Robust Deadbeat Control of Twin Rotor Multi Input Multi Output System

Robust Deadbeat Control of Twin Rotor Multi Input Multi Output System

Abhishek J. Bedekar Department of Electrical Engg. Walchand College of Engineering, Sangli (Maharashtra), India.

Santosh Shinde

Department of Electrical Engg. Walchand College of Engineering, Sangli (Maharashtra), India.

AbstractThis paper deals with a deadbeat PID control of Non-linear higher order system. The method used in this paper uses feedback arrangement which helps to improve system response. TheDeadbeat control method is applied to Non-linear model of Twin Rotor MIMO Laboratory setup (Helicopter model).This helicopter model is first simulated in matlab with simple PID controller and then with deadbeat controller. Comparative study shows that Deadbeat PID controller is capable of providing robust performance.

Keywords MIMOsystem, TRMS, PID, Deadbeat control.

1. INTRODUCTION

   The TRMS system is nonlinear system similar to helicopter as shown in Fig. 1.The conventional controller like PID is used to control such nonlinear system[1-4], as it has simple structure and easy to use. In recent years new methods like feedback linearization [5] and gain scheduling [6] have been applied with good outcomes. Intelligent control theories like Fuzzy control , Neural network and GA applied, but designing logic for such intelligent controller is not easy[7,8,9].

   Many methods are tested in order to get an acceptable performance of the controller. This includes minimum settling time, rise time, zero steady state error, less oscillation, robustness against disturbances. TRMS system is very complex and it includes uncertainties thus finding gains of PID become difficult task. Also modeling of TRMs system is difficult [10, 11, and 12]. Even if we get system model, for entire input range it will not represent system exactly. In order to control TRMS system here, we use the method proposed in [13, 14] which include PID and Deadbeat controller. In [13] writer claims that response will remain unchanged when all parameters change by as much as 50%.

   The main problem with TRMS is that of coupling effect between two rotors, to encounter this problem we are going to decouple [15] into two SISO system and effect is considered as disturbances to each of the SISO system. Then the Deadbeat control scheme is applied to the SISO system.

   This paper is organized as follows. Section 1 introduction, Section 2 system model and its mathematical representation, section 3 Deadbeat control method and controller design, section 4 then system is simulated in MATLAB. Finally, comparative analysis between PID and Deadbeat PID controller and this study is summarized in conclusion.

   Fig.1 Schematic diagram of TRMS control system

2. TWIN ROTOR SYSTEM

   The TRMs is a laboratory platform designed for control experiment by Feedback Instruments Ltd [16]. It consists of a beam pivoted on its base in such way that it can rotate in both vertical and horizontal planes. Two DC motors are used for two rotors (main and tail rotor), at each end of the beam. In actual Helicopter, aerodynamic force is controlled by changing the angle of attack of the blades but TRMS model designed such that force is controlled by controlling speed of motors. The main rotor produces force, which allows the beam to raise vertically making rotation along pitch axis. While tail rotor produces horizontal force to make the beam turn left or right around the yaw axis.

   Usually, phenomenological models tending to be nonlinear, this means that at least one of the states is an argument of a nonlinear function. So as to present such a model as a transfer function (a form of linear plant dynamics representing a control system), it has to be linearized. The following non- linear model equations can be derived.

   Mathematical equation in vertical plane is given as

   1. = 1 (1) Where

   1

   1 = 1. 2 + 1. 1Nonlinear static characteristic(2)

   = sin Gravity momentum (3) Friction forces momentum

   = 1 . + 2 . sin( ) (4)

   = . 1. . cos Gyroscopic momentum (5)

   The motor and the electrical control circuit is approximated as a first order transfer function, thus the rotor momentum in Laplace domain is described as

   1 1

   = 1 (6)

   11 +10

   Mathematical equation in horizontal plane is given as

   2. = 2 (7) Where

   2

   2 = 2. 2 + . 2 Nonlinear static characteristic (8)

   Friction forces momentum

   = 1 . + 2 sin (9)

   Table 1 gives approximated values of parameters.

