos control for under-actuated mechanical systems. • Real Time Control using Matlab-Simulink R .
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 4 Outline 1 Introduction 2 Linear ultrasonic motors 3 Model of a Linear ultrasonic motor 4 Parametric Identification 5 DOB applied to a Linear Ultrasonic Motor 6 GPI observer applied to a Linear Ultrasonic Motor 7 Experiments 8 Conclusions 9 References
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 5 Ultrasonic Motors (a) Linear Ultrasonic Motors (LUM). Figure 1: Ultrasonic Motors (UMs). UMs base their working principle on • The inverse piezoelectric effect. • The ability of piezoelectric materials to vibrate in the ultrasonic frequency. • The movement is obtained by means of the frictional contact force between the fixed and mobile parts.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 6 Some applications of Ultrasonic Motors (a) Satellites (b) Nanotechnology (c) Cellular puncture Figure 2: Ultrasonic motors used in a tumor ablation instrument that operates inside a magnetic resonance instrument that has a magnetic field of 3T.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 7 Principle of operation of Linear ultrasonic motor Figure 3: Principle of operation.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 8 The control strategies applied to the UMs reported in the literature are: • Proportional Integral Derivative (PID) Control law with fixed and variable gains. • Adaptive control techniques. • Controllers based on neural networks. • Sliding modes control.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 9 Mathematical Model of a Linear ultrasonic motor Mÿ + Kfvẏ = Kf (VIN − Voffset) + ϕ(t) (1) where VIN := u + V̂offset. Alternative written of the model ÿ + aẏ = bu + ˜ d(t) (2) where the following terms are defined a := Kfv M b := Kf M ˜ d(t) := ¯ d(t) M ¯ d(t) := Kf (V̂offset − Voffset) + ϕ(t) If the friction torques are unknown, then they are lumped with the disturbance ¯ d. Simplified Model of a LUM ÿ = bu + d(t) where d(t) = ˜ d(t) − aẏ (3)
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 10 The main features of the HR4-K-S3 motor from NANOMOTION-LTD are the following: • It is bidirectional and with direct contact between the stator and the moving part. • Motor speed limit 250 mm/s = 25 cm/s. • The nominal platform preload is 72 N. • Optical linear encoder Renishaw with resolution of 50 nm. • The HR4-K-S3 powered by an AB5 amplifier, configured in servo mode, which generates a sine wave at 39.6 KHz whose amplitude is a function of the control signal that operates in a range of ±10 V.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 11 Parametric Identification Model used for parametric identification without the term d(t) ÿ + aẏ = bu (4) To perform the parametric identification the Least Squares method was used θ̂ = (A> A)−1 A> z (5) where θ̂ = â b̂ is an estimate of θ = a b . A =    −ẏf (t1) uf (t1) . . . . . . −ẏf (tn) uf (tn)    z =    ÿf (t1) . . . ÿf (tn)    (6)
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 12 Experimental setup Control Computer Position Measurement Linear Ultrasonic Motor Position Sensor Galvanic Insulator Differential Analog Input Connection Box Driver AB5 Voltage Source 24 V Data Acquisition Card Control Signal ± 10 V Linear Ultrasonic Motor With Coupled Load Control Signal ± 10 V Maximal Voltage 270 Vrms, 39.6 KHz, Sine Wave Figure 4: Experimental setup for the LUM The control algorithms are implemented using the MATLAB/Simulink programming platform under the QUARC real-time control envi- ronment and a QPID data acquisition card, the latter two of the Quanser Consulting brand.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 13 Experimental results A Proportional-Derivative (PD) control is used to stabilize the LUM in closed loop without prior knowledge its parameters. u = Kpe − Kdẏe (7) where e := r − y. The following transfer function G1(s) allows obtaining estimates of the velocity ẏe from the position measurements y. G1(s) = 450s s + 450 400 s + 400 (8) • A mass of 1 kg was added to the motor as a load. Kp Kd Estimate â of a Estimate b̂ of b 1500 25 56.6982 1.0213 Table 1: Experimental results of the parametric identification.