arXiv subjects

Iman Shames

Publications and source records attributed to Iman Shames.

At least 73 records · Page 4Linked to original sources

arXiv2022

Global convergence and asymptotic optimality of the heavy ball method for a class of non-convex optimization problems

In this letter we revisit the famous heavy ball method and study its global convergence for a class of non-convex problems with sector-bounded gradient. We characterize the parameters that render the method globally convergent and yield the best $R$-convergence factor. We show that for this family of functions, this convergence factor is superior to the factor obtained from the triple momentum method.

math.OC↗

arXiv2022

Probabilistic Data Association for Semantic SLAM at Scale

With advances in image processing and machine learning, it is now feasible to incorporate semantic information into the problem of simultaneous localisation and mapping (SLAM). Previously, SLAM was carried out using lower level geometric features (points, lines, and planes) which are often view-point dependent and error prone in visually repetitive environments. Semantic information can improve the ability to recognise previously visited locations, as well as maintain sparser maps for long term SLAM applications. However, SLAM in repetitive environments has the critical problem of assigning measurements to the landmarks which generated them. In this paper, we use k-best assignment enumeration to compute marginal assignment probabilities for each measurement landmark pair, in real time. We present numerical studies on the KITTI dataset to demonstrate the effectiveness and speed of the proposed framework.

cs.RO↗

arXiv2022

Approximate Midpoint Policy Iteration for Linear Quadratic Control

We present a midpoint policy iteration algorithm to solve linear quadratic optimal control problems in both model-based and model-free settings. The algorithm is a variation of Newton's method, and we show that in the model-based setting it achieves cubic convergence, which is superior to standard policy iteration and policy gradient algorithms that achieve quadratic and linear convergence, respectively. We also demonstrate that the algorithm can be approximately implemented without knowledge of the dynamics model by using least-squares estimates of the state-action value function from trajectory data, from which policy improvements can be obtained. With sufficient trajectory data, the policy iterates converge cubically to approximately optimal policies, and this occurs with the same available sample budget as the approximate standard policy iteration. Numerical experiments demonstrate effectiveness of the proposed algorithms.

math.OC↗

arXiv2022

Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation

We study numerical optimisation algorithms that use zeroth-order information to minimise time-varying geodesically-convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalisation of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimise the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimisation setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem.

math.OC↗

arXiv2022

Sensitivity Analysis for Bottleneck Assignment Problems

In assignment problems, decision makers are often interested in not only the optimal assignment, but also the sensitivity of the optimal assignment to perturbations in the assignment weights. Typically, only perturbations to individual assignment weights are considered. We present a novel extension of the traditional sensitivity analysis by allowing for simultaneous variations in all assignment weights. Focusing on the bottleneck assignment problem, we provide two different methods of quantifying the sensitivity of the optimal assignment, and present algorithms for each. Numerical examples as well as a discussion of the complexity for all algorithms are provided.

math.OC↗

arXiv2022

On modularity in reactive control architectures, with an application to formal verification

Modularity is a central principle throughout the design process for cyber-physical systems. Modularity reduces complexity and increases reuse of behavior. In this paper we pose and answer the following question: how can we identify independent `modules' within the structure of reactive control architectures? To this end, we propose a graph-structured control architecture we call a decision structure, and show how it generalises some reactive control architectures which are popular in Artificial Intelligence (AI) and robotics, specifically Teleo-Reactive programs (TRs), Decision Trees (DTs), Behavior Trees (BTs) and Generalised Behavior Trees ($k$-BTs). Inspired by the definition of a module in graph theory, we define modules in decision structures and show how each decision structure possesses a canonical decomposition into its modules. We can naturally characterise each of the BTs, $k$-BTs, DTs and TRs by properties of their module decomposition. This allows us to recognise which decision structures are equivalent to each of these architectures in quadratic time. Our proposed concept of modules extends to formal verification, under any verification scheme capable of verifying a decision structure. Namely, we prove that a modification to a module within a decision structure has no greater flow-on effects than a modification to an individual action within that structure. This enables verification on modules to be done locally and hierarchically, where structures can be verified and then repeatedly locally modified, with modules replaced by modules while preserving correctness. To illustrate the findings, we present an example of a solar-powered drone controlled by a decision structure. We use a Linear Temporal Logic-based verification scheme to verify the correctness of this structure, and then show how one can modify modules while preserving its correctness.

