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This paper presents advanced signal processing methods and command synthesis for memory-limited complex systems. For accurate measurements performed on limited time interval, some specific methods should be added. For signal processing, a robust filtering and sampling procedure performed on a specific working interval is required, so as the influence of low-amplitude and high-frequency fluctuations to be diminished. This study shows that such a signal processing method for the case of memory-limited complex systems requires the use of certain differentiation/integration procedures performed by oscillating systems, so as robust results suitable for efficient command synthesis to be available. A brief comparison with uncertainty aspects in modern physics (where quantum aspects can be considered as features of complex systems) is also presented.

As it is known, an important aspect in observing and modeling dynamic environmental phenomena consists in measuring with higher accuracy some physical quantities corresponding to changes in the environment. Yet for accurate measurements performed on limited time interval, for memory-limited complex systems, some specific methods should be used. Sudden (sharp) changes in the environment require a pair of consecutive values for the measured quantity so as any difference to be detected as soon as possible. Moreover, any value taken into consideration by the complex system should be established using a robust filtering and sampling procedure performed on a specific working interval, so as the influence of low-amplitude and high-frequency fluctuations to be decreased in a significant manner. Being quite possible for sharp (sudden) changes in the environment to appear during such a working interval (on which filtering and sampling procedures are performed), it results that specific signal processing methods based on the values achieved on a set of succesive time intervals are necessary.

Filtering and sampling devices usually consist of asymptotically stable systems, sometimes an integration of the output over a certain time interval being added. Yet such structures are very sensitive at random variations of the integration period, being recommended for the signal which is integrated to be approximately equal to zero at the end of the integration period. For this reason, oscillating systems for filtering the received signal should be used, so as the filtered signal and its slope to be approximately zero at the end of a certain time interval (at the end of an oscillation). For avoiding instability of such oscillating systems on extended time intervals, certain electronic devices (gates) controlled by computer commands should be added, so as to restore the initial null conditions for the oscillating system before a new working cycle to start [

The filtering performances of asymptotically stable systems are determined by their transfer function. a Filtering and sampling devices consisting of low-pass filters of first or second order having the transfer function

But such structures are very sensitive at the random variations of the integration period (for unity-step input, the signal which is integrated is equal to unity at the sampling moment of time), and the use of oscillators with a very high accuracy cannot solve the problem due to switching phenomena appearing at the end of the integration period (when an electric current charging a capacitor is interrupted).

These random variations cannot be avoided if we use asymptotically stable filters. For robustness, the signal processing structure based on an integration procedure should provide a null value for the integrating signal at the end of a certain working interval. This property is similar to wavelets aspects presented in [

Mathematically, an ideal solution could consist in using an extended Dirac function for multiplying the received signal before the integration (see [

A heuristic algorithm for generating practical test functions using MATLAB procedures was presented in [

It was shown that the simplest structure possessing such properties is represented by an oscillating second-order system having the transfer function

This oscillating system attenuates about

These results have shown that such a structure provides practically the same performances as a structure consisting of an asymptotically stable second-order system and an integrator (response time of about

In [

Thus, the output

However, we must notice that, usually, such a filtering and sampling structure receives an electronic signal presenting possible step changes from an already measured value to a final unknown value. Since the previously measured value can be substracted from the received signal during subsequent working intervals, the analysis of sudden (sharp) changes in the environment could start by considering that the input of the second-order oscillating system is represented by a null signal for

We will continue the analysis of this structure by considering that the input is represented by a short-step pulse which differs to zero on the time interval

The transfer function of the second-order oscillating system is

By denoting with

It can be easily noticed that

This result shows that the sampled values for

For this purpose, we can divide the sampled value for

On the subsequent working interval, we can consider that the input of the second-order oscillating system equals

However, this result is far of being useful for practical applications. Since the differential equation of the second-order oscillating system is

The previously presented algorithm could be accepted if the integral can be performed on an extended time interval. In this case, we can simply estimate the quantity

In case of electric drives, an extended integration interval for quantity

For this purpose, we could notice that quantity

Since

While

By adjusting the relation between

This intuitive model is also valid for any complex (biological) system which should maintain its position or the velocity of certain components at a specific value.

