1 INDIRECT ADAPTIVE CONTROL OF MISSILES Ufuk Deirci* and Feza Kerestecioglu** *Turkish Navy Guided Missile Test Station, Beykoz, Istanbul, TURKEY **Departent of Electrical and Electronics Engineering, Bogazici University, Bebek, Istanbul,80815, TURKEY Abstract: An indirect adaptive controller is designed for aerodynaically driven issiles. The design is developed using a linearized odel of a issile. Recursive least squares estiation ethod with exponential forgetting is used to estiate the tievarying issile paraeters. A covariance anageent algorith is suggested to prevent the exponential increase in covariance gain due to lack of persistently exciting inputs produced by the guidance unit. Keywords: Missile, self-tuning control, tie-varying systes, recursive least squares. 1. INTRODUCTION The purpose of this paper is to present soe results obtained for the indirect adaptive control of a linear issile odel. The objective of the controller is to follow a preprograed trajectory aerodynaically without saturation at the output of the actuator syste. The issile dynaics is tie-varying with respect to the Mach nuber profile and changes in the environental conditions. The issile is assued having no roll otion and, hence, is controlled in the pitch and yaw directions independently. The sae indirect adaptive controller is used for both directions. 2. MISSILE DYNAMICS The equations of otion of a issile consist of six kineatic and six dynaic first order nonlinear differential equations. Kineatic equations are divided into two groups; three translational kineatics and three rotational kineatics. The kineatic equations of otion ay be derived fro the geoetric relations between the earth and issile axes. Dynaics equations are also divided into two groups; naely, three translational and three rotational dynaics. The dynaic equations of otion ay be derived fro Newton s law of otion for a rigid body. A detailed derivation of these equations are given by Mahutyazicioglu and Eskinat (1994). Missile dynaics also consist of odelling of variation of issile ass and oent of inertia, control surfaces, control actuation syste and dynaics of easuring instruents. In addition to these, aerodynaic forces act on the issile should be taken into consideration during the derivation of a issile odel. 3. LINEARIZATION OF THE MISSILE MODEL The equations of otion of a issile for a coplicated set of coupled nonlinear differential equations. These nonlinear equations are too coplicated for the purpose of the controller design. Therefore, they need to be siplified and linearized. 1 This work has been supported under the grants of The Scientific and Technical Research Council of Turkey (TUBITAK-SAGE) and B.U. Research Fund (Grant No. 98A201).
No rolling otion: With this assuption, the otions in the pitch and yaw directions becoe uncoupled and can be controlled independently. Constant speed, Mach nuber and altitude: With these assuptions, aerodynaic forces acting on the issile turn out to be constant. Sall angle of attack and sideslip angles: This assuption iplies that, the velocities in pitch and yaw directions reain sall and, therefore, brings linearity in the odel. In addition to that the equations of otion can be written in ters of angle of attack and sideslip angles of the issile. gyroscope. By using these two easured signals and the coand acceleration, a nc, coing fro the guidance loop, which is the reference signal, to be followed, the pitch autopilot deterines the necessary fin deflection, e, to control the issile in the pitch direction. The specification for the pitch autopilot is given in the for of a desired closed-loop transfer function G. This desired transfer function is chosen as B ( z) G ( z) (3) A ( z) After all these assuptions, the equations of otion of a issile can be linearized around an equilibriu point under a certain flight regie. The discrete-tie transfer functions describing this linearized dynaics of the issile can be expressed as follows (Mahutyazıcıoğlu and Eşkinat,1994): In the pitch direction, one has anz( z) G ( z) e z p( z) ( z) and G ( z) e p ( z), (1) where a nz is the noral acceleration in the pitch direction p is the pitch-rate and e is the elevator deflection. On the other hand, in the yaw direction, the linearized dynaics can be expressed by an y ( z) G ( z) r y r( z) ( z) and G ( z) r r ( z), (2) where a n y is the noral acceleration in the yaw direction r is the yaw-rate and r is the rudder deflection. As entioned previously, the controller design has been developed independently in the pitch and yaw directions due to the assuption that there is no roll otion. The sapling-tie T s is chosen as 0.07s. This choice is based on the requireent that the sapling-rate should be less than the bandwidth of the control actuation syste (CAS) and larger than the bandwidth of the desired closed-loop response specification (Åströ and Wittenark, 1997). 