International Journal Of Advances in Engineering and Management (IJAEM) Page 141 Volume 1, Issue 5, November - 214. Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus 1 Rami Ali Abdalla, 2 Muawia Mohamed Ahmed, 3 Eltahir Mohamed Hussein 1 P.G. Student, Department of Control Engineering, Faculty of Engineering College, El.Neelain University Kh, Sudan (E-mail id: rami_2236@hotmail.com) 2 Associate Professor, Department of control Engineering, Faculty of Engineering, El.Neelain University, Kh, Sudan (E-mail id: muawia15858@hotmail.com) 3 Associate Professor, Department of Biomedical Engineering, College of Engineering, Sudan University of Since and Technology, Kh, Sudan (E-mail id: Altahir_33@yahoo.com) -------------------------------- -------------------------------- Abstract: The present paper aims at designing of Automatic Landing System for MISSILE based on Root Locus modern control system. The control method is used to determine the gains of the controllers to apply the Root Locus method. The block diagram of the proposed control system with required controller gains is established. The transfer functions for open loop and then the closed loop are obtained based on automatic control principles. The Root Locus for open loop is drawn and then gain (K) values are found for given damping ratios. Finally the step responses of the closed loop system with automatic landing system controller auto-pilot were drawn. The digital simulation results have proven the effectiveness of proposed control system in terms of fast response in the presence of external disturbances. Keywords: MISSILE, Two-loop Auto-pilot, Pitch Attitude, Root-Locus. -------------------------------- -------------------------------- 1. Introduction The missile flight control system is one element of the overall homing loop [2]. Due to the wide parameter variation and stringent performance requirements, missile auto-pilot design is a challenging task. The traditional method of guaranteeing stability in the presence of aerodynamic parameter variation or uncertainty is the gain scheduling control strategy. Modern air-to-air or surface-toair missiles need large and uncertain flight envelopes, for which accurate aerodynamic parameters are difficult or extremely expensive to obtain from wind tunnel tests; also, the gain scheduling controllers need more operating points. The control objective for these missiles is to ensure accurate interception, with guaranteed robustness, without sacrificing maneuverability. For this purpose, many advanced modern control theories have been extensively studied by numerous researchers to address this problem[3]. A Guided missile is one which receives steering commands from the guided system to improve its accuracy. Guidance system actually gives command to the auto-pilot to activate the controls to achieve the correction necessary. Auto-pilot is an automatic control mechanism for keeping the spacecraft in desired flight path. An auto-pilot in a missile is a close loop system and it is a minor loop inside the main guidance loop. If the missile carries accelerometer and rate gyros to provide additional feedback into the missile servos to modify the missile motion then the missile control system is usually called an autopilot. When the auto-pilot controls the motion in the pitch and the yaw plane, they are called lateral auto-pilot. For a symmetrical cruciform missile pitch and the yaw auto-pilots are identical [4]. As with phase lag and phase lead compensation, the purpose of lag-lead compensator design in the frequency domain generally is to satisfy specifications on steady-state accuracy and phase margin. Typically, there is also a specification (implicitly or explicitly) on gain crossover frequency or closed-loop bandwidth. A phase margin specification can represent a requirement on relative stability due to pure time delay in the system, or it can represent desired transient response characteristics that have been translated from the time domain into the frequency domain. A specification on bandwidth or crossover frequency can represent a requirement on speed of response in the time domain or a frequency-domain requirement on which sinusoidal frequencies will be passed by the system without significant attenuation [5]. 2. Missile Autopilot Overview Guided missiles have assumed much importance in recent years. A guided missile is one which receives steering commands from the guided system to improve its accuracy. Guided action for guided missiles may be defined as the process of gathering information concerning the flight of a missile towards a given objective or target and utilising this information to develop manoeuvring commands to the control system of the missile. Guidance system functions by comparing the actual path of the missile with the desired path and providing commands to the control system which will result in manoeuvring the missile to its desired path. Guidance system actually gives command to the auto-pilot to activate the controls to achieve the correction necessary. Auto-pilots are closed loop system and these are minor loops inside the main guidance loop. An auto-pilot may be defined as the missile control system which modify the missile motion according to accelerometer and / or gyros feedback which provide information about the missile acceleration and body rate respectively. A lateral auto-pilot receives guidance command from the guided system of the missile to produce desired missile acceleration in lateral planes to follow the guidance path needed to the target. The autopilot responses to guidance system demand by deflecting the control surfaces of the missile for aerodynamic controlled missiles. The deflections in
