Control of nonlinear chained systems: from the Routh-Hurwitz stability criterion to time-varying exponential stabilizers

141 To generate identification data, we simulated this system using the feedback law u(t) = r(t) 0 (00:95q 02)y(t) = r(t) + 0:95y(t 0 2) (43) which places the closed-loop poles in 0.8618 and 0.6382. In the simulation we used independent, zero-mean, Gaussian white noise reference and noise signals fr(t)g and fe(t)g with variances 1 and 0.01, respectively. N = 200 data samples were used. In Table I, we have summarized the results of the identification, the numbers shown are the estimated parameter values together with their standard deviations. For comparison, we have, apart from the model structure (24), used a standard output error model model structure and a second-order ARMAX model structure. As can be seen, the standard output error model structure gives completely useless estimates, and the modified output error and the ARMAX model structures give similar and accurate results. V. AN ALTERNATIVE BOX–JENKINS MODEL STRUCTURE The trick to include a modified noise model in the output error model structure is of course also applicable to the Box–Jenkins model structure. The alternative form will in this case be y(t) = B(q) F (q) u(t) + F 3 a (q)C(q) Fa(q)D(q) e(t): (44) An explicit expression for the gradient filters for this predictor can be derived similarly as in the output error case, albeit the formulas will be even messier. We leave the details to the reader. In this paper, we have proposed new versions of the well-known output error and Box–Jenkins model structures that can also be used for identification of unstable systems. The new model structures are equivalent to the standard ones, as far as number of parameters and asymptotical results are concerned, but guarantee stability of the pre-dictors. An indirect method for transfer function estimation from closed loop data, " Automatica, vol. Identification of normalized coprime factors from closed-loop experimental data, " Eur. Abstract—We show how any linear feedback that stabilizes the origin of a linear chain of integrators induces a simple, continuous time-varying feedback that exponentially stabilizes the origin of a nonlinear chained-form system. The design method is related to a method developed by M'Closkey and Murray to transform smooth feedback yielding slow polynomial convergence into continuous homogeneous ones that give exponential convergence .