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

Development of U-model enhansed nonlinear systems

Nonlinear control system design has been widely recognised as a challenging issue where the key objective is to develop a general model prototype with conciseness, flexibility and manipulability, so that the designed control system can best match the required performance or specifications. As a generic systematic approach, U-model concept appeared in Prof. Quanmin Zhu’s Doctoral thesis, and U-model approach was firstly published in the journal paper titled with ‘U-model based pole placement for nonlinear plants’ in 2002.The U-model polynomial prototype precisely describes a wide range of smooth nonlinear polynomial models, defined as a controller output u(t-1) based time-varying polynomial models converted from the original nonlinear model. Within this equivalent U-model expression, the first study of U-model based pole placement controller design for nonlinear plants is a simple mapping exercise from ordinary linear and nonlinear difference equations to time-varying polynomials in terms of the plant input u(t-1). The U-model framework realised the concise and applicable design for nonlinear control system by using such linear polynomial control system design approaches.Since the first publication, the U-model methodology has progressed and evolved over the course of a decade. By using the U-model technique, researchers have proposed many different linear algorithms for the design of control systems for the nonlinear polynomial model including; adaptive control, internal control, sliding mode control, predictive control and neural network control. However, limited research has been concerned with the design and analysis of robust stability and performance of U-model based control systems.This project firstly proposes a suitable method to analyse the robust stability of the developed U-model based pole placement control systems against uncertainty. The parameter variation is bounded, thus the robust stability margin of the closed loop system can be determined by using LMI (Linear Matrix Inequality) based robust stability analysis procedure. U-block model is defined as an input output linear closed loop model with pole assignor converted from the U-model based control system. With the bridge of U-model approach, it connects the linear state space design approach with the nonlinear polynomial model. Therefore, LMI based linear robust controller design approaches are able to design enhanced robust control system within the U-block model structure.With such development, the first stage U-model methodology provides concise and flexible solutions for complex problems, where linear controller design methodologies are directly applied to nonlinear polynomial plant-based control system design. The next milestone work expands the U-model technique into state space control systems to establish the new framework, defined as the U-state space model, providing a generic prototype for the simplification of nonlinear state space design approaches.The U-state space model is first described as a controller output u(t-1) based time-varying state equations, which is equivalent to the original linear/nonlinear state space models after conversion. Then, a basic idea of corresponding U-state feedback control system design method is proposed based on the U-model principle. The linear state space feedback control design approach is employed to nonlinear plants described in state space realisation under U-state space structure. The desired state vectors defined as xd(t), are determined by closed loop performance (such as pole placement) or designer specifications (such as LQR). Then the desired state vectors substitute the desired state vectors into original state space equations (regarded as next time state variable xd(t) = x(t) ). Therefore, the controller output u(t-1) can be obtained from one of the roots of a root-solving iterative algorithm.A quad-rotor rotorcraft dynamic model and inverted pendulum system are introduced to verify the U-state space control system design approach for MIMO/SIMO system. The linear design approach is used to determine the closed loop state equation, then the controller output can be obtained from root solver. Numerical examples and case studies are employed in this study to demonstrate the effectiveness of the proposed methods

Advanced Feedback Linearization Control for Tiltrotor UAVs: Gait Plan, Controller Design, and Stability Analysis

Three challenges, however, can hinder the application of Feedback Linearization: over-intensive control signals, singular decoupling matrix, and saturation. Activating any of these three issues can challenge the stability proof. To solve these three challenges, first, this research proposed the drone gait plan. The gait plan was initially used to figure out the control problems in quadruped (four-legged) robots; applying this approach, accompanied by Feedback Linearization, the quality of the control signals was enhanced. Then, we proposed the concept of unacceptable attitude curves, which are not allowed for the tiltrotor to travel to. The Two Color Map Theorem was subsequently established to enlarge the supported attitude for the tiltrotor. These theories were employed in the tiltrotor tracking problem with different references. Notable improvements in the control signals were witnessed in the tiltrotor simulator. Finally, we explored the control theory, the stability proof of the novel mobile robot (tilt vehicle) stabilized by Feedback Linearization with saturation. Instead of adopting the tiltrotor model, which is over-complicated, we designed a conceptual mobile robot (tilt-car) to analyze the stability proof. The stability proof (stable in the sense of Lyapunov) was found for a mobile robot (tilt vehicle) controlled by Feedback Linearization with saturation for the first time. The success tracking result with the promising control signals in the tiltrotor simulator demonstrates the advances of our control method. Also, the Lyapunov candidate and the tracking result in the mobile robot (tilt-car) simulator confirm our deductions of the stability proof. These results reveal that these three challenges in Feedback Linearization are solved, to some extents.Comment: Doctoral Thesis at The University of Toky