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 14 Disturbance Observer (DOB)-based control • Disturbance Observer is employed for rejecting internal and external disturbances acting on a plant. • It relies on input and output measurements on a nominal model of a perturbed plant to estimate the disturbances [Ohishi et al., 1988, Ohnishi et al., 1996]. • The disturbance estimate is injected to the plant input to counteract the effects of the real disturbance.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 15 Dynamic model of the LUM s2 Y (s) = bU(s) + D(s) ⇒ D(s) = s2 Y (s) − bU(s) The disturbance estimation is performed as follows D̂(s) = s2 Y (s) − bU(s) F(s) = sY (s) sβ s + β − U(s) bβ s + β (9) The DOB filter is defined as F(s) = β s + β where β 0 + 1 s b 1 s b s b b + s s b b + + - Linear Ultrasonic Motor - DISTURBANCE OBSERVER - + PD CONTROLLER + - 1 b p K d K + Y(s) ࢙૛Y(s) sY(s) ࡰ ෡ሺ࢙ሻ ࡰሺ࢙ሻ E(s) U(s) R(s) Figure 5: PD+DOB controller block diagram.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 16 Generalized Proportional Integral (GPI) Observer • GPI observers are based on the exact linearization of a state space model [Sira-Ramírez et al., 2010]. • GPI Observer is employed for rejecting internal and external disturbances acting on a plant. • Is a high-gain linear observer that estimates the variables related to the plant. • It has the ability to estimate on-line the disturbance and a certain number of its time-derivatives. • The disturbance estimate is injected to the plant input to counteract the effects of the real disturbance.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 17 State-space expression for the LUM ẋ1 = ẏ = x2 ẋ2 = ÿ = bu + d(t) (10) Considering d(t) as a new state, and assuming that its time deriva- tive exists ẋ1 = x2 ẋ2 = bu + x3 ẋ3 = ϕ(t) (11) Thus, the GPI observer takes the form ẋ1e = x2e + k1(x1 − x1e) ẋ2e = b̂u + x3e + k2(x1 − x1e) ẋ3e = k3(x1 − x1e) (12) Setting k1 = 3θ, k2 = 3θ2 , k3 = θ3 yields the next GPI char- acteristic polynomial s3 + 3θs2 + 3θ2 s + θ3 (13)
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 18 Experimental setup • The reference is a filtered pulse train of 1 mm of amplitude. • The performance is measured using the Integral Squared Error (ISE), the Integral of the Absolute value of the Control (IAC) and the Integral of the Absolute value of the Control Variation (IACV ) indexes, which are evaluated at T = 5s. ISE = R T 0 100 [e(t)] 2 dt, IAC = R T 0 |u(t)| dt IACV = R T 0 du(t) dt dt
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 19 Experiments PD+DOB control law U(S) = 1 b h KpE(s) − KdsY (s) − D̂(s) i D̂(s) = sY (s) sβ s + β − U(s) bβ s + β The corresponding time domain expression is uDOB = 1 b̂ h Kp(r − y) − Kdẏe − ˆ d(t) i PD+GPI observer control law uGP I = 1 b̂ [Kp(r − x1) − Kdx2e − x3e] (14)
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 20 PD+DOB and PD+GPI controllers. Figure 6: Responses of the PD+DOB and PD+GPI controllers.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 21 (a) Error position. (b) Control signal.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 22 Table 2: Experimental results for regulation using Kp = 4225 and Kd = 130. Error in steady state ±essp is equivalent to 50 nanometers per pulse. Controller +essp −essp β ISE IACV IAC PD+DOB -1 0 20 3.1709 6.7716 0.7364 +ess −ess θ ISE IACV IAC PD+GPI -1 2 90 5.2373 6.7530 0.3275 +ess −ess ISE IACV IAC PD 995 -1594 5.9649 7.5324 1.0106
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter. • An important issue is that in the case of the GPI observer velocity and disturbance are simultaneously estimated, which could be seen as an advantage.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter. • An important issue is that in the case of the GPI observer velocity and disturbance are simultaneously estimated, which could be seen as an advantage. • In the case of the DOB, the dynamics of the disturbance and the velocity estimation are decoupled. Note also that the value of cutoff frequency β in the case of DOB filter is lower than the gain value θ used in the GPI observer. Therefore the peaking phenomenon in the DOB is less likely to happen.