control surfaces produce change in missile angle of incidence. If the incidence angle is changed, the forces acting on the missile body changes and it results in change in missile acceleration. The two-loop auto-pilot system uses two-loops to feedback information of missile motion to the forward path of the autopilot. One loop is involved with body rate information which is fed back using one rate gyro. The other is the missile acceleration, sensed using accelerometer and provides the main feedback. The auto-pilot system results in change in missile motion. So, modelling of missile airframe dynamics is an important part of configuring an autopilot system. Missile dynamics is of non linear type. For configuring missile dynamics in transfer function form the missile airframes are trimmed and then linearized[1].the following block diagrams (fig. 1 represents the transfer function model of flight path rate demand two- loop auto-pilot in pitch plane).[5] Fig. 1 Conventional Flight Path Rate Demand Two-Loop Auto-pilot Configuration Where G1 and G2 are the Aerodynamic transfer function, G3 is the Actuator transfer function, γ is flight path rate;q is pitch rate, wa is natural frequency ofactuator, ζa is damping ratio of actuator, KP,Kq,Kb are the control gains, wb is weather cock frequency, Ta is the incidence lag of the air frame, η is Elevator deflection; σ is aquantity whoose invese determines the locations ofnon minimum phase zeros, wi integrator gain. The open loop model i.e. the cascaded combination- G1G2G3 of fig. 1 can be converted to the corresponding state space model given by X = Ax = Bu & y = Cx (discussed in the next section) and the conventional two-loop configuration can be converted to an equivalent state space model. The open loop model of two loop autopilot (Fig. 1) Fig. 2 Four State Variables can be converted to state variable form based on the following four state variables: x1=γ Flight path rate demand ; x2=q pitch rate ; x3= η elevator deflection ; x4=η (rate of chance of elevator deflection Out of them x1 and x2 have been considered to be as outputs. Thus two-loop autopilot model is a SIMO (single input multiple outputs) system. Such that the A, B & C matrices become (taken from [5]). 3. Methodology The methodology followed to achieve the desired system in this paper is could be stated as, first of all it was the modeling of the system this includes, Setting of the aircraft equations of the motion and Derivation of the transfer function and state space, then comes the setting the requirement and testing the controller with different gains and analyzing the result to obtain the final desired controller. A. Transfer Function and State-Space Two-Loop Auto-pilot Recognizing the fact that the modeling equations above are already in the state-variable form, they can be rewrite as matrices as shown below. 1 (1 + σ 2 w 2 b ) K bσ 2 2 w b T a T a T a K b σ 2 2 w b A = (1 + w 2 b T 2 a ) 1 K b W b T a (1 σ 2 T 2 a ) T a (1 + σ 2 w 2 b ) T a (1 + σ 2 w 2 b ) 1 2 W a 2ζ a W a ; B = K q w b 2 (1a) and C = 1 1 (1b) Page 142
Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus State Space Now the state space equivalent model of conventional two-loop auto-pilot (Fig.1) can be represented by the state space equation given by (1a & 1b). A = 2.77 1.186 2.8894.4269 1 5.6161 58.388 2.77 324 216. x 1 x 2 x 3 x 4 x 5 + 38881 u (2a) Y = 1 1 x 1 x 2 x 3 x 4 x 5 (2b) Numerical Values The following numerical data for a class of guided missile have been considered for MATLAB simulation [5]. Transfer Function Of Two-Loop Auto-pilot In Pitch Plane Dynamical stability analysis is performed in the following section using Matlab software for examining the roots of the characteristic equation. The function ss is used for creating state-space model within the Matlab enviroment, which takes the model data as input and produces ss object that store this data in a single Matlab variable. Transfer function is by matlab in the following manner: Wo=tf(system). 1: 2: 166 S 2 3.648e8 S 2.344e6 S 4 + 727.2 S 3 + 1.12e5 S 2 + 7.585e4 S 167 3888 S 2 1.987e6 S 6.44e6 S 4 + 727.2 S 3 + 1.12e5 S 2 + 7.585e4 S 167 (3) (4) The resulting closed-loop step response is shown Figure 2. Fig. 2 Step response of original Two loop autopilot Design requirements The next step is to choose some design criteria. In this example we will design a feedback controller so that in response to a step command of pitch angle the actual pitch angle the design requirements are the following. Overshoot less than 1% Rise time less than 2 seconds Settling time less than 1 seconds Steady-state error less than 2% Page 143
