Robust Nonlinear Design of Three Axes Missile Autopilot via Feedback Linearization

Transcription

1 Robust Nonlinear Design of Three Axes Missile Autopilot via Feedback Linearization Abhijit Das, Ranajit Das and Siddhartha Mukhopadhyay, Amit Patra 1 1 Abstract The nonlinearity and coupling of the missile dynamics can be linearized and decoupled by well known feedback linearization techniques. The fundamental problem of feedback linearization technique is not able to cater the model uncertainty. In accordance to this, a new robust nonlinear controller structure using feedback linearization technique is presented here. It linearizes the nonlinear dynamics, decouples the longitudinal and lateral motions and caters the model uncertainty. The design is carried out in two steps. The first one is to design a robust feedback linearization controller. It linearizes the fast dynamics of missile and decouples the axes using input output feedback linearization technique. In the next step a robust linear controller has been proposed almost similar to a model reference controller which gives the desired tracking performances. The performances of the design in terms of stability, robustness, minor coupling between the longitudinal motion and lateral motion and guaranteed tracking are shown in a realistic 6-DOF simulation. 2 Introduction The basic requirement for an autopilot is fast response because of the short amount of time involved in the end game. Secondly minimum error is an obvious requirement if the missile is to achieve a hit to kill miss. Finally robustness of model uncertainties and decoupling between longitudinal and lateral motion are important in order for the missile to achieve its objective in the physical environment. The highly nonlinear nature of the missile dynamics [1],[2] due to the severe kinematics and inertial coupling of the missile airframe as well as the aerodynamics are always remain a challenge to the autopilot design that is required to have satisfactory performance for all flight conditions in probable engagements. 1 Authors are with the Systems and Information Lab., Dept. of Electrical Engineering, IIT Kharagpur and express their gratirude to DRDO, INDIA for financial and technical support. - abhijit.das@ieee.org

2 Classically, missile autopilots are designed using linear control approaches. Traditional approach is to first linearize the missile short period dynamics for each axis about an operating condition, and then to apply linear control theory to synthesize a feedback controller. This process is repeated at multiple operating conditions and the controllers are then scheduled with respect to the flight conditions. The assumptions hold insignificant at high angle of attack because of lack of knowledge in huge model i.e. aerodynamic parameter uncertainties and coupling. Aerodynamic considerations such as high maneuverability is must to intercept the modern threat profiles. Severe coupling due to high unwanted roll disturbance torque at high angle of attack is the most concern in homing head missile in view of: 1. May cause track loss of the missile homing head as seeker is having limited gimbal freedom from physical considerations. 2. High side force disturbance may lead to control saturation of the actuation servo system. A natural idea for handling the nonlinear dynamics is to design the autopilot on the basis of the more accurate nonlinear model, thus leading to nonlinear autopilots. One of such strategies is well known feedback linearization approach which uses the feedback and or coordinates transform to linearize the nonlinear system. Then it addresses the design issues on the linearized system thus obtained. Background on the application of feedback linearization technique to missile autopilot problem can be found in References. [3],[4] present a three axe missile autopilot design using classical linear time invariant control and static and dynamic approximate input output linearizing feedback control. [5] examined the closed loop stability of an air to air missile with a dynamic inversion controller using a two timescale separation. [6],[7] and [8] presented the linearizing transformation technique to the autopilot design of air to air missile systems. [9] investigates the feasibilities of autopilot design for highly maneuverable bank to turn missile using feedback linearization based approach. All these paper works show the comparable performances for well known model. But model uncertainty is a challenge. In the approach of the research, the nonlinear controller is designed corresponds to a basic application of linearizing techniques combined with input output linearization and time scale separation. The fast dynamics (inner loop) are linearized based on the input- output feedback linearization of the missile dynamics eventually the slow dynamics (outer loop) is designed in order to be robust with regards to uncertainties. The main feature of the research is the consideration of a three-axis time variant nonlinear missile model. The aerodynamic parameters are used into a look up table and as a function of mach no, angle of attack and maneuver plane roll orientation. Sensors and actuators modeling with physical limit are also taken into account. The objective of the present research is to design a three axes nonlinear autopilot that decoupled the roll pitch and yaw 2

