IntroductionuOttawa.ca Introduction The three degree of free dome Twin Rotor Control System presented in this paper is an under actuated system with two actuators and three degrees of freedom. As shown from the simulated model and the open loop response, this system is unstable and nonlinear, given that one of the three outputs is unstable. Under this circumstance, a robust controller will be needed in future for this system in order to ensure the stability.
The State ParametersuOttawa.ca The State Parameters X = [∅ ∅ ∅ �� �� �� ℎ ℎ ℎ �� 1 �� 1 �� 1 �� 2 �� 2 �� 2 ] ∅: The angle of moment given by the aerodynamic forces. ∅ ∅ : The angular velocity of ∅. ��: The angle of moment given by the aerodynamic forces. ��: ��: The angular velocity of ��. ℎ: Distance ℎ ℎ : Velocity �� 1 �� 1 �� 1 : The angular velocity to the main rotor. �� 2 �� 2 �� 2 : The angular velocity to the tail rotor.
Inputs and OutputsuOttawa.ca Inputs and Outputs U = �� 1 �� 2 �� �� 1 �� 2 �� 1 �� 1 �� 1 �� 2 �� 2 �� 2 �� 1 �� 2 �� 1 �� 2 �� �� 1 �� 2 �� �� 1 �� 1 �� 1 : The voltage to main rotor. �� 2 �� 2 �� 2 : The voltage to tail rotor. Y= [∅ �� ℎ] ∅: The angle of moment given by the aerodynamic forces. ��: The angle of moment given by the aerodynamic forces. ℎ: Distance
Controllability And ObservabilityuOttawa.ca Controllability And Observability >> rank(ctrb(A,B)) ans = 8 = # of variables So, the system is controllable. >>rank(obsv(A,C)) ans = 8 = # of variables So, the system is observable.
Open-Loop ModeluOttawa.ca Open-Loop Model We simulated the state-space model of Twin Rotor System. In this model, we put 2 inputs and 3 outputs and applied the same state-space values. After adding four scopes, we observed the output values.
INPUTS (1-2)uOttawa.ca INPUTS (1-2) Voltage to the main rotor( �� 1 �� 1 �� 1 ) & Voltage to the tail rotor (�� 2 (�� 2 (�� 2 ). It means that the voltages to both the main and tail rotors were set to be 1 unit.
Outputs (1-2-3)uOttawa.ca Outputs (1-2-3) The output h (Distance) increases with the input 1 unit. The results of output 2 and 3 were observed to be stable.
State-space Model DatauOttawa.ca State-space Model Data We created a state-space model object representing the continuous-time state- space model from the data of matrix A, B, C, and D, where �� �� = Ax + Bu, y = Cx + Du.
Controller DesignuOttawa.ca Controller Design We determined the state feedback gain matrix K of the Ackermann’s Formula. Given parameters for K were [-10 -11 -13 -15 -17 -18 -19 -20].
Slide24uOttawa.ca We created new A with our K value to find new controller design first.
Pole Placement DesignuOttawa.ca Pole Placement Design K = place(A,B,p) [K,prec,message] = place(A,B,p) Given the single- or multi-input system ˙x=Ax+Bu and a vector p of desired self-conjugate closed-loop pole locations, place computes a gain matrix K such that the state feedback u = –Kx places the closed-loop poles at the locations p. In other words, the eigenvalues of A – BK match the entries of p (up to the ordering).
Slide29uOttawa.ca >> step(sys_closedloop) >> step(sys_openloop) It shows that the system is now stable.
The Open Loop Results Vs The Closed Loop ResultsuOttawa.ca The Open Loop Results Vs The Closed Loop Results •The open loop response shows that the one of the tree outputs is unstable, while the closed loop response shows that the system is stable after the controller is designed. •We can see that the open loop control system response is completely based on input and outputs do not affect the control action. For the closed loop control system, current outputs are under consideration and altered to the desired condition. In another words, the control action acts on the basis of outputs. •Given that an open loop control system works on fixed operation condition, no disturbance is considered. However, a closed loop control system has reaction on external disturbances.
ConclusionuOttawa.ca Conclusion Three degree of freedom Helicopter/Twin rotor control system has two inputs and three outputs is described in detail in this report. The modeling process starts with an open loop control system, whose response is simulated by SIMULINK. The results show that the system is not stable. In this case, we designed a controller and simulate the closed loop control system with MATLAB, which ensures the stability for the system. In summary, a closed loop control system is preferred to an open loop control system it terms of stability and sensibility.
ReferencesuOttawa.ca References 1.Magnus Gäfvert, “Modeling of ETH Helicopter Laboratory Process,” Department of Automatic Control Lund Institute of Technology, Box 118, SE-221 00 Lund Sweden, ISRN LUTFD2/TFRT--7596—SE, November 2001. 2.Thomas Bevan, The Theory of Machines, 3rd edition, CBS publishers and Distributors, 4596/1A, 11 Derya Ganj, New Delhi, India, 2003 3.Gary Fay, “Derivation of the Aerodynamic Forces for Mesicopter Simulation”. http://adg.standford.edu/mesicopter/progressRe ports/mesicopteraeromodel.pdf 4.Alberto Isidori, Nonlinear Control System, 3rd edition, 1996, Springer-verlag London Limited, Great Britain. 5.Martin Hepperle, D-38108 Braunshweig 1, “Aerfoil & Propeller Design”. http://www.mh-aerotools.de 6.Katsuhiko Ogata, Modern Control Engineering, 4th edition, Prentice hall of India Private Limited, M-97, Connaught Circus New Delhi-110001. 7.Abdul Qayyum Khan, Naeem Iqbal ‘‘Modeling And Design Of An Optimal Regulator For Three Degree Of Freedom Helicopter/ Twin Rotor Control System’’