JOURNAL OF TELECOMMUNICATIONS, VOLUME 20, ISSUE 2, JUNE 2013 22

Optimisation of Settling Time for PID Po-sition Control in Mobile Satellite Dish

Network within Nigeria A. T. Ajiboye, T. S. Ibiyemi, and J. A. Falade

Abstract— The Proportional-Integral-Derivative (PID) position controller parameters that yields optimum settling time for mobile satellite dish network within Nigeria when the central control office (CCO) was at Abuja were determined. For the determined value of proportional gain value, Kp, the system acceptable stability region was graphically determined in the integral gain value, Ki - derivative gain value, Kd plane. Using Linear Programming, values for Ki and Kd that yielded optimal settling time, ts were determined within the region of acceptable stability. The determined values of Kp , Ki, and Kd were used in compensating the system and the composite transfer function of the com-pensated system was subjected to a unit step forcing function to test the designed system for robustness by varying the time delay from min-imum to maximum values. The test was done using MATLAB as a simulation tool. The result analysis confirmed remarkable improvement in the settling-time of the system with optimal settling-time over the uncompensated system.

Index Terms— Nigeria, optimal settling-time, PID position controller, satellite dish network, stability region.

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1 INTRODUCTIONHIS work is concerned with determination of controller parame-ters for centrally commanding and controlling network of satel-lite dishes mounted on mobile vehicles cruising within Nigeria at

a maximum speed of 240 km/h to point and lock onto satellite of in-terest quickly and precisely with optimum settling-time. The satellite of interest is the Nigerian’s geostationary communication satellite named NigComSat-1R located at 42.5oE in ITU’s region1 and the central control office, CCO is at Abuja (7.18°E, 9.2°N)..

Quite a number of civil and military systems such as tele-medicine, military intelligent systems rely on effective real-time communication via global area network such as geostationary satel-lite. To obtain best results from these systems, it is necessary that the satellite dishes in the network, either mobile or in fixed position, re-main pointed and locked to a desired geostationary satellite at all time. This condition can only be met by incorporation of position controller into the system. For the purpose of this work the command and control is based on Proportional-Integral-Derivative, PID, con-troller algorithm because of its popularity and simple structure [1], [2], [3], [4], [5], [6].

The determination of the region of stability in the integral gain, Ki and derivative gain Kd plane for a given Kp value that yielded robust and effective PID position controller in a supervisory control configu-ration of network of mobile satellite dishes incurring large and varia-

ble time delays within Nigeria was presented in [7]. It was established by Ibiyemi and Ajiboye in [7] that any combined values of the con-troller gains in this stability region gives robust and effective control-ler performance when the time delay is between the minimum and maximum possible values obtainable within Nigeria, but optimization of the settling-time was not considered.

As explained in [8], the effectiveness of PID, controller is largely determined by the amount of delay between formulation of PID con-trol law and its delivery to the controller’s actuator. Also the perfor-mance of PID controller degrades with increase in time delay because the formulated corrective action is based on the past output and not the current output being corrected [7]. The two major sources of time delay are the position within Nigeria of a vehicle with a satellite dish mounted on it and the vehicle speed of up to 240 km/h.

The parameters needed to formulate the control law are the round trip delay, and the plant’s transfer function. This round trip time de-lay is the sum total of delays from the plant to the satellite, the satel-lite to the control office in Abuja, to the satellite, and the satellite back to the node; or vice versa. Therefore, a model for predicting the end-to-end delays was developed and the plant transfer function was empirically determined.

The equations for determining the stability boundary in the Ki - Kd plane for a given value of Kp was obtained from the system and con-troller transfer functions using the method due to [1]. Since the result-ing closed loop transfer function for the uncompensated system is of order four, there is the need for approximating it with the lower order by determine the system dominant pole(s). From the approximated system transfer function and controller transfer function the equation that relates the settling time to the controller parameters was obtained. The settling time equation and equations for determining the stability

T

———————————————— • A. T. Ajiboye is with theEngineering and Scientific Services Department,

National Centre for Agricultural Mechanization, km 20 Ilorin-Lokoja highway, Idofian, PMB 1525, Ilorin, Kwara State. Nigeria

• T. S. Ibiyemi is with the Department of Electrical and Electronics Engi-neering, University of Ilorin, Ilorin, Nigeria.

• J. A. Falade is with the Department of Electrical and Electronics Engineer-ing, University of Ilorin, Ilorin, Nigeria.

© 2013 JOT www.journaloftelecommunications.co.uk

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boundary serve as the objective function and constraints respectively. The three PID parameters required are the proportional gain value,

Kp which was obtained using root locus method; integral gain value, Ki; and the derivative gain value, Kd that guarantee optimum settling time in spite the control action. Then for the determined value of Kp the values for Ki and Kd were determined for an optimum settling time, ts using linear programming as a novel method.