3 axis and cater significant model uncertainties. At the same time, the autopilot performances in terms of BW, rise time and damping are satisfactory. 3 Plant Dynamics The nonlinear differential equations for the missile model are given as [1],[2], { u=rv qw+ 1 m [T X QSC DO ]+g x v=pw ru+ 1 m [T Y +QS{C S C lς δ Y + D ( C 2V M y β β Cyrr)}]+gy (1) ẇ=wu pv+ 1 m [T z+qs{c N C lη δ P + D ( C 2V zqq C z α α)}]+g z M {ṗ= 1 [ I I XX p+t MX +QSD{ D C XX 2V LP p C lξ δ R +C l }] M q= 1 [ I I Y Y q+t MY (I XX I Y Y )pr+qsd{ C mηδ P +C mp+ D (C Y Y 2V m q q+c m α α)}] M (2) ṙ= 1 Izz [ I zzr+t MZ (I Y Y I XX )pq+qsd{c mςδ Y +C ny+ D (C 2V nrr C M m β β)}] Where, the aerodynamic coefficients are used in terms of look up table and the nomenclatures are given as: C D0 = axial force or drag coefficient, C N = Normal Force Coefficient, C mp = Pitching Moment Coefficient, C S = Side Force Coefficient and C L = Rolling Moment Coefficient. All are used as a function of f (M, α R, φ a ). Here, α R = Resultant angle of attack and φ a = Maneuver plane roll orientation. The control force and moment effectiveness are function of Mach number and control deflection δ and used as a lookup table. C Lξ = Roll control moment effectiveness, C Lη and C Lζ are Pitch and Yaw control force effectiveness respectively. (X And moment coefficients C mς = C cpδ X cg) (X Lς and C D mη = C cpδ X cg) Lη. Let the D state, output and the input vectors be defined as X = [ u v w p q r ] T, Y = [ p q r ] T [ ] T and U = δr δ P δ Y The plant model can be represented as Ẋ = f (X) + g (X) U (3) Y = h(x) Where, f (X) = f 1 f 2 f 2 f 4 f 5 f 6 = rv qw + 1 [T m X QSC DO ] + g x pw ru + 1 [T m Y + QS{C S + D 2V M ( C y β β Cyr r)}] + g y wu pv + 1 [T z + QS{C m N + D 2V M ( C zq q C z α α)}] + g z 1 I XX [ I XX p + T MX + QSD{ D 2V M C LP p + C l }] 1 I Y Y [ I Y Y q + T MY (I XX I Y Y )pr + QSD{C mp + D 2V M (C m q q + C m α α)}] 1 I zz [ I zz r + T MZ (I Y Y I XX )pq + QSD{C ny + D 2V M (C nr r C m β β)}] 3

4 and QSC lς / m 0 QSC lη / m 0 g (X) = QSDC lξ / IXX QSDC mη / IY Y QSDC mς / IZZ 3.1 Application of feedback linearization The central idea of the feedback linearization is to algebraically transform a nonlinear system dynamics into a linear form by using state feedback, with input state linearization corresponding to complete linearization and input output linearization to partial linearization. The theory is well known and available in text books [10],[11]. Here the design methodology is presented only. The system under consideration is described by Ẋ(t) = f(x(t)) + g(x(t))u(t) Y (t) = h(x(t)) Where, X(t) is the n dimensional plant state, U(t) is m dimensional plant input and Y (t) is m dimensional plant output. f : R n R n, g : R n R n R m is m and h : R n R m are smooth functions. The input output feedback linearization of 3.1 can be performed if there exist constants ρ 1, ρ 2,..., ρ m and input output mapping of the form y (ρ 1) y (ρ 2) 1 (t) 2 (t).. y (ρm) m (t) = B (X (t)) + A (X (t)) u 1 (t) u 2 (t).. u m (t) Where, u i, y i, i = 1,..., m are components of U and Y, B : R n R m and A : R n R m R m are smooth with A(X(t)) invertible. For if this is the case, the following feedback control U (t) = A 1 (X (t)) [V (t) B (X (t))] (6) Where, V (t) = [ V 1 (t) V 2 (t).. V m (t) ] T R m yields (4) (5) y (ρ i) i (t) = V i (t), i = 1, 2,..., m (7) Since the new input V i only affects the output Y i, (3.1) is called a decoupling control law and the invertible matrix A(X) is called the decoupling matrix of the system. On the basis of it, single input, single output linear control techniques can be utilized to achieve the desired objectives. 4

5 3.1.1 Design formulation The design is carried out using a two time scale assumption to separate the dynamics as shown in Fig.1. Fig. 1: Block diagram of proposed control configuration 1. The inner loop (body rate loop) is design by a nonlinear controller using input output feedback linearising control 2. The outer loop (flight path rate loop) is design by a linear controller. In this paper, the inner loop nonlinear controller design methodology is mainly highlighted Inner loop control design The purpose of the inner loop design is to decouple the pitch roll and yaw channel and the equivalent transfer function between the control input V i and output Y i (body rates p, q and r) is that of an integrator. Here, in our problem, there are three inputs and three outputs available. X = [ u v w p q r ] T is the 6 1 state vector, U = [ ] T [ ] T δ R δ P δ Y is the 3 1 input vector and Y = p q r is the 3 1 output. Referring to the governing equations (1),(2), it may be redefined that where, ṗ q ṙ = M(X) + E(X) M(X) = [ ] Lf h 1 (X) L f h 2 (X) = L f h 3 (X) f 4 f 5 f 6 δ R δ P δ Y ] (8) 5

