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

Reach Control on Simplices by Piecewise Affine Feedback

We study the reach control problem for affine systems on simplices, and the focus is on cases when it is known that the problem is not solvable by continuous state feedback. We examine from a geometric viewpoint the structural properties of the system which make continuous state feedbacks fail. This structure is encoded by so-called reach control indices, which are defined and developed in the paper. Based on these indices, we propose a subdivision algorithm and associated piecewise affine feedback. The method is shown to solve the reach control problem in all remaining cases, assuming it is solvable by open-loop controls

Unified Approach to Convex Robust Distributed Control given Arbitrary Information Structures

We consider the problem of computing optimal linear control policies for linear systems in finite-horizon. The states and the inputs are required to remain inside pre-specified safety sets at all times despite unknown disturbances. In this technical note, we focus on the requirement that the control policy is distributed, in the sense that it can only be based on partial information about the history of the outputs. It is well-known that when a condition denoted as Quadratic Invariance (QI) holds, the optimal distributed control policy can be computed in a tractable way. Our goal is to unify and generalize the class of information structures over which quadratic invariance is equivalent to a test over finitely many binary matrices. The test we propose certifies convexity of the output-feedback distributed control problem in finite-horizon given any arbitrarily defined information structure, including the case of time varying communication networks and forgetting mechanisms. Furthermore, the framework we consider allows for including polytopic constraints on the states and the inputs in a natural way, without affecting convexity

New computer-based search strategies for extreme functions of the Gomory--Johnson infinite group problem

We describe new computer-based search strategies for extreme functions for the Gomory--Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions.Comment: 54 pages, many figure

A control problem for affine dynamical systems on a full-dimensional simplex

Given an affine system on a simplex, the problem of reaching a particular facet of the simplex, using affine state feedback is studied. Necessary and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the simplex. If these conditions are met, a constructive procedure yields an affine feedback control law, that solves this reachability problem

Predictive Control of Linear Uncertain Systems

Predictive control is a very useful tool in controlling constrained systems, since the constraints can be satisfied explicitly by the optimisations. Sets, namely, reachable sets, controllable sets, invariant sets, etc, play fundamental roles in designing predictive control strategies for uncertain systems. Meanwhile, in addition to the commonly assumed boundedness of the uncertainty, the explicit use of its stochastic properties can lead to imprq\fement in system response. This thesis is concerned with robust set theories, mainly for reachable sets, with applications to time-optimal control; and the use of stochastic properties of the uncertainty to achieve less conservative controls. In the first part of this thesis, we focus on LTI systems subject to, additional to the usual constraints, a constraint on the control change between sample times. One key ingredient in controlling such constrained systems is the initial control value, which, via analyses and simulations, is shown to be a useful extra degree of freedom. Reachable sets that incorporate this influential initial control value are derived and analyzed, with theoretical as well as computational algorithms developed for both nominal and uncertain systems under different types of feedback policy. Following this, the reachable set is discussed in connection with time-optimal control to obtain desired control laws. In addition, controllable sets, stabilisable sets and invariant sets for such constrained uncertain systems are studied. In the second part, the uncertainties are assumed to have stochastic properties. They are exploited in three different ways: the expected worst-case is used instead of the worst-case to achieve less conservative control even when the uncertainty is relatively large; the stochastic invariant set is proposed to provide alternative methods for approximating disturbance invariant sets; the relaxed set difference is developed to obtain less restrictive controls and/or replacing probabilistic constraint or slack variables.Imperial Users onl

Walking in the Shadow: A New Perspective on Descent Directions for Constrained Minimization

Descent directions such as movement towards Frank-Wolfe vertices, away steps, in-face away steps and pairwise directions have been an important design consideration in conditional gradient descent (CGD) variants. In this work, we attempt to demystify the impact of movement in these directions towards attaining constrained minimizers. The best local direction of descent is the directional derivative of the projection of the gradient, which we refer to as the $\textit{shadow}$ of the gradient. We show that the continuous-time dynamics of moving in the shadow are equivalent to those of PGD however non-trivial to discretize. By projecting gradients in PGD, one not only ensures feasibility but is also able to "wrap" around the convex region. We show that Frank-Wolfe (FW) vertices in fact recover the maximal wrap one can obtain by projecting gradients, thus providing a new perspective on these steps. We also claim that the shadow steps give the best direction of descent emanating from the convex hull of all possible away-steps. Viewing PGD movements in terms of shadow steps gives linear convergence, dependent on the number of faces. We combine these insights into a novel $S\small{HADOW}$-$CG$ method that uses FW steps (i.e., wrap around the polytope) and shadow steps (i.e., optimal local descent direction), while enjoying linear convergence. Our analysis develops properties of the curve formed by projecting a line on a polytope, which may be of independent interest, while providing a unifying view of various descent directions in the CGD literature

Compositional Synthesis via a Convex Parameterization of Assume-Guarantee Contracts

We develop an assume-guarantee framework for control of large scale linear (time-varying) systems from finite-time reach and avoid or infinite-time invariance specifications. The contracts describe the admissible set of states and controls for individual subsystems. A set of contracts compose correctly if mutual assumptions and guarantees match in a way that we formalize. We propose a rich parameterization of contracts such that the set of parameters that compose correctly is convex. Moreover, we design a potential function of parameters that describes the distance of contracts from a correct composition. Thus, the verification and synthesis for the aggregate system are broken to solving small convex programs for individual subsystems, where correctness is ultimately achieved in a compositional way. Illustrative examples demonstrate the scalability of our method

Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach

International audienceIn this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov-Metzler condition for stabilizability does not hold