TEOREMA DE LYAPUNOV- DEMOSTRACIÓN. BÚSQUEDA DE FUNCIONES DE LYAPUNOV. BÚSQUEDA DE FUNCIONES DE LYAPUNOV. BÚSQUEDA. This MATLAB function solves the special and general forms of the Lyapunov equation. funciones de Lyapunov; analisis númerico. 1 Introduction. The synchronization of electrical activity in the brain occurs as the result of interaction among sets of.

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In the case where the uncertainties come from parametric perturbations, we have a family of closed-loop characteristic polynomials instead of a single nominal characteristic polynomial. The necessary and sufficient condition for the control input to make stable is that all the roots of the polynomial: The self-tuning control based on GMVC algorithm is given by the following recursive estimation equations: By using this site, you agree to the Terms of Use and Privacy Policy.

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This is dr translation Translated by. You must use empty square brackets [] for this function. In the theory of ordinary differential equations ODEsLyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. The uncertainty in system characteristics leads to a certain family of models rather than to a single system model to be considered.

The controller design with the GMVC based on the sliding mode control concept, in the case of time-invariant systems [3, 6, 7], is reviewed in this section. Using the control law in Eq. For robust stability analysis, we may use the method by Tsypkin [10] for closed-loop discrete-time systems, which involves the modified characteristic locus criterion.

For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. The validity of the proposed algorithm was also demonstrated through simulation results. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known.

Recursive estimates of controller parameters based on generalized minimum variance criterion with a forgetting factor: Then, by using of the fucniones equations 13 and 14 for a positive boundedis proved negative semi-definite, i. Equation 25 yields Eq. Initiallythenlyapknov gives. Generalized minimum variance control.

The principal contribution of the obtained stability results is to assure the overall stability if the presented control algorithm is implemented on a real system with time-varying parameters, even in the presence of system and measurement noises. From the viewpoint of sliding mode control SMCPatete [6, 7] gave a complete proof for the stability of implicit self-tuning controllers based on GMVC for minimum or non-minimum phase systems by the use of a Lyapunov function.

The time difference of Eq. The automated translation of this page is provided by a general purpose third party translator tool.

Using the definition of given in Eq. MathWorks does not warrant, and disclaims all liability pyapunov, the accuracy, suitability, or fitness for purpose of the translation. Whereandare the upper and lower bounds of andrespectively. Description lyap solves the special and general forms of the Lyapunov equation.

Definingwhich maps the stable zone inside the unit circle lyapuniv the outside in the z-plane, then is defined as: Instead, each element of approaches to constant values in the sense of expectation with respect to. Retrieved from ” https: Equation 31 implies that approaches to zero as N goes to infinity; then the left-hand side of Eq.

The main results are the theorems which assure the overall stability of the closed-loop system, which are proved in a straight way compared with previous stability analysis results. Based on key technical lemmas, the global convergence of implicit self-tuning controllers was studied for discrete-time minimum phase linear systems in a seminal paper by Goodwin [4] and for explicit self-tuning controllers in the case of non-minimum phase systems by Goodwin [5].

Trial Software Product Updates. The robust stability analysis of the closed-loop system in presence of parametric interval uncertainties is shown in Figure 1.

For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical funfiones.

### Continuous Lyapunov equation solution – MATLAB lyap

Finally, by taking the expectation with respect to in 28Eq. The proof follows as the given proof in Theorem 1, using equations 33 – 36combined with the proof given in Patete [11]-Theorem 2 for auto regressive time-invariant systems. For robust stability of closed-loop discrete-time parametric systems, it is sufficient that.

Choose a web site to get translated content where available and see local events and offers. This page was last edited on 6 Octoberat The purpose of this paper is to analyze the stability of the implicit self-tuning controller for discrete time-varying systems TVS and discrete time-varying systems subject to system and measurement noises. Self-tuning control of time-varying systems based on GMVC.

### Lyapunov function – Wikipedia

In order to derive the nominal control law the polynomials and are assumed to have constant and known parameters, represented by: Select the China site in Chinese or English for best site performance.

The analysis is extended to the case where the system model is subject to system and measurement noises. In the first, for the simulation example, the real system model is assumed to be represented by: Click here to see To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

If Q is a symmetric matrix, the solution X funcionse also a symmetric matrix. The A matrix is stable, and the Q matrix is positive definite. Thus, and vanish as N approaches to infinity.

The nominal system model and the family of system models to be considered in this section are represented as: The degree of polynomial iswhich implies that depends only on future states of x. Self-tuning control, generalized minimum variance control, sliding-mode control, discrete-time systems, time-varying systems, Lyapunov function.

The parameters of the control law for the real systems with unknown parameters are estimated using a recursive least-squares RLS algorithm. The representation of the nominal system with lyapunoc and output is given by: The criterion considered is the minimization of an auxiliary controlled variable based on the concept of sliding mode control to yield the system stability.