This paper investigates the design of a robust fixed-order controller for single-input-single-output (SISO) polytopic systems with interval uncertainties, with the aim that the closed-loop stability is appropriately ensured and the performance specifications on sensitivity shaping are conformed in a specific finite frequency range. Utilizing the notion of generalized common Lyapunov strictly positive realness (CL-SPRness), the equivalence between strictly positive realness (SPRness) and strictly bounded realness (SBRness) is established; and then the specifications on robust stability and performance are transformed into the SPRness of newly constructed systems and further characterized in the framework of linear matrix inequality (LMI) conditions. The proposed methodology avoids the tedious yet mandatory evaluations of the specifications on all vertices of the uncertain polytopic system in an explicit form. Instead, solving five LMIs exclusively suffices for ensuring the robust stability and performance regardless of the number of vertices, and thus the typically heavy computational burden is considerably alleviated. It is also noteworthy that the proposed methodology additionally provides the necessary and sufficient conditions for this robust controller design with the consideration of a prescribed finite frequency range, and therefore significantly less conservatism is attained in the system performance.
Typically, it is desirable to design a control system that is not only robustly stable in the presence of parametric uncertainties but also guarantees an adequate level of system performance. However, most of the existing methods need to take all extreme models over an uncertain domain into consideration, which then results in costly computation. Also, since these approaches attempt (rather unrealistically) to guarantee the system performance over a full frequency range, a conservative design is always admitted. Here, taking a specific viewpoint of robust stability and performance under a stat…
We present a new continuous Lyapunov Redesign (LR) methodology for the robust stabilization of a class of uncertain time-delay systems that is based on the so-called Super Twisting Algorithm. The main feature of the proposed approach is that allows one to simultaneously adjust the chattering effect and achieve asymptotic stabilization of the uncertain system, which is lost when continuous approximation of the unit control is considered. At the basis of the Super Twisting based LR methodology is a class of Lyapunov-Krasovskii functionals, whose particular form of its time derivative allows one …
Stabilizing controller design and region of attraction (RoA) estimation are essential in nonlinear control. Moreover, it is challenging to implement a control Lyapunov function (CLF) in practice when only partial knowledge of the system is available. We propose a learning framework that can synthesize state-feedback controllers and a CLF for control-affine nonlinear systems with unstructured uncertainties. Based on a regularity condition on these uncertainties, we model them as bounded disturbances and prove that a CLF for the nominal system (estimate of the true system) is an input-to-state s…
Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室）
Safety and robustness are two desired properties for any reinforcement learning algorithm. CMDPs can handle additional safety constraints and RMDPs can perform well under model uncertainties. In this paper, we propose to unite these two frameworks resulting in robust constrained MDPs (RCMDPs). The motivation is to develop a framework that can satisfy safety constraints while also simultaneously offer robustness to model uncertainties. We develop the RCMDP objective, derive gradient update formula to optimize this objective and then propose policy gradient based algorithms. We also independentl…
This paper presents adaptive robust quadratic program (QP) based control using control Lyapunov and barrier functions for nonlinear systems subject to time-varying and state-dependent uncertainties. An adaptive estimation law is proposed to estimate the pointwise value of the uncertainties with pre-computable estimation error bounds. The estimated uncertainty and the error bounds are then used to formulate a robust QP, which ensures that the actual uncertain system will not violate the safety constraints defined by the control barrier function. Additionally, the accuracy of the uncertainty est…
Modern control systems must operate in increasingly complex environments subject to safety constraints and input limits, and are often implemented in a hierarchical fashion with different controllers running at multiple time scales. Yet traditional constructive methods for nonlinear controller synthesis typically "flatten" this hierarchy, focusing on a single time scale, and thereby limited the ability to make rigorous guarantees on constraint satisfaction that hold for the entire system. In this work we seek to address the stabilization of constrained nonlinear systems through a \textit{multi…
This paper addresses to Sliding Mode Learning Control (SMLC) of uncertain nonlinear systems with Lyapunov stability analysis. In the control scheme, a conventional control term is used to provide the system stability in compact space while a Type-2 Neuro-Fuzzy Controller (T2NFC) learns system behavior so that the T2NFC takes the overall control of the system completely in a very short time period. The stability of the sliding mode learning algorithm was proven in literature; however, it is so restrictive for systems without the overall system stability. To address this shortcoming, a novel con…
In this work we addressed the problem of stability analysis for an uncertain piecewise affine model of a genetic regulatory network. In particular we considered polytopic parameter uncertainties on the proteins production rate functions, giving conditions for the existence of a piecewise quadratic Lyapunov function for any possible realization of the system. In the spirit of other works in literature, the resulting conditions will be given on the vertices of the parameter polytope, while still taking into consideration the piecewise nature of the Lyapunov function and the presence, in general,…
Massachusetts Institute of Technology（麻省理工学院）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
Safety and stability are common requirements for robotic control systems; however, designing safe, stable controllers remains difficult for nonlinear and uncertain models. We develop a model-based learning approach to synthesize robust feedback controllers with safety and stability guarantees. We take inspiration from robust convex optimization and Lyapunov theory to define robust control Lyapunov barrier functions that generalize despite model uncertainty. We demonstrate our approach in simulation on problems including car trajectory tracking, nonlinear control with obstacle avoidance, satell…
In this report, we analyze and design feedback policies for discrete-time Piecewise-Affine (PWA) systems with uncertainty in both the affine dynamics and the polytopic partition. The main idea is to utilise the Difference-of-Convex (DC) decomposition of continuous PWA systems to derive quadratic Lyapunov functions as stability certificates and stabilizing affine policies in a higher dimensional space. When projected back to the state space, we obtain time-varying PWQ Lyapunov functions and time-varying PWA feedback policies.