The signal processing method presented in previous paragraph is based on sampled values for three successive working intervals of the second-order oscillating system. Considering that a step change for the input is detected on a certain working interval, the previous value for the input is determined on an initial working interval (on which the input equals a certain value

However, the linearity of this second-order oscillating system allows a more efficient and robust procedure to be used. The algorithm presented in the previous paragraph requires two identical oscillating second-order systems working at the same time: one for processing the input (so as to determine the estimated values for an input considered to be constant on that interval) and another for processing the difference between the received signal and the previously sampled value (so as to detect a possible step change during this interval by determining the quantity

In previous paragraph, it was shown that filtering properties and robustness require an extended time interval for processing the input signal (received from transducers). For this purpose, we can use either a second-order oscillating system with a period equal to the working interval (which means

This suggests the possibility of sampling

Thus, the successive values

A quick analysis for first pair of values for

The phase difference between

It can be easily noticed that the ratio

A quick comparison of

In the third section of this paper has been presented an algorithm for a preliminary estimation of a step change in the environment based on two state variables (two successive values for a

It can be noticed that measurements performed for sinusoidal functions at time moments when the phase difference equals

It results that sampled values correspond to the pair

This pair can be considered also as a value for a sine function and a value for its derivative for the same phase

This aspect is similar to the uncertainty principle in physics, when measurements corresponding to a certain physical variable and to its conjugated variable (usually corresponding to a derivative of a function in respect to the previous variable) are involved. Moreover, it should be noticed that for a cosine function

Quite similar, the action of the operator

It can be noticed that the cosine function

As a conclusion for memory-limited complex systems, we can notice that the use of second-order oscillating systems allows ot just robust sampling procedures on extended time intervals for certain quantities corresponding to step changes in the environment, but also the use of just two state variables (corresponding to sampled values for sine and cosine function at certain moments of time) for a preliminary estimation of such step changes during the working interval. For an extended working interval which includes several alternances of the oscillations generated by the received signal, this implies the possibility of transmitting quick preliminary commands towards the actuators, a final adjustment being determined at the end of the whole working interval, based on the difference between required action and the already-performed action. Similar to aspects presented in previous section, these preliminary commands

For this reason, the algorithm previously presented is suitable for memory-limited complex systems since it performs both a preliminary analysis of signal received from environment for detecting step changes (with preliminary commands transmitted towards actuators) and a final accurate estimation for the required action on next extended time intervals (computed as a difference between the whole action required by the step change and the action already performed by preliminary commands).

As in case of biological systems, this algorithm is based on values sampled at some successive moments of time. It generates a sequence of certain commands towards the environment as a sequence of pulses, analyzes the difference between expected values and real values for the signal received from the environment, and adjusts the command with higher accuracy after an extended time interval (a kind of multilevel control and command). Another important similarity between this algorithm and behaviour of biological systems should be noticed; the sampling moments of time (when the processed signal is recorded) differ to the time moments when the filtered (processed) signal has a great slope (considered as modulus) so as to allow a robust estimation using just two sampled (recorded) values. The fact that less memory is involved is essential for complex systems which have to survey a great number of parameters (motion parameters, for instance) in the environment and to check the effect of commands transmitted towards a great number of actuators, see the case of vision processing studied in [

This study has presented advanced signal processing methods and command synthesis for memory-limited complex systems. It was shown that for observing, modeling, and controlling dynamic environmental phenomena in case of memory-limited complex systems, some specific methods based on accurate measurements performed on limited time intervals are required. Starting from the necessity of a set of consecutive measurements performed in a robust manner for detecting step changes in the environment, it was shown that an extended time interval for processing the input signal is necessary. For this reason, the use of second-order oscillating systems was improved by adding a supplementary algorithm so as preliminary values for step changes in the environment to be available for control and command during the signal processing interval. A method for generating a robust command towards the control equipment on a limited time interval in order to compensate the step changes detected on previous working interval was also presented. Finally, similarities between measurements of a certain quantity and of its derivative for a sine function, by one side, and the uncertainty principle in physics (by the other side) were briefly mentioned.