Predictive Control of Multibody Systems for the Simulation of Maneuvering Rotorcraft

1. Predictive Control of Multibody Systems for the Simulation of Maneuvering Rotorcraft YALCIN FAIK SUMER Georgia Institute of Technology MS Thesis Defense Comitee members Dr. C.L. Bottasso Dr. O.A. Bauchau Dr. D.H. Hodges

2. OUTLINE Introduction and Motivation Maneuvering Multibody Systems with MMSA General information about Neural Networks Detailed description of the methodology Reduced model – Detailed model Path Planning (trajectory optimization) Path Tracking (receding horizon model predictive control) Model Adaption with Neural Network Numerical Example Design of optimal Neural Network Conclusion and suggestions for future studies

3. Maneuvering Complex Models During the maneuvering flight, some limiting factors are encountered such as max. loads, max. turning rates, noise, vibration, etc. near the proximity of the flight envelope. It is impossible to guess the controls that will fly a complex maneuver of long duration guaranteeing to stay within the flight envelope boundaries. • Clear the obstacle by violent pull –up and accordingly violent pull-down • Recover the mission altitude and speed in minimum time.

4. Maneuvering Complex Models Maneuvers can be formulated asOptimal Control Problems + Constraints Cost Function Vehicle equations of motion Performance index Physical limitations Limited control authority Flight envelope boundaries etc. Limitations based on the given task Solution Controls (fly the vehicle along this trajectory) Trajectory • Solution of optimal control problems with large comprehensive models are not feasible!!!

5. The Multi - Model Steering Algorithm (MMSA) Comprehensive model: many degrees of freedom, captures fine scale solution details. Reduced model:few degrees of freedom, captures flight mechanics solution. • Motivation: • Solve expensive optimal control problems with reduced model • Use comprehensive model only for initial value problems (known control inputs).

6. The Multi - Model Steering Algorithm (MMSA) Reduced model Path planning level To get the Reference Trajectory solve Maneuver Optimal Control Problem Path tracking level • Use non-linear model predictive control(NMPC) to track reference trajectory. Tracking problem formulated as optimal control problem on a shifting prediction window (BVP); • Apply computed control inputs to comprehensive model until • next prediction (Initial Value Problem, IVP). Iteratively, reduced model is adapted to the comprehensive one in order to minimize tracking errors.

7. The Multi - Model Steering Algorithm (MMSA) 1- Maneuver optimal control problem (reduced model) Reference trajectory 2- Tracking Problem (reduced model) 4- Adaption of reduced model by system identification (Neural Network based) 3- Initial Value Solution (steering comprehensive model) Predictive solution Trajectory flown by comprehensive model

8. The Multi - Model Steering Algorithm (MMSA) Advantages • Computationally feasible:reduced model for expensive BVP, • comprehensive model for IVP • Applicable to any multibody code • MPCis based on non-linearflight mechanics (reduced) models • MPC can deal with input(limited authority) and • output (flight envelope boundary, procedures, etc.) constraints • MPC isstable for large classes of problems • Applicable to unstable vehicles • Adaptivity ensures convergence

9. Bias Activation function Output Input signals Summing junction Synaptic weights Neural Networks Architecture Fully connected 3-5-2 feedforward structure output layer input layer hidden layer

10. Neural Networks Training modes Incremental (sequential) mode Batch mode examples One epoch is completed Training set NN

11. Comparison between two modes The incremental mode requires less local storage for each synaptic connection. The incremental mode makes the search in weight space stochastic in nature. This makes it less likely for the algorithm to be trapped in local minimum. The batch mode of training provides accurate estimate of gradient vector, convergence to a local minimumis guaranteed under simple conditions. However, incremental mode makes it difficult to establish theoretical conditions for convergence. The incremental mode is simple to implement. The incremental mode provides the effective solutions to large and difficult problems. For the on-line training, the incremental mode is preferred over the batch mode.

12. Neural Networks Algorithm The error back-propagation algorithms are commonly used for feedforward structures because of their robustness. The basic methodology used with this algorithm is method of steepest descent. The successive adjustments to the weight vector w is in a direction opposite to the gradient vector. In one dimension

13. Weight adjustments for the next step Learning rate When learning rate is smallthe response of the algorithm overdamped. For this case the algorithm will take too long to converge. When learning rate is large the response of the algorithm underdamped. For this case trajectory of w follows has a oscillatory path. When learning rateexceeds a certain critical value, the algorithm becomes unstable.

14. Reduced Model Identification Non-linear Reduced model represented by set of States Controls Neural Network parameters (outputs) (weights and biases) Full Model represented by We are trying to satisfy a proper matching between reduced model outputs and full model outputs Controls of multibody model might have different correspondance on the flight mechanics controls It is always possible to map one set of controls to another For simplicity we will consider We can define a Reference model augmented by a Neural Network We can define a Reference model Reference model error (defect)

15. Reduced Model Identification Reconstruction error Parameters of the reduced model are weights and biases of NN p The error we want to minimize is defined as Plant output E Control input Network output Desired output Minimizing error Good approximation of defect Matching We can express the reduced model in a compact notation

16. Path Planning – Trajectory Optimization (Flight Mechanics Level) Maneuver Optimal Control Problem Cost function Minimize this cost function : Boundary conditions Constraints A maneuver can be defined mathematically in a clear way by cost function, constraints and bounds In order to make the problem finite dimensional we can discretize the equations of equilibrium, constraints and cost function, then solve this discrete problem (direct approach).

