# PID Controller Design for a Real Time Ball and Beam System – A Double Integrating Process with Dead Time

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Published on February 17, 2014

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## Description

In this paper, the authors have discussed and shown
how to tune the PID controller in closed loop with time-delay
for the double integrator systems for a particular stability
margins. In math model it is assumed that time delay (ô) of
the plant is known. As a case study the authors have consid-
ered the mathematical model of the real-time beam and ball
system and analyzed the simulation and real time response.

Poster Paper Proc. of Int. Conf. on Advances in Signal Processing and Communication 2013 PID Controller Design for a Real Time Ball and Beam System – A Double Integrating Process with Dead Time I.Thirunavukkarasu1, Marek Zyla2, V.I.George3and Shanmuga Priya4 1& 4 Associate Professor, Dept. of ICE & Chemical Engg, MIT 3 IAESTE Student, AGH Univ. of Science and Tech, Poland. 2 Registrar, Manipal University Jaipur, Rajasthan. Mail: it.arasu@manipal.edu Abstract— In this paper, the authors have discussed and shown how to tune the PID controller in closed loop with time-delay for the double integrator systems for a particular stability margins. In math model it is assumed that time delay (ô) of the plant is known. As a case study the authors have considered the mathematical model of the real-time beam and ball system and analyzed the simulation and real time response. or ~ ~ K i  ~K p s ~)  T (s ~ ~ s ~ s ~ 3  ~K e ~  K e ~  ~ 2 K e ~ s s p s s i d (2) where ~  s s ~ K i  K i m 3 ~ K p  K p m 2 Index Terms— double integrator, PID, stability, time delay I. INTRODUCTION JWATKINS [1] worked with the PD control for double integrator systems with time delay. This paper is an extension in which PID control is analyzed in simulation as well as in real time. Integral part of the controller eliminates steadystate error, which can be necessary in this kind of systems. Equations delivered in this paper and m-files based on them can be helpful in tuning a PID controlled real-time model of beam and ball system, which is an example of double integrator system with time delay. ~ K d  K d m (3) (4) (5) (6) The characteristic equation of system (2) can be written as ~ ~~ ~2 ~ s ~ )  1  s K p  s K d  K i e ~ 1  L( s (7) ~3 s ~ ~ By setting    magnitude and phase of L ( j  ) in frequency domain can be written as II. STABILITY Consider the feedback control system shown in Fig. 1. The closed-loop transfer function can be written as T (s)  ~ L ( j )  ~ ~ ~ 2 K 2  K d Ki  p   ~  ~  3  ~ 4     (8) sK p  K i 3 s  sK p e  s  K i e  s  s 2 K d e  s m ~ ~ ~  2 K d  K i  ~ ~  L ( j )  tan 1     ~~  K p    (1) (9) Fig. 1. Feedback control system with PID controller and double integrating plant with time delay. It is assumed that velocity (derivative of controlled value) is known. © 2013 ACEEE DOI: 03.LSCS.2013.3.511 96

Poster Paper Proc. of Int. Conf. on Advances in Signal Processing and Communication 2013 ~ By setting K d as a parameter, (20) and (21) can be used to choose gain margin (GM) of the system. The stability boundary (for gain) can be plotted by setting GM  1 . Substituting (3), (4), (5), and (6) into (13), (14), (15), (19), (20), and (21) gives At gain crossover frequency ~ L ( j gc ) (10)  1 ~ PM   L ( j  gc )   (11) Substituting (9) into (11) gives 0  PM   gc  ~ ~ ~  2 K d  K i  ~ PM  tan    ~~  K p    1 Kp  (12) Rearranging (12) it can be shown that K i (K d )   ~ 0  PM   gc  2 (22)  gc (23) m  1  tan 2 ( PM   gc )  2 ~  gc K d m   gc 1  cos 2 ( PM   gc )  m (13) (24) Using (8), (10), and (12) it can be shown that for phase margin plot and 0   pc  ~  gc ~ Kp   2 (14) ~ 1  tan 2 ( PM   gc ) Kp    ~ ~ ~2 ~ ~ ~ K i ( K d )   gc K d   gc 1  cos 2 ( PM   gc ) (15) 2  pc (26) 1  tan 2  pc mGM 3  pc  m  (27) for gain margin plot. Combining PM and GM plots for  that satisfy (22) and (25) stability margins for both phase and gain can be observed (16) ~  L ( j  pc )   (25) ~ 2 K m2 K p 1  d  K i (K d )  ~  4   pc GM 2  pc  ~ By setting K d as a parameter, (14) and (15) can be used to choose phase margin of the system. The stability boundary (as far as phase is concerned) can be plotted by setting PM  0 . At the phase crossover frequency 1 ~ L ( j  pc )  2 (17) GM  for fixed  , m , and K d as on Fig. 2 and 3. Fig. Substituting (9) into (17) gives ~ ~2 ~ ~  tan 1   pc K d  K i   pc  ~ ~    pc K p    (18) From (18) it can be shown that  ~ 0   pc  2 (19) Using (8), (16), and (18) it can be shown that ~ Kp  ~2  pc (20) ~ GM 1  tan 2  pc ~ K ~ ~ K i (K d )   ~ d    pc © 2013 ACEEE DOI: 03.LSCS.2013.3.511 1 GM 2 ~2 Kp  ~4  pc Fig. 2. Stability margins of the system. Solid line states for gain and dashed line for phase. Bold lines are the borders of stability. Below both bold lines the system is stable. Each step from higher dashed line   3 ~  pc  to a lower one increases phase margin by (21)  20 . Each step from higher solid line to a lower one increases gain margin by 0 .2 highest stability margins are obtained in lower left corner. 97 . The

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