cs.AI↗

arXiv2022

Sampling Polynomial Trajectories for LTL Verification

This paper concerns the verification of continuous-time polynomial spline trajectories against linear temporal logic specifications (LTL without 'next'). Each atomic proposition is assumed to represent a state space region described by a multivariate polynomial inequality. The proposed approach samples a trajectory strategically, to capture every one of its region transitions. This yields a discrete word called a trace, which is amenable to established formal methods for path checking. The original continuous-time trajectory is shown to satisfy the specification if and only if its trace does. General topological conditions on the sample points are derived that ensure a trace is recorded for arbitrary continuous paths, given arbitrary region descriptions. Using techniques from computer algebra, a trace generation algorithm is developed to satisfy these conditions when the path and region boundaries are defined by polynomials. The proposed PolyTrace algorithm has polynomial complexity in the number of atomic propositions, and is guaranteed to produce a trace of any polynomial path. Its performance is demonstrated via numerical examples and a case study from robotics.

cs.LO↗

arXiv2022

A Greedy and Distributable Approach to the Lexicographic Bottleneck Assignment Problem with Conditions on Exactness

Solving the Lexicographic Bottleneck Assignment Problem (LexBAP) typically relies on centralised computation with order quartic complexity. We consider the Sequential Bottleneck Assignment Problem (SeqBAP), which yields a greedy solution to the LexBAP and discuss the relationship between the SeqBAP, the LexBAP, and the Bottleneck Assignment Problem (BAP). In particular, we reexamine tools used to analyse the structure of the BAP, and apply them to derive an algorithm that solves the SeqBAP with cubic complexity. We show that the set of solutions of the LexBAP is a subset of the solutions of the SeqBAP and analyse the conditions for which the solutions sets are identical. Furthermore, we provide a method to verify the satisfaction of these conditions. In cases where the conditions are satisfied, the proposed algorithm for solving the SeqBAP solves the LexBAP with computation that has lower complexity and can be distributed over a network of computing agents. The applicability of the approach is demonstrated with a case study where mobile robots are assigned to goal locations.

math.OC↗

arXiv2022

Risk Bounded Nonlinear Robot Motion Planning With Integrated Perception & Control

Robust autonomy stacks require tight integration of perception, motion planning, and control layers, but these layers often inadequately incorporate inherent perception and prediction uncertainties, either ignoring them altogether or making questionable assumptions of Gaussianity. Robots with nonlinear dynamics and complex sensing modalities operating in an uncertain environment demand more careful consideration of how uncertainties propagate across stack layers. We propose a framework to integrate perception, motion planning, and control by explicitly incorporating perception and prediction uncertainties into planning so that risks of constraint violation can be mitigated. Specifically, we use a nonlinear model predictive control based steering law coupled with a decorrelation scheme based Unscented Kalman Filter for state and environment estimation to propagate the robot state and environment uncertainties. Subsequently, we use distributionally robust risk constraints to limit the risk in the presence of these uncertainties. Finally, we present a layered autonomy stack consisting of a nonlinear steering-based distributionally robust motion planning module and a reference trajectory tracking module. Our numerical experiments with nonlinear robot models and an urban driving simulator show the effectiveness of our proposed approaches.

