On stability of switched homogeneous nonlinear systems

Abstract

In this paper, the problem of stability of switched homogeneous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogeneous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogeneous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogeneous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planar switched homogeneous systems, an LMI approach to stability of planar switched homogeneous systems is presented. Similar to the condition for linear systems, the LMI-type condition is easily verifiable. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.

Sum of squares decomposition: theory and applications in control

“…Ejemplo numérico: sistema conmutado. Se considera el sistema conmutado siguiente: ˙ x = fi(x), x = [x1 x2]T, con Este sistema está considerado como conmutado homogéneo presentado para análisis de estabilidad (Zhang et al, 2007). Con el propósito de probar estabilidad bajo conmutaciones arbitrarias, se usa la representación equivalente polinomio Usando estas matrices poligonales se obtiene, Utilizando SOSTOOLS se obtiene una función de Lyapunov de grado sexto, i. e.: la cual, a través del teorema 3.5, prueba que el origen del sistema conmutado homogéneo no lineal reformulado como una DAE polinomio es estable.…”

On Finite-Time Stability of Switched Systems with Hybrid Homogeneous Degrees

“…It is worth noting that homogeneous theory [13,14] can give simplified conditions for stability analysis of switched nonlinear systems, where the value of is obtained accurately. In [15], stability problem of switched homogeneous systems is addressed using semitensor product of matrices and LMI conditions are achieved. In [16], homogeneous Lyapunov function is constructed and stability analysis via both CLF and MLFs is given.…”

L2 stability, H∞ control of switched homogeneous nonlinear systems and their semi‐tensor product of matrices representation

“…Many engineering systems, such as robot manipulators, traffic management, and power systems, are essentially switched systems. Studies on switched systems mainly focus on stability, stabilization, controllability, optimal control, H 1 control, and so on (see [1][2][3][4][5][6][7][8][9][10][11]). Various methods, such as common Lyapunov function, LMI approach, and nonlinear programming, have been widely used to handle the corresponding problems of switched systems (see [2][3][4]).…”

“…It is a useful tool to represent high dimensional arrays, multilinear mappings and multivariable polynomials. Nowadays, STP has been applied to many fields, such as nonlinear system control and Boolean networks (see [1,[18][19][20], and references therein).…”

“…Homogeneous nonlinear systems keep some good dynamics such as linear systems because of their homogeneous properties, so they are regarded as a kind of nonlinear systems that can be deeply investigated like linear systems. So far, many researchers pay their attention to studies of homogeneous systems, for example, , and so on. L 2 stability problem and H ∞ control problem are generally solved via finding a smooth solution of Hamilton–Jacobi–Isaacs (HJI) inequalities, see .…”