4. INDIRECT ADAPTIVE CONTROL The following autopilot design is considered in the pitch direction. The pitch autopilot requires two easureents; noral acceleration in z direction, a nz, easured by an acceleroeter and the angular rate in the pitch direction, p, easured by a rate For the case where the syste paraeters are known, the controller equation will be R( ( T( a ( S( a (. (4) e nc nz where q denotes forward shift operator. Here, R( and S( are obtained fro the well-known Diophantine equation. A( R( B( S( A ( A ( (5) where A and B are denoinator and nuerator of G, respectively. z 4.1 Causality Conditions for the Autopilot Design. The standard indirect adaptive controller (Wittenark and Åströ, 1984) based on RLS is ipleented and to have a casual controller, soe degree conditions has to be iposed on the polynoials related to design specifications, i.e., A o, A, B. For the controller polynoials R, S, and T in (4), the following conditions ust hold to have a causal pitch autopilot. deg S deg R and deg T deg R (6) A specification for the pitch autopilot is the desired observer polynoial A o. Before defining an observer polynoial, the causality conditions for the observer polynoial ust be checked out as (Åströ and Wittenark, 1995) hence, o deg A 2deg A deg A 1 (7) o deg A o 4 21 1. To have a strictly proper syste deg A 0 can be selected as 2. That is, denoting the reciprocal polynoials with a superscript (*), 1 2 A o 1 a01q a02q.
Note that (6) also iplies that deg A deg B which is satisfied by (3). deg Adeg B 2 2 0 output of the issile, which is the noral acceleration in the pitch direction winds up as can be seen in Fig. 1 due to estiation proble of the issile coefficients. Equation (5) can be solved by choosing deg R = 2, deg S = 1, i.e., and R ( q S ( q 1 1 2 ) 1 r1 q r2q 1 1 2 ) s0q s1q Note that R polynoial is assued to be onic without loss of generality. The T polynoial can be calculated (Åströ and Wittenark, 1995) to assure zero steady-state error as, A (1) T Ao, (8) B(1) of the issile transfer function. 4.2 Estiation of the Missile Paraeters Fig. 1. Wind-up of the issile output. Estiates of the issile paraeters can be seen in Fig. 2. when the input signal is not persistently exciting. The autopilot design by ipleenting the indirect self-tuning ethod requires satisfactory estiates of paraeters. There are five paraeters to be estiated in every sapling instant and the dynaics of the issile changes with respect to the environental conditions and the Mach nuber. It is well-known that the paraeters of ost deterinistic tie-varying systes can be estiated satisfactorily by ipleenting the RLS estiator with exponential forgetting. For that, a persistently exciting input signal and a properly selected forgetting factor are required. To do so, the following steps are taken: Firstly, the forgetting factor of the RLS estiator is selected by running siulations on the issile odel, and using a persistently exciting signal and various forgetting factors between 0.85-0.99. Then, the forgetting factor selection can be done for the RLS estiator by coparing the estiation results for each siulation done with different forgetting factors but the sae persistently exciting signal. As a result of this, the forgetting factor is found to be 0.96 suitable for the paraeter tracking during the whole flight period. After the selection of the forgetting factor, a typical coand acceleration (a nc ) data generated by the guidance loop has been used to siulate the issile aerodynaic flight in a ore realistic way. It has been observed that the coand deflection signal generated by the given coand acceleration data can only excite the RLS estiator for the initial period of the whole issile flight. Besides that, the Fig. 2. Estiates of the issile odel. It has been found that the reason for this wind-up is the unsatisfactory input to the issile. That causes the covariance gain of the RLS estiator to increase exponentially. This, in return leads to the loss of paraeter identifiability. To overcoe this proble, the paraeters are estiated by adopting the idea of conditional updating (Åströ and Wittenark, 1995). It is well-known that, if there is sufficient excitation, then RLS ensures that agnitude of the eleents of the covariance atrix, P(, decreases in tie. Thus