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter. • An important issue is that in the case of the GPI observer velocity and disturbance are simultaneously estimated, which could be seen as an advantage. • In the case of the DOB, the dynamics of the disturbance and the velocity estimation are decoupled. Note also that the value of cutoff frequency β in the case of DOB filter is lower than the gain value θ used in the GPI observer. Therefore the peaking phenomenon in the DOB is less likely to happen.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 24 References I [Ohishi et al., 1988] Ohishi, K., Ohnishi, K., and Miyachi, K. (1988). Adaptive dc servo drive control taking force disturbance suppression into account. IEEE Transactions on Industry Applications, 24(1):171–176. [Ohnishi et al., 1996] Ohnishi, K., Shibata, M., and Murakami, T. (1996). Motion control for advanced mechatronics. IEEE/ASME Transactions On Mechatronics. [Sira-Ramírez et al., 2010] Sira-Ramírez, H., Ramírez-Neria, M., and Rodríguez-Angeles, A. (2010). On the linear control of nonlinear mechanical systems. In Decision and Control (CDC), 2010 49th IEEE Conference on. IEEE. [Zhao, 2011] Zhao, C. (2011). Ultrasonic motors: technologies and applications. Springer Science Business Media.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 25 Journal Publications • Robust ultra-precision motion control of linear ultrasonic motors: A combined ADRC-Luenberger observer approach. Control Engineering Practice Vol 111, 2021. Status: Accepted and Published. • A Teaching Methodology Based on an Educational Experimental Platform. IEEE Latin America Transactions (ISSN: 1548-0992) Vol 17, Issue 08, 2019, Status: Accepted and Published. • Delay-based nonlinear controllers for motion control: ultrasonic motor application. Status: Under Review. • Fast Parameter Identication of Perturbed Servo Systems: A Least Squares of Orthogonal Distances Approach. Status: Under Review. • Adaptive Active Disturbance Rejection Control for Velocity Tracking Trajectory Tasks With Noise-Free Desired Values. Status: Under Review.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 26 WORKSHOPS CONFERENCES • On the equivalence between PD+DOB and PID controllers applied to servo drives. 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control PID 2018. • Position control of a linear ultrasonic motor: An active disturbance rejection approach. 15th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) 2018. • Parameter Estimation of a Linear Ultrasonic Motor Using the Least Squares of Orthogonal Distances Algorithm. 16th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) 2019. • Micro-posicionamiento de un motor piezoelectrico ultrasónico lineal basado en un controlador PID. Congreso Internacional en Robótica y Computación, 2017. • Micro-posicionamiento de un motor piezoeléctrico ultrasónico lineal basado en Observadores Proporcionales Integrales Generalizados. Congreso Nacional de Control Automatico 2018.
Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 27 Thanks!

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Presentacion CCE_2018.pdf

  • 1. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 1 Center for Research and Advanced Studies- National Polytechnic Institute (CINVESTAV-IPN) Position control of a linear ultrasonic motor: An active disturbance rejection approach. Jose Luis Luna CINVESTAV-Departamento de Control Automático, México México City, México November 22, 2022
  • 2. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 2 JOSE LUIS LUNA PINEDA • I received the B.Sc. (2013) in Electrical Engineering from the Instituto Tecnologico de Pachuca, Hidalgo, Mexico. • The M.Sc. and the Dr. Sc (2017 and 2021) in Automatic Control from the Center for Research and Advanced Studies (CINVESTAV), Mexico City, Mexico. • Professor, Anahuác University, State of Mexico (Mexico). Teaching classes of Industrial Robotics with UR5, Fanuc and Mitsubishi Robots, Classical Control Theory, Basic Mathematics, Advanced Linear Algebra, Discrete Mathematics, Numerical Methods. • Professor at National Polytechnic Institute, Mexico City, Mexico. Teaching classes of Physics, Arithmetic, Algebra, Statistics, Probability, Geometry, Trigonometry, Differential and Integral Calculus.
