Novel analysis of limit cycle

Transcription

1 Novel analysis of limit cycle for PWM signal of PD control system Sang-Woon Jeon 1a) and Seul Jung 2b) 1 Control Dept., Korea Aerospace Research Institute, Yusung, Daejon, Korea 2 Mechatronics Dept., Chungnam National University, Yusung, Daejon, Korea a) swjeon@kari.re.kr b) jungs@cnu.ac.kr Abstract: In this paper a limit cycle analysis of attitude control for a launch vehicle is described. PWM signals from PD control laws are applied to attitude control and time delay in solenoid valve is considered. A novel phase plane method is proposed for describing characteristics of limit cycle. Important characteristics of resultant limit cycle such as frequency, amplitude, maximum rate, and duty ratio could be analytically solved by the proposed phase plane method. Keywords: PWM control, attitude control, limit cycle analysis Classification: Science and engineering for electronics References [1] L. M. Tolbert, F. X. Peng, and T. G. Habetler, Multilevel PWM methods at low modulation indices, IEEE Trans. Power Electron., no. 4, pp , [2] X. Shen, J. Zhang, E. J. Barth, and M. Goldfarb, Nonlinear model-base control of pulse width modulated pneumatic servo system, ASME Journal of Dynamics Systems, Measurement, and Control, vol. 128, pp , [3] T. Ieko, Y. Ochi, K. Kanai, N. Hori, and P. N. Nikiforuk, Design of a pulse-width-modulation spacecraft attitude control system via digital redesign, IFAC 14 Triennial World Congress, pp , [4] B.-F. Wu, J.-W. Perng, and H.-I. Chin, Limit cycle analysis of nonlinear sampled-data system by gain-phase margin approach, Journal of the Franklin Institute, pp , [5] V. E. Haloulakos, Thrust and impulse requirements for jet attitudecontrol systems, Journal of Spacecraft, pp , [6] S. J. Dodds, A predicted signed switching time high precision satellite attitude control law, INT. J. CONTROL, pp , [7] H. B. Hablani, Target acquisition, tracking, spacecraft attitude control, and vibration suppression with IPFM reaction jet controllers, AIAA Guidance, Navigation and Control Conf., Hilton Head Island, SC, Bell Syst. Tech. J. pp , [8] V. E. Haloulakos, Analysis of jet attitude control systems subject to varying magnitudes of external disturbing torques, AIAA, Guidance, Control and Flight Dynamics Conf., pp. 1 11,

2 1 Introduction The pulse width modulator (PWM) control method is used most to control electric systems, pneumatic systems, and thruster-controlled spacecrafts [1, 2, 3]. Since the PWM has nonlinear characteristics such as hysteresis and dead bands, a limit cycle occurs under the presence of disturbing moments. Properties of the limit cycle are determined primarily by design of control parameters, level of the thrust, characteristics of the inertia, magnitudes of disturbance as well as time delays caused by the opening and closing operation of the solenoid valves. The limit cycle frequency is directly related to the fuel consumption of the system to be controlled. The higher the frequency, the more fuel is required. Thus, it is desirable to reduce the limit cycle frequency. The limit cycle can be analyzed theoretically by using a describing function method [4] and a phase plane method [5, 6, 7, 8]. The describing function method is an approximate procedure for analyzing nonlinear control based on quasilinearization which is an approximation of a nonlinear system by a specific family of input waveforms. As the phase plane method plots the velocity vs. displacement of control system in phase plane, this method can be deduced the property of the limit cycle by analysis. The phase plane method along with analysis of limit cycle is usually used for thrust selection for jet attitude control system [5]. Although analysis of the limit cycle for the integral pulse frequency modulation control has been used in controller design, control parameters and time delay effect are not considered [7]. Equations of the limit cycle with the phase plane method have been derived in [5] and [8]. Haloulakos s method is applicable to a control system, whose output is described by on-off state for thruster. But it is not applicable to a PWM control since its output should be described by pulse width. Unlike these prior works, this paper presents the limit cycle analysis using a phase plane method with switching line for a traditional PD feedback input-based PWM control of roll attitude of a launched vehicle. Frequency, amplitude, duty ratio, and switching line of the limit cycle are analytically solved with control parameters, external disturbances, and time delays of the thruster. This approach is verified with the simulation results of a single degree-of-freedom attitude control system. 2 System modelling The attitude of the rocket should be controlled in three mutually perpendicular axes (uncoupled), each with two degrees of rotational freedom (clockwise and counter-clockwise rotation). Generally, the equation of motion for roll is given by Eq. (1) dθ dt = ω(t) dω dt = F (1) jetδ u(t) J 788

3 where θ, ω, F jet,δ,u(t), and J are a roll angle, an angular velocity, PWM control force, moment arm, PWM output, and inertia of launch vehicle, respectively. The block diagram for the PD feedback input-based PWM control system of a roll attitude is shown in Fig. 1. Fig. 1. Roll attitude control system The control command β(t) which is a switching line in the phase plane is given in equation (2) when θ c =0 β(t) = (K p θ(t)+k d ω(t)) (2) where K p is a proportional gain and K d is a derivative gain. On the other hand, the pulse width of thruster operating is expressed in Eq. (3) β(t) t on = F jet Δ t ACS ;( β(t) F jet Δ) sgn(β(t))t ACS :( β(t) >F jet Δ) where t on is an on time pulse, t ACS is a basic PWM sampling time and Δ is moment arm [3]. (3) 3 Limit cycle analysis In the case of a roll attitude control system, the main parameters for controller design are maximum angle, maximum angular rate, frequency of the limit cycle, and duty ratio of the limit cycle. At first, the magnitude of the angle and the angular velocity should be set at a limit level of the system requirements. The frequency of limit cycle must be isolated with natural frequency of system vibration. Low duty ratio is needed for a long life span of the thrusters and for the low gas consumption. The frequency and amplitude of the limit cycle are related to the MTIB (minimum torque impulse bit), which is the minimum operating time for the thruster. The MTIB for PWM control system is given by Eq. (4) MTIB = t min F jet Δ (4) where t min is the minimum pulse time. Combining PD control law Eq. (2) and PWM control Eq. (3), switching lines in the phase plane can be defined as Eq. (5). ω = K p K d θ ± ( ) tmin F jet Δ K d t ACS (5) 789

