Integrated Estimation, Guidance & Control II

Transcription

1 Optimal Control, Guidance and Estimation Lecture 32 Integrated Estimation, Guidance & Control II Prof. Radhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Motivation To fuse the estimation, guidance and control loops at various levels Benefits arise because integrated designs are capable of retaining and exploiting the synergy between various subsystems Integrated design approaches proposed in literature can be broadly classified into three groups: Integrated guidance and control (IGC) Integrated estimation and guidance (IEG) Integrated estimation guidance and control (IEGC) 2

2 Philosophy IEG IEGC (IGC&E) 3 Philosophy 4

3 Topics Integrated Estimation and Guidance P. N. Dwivedi, S. N. Tiwari, A. Bhattacharya and Radhakant Padhi, A ZEM Based Effective Integrated Estimation and Guidance of Interceptors in Terminal Phase, AIAA Guidance, Navigation and Control Conference, 2010, Toronto, Canada. Integrated Guidance, Control & Estimation P. N. Dwivedi, S. N. Tiwari, A. Bhattacharya and Radhakant Padhi, A ZEM Dynamics Based Integrated Estimation Guidance and Control of Interceptors for High Speed Targets, AIAA Guidance, Navigation and Control Conference, 2011, Portland, USA. 5 A ZEM Based Effective Integrated Estimation and Guidance of Interceptors in Terminal Phase Prof. Radhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

4 Reference P. N. Dwivedi, S. N. Tiwari, A. Bhattacharya and Radhakant Padhi, A ZEM Based Effective Integrated Estimation and Guidance of Interceptors in Terminal Phase, AIAA Guidance, Navigation and Control Conference, 2010, Toronto, Canada. 7 Concept: Lateral acceleration commands have been obtain as an algebraic function of the estimated states (with no delay). Guidance parameters ZEM (all three components) and T go has been chosen as a state of estimator itself. Note : In PN Guidance, a = ZEM T 2 go 8

5 Summary of Extended KF: Continuous-Discrete Formulation Model Initialization Gain Computation Xɺ ( t) = f ( X, U, t) + G( t) W ( t) Y = h( X ) + V k ˆ X ( t ) = X 0 0 T P0 = E Xɶ ( t 0 ) Xɶ ( t 0 ) k T T K e ( t ) = P k k C k C k Pk C k + R h w here, C k = X ˆ X k 1 9 Summary of Extended KF: Continuous-Discrete Formulation Updation Propagation ( ) k ( ) ( ) Xˆ = Xˆ + K Y h Xˆ + k k e k k T + T k = ek k k ek k + ek k ek P I K C P I K C K R K ( preferable ) ( I K e C ) ( not preferable ) k k Pk = ˆɺ X ( t) = f ( Xˆ, U, t); Xˆ ˆ k X k T T Pɺ ( t) = AP + PA + GQG ; Pˆ Pˆ f where A( t) = X Xˆ ( t ) k k

6 Problem Formulation State, Measurement & Desired Output Equations States variables are The measurement variables are the direct output of seeker ( range, range rate, gimbal angles and gimbal angles rates) given as Output variables are 11 ZEM-Dynamics in 1-D & Time-to-go Dynamics ZEM in y-direction R ZEM = y + V T y y go d ( ZEM ) = y ɺ + V ɺ T + V T ɺ dt Vɺ = a a y Ty M y y y go y go Time-to-go (T go ) go ( / ) T = R R ɺ Tɺ go Rɺ Rɺ R Rɺɺ R Rɺɺ = = Rɺ Rɺ 12

7 System Dynamics in 3-D ZEM = X + V T d ( ZEM ) = X ɺ + V ɺ T go + V T ɺ dt go go Inertial Components 13 Measurement Equation (in LOS frame) Using Note: Measurements are available in seeker gimbal frame, LOS frame information is obtained through a series of transformations. 14

8 Measurement Equation (in Seeker Gimbal Frame) r = x + y + z + η 1 Where r m = range Φ ym = gimbal angle along y Φ zm = gimbal angle along z ω gym = gimbal angle rate along y ω gzm = gimbal angle rate along z C : Transformation matrix DCM matrix of LOS to gimbal frame 15 Unit LOS Vector in Body Frame & Transformation Matrices where, 16

9 Output Equation: Guidance Commands Body Components Inertial Components C i b where is the DCM matrix of inertial to body. Here N = 3 is the navigation constant. Note: (i) No lag between Estimation & Guidance! (ii) Lateral acceleration along x-axis cannot be realized. 17 Nonlinear Controller (Using Dynamic Inversion) 18

10 OUTER LOOP: Command Transfer This body rate command is going to inner loop 19 INNER LOOP: Body Rate Tracking This commands are decomposed to fin deflection commands and fed to the actuators. (passed though second-order dynamics, rate and magnitude bounds, and then combined Back to feed into the 6-DOF dynamics for realistic simulations) 20

