International audiencea smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy certain further increase or decrease conditions. These cells have fibers of actin and myosin which run through the cell and are supported by a framework of other proteins. Learning control lyapunov function to ensure stability of. Lyapunovs indirect method and linking lyapunov function. In the theory of ordinary differential equations odes, lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ode. Examples of lyapunov functions generated by linear programming 78 9.
Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunovs second method for stability are important to stability theory of dynamical systems and control. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1. Lecture 12 basic lyapunov theory stanford university. A lyapunov function is a nonnegative scalar measure of this multidimensional state. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy certain further increase or decrease conditions. Lyapunov stability various types of stability may be discussed for the solutions of differential equations describing dynamical systems. Singleunit smooth muscle in the walls of the viscera, called visceral muscle, has a stressrelaxation response that permits muscle to stretch, contract, and relax as the organ expands. An almost strict smooth patchy control lyapunov function for in the sense ofdefinition 5.
Smooth muscle is also present in the eyes, where it functions. Computation of lyapunov functions and stability of. Common polynomial lyapunov functions for linear switched. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Smooth lyapunov functions for discontinuous stable systems 377 of a continuous weak lyapunov function in the small v vt,x for 1. It is a global control lyapunov function if it is radially unbounded and. This paper develops nonsmooth lyapunov stability theory and lasalles invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories. Lyapunovs direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. This video covers the prerequisites for studying lyapunov s second method. Smooth muscle sonamed because the cells do not have striations is present in the walls of hollow organs like the urinary bladder, uterus, stomach, intestines, and in the walls of passageways, such as the arteries and veins of the circulatory system, and the tracts of the respiratory, urinary, and reproductive systems ab.
Amplificador logaritmico y antilogaritmo pdf download effettuare diverse aggiunte di titolante registrando i valori smplificador di potenziale in mv e calcolare il valore dell antilogaritmo. Smooth lyapunov functions for discontinuous stable systems. The method of lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. Skeletal muscle is capable of spontaneous contraction. Smooth muscle is a type of muscle tissue which is used by various systems to apply pressure to vessels and organs. Singleunit smooth muscle tissue contains gap junctions to synchronize membrane depolarization and contractions so that the muscle contracts as a single unit. The method is a generalization of the idea that if there is some measure of energy in a system, then. In this chapter, basic notions of lyapunov and lyapunovlike functions will be presented. A generalization to a class of piecewisepolynomial lyapunov functions is given.
Lyapunov function from a set of demonstrations, 3 proposing an optimal control problem based on the learned lyapunov function to ensure stability of nonlinear autonomous ds, and 4 extending the classical clf control scheme and present it in the context of learning robot discrete motions from demonstrations. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the. Video created by university of colorado boulder for the course control of nonlinear spacecraft attitude motion. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong. Lyapunov stability theory of nonsmooth systems ieee. This result answers a question raised by dayawansa and martin. When a woman gives birth, the smooth muscle cells found in the uterus contract to push the. Defining stability using lyapunov functions energy functions. Amplificadores operacionales logaritmico diego guevara. The most important type is that concerning the stability of. Get a printable copy pdf file of the complete article 548k, or click on a page image below to browse page by page. For instance, one of the useful characteristic of smooth clfs is the existence of universal formulas for stabilization. We prove that such a control lyapunov function exists for any asymptotically controllable nonlinear system. Smooth patchy control lyapunov functions sciencedirect.
Suchafunctionisasmooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. The 3dimages are best shown live wiggle w the mouse, but can also be. In the urinary system, smooth muscle cells contract to push urine into the urethra and out of the body. The main purpose is to provide powerful images to be used as backdrops for classdiscussions. Lyapunov stability game the adversary picks a region in the state space of radius. Smooth approximation of patchy lyapunov functions for. Links to pubmed are also available for selected references. In his book 6, filipov studies the equilibria of differential equations with discontinuous righthand sides, but deals with smooth lyapunov functions. Smooth muscle is found in the walls of hollow organs.
Smooth muscle definition, function and location biology. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunov s second method for stability are important to stability theory of dynamical systems and control. Lyapunov functions are crucial in the present book aims, given the strict relation between lyapunov functions and invariant sets. Starting with a locally lipschitz patchy lyapunov function for a given switched system, we provide the construction of a continuously di erentiable smooth lyapunov function, obtained via a convolutionbased approach. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth. There is a small amount of smooth muscle in the heart. Then our main result is that the degree of that common polynomial lyapunov function is not uniformly bounded over all the uas systems. The idea of lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability, which concerns the behavior of different but nearby solutions to differential equations. Smooth muscle is composed of sheets or strands of smooth muscle cells. Citeseerx smooth patchy control lyapunov functions.
The state of a system at a particular time is often described by a multidimensional vector. This result is extended in 6 to discontinuous systems, i. Lyapunov s theorem will be covered in the next lecture. Lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. In the opposite case, we will show that pa 1 and pa 2 of proposition 5. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. Lyapunov function and relate it to these v arious stabilit y notions.
Because vertebrate smooth muscle is located in the walls of many hollow organs, the normal functioning of the cardiovascular, respiratory, this article was most recently revised and updated by richard. Full text full text is available as a scanned copy of the original print version. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a decrease condition when the domains overlap. The primary use of these results is in analyzing the.
Furthermore, contraction metrics and finslerlyapunov functions as a related method to study. Lecture 14 lyapunov theory with inputs and outputs systems with inputs and outputs reachability bounding bounds on rms gain boundedreal lemma feedback synthesis via controllyapunov functions 141. R n is an equilibrium point of the system if fxe 0 xe is an equilibrium point xt xe is a trajectory suppose xe is an equilibrium point system is globally asymptotically stable g. Lyapunov functions are used extensively in control theory to ensure different forms of system stability. Smooth muscle cells are responsible for helping food pass through the digestive system and for pushing food up into the esophagus when vomiting occurs. Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1.
Inputtostate stability iss applies lyapunov notions to. Lyapunov functions and feedback in nonlinear control. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunov s second method for stability are important to stability theory of dynamical systems and control theory. This smooth function approximates the patchy function when working with clarkes generalized gradient. Lyapunovs second or direct method provides tools for studying asymp. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open. We prove that such a control lyapunov function exists for any. Yoshizawa 14 developed the lyapunov theory for lipschitz potential functions, but this work assumed a continuous vector field and smooth trajectories. Computable tests based on filipovs differential inclusion and clarkes generalized gradient are derived. Controllyapunov functions for systems satisfying the. Smooth control lyapunov functions clfs are instrumental in many feedback control designs and can be traced back to artstein who introduced this lyapunov characterization of asymptotic controllability in 4. Ppt amplificador operacional ii powerpoint presentation. Systems with inputs we now consider systems with inputs, i. Thisv 2isa smoothcontrollyapunovfunctionono 2,asverifiedbyud1.
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