Since the principal performance index for the formulated control action is based on the settling time, ts, of the composite system dy-namic’s time response. After putting together the composite system transfer function, it was then subjected to a step input forcing func-tion which yielded an output with a settling time value for the un-compensated system assume to be the worse case settling time. This worse case settling time, form the basis for the determination of PID controller performance.

The obtained values of Kp , Ki, and Kd for optimum settling time were used for the system compensation. The designed system compo-site transfer function was subjected to a step input forcing function to test for robustness by varying the time delay from minimum to max-imum values. The testing was done using MATLAB as a simulation tool. The performance indices are the settling time, time to peak over-shoot, percentage overshoot and rise time. The result analysis con-firmed remarkable improvement in the settling-time of the system with optimized settling-time over the other systems particularly the uncompensated system.

2 SYSTEM MODELLING AND DESIGN  2.1 System Transfer Functions The  transfer  functions  of  (1),  (2),  (3)  and  (4)  are  that  of  satellite  dish  dynamic,  drive  unit,  gear  train  and  round  trip  time-­‐‑delay  [7],  [8].  Where       and    are  dish  actual  position,  geared  rotor   actual   position,  motor   shaft   angular  position   and  motor  applied  voltage.                            

 

                                                                                                                 (1)  

(2)                                                                                                                                                                                                                                        

(3)  

N  is  the  gear  ratio  and  was  empirically  determined  to  be  30.   (4)  

   is  the  round-­‐‑trip  delay  and  was  determined  to  be  between    

0.4938  seconds  and  0.4982  seconds  [9].    2.2 Determination of stability region in the Ki - Kd plane The   networked   control   system   is   represented   with   the   block  diagram   of   Fig.1,   assuming   unit   negative   feedback.   The   PID  controller   open   loop   transfer   function   is   as   given   in   (5)  while  the   plant   transfer   function   was   obtained   by   combining   the  transfer   functions   of   (1),   (2)   and   (3)   in   cascade   and   is   as   ex-­‐‑pressed   in  equation   (6).  The  system  closed   loop   transfer   func-­‐‑tion  was  obtained  From  Fig.  1  and  is  as  given  in  (7).          

                                               Fig.  1.  System  block  diagram  

   

                                                                                                                     (5)    

                                                                                     (6)  

                                                                             (7)  

Where       and     are  PID,  plant,   forward  path  time  delay  and   feedback   time  delay   transfer   function   re-­‐‑spectively  and    is  the  dish  reference  position.    

From   the   closed   loop   characteristic   equation   using   the  graphical  method  explained  in  [1]  (8)  to  (11)  for  the  generation  of  the  four  lines  that  define  the  sides  of  the  stability  polygon  in  Ki  -­‐‑  Kd  plane  were  obtained.  

(8)  

(9)  

(10)  

  (11)  

   

 The   stability   polygon   of   Fig.2   in   the  Ki   and  Kd   plane  was   ob-­‐‑tained   for   a   fixed   value   of   Kp   in   this   case   20   by   plotting   the  graph  of  (8)  to  (11)  on  the  same  coordinate.  The  system  will  be  stable  for  appropriate  combination  of      Ki  and  Kd  in  the  stability  polygon  with   the  Kp=20  but   the   response  may  not  necessarily  meet   the   acceptable   and   optimal   system   performance.   Deter-­‐‑mination   of   the   controller   gain   values   that   results   in   optimal  settling  time  are  considered  in  the  section  that  follows.    

 

           Fig.  2.  Stability  region  in  the  (Ki,  Kd)  plain  for  Kp  =  20.    

E(s)      

θr(s)        +    

                           -­‐‑                    θm(s)  

 

θA(s)              

Gpid(s)   Gd1(s)   Gp(s)  

Gd2(s)  

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3 OPTIMISATION OF SETTLING TIME Since  the  system  we  are  considering  in  this  research  work  is  of  fourth  order   there   is   the  need   for   approximating   it  with   lower  order.  This  will  simplify  the  analysis  involved  in  the  determina-­‐‑tion  of  system  time  domain  performance  specifications  because  formulas  have  been  established  for  lower  order  system  particu-­‐‑larly   the   second-­‐‑order   systems.  The  minimum  possible   settling  time  and   the  values  of  controller  gains  at  which   it  occurs  were  determined  in  this  section  using  Linear  Programming.  

3.1 Determination of System Dominant Roots The   original   system   closed-­‐‑loop   transfer   function   is   given   in  (12).  