This paper studies the problem of utilizing data-driven adaptive control techniques to guarantee stability and safety of uncertain nonlinear systems with high relative degree. We first introduce the notion of a High Order Robust Adaptive Control Barrier Function (HO-RaCBF) as a means to compute control policies guaranteeing satisfaction of high relative degree safety constraints in the face of parametric model uncertainty. The developed approach guarantees safety by initially accounting for all possible parameter realizations but adaptively reduces uncertainty in the parameter estimates levera…
In this paper, a position and velocity estimation method for robotic manipulators which are affected by constant bounded disturbances is considered. The tracking control problem is formulated as a disturbance rejection problem, with all the unknown parameters and dynamic uncertainties lumped into disturbance. Using adaptive robust extended Kalman filter(AREKF) the movement and velocity of each joint is predicted to use in discontinuous Lyapunov-based controller structure. The parameters of the error dynamics have been validated off-line by real data. Computer simulation results given for a two…
This technical note studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefined-time stability. The origin of a dynamical system is predefined-time stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow to demonstrate equivalence between previous Lyapunov theorems for predefined-time stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing th…
Never Too Rigid to Reach: Adaptive Virtual Model Control with LLM- and Lyapunov-Based Reinforcement Learning
Department of Engineering, University of Cambridge（剑桥大学工程系）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程与计算机科学系）；Zhejiang University-University of Illinois Urbana-Champaign Institute, Zhejiang University（浙江大学-伊利诺伊大学厄巴纳-香槟分校联合研究所）；Tsinghua Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际 Graduate School）；Department of Data Science, New Jersey Institute of Technology（新泽西理工学院数据科学系）
This paper proposes a Lyapunov approach to the design of a multivariable generalized Super-Twisting algorithm (MGSTA), which is able to control a system with perturbations and uncertain control matrix, both depending on time and the system states. The presented procedure shows that, under reasonable assumptions for the uncertainties, it is always possible to find a set of constant gains for the MGSTA in order to ensure global and robust finite-time stability of the system's outputs. Simulation results on an omnidirectional mobile robot illustrate the performance of the MGSTA.
The spacecraft attitude tracking problem is addressed with actuator faults and uncertainties among inertias, external disturbances, and, in particular, state estimates. A continuous sliding mode attitude controller is designed using attitude and angular velocity estimates from an arbitrary stable stand-alone observer. Rigorous analysis shows that the controller ensures robust stability of the entire closed-loop system as long as the observer yields state estimates with uniformly ultimately bounded estimation errors. In addition, a sequential Lyapunov analysis is utilized to obtain a convergent…
This paper proposes a framework for automatic formal controller synthesis for general hybrid systems with a subset of safety and reachability specifications. The framework uses genetic programming to automatically co-synthesize controllers and candidate Lyapunov-like functions. These candidate Lyapunov-like functions are used to formally verify the control specification, and their correctness is proven using a Satisfiability Modulo Theories solver. The advantages of this approach are: no restriction is made to polynomial systems, the synthesized controllers are expressed as compact expressions…
A continuous adaptive control design is developed for nonlinear dynamical systems with linearly parameterizable uncertainty involving time-varying uncertain parameters. The key feature of this design is a robust integral of the sign of the error (RISE)-like term in the adaptation law which compensates for potentially destabilizing terms in the closed-loop error system arising from the time-varying nature of uncertain parameters. A Lyapunov-based stability analysis ensures asymptotic tracking, and boundedness of the closed-loop signals.
