Showing papers on "Control-Lyapunov function published in 2013"
TL;DR: It is proved that for any initial condition within any given closed set the minimal inter-sampling time is proved to be below bounded avoiding the infinitely fast sampling phenomena.
Abstract: In this technical note, a universal formula is proposed for event-based stabilization of general nonlinear systems affine in the control. The feedback is derived from the original one proposed by E. Sontag in the case of continuous time stabilization. Under the assumption of the existence of a smooth Control Lyapunov Function, it is proved that an event-based static feedback, smooth everywhere except at the origin, can be designed so to ensure the global asymptotic stability of the origin. Moreover, the inter-sampling time can be proved not to contract at the origin. More precisely, it is proved that for any initial condition within any given closed set the minimal inter-sampling time is proved to be below bounded avoiding the infinitely fast sampling phenomena. Moreover, under homogeneity assumptions the control can be proved to be smooth anywhere and the inter-sampling time bounded below for any initial condition. In that case, we retrieve a control approach previously published for continuous time stabilization of homogeneous systems.
170 citations
TL;DR: The proposed methodology uses recent advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition to use an algebraic reformulation technique to recast the system's dynamics into a set of polynomial differential algebraic equations.
Abstract: We present a methodology for the algorithmic construction of Lyapunov functions for the transient stability analysis of classical power system models. The proposed methodology uses recent advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition, which have been powerful tools for the analysis of systems with polynomial vector fields. In order to apply these techniques to power grid systems described by trigonometric nonlinearities we use an algebraic reformulation technique to recast the system's dynamics into a set of polynomial differential algebraic equations. We demonstrate the application of these techniques to the transient stability analysis of power systems by estimating the region of attraction of the stable operating point. An algorithm to compute the local stability Lyapunov function is described together with an optimization algorithm designed to improve this estimate.
141 citations
TL;DR: This work develops the first systematic design of ES controllers for unstable plants, a simple non-model based universal feedback law that emulates, in an average sense, the “L g V controllers” for stabilization with inverse optimality, and a new strategy for stabilization of systems with unknown control directions.
Abstract: Employing extremum seeking (ES) for seeking minima of control Lyapunov function (CLF) candidates, we develop 1) the first systematic design of ES controllers for unstable plants, 2) a simple non-model based universal feedback law that emulates, in an average sense, the “ $L_{g}V$ controllers” for stabilization with inverse optimality, and 3) a new strategy for stabilization of systems with unknown control directions, as an alternative to Nussbaum gain controllers that lack exponential stability, lack transient performance guarantees, and lack robustness to changes in the control direction. The stability analysis that underlies our designs is inspired by an analysis approach synthesized in a recent work by Durr, Stankovic, and Johansson, which combines a Lie bracket averaging result of Gurvits and Li with a semiglobal practical stability result under small parametric perturbations by Moreau and Aeyels.
117 citations
Posted Content•
TL;DR: In this paper, a novel method for directly incorporating user-defined control input saturations into the calculation of a control Lyapunov function (CLF)-based walking controller for a biped robot is presented.
Abstract: This paper presents a novel method for directly incorporating user-defined control input saturations into the calculation of a control Lyapunov function (CLF)-based walking controller for a biped robot. Previous work by the authors has demonstrated the effectiveness of CLF controllers for stabilizing periodic gaits for biped walkers, and the current work expands on those results by providing a more effective means for handling control saturations. The new approach, based on a convex optimization routine running at a 1 kHz control update rate, is useful not only for handling torque saturations but also for incorporating a whole family of user-defined constraints into the online computation of a CLF controller. The paper concludes with an experimental implementation of the main results on the bipedal robot MABEL.
105 citations
TL;DR: A strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media and derive a control law to preserve the ISS properties of the system and improve its performance.
Abstract: In this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as close as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances.
100 citations
TL;DR: The authors propose a particle swarm optimization (PSO) for a discrete-time inverse optimal control scheme of a doubly fed induction generator (DFIG) and demonstrates the comparison between both mechanisms.
Abstract: In this paper, the authors propose a particle swarm optimization (PSO) for a discrete-time inverse optimal control scheme of a doubly fed induction generator (DFIG). For the inverse optimal scheme, a control Lyapunov function (CLF) is proposed to obtain an inverse optimal control law in order to achieve trajectory tracking. A posteriori, it is established that this control law minimizes a meaningful cost function. The CLFs depend on matrix selection in order to achieve the control objectives; this matrix is determined by two mechanisms: initially, fixed parameters are proposed for this matrix by a trial-and-error method and then by using the PSO algorithm. The inverse optimal control scheme is illustrated via simulations for the DFIG, including the comparison between both mechanisms.