Controller synthesis can be done using different frequency or time-domain methods. The root locus technique is applied in this paper, the discussion of which is presented below. This technique provides graphical information in the complex plane on the trajectory of the roots of the characteristic equation for variations in one or more system parameters. Since the roots placement in the complex plane governs the type of the response that can be expected to occur, the ability to view the movement of the roots in the complex plane, as one or more system parameters are varied, turns out to be very useful. The roots of the closed-loop characteristic equation are obtained using graphical-user interface (GUI) within Matlab named SISO Design Tool, designated for controller design using root locus technique, among others. At the beginning, a designer can pick one of the feedback control system structures, and import transfer functions from Matlab s workspace. The root locus and the closed-loop step response plot of the transfer function 1 defined in Fig.3. Fig.3 root locus and the closed-loop step response plot of the transfer function 1. The root locus and the closed-loop step response plot of the transfer function 2 defined in Fig. 4 Fig. 4 Root Locus and the closed-loop step response plot of the transfer function 2. B. Lead Compensation The transfer function of a typical lead compensator is the following, where the zero is smaller than the pole, that is, it is closer to the imaginary axis in the complex plane. S Z C S = K (5) S P In order to see the effect of moving the pole of the lead compensator, you can enter different numerical values under the Compensator Editor tab. Any changes in the compensator here will be reflected in the root locus plot. Alternatively, you can tune the compensator graphically directly from the root locus plot. Specifically, if you click on the open-loop pole at -3 (marked by a red x) you can then slide the pole along the real axis and observe how the root locus plot changes. Specifically, you should see that as you move the pole to the left, the root locus gets pulled farther to the left (and further into our desired region)[7]. Then by clicking the Show Analysis Plot button a window entitled LTI Viewer for SISO Design Task displaying the system's closed-loop step response will open. You can also identify some characteristics of the step response. Specifically, right-click on the figure and under Characteristics choose Settling Time. Then repeat for Rise Time. 4. Simulation Results The simulations are carried out in MATLAB environment and the results obtained are shown in Fig.5, Fig.6, Fig.7, Fig.8, Fig.9 and Fig.1. Page 144
Fig. 5 control and estimation tool manager of the transfer function 1. Fig. 6 SISO design for the transfer function 1. Fig.7 Closed-loop step response of the transfer function 1. Page 145
Fig. 8 control and estimation tool manager of the transfer function 2. Fig. 9 SISO design for the transfer function 2. Fig.1 Closed-loop step response of the transfer function 2. 5. Conclusions The accomplished dynamical behavior of the missile two-loop auto-pilot, with selected prefilter and compensator transfer function parameters completely satisfies the design requirements. It can be concluded that steady-state error was completely eliminated and the overshoot value of 9.8, while settling time value around.786 s and rise time value of.55 s for transfer function1. steady-state error completely eliminated and overshoot value of 3.42, while settling time value of.236 s and rise time value of.236 s for the second transfer function. REFERENCES [1] Jyoti Prasad Singha Thakur, Amit Mukhopadhyay, 213, A Simple Design Approach In Yaw Plane For Two Loop Lateral Autopilots, International Journal of Innovative Research in Science, Engineering and Technology. [2] Paul B. Jackson, 21, Overview of Missile Flight Control Systems, Johns Hopkins Apl Technical Digest. [3] He Shao-ming, Lin De-fu, 214, Missile two-loop acceleration autopilot design based on L1 adaptive output feedback control, Intl J. of Aeronautical & Space Sci. [4] A. Chowdhury and S. Das, 213, Analysis and Design of Missile Two Loop Autopilot, Advance in Electronic and Electric Engineering. [5] Parijat Bhowmick, Prof. Gourhari Das, 212, Modification of Classical Two Loop Autopilot Design using PI Controller and Reduced Order Observer (DGO), International Journal of Engineering Research and Development. [6] Branimir Stojiljković - Ljubiša Vasov, Časlav Mitrović, Dragan Cvetković, 29, The Application of the Root Locus Method for the Design of Pitch Controller of an F-14A Aircraft, Strojniški vestnik - Journal of Mechanical Engineering. Page 146