Brief history on reduction methods

The majority ofreduction techniques in the Iiterature  are based mainly on the knowledge of the transfer function of the system that should be reduced. These reduced methods involve a lot of mathematical equations manipulations to find an identical reduced system. In most cases, this reduced system is a second order system. Once the transfer function is reduced, a controller is developed based on the reduced function, and then applied to the real system.

Integral square error method

Minimization of integral square error (ISE) [1], [13] is based on minimizing the squared error area between a known real system transfer function output and unknown reduced second order transfer function output. This technique involves a lot of mathematical equations and computer analysis to find the unknown coefficients of the second order system that makes the squared error area as close as possible to zero. The reduced system is then achieved.

Pade approximation method

Pade approximation and its improvements [2], [3], [8], [10] is another reduction method based on Taylor-McLaurin series. The real system transfer function should be known prior to using this technique to find a reduced second order transfer function identical to the real system. This method also uses a lot of mathematical equations using McLaurin series to find the numerator and the denominator coefficients of the reduced system.

Other reduction methods

Dominant pole retention method [1], pole clustering [3], Big Bang big Crunch reduction method [11] and many other methods [5], [14-18] have the same objective, finding an identical reduced system with known original transfer function but with different mathematical approaches. The objective of ail these reduction methods is not to find a reduced identical system only, but to design a controller in closed loop that controls the real system perfectly in the same way as the approximated mode!.

Research goals 

Main objective 

he main objective of this research is to design a controller for a high order complex system with no need of its transfer function and to avoid heavy differential equations to find an identical approximated system. This new proposed reduction method is based on a similar system concept. This similarity is achieved by using weighted elements that characterize the real system. Those elements are deterrnined from the output response only. The complex system transfer function is not needed to find a reduced similar system.

Contribution ofthis thesis 

The main contribution of this thesis is to develop a new approach based on a new concept called « similarity ». This new method uses only the output response of the real system. Output response data can be obtained either by experimentation or by simulation. Based on this notion of similarity, the real system output is reduced to a first order system output that is similar in its weighted elements. These elements must be clearly and carefully defined. The controller design is based on this reduced similar first order system and will be applied to the black box or higher order system. Simulation is conducted to highlight the performance of this proposed new method compared to existing classical reduction methods.

The second contribution of this research is to introduce new irreducible high order transfer functions systems that c1assical reduction methods cited above are unable to reduce to a lower order especially second order systems. This will highlight the limits of the c1assical reduction methods. In the meantime, the application of the new approach based on first order original or similar system should be still valid to design a controller for such high order irreducible systems.

The last contribution of this research is to find sorne applications to these theoretical irreducible systems. New electrical circuit as an application of Fibonacci wave functions (FWFs) called Fibonacci electrical circuits (FECs), are introduced for the first time to model perfectly the recursive LC ladder network. These FECs can be used to model transmission cables [21], [22], [25], the neural dynamic in biology [24] and the behavior and interaction of the infinitely small partic1es using the infinite LC networks [23] in quantum mechanics. These Fibonacci systems have irreducible transfer function and cannot be reduced to an equivalent second order system.

Spring mass chain as another application of these irreducible Fibonacci wave functions is also introduced in this research. This Fibonacci spring mass chain (FSMC) is also used in many applications to model the behavior and interaction of particles from mechanical view, especially in fluid mechanics and quantum mechanics.