Complementarity problems, a class of mathematical optimization problems with orthogonality constraints, are widely used in many robotics tasks, such as locomotion and manipulation, due to their ability to model non-smooth phenomena (e.g., contact dynamics). In this paper, we propose a method to analyze the stability of complementarity systems with neural network controllers. First, we introduce a method to represent neural networks with rectified linear unit (ReLU) activations as the solution to a linear complementarity problem. Then, we show that systems with ReLU network controllers have an equivalent linear complementarity system (LCS) description. Using the LCS representation, we turn the stability verification problem into a linear matrix inequality (LMI) feasibility problem. We demonstrate the approach on several examples, including multi-contact problems and friction models with non-unique solutions.
This paper presents an algebraic technique to compute the maximum time-delay that can be accepted in the control loop of a Brushless DC Motor (BLDCM) speed controller before the closed loop response becomes unstable. Using a recently proposed time-delay stability analysis methodology, we derive accurate stability conditions for the BLDCM speed controller. The results of applying the new method show that tuning the PI controller for very fast response in the order of magnitude of the BLDCM mechanical time constant cause the time-delay to significantly affect the system stability.
University of California, Berkeley（加州大学伯克利分校）；University of Michigan（密歇根大学）；Virginia Tech（弗吉尼亚理工大学）
Neural network controllers have become popular in control tasks thanks to their flexibility and expressivity. Stability is a crucial property for safety-critical dynamical systems, while stabilization of partially observed systems, in many cases, requires controllers to retain and process long-term memories of the past. We consider the important class of recurrent neural networks (RNN) as dynamic controllers for nonlinear uncertain partially-observed systems, and derive convex stability conditions based on integral quadratic constraints, S-lemma and sequential convexification. To ensure stabil…
Saturations in control signal can challenge the stability proof of a feedback-linearization-based controller, even leading the system unstable [1]. Thus, several approaches are established to avoid reaching the saturation bound [2,3]. Meanwhile, to help design the controller for a quad-tilt-rotor, [1] modeled a tilt vehicle with implementing the feedback-linearization-based controllers. In this article, we provide a gait plan for this tilt vehicle and control it utilizing the feedback linearization. Since saturations exist in the control signals, we study the stability based on Lyapunov theory…
This paper studies an attitude control system design based on modified Rodrigues parameters feedback. It employs a linear continuous sliding mode controller. The sliding mode controller is able to bring the existence of the sliding motion asymptotically. Besides, the attitude control system equilibrium point is proved to have an asymptotic stability guarantee through further analysis. This stability analysis is conducted since the sliding mode existence on the designed sliding surface does not imply the stability guarantee of the system's equilibrium. This paper ends with some numerical exampl…
Current commercial adaptive cruise control (ACC) systems consist of an upper-level planner controller that decides the optimal trajectory that should be followed, and a low-level controller in charge of sending the gas/brake signals to the mechanical system to actually move the vehicle. We find that the low-level controller has a significant impact on the string stability (SS) even if the planner is string stable: (i) a slow controller deteriorates the SS, (ii) slow controllers are common as they arise from insufficient control gains, from a "weak" gas/brake system or both, and (iii) the integ…
Vehicle stability control systems are important components of active safety systems for road transport. The problem of vehicle lateral stability control is addressed in this paper using active front wheel steering and individual braking. Vehicle lateral stability control means keeping the vehicle yaw rate and the vehicle side slip angle in desired values. For this reason, a model-based controller is designed. The desired yaw rate is obtained from the single track vehicle model and the desired side slip angle is chosen as zero. Controller design consists of two parts, lower and upper controller…
In this paper, modification of the classical PID controller and development of open-loop control mechanisms to improve stability and robustness of a differential wheeled robot are discussed. To deploy the algorithm, a test platform has been constructed using low-cost and off-the-shelf components including a microcontroller, reflectance sensor, and motor driver. This paper describes the heuristic approach used in the identification of the system specifications as well as the optimization of the controller. The PID controller is analyzed in detail and the effect of each term is explained in the …