                                                                                                                                                                                         (12)                                                                          

The  dominance  of  a  pole  or  a  pair  of  complex  conjugate  poles  of  the  closed-­‐‑loop  transfer  function  of  (12)  depends  on  two  ma-­‐‑jor  factors  [10]:  

(i) the  relative  magnitude  of  the  residue  at  the  pole  which  determine  the  percentage  of   the   total  response  due  to  that  particular  pole  and  

(ii) the   size   of   real   part   of   the   pole  which   determine   the  relative   rate  of  decay  of   the   transient   term  due   to   the  pole.    

The  values  of  residues  r,  poles  p  and  direct  term  k  obtained  for  the  system  using  MATLAB  are:  r  =      0.0000,  0.0050  +  0.0074i,  0.0050  -­‐‑  0.0074i,  -­‐‑1.0101,  1.0000                      p  =  -­‐‑66.6583,  -­‐‑0.4382  +  1.4294i,  -­‐‑0.4382  -­‐‑  1.4294i,  -­‐‑0.0252,  0                k  =  [0]  From  equation   (12)   the   closed-­‐‑loop  poles  are  at   s   =   -­‐‑66.6583,   -­‐‑0.4382  +  1.4294i,   -­‐‑0.4382   -­‐‑   1.4294i,   -­‐‑0.0252  and   the   correspond-­‐‑ing  residue  are  0.0000,  0.0050  +  0.0074i,  0.0050  -­‐‑  0.0074i,  -­‐‑1.0101.    It  can  be  seen  that  the  root  at  -­‐‑0.0252  is  the  dominant  pole  be-­‐‑cause  is  less  than  1/10  of  the  real  part  of  any  of  the  other  poles  and  it  has  the  highest  residue  magnitude.  Therefore,   the   system   can   be   approximated   by   the   first-­‐‑order  closed-­‐‑loop  transfer  function  of  (13).    

                                                                                                                                           (13)  

3.2 Minimization of Settling Time The  minimum  possible   value   for   the   settling   time  was   deter-­‐‑mined  using  Linear  Programming  with  application  of  simplex  method.  The  objective  function  was  obtained  as  follows:  From   the   approximated   closed-­‐‑loop   transfer   function   of   (13)  the  open-­‐‑loop  transfer  function,          is  giv-­‐‑ en  by  (14)    

(14)

The   system   closed-­‐‑loop   transfer   function   with   PID   controller  can  be  express  as;    

(15)

Substituting    and    in  (15)  gives;    

                                       

(16)  Comparing the closed-loop characteristic equation of (16) with that of standard second-order system we have;

(17)

where and are system damping ratio and natural frequen-cy respectfully. If the settling time , is taking to be the time required for the system to settle within 2% of the input amplitude, then;

(18)

Substitute (17) in (18) gives (19)

(19)

Let Then

(20)

Equation (20) is the objective function required to be mini-mized. The objective function is subjected to the constraints obtained from the equations of the lines that form the stability polygon and are as given by (8) to (11) [1]. This is not a standard maximization problem because the con-straints are not “less-than-or-equal-to” inequalities. Therefore, in order to apply simplex method the objective function were modified and the constraints were converted to equations by introducing slack variables as follows;

(21)

(21)

(21)

(21)

(21)

Using (21), (22), (23), (24) and (25) linear programming method was used to optimise P and the maximum value of P was ob-tained when P = 0, which occurred when Kd = 0 and Ki = 4. But, Therefore,

and Therefore, the minimum possible settling time is 15.81 seconds and it occur when Kd = 0 and Ki = 4. The point at which the minimum settling time occurred inside the stability polygon is as indicated in Fig. 3.

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           Fig.   3.   Point   of  minimum   settling   time   inside   the   stability  polygon  

4 SIMULATION AND DISCUSSION OF RESULTS The   principal   performance   index   for   the   formulated   control  action  is  based  on  the  settling  time,  ts,  of  the  composite  system  dynamic   time   response.   After   putting   together   the   composite  system  transfer   function,   it  was   then  subjected  to  a  step   input  forcing   function  which  yielded  an  output  with  a   settling   time  value   for   the   uncompensated   system   assume   to   be   the  worst  case  settling  time.  This  worse  case  settling  time,  form  the  basis  for  the  determination  of  PID  controller  performance.    

The   determined   values   of  Kp,  Ki,   and  Kd   that   yielded   opti-­‐‑mum   settling   time   were   used   to   compensate   the   system,   the  compensated  system  composite  transfer  function  was  then  sub-­‐‑jected   to   a   unit   step   forcing   function   to   evaluate   the   perfor-­‐‑mance   of   the   controller.   The   compensated   system  was   tested  for   robustness   through   simulation   by   varying   the   time   delay  from  minimum  to  maximum  value.  The  simulations  were  car-­‐‑ried  out  using  MATLAB  as  a  simulation  tool.  