77 citations
Book•
08 Apr 2013
TL;DR: In this article, the authors proposed a novel inverse optimal control scheme for stabilization and trajectory tracking of discrete-time nonlinear systems, which avoids the need to solve the associated Hamilton-Jacobi-Bellman equation.
Abstract: Discrete-Time Inverse Optimal Control for Nonlinear Systems proposes a novel inverse optimal control scheme for stabilization and trajectory tracking of discrete-time nonlinear systems. This avoids the need to solve the associated Hamilton-Jacobi-Bellman equation and minimizes a cost functional, resulting in a more efficient controller. Design More Efficient Controllers for Stabilization and Trajectory Tracking of Discrete-Time Nonlinear Systems The book presents two approaches for controller synthesis: the first based on passivity theory and the second on a control Lyapunov function (CLF). The synthesized discrete-time optimal controller can be directly implemented in real-time systems. The book also proposes the use of recurrent neural networks to model discrete-time nonlinear systems. Combined with the inverse optimal control approach, such models constitute a powerful tool to deal with uncertainties such as unmodeled dynamics and disturbances. Learn from Simulations and an In-Depth Case Study The authors include a variety of simulations to illustrate the effectiveness of the synthesized controllers for stabilization and trajectory tracking of discrete-time nonlinear systems. An in-depth case study applies the control schemes to glycemic control in patients with type 1 diabetes mellitus, to calculate the adequate insulin delivery rate required to prevent hyperglycemia and hypoglycemia levels. The discrete-time optimal and robust control techniques proposed can be used in a range of industrial applications, from aerospace and energy to biomedical and electromechanical systems. Highlighting optimal and efficient control algorithms, this is a valuable resource for researchers, engineers, and students working in nonlinear system control.
70 citations
TL;DR: The 2-D fuzzy system model is established based on the Fornasini–Marchesini local state-space model, and a control design procedure is proposed based on a relaxed approach in which basis-dependent Lyapunov functions are used.
Abstract: This paper investigates the problem of stability analysis and stabilization for two-dimensional (2-D) discrete fuzzy systems. The 2-D fuzzy system model is established based on the Fornasini---Marchesini local state-space model, and a control design procedure is proposed based on a relaxed approach in which basis-dependent Lyapunov functions are used. First, nonquadratic stability conditions are derived by means of linear matrix inequality (LMI) technique. Then, by introducing an additional instrumental matrix variable, the stabilization problem for 2-D fuzzy systems is addressed, with LMI conditions obtained for the existence of stabilizing controllers. Finally, the effectiveness and advantages of the proposed design methods based on basis-dependent Lyapunov functions are shown via two examples.
56 citations
TL;DR: An energy-related Lyapunov function is used to guide the input selection process by making switching decisions based on the stabilizing effect each input has on the closed-loop system, leading to a stability proof for a quantized input control law.
Abstract: We present a new method for the generation of input switching sequences in a synchronous motor control system based on the evaluation of a control Lyapunov function over a discrete set of realizable inputs. Typical reference input realization methods, such as space vector modulation, rely on high-frequency state space averaging which can yield unnecessary switching events and increased switching losses. Alternative input selection strategies, such as lookup-table-based direct torque control, rely on heuristically chosen hysteresis bands to determine switching instants, which often results in a suboptimal choice between switching frequency and other performance measures. In this paper, we use an energy-related Lyapunov function to guide the input selection process by making switching decisions based on the stabilizing effect each input has on the closed-loop system. We provide a theoretical analysis of a motor-inverter system, leading to a stability proof for a quantized input control law. The controller performance is verified through computer simulations and experimental results.
53 citations
TL;DR: This paper proposes an observer structure with a pre-observer or internal model augmented by an equivariant innovation term that leads to autonomous error evolution and a control Lyapunov function construction is used to design the observer innovation.
51 citations
TL;DR: A backstepping-based inverse optimal attitude controller (BIOAC) is derived which has the property of a maximum convergence rate in the sense of a control Lyapunov function under input torque limitation.
