Real-time dynamic optimization of nonlinear systems: A flatness-based approach

In this paper, a real-time optimizing controller is developed to steer a differentially flat nonlinear control system to closed-loop trajectories that optimize a cost functional of interest. An interior point optimization method with penalty function is used to formulate a real-time optimization scheme. The problem is posed as a real-time optimal trajectory generaction problem in which the optimal trajectories are computed using an adaptive extremum-seeking approach. A fed-batch bioreactor simulation example is used to demonstrate the application of the technique to a finite time dynamic optimization problems that arise in batch process control.

This paper presents a systematic method for generating optimal helicopter flight trajectories in real time for cockpit display. Helicopter flights in the event of a single engine failure are critical, and display of optimal flight paths on-board is desirable. Due to the large number of states and constraints, however, numerical solutions on a mainframe computer can take a long tune. In this paper, optimal solutions are first obtained off-line using a ground computer. For a given set of initial helicopter states and flight parameters, optimal solutions are fitted by a Fourier series. The coefficients of the Fourier approximations are then interpolated as functions of initial flight states and parameters. The approximate optimal trajectories can be quickly generated on-board using the proposed method. Introduction Engine failure is a major safety issue for all helicopters, especially during takeoff and landing operations. Due to the complexity of the situation and the dependence of safety on many parameters, optimal flight trajectories can provide useful guidance for pilot operation. In particular, on-board display of optimal trajectories is desirable. This paper presents a method for generating optimal flight trajectories in real-tune that can be used for on-board display. Depending on its capability and performance during an engine failure, a transport helicopter can be certified as Category A or Category B. The Federal Aviation Administration (FAA) grants Category B certification to a single or multi-engine helicopter with gross weight less than 20,000 Ib and capable of landing safely in case of one or all engine failures. Category A certification, on the other hand, is granted to multi-engine helicopters with independent engine systems. Category A helicopters must be able to either land safely or continue flight in case of One Engine Inoperation (OEI). This makes Category A helicopters uniquely suited for remote places where emergency landing sites are not available. I Ph.D. Candidate, Center for Control Sciences, Member AIAA * Assistant Professor, Dept. of Aerospace Eng. and Mechanics, Member AIAA + Civil Rotorcraft Group Leader, Rotorcraft and Powered Lift Branch The decision to continue the flight or to land represents an energy management problem. It has been shown''' that this decision depends on initial conditions at the time of failure as well as flight parameters. Due to the numerous initial variables involved in this decision, a method to communicate all this information to the pilot instantaneously, at engine failure, is needed. This method must keep the emergency procedure simple and manageable. In particular, on-board cockpit display of optimal flight trajectories is useful. This requires a reliable real-tune solution of optimal trajectories, on board, with limited time, limited computing power, and limited memory available. If possible, an analytical solution of optimal trajectories is desirable. Unfortunately, this can only be achieved for systems of low order and where the equations of motion and the cost function are simple. Therefore, for a complex system, an open-loop numerical procedure becomes necessary. Research efforts were made to solve optimal control problems in real-time. Schultz et al. used calculus of variation and energy state method to generate optimal trajectories on-board for the National Aerospace Plane (NASP). In his study, the equations of motion of the NASP are less complex than those of the helicopter, and an interval of time of few minutes is allowed. Bryson used neighboring control theory to generate near-optimal solutions that could be used in real-time. This method is elegant but requires a large amount of memory to store the needed perturbation matrices. Also, it is valid on a relatively small neighborhood of the nominal trajectory. Perturbation methods and table lookups are often used if only few parameters are involved. In the current problem, the equations of motion of the helicopter are complex and too many path equality and inequality constraints are present. Also, limitations on tune, computing power, and memory available make all the above methods unsuitable for this problem. This paper proposes a technique of real-time optimal trajectory generation based on approximation and

Real-time trajectory optimization for collision-free asteroid landing based on deep neural networks