11 Simulation Results The R matrix is chosen as square of 1-σ measurement noise as a diagonal element 21 RESULTS 1. Maneuvering Aircraft Step Maneuver Sinusoidal Maneuver 2. Ballistic Target 22

12 Result with Maneuvering Aircraft with Step Maneuver 23 Result with Maneuvering Aircraft with Step Maneuver Interception Trajectory in Azimuth and Elevation plane 24

13 Result with Maneuvering Aircraft with Step Maneuver Error in Estimated Zero effort Miss and velocity with 1σ Bound 25 Result with Maneuvering Aircraft with Step Maneuver Error in Estimated Target acceleration and T go with 1-σ Bound 26

14 Result with Maneuvering Aircraft with Step Maneuver Estimated Range, Range Rate, Gimbal angles and LOS rates Error 27 Result with Maneuvering Aircraft with Step Maneuver Command and achieved acceleration, body rates and fin deflections 28

15 Result with Maneuvering Aircraft with Step Maneuver Histogram plot for miss distance for 100 MC run 29 Result with Maneuvering Aircraft with Step Maneuver Results of 100 MC runs with perturbation cases 30

16 Results with Maneuvering A/C with Sinusoidal Maneuver Interception Trajectory in Azimuth and Elevation plane 31 Results with Maneuvering A/C with Sinusoidal Maneuver Error in Estimated Zero effort Miss and velocity with 1σ Bound 32

17 Results with Maneuvering A/C with Sinusoidal Maneuver Error in Estimated Target acceleration and Tgo with 1σBound 33 Results with Maneuvering A/C with Sinusoidal Maneuver Estimated Range, Range Rate, Gimbal angles and LOS rates Error 34

18 Results with Maneuvering A/C with Sinusoidal Maneuver Command and achieved acceleration, body rates and fin deflections 35 Results with Maneuvering A/C with Sinusoidal Maneuver Histogram plot for miss distance for 100 MC run 36

19 Results with Maneuvering A/C with Sinusoidal Maneuver Results of 100 MC runs with perturbation cases 37 Results with Ballistic Target 3-D Interception Trajectory 38

20 Results with Ballistic Target Interception Trajectory in Azimuth and Elevation plane 39 Results with Ballistic Target Error in Estimated Zero effort Miss and velocity with 1σ Bound 40

21 Results with Ballistic Target Error in Estimated Target acceleration and Tgo with 1σBound 41 Results with Ballistic Target Estimated Range, Range Rate, Gimbal angles and LOS rates Error 42

22 Results with Ballistic Target Command and achieved acceleration, body rates and fin deflections 43 Results with Ballistic Target Histogram plot for miss distance for 100 MC run 44

23 Results with Ballistic Target Results of 100 MC runs with perturbation cases 45 Conclusions A new Integrated Estimation and Guidance algorithm is introduced for Interceptors in Terminal Phase. The design implicitly uses the time-to-go as well as guidance parameter like zero-effort-miss in the estimation process to reduce the consequences of estimation errors and delay on guidance performance. This new IEG formulation has been applied for various targets, which demonstrated a substantial improvement in the result This scheme also possesses the potential to satisfy the hit-to-kill requirement. 46

24 A ZEM Dynamics Based Integrated Estimation Guidance and Control of Interceptors Prof. Radhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Reference P. N. Dwivedi, S. N. Tiwari, A. Bhattacharya and Radhakant Padhi, A ZEM Dynamics Based Integrated Estimation Guidance and Control of Interceptors for High Speed Targets, AIAA Guidance, Navigation and Control Conference, 2011, Portland, USA. 48

25 Features of Proposed IEGC Integration of Estimation, Guidance and Outer loop of Control It estimates the following guidance parameters and utilizes them to compute the necessary body rate commands directly Zero effort miss (ZEM) Relative velocity ( V ) Target acceleration Time-to-go (T go ) Inner loop of the control is still synthesized in a separate loop: Time scale separation between G & C loops is preserved explicitly (similar to the concept of Partial IGC ) Benefits: Overall reduction of loop delay Optimality of the overall system 49 Conventional Approach 50

26 Philosophy of Proposed IEGC Estimation, Guidance and Outer loop of control is integrated as a single loop. 51 Mathematical Formulation of IEGC State Equation In State formulation zero- effort-miss, relative velocity, target acceleration and time-to-go have been taken as a state variable, all in inertial frame. Measurement Equation The measurement variables are the direct output of seeker in seeker polar frame (range, range rate, gimbal angles and gimbal angles rates) Objective: To compute the body rates directly from these estimated variables [ ] T Z = q r c c 52