Provably Safe Reinforcement Learning via Action Projection using Reachability Analysis and Polynomial Zonotopes

While reinforcement learning produces very promising results for many applications, its main disadvantage is the lack of safety guarantees, which prevents its use in safety-critical systems. In this work, we address this issue by a safety shield for nonlinear continuous systems that solve reach-avoid tasks. Our safety shield prevents applying potentially unsafe actions from a reinforcement learning agent by projecting the proposed action to the closest safe action. This approach is called action projection and is implemented via mixed-integer optimization. The safety constraints for action projection are obtained by applying parameterized reachability analysis using polynomial zonotopes, which enables to accurately capture the nonlinear effects of the actions on the system. In contrast to other state-of-the-art approaches for action projection, our safety shield can efficiently handle input constraints and dynamic obstacles, eases incorporation of the spatial robot dimensions into the safety constraints, guarantees robust safety despite process noise and measurement errors, and is well suited for high-dimensional systems, as we demonstrate on several challenging benchmark systems

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Contributions to fuzzy polynomial techniques for stability analysis and control

The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees. The contributions of the thesis are: ¿ Improved domain of attraction estimation of nonlinear systems for both continuous-time and discrete-time cases. An iterative methodology based on invariant-set results is presented for obtaining polynomial boundaries of such domain of attraction. ¿ Extension of the above problem to the case with bounded persistent disturbances acting. Different characterizations of inescapable sets with polynomial boundaries are determined. ¿ State estimation: extension of the previous results in literature to the case of fuzzy observers with polynomial gains, guaranteeing stability of the estimation error and inescapability in a subset of the zone where the model is valid. ¿ Proposal of a polynomial Lyapunov function with discrete delay in order to improve some polynomial control designs from literature. Preliminary extension to the fuzzy polynomial case. Last chapters present a preliminary experimental work in order to check and validate the theoretical results on real platforms in the future.Pitarch Pérez, JL. (2013). Contributions to fuzzy polynomial techniques for stability analysis and control [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34773TESI

Robust post-stall perching with a simple fixed-wing glider using LQR-Trees

Birds routinely execute post-stall maneuvers with a speed and precision far beyond the capabilities of our best aircraft control systems. One remarkable example is a bird exploiting post-stall pressure drag in order to rapidly decelerate to land on a perch. Stall is typically associated with a loss of control authority, and it is tempting to attribute this agility of birds to the intricate morphology of the wings and tail, to their precision sensing apparatus, or their ability to perform thrust vectoring. Here we ask whether an extremely simple fixed-wing glider (no propeller) with only a single actuator in the tail is capable of landing precisely on a perch from a large range of initial conditions. To answer this question, we focus on the design of the flight control system; building upon previous work which used linear feedback control design based on quadratic regulators (LQR), we develop nonlinear feedback control based on nonlinear model-predictive control and 'LQR-Trees'. Through simulation using a flat-plate model of the glider, we find that both nonlinear methods are capable of achieving an accurate bird-like perching maneuver from a large range of initial conditions; the 'LQR-Trees' algorithm is particularly useful due to its low computational burden at runtime and its inherent performance guarantees. With this in mind, we then implement the 'LQR-Trees' algorithm on real hardware and demonstrate a 95 percent perching success rate over 147 flights for a wide range of initial speeds. These results suggest that, at least in the absence of significant disturbances like wind gusts, complex wing morphology and sensing are not strictly required to achieve accurate and robust perching even in the post-stall flow regime.United States. Office of Naval Research. Multidisciplinary University Research Initiative (N00014-10-1-0951)National Science Foundation (U.S.) (Award IIS-0915148