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Design of a Digital Controller for a Sub-micrometric Positioning System Projeto de um Controlador Digital para um Sistema de Posicionamento Submicrométrico WALTER LINDOLFO WEINGAERTNER Universidade Federal de Santa Catarina
[email protected]
RICARDO COMPIANI TAVARES Universidade Federal de Santa Catarina
[email protected]
CZESLAU BARCZAK Universidade Federal de Santa Catarina
[email protected]
CARLOS A. MARTIN Universidade Federal de Santa Catarina
[email protected]
RESUMO – Neste artigo é proposto o projeto de um controlador digital de alta resposta dinâmica, projetado para o controle de posição de um sistema de posicionamento submicrométrico. Tal sistema posicionador simulado é constituído de uma mesa guiada por mancais de molas (flexural spring guides), acionada por um atuador piezelétrico (piezoelectric actuator) e com extensômetros dispostos em ponte para o sensoreamento de sua posição linear. A discretização do sistema de posicionamento submicrométrico, compreendendo o atuador piezelétrico e seus subsistemas mecânicos e eletrônicos, foi executada com sucesso. A resposta dinâmica da planta e do grampeador de ordem zero (zero-order hold) foi muito boa, com uma sobrepassagem (overshoot) menor em aproximadamente 20%, comparada com a resposta dinâmica do controlador e da planta contínuos no tempo. O projeto do controlador PD (Proporcional-Derivativo) discreto resultou em um sistema com desempenho dinâmico excelente e com tempo de resposta e tempo de estabilização bem menores do que os resultados obtidos pela discretização do sistema contínuo. Palavras-chave: CONTROLADOR DIGITAL – POSICIONAMENTO SUBMICROMÉTRICO – DISCRETIZAÇÃO – ATUADOR PIEZELÉTRICO – CONTROLADOR PD. ABSTRACT – This paper concerns a digital controller design for position control of a sub-micrometric system of high dynamic response. The system is assembled on a table with flexural spring guides driven by a piezoelectric actuator, and with extensometers for linear position measurement. The discretization of the sub-micrometric positioning system through the piezoelectric actuator, and its mechanical sub-system was successfully performed. The dynamic response of the plant with zero-order hold was very good, with an overshoot of less than 20%, same as response of the continuous system. The design of a PD discreet controller resulted in a even better dynamic performance, with rise and settling times less than those obtained with the discretization of the continuous system. Keywords: DIGITAL CONTROLLER – SUB-MICROMETRIC POSITIONING – DISCRETIZATION – PIEZOELECTRIC ACTUATOR – PD CONTROLLER.
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INTRODUCTION
T
his paper presents the digital controller design for a sub-micrometric positioning system with a single translation axis, using a piezoelectric actuator and flexural spring guides, directly driven. The design of the controller was based in two methods: i) design of a digital controller by discretization of the continuous system; ii) design of a digital controller by discretization in time (discret PID controller in z-plane). The piezoelectric actuator used in the sub-micrometric positioning is a P-172 of Physik Instrumente. The P-172 have four extensometers for the sub-micrometric positioning feedback control of the table. The following list provides technical data of P-172. List 1. Technical data of P-172. Technical characteristics
P-172
Nominal Voltage
–1000 V
Maximum Operational Voltage
–1500 V
Pole
negative
Maximum Expansion Strength
1000 N
Maximum Contraction Strength
100 N
Nominal Expansion (-1000 V)
20 µm
Maximum Expansion (-1500 V)
30 µm
Rigidity
35 N/µm
Electric Capacitance
38 nF
Ressonance Frequency
9 kHz
Thermal Dilatabilty Coeficient
0.4 µm/K
Weight
0.08 N
The positioning system have the following characteristics, which must be satisfied by the controller: • precision of positioning ± 50 nm; • resolution 0.1 µm; • settling time maximum 10 ms; • rise time maximum 0.5 ms.
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DESIGN OF THE CONTINUOUS CONTROLLER The sub-micrometric positioning system model was established considering the piezoelectric actuator with its electronics, and the mechanics, resulting in the following system transfers functions (Ogata, 1998; Tavares, 1995; e Troncoso et al., 1994). The transfer function of the piezoelectric actuator with its electronics is given by: K ⋅ ω 2 n1 G(s) 1 = -------------------------------------------s 2 s + 2ξ 1 ω n1 + ω n1
(1).
From the references (Tavares, 1995; Troncoso et al., 1994; Lutrell et al., 1987) one may consider that the natural frequency ωn1 is 5.03 x 104 rad/s, the damping ratio ξ1 is 0.7, and the gain K is 1 x 103 µm/V. Entering the values in equation (1), results: 6
2.53x10 G ( s ) 1 = -------------------------------------------------------------2 4 9 s + 7.04x10 s + 2.5x10
(2).
The positioning system mechanics transfer function is defined by: 2
ω n2 G ( s ) 2 = --------------------------------------2 2 s + 2ξ 2 ω 2 + ω 2
(3).