4 A switching line divides the phase plane into two regions: a switch-on region and a switch-off region. Usually, there is a delay time associated with both the rising and falling edges of the thrust pulse. In a real reaction control system, there are also oscillations associated with the rising and falling edges of the thrust pulse. The rising delay time (t don ) is defined as the elapsed time from the on command to the initiation of thrust level. The falling delay time (t doff ) is defined as the elapsed time from the off command to the on thrust. 3.1 Zero External Torque The case with no external disturbing torque is described by constant velocity trajectories in the dead region and constant acceleration trajectories outside the dead zone. Three trajectories by different thruster operations with a positive torque, a negative torque or zero torque are considered. The behavior of motions can be described in terms of phase plane coordinates and its portraits are plotted to determine dynamical characteristics. In this case, the trajectories of limit cycle are shown in Fig. 2 (a). Fig. 2. Trajectories of limit cycle in the phase plane: (a) zero external torque; and (b) constant external torque. Maximum angular velocity of the limit cycle and angle of point 1 can be derived by switching lines, total operating time for thruster, and trajectories relationship of limit cycle. We use a real operating time (t min t don +t doff )for thruster and switching line equations in calculation of the maximum angular velocity. Different points are compared with Haloulakos s method described in [5] and [8]. ω max = ω 1 = T jetδ J ( ) tmin t don + t doff θ 1 = K d ω max + t minf jet Δ + ω m t don K p K p t ACS 2 (6) We can derive the amplitude of limit cycle after a minor transformation of 790

5 Eq. (1) and Eq. (6) as follows: θ max = θ 1 + ω2 maxj 2F jet Δ (7) The period, the frequency, and the duty ratio of limit cycle become as follows: T L =2t on +4 θ 1 ω max = 1 f DR = 2(t min t don + t doff ) T L (8) 3.2 Constant external torque Under a constant external torque, a reaction jet controller exhibits a limit cycle behavior shown in Fig. 2 (b) (for > 0, control torque T c < 0, and T c > ). Under the constant external torque the switching line and the minimum pulse time are changed with the magnitude of disturbance. The minimum pulse time in the presence of the external torque can be derived as follows: F jet Δ t ACS ; t min D = t min ; F jet Δ t ACS >t min F jet Δ t ACS t min Therefore, the switching line in the presence of a constant external torque is given by Eq. (10). ω = K p θ ± t min DF jet Δ (10) K d t ACS K d Then the maximum angular velocity of the limit cycle and angle of point 1 can be derived when we use a real operating time (t mind t don + t doff )for thruster and switching line equation. (9) ω max = ω 1 = (F jetδ ) (t mind t don +t doff ) 2J ( ) ( Kd θ 1 = t don ω max T ) d K p J t don + t min DF jet Δ + t ACS K p 2J t2 don (11) We can derive the amplitude of limit cycle as follows: θ max = θ 1 + ω 2 m 2(F jet Δ ) (12) Therefore, the period, the frequency, and the duty ratio of limit cycle become as follows. T LD = (F jetδ ) (t min D t don + t doff )= 1 f D DR D = (t min t don + t doff ) T LD (13) 791

6 4 Simulation results Simulation results are presented for a roll control system of the KSLV-I (Korea Space Launch Vehicle-I). The Korean launch vehicle is capable of injecting a 100 kg class satellite into a low earth orbit. The simulation parameters are as follows. Flight time is 70.2 sec; thrust level for thrusters is 22N 4(EA); moment of inertia I xx (kgm 2 ) = 501; moment arm of roll axes is 1 m; sampling time t ACS = 0.3 sec; rising delay time t don = 0.05 sec; falling delay time t doff = 0.05 sec; proportional gain K p = 987.6; derivative gain K d = ; constant external disturbance is 0Nm or 44Nm; initial condition for the angular velocity error is zero; the initial condition for the angle error is 10 deg. First, no external torque is applied. The corresponding simulation result is established Fig. 3 (a). The maximum angular velocity and angle of point 1 become (rad/sec) and (rad ) by the derived Eq. (6). If we use Halulakos s method, the maximum angular velocity and angle of point 1are (rad/sec) and (rad ). Both methods show only minor difference with simulation results. Second, Fig. 3 (b) shows the simulation results when a constant external torque is applied. The maximum angular velocity and angle of point 1 become (rad/sec)and (rad ) by the derived Eq. (11), which shows no difference with simulation results. However, if we use the Haloulakos method, the maximum angular velocity and angle of pont 1 are calculated as (rad/sec) and (rad ), which show significant differences with simulation results. Fig. 3. Simulation results: (a) undisturbed trajectory; and (b) disturbed trajectory 5 Conclusion In this paper a novel limit cycle analysis of the attitude control system using jet thrusters is presented based on a phase plane method. The limit cycle parameters are analyzed and obtained in terms of external disturbance, control moment, proportional gain, derivative gain, and the delay times of the thruster. It is shown in simulation results that the analysed results of the 792

7 limit cycle are more accurate than those of the Haloulakos method. The analyzed results can be used for the design of the attitude controller without Monte-Carlo simulation. Another advantage is that the control frequency that could be designed using equations related to the limit cycle can be separated from natural frequency of the system. 793