eess.SY↗

arXiv2021

A Sequential Learning Algorithm for Probabilistically Robust Controller Tuning

We introduce a sequential learning algorithm to address a robust controller tuning problem, which in effect, finds (with high probability) a candidate solution satisfying the internal performance constraint to a chance-constrained program which has black-box functions. The algorithm leverages ideas from the areas of randomised algorithms and ordinal optimisation, and also draws comparisons with the scenario approach; these have all been previously applied to finding approximate solutions for difficult design problems. By exploiting statistical correlations through black-box sampling, we formally prove that our algorithm yields a controller meeting the prescribed probabilistic performance specification. Additionally, we characterise the computational requirement of the algorithm with a probabilistic lower bound on the algorithm's stopping time. To validate our work, the algorithm is then demonstrated for tuning model predictive controllers on a diesel engine air-path across a fleet of vehicles. The algorithm successfully tuned a single controller to meet a desired tracking error performance, even in the presence of the plant uncertainty inherent across the fleet. Moreover, the algorithm was shown to exhibit a sample complexity comparable to the scenario approach.

math.OC↗

arXiv2021

Toward nonlinear dynamic control over encrypted data for infinite time horizon

Recent studies on encrypted control using homomorphic encryption allow secure operation by directly performing computations on encrypted data without decryption. Implementing dynamic controllers on encrypted data presents unique challenges due to limitations on the number of operations on an encrypted message. Hence, it may not be possible to perform the recursive operations for an infinite time horizon. In this note, we demonstrate that it is possible to run a dynamic controller over encrypted data for an infinite time horizon if the output of the controller can be represented as a function of a fixed number of previous inputs and outputs. The presented implementation requires encryption at both input and output of the plant. We identify a class of nonlinear systems that can accommodate the proposed implementation. The closed-loop performance can be guaranteed using the proposed encrypted controller by ensuring that quantization error is made arbitrarily small with appropriate choice of parameters. We show that the proposed method is amenable to linear systems (as a subset of the said nonlinear systems) with performance guarantees.

eess.SY↗

arXiv2021

Zeroth-order optimisation on subsets of symmetric matrices with application to MPC tuning

This paper provides a zeroth-order optimisation framework for non-smooth and possibly non-convex cost functions with matrix parameters that are real and symmetric. We provide complexity bounds on the number of iterations required to ensure a given accuracy level for both the convex and non-convex case. The derived complexity bounds for the convex case are less conservative than available bounds in the literature since we exploit the symmetric structure of the underlying matrix space. Moreover, the non-convex complexity bounds are novel for the class of optimisation problems we consider. The utility of the framework is evident in the suite of applications that use symmetric matrices as tuning parameters. Of primary interest here is the challenge of tuning the gain matrices in model predictive controllers, as this is a challenge known to be inhibiting industrial implementation of these architectures. To demonstrate the framework we consider the problem of MIMO diesel air-path control, and consider implementing the framework iteratively ``in-the-loop'' to reduce tracking error on the output channels. Both simulations and experimental results are included to illustrate the effectiveness of the proposed framework over different engine drive cycles.

math.OC↗

arXiv2021

Temporal Logic Planning for Minimum-Time Positioning of Multiple Threat-Seduction Decoys

Reusable decoys offer a cost-effective alternative to the single-use hardware commonly applied to protect surface assets from threats. Such decoys portray fake assets to lure threats away from the true asset. To deceive a threat, a decoy first has to position itself such that it can break the radar lock. Considering multiple simultaneous threats, this paper introduces an approach for controlling multiple decoys to minimise the time required to break the locks of all the threats. The method includes the optimal allocation of one decoy to every threat with an assignment procedure that provides local position constraints to guarantee collision avoidance and thereby decouples the control of the decoys. A crude model of a decoy with uncertainty is considered for motion planning. The task of a decoy reaching a state in which the lock of the assigned threat can be broken is formulated as a temporal logic specification. To this end, the requirements to complete the task are modelled as time-varying set-membership constraints. The temporal and logical combination of the constraints is encoded in a mixed-integer optimisation problem. To demonstrate the results a simulated case study is provided.