Abstract: Input saturation must be taken into account for applying rapid reorientation in the large angle manoeuvre of a quadrotor In this paper, a backstepping-based inverse optimal attitude controller (BIOAC) is derived which has the property of a maximum convergence rate in the sense of a control Lyapunov function (CLF) under input torque limitation In the controller, a backstepping technique is used for handling the complexity introducing by the unit quaternion representation of the attitude of a quadrotor with four parameters Moreover, the inverse optimal approach is employed to circumvent the difficulty of solving the Hamilton-Jacobi-Bellman (HJB) equation The performance of BIOAC is compared with a PD controller in which the input torque limitation is not considered under the same unit quaternion representation using numerical simulation while the results show that BIOAC gains faster convergence with less control effort Next, BIOAC is realized on a test bed and the effectiveness of the control law is verified by experimental studies
TL;DR: A Rifford–Sontag-type discontinuous static state feedback controller for asymptotic stabilization with the disassembled differential of the locally semiconcave practical CLF (LS-PCLF) is proposed by means of sample stability and shown to have a special advantage with which one can easily design a controller in the case of the minimum projection method.
TL;DR: For a class of hybrid systems given in terms of constrained differential and difference equations/inclusions, this work studies the existence of control Lyapunov functions when compact sets are asymptotically stable as well as the stabilizability properties guaranteed when control Lyapsinov functions exist.
Abstract: For a class of hybrid systems given in terms of constrained differential and difference equations/inclusions, we study the existence of control Lyapunov functions when compact sets are asymptotically stable as well as the stabilizability properties guaranteed when control Lyapunov functions exist. An existence result asserting that asymptotic stabilizability of a compact set implies the existence of a smooth control Lyapunov function is established. When control Lyapunov functions are available, conditions guaranteeing the existence of stabilizing continuous state-feedback control laws are provided.
05 Nov 2013
TL;DR: It is shown that the resulting closed-loop system without disturbance is associated with a nonlinear semigroup and is asymptotically stable and robust to the external disturbance.
Abstract: In this paper, we are concerned with the boundary output feedback stabilization of an Euler-Bernoulli beam equation with free boundary at one end and control and disturbance at the other end. A variable structure output feedback stabilizing controller is designed by the Lyapunov function approach. It is shown that the resulting closed-loop system without disturbance is associated with a nonlinear semigroup and is asymptotically stable. In addition, we show that this controller is robust to the external disturbance in the sense that the vibrating energy of the closed-loop system is also convergent to zero as time goes to infinity in the presence of finite sum of harmonic disturbance at the control end.
TL;DR: This paper first derive an algebraizable sufficient condition for the existence of a polynomial Lyapunov function, then applies a real root classification based method step by step to under-approximate this derived condition as a semi-algebraic system such that the semi-Al algebraic system only involves the coefficients of the pre-assumed polynometric.
TL;DR: In this article, a centralized nonlinear control strategy for coordinating several point-to-point and multiterminal HVDC systems based on Lyapunov theory is proposed.
Abstract: The security region of a power system is an important and timely issue; different stability criteria may be limiting. Rotor-angle stability can be improved by modulating active power of installed high-voltage direct current (HVDC) links. This paper proposes a new centralized nonlinear control strategy for coordinating several point-to-point and multiterminal HVDC systems based on Lyapunov theory. The proposed control Lyapunov function is negative semi-definite along the trajectories and uses the internal node representation of the system. The proposed control Lyapunov function increases the domain of attraction and, thus, improves the rotor-angle stability. Nonlinear simulations are performed on the IEEE 10-machine 39-bus system which shows the effectiveness of the controller. In comparison, simulations using the conventional lead-lag controller are also run.
TL;DR: In this paper, the authors consider continuous Lyapunov functions and reveal a relation between existence of a continuous LyAPunov function and inherent input-to-state stability on compact sets with respect to both inner and outer perturbations.
Abstract: This technical note considers input-to-state stability analysis of discrete-time systems using continuous Lyapunov functions. The main result reveals a relation between existence of a continuous Lyapunov function and inherent input-to-state stability on compact sets with respect to both inner and outer perturbations. If the Lyapunov function is K∞-continuous, the result applies to unbounded sets as well.
TL;DR: Simulations, simulations and experimental results of speed sensing feedback in BLDC motors of the Quad-rotor aircraft in the presence of disturbances are presented and it is concluded that the system is asymptotically stable.
Abstract: A real-time attitude stabilization control scheme is proposed for the efficient performance of a mini-UAV Quad-Rotor. Brushless DC (BLDC) motor speed sensing is performed by reflective sensors in order to obtain a robust stabilization of the vehicle in hovering mode both indoor and outdoor. The speed measurement has the advantage of introducing this state information directly in the closed loop control which should be very useful for achieving robust stabilization of the mini-UAV. Furthermore a stabilizing control strategy based on Control Lyapunov Function (CLF) is proposed. The control scheme contains two control loops. The inner loop is devoted to control the motors speed while the outer loop is devoted to control the attitude stabilization of a mini-UAV. Assuming that the motors can be considered as a disturbance of the system, then by the standard singular perturbation theory, we may conclude that the system is asymptotically stable. Finally, to verify the satisfactory performance of proposed embedded controller, simulations and experimental results of speed sensing feedback in BLDC motors of the Quad-rotor aircraft in the presence of disturbances are presented.
TL;DR: This paper constructs explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: the general 2-species case, a 3-species model, and the model of May-Leonard.
Abstract: Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: $(1)$ the general 2-species case, $(2)$ a 3-species model, and $(3)$ the model of May-Leonard. The basins of attraction for these examples are demonstrated, including cases with bistability and cyclical behavior. The first two examples are the generalized gradient system, where the energy dissipation may not follow the gradient of the Lyapunov function. In addition, under a new type of stochastic interpretation, the Lyapunov function also leads to the Boltzmann-Gibbs distribution on the final steady state when multiplicative noise is added.
01 Dec 2013
TL;DR: In this article, it was shown that for a monotone system on a compact state space, asymptotic stability implies existence of a max-separable Lyapunov function.
Abstract: Separable Lyapunov functions play vital roles, for example, in stability analysis of large-scale systems. A Lyapunov function is called max-separable if it can be decomposed into a maximum of functions with one-dimensional arguments. Similarly, it is called sum-separable if it is a sum of such functions. In this paper it is shown that for a monotone system on a compact state space, asymptotic stability implies existence of a max-separable Lyapunov function. We also construct two systems on a non-compact state space, for which a max-separable Lyapunov function does not exist. One of them has a sum-separable Lyapunov function. The other does not.
TL;DR: In this paper, the authors consider a control problem where the state must approach asymptotically a target C while paying an integral cost with a non-negative Lagrangian l. Through an inequality involving a positive number p¯0 and a Minimum Restraint Function U=U(x) -a special type of Control Lyapunov Function, they provide a condition implying that the system is asymptonically controllable, and the value function is bounded by U/p¯0.
08 Apr 2013
TL;DR: The end result is the generation of bipedal robotic walking that is remarkably human-like and is experimentally realizable, as evidenced by the implementation of the resulting controllers on multiple robotic platforms.
Abstract: This paper briefly presents the process of formally achieving bipedal robotic walking through controller synthesis inspired by human locomotion. Motivated by the hierarchical control present in humans, we begin by viewing the human as a "black box" and describe outputs, or virtual constraints, that appear to characterize human walking. By considering the equivalent outputs for the bipedal robot, a nonlinear controller can be constructed that drives the outputs of the robot to the outputs of the human; moreover, the parameters of this controller can be optimized so that stable robotic walking is provably achieved while simultaneously producing outputs of the robot that are as close as possible to those of a human. Finally, considering a control Lyapunov function based representation of these outputs allows for the class of controllers that provably achieve stable robotic walking can be greatly enlarged. The end result is the generation of bipedal robotic walking that is remarkably human-like and is experimentally realizable, as evidenced by the implementation of the resulting controllers on multiple robotic platforms.
TL;DR: An approach to avoid the computation of an explicit solution of the Lyapunov partial differential inequality is proposed, introducing the concept of Dynamic Lyap unov function, which allow to study stability properties of equilibrium points, similarly to standard Lyap Unov functions.
17 Jun 2013
TL;DR: A sampled-data event detection strategy is proposed for linear continuous-time systems, whose output is sampled periodically to be tested for determining the transmission, and dynamic logic conditions are provided to characterize the asymptotic stability property for event-triggered control systems by defining discrete Lyapunov functions.
Abstract: In this paper, a sampled-data event detection strategy is proposed for linear continuous-time systems, whose output is sampled periodically to be tested for determining the transmission. Dynamic logic conditions are provided to characterize the asymptotic stability property for event-triggered control systems by defining discrete Lyapunov functions. State and output feedback control laws with event-driven communication are proposed such that the states of event based control systems converge to zero eventually, in the regulation setting. Some examples are presented which demonstrate the utility of this new definition.
TL;DR: A stochastic asymptotic stabilization method is proposed for deterministic input-affine control systems, which are randomized by including Gaussian white noises in control inputs, and the sufficient condition is derived for the diffucion coefficients so that there exist Stochastic control Lyapunov functions for the systems.
