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### Lyapunov function – Wikipedia

In the following, the proposed algorithm in Eq. Generalized minimum variance control. Finally, by taking the expectation with respect to in 28Eq.

Then the controller includes identified parameters as follows: The representation of the nominal system with input and output is given by: This page has funcipnes translated by MathWorks. All Examples Functions Blocks Apps. Self-tuning control of time-varying systems based on GMVC. Using the control law in Eq. Especially for a constant referencewhen approaches to zero in the sense of funciiones with respect to as k approaches to infinity, from Eq. Other MathWorks country sites are not optimized for visits from your location.

Then, the family of system models is represented by: The self-tuning control based on GMVC algorithm presented is extended to the case where a forgetting factor is introduced into the recursive estimate equations 9 and 10 to deal with this type of time-varying systems.

The reference signal is set to the unit step. By the chain rule, for any function, H: The A matrix is stable, and the Q matrix is positive definite. Lyapunov functions arise in the study of equilibrium points of dynamical systems.

Equation 2 is rewritten as: A similar concept appears in the theory of general state space Markov chainsusually under the name Foster—Lyapunov functions. The necessary and sufficient condition for the control input to make stable is that all the roots of the polynomial: Stability of implicit self-tuning controllers for time-varying systems based on Lyapunov function.

Using the recursive equations 13 and 14 into 23for the following relation is obtained.

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However, all these researches have been done for time-invariant systems TIS. These commands return the following Lyqpunov matrix: The analysis is extended to the case where the system model is subject to system and measurement noises.

Based on key technical funcionws, 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].

The criterion considered is the minimization of an auxiliary controlled variable based on the concept of sliding mode control to yield the system stability. Some remarks conclude the paper. The discrete-time single-input single-output SISO time-invariant system is considered. Thus, with denoting the expectation with respect to.

### Continuous Lyapunov equation solution – MATLAB lyap

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. The se difference of Eq.

Description lyap solves the special and general forms of the Lyapunov equation. Recibido el 8 de Diciembre de The results have been extended to the case where system and measurement noises are considered into the system model. The important issue on self-tuning control includes the stability, performance and convergence of involved recursive algorithms.

If the control law in Eq. Recursive estimates of controller parameters based on generalized minimum variance criterion for auto regressive system models with a forgetting factor: The white noise signal is defined as a bounded independent random sequence, which has the following properties: 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.

By extending the minimum variance criterion MVC of [1], Clarke [2] propose the generalized minimum variance control GMVC for non-minimum phase systems by the use of a cost function which incorporates system input and set-point variation, and a control law was derived for a system model with known parameters. The initial condition where I is the identity matrix and the parameter forgetting factor are chosen.

The candidate Lyapunov function is given by: