Field programmable gate array based predictive control system for spacecraft rendezvous in elliptical orbits

Copyright © 2014 John Wiley & Sons, Ltd. Copyright © 2014 John Wiley & Sons, Ltd. Summary A field programmable gate array (FPGA) based model predictive controller for two phases of spacecraft rendezvous is presented. Linear time-varying prediction models are used to accommodate elliptical orbits, and a variable prediction horizon is used to facilitate finite time completion of the longer range manoeuvres, whilst a fixed and receding prediction horizon is used for fine-grained tracking at close range. The resulting constrained optimisation problems are solved using a primal-dual interior point algorithm. The majority of the computational demand is in solving a system of simultaneous linear equations at each iteration of this algorithm. To accelerate these operations, a custom circuit is implemented, using a combination of Mathworks HDL Coder and Xilinx System Generator for DSP, and used as a peripheral to a MicroBlaze soft-core processor on the FPGA, on which the remainder of the system is implemented. Certain logic that can be hard-coded for fixed sized problems is implemented to be configurable online, in order to accommodate the varying problem sizes associated with the variable prediction horizon. The system is demonstrated in closed-loop by linking the FPGA with a simulation of the spacecraft dynamics running in Simulink on a PC, using Ethernet. Timing comparisons indicate that the custom implementation is substantially faster than pure embedded software-based interior point methods running on the same MicroBlaze and could be competitive with a pure custom hardware implementation.

Bypass transition to sustained thermoacoustic oscillations in a linearly stable rijke tube

In this paper we examine triggering in a simple linearly-stable thermoacoustic system using techniques from flow instability and optimal control. Firstly, for a noiseless system, we find the initial states that have highest energy growth over given times and from given energies. Secondly, by varying the initial energy, we find the lowest energy that just triggers to a stable periodic solution. We show that the corresponding initial state grows first towards an unstable periodic solution and, from there, to the stable periodic solution. This exploits linear transient growth, which arises due to nonnormality in the governing equations and is directly analogous to bypass transition to turbulence. Thirdly, we introduce noise that has similar spectral characteristics to this initial state. We show that, when triggering from low noise levels, the system grows to high amplitude self-sustained oscillations by first growing towards the unstable periodic solution of the noiseless system. This helps to explain the experimental observation that linearly-stable systems can trigger to self-sustained oscillations even with low background noise. © 2010 by University of Cambridge. Published by the American Institute of Aeronautics and Astronautics, Inc.

Measurement-Integrated simulations and Kalman filter applied to a co-flowing jet

This paper deals with the experimental evaluation of a flow analysis system based on the integration between an under-resolved Navier-Stokes simulation and experimental measurements with the mechanism of feedback (referred to as Measurement-Integrated simulation), applied to the case of a planar turbulent co-flowing jet. The experiments are performed with inner-to-outer-jet velocity ratio around 2 and the Reynolds number based on the inner-jet heights about 10000. The measurement system is a high-speed PIV, which provides time-resolved data of the flow-field, on a field of view which extends to 20 jet heights downstream the jet outlet. The experimental data can thus be used both for providing the feedback data for the simulations and for validation of the MI-simulations over a wide region. The effect of reduced data-rate and spatial extent of the feedback (i.e. measurements are not available at each simulation time-step or discretization point) was investigated. At first simulations were run with full information in order to obtain an upper limit of the MI-simulations performance. The results show the potential of this methodology of reproducing first and second order statistics of the turbulent flow with good accuracy. Then, to deal with the reduced data different feedback strategies were tested. It was found that for small data-rate reduction the results are basically equivalent to the case of full-information feedback but as the feedback data-rate is reduced further the error increases and tend to be localized in regions of high turbulent activity. Moreover, it is found that the spatial distribution of the error looks qualitatively different for different feedback strategies. Feedback gain distributions calculated by optimal control theory are presented and proposed as a mean to make it possible to perform MI-simulations based on localized measurements only. So far, we have not been able to low error between measurements and simulations by using these gain distributions.

Verifying stability of approximate explicit MPC

Several authors have proposed algorithms for approximate explicit MPC [1],[2],[3]. These algorithms have in common that they develop a stability criterion for approximate explicit MPC that require the approximate cost function to be within a certain distance from the optimal cost function. In this paper, stability is instead ascertained by considering only the cost function of the approximate MPC. If a region of the state space is found where the cost function is not decreasing, this indicates that an improved approximation (to the optimal control) is required in that region. If the approximate cost function is decreasing everywhere, no further refinement of the approximate MPC is necessary, since stability is guaranteed. ©2009 IEEE.

Robust multiplexed MPC for distributed multi-agent systems

This paper extends the authors' earlier work which adapted robust multiplexed MPC for application to distributed control of multi-agent systems with non-interacting dynamics and coupled constraint sets in the presence of persistent unknown, but bounded disturbances. Specifically, we propose exploiting the single agent update nature of the multiplexed approach, and fix the update sequence to enable input move-blocking and increased discretisation rates. This permits a higher rate of individual policy update to be achieved, whilst incurring no additional computational cost in the corresponding optimal control problems to be solved. A disturbance feedback policy is included between updates to facilitate finding feasible solutions. The new formulation inherits the property of rapid response to disturbances from multiplexing the control and numerical results show that fixing the update sequence does not incur any loss in performance. © 2011 IFAC.

Lyapunov theory for zeno stability

Zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time. This behavior commonly arises in mechanical systems undergoing impacts and optimal control problems, but its characterization for general hybrid systems is not completely understood. The goal of this paper is to develop a stability theory for Zeno hybrid systems that parallels classical Lyapunov theory; that is, we present Lyapunov-like sufficient conditions for Zeno behavior obtained by mapping solutions of complex hybrid systems to solutions of simpler Zeno hybrid systems defined on the first quadrant of the plane. These conditions are applied to Lagrangian hybrid systems, which model mechanical systems undergoing impacts, yielding simple sufficient conditions for Zeno behavior. Finally, the results are applied to robotic bipedal walking. © 2012 IEEE.

CLF based designs with robustness to dynamic input uncertainties

The problem of robust stabilization of nonlinear systems in the presence of input uncertainties is of great importance in practical implementation. Stabilizing control laws may not be robust to this type of uncertainty, especially if cancellation of nonlinearities is used in the design. By exploiting a connection between robustness and optimality, "domination redesign" of the control Lyapunov function (CLF) based Sontag's formula has been shown to possess robustness to static and dynamic input uncertainties. In this paper we provide a sufficient condition for the domination redesign to apply. This condition relies on properties of local homogeneous approximations of the system and of the CLF. We show that an inverse optimal control law may not exist when these conditions are violated and illustrate how these conditions may guide the choice of a CLF which is suitable for domination redesign. © 1999 Elsevier Science B.V. All rights reserved.

Minimum maneuver time calculation using convex optimization

The problem of calculating the minimum lap or maneuver time of a nonlinear vehicle, which is linearized at each time step, is formulated as a convex optimization problem. The formulation provides an alternative to previously used quasi-steady-state analysis or nonlinear optimization. Key steps are: the use of model predictive control; expressing the minimum time problem as one of maximizing distance traveled along the track centerline; and linearizing the track and vehicle trajectories by expressing them as small displacements from a fixed reference. A consequence of linearizing the vehicle dynamics is that nonoptimal steering control action can be generated, but attention to the constraints and the cost function minimizes the effect. Optimal control actions and vehicle responses for a 90 deg bend are presented and compared to the nonconvex nonlinear programming solution. Copyright © 2013 by ASME.