Abstract: In this paper, a stochastic asymptotic stabilization method is proposed for deterministic input-affine control systems, which are randomized by including Gaussian white noises in control inputs. The sufficient condition is derived for the diffucion coefficients so that there exist stochastic control Lyapunov functions for the systems. To illustrate the usefulness of the sufficient condition, the authors propose the stochastic continuous feedback law, which makes the origin of the Brockett integrator become globally asymptotically stable in probability.
Posted Content•
TL;DR: It is shown that deciding local or global asymptotic stability of cubic vector fields is strongly NP-hard, and that if the degree of the polynomial Lyapunov function is fixed, then sos programming may fail to find a valid Lyap unov function even though one exists.
Abstract: We consider polynomial differential equations and make a number of contributions to the questions of (i) complexity of deciding stability, (ii) existence of polynomial Lyapunov functions, and (iii) existence of sum of squares (sos) Lyapunov functions.
(i) We show that deciding local or global asymptotic stability of cubic vector fields is strongly NP-hard. Simple variations of our proof are shown to imply strong NP-hardness of several other decision problems: testing local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, invariance of the unit ball, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, local collision avoidance, and existence of a stabilizing control law.
(ii) We present a simple, explicit example of a globally asymptotically stable quadratic vector field on the plane which does not admit a polynomial Lyapunov function (joint work with M. Krstic). For the subclass of homogeneous vector fields, we conjecture that asymptotic stability implies existence of a polynomial Lyapunov function, but show that the minimum degree of such a Lyapunov function can be arbitrarily large even for vector fields in fixed dimension and degree. For the same class of vector fields, we further establish that there is no monotonicity in the degree of polynomial Lyapunov functions.
(iii) We show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming may fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a planar or a homogeneous vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos.
01 Jan 2013
TL;DR: In this article, a novel method for directly incorporating user-defined control input saturations into the calculation of a control Lyapunov function (CLF)-based walking controller for a biped robot is presented.
Abstract: This paper presents a novel method for directly incorporating user-defined control input saturations into the calculation of a control Lyapunov function (CLF)-based walking controller for a biped robot. Previous work by the authors has demonstrated the effectiveness of CLF controllers for stabilizing periodic gaits for biped walkers (2), and the current work expands on those results by providing a more effective means for handling control saturations. The new approach, based on a convex optimization routine running at a 1 kHz control update rate, is useful not only for handling torque saturations but also for incorporating a whole family of user-defined constraint s into the online computation of a CLF controller. The paper concludes with an experimental implementation of the main results on the bipedal robot MABEL.
TL;DR: The proposed control law of a homogeneous semiconcave control Lyapunov function for a two-wheeled mobile robot, applied to a Roomba, is confirmed and the effectiveness of the proposed method is confirmed.
TL;DR: The conditions for the gas of affine systems provided an appropriate clf is known are studied, and an explicit formula for a one-parameterized family of bounded regular feedback global stabilizers is proposed.
Abstract: Our main purpose in this paper is to address the problem of synthesis of regular feedback controls for the global asymptotic stabilization (gas) of nonlinear systems with controls taking values in the $m$-dimensional $\mathbf{r}$-weighted hyperbox $\mathcal{B}_{\mathbf{r}}^{m}(\infty):=[-r_{1}^{-},r_{1}^{+}]\times\cdots\times[-r_{m}^{-},r_{m}^{+}]$. Working along the line of Artstein and Sontag's control Lyapunov function (clf) approach, we study the conditions for the gas of affine systems provided an appropriate clf is known, and propose an explicit formula for a one-parameterized family of bounded regular feedback global stabilizers. The case of scalar bounded positive feedback controls ($r^{-}=0$) is also included. Finally, the problem of designing a marginally robust control function is addressed.
TL;DR: In this article, a multilayer minimum projection method is proposed for control Lyapunov function (CLF) design on a manifold, which is based on desingularization of the functions on the manifolds.
Abstract: Control Lyapunov function (CLF) design on a manifold is a difficult problem in control theory. To address this problem, we have proposed the multilayer minimum projection method. The method requires CLFs on different manifolds from the manifold where the control problem is defined. In this paper, we relax the requirement by desingularization of the functions on the manifolds.
The paper focuses on the problem of desingularization in the multilayer minimum projection method. We show that the functions on other manifolds need not be CLFs by consideration of desingularization. Moreover, we propose a CLF design method by singular point assignment based on the advantage of desingularization. The method enables us to merge local CLFs into the global CLF.
This paper proposes two CLF design methods: desingularization and singular point assignment. A CLF design example is provided for each method; the advantages of the proposed methods are confirmed by those two examples.