This paper presents a practical optimization framework for improving trajectory feasibility in constrained nonlinear optimal control problems for agile unmanned aerial vehicles (UAVs). The proposed method addresses trajectory optimization problems with non-convex geometric constraints, where gradient-based solvers often fail to converge to feasible solutions. The framework combines Model Predictive Path Integral (MPPI) control and the Augmented Lagrangian iterative Linear Quadratic Regulator (AL-iLQR). MPPI is employed as a fast sampling-based guidance mechanism to explore feasible regions of the trajectory space, while AL-iLQR is used to efficiently refine locally optimal solutions with high numerical accuracy. By decoupling feasibility exploration from local optimal refinement, the proposed method mitigates the sensitivity of gradient-based trajectory optimization to initialization in highly constrained environments. Numerical simulations involving both simplified two-dimensional dynamics and full quadrotor models demonstrate that the proposed approach significantly improves the probability of converging to feasible and dynamically consistent trajectories compared with AL-iLQR alone. The proposed method does not aim to provide theoretical guarantees of global optimality; instead, it offers a practical and computationally efficient strategy for enhancing feasibility and robustness in real-time UAV trajectory optimization.

Real-time dynamic optimization of batch systems

As gas and oil fields mature their operation becomes increasingly more complex, due to complex process dynamics, like slugging, gas coning, water breakthrough, salt or hydrate deposition. Moreover these phenomena also lead to production constraints in the upstream facilities. This complexity asks for operator support in how to optimally operate such an asset. In this paper we explore how, based on physical models of reservoirs, wells and top side separation and compression equipment, real-time optimization may be used in optimal operation of a maturing asset showing complex dynamic behavior and top side compressor constraints. In a joint industry project of Wintershall, GDF, EBN, Siemens and TNO the applicability of Real-Time Optimization is explored. Real-time optimization will be demonstrated on a maturing North Sea off-shore gas production asset consisting of eight wells and two separator and compression trains. The current results show how the optimization problem has been defined and implemented in a real-time architecture. First results are shown realizing the gas production set point and minimizing compressor power, while the production of individual wells suffer from degrading performance and (unplanned) shut-ins. The optimizer computes how the production must be dynamically distributed over the wells and how the compressor trains must be operated. Dynamic optimization strategies have been exploited successfully in downstream operations. A similar development is happening in upstream industry, while having to cope with the higher uncertainties typically present in upstream processes. As the paper shows Dynamic Optimization may lead to large benefits in Real-Time Operation of upstream production facilities.

On interfacing model predictive controllers with a real-time optimizer

In this note, an extremum-seeking controller is developed to steer a periodic system to orbits that maximize a functional of interest for a class of differentially flat nonlinear systems. The problem is posed as a real-time optimal trajectory generation problem in which the optimal orbit is computed using an extremum-seeking approach. Using the flatness property of the dynamics, the original dynamic optimization problem is transformed to a parameterized optimization problem which can be solved in real-time to approximate the optimal orbit. The control algorithm provides tracking of the optimal orbit. A drug delivery control problem is considered to demonstrate the application of the technique.

Distributed trajectory optimization for multiple solar-powered UAVs target tracking in urban environment by Adaptive Grasshopper Optimization Algorithm

Distributed real-time dynamic cooperative optimization with unknown performance function form under resources constraints.

Reservoir computing refers to a computational framework based on recurrent neural networks that can process time-series data. In an echo state network (ESN), which is a type of reservoir computing framework, the reservoir consists of a recursive network of artificial neurons with nonlinear activation functions. A model predictive control (MPC) technique can determine the control signals by solving the optimization problem of a system using the finite-time domain of each control period. However, real-time optimization cannot be achieved unless the optimal control problem can be solved within the next control period. To overcome this limitation, we propose a new control method based on MPC that explicitly incorporates the predicted disturbance of a time-varying trajectory using ESN to achieve the active vibration control of hybrid electric vehicle (HEV) powertrains. Once the ESN has been trained, the associated MPC explicitly satisfies the constraints over a moving horizon without further training. Instead of completing the real-time optimization within the control period, ESN predicts the future disturbance and applies it to the MPC in the future control period. Based on the predicted future disturbance, the system calculates the optimal control signals required for the future. Thus, real-time control can be realized because the optimal signals are determined before the subsequent control period occurs. The proposed method can be implemented in MPC even if the control period is too short to optimize as long as the disturbance can be reasonably measured and predicted. In this study, the simulation approach was demonstrated using the engine start condition in an HEV powertrain. The importance of this study is that the limitation of MPC relevant to real-time optimization can be relaxed by applying our proposed method.