6 and the decoupling matrix E(X) = L g1 h 1 L g2 h 1 L g3 h 1 L g1 h 2 L g2 h 2 L g3 h 2 L g1 h 3 L g2 h 3 L g3 h 3 = QSDC lξ / IXX QSDC mη / IY Y QSDC mς / IZZ The 3 3 matrix E(X) is invertible over the region. Then the input transformation [12],[13]. δ R v 1 L r 1 U = δ P = E 1 f h 1 v 2 L r 2 f h 2 δ Y v 3 L r 3 f h 3 or δ R δ P δ Y [ = E 1 M + E 1 v1 ] v 2 v 3 The above three equations of the simple form [ ṗ q ṙ ] T = [ v1 v 2 v 3 ] T. Hence the control law decouples the longitudinal motion and lateral motion. In practice, the system description is rarely known exactly. The model uncertainty is not able to cater by the feedback linearization control law as shown in 9. The feedback linearization can only be performed based on nominal plant. Now our objective is how to control the plant uncertainty. After a lot of literature survey, it is found that various robust schemes have been proposed to account for the model uncertainty for the state input linearizable system and input output linearizable system. Basically after feedback linearized plant, a robust controller is designed to handle the plant uncertainty. In this research, a robust feedback linearization scheme based on model reference adaptive control is proposed to cater the model uncertainty. The scheme shows significant performances and this is one of the main contributions of this research. The derivation of the scheme is presented in the following section. The robustness of the scheme will be presented through a lot of 6 DOF simulation. 3.2 Robust feedback linearization control For simplicity, let we denote the nominal plant (model) output is Y N and actual plant output is Y. The output equation may be expressed as (9) Ẏ = M (X) + E (X) U Ẏ N = M N (X N ) + E N (X N ) U (10) The model uncertainty is included in M(X) and E(X). The input output relationship of the nominal plant as discussed earlier is ẎN = V N, where the relationship between the linear control V N and the plant input U is given by U = E 1 N [V N M N ]. Our objective is to derive a new linear control V and the 6

7 relationship between this new input V to FBLC and actual plant input U such that it will linearize and decouple the output equation in the form Ẏ = V. Now doing some mathematical formulation, we can express V in terms of V N and as well as in terms of U as follows ) V V N = Ẏ ẎN V N = V (Ẏ Ẏ N (11) Hence, the control input is obtained as [ ) ] U = E 1 N V (Ẏ Ẏ N M N (12) The inner loop block diagram with robust feedback linearization control law is shown in Fig.2. The same input signal U goes to model as well as plant. The actual ) output is compared with the expected output and the error signal (Ẏ Ẏ N is feedback to the reference input V. Hence, the equivalent transfer function between the input signal V and the actual output Y is just like an integrator. It may be noted that under perturbed cases also, the robust scheme holds the linearity and decoupling properties. Fig. 2: Modified robust feedback linearization controller The inner loop of the dynamic inversion control law controls the fast states p, [ q and r. This ] loop calculates the virtual control [ surface deflection ] commands v1 v 2 v 3 from the rate commands given pd q q r d by the slow dynamics [ ] φ f z f y. The desired dynamics used are given by ṗ = k4 (p d p), q = k 5 (q d q) and ṙ = k 6 (r d r), where k 4, k 5 and k 6 are design parameters. Hence may be written that v 1 v 2 v 3 = k 4 (p d p) k 5 (q d q) k 6 (r d r) (13) 7

8 3.3 Outer loop design Here the basic assumption is that the coupling between the force and control deflections is relative small in comparison with the coupling between the moment and control deflection [1]. Hence, it is neglected. The linearized plant transfer functions between rate and latax are derived as ( V m 1 s2 q = ωz 2 (1 + T α s) f z ) (, f V m 1 s2 y r = ωz 2 (1 + T α s) ) and φ p = 1 s The gains k 1, k 2, k 3 are designed to satisfy the desired specifications of autopilot as demanded by the guidance loop. 4 Simulation results The design is thoroughly validated in a full scale 6 DOF platform considering all the plant parameter variations. A conventional linear three loop lateral autopilot and a PI type roll autopilot are also designed to show the potential of the nonlinear autopilot performance as a comparison. All the cases, the nominal as well as off nominal cases, the performances of the nonlinear autopilot are carried out in two stages. 1. Performances of the nonlinear autopilot by applying step body rate demands (inner loop performances). 2. Performance of the nonlinear autopilot by applying step latax demands (outer loop performances). The performances are compared with linear controller and discussed in the following section. 4.1 Autopilot performances by applying step body rate demands In 6-DOF, body rate is forced and are applied to inner loop as demand for different time intervals for different channels. Fig.3 shows that body rate response of the nonlinear controller for nominal condition when the pitch body rate is demanded at 10.5 sec, yaw rate is demanded at 12.3 sec and roll rate is demanded at 11.3 sec. It may be pointed out from the Figure that the ignition of body rate in one channel is not affects the other two channels. It proves the evidence of decoupling. The autopilot response is almost critically damped in nature and time constant (0.05 sec) indicates the band width of the order of 4Hz. It again validates the linear controller design. Fig.4 shows the same with disturbance condition and compared with the performances obtained with conventional linear controller. Here also it may found that the nonlinear controller decouples the longitudinal 8