This paper proposes a novel control architecture for state and input constrained Euler-Lagrange (E-L) systems with parametric uncertainties. A simple saturated controller is strategically coupled with a Barrier Lyapunov Function (BLF) based controller to ensure state and input constraint satisfaction. To the best of the authors' knowledge, this is the first result for E-L systems that guarantee asymptotic tracking with user-specified state and input constraints. The proposed controller also ensures that all the closed-loop signals remain bounded. The efficacy of the proposed controller in term…
In this paper, we investigate the consensus problem of second-order multiagent systems under directed graphs. Simple yet robust consensus algorithms that advance existing achievements in accounting for velocity and input constraints, agent uncertainties, and lack of neighboring velocity measurements are proposed. Furthermore, we show that the proposed method can be applied to the consensus control of uncertain robotic manipulators with velocity and control torque constraints. We rigorously prove that the velocity and control inputs stay within prespecified ranges by tuning design parameters a …
Robust Reduced-Order Model Stabilization for Partial Differential Equations Based on Lyapunov Theory and Extremum Seeking with Application to the 3D Boussinesq Equations
Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；Interdisciplinary Center for Applied Mathematics, Virginia Tech（弗吉尼亚理工大学跨学科应用数学中心）；Department of Aeronautics and Astronautics, Massachusetts Institute of Technology（麻省理工学院航空航天系）
Robust linear static anti-windup with probabilistic certificates
Politecnico di Milano（米兰理工大学）；CNR-IEIIT（意大利国家研究委员会信息与通信技术研究所）；CNRS, LAAS（法国国家科学研究中心，图卢兹实验室）；Univ. de Toulouse, LAAS（图卢兹大学，图卢兹实验室）；University of Trento（特伦托大学）
A novel MIMO homogeneous Super-Twisting Algorithm is proposed in this paper for nonlinear systems with relative degree one, having a time and state-varying uncertain control matrix. The uncertainty is represented by a constant but unknown left matrix factor. Sufficient conditions for stability with full-matrix control gains are established, in contrast to the usual scalar gains. For this a smooth Lyapunov function, based on a passivity interpretation, is used. Moreover, continuous and homogeneous approximations of the classical discontinuous Super-Twisting algorithm are obtained, using a unifi…
The ever increasing complexity of real-time control systems results in significant deviations in the timing of sensing and actuation, which may lead to degraded performance or even instability. In this paper we present a method to analyze stability under mostly-periodic timing with bounded uncertainty, a timing model typical for the implementation of controllers that were actually designed for strictly periodic execution. In contrast to existing work, we include the case of multiple sensors and actuators with individual timing uncertainty. Our approach is based on the discretization of a linea…
Adaptive type-2 fuzzy second order sliding mode control for nonlinear uncertain chaotic system
QUERE Laboratory, Engineering Faculty, University of Setif 1（塞提夫第一大学工程学院QUERE实验室）；CReSTIC of Reims Champagne-Ardenne University,IUT de Troyes（兰斯香槟-阿登大学CReSTIC中心特鲁瓦高等师范学院）
This paper presents a method to design a min-norm Control Lyapunov Function (CLF)-based stabilizing controller for a control-affine system with uncertain dynamics using Gaussian Process (GP) regression. In order to estimate both state and input-dependent model uncertainty, we propose a novel compound kernel that captures the control-affine nature of the problem. Furthermore, by the use of GP Upper Confidence Bound analysis, we provide probabilistic bounds of the regression error, leading to the formulation of a CLF-based stability chance constraint which can be incorporated in a min-norm optim…
In this paper, we consider controlling a class of single-input-single-output (SISO) commensurate fractional-order nonlinear systems with parametric uncertainty and external disturbance. Based on backstepping approach, an adaptive controller is proposed with adaptive laws that are used to estimate the unknown system parameters and the bound of unknown disturbance. Instead of using discontinuous functions such as the $\mathrm{sign}$ function, an auxiliary function is employed to obtain a smooth control input that is still able to achieve perfect tracking in the presence of bounded disturbances. …
Direct Adaptive Controller for Uncertain MIMO Dynamic Systems with Time-varying Delay and Dead-zone Inputs
South China University of Technology（华南理工大学）；Massachusetts Institute of Technology (MIT)（麻省理工学院）；Northeastern University（东北大学）；University of Science and Technology Beijing（北京科技大学）