This paper proposes a stabilization method for dynamic walking of a bipedal robot with real-time optimization of capture point trajectories. We used the capture point trajectories to generate the control input, which is the desired zero moment point (ZMP) with a sliding-mode ZMP controller to follow the desired ZMP. This method enables the robot to implement various dynamic walking commands, such as forward stride, lateral stride, walking direction, single support time, and double support time. We also adopted enhanced dynamics with the three mass linear inverted pendulum model (3M-LIPM). First, the compensated ZMP is calculated by both walking commands and kinematic configuration of the robot in closed form. Then, the walking pattern is obtained by using initial and boundary conditions of the 3M-LIPM, which satisfies the walking commands. The capture point (CP) trajectory is optimized in real time to control the walking stability and a capture point tracking controller is used for tracking the optimized CP trajectory, which generates an optimal control input that is near the center of the support polygon. The performance of the proposed stabilization method was verified by a dynamics simulator, Webots, and comparison with the original capture point controller-based walking algorithm is presented.

This paper considers the problem of real-time trajectory generation and tracking for nonlinear control systems. We employ a two-degree-of-freedom approach that separates the nonlinear tracking problem into real-time trajectory generation followed by local (gain-scheduled) stabilization. The central problem which we consider is how to generate, possibly with some delay, a feasible state space and input trajectory in real time from an output trajectory that is given online. We propose two algorithms that solve the real-time trajectory generation problem for differentially flat systems with (possibly non-minimum phase) zero dynamics. One is based on receding horizon point to point steering, the other allows additional minimization of a cost function. Both algorithms explicitly address the tradeoff between stability and performance and we prove convergence of the algorithms for a reasonable class of output trajectories. To illustrate the application of these techniques to physical systems, we present experimental results using a vectored thrust flight control experiment built at Caltech. A brief introduction to differentially flat systems and its relationship with feedback linearization is also included. © 1998 John Wiley & Sons, Ltd.

Real-Time Optimization for Kites

This paper explores the potential of integrating Global Navigation Satellite System (GNSS) Avionics Based Integrity Augmentation (ABIA) functionalities in Unmanned Aerial Systems (UAS) to perform mission planning and real-time trajectory optimisation tasks. In case of mission planning, a pseudo-spectral optimization technique is adopted. For real-time trajectory optimisation a Direct Constrained Optimisation (DCO) method is employed. In this method the aircraft dynamics model is used to generate a number of feasible flight trajectories that also satisfy the GNSS integrity constraints. The feasible trajectories are calculated by initialising the aircraft dynamics model with a manoeuvre identification algorithm. The performance of the proposed GNSS integrity augmentation and trajectory optimisation algorithms was evaluated in representative simulation case studies. Additionally, the ABIA performance was compared with Space-Based and Ground-Based Augmentation Systems (SBAS/GBAS). Simulation results show that the proposed integration scheme is capable of performing safety-critical UAS tasks (CAT III precision approach, UAS Detect-and-Avoid, etc.) when GNSS is used as the primary source of navigation data. There is a synergy with SBAS/GBAS in providing suitable (predictive and reactive) integrity flags in all flight phases. Therefore, the integration of ABIA with SBAS/GBAS is a clear opportunity for future research towards the development of a Space-Ground-Avionics Augmentation Network (SGAAN) for UAS SAA and other safety-critical aviation applications.