9 Fig. 3: Body rate profiles with forced body rate demand Fig. 4: A comparative performances with forced body rate demand and lateral motion. Where, linear controller fails to decouple the motions. We 9

10 may note from Fig.4 that severe coupling for linear controller when simultaneous pitch and yaw maneuver is present. 4.2 Autopilot performances by applying step latax demands Latax demand is forced as a outer loop demand for pitch and yaw channel, whereas, roll channel demand is zero. A step Pitch latax demand of 40m/s2 is applied at 10.5 sec where as, a step yaw latax demand of 5m/s2 is applied at 12.3 sec. The corresponding latax profiles are shown in Fig.5 The body rate Fig. 5: Autopilot Performances for step latax demand profiles are shown in Fig.6. Here, also significant amount of decoupling among the three axes may be found compared to linear controller. The roll rate is almost silent for the nonlinear controller which is the main objective of the design. The control deflection profiles are shown in Fig.7. 5 Conclusion This paper demonstrates the design of a nonlinear controller using feedback linearization technique for the missile control problem associated with severe coupling of longitudinal and lateral motion and significant plant uncertainty. The nonlinear autopilot design methodology is carried out in two time scale separation. The 10

11 Fig. 6: Body rate profiles for step latax demand Fig. 7: Control deflection for step latax demand 11

12 design is validated here through a full scale 6 DOF simulation. A comparison is also made with linear traditional controller. Comparing the 6 DOF simulation results, the following salient features of the new nonlinear controller design may be noted: 1. It decouples the roll, pitch and yaw channel. 2. The roll rate is fairly close to zero during maneuver phases. The FBLC reduces the roll induced yaw rate and control requirement and hence control saturation. 3. The autopilot response (both latax as well as body rate) is significantly faster. Quick correction is possible. REFERENCES [1] P. Garnell and D. J. East, Guided Weapon Control System. Pergamon Press,2nd Ed.,Oxford, [2] A. L. Greensite, Analysis and Design of Space Vehicle Flight Control Systems. Pergamon Press,2nd Ed.,Oxford, [3] E. Devaud, J.-P. Harcaut, and H. Siguerdidjane, Three-axes missile autopilot design: From linear to nonlinear control strategies, Journal of Guidance, Control, and Dynamics, vol. 24, no. 1, pp , January-February [4], Various strategies to design a three axes missile autopilot, AIAA, no. 3976, [5] C. Schumacher and P. Khargonekar, Stability analysis of a missile control system with a dynamic inversion controller, Journal of Guidance, Control and Dynamics, vol. 21, June [6] M. Tahk, M. M. Brigges, and P. Menon, Application of plant inversion via state feedback to missile autopilot design, Procedings of the 27 th IEEE Conference on Decesion and Control, pp , December [7] P. Menon and M. Yousefpor, Design of nonlinear autopilots for high angle of attack missiles, Optimal Synthesis, [8] P. Menon, V. Iragavarapu, and E. Ohlmeyer, Nonlinear missile autopilot using time scale separation, AIAA, vol , [9] J. Huang, C. F. Lin, J. R. Cloutier, J. H. Evers, and C. D. Souza, Robust feedback linearization approach to autopilot design, Proc. IEEE Conf. Contr. Applicat., vol. 1, pp ,

13 [10] J.-J. E. Slotine and W. Li, Applied Nonlinear Control. Prentice Hall, [11] A. Isidori, Nonlinear Control Systems: an introduction. Berlin: Springer- Verlag, [12] A. Das, R. Das, S. Mukhopadhyay, and A. Patra, Nonlinear autopilot and observer design for a surface-to-surface, skid-to-turn missile, Second India annual conference, Proceedings of the IEEE INDICON 2005, pp , December [13], Sliding mode controller along with feedback linearization for a nonlinear missile model, 1st International Symposium on Systems and Control in Aerospace and Astronautics, January