   = 0+1

   Cross reaction momentum (10)

   Parameters	Value
   1- moment of inertia of vertical rotor	6.8 102 2
   2- moment of inertia of horizontal rotor	2 102 2
   1-static characteristic parameter	0.0135
   1-static characteristic parameter	0.0924
   2-static characteristic parameter	0.02
   2-static characteristic parameter	0.09
   -gravity momentum	0.32
   1 -friction momentum function parameter	6 103 /
   2 – friction momentum function parameter	1 103 2 /
   1- friction momentum function parameter	1 101 /
   2 – friction momentum function parameter	1 102 2 /
   – gyroscopic momentum parameter	0.05 /
   1-motor 1 gain	1.1
   2-motor 2 gain	0.8
   11-motor 1 denominator parameter	1.1
   10- motor 1 denominator parameter	1
   21- motor 2 denominator parameter	1
   20- motor 2 denominator parameter	1
   11- cross reaction momentum parameter	2
   11- cross reaction momentum parameter	3.5
   1-cross reaction momentum gain	-0.2

   +1 1

   Rotor momentum in Laplace domain is given as

   = (11)

   +

   The model parameters used in above (1)-(11) equation are chosen experimentally, which makes the TRMS non-linear model a semi-phenomenological model.

   The boundary for the control signal is set to [-2.5 to +2.5].

   Fig.2

   Table 1

   As we know there is cross coupling between main and tail rotor. With the help of model equation nonlinear Simulink model of TRMS is obtained as shown in Fig.3

   Fig. 3 Nonlinear simulink model of TRMS

3. DEADBEAT CONTROLLER DESIGN METHODOLOGY

   H1(s) =1 for = 2

   H1(s) =1+ for = 3 4

   H1(s) = 1 + + 2 for = 5

   The main goal of control system is to reach the desired value6. select gain, using the coefficient from table 2 to achieve

   with zero steady state error within specified settling time. A deadbeat response should have zero steady state error, minimum rise time, controllable settling time, % overshoot from 0 to 2%, % undershoot < 2% . Also have robustness against external disturbance and parameter uncertainty.Rfer following fig.4 for the basic structure of the controlled system design.

   response with the following requirement: (a)set k=1

   (b)set =

   80%

   (c)set C.E. of closed loop equal to:

   + 1 + 2 2+. . . +

   1. The roots of H(s) must be real & negative

   2. Smallest root of H(s) will set the desired

Fig.4 Basic structure

1. Use PID controller as ().

   4

   80%

2. Add cascade gain K before the PID controller.

3. Add state variable feedback gain Ka. This will make the system over specified by at least one variable

4. Determine for (), where equals the no of poles in ().

5. Add the feedback

1. Increase K until the response becomes deadbeat and the settling time is approximately equal to the desired value.

   Order
   2nd	1.82	–	–	–	–	4.82
   3rd	1.90	2.20	–	–	–	4.04
   4th	2.20	3.50	2.80	–	–	4.81
   5th	2.70	4.90	5.40	3.40	–	5.43
   6th	3.15	6.50	8.70	7.55	4.05	6.04

   Table 2 Deadbeat coefficients and response time

   1. CONTROLLER DESIGN

      First using the decouple techniques we separate the system into two SISO ones as shown below

      Vertical part (Tail Rotor)

      This TF will be utilized throughout this work.

      1. Tail rotor controller design

         Fig.7 Tail rotor control

         The close loop transfer function

         () = 1 ()2 ()

         ()

         1+2 2 +1 ()2 ()1 () (14)

         Where 1 = and 2 =

         Equals the no of poles in () = 4 Thus

         1 = 1 + And2 =

         Finally

         Fig.5

         ()

         15.02[3(2 + + )]

         Horizontal part (Main Rotor)

         () = 4 + 3.458 + 15.02 3 3 + 2.225 + 15.023 + 15.02 3 2 +

         {15.02 + 15.023 + 15.02 3} + {15.023}

         (15)

         Then CE of above transfer function compared with Standard CE

         4 + 3 + 2 2 + 3 + 4 (16)

         From design procedure we get coefficient values from Table 2, also we get value of