The  performance  indices  are  the  settling  time,  time  to  peak  overshoot,  percentage  overshoot  and  rise  time.  The  simulation  results   for   the  uncompensated   system  obtained   from   the  unit  step   response   of   Fig.   4   gives   settling   time  of   158   seconds.  On  the  other  hand,  the  unit  step  response  of  Fig.5  shows  the  simu-­‐‑lation  results   for   the  compensated  system  from  where  the  set-­‐‑tling  time  was  determined  to  be  19.1seconds.  

 

Fig.   4.   Step   response   of   uncompensated   system   to   unit   step  forcing  function  

The  difference  between  the  value  of  settling  time  obtained  from  system   simulation,   19.1   seconds   and   that   of   optimised   value,  15.9  seconds  is  due  to  the  error  of  approximation  of  higher  or-­‐‑der   system   with   lower   order.   The   compensated   system   was  

tested   for   robustness   by   varying   the   time   delay   from   0.2469  seconds   to   0.2491   seconds.   The   corresponding   performance  indices   obtained   from   Fig.   6   were   19.1   seconds   settling   time,  4.38  seconds  time  to  peak  overshoot,  69%  percentage  overshoot  and  2.82  seconds  rise   time.  These  performance   indices   remain  constant  over  the  entire  time  delay  range,  thus  confirming  the  robustness   of   the  developed   system.  The   system   settling   time  of  19.1  seconds  and  158  seconds  obtained  for  the  compensated  and   uncompensated   systems   respectively   show   an   improve-­‐‑ment   in   the   steady   state   performance   of   the   system  with  PID  controller  due  to  reduction  in  settling  time  value.                                Fig.   5.   Step   response   of   compensated   system   to   unit   step  

forcing  function  

5 CONCLUSION The  system  stability   region   in   the  Ki   and  Kd  plane  was  graph-­‐‑ically  determined.  PID  position  controller  algorithm  that  yield-­‐‑ed  optimum  settling  time  for  controlling  centrally  the  position  of   network   of   mobile   satellite   dishes   within   Nigeria   to   point  and   lock   onto   NigComSat-­‐‑1R   satellite   quickly   and   precisely  with   central   control  office  at  Abuja  has  been  determined.  The  values  of  Kp   ,  Ki   and  Kd  were  determined   for  an  optimum  set-­‐‑tling   time,   ts   using   linear   programming.   When   the   compen-­‐‑sated  and  uncompensated  system  composite  transfer  functions  were   subjected   to   a   unit   step   function,   the   obtained   result  shows  remarkable  improvement  in  the  settling-­‐‑time  of  the  sys-­‐‑tem  with  optimized  settling-­‐‑time  over  the  uncompensated  sys-­‐‑tem.  

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A. T. Ajiboye received the B.sc. degree in electrical engineering from Uni-versity of Ibadan, Ibadan, Nigeria, in 1989. He obtained M.Eng. and Ph.D degrees, all in electrical engineering from University of Ilorin, Ilorin, Nigeria, in 2005, and 2012, respectively. He was with the Department of Electrical and Electronics Engineering, Kwara state polytechnic, Ilorin as a lecturer between 1992 and 2002. He has been with the National Centre for Agricul-tural Mechanisation, Ilorin, Kwara state as research electrical engineer since 2002 till date and he is currently an Assistant Director Electrical En-gineering in the Centre. His research interests have been in empirical mod-elling of control systems, PID controller design, control systems simulation and control system applications. He has applied control technologies, in corporation with agricultural engineers, to solve agricultural problems. He is also currently working on optimisation of controller parameters in distribut-ed control systems particularly satellite systems. Dr. Ajiboye has numbers of publications in reputable journals and conferences,both local and inter-national. He is a coporate member of Nigeria Society of Engineers (NSE)and registered with the Council for Regulation of Engineering in Nige-ria (COREN).

T. S. Ibiyemi obtained PhD control engineering from university of Bradford, Bradford, UK in 1982. He is presently a full professor of electrical engineer-ing (computer & control) at department of electrical and electronics engineer-ing, university of Ilorin, Ilorin, Nigeria. His research interest is in biometric signal processing and satellite system development. J. A. Falade obtained Ph.D in electronics and instrumentation from the University of Ibadan, Ibadan, Nigeria in 2005. He is member of Nigerian Society of Engineer (NSE) and the Institute of Electrical and Electronic Engineering (IEEE).