On the Construction of Safe Controllable Regions for Affine Systems with Applications to Robotics

This paper studies the problem of constructing in-block controllable (IBC) regions for affine systems. That is, we are concerned with constructing regions in the state space of affine systems such that all the states in the interior of the region are mutually accessible through the region's interior by applying uniformly bounded inputs. We first show that existing results for checking in-block controllability on given polytopic regions cannot be easily extended to address the question of constructing IBC regions. We then explore the geometry of the problem to provide a computationally efficient algorithm for constructing IBC regions. We also prove the soundness of the algorithm. We then use the proposed algorithm to construct safe speed profiles for different robotic systems, including fully-actuated robots, ground robots modeled as unicycles with acceleration limits, and unmanned aerial vehicles (UAVs). Finally, we present several experimental results on UAVs to verify the effectiveness of the proposed algorithm. For instance, we use the proposed algorithm for real-time collision avoidance for UAVs.Comment: 17 pages, 18 figures, under review for publication in Automatic

Control and stabilization of systems with homoclinic orbits

In this paper we consider the control of two physical systems, the near wall region of a turbulent boundary layer and the rigid body, using techniques from the theory of nonlinear dynamical systems. Both these systems have saddle points linked by heteroclinic orbits. In the fluid system we show how the structure of the phase space can be used to keep the system near an (unstable) saddle. For the rigid body system we discuss passage along the orbit as a possible control manouver, and show how the Energy-Casimir method can be used to analyze stabilization of the system about the saddles

Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases

Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizable and $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting, i.e., there exist two positive constants $\mu,T$ such that, for every $t\geq 0$, $\int_t^{t+T}\alpha(s)ds\geq \mu$. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback $u=Kx$, with $K$ only depending on $(A,B)$ and possibly on $\mu,T$, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator

Recovering Linear Controllability of an Underactuated Spacecraft by Exploiting Solar Radiation Pressure

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140657/1/1.G001446.pd

Further results on saturated globally stabilizing linear state feedback control laws for single-input neutrally stable planar systems

It is known that for single-input neutrally stable planar systems, there exists a class of saturated globally stabilizing linear state feedback control laws. The goal of this paper is to characterize the dynamic behavior for such a system under arbitrary locally stabilizing linear state feedback control laws. On the one hand, for the continuous-time case, we show that all locally stabilizing linear state feedback control laws are also globally stabilizing control laws. On the other hand, for the discrete-time case, we first show that this property does not hold by explicitly constructing nontrivial periodic solution for a particular system. We then show for an example that there exists more globally stabilizing linear state feedback control laws than well known ones in the literature

Geometric algorithms for input constrained systems with application to flight control.

In this thesis novel numerical algorithms are developed to solve some problems of analysis and control design for unstable linear dynamical systems having their input constrained by maximum amplitude and rate of the control signals. Although the results obtained are of a general nature, all the problems considered are induced by flight control applications. Moreover, all these problems are stated in terms of geometry, and because of this their solution in the thesis was effectively achieved by geometrically-oriented methods. The problems considered are mainly connected with the notions of the controllable and stability regions. The controllable region is defined as the set of states of an unstable dynamical system that can be stabilized by some realizable control action. This region is bounded due to input constraints and its size can serve as a controllability measure for the control design problem. A numerical algorithm for the computation of two-dimensional slices of the region is proposed. Moreover, the stability region design is also considered. The stability region of the closed-loop system is the set of states that can be stabilized by a particular controller. This region generally utilizes only a part of the controllable region. Therefore, the controller design objective may be formulated as maximizing this region. A controller that is optimal in this sense is proposed for the case of one and two exponentially unstable open-loop eigenvalues. In the final part of the thesis a linear control allocation problem is considered for overactuated systems and its real-time solution is suggested. Using the control allocation, the actuator selection task is separated from the regulation task in the control design. All fault detection and reconfiguration capabilities are concentrated in one special unit called the control allocator, while a general control algorithm, which produces 'virtual' input for the system, remains intact. In the case of an actuator fault, only the control allocation unit needs to be reconfigured and in many cases it can generate the same 'virtual' input using a different set of control effectors. A novel control allocation algorithm, which is proposed in the thesis, is based on multidimensional interval bisection techniques