         Fig.6

         =

         =

         80%

         4.81

         1.6

         = 3.00625

         TF of theTRMS in vertical and horizontal movement are given as

         =

         15.02 . Tail rotor (12)

         3 +3.4582 +2.225

         =

         1.519 . Mainrotor (13)

         3 +0.7482 +1.533+1.046

         Therefore CE of deadbeat TF is:

         4 + 6.6133 + 31.63142 + 76.07355 + 81.6771 (17)

         Comparing characteristic equation (16) and (17) we have

         3.485 + 15.02 3 = 6.6138

         2.225 + 15.023 + 15.02 3 = 31.6314

         15.02 + 15.023 + 15.02 3 = 76.0735

         15.023 = 81.6771

         Then cascade gain K is set equal to 1. After some trial and error we get values for 3 = 17

         = 0.243 = 45.848

         = 14.21 = 38.65

         Select k until deadbeat response not reached.

      2. Main rotor control

      Similarly,

      () = 1()2()

      () 1 + 2 2 + 1()2()1()

   2. SIMULATION AND RESULTS

      1. For Step input

         Fig. 8 PID response to main rotor(pitch)

         Where 1 = and 2 =

         Equals the no of poles in () = 4 Thus

         1 = 1 + And2 =

         ()

         [3(2 + + )]

         () = 4 + 0.748 + 1.519 3 3 + 1.533 + 0.15493 2 +

         {1.046 + 1.519 + 0.1549 3} + {1.5193}

         Then CE of above transfer function compared with Standard CE

         4 + 3 + 2 2 + 3 + 4 (18)

         Fig. 9 Deadbeat PID response to main rotor(pitch)

         From design procedure we get coefficient values from Table 2, also we get value of

         =

         =

         80%

         4.81

         1.6

         = 3.00625

         Therefore CE of deadbeat TF is:

         4 + 6.6133 + 31.63142 + 76.07355 + 81.6771(19)

         Comparing characteristic equation (18) and (19) we have

         0.748 + 1.519 3 = 6.6138

         1.533 + 0.15493 = 31.6314

         1.046 + 1.519 + 0.1549 3 = 76.0735

         1.5193 = 81.6771

         Then cascade gain K is set equal to 1. After some trial and error we get values for 3 = 7.723

         = 0.5 = 2.5453

         = 3.131 = 6.963

         Fig. 10 PID response to tail rotor

         Fig.11 Deadbeat PID response to tail rotor

      2. For sin wave input

         Fig. 12 PID response to tail rotor (yaw)

         Fig.13 Deadbeat PID response to tail rotor (yaw)

      3. For square input

      Fig. 14 PID response to main rotor(pitch)

      Fig. 15 Deadbeat PID response to main rotor(pitch)

      Above Figures shows the controller effect for different type of inputs. From results it is clear that traditional PID controller responses have overshoots and its settling time is more than designed Deadbeat controller. Here with Deadbeat control approach result shows remarkable improvement in system behavior.

   3. CONCLUSION

      In this study, we have successfully modeled TRMS system and successfully applied robust Deadbeat controller scheme. In system performance, the settling time has been reduced also overshoot has been decreased.Implemented scheme does not deal with much harder mathematics. it is easily understood method .

      But as this method is model based, we require accurate system transfer function.

      Fig. 16 simulation diagram of PID controller scheme

      Fig. 17 simulation diagram of deadbeat controller scheme

   4. REFERENCES

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Abhishek Jayant Bedekar has obtained

B.E. in Electrical Engineering from Rajarambapu Institute of Tech. sakharale

, Islampur, Maharashtra, India in 2012. He is currently pursuing his M.Tech in Electrical Control System from Walchad College of Engg. Sangli under the guidence of Prof. Mr. N. V. Patel. His areas of interests are Controller Design, Control Systems.

Santosh Shinde has obtained B.E. in Electrical Engineering from Rajaram shinde college of engineering pedhambe chiplun Maharashtra, India in 2012. He is currently pursuing his M.Tech in Electrical Control System from Walchad College of Engg. Sangli under the guidence of Prof. Mr. N. V. Patel. His areas of interests are Controller Design, Control Systems.