Trajectory-based safety controller synthesis

Winn, Andrew Kevin
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Julius, Anak Agung
Wen, John T.
Mishra, Sandipan
Wu, Wencen
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Electrical engineering
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Attribution-NonCommercial-NoDerivs 3.0 United States
This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.
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We next examine methods for improving the controller synthesis process or improving the resulting controller. We present two methods of optimizing the trajectories. In the first, we use a gradient descent approach to minimize a cost functional while maintaining a desired robustness using potential functions. In the second, we use a tool from nonsmooth optimization to optimize the robustness directly. These methods are then used to automatically generate new trajectories from given trajectories. We then present a method for learning a piecewise affine feedback control law from the trajectory-based controller. Finally, we will examine methods of performing distance computations, a calculation vital for any implementation of our approach, and present a novel method for calculating the distance to the boundary of an ellipsoid with better robustness and efficiency than existing methods.
As technology advances, the control systems that are used to drive devices are growing more and more complicated. In many situations the PID controllers of the past are replaced with complicated high-level controllers that allow complex systems to satisfy complex spacial and temporal specifications. It is important to have methods for designing controllers that guarantee that these tasks are performed correctly.
Our research focuses on generating controllers for hybrid systems that satisfy safety and temporal logic specifications. A hybrid system is a dynamical system that incorporates both continuous and discrete elements. Safety specifications involve avoiding a set of unsafe states while eventually reaching a set of goal states, and temporal logic specifications involve reaching or avoiding sets of states depending on the the system's history. Some existing methods solve this problem globally, but either grow computationally intractable as the dimension of the dynamical system grows, or requires the system to have very simple dynamics. Other methods perform an on-line search for a feasible trajectory, but without a guarantee that a satisfying trajectory will be found in an acceptable time frame. The proposed method lies in between these approaches in that it guarantees satisfaction of the specifications for a local set of initial states, and can be applied to high dimensional and/or complex dynamical systems.
In our approach we first introduce the control autobisimulation function, which is the analogue of the control Lyapunov function for approximate bisimulation. We use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness, a property that ensures that any trajectory of the closed-loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input. This feedback control and input can be used as the controller for some subset of the initial states. We next derive a method to compute the largest tube such that any trajectory inside the tube is guaranteed to satisfy the desired specifications. By generating feasible trajectories, say by a human playing a video game simulation of the system, we can create a set of trajectories such that for any allowable initial state there is a trajectory whose feedback law and input will meet the desired specifications.
May 2015
School of Engineering
Dept. of Electrical, Computer, and Systems Engineering
Rensselaer Polytechnic Institute, Troy, NY
Rensselaer Theses and Dissertations Online Collection
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