Exponential stability of a PI plus reset integrator controller by a sampled-data system approach
Gipsa-lab（吉普萨实验室）；CNRS, LAAS（法国国家科学研究中心，图卢兹航空航天实验室）；Univ de Toulouse, UPS, LAAS（图卢兹大学，国立高等邮电学校，图卢兹航空航天实验室）；Dpto. Informática y Sistemas, University of Murcia（穆尔西亚大学，计算机与系统系）；Univ de Toulouse, LAAS（图卢兹大学，图卢兹航空航天实验室）
This paper introduces an analytical framework for the derivation of hybrid equations of motion of a flexible quadrotor. This approach helps obtain rigid and elastic equations of motion simultaneously, in a decoupled form, which facilitates the controller design. A delay-dependent stability condition is obtained for the overall system dynamics, including the operator with reaction time delay, the adaptive controller and the flexible quadrotor dynamics. Two different adaptive controllers are implemented to address the uncertainties. It is demonstrated via simulations that the flexible arm tip os…
A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal inner-approximation to the region of attraction (ROA). The first theorem addresses linear time-invariant systems, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the system to include perturbations such as unmodeled dynamics, slope-restricted nonlinearities, and time delay, using integral qua…
Neural networks can be used as approximations of several complex control schemes such as model predictive control. We show in this paper which properties deep neural networks with rectifier linear units as activation functions need to satisfy to guarantee constraint satisfaction and asymptotic stability of the closed-loop system. To do so, we introduce a parametric description of the neural network controller and use a mixed-integer linear programming formulation to perform output range analysis of neural networks. We also propose a novel method to modify a neural network controller such that …
Aiming at the difficulty of stability analysis in practical application of existing control methods, a controller strategy based on lyapunov stability theory is proposed to realize stable control for any control method. In order to illustrate the effectiveness of this control scheme, this paper lists two different lyapunov functions to test the control effect of different system models in tracking the target function. The simulation results show that the proposed method has excellent control effect on different types of systems and is advanced.
Today, linear PID controllers cannot satisfy requirements of high-precision industry due to the development of technology. According to the literature, reset controllers can overcome this important barrier. However, similar to other non-linear controllers, stability analysis for these controllers is complex and needs parametric models of the systems. Consequently, the applicability of these controllers may diminish in industry. The well-known Hb method is one of the solutions of this significant problem. Nevertheless, assessing the Hb condition in the frequency-domain is complex, particularly …
Recent advances in learning for control allow to synthesize vehicle controllers from learned system dynamics and maintain robust stability guarantees. However, no approach is well-suited for training linear time-invariant (LTI) controllers using arbitrary learned models of the dynamics. This article introduces a method to do so. It uses a robust control framework to derive robust stability criteria. It also uses simulated policy rollouts to obtain gradients on the controller parameters, which serve to improve the closed-loop performance. By formulating the stability criteria as penalties with …
Nowadays, the demand for an alternative to linear PID controllers has increased because of the rising expectations of the high-precision industry. The potential of reset controllers to solve this important challenge has been extensively demonstrated in the literature. However, similarly to other non-linear controllers, the stability analysis for these controllers is complex and relies on parametric models of the systems which may hinder the applicability of these controllers in industry. The well-known Hbeta method tries to solve this significant issue. However, assessing the H\b{eta} conditio…
High voltage direct current links based on voltage source converters (VSC-HVDC) embedded in alternating current (AC) systems are receiving a great deal of attention recently because they can contribute positively to the flexibilisation of modern power systems. Among several possibilities, AC-line-emulation control has been highlighted as an simple-but-useful alternative for these type of systems. With this strategy, the power flow through the link is controlled proportionally to the angle difference between its two AC terminals and this provides self-adaptation of the power flow in case of con…