  • 3. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 3 Research interest: • Design of Active Disturbance Rejection Control. • Proportional Integral Derivative controllers. • Parameter identification and Time Delay Based-Control applied to servodrives, Linear Piezoelectric Ultrasonic Motors and robots. • Adaptive and Chaos control for under-actuated mechanical systems. • Real Time Control using Matlab-Simulink R .
  • 4. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 4 Outline 1 Introduction 2 Linear ultrasonic motors 3 Model of a Linear ultrasonic motor 4 Parametric Identification 5 DOB applied to a Linear Ultrasonic Motor 6 GPI observer applied to a Linear Ultrasonic Motor 7 Experiments 8 Conclusions 9 References
  • 5. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 5 Ultrasonic Motors (a) Linear Ultrasonic Motors (LUM). Figure 1: Ultrasonic Motors (UMs). UMs base their working principle on • The inverse piezoelectric effect. • The ability of piezoelectric materials to vibrate in the ultrasonic frequency. • The movement is obtained by means of the frictional contact force between the fixed and mobile parts.
  • 6. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 6 Some applications of Ultrasonic Motors (a) Satellites (b) Nanotechnology (c) Cellular puncture Figure 2: Ultrasonic motors used in a tumor ablation instrument that operates inside a magnetic resonance instrument that has a magnetic field of 3T.
  • 7. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 7 Principle of operation of Linear ultrasonic motor Figure 3: Principle of operation.
  • 8. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 8 The control strategies applied to the UMs reported in the literature are: • Proportional Integral Derivative (PID) Control law with fixed and variable gains. • Adaptive control techniques. • Controllers based on neural networks. • Sliding modes control.
  • 9. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 9 Mathematical Model of a Linear ultrasonic motor Mÿ + Kfvẏ = Kf (VIN − Voffset) + ϕ(t) (1) where VIN := u + V̂offset. Alternative written of the model ÿ + aẏ = bu + ˜ d(t) (2) where the following terms are defined a := Kfv M b := Kf M ˜ d(t) := ¯ d(t) M ¯ d(t) := Kf (V̂offset − Voffset) + ϕ(t) If the friction torques are unknown, then they are lumped with the disturbance ¯ d. Simplified Model of a LUM ÿ = bu + d(t) where d(t) = ˜ d(t) − aẏ (3)
  • 10. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 10 The main features of the HR4-K-S3 motor from NANOMOTION-LTD are the following: • It is bidirectional and with direct contact between the stator and the moving part. • Motor speed limit 250 mm/s = 25 cm/s. • The nominal platform preload is 72 N. • Optical linear encoder Renishaw with resolution of 50 nm. • The HR4-K-S3 powered by an AB5 amplifier, configured in servo mode, which generates a sine wave at 39.6 KHz whose amplitude is a function of the control signal that operates in a range of ±10 V.
  • 11. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 11 Parametric Identification Model used for parametric identification without the term d(t) ÿ + aẏ = bu (4) To perform the parametric identification the Least Squares method was used θ̂ = (A> A)−1 A> z (5) where θ̂ = â b̂ is an estimate of θ = a b . A =    −ẏf (t1) uf (t1) . . . . . . −ẏf (tn) uf (tn)    z =    ÿf (t1) . . . ÿf (tn)    (6)
  • 12. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 12 Experimental setup Control Computer Position Measurement Linear Ultrasonic Motor Position Sensor Galvanic Insulator Differential Analog Input Connection Box Driver AB5 Voltage Source 24 V Data Acquisition Card Control Signal ± 10 V Linear Ultrasonic Motor With Coupled Load Control Signal ± 10 V Maximal Voltage 270 Vrms, 39.6 KHz, Sine Wave Figure 4: Experimental setup for the LUM The control algorithms are implemented using the MATLAB/Simulink programming platform under the QUARC real-time control envi- ronment and a QPID data acquisition card, the latter two of the Quanser Consulting brand.
