Brief PaperRobust exact differentiation via sliding mode technique*
Introduction
Differentiation of signals given in real time is an old and well-known problem. Construction of a special differentiator may often be avoided. For example, if the signal satisfies a certain differential equation or is an output of some known dynamic system, the derivative of the given signal may be calculated as a derivative with respect to some known dynamic system. Thus, the problem is reduced to the well-known observation and filtration problems. In other cases construction of a differentiator is inevitable. However, the ideal differentiator could not be realized. Indeed, together with the basic signal it would also have to differentiate any small high-frequency noise which always exists and may have a large derivative.
The main approach to linear-differentiator construction is to approximate the transfer function of the ideal differentiator on a definite signal frequency band (Pei and Shyu, 1989; Kumar and Roy, 1988; Rabiner and Steiglitz, 1970). The frequency band of the noise being known, low-pass filters are used to damp noises. Stochastic features of the signal and the noise may also be considered (Carlsson et al., 1991). In the latter case, the stochastic models of both the signal and the noise are presumed to be known. Linear observers (Luenberger, 1971) may be used. In any case a linear differentiator with constant coefficients may provide for asymptotically exact differentiation for a rather thin class of inputs and does not calculate exact derivatives of other noise-free signals.
If nothing is known on the structure of the signal except some differential inequalities, then sliding modes (Utkin, 1992) are used. In the absence of noise the exact information on the signal derivative may be obtained by averaging high-frequency switching signals. Also, sliding observers (Slotine et al., 1987) or observers with large gains (Nicosia et al., 1991) are success fully employed. However, in all these cases the exact differentiation is provided only when some differentiator parameters tend to inadmissible values (like infinity). Thus, here too the resulting differentiator cannot calculate exact derivatives of noise-free signals.
The performance of the known differentiators follows the following principle: only approximate differentiation is provided in the absence of noise, at the same time the differentiator is insensitive to any high-frequency signal components considered to be noises. Thus, differentiation is robust but not exact, the error does not tend to zero in the presence of vanishing noise at any fixed time, and no asymptotic error analysis is sensible for any fixed differentiator parameters and time.
Another principle employed here combines exact differentiation (with finite transient time) for a large class of inputs with robustness in respect to any small noises. A known approach (Golembo et al., 1976) is chosen: high-quality tracking of f(t) by , having been provided, control u(t) may be used for evaluation of . The new result is attained here due to application of a two-sliding algorithm (Levantovsky, 1985; Emelyanov et al., 1986; Levant (Levantovsky), 1993; Fridman and Levant, 1996) which forms continuous control u(t) providing for keeping the equalities after a finite-time transient process. The purpose of this paper is
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• to clear some inherent restrictions on exact robust differentiation and its error asymptotics;
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• to propose a robust first-order differentiator that is exact on signals with a given upper bound for Lipschitz’s constant of the derivative;
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• to ensure the best-possible error asymptotics order wh en the input noise is a measurable (Lebesgue) bounded function of time.
Section snippets
Robust exact differentiation limitations
Let input signals belong to the space of measurable functions bounded on a segment and let . Define abstract differentiator as a map associating an output signal with any input signal. A differentiator is called exact on some input if the output coincides with its derivative. The differentiator order is the order of the derivative which it produces. Differentiator D is called robust on some input f(t) if the output tends uniformly to Df(t) while the input signal tends
Practical first-order robust exact differentiator
The above abstract differentiators were not intended for realization. Consider now a practical real-time differentiation problem. Let input signal f(t) be a measurable locally bounded function defined on [0,∞) and let it consist of a base signal having a derivative with Lipschitz’s constant C>0 and a noise. In order to differentiate the unknown base signal, consider the auxiliary equationApplying a modified two-sliding algorithm (Levant ((Levantovsky), 1993) to keep x−f(t)=0, obtain
Computer simulation
It was taken that t0=0, initial values of the internal variable x(0) and the measured input signal f(0) coincide, initial value of the output signal u(0) is zero. The simulation was carried out by the Euler method with measurement and integration steps equaling 10-4.
Compare the proposed differentiator Eq. (3), Eq. (4)with a simple linear differentiator described by the transfer function p/(0.1p+1)2. Such a differentiator is actually a combination of the ideal differentiator and a low-pass
Conclusions
Inherent restrictions on exact robust differentiation and its error asymptotics were found. The existence of an arbitrary-order robust differentiator with the optimal order of error asymptotics was established.
A first-order robust exact differentiator was proposed providing for maximal derivative error to be proportional to the square root of the input noise magnitude after a finite-time transient process. This asymptotics order was shown to be the best attain able in the case when the only
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This paper was recommended for publication in final form by Associate Editor Hassan Khalil under the direction of Editor Tamer Basar.