Where, from Tavares (1995), one obtaIn: ωn2 = 2 x 104 rad/s; ξ2 = 0.08. Entering the values in equation (3), results: 8
4x10 G ( s ) 2 = -----------------------------------------------------2 3 8 s + 3.2x10 s + 4x10
(4).
The figure 1 shows the block diagram of the continuous system. Making G(s)=G(s)1 . G(s)2 (5), results:
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1.01 × 10 G ( s ) = ----------------------------------------------------------------------------------------------------------------------------------------------------4 4 3 9 2 13 18 s + 7.36 × 10 s + 3.16 × 10 s + 3.64 × 10 s + 1.01 × 10
(6).
Fig. 1. Block diagram of the positioner feedback control
The figure 2 shows the step response of the system and figure 3 show a Bode Diagram (or frequency response) of the continuous system. Fig. 2. Step Response of the System. -3
1.8
x 10
1.6 1.4
The sub-micrometric positioning system showed a excellent rise time of 0.25 ms, less than the specified 0.50 ms. The system showed a settling time is 5 ms, less than the specified 10 ms. The gain margin is 60dB, corroborating the excellent dynamics of the sub-micrometric positioning system. The sub-micrometric positioning system showed initially a overshoot of 70%, however the system stabilized rapidly.
1.2
CONTINUOUS PLANT DISCRETIZATION
1 0.8
This item will present the discreet transfer function of the plant for the sub-micrometric positioning system, with the selection and analysis of zero-order hold (ZOH) continuous in time. A discretization of the continuous transfer function G(s), according to Franklin (1998), Bishop (1997) and Ogata (1995), will then made.
0.6 0.4 0.2 0 0
1
2
3
4
5 -3
x 10
Fig. 3. Frequency Response of the System.
SELECTION AND ANALYSIS OF ZERO-ORDER HOLD Goz(s) INFLUENCE
Gain dB
0
-100
By the aproximation gived by equation (6) with sampling time T = 0.0001s = 0.1 ms, results:
-200 2 10
10
3
4
10 Frequency ( rad/sec)
10
5
10
6
Phase deg
0
-180
-360 10
2
10
3
4
10 Frequency ( rad/sec)
REVISTA DE CIÊNCIA & TECNOLOGIA • V. 8, Nº 16 – pp. 57-64
10
5
10
6
2⁄T G oz ( S ) = ------------------s+2⁄T
(7),
2 ---------------0.0001 20000 G oz ( S ) = ------------------------- ⇒ G oz ( S ) = ----------------------2 s + 20000 s + ---------------0.0001
(8).
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Fig. 4. Block diagram of the continuous system with Goz(s).
Fig. 5. Root Locus of the system with.
dynamics, since this pole tends to negative infinit. The gain margin of system is 60 dB (fig. 6). Goz(s)Goz(s) The figure 7 presents the step response of the sub-micrometric positioner. The overshoot of the system with Goz(s) is the approximately 12% less than the continuous system response without zeroorder hold (fig. 2). The rise time with Goz(s) is 3 ms. The inclusion of zero-order hold Goz(s) in the continuous system increased he dynamic performance of the sub-micrometric positioner. Fig. 7. Step response of the sub-micrometric positioner with Goz(s).
Fig. 6. Bode Diagrams of the system with.
PLANT DISCRETIZATION The sub-micrometric positioner with zeroorder hold Goz(s) is represented in figure 4. The figures 5 and 6 show the results of system simulation. The figure 5 indicates that additional pole of the zero-order hold not affect the sytem
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The Tustin’s bilinear aproximation was used to obtain the discreet transfer function of the G’(z) plant. The value of G’(z) was obtained from the continuous system G(s). The transfer function for the sub-micrometric positioner G(s) is:
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1.01 × 10 G ( s ) = ----------------------------------------------------------------------------------------------------------------------------------------------------4 4 3 9 2 13 18 s + 7.36 × 10 s + 3.16 × 10 s + 3.64 × 10 s + 1.01 × 10
(9).
From this equation, was obtained the discret transfer function G’(z). –1
3
–2
–3
–4
1 × 10 ( 0.07 + 0.27z + 0.41z + 0.27z + 0.07z ) G′ ( z ) = ------------------------------------------------------------------------------------------------------------------------------------–1 –2 –3 –4 1 – 86z + 0.76z + 0.02z + 0.17z
(10).
THE ZERO-ORDER HOLD DISCRETIZATION The discreet transfer function of Goz(s) is generated by Tustin’s bilinear transformation, resulting in the discreet time function Goz(z). Therefore: –1
0.33 + 0.33z G oz ( Z ) = ----------------------------------–1 1 – 0.33z
(11).
DISCRET TRANSFER FUNCTION OF THE PLANT G(z) WITH ZERO-ORDER HOLD Goz(s) With discreet transfer functions of the zero-order hold and of the plant, Goz(z) and of G’(z) respectively, developed in Plant Discretization and The Zero-Order Hold Discretization, is possible to determinate the discreet transfer function of the sub-micrometric positioner system G(z). G(z) is the produt of Goz(z) and G’(z), as in (12). According to figure 8, G(z) = Goz(z) .G’(z)(12). Substituting both Goz(z) and G’(z) in the equation (12), results In: –1
–3
–1
–2
–3
–4
1 × 10 ( 007 + 0.27z + 0.41z + 0.27z + 0.07z ) 0.33 + 0.33z - ⋅ -------------------------------------------------------------------------------------------------------------------------------------G ( z ) = ---------------------------------–1 –1 –2 –3 –4 1 – 0.33z 1 – 0.86z + 0.76z + 0.02z + 0.17z
(13).
