In this paper, we focus on the problem about direct way to design a stable controller for nonlinear system. A framework of learning controller with Lyapunov-based constraint is proposed, which is intended to transform designing and analyis of a controller to straightforward way to make controller by solving an optimization with the Lyapunov constraint, and which can be a novel way to design a global stability guaranteed controller directly. Firstly, an optimization problem subject to Lyapunov-based constraints is formulated, in which the tracking error is the objective function to minimize. Secondly, the controller combines with PID and feedforward is given in form of neural networks, Finally, the solution of controller method to the optimization problem is analyzed, in which we leverage some deep learning technologies to boost the capbility of solution. The results of two simulations of 2 order linear and nonlinear systems demonstrate that the method proposed has some high performance in speed of convergence, tracking error and smoothness and amplitude of control output. The results of simulation to the nonlinear system with disturbance, noise, uncertainty of parameters and the difference of reference output demostrate that our method has high performance in robustness and generalization.
A controller synthesis method for state- and input-constrained nonlinear systems is presented that seeks continuous piecewise affine (CPA) Lyapunov-like functions and controllers simultaneously. Non-convex optimization problems are formulated on triangulated subsets of the admissible states that can be refined to meet primary control objectives, such as stability and safety, alongside secondary performance objectives. A multi-stage design is also given that enlarges the region of attraction (ROA) sequentially while allowing exclusive performance for each stage. A clear boundary for an invarian…
Adaptive control architectures often make use of Lyapunov functions to design adaptive laws. We are specifically interested in adaptive control methods, such as the well-known L1 adaptive architecture, which employ a parameter observer for this purpose. In such architectures, the observation error plays a critical role in determining analytical bounds on the tracking error as well as robustness. In this paper, we show how the non-existence of coercive Lyapunov operators can impact the analytical bounds, and with it the performance and the robustness of such adaptive systems.
This paper presents a new Lyapunov-based nonlinear model predictive controller (LNMPC) for the attitude control problem of unmanned aerial vehicles (UAVs), which is essential for their functioning operation. The controller is designed based on a quadratic cost function integrating UAV dynamics and system constraints. An additional contraction constraint is then introduced to ensure closed-loop system stability. That constraint is fulfilled via a Lyapunov function derived from a sliding mode controller (SMC). The feasibility and stability of the LNMPC are finally proved. Simulation and comparis…
This paper presents a method to stabilize state and input constrained nonlinear systems using an offline optimization on variable triangulations of the set of admissible states. For control-affine systems, by choosing a continuous piecewise affine (CPA) controller structure, the non-convex optimization is formulated as iterative semi-definite programming (SDP), which can be solved efficiently using available software. The method has very general assumptions on the system's dynamics and constraints. Unlike similar existing methods, it avoids finding terminal invariant sets, solving non-convex o…
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…
Toyota Research Institute（丰田研究院）；Stanford University（斯坦福大学）；Massachusetts Institute of Technology（麻省理工学院）
Deep learning has had a far reaching impact in robotics. Specifically, deep reinforcement learning algorithms have been highly effective in synthesizing neural-network controllers for a wide range of tasks. However, despite this empirical success, these controllers still lack theoretical guarantees on their performance, such as Lyapunov stability (i.e., all trajectories of the closed-loop system are guaranteed to converge to a goal state under the control policy). This is in stark contrast to traditional model-based controller design, where principled approaches (like LQR) can synthesize stabl…
The paper introduces novel families of cruise controllers for autonomous vehicles on lane-free ring-roads. The design of the cruise controllers is based on the appropriate selection of a Control Lyapunov Function expressed on measures of the energy of the system with the kinetic energy expressed in ways similar to Newtonian or relativistic mechanics. The derived feedback laws (cruise controllers) are decentralized (per vehicle), as each vehicle determines its control input based on: (i) its own state; (ii) either only the distance from adjacent vehicles (inviscid cruise controllers) or the sta…
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.