  • 13. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 13 Experimental results A Proportional-Derivative (PD) control is used to stabilize the LUM in closed loop without prior knowledge its parameters. u = Kpe − Kdẏe (7) where e := r − y. The following transfer function G1(s) allows obtaining estimates of the velocity ẏe from the position measurements y. G1(s) = 450s s + 450 400 s + 400 (8) • A mass of 1 kg was added to the motor as a load. Kp Kd Estimate â of a Estimate b̂ of b 1500 25 56.6982 1.0213 Table 1: Experimental results of the parametric identification.
  • 14. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 14 Disturbance Observer (DOB)-based control • Disturbance Observer is employed for rejecting internal and external disturbances acting on a plant. • It relies on input and output measurements on a nominal model of a perturbed plant to estimate the disturbances [Ohishi et al., 1988, Ohnishi et al., 1996]. • The disturbance estimate is injected to the plant input to counteract the effects of the real disturbance.
  • 15. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 15 Dynamic model of the LUM s2 Y (s) = bU(s) + D(s) ⇒ D(s) = s2 Y (s) − bU(s) The disturbance estimation is performed as follows D̂(s) = s2 Y (s) − bU(s) F(s) = sY (s) sβ s + β − U(s) bβ s + β (9) The DOB filter is defined as F(s) = β s + β where β 0 + 1 s b 1 s b s b b + s s b b + + - Linear Ultrasonic Motor - DISTURBANCE OBSERVER - + PD CONTROLLER + - 1 b p K d K + Y(s) ࢙૛Y(s) sY(s) ࡰ ෡ሺ࢙ሻ ࡰሺ࢙ሻ E(s) U(s) R(s) Figure 5: PD+DOB controller block diagram.
  • 16. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 16 Generalized Proportional Integral (GPI) Observer • GPI observers are based on the exact linearization of a state space model [Sira-Ramírez et al., 2010]. • GPI Observer is employed for rejecting internal and external disturbances acting on a plant. • Is a high-gain linear observer that estimates the variables related to the plant. • It has the ability to estimate on-line the disturbance and a certain number of its time-derivatives. • The disturbance estimate is injected to the plant input to counteract the effects of the real disturbance.
  • 17. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 17 State-space expression for the LUM ẋ1 = ẏ = x2 ẋ2 = ÿ = bu + d(t) (10) Considering d(t) as a new state, and assuming that its time deriva- tive exists ẋ1 = x2 ẋ2 = bu + x3 ẋ3 = ϕ(t) (11) Thus, the GPI observer takes the form ẋ1e = x2e + k1(x1 − x1e) ẋ2e = b̂u + x3e + k2(x1 − x1e) ẋ3e = k3(x1 − x1e) (12) Setting k1 = 3θ, k2 = 3θ2 , k3 = θ3 yields the next GPI char- acteristic polynomial s3 + 3θs2 + 3θ2 s + θ3 (13)
  • 18. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 18 Experimental setup • The reference is a filtered pulse train of 1 mm of amplitude. • The performance is measured using the Integral Squared Error (ISE), the Integral of the Absolute value of the Control (IAC) and the Integral of the Absolute value of the Control Variation (IACV ) indexes, which are evaluated at T = 5s. ISE = R T 0 100 [e(t)] 2 dt, IAC = R T 0 |u(t)| dt IACV = R T 0 du(t) dt dt
  • 19. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 19 Experiments PD+DOB control law U(S) = 1 b h KpE(s) − KdsY (s) − D̂(s) i D̂(s) = sY (s) sβ s + β − U(s) bβ s + β The corresponding time domain expression is uDOB = 1 b̂ h Kp(r − y) − Kdẏe − ˆ d(t) i PD+GPI observer control law uGP I = 1 b̂ [Kp(r − x1) − Kdx2e − x3e] (14)
  • 20. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 20 PD+DOB and PD+GPI controllers. Figure 6: Responses of the PD+DOB and PD+GPI controllers.
  • 21. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 21 (a) Error position. (b) Control signal.
  • 22. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 22 Table 2: Experimental results for regulation using Kp = 4225 and Kd = 130. Error in steady state ±essp is equivalent to 50 nanometers per pulse. Controller +essp −essp β ISE IACV IAC PD+DOB -1 0 20 3.1709 6.7716 0.7364 +ess −ess θ ISE IACV IAC PD+GPI -1 2 90 5.2373 6.7530 0.3275 +ess −ess ISE IACV IAC PD 995 -1594 5.9649 7.5324 1.0106
  • 23. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger.