Developind the above expression, results in (14): –5
–4 –1
–4 –2
–4 –3
–4 –4
–5 –5
2.31 × 10 + 1.12 × 10 z + 2.21 × 10 z + 2.21 × 10 z + 1.12 × 10 z + 2.31 × 10 z G ( z ) = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------–1 –2 –3 –4 –5 1 – 1.19z + 1.04z – 0.23z + 0.16z – 0.06z
Figures 9 and 10 show the step response and frequency response diagrams of the discreet sub-micrometric positioner system simulation, using G(z). Fig. 8. Block diagram of the discret system.
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Fig. 9. Step response of the discreet system.
positioner system’s dynamics. The figure 11 shows the system and digital controller squematic diagram.
PROPORTIONALDERIVATIVE CONTROLLER (PD)
D(z) = KP + KD (1 – z–1) (15) the proportional-derivative controller is described by, where KP is the proportional gain and KD is the derivative gain, resulting: ( K P + K D )z – K D D ( z ) = -----------------------------------------z
Fig. 10. Frequency response of the discreet system.
(16).
Algebraically working the equation (16), results In: k(z – a) D ( z ) = ------------------z
(17), with 0 ≤ a ≤ 1.
Comparing equations (16) and (17), results In: k = KP + KD
and
KD -. a = -------------------KP + KD
Transforming the equation (17) to negative powers from z, results: D(z) = k (1 – az-1)
The discreet system presented an overshoot of 20% less than the continuous system. The design specifications are attained when the discret system is settling in stationary state in k = 70. The gain margin is 20 dB.
DESIGN THE PID DISCREET CONTROLLER With the discreet transfer function of the system G(z) given by (14), a discreet controller was designed, whith the purpose of improving the sub-micrometric
(18).
The figures 12 and 13 show the step response and frequency response diagrams of the system with a proportional derivative controller (PD), having KP = 0.5 and KD = 0.5 resulting in a = 1. Figure 12 indicates that the system’s overshoot is, approximately, one thousand times less than the continuous system. The digital controller PD increased the system’s dynamics. The figure 13 shows a Bode diagram frequency response of the sub-micrometric positioning system with the digital controller. A gain margin of 80 dB was considered excellent.
Fig. 11. Bock diagram of the discreet positioner system with digital controller.
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Fig. 12. Step response of the discreet system and controller.
Fig. 13. Frequency response of the discreet and controller.
CONCLUSIONS The discretization of a sub-micrometric positioning system assembled with a PZT, its electronics and the mechanical sub-system, continuous in time, was performed with success. The dynamic response of the plant G(z) with the zero-order hold Goz(z) was very good, with an overshoot 20% less than the continuous system response. The proportional-derivative (PD) discreet controller design resulted in the system with dynamic performance even better, with rise and settling times less than those obtained with the discretization of the continuous system Goz(z) without controller. The overshoot is approximaly one thousand times less than the continuous system. This resulted in better positioning precision of the sub-micrometric positioning system.
Agradecimentos Os autores agradecem ao apoio financeiro do CNPq (bolsa de doutorado) para a realização deste trabalho.
REFERENCES BISHOP, R.H. Modern Control Systems. Analysis & Design: Using MATLAB and SIMULYNK. Menlo Park: Addison-Wesley, 1997. FRANKLIN, G.F. et al. Feedbak Control of Dynamic Systems, 3rd ed. Reading: Addison-Wesley, 1994. ___________. Digital Control of Dynamic Systems, 3rd. ed. Reading: Addison-Wesley, 1998. LEVINE, W.S. (org). The Control Handbook. Boca Raton: CRC Press, 1996. LUTTRELL, D.E. & DOWN, T.A. Development of a High Speed System to Control Dynamics Behaviour of Mechanical Structures. Precision Engineering, 9 (4), Surrey, UK, Butterworth Scientific Ltd., October/1987. OGATA, K. Discrete-Time Control Systems, 2nd ed., Englewood Cliffs, NJ Prentice-Hall, 1995. OGATA, K.. System Dynamics, 3rd. ed., Englewood Cliffs, NJ, Prentice-Hall, 1998. TAVARES, Ricardo C. Projeto de um Posicionador Submicrométrico para Litografia Óptica. Florianópolis, 1995. [Dissertação de mestrado, Pós-Graduação em Engenharia Mecânica/UFSC]. TRONCOSO, L.S. et al. Sistema Activo de Estabilizacion Aplicado à la Holografia Electronica. Laboratório de Metrologia e Automatização, Departamento de Engenharia Mecânica/UFSC. Anales VI Congresso Nacional de Ingenieria Mecànica. Santiago, 1994.
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