This paper presents an efficient, offline method to simultaneously synthesize controllers and seek closed-loop Lyapunov functions for constrained piecewise affine systems on triangulated subsets of the admissible states. Triangulation refinements explore a rich class of controllers and Lyapunov functions. Since an explicit Lipschitz Lyapunov function is found, an invariant subset of the closed-loop region of attraction is obtained. Moreover, it is a control Lyapunov function, so minimum-norm controllers can be realized through online quadratic programming. It is formulated as a sequence of sem…
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 perf…
Transient stability of power systems is becoming increasingly important because of the growing integration of renewable resources. These resources lead to a reduction in mechanical inertia but also provide increased flexibility in frequency responses. Namely, their power electronic interfaces can implement almost arbitrary control laws. To design these controllers, reinforcement learning (RL) has emerged as a powerful method in searching for optimal non-linear control policy parameterized by neural networks. A key challenge is to enforce that a learned controller must be stabilizing. This pape…
In this paper, we consider the problem of platooning control with mismatched disturbances using the distributed adaptive backstepping method. The main challenges are: (1) maintaining the compositionality and the distributed nature of the controller, and (2) ensuring the robustness of the controller with respect to general types of disturbances. To address these challenges, we first propose a novel notion that we named "Vector String Lyapunov Function", whose existence implies l2 weak string stability. This notion is based on the vector Lyapunov function-based stability analysis, which depends …
Controller design for nonlinear systems with Control Lyapunov Function (CLF) based quadratic programs has recently been successfully applied to a diverse set of difficult control tasks. These existing formulations do not address the gap between design with continuous time models and the discrete time sampled implementation of the resulting controllers, often leading to poor performance on hardware platforms. We propose an approach to close this gap by synthesizing sampled-data counterparts to these CLF-based controllers, specified as quadratically constrained quadratic programs (QCQPs). Assumi…
ETH Zürich（苏黎世联邦理工学院）；California Institute of Technology（加州理工学院）
The theoretical unification of Nonlinear Model Predictive Control (NMPC) with Control Lyapunov Functions (CLFs) provides a framework for achieving optimal control performance while ensuring stability guarantees. In this paper we present the first real-time realization of a unified NMPC and CLF controller on a robotic system with limited computational resources. These limitations motivate a set of approaches for efficiently incorporating CLF stability constraints into a general NMPC formulation. We evaluate the performance of the proposed methods compared to baseline CLF and NMPC controllers wi…
In this paper, we consider the tracking control problem for robot manipulators which are affected by constant bounded disturbances. Three control schemes are applied for the problem, which composed of integral action and tracking controllers. The goal is improving the accuracy of tracking procedure for a robot manipulator to track a specified reference signal in the presence of constant bounded disturbances. Inverse dynamics controller, improved Lyapunov-based controller with integral action and discontinuous Lyapunov-based controller are three schemes that are evaluated in this paper. Third o…
A constructive tool of nonlinear control systems design, the method of Control Lyapunov Functions (CLF) has found numerous applications in stabilization problems for continuous time, discrete-time and hybrid systems. In this paper, we address the fundamental question: given a CLF, corresponding to the continuous-time controller with some predefined (e.g. exponential) convergence rate, can the same convergence rate be provided by an event-triggered controller? Under certain assumptions, we give an affirmative answer to this question and show that the corresponding event-based controllers provid…
Design of robust H_inf fuzzy output feedback controller for affine nonlinear systems:Fuzzy Lyapunov function approach
University of Technology Malaysia（技术大学马来西亚）；Shiraz University of Technology（谢尔兹技术大学）；Nanyang Technological University（南洋理工大学）
本文提出了一种基于非二次Lyapunov函数和引入松弛矩阵技术的新系统方法，用于具有扰动的仿真非线性系统。首先，将仿真非线性系统表示为Takagi-Sugeno（T-S）模糊双线性模型。随后，基于并行分布式补偿（PDC）方案设计鲁棒H_∞控制器。通过利用Lyapunov函数推导出稳定性条件，以线性矩阵不等式（LMIs）形式表达。此外，提出一些松弛矩阵以减少LMIs稳定性条件的保守性。最后，通过详细讨论等温连续搅拌釜反应器（CSTR）用于Van de Vusse反应器的应用，来说明所提方法的优点并验证其有效性。