as the estiates becoe ore accurate they require less adjustent. Then it can be stated that in a recursive estiator, P( is the basic variable for controlling the adaptive capabilities of the estiator with a correct selection of the forgetting factor (Ljung and Söderströ, 1983). Therefore, for paraeter tracking and satisfactory estiates of the issile paraeters, the size of the P( atrix should be controlled. 5. SIMULATION RESULTS By ipleenting the above algorith, it can be assured that the covariance atrix doesn t grow exponentially when there is not sufficient excitation and the estiates of the issile paraeters are anaged properly. This can be seen in Fig. 3 and 4. This can be done by using the estiation error as the reference point for the conditional updating. This algorith can be expressed as (Deirci, 1998) If (, T L If T ( T, U θ ˆ( θˆ( k 1), P ( P( k 1), L θˆ ( θˆ( k 1) K( (, P ( (1 K( ( k 1)) P( k 1) /, If (, T U Fig. 3. Estiates of the Nuerator Paraeters. θˆ ( θˆ (k1) K( (, P( k ) 10 6 I where, T U and T L are predeterined thresholds,, T ( y( ( k 1) θˆ( k 1) is the prediction error and K( P( (. Also, note that θ ˆ( k ) and ( are the estiate and regressor vectors as defined for the RLS in a standard way. 4.3 Calculation of the Controller Paraeters. After the estiation of the paraeters of the linear issile odel, control input can be calculated (Åströ, 1983) in the pitch direction as in (4). The controller polynoials R and S can be calculated fro (5), save that, instead of A and B, their estiates are used. Notice that, the controller does not attept to cancel the plant zeros. Because by solving the nuerators of the transfer functions Gz and Gy for typical flight regies, it can be seen that the zeros near the coplex right-half plane. Fig. 4. Estiates of the Denoinator Paraeters. Subsequently, an interesting result was observed: The estiated paraeters have not been converging to the true paraeters of the linear issile odel as in Fig. 3 and 4. Nevertheless, the response of the linear issile odel followed the coand signal satisfactorily with respect to the desired closed-loop response specification as can be seen in Fig. 5. The coand signal generated by the guidance loop in every sapling instant can be seen in Fig. 6.
Fig. 5. Noral Acceleration in the Pitch Direction. Fig. 8 Elevator Deflection at the Input of the CAS. Fig. 6. Coand Acceleration in the Pitch Direction. The pitch-rate of the issile is also regulated with the sae controller as can be seen in Fig. 7. Fig. 8 Coand Elevator Deflection at the Output of the CAS. It is also observed that the choice of the sapling interval is also critical for a satisfactory closed-loop response. 6. CONCLUSIONS In this work a standard pole-placeent indirect adaptive control technique is applied to the aerodynaic control of a guided issile. Fig. 7. The Pitch-rate of the Missile. The siulations under typical flight regies show that the control actuation syste can generate the required deflections to follow the predeterined trajectory aerodynaically without any rate or position saturation as can be seen in Fig. 8 and 9. There has been an estiation proble for the linear issile odel due to the fact that the input ay not be persistently exciting throughout the whole flight trajectory. Therefore, the estiates of the issile paraeters have been winding-up because of exponentially increasing covariance gain. Hence, an algorith based on covariance anageent has been suggested to solve the proble. This paper is a part of a continuing project and the next steps to be taken will be to validate the controller design in siulations using the nonlinear issile odel and hardware-in-the-loop siulations which are in progress.
REFERENCES Åströ, K.J. (1983). Theory and Applications of. Adaptive Control, Autoatica, 19, 471-486. Åströ, K.J. and B. Wittenark (1984). Practical Issues in the Ipleentation of Self-tuning Control, Autoatica, 20, 501-517. Åströ, K.J. and B. Wittenark (1995). Adaptive Control, Addison-Wesley Publishing. Åströ, K.J. and B. Wittenark (1997). Coputer- Controlled Systes-Theory and Design, Prentice Hall, New Jersey. Deirci, U. (1998). Adaptive Aerodynaic Control of Guided Missiles, M.S. Thesis, Bogaziçi University. Ljung, L. and T. Söderströ (1983). Theory and Practice of Recursive Identification, MIT Press, Cabridge. Mahutyazıcıoğlu, G. and E. Eşkinat (1994). A general validation of guided and unguided issile siulations done by TÜBÍTAK-SAGE, 22121, TÜBÍTAK, ANKARA.