eess.SY↗

arXiv2021

Ordinal Optimisation and the Offline Multiple Noisy Secretary Problem

We study the success probability for a variant of the secretary problem, with noisy observations and multiple offline selection. Our formulation emulates, and is motivated by, problems involving noisy selection arising in the disciplines of stochastic simulation and simulation-based optimisation. In addition, we employ the philosophy of ordinal optimisation - involving an ordinal selection rule, and a percentile notion of goal softening for the success probability. As a result, it is shown that the success probability only depends on the underlying copula of the problem. Other general properties for the success probability are also presented. Specialising to the case of Gaussian copulas, we also derive an analytic lower bound for the success probability, which may then be inverted to find sufficiently large sample sizes that guarantee a high success probability arbitrarily close to one.

math.OC↗

arXiv2021

A principled analysis of Behavior Trees and their generalisations

As complex autonomous robotic systems become more widespread, the need for transparent and reusable Artificial Intelligence (AI) designs becomes more apparent. In this paper we analyse how the principles behind Behavior Trees (BTs), an increasingly popular tree-structured control architecture, are applicable to these goals. Using structured programming as a guide, we analyse the BT principles of reactiveness and modularity in a formal framework of action selection. Proceeding from these principles, we review a number of challenging use cases of BTs in the literature, and show that reasoning via these principles leads to compatible solutions. Extending these arguments, we introduce a new class of control architectures we call generalised BTs or $k$-BTs and show how they can extend the applicability of BTs to some of the aforementioned challenging BT use cases while preserving the BT principles.

cs.AI↗

arXiv2021

Ordinal Optimisation for the Gaussian Copula Model

We present results on the estimation and evaluation of success probabilities for ordinal optimisation over uncountable sets (such as subsets of $\mathbb{R}^{d}$). Our formulation invokes an assumption of a Gaussian copula model, and we show that the success probability can be equivalently computed by assuming a special case of additive noise. We formally prove a lower bound on the success probability under the Gaussian copula model, and numerical experiments demonstrate that the lower bound yields a reasonable approximation to the actual success probability. Lastly, we showcase the utility of our results by guaranteeing high success probabilities with ordinal optimisation.

math.OC↗

arXiv2021

Safe Learning of Uncertain Environments

In many learning based control methodologies, learning the unknown dynamic model precedes the control phase, while the aim is to control the system such that it remains in some safe region of the state space. In this work, our aim is to guarantee safety while learning and control proceed simultaneously. Specifically, we consider the problem of safe learning in nonlinear control-affine systems subject to unknown additive uncertainty. We first model the uncertainty as a Gaussian noise and use state measurements to learn its mean and covariance. We provide rigorous time-varying bounds on the mean and covariance of the uncertainty and employ them to modify the control input via an optimization program with potentially time-varying safety constraints. We show that with an arbitrarily large probability we can guarantee that the state will remain in the safe set, while learning and control are carried out simultaneously, provided that a feasible solution exists for the optimization problem. We provide a secondary formulation of this optimization that is computationally more efficient. This is based on tightening the safety constraints to counter the uncertainty about the learned mean and covariance. The magnitude of the tightening can be decreased as our confidence in the learned mean and covariance increases (i.e., as we gather more measurements about the environment). Extensions of the method are provided for non-Gaussian process noise with unknown mean and covariance as well as Gaussian uncertainties with state-dependent mean and covariance to accommodate more general environments.

cs.LG↗

arXiv2021

Fast Spline Trajectory Planning: Minimum Snap and Beyond

In this paper, we study spline trajectory generation via the solution of two optimisation problems: (i) a quadratic program (QP) with linear equality constraints and (ii) a nonlinear and nonconvex optimisation program. We propose an efficient algorithm to solve (i), which we then leverage to use in an iterative algorithm to solve (ii). Both the first algorithm and each iteration of the second algorithm have linear computational complexity in the number of spline segments. The scaling of each algorithm is such that we are able to solve the two problems faster than state-of-the-art methods and in times amenable to real-time trajectory generation requirements. The trajectories we generate are applicable to differentially flat systems, a broad class of mechanical systems, which we demonstrate by planning trajectories for a quadrotor.

eess.SY↗