In mathematical psychology, decision makers are modeled using the Lindbladian equations from quantum mechanics to capture important human-centric features such as order effects and violation of the sure thing principle. We consider human-machine interaction involving a quantum decision maker (human) and a controller (machine). Given a sequence of human decisions over time, how can the controller dynamically provide input messages to adapt these decisions so as to converge to a specific decision? We show via novel stochastic Lyapunov arguments how the Lindbladian dynamics of the quantum decisio…
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…
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…
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…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；ETH Zürich（苏黎世联邦理工学院）；Google Research（谷歌研究院）
Learning controllers merely based on a performance metric has been proven effective in many physical and non-physical tasks in both control theory and reinforcement learning. However, in practice, the controller must guarantee some notion of safety to ensure that it does not harm either the agent or the environment. Stability is a crucial notion of safety, whose violation can certainly cause unsafe behaviors. Lyapunov functions are effective tools to assess stability in nonlinear dynamical systems. In this paper, we combine an improving Lyapunov function with automatic controller synthesis in …
This letter investigates the prescribed-instant stabilization problem for high-order integrator systems. In anothor word, the settling time under the presented controller is independent of the initial conditions and equals the prescribed time instant. The controller is designed with the concept of backstepping. A strict proof based on the Lyapunov method is presented to clamp the settling time to the prescribed time instant from both the left and right sides. This proof serves as an example to present a general framework to verify the designed stabilization property. It should be emphasized th…
As more inverter-connected renewable resources are integrated into the grid, frequency stability may degrade because of the reduction in mechanical inertia and damping. A common approach to mitigate this degradation in performance is to use the power electronic interfaces of the renewable resources for primary frequency control. Since inverter-connected resources can realize almost arbitrary responses to frequency changes, they are not limited to reproducing the linear droop behaviors. To fully leverage their capabilities, reinforcement learning (RL) has emerged as a popular method to design n…
The introduction of unexpected system disturbances and new system dynamics does not allow guaranteed continuous system stability. In this research we present a novel approach for detecting early failure indicators of non-linear highly chaotic system and accordingly predict the best parameter calibrations to offset such instability using deep machine learning regression model. The approach proposed continuously monitors the system and controller signals. The Re-calibration of the system and controller parameters is triggered according to a set of conditions designed to maintain system stability…
In this technical note, we generalize the well-known Lyapunov-based stabilizability and detectability tests for linear time-invariant (LTI) systems to the context of discrete-time (DT) polytopic linear parameter-varying (LPV) systems. To do so, we exploit the mathematical structure of the class of poly-quadratic Lyapunov functions, which enables us to formulate conditions in the form of linear matrix inequalities (LMIs). Our results differ from existing conditions in that we adopt weaker assumptions on the parameter dependence of the controllers/observers and our method does not require explic…
Control Lyapunov functions (CLFs) and control barrier functions (CBFs) are widely used tools for synthesizing controllers subject to stability and safety constraints. Paired with online optimization, they provide stabilizing control actions that satisfy input constraints and avoid unsafe regions of state-space. Designing CLFs and CBFs with rigorous performance guarantees is computationally challenging. To certify existence of control actions, current techniques not only design a CLF/CBF, but also a nominal controller. This can make the synthesis task more expensive, and performance estimation …
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…
Recent advances in the reinforcement learning (RL) literature have enabled roboticists to automatically train complex policies in simulated environments. However, due to the poor sample complexity of these methods, solving RL problems using real-world data remains a challenging problem. This paper introduces a novel cost-shaping method which aims to reduce the number of samples needed to learn a stabilizing controller. The method adds a term involving a Control Lyapunov Function (CLF) -- an `energy-like' function from the model-based control literature -- to typical cost formulations. Theoreti…
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…
