Michael P. Vitus

Exponential stabilization of discrete-time switched linear systems

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal policy of a related switched linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.

Design of guaranteed safe maneuvers using reachable sets: Autonomous quadrotor aerobatics in theory and practice

For many applications, the control of a complex nonlinear system can be made easier by modeling the system as a collection of simplified hybrid modes, each representing a particular operating regime. An example of this is the decomposition of complex aerobatic flights into sequences of discrete maneuvers, an approach that has proven very successful for both human piloted and autonomously controlled aircraft. However, a critical step when designing such control systems is to ensure the safety and feasibility of transitions between these maneuvers. This work presents a hybrid dynamics framework for the design of guaranteed safe switching regions and is applied to a quadrotor helicopter performing an autonomous backflip. The regions are constructed using reachable sets calculated via a Hamilton-Jacobi differential game formulation, and experimental results are presented from flight tests on the STARMAC quadrotor platform.

Applications of hybrid reachability analysis to robotic aerial vehicles

The control of complex non-linear systems can be aided by modeling each system as a collection of simplified hybrid modes, with each mode representing a particular operating regime defined by the system dynamics or by a region of the state space in which the system operates. Guarantees on the safety and performance of such hybrid systems can still be challenging to generate, however. Reachability analysis using a dynamic game formulation with Hamilton—Jacobi methods provides a useful way to generate these types of guarantees, and the technique is flexible enough to analyze a wide variety of systems. This paper presents two applications of reachable sets, both focused on guaranteeing the safety and performance of robotic aerial vehicles. In the first example, reachable sets are used to design and implement a backflip maneuver for a quadrotor helicopter. In the second, reachability analysis is used to design a decentralized collision avoidance algorithm for multiple quadrotors. The theory for both examples is explained, and successful experimental results are presented from flight tests on the STARMAC quadrotor helicopter platform.

On efficient sensor scheduling for linear dynamical systems

Consider a set of sensors estimating the state of a process in which only one of these sensors can operate at each time-step due to constraints on the overall system. The problem addressed here is to choose which sensor should operate at each time-step to minimize a weighted function of the error covariance of the state estimation at each time-step. This work investigates the development of tractable algorithms to solve for the optimal and suboptimal sensor schedule. First, a condition on the non-optimality of an initialization of the schedule is presented. Second, using this condition, both an optimal and a suboptimal algorithm are devised to prune the search tree of all possible sensor schedules. This pruning enables the solution of larger systems and longer time horizons than with enumeration alone. The suboptimal algorithm trades off the quality of the solution and the complexity of the problem through a tuning parameter. Third, a hierarchical algorithm is formulated to decrease the computation time of the suboptimal algorithm by using results from a low complexity solution to further prune the tree. Numerical simulations are performed to demonstrate the performance of the proposed algorithms.

Tunnel-MILP: Path Planning with Sequential Convex Polytopes

This paper focuses on optimal path planning for vehicles in known environments. Previous work has presented mixed integer linear programming (MILP) formulations, which suer from scalability issues as the number of obstacles, and hence the number of integer variables, increases. In order to address MILP scalability, a novel three-stage algorithm is presented which rst computes a desirable path through the environment without considering dynamics, then generates a sequence of convex polytopes containing the desired path, and nally poses a MILP to identify a suitable dynamically feasible path. The sequence of polytopes form a safe tunnel through the environment, and integer decision variables are restricted to deciding when to enter and exit each region of the tunnel. Simulation results for this approach are presented and reveal a signicant increase in the size and complexity of the environment that can be solved.

Closed-loop belief space planning for linear, Gaussian systems

This paper considers the problem of motion planning for linear, Gaussian systems, and extends existing chance constrained optimal control solutions [1], [2] by incorporating the closed-loop uncertainty of the system and by reducing the conservativeness in the constraints. Due to the imperfect knowledge of the system state caused by motion uncertainty and sensor noise, the constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically. In this work, they are formulated as convex constraints on a univariate Gaussian random variable, with the violation probability of all the constraints guaranteed to be below a threshold. This threshold is a tuning parameter which trades off the performance of the system and the conservativeness of the solution. In contrast to similar methods, the proposed work considers the specific estimator and controller used in the closed-loop system in order to directly characterize the a priori distribution of the closed-loop system state. Using this distribution, a convex optimization program is formulated to solve for the optimal solution for the closed-loop system. The performance of the algorithm is demonstrated through several examples.

Sensor Placement for Improved Robotic Navigation.

Consider a set of fixed sensors used to estimate the state of a vehicle (e.g. position, orientation and velocity) while it attempts to follow a pre-planned trajectory. Since the sensor can only provide a measurement to the vehicle when it is within range, the deployment of the sensors will have a major impact on the ability of the vehicle to follow the trajectory. The problem addressed here is to optimally place the sensors in the environment such that the weighted function of the estimation error at each time-step is minimized. An optimization formulation is proposed that accounts for the uncertainty of the vehicle’s state in determining whether it can receive a measurement from a sensor. A confidence level is introduced as a tuning parameter that controls the conservativeness of the solution. Consequently, the resulting solution increases the likelihood of the vehicle successfully following its intended trajectory. Finally, due to the interdependence between the sensors’ positions, a novel incremental optimization algorithm is presented which significantly outperforms a standard nonlinear optimization procedure. Experimental and simulation results are shown which characterize the performance of the proposed algorithm.

On the optimal solutions of the infinite-horizon linear sensor scheduling problem

This paper studies the infinite-horizon sensor scheduling problem for linear Gaussian processes with linear measurement functions. Several important properties of the optimal infinite-horizon schedules are derived. In particular, it is proved that under some mild conditions, both the optimal infinite-horizon average-per-stage cost and the corresponding optimal sensor schedules are independent of the covariance matrix of the initial state. It is also proved that the optimal estimation cost can be approximated arbitrarily closely by a periodic schedule with a finite period, and moreover, the trajectory of the error covariance matrix under this periodic schedule converges exponentially to a unique limit cycle. These theoretical results provide valuable insights about the problem and can be used as general guidelines in the design and analysis of various infinite-horizon sensor scheduling algorithms.

Design and Analysis of Hybrid Systems, with Applications to Robotic Aerial Vehicles

Decomposing complex, highly nonlinear systems into aggregates of simpler hybrid modes has proven to be a very successful way of designing and controlling autonomous vehicles. Examples include the use of motion primitives for robotic motion planning and equivalently the use of discrete maneuvers for aggressive aircraft trajectory planning. In all of these approaches, it is extremely important to verify that transitions between modes are safe. In this paper, we present the use of a Hamilton-Jacobi differential game formulation for finding continuous reachable sets as a method of generating provably safe transitions through a sequence of modes for a quadrotor performing a backflip maneuver.

Hybrid Systems in Robotics

Robotics has provided the motivation and inspiration for many innovations in planning and control. From nonholonomic motion planning [1] to probabilistic road maps [2], from capture basins [3] to preimages [4] of obstacles to avoid, and from geometric nonlinear control [5], [6] to machine-learning methods in robotic control [7], there is a wide range of planning and control algorithms and methodologies that can be traced back to a perceived need or anticipated benefit in autonomous or semiautonomous systems.