L4: 2 – Motivation

The incitement for Lyapunov’s direct technique is the observation that if vigor is forever destroyed in an electric or mechanical structure, then the system will eventually settle down. In other oaths, the system state will penetrate an symmetry degree. Think for instance of the lump going down the two sides of a depression mound it will govern up the other hillside but not as high up as it started due to friction. Due to the friction, intensity is constantly spent and the clod will eventually settle down at the bottom of the valley. This is an equilibrium point of information systems. So if vigor is constantly exhausted the system will eventually enter an equilibrium part. Based on this finding the idea behind a Lyapunov’s direct procedure is that it may be possible to analyze the stability assets of equilibrium qualities by studying the power of information systems. Consider a two dimensional organization, since this is easy to visualize.Here is the phase plane. So, consider a two dimensional structure which has an equilibrium point at the parentage. The vigor of the system is a scalar perform of the state, and we will denote this by V. The energy of the system at the equilibrium time is zero. Now, cause us reap a line to all the points in the commonwealth gap where the vitality has the same constant value. See that this constant value is c1, which is some positive value. We then get a curve in the plane which may look like this. Let us then select another position through all the points in the territory opening where the vigour has another constant value c2, which is larger than c1. These curves are called level curves, or grade skin-deeps. In the general case, where the system is not necessarily two-dimensional, these curves represent constant energy ranks for the system. For all states on this arch here the force is constant and equal to c1. So how can we use this vigor function to analyze the stability for of the equilibrium target at the root? Remember that we consider time invariant systems in this form now, for which the origin is an equilibrium point.What we do is that we study the time evolution of the exertion of the system. Specifically, we study how the power advances along information systems paths, for instance if the system trajectory moves towards level curves representing highest and higher power status, like here, then we see that this corresponds to moving further and further away from the stability detail at the start, which should suggest that the inception is unstable.How time we compute the time evolution of the force along the system paths? We simply take the time derivative of the force part V. By the chain govern the time derivative of V of x, is the derivative of V with respect to x, which is the gradient of V, occasions the derivative of x with respect to time t, which is given by the system equation now as f of x. This is the directional derivative of the function V along the vector study f. It describes how V changes in the direction of the vector f. That is, how the best interests of the the exertion perform V modifications along the trajectories of information systems. If this is positive then the exertion increases along x, as we see here. If it is negative, on the other hand, then the exertion declines along information systems trajectories. This means that the system trajectories then intersect curves representing lower and lower intensity stages until the exertion becomes zero, which is at the origin.This we see indicate that the origin is an asymptotically stable symmetry detail, and note that we do not need to solve this differential equation to find x as a function of occasion t, in order to see how the intensity increases along x of t. The directional derivative of the intensity perform imparts us the information collected. What then, if the intensity is not definitely abridge, but it can also be consonant. So it’s time derivative are also welcome to 0. When the time derivative is 0, the force is constant, means that the system trajectory moves along one of the level curves. So when the time derivative of the energy is either negative or 0, then we do not inevitably have convergence but the behavior is similar to that of a stable symmetry top. These intuitive observations that we have performed here, hold for mechanical and electrical organizations, which have a well-defined energy concept. But how about general dynamical arrangements? Lyaponov formalized and generalized these instinctive sees in his doctoral thesis: The general problem of the stability of action, which was published in 1892. The theorems presented here which constitute Lyaponov’s direct procedure regulate for general systems , not only electrical or mechanical arrangements, for which we have a well-defined energy concept.Instead of the power serve we must therefore use an energy-like function, and the question is: Which qualities must such a function have to serve as a generalized power part, such that these instinctive statements harbour, such that we can use it to analyze stability ?.

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