17. Model Predictive Tracking Reference Trajectory Model predictive control problem is defined by the following equations with dt s.t

18. Accordingly we use the controls that we have already found to steer the plant. This will be a standart initial value problem with the given formulation. The solution we get at the end of the steering window provides the new initial condition for the next step. Iteratively tracking and steering procedures are kept until end of the maneuver is reached. Remarks : Shorter tracking windows require small computational costs, on the other hand longer windows will provide improved stability and performance. Longer steering windows cause the system to shift away more. However, they will decrease computational costs by reducing the number of tracking problems that need to be solved.

19. MODEL ADAPTION Aim : Minimize the mismatch between the reduced model and the plant using a local adaption. Augment the NN based model parameters iteratively. Assume that after steering the multibody model we have Given controls Resulting outputs The model adaption problem can be formulated as E Current parameters are adjusted as :

20. Rotorcraft Flight Mechanics Models Classical 2D longitudinal model for helicoptersand tilt-rotors: (MR = Main Rotor; TR = Tail Rotor) Power balance equation:

21. Rotorcraft Flight Mechanics Models For helicopters enforce yaw, roll and lateral equilibrium Rotor aerodynamic forces are based on classical blade element theory y = where (states) u = (controls) (model parameters) but no For tilt rotor case additional controls are

22. Minimum Time Obstacle Avoidance Optimal Control Problem (with unknown internal event at ) Cost function Constraints and bounds Initial trimmed conditions at 50 m/s Power limitations Window size

23. NMPC with feedback loop

24. Optimal design of the NN architecture Input layer Hidden layer Most of the techniques employ trail and error methods to find optimal number of neurons. Too many neurons degrade the effectiveness of the model (cause overfitting). Too few neurons may not capture the full complexity of the data (cause underfitting). It is started with 20hidden neurons and decreased to 15and 10 . No change is observed in the performance of the algorithm until 10. Below 10, the learning started to slow down.

25. Output layer Output layer of the NN is defined in terms of the defect of equilibrium equations, which are equations. Strategies for Training Training Mode More than one example is introduced to the NN by the information supplied at the end of each steering window. The procedure doesn’t only include the incremantal mode. This training mode can be called as batch-incremental mode.

26. Adaptive Learning rate Each component of the error tried to be decreased at the same time. Not successful Successful Learning rate decreased by 30% Learning rate increased by 5% Learning rate hit zero level . MSE error tried to be decreased. In this case, learning rate increased to the upper bound quickly and stayed there during the rest of the iterations. It behaved like a constant learning rate.

27. Constant Learning rate Different constant learning rates are tried. The optimal value found : 0.02 The critical value found : 0.08 Optimal NN is defined as: 11-10-3 fully connected feedforward structure Backpropagation algorithm with steepest descent method Constant learning rate is 0.02 Batch-incremental training mode

28. Trajectory flown by the reduced ( line) and multibody (solid line) models, before (left) , after (right) fourteen planning / tracking steering / adaption iterations. Fuselage pitch for the reduced ( line) and multibody (solid line) models, before (right) , after (left) fourteen planning / tracking steering / adaption iterations.

29. Airspeed of the reduced ( line) and multibody (solid line) models, before (left) , after (right) fourteen planning / tracking steering / adaption iterations. Longitudinal of the reduced ( line) and multibody (solid line) models, before (left) , after (right) fourteen planning / tracking steering / adaption iterations.

30. Contributions of weights The largest values of the weights are observed on the weights connecting input variable and the defect of the pitch moment equation to the hidden layer.

31. Conclusion and Suggestions Multi-model approach allows reasonable computational costs even for verylarge aeroelastic models. We can compute compatible optimal trajectories with vehicle dynamics based on a given task. Receding horizon formulation of MMSA allows for the analysis of unstable systems, such as helicopters. Using NMPC in solving the tracking problem provides improved tracking performance with respect to the other control strategies. Same software can be used for planning and tracking phase since they are both optimal control problems. Using neural network as an adaptive element increased the predictive capabilities of the reduced model in a robust way. Optimal neural network architecture is achieved for this application. Constant learning rate strategy is preffered over the adaptive learning rate. It is ensured that the created NN is robust. It performed well with different hidden layer neurons. This means basics of the given architecture can be applied to further studies.

32. Future studies Other training algorithms can be tried to increase the performance of the neural network. Adaptive learning rate strategy can be augmented with possible ideas to prevent the learning rate hitting zero level. If it can be implemented in a right way, it should be expected to provide better performance rather than constant learning rate. Pretraining (off-line) training can still be an option to start with better initialized values. In this case, ranges of control and states must be well-determined to capture the bounds, also high computational costs must be decreased in creating the pre-training data.

33. QUESTIONS ?