We present a full review of PID passivity-based controllers (PBC) applied to power electronic converters, discussing limitations, unprecedented merits and potential improvements in terms of large-signal stability, robustness and performance. We provide four main contributions. The nominal case is first considered and it is shown, under the assumption of perfect knowledge of the system parameters, that the PID-PBC is able to guarantee global exponential stability of a desired operating point for any positive gains. Second, we analyze robustness of the controller to parameters uncertainty for a …
This paper proposes a novel unit-quaternion observer-based controller for attitude tracking (attitude and angular velocity) with guaranteed transient and steady-state performance. The proposed approach is computationally cheap and can operate based on measurements provided, for instance by a typical low-cost inertial measurement unit (IMU) or magnetic, angular rate, and gravity (MARG) sensor without the knowledge of angular velocity. First, an observer evolved on $\mathbb{S}^{3}\times\mathbb{R}^{3}$ is developed guaranteeing asymptotic stability of the closed loop error signals starting from a…
Small-Signal Stability Manifolds in Converter-Dominated Power Systems
Università Campus Bio-Medico di Roma（罗马生物医学大学）；University of Colorado Denver（科罗拉多大学丹佛分校）；Università degli Studi di Genova（热那亚大学）；Politecnico di Milano（米兰理工大学）
In this paper, the stability and stabilization problem of positive nonlinear systems, described by the Takagi-Sugeno discrete-time fuzzy model, is studied. The proposed approach is based on the linear co-positive Lyapunov function and parallel distributed controller. Necessary and sufficient conditions for the existence of a state feedback controller, which guarantees the positivity and stability of the closed-loop system, are obtained by a new approach as linear programming, and to solve it and obtain the controller coefficients, the optimal Convex Optimization algorithm is used. Moreover, th…
We design a reduced attitude controller for reorienting the spin axis of a gyroscope in a geometric control framework. The proposed controller preserves the inherent gyroscopic stability associated with a spinning axis-symmetric rigid body. The equations of motion are derived in two frames: a non-spinning frame to show the gyroscopic stability, and a body-fixed spinning frame for deriving the controller. The proposed controller is designed such that it retains the gyroscopic stability structure in the closed loop and renders the desired equilibrium almost-globally asymptotically stable. Due to…
This manuscript describes several approaches to tune the parameters of a class of passivity-based controllers for standard nonlinear mechanical systems. In particular, we are interested in controllers that preserve the mechanical system structure in closed-loop. To this end, first, we provide tuning rules for stabilization, i.e., the rate of convergence (exponential stability) and stability margin (input-to-state stability). Then, we provide guidelines to remove the overshoot while prescribing the rise time. Additionally, we propose a methodology to tune the gyroscopic-related parameters. We a…
Stability-Preserving Online Adaptation of Neural Closed-loop Maps
Institute of Mechanical Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院机械工程学院）；Department of Engineering Science, University of Oxford（牛津大学工程科学系）
This paper considers the problem of voltage regulation in distribution networks. The primary motivation is to keep voltages within preassigned operating limits by commanding the reactive power output of distributed energy resources (DERs) deployed in the grid. We develop a framework for developing local Volt/Var control that comprises two main steps. In the first, by exploiting historical data and for each DER, we learn a function representing the desirable equilibrium points for the power network. These points approximate solutions of an Optimal Power Flow (OPF) problem. In the second, we pro…
This paper considers online switching control with a finite candidate controller pool, an unknown dynamical system, and unknown cost functions. The candidate controllers can be unstabilizing policies. We only require at least one candidate controller to satisfy certain stability properties, but we do not know which one is stabilizing. We design an online algorithm that guarantees finite-gain stability throughout the duration of its execution. We also provide a sublinear policy regret guarantee compared with the optimal stabilizing candidate controller. Lastly, we numerically test our algorithm…
An eigenvalue based framework is developed for the stability analysis and stabilization of coupled systems with time-delays, which are naturally described by delay differential algebraic equations. The spectral properties of these equations are analyzed and their stability properties are studied, taking into account the effect of small delay perturbations. Subsequently, numerical methods for stability assessment and for designing stabilizing controllers with a prescribed structure or order, based on a direct optimization approach, are briefly addressed. The effectiveness of the approach is ill…
