Experimental feedback linearisation of a vibrating system with a non-smooth nonlinearity

Experimental feedback linearisation of a non-smooth nonlinear system by the method of receptances

“…The stability of the zero dynamics can alternatively be verified using the receptance method with a describing function (DF) approach [21], either using an analytical DF (as in the present case) or by carrying out a series of slow-sweep amplitude-controlled sine-excitation tests. An exposition of the theory of feedback linearisation by the receptance method was recently presented by Zhen et al [22], and the purpose of the present paper is to demonstrate experimentally their findings, using the approach presented in that paper to replicate the closed-loop results already obtained for the three degrees of freedom test-rig with classical feedback linearisation by Lisitano et al [20].…”

“…The application of classical input-output partial feedback linearisation was extended to systems with non-smooth nonlinearities by Jiffri et al [18]. Experimental application of the method to the three degrees of freedom system discussed in this paper was successfully achieved by the present authors [19, 20], but significant effort was required to tune the numerical model to an adequate level of accuracy, in both the linear and nonlinear parameters, to successfully achieve the desired closed-loop dynamics. A special case was considered by Lisitano et al [19] whereby control was applied to the same degree of freedom as the nonlinearity.…”

“…are stable for the range of all amplitudes Z of vibrations of the system. The measured receptance matrix of the linear system may be combined with the DF approximation of the system nonlinearities to compute the amplitude-dependent nonlinear receptance matrix, as in equations (20) and (13), from which the expression (26) may be found.…”

“…Therefore, B in equation (14) is a 1 × 3 vector, b = ( 1 0 0 ) T and the transformation matrix in equation (17) may be written as the identity matrix, T = I 3 × 3 so that the two systems of coordinates referred to earlier are identical, that is z = x . Parameters ε and k 2 are chosen to take values of 0.3954 and 1722.6 N/m, having been identified previously as reported by the present authors [20].…”

“…In such a case, the system is linearised completely and the problem of determining the zero dynamics is trivialised to a check that the system is minimum phase. Lisitano et al [20] considered the case when the nonlinearity was located at a different position, requiring a detailed and complex analysis to be carried out to establish the zero dynamics. The stability of the zero dynamics can alternatively be verified using the receptance method with a describing function (DF) approach [21], either using an analytical DF (as in the present case) or by carrying out a series of slow-sweep amplitude-controlled sine-excitation tests.…”