  • 24. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter.
  • 25. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter. • An important issue is that in the case of the GPI observer velocity and disturbance are simultaneously estimated, which could be seen as an advantage.
  • 26. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter. • An important issue is that in the case of the GPI observer velocity and disturbance are simultaneously estimated, which could be seen as an advantage. • In the case of the DOB, the dynamics of the disturbance and the velocity estimation are decoupled. Note also that the value of cutoff frequency β in the case of DOB filter is lower than the gain value θ used in the GPI observer. Therefore the peaking phenomenon in the DOB is less likely to happen.
  • 27. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 23 Conclusions • Both controllers have an steady-state error in pulses equivalent to 50 nm. It is observed that the ISE index for the PD+DOB controller has a lower value, and both responses have an overshoot, in the case of PD+GPI controller the overshoot is larger. • The IAC value for the PD+DOB controller is twice the corresponding value of the PD+GPI controller which means that the former requires more energy compared with the latter. • An important issue is that in the case of the GPI observer velocity and disturbance are simultaneously estimated, which could be seen as an advantage. • In the case of the DOB, the dynamics of the disturbance and the velocity estimation are decoupled. Note also that the value of cutoff frequency β in the case of DOB filter is lower than the gain value θ used in the GPI observer. Therefore the peaking phenomenon in the DOB is less likely to happen.
  • 28. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 24 References I [Ohishi et al., 1988] Ohishi, K., Ohnishi, K., and Miyachi, K. (1988). Adaptive dc servo drive control taking force disturbance suppression into account. IEEE Transactions on Industry Applications, 24(1):171–176. [Ohnishi et al., 1996] Ohnishi, K., Shibata, M., and Murakami, T. (1996). Motion control for advanced mechatronics. IEEE/ASME Transactions On Mechatronics. [Sira-Ramírez et al., 2010] Sira-Ramírez, H., Ramírez-Neria, M., and Rodríguez-Angeles, A. (2010). On the linear control of nonlinear mechanical systems. In Decision and Control (CDC), 2010 49th IEEE Conference on. IEEE. [Zhao, 2011] Zhao, C. (2011). Ultrasonic motors: technologies and applications. Springer Science Business Media.
  • 29. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 25 Journal Publications • Robust ultra-precision motion control of linear ultrasonic motors: A combined ADRC-Luenberger observer approach. Control Engineering Practice Vol 111, 2021. Status: Accepted and Published. • A Teaching Methodology Based on an Educational Experimental Platform. IEEE Latin America Transactions (ISSN: 1548-0992) Vol 17, Issue 08, 2019, Status: Accepted and Published. • Delay-based nonlinear controllers for motion control: ultrasonic motor application. Status: Under Review. • Fast Parameter Identication of Perturbed Servo Systems: A Least Squares of Orthogonal Distances Approach. Status: Under Review. • Adaptive Active Disturbance Rejection Control for Velocity Tracking Trajectory Tasks With Noise-Free Desired Values. Status: Under Review.
  • 30. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 26 WORKSHOPS CONFERENCES • On the equivalence between PD+DOB and PID controllers applied to servo drives. 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control PID 2018. • Position control of a linear ultrasonic motor: An active disturbance rejection approach. 15th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) 2018. • Parameter Estimation of a Linear Ultrasonic Motor Using the Least Squares of Orthogonal Distances Algorithm. 16th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) 2019. • Micro-posicionamiento de un motor piezoelectrico ultrasónico lineal basado en un controlador PID. Congreso Internacional en Robótica y Computación, 2017. • Micro-posicionamiento de un motor piezoeléctrico ultrasónico lineal basado en Observadores Proporcionales Integrales Generalizados. Congreso Nacional de Control Automatico 2018.
  • 31. Introduction Linear ultrasonic motors Model of a Linear ultrasonic motor Parametric Identification DOB applied to a Linear Ultrasonic Motor GPI observer applied to a Linear Ultrasonic Motor Experiments Conclusions References 27 Thanks!