Multirate Sampling and Zero Dynamics: from linear to nonlinear
It has recently been shown that the concept of zero dynamics plays a central role in the design of some nonlinear control systems. As in the linear context, where the property of stable zeros is necessary in the use of direct design techniques, in the solution of nonlinear control problems such as input-output linearization, tracking or input-output decoupling, the stability of part or of the whole zero dynamics constitutes a basic requirement.
When solving the above-mentioned nonlinear control problems by means of a digital scheme, where the design of the control law is based on the sampled model of the plant, some extra problems appear since the zero dynamics stability is not preserved under sampling. In fact, for small sampling intervals, the zero dynamics of the sampled model is always unstable if the relative degree of the plant is greater than one.
The purpose of this paper is to show how this drawback can be avoided by using a discretization technique on a time scale on the output which is a multiple of the time scale on the control (multirate sampling), the order being equal to the relative degree of the continuous given single input-single output model.
Multirate discretization techniques are known in the literature; in the linear case this technique at order n allows the arbitrary positioning of zeros of the sampled transfer function. A different point of view is taken here where the proposed multirate sampled model results in a square system of dimension equal to the multirate order. The paper studies the properties of the zero dynamics of the multirate sampled model of a given nonlinear plant in the SISO and MIMO cases. It is shown that the multirate control strategies based on such a sampling technique allows us to obtain results which maintain and also improve the performances of the continuous control scheme.
KeywordsUnder Sampling Minimum Phase Zero Dynamic Continuous Plant Digital Feedback
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