27 ZEM-Dynamics in 1-D & Time-to-go Dynamics ZEM in y-direction R ( Z y : Zero-Effort-Miss in y - direction ) Z = y + V T y y go Z ɺ = yɺ + V ɺ T + V T ɺ y y go y go Vɺ = a a y Ty M y Time-to-go (T go ) go ( / ) T = R R ɺ Tɺ go Rɺ Rɺ R Rɺɺ R Rɺɺ = = Rɺ Rɺ 53 System Dynamics in 3-D ZEM = X + V T d ( ZEM ) = X ɺ + V ɺ T go + V T ɺ dt go go Inertial Components 54

28 States Equations Where Z : ZERO effort miss V : Relative velocity a t : Target acceleration a m : Interceptor acceleration T go : Time to go R : Relative range 55 Measurement Equation (in LOS frame) Using Note: Measurements are available in seeker gimbal frame, LOS frame information is obtained through a series of transformations. 56

29 Measurement Equation (in Seeker Gimbal Frame) Where r m = range Φ ym = gimbal angle along y Φ zm = gimbal angle along z ω gym = gimbal angle rate along y ω gzm = gimbal angle rate along z C : Transformation matrix 57 IGC&E Formulation: Outer loop Enforced Error Dynamics: However, 0 Hence, from the enforced error dynamics, Note: This has three components 58

30 Next IGC&E Formulation: Outer loop where However Hence Note: Roll rate command (p c ) is chosen to be zero. 59 Inner Loop: DI Based Nonlinear Autopilot For tracking of demanded body rate feedback linearized Dynamic Inversion Autopilot has been used as: 60

31 Summary of IGC&E Philosophy Guidance and outer loop of controller has been integrated with estimator. Demanded body rates are coming from output of estimator Inner loop is synthesized separately: Time scale separation is preserved 61 RESULTS Six-DOF simulation platform has been used for showing the results (with many practical constraints) A second order actuator model has been used as: 2 δ o ω = 2 2 δ i s + 2 ξω s + ω For sensing the body rates and acceleration, a second order gyro and accelerometer model has been used Realistic seeker model has been used for generating seeker measurements. 62

32 RESULTS Three Target Scenarios: o Case1: Maneuvering Aircraft with step maneuver o Case2: Maneuvering Aircraft with sinusoidal maneuver o Case3: Ballistic Missile Target 63 P 0,Q & R matrix R matrix : Diagonal matrix with square of 1σ measurement noise 64

33 Case1:Result with Maneuvering Aircraft with step maneuver Azimuth& Elevation Plane Trajectory 3-D Trajectory 65 Case1: Estimated Zero effort miss & Velocity 66 66

34 Case1: Estimated acceleration and Time-to-go Demanded & Achieved body rates & fin deflections 67 Case2:Result with Maneuvering A/C with sinusoidal maneuver 68

35 Case2: Estimated Zero effort miss & Velocity 69 Case2: Estimated acceleration & Time to go Demanded & Achieved body rates & fin deflection 70

36 Case3 :Results with Ballistic Target 71 Case3: Estimated Zero effort miss & Velocity 72

37 Case3: Estimated acceleration & Time-to-go Demanded & Achieved body rates & fin deflection 73 Result with Randomly Perturbed Plant Parameters (100 cases) Case1: Case2: Case3: 74

38 Comparison of IEGC with conventional Three-loop approach ZEM a_c=n T 2 go 75 Conclusions A new Integrated Estimation, Guidance and Control algorithm is developed. The design implicitly uses the time-to-go as well as guidance parameter like zero effort miss in the estimation process. Second order ZEM dynamics has been used for the integration purpose. New IEGC formulation has shown a substantial improvement for various interception scenarios 76

39 Conclusions Integrated designs are more natural to the flight vehicles Integrated designs bring more synergy between various subsystems Necessity of having a compatible point mass equation in parallel is completely avoided Integrated designs lead to better performance in general. Integrated designs can be proposed following various philosophies: IGC, IEG, IEGC etc. 77 Topics Integrated Estimation and Guidance P. N. Dwivedi, S. N. Tiwari, A. Bhattacharya and Radhakant Padhi, A ZEM Based Effective Integrated Estimation and Guidance of Interceptors in Terminal Phase, AIAA Guidance, Navigation and Control Conference, 2010, Toronto, Canada. Integrated Guidance, Control & Estimation P. N. Dwivedi, S. N. Tiwari, A. Bhattacharya and Radhakant Padhi, A ZEM Dynamics Based Integrated Estimation Guidance and Control of Interceptors for High Speed Targets, AIAA Guidance, Navigation and Control Conference, 2011, Portland, USA. 78

40 Thanks for the Attention.! 79