Order Reduction of z-Domain Interval Systems by Advanced Routh Approximation Method

Abstract

Since decades mathematicians have been designing the transfer function for the available physical models followed by the involvement of control engineers to work on it. Through the study of the offered representations, many systems were found to be of higher order which are nevertheless not easy to study and analyze in their core form. Furthermore, again uncertainties within the system was found that cannot be ignored. All these increases the complexities for analysis of the physical systems. This demands a technique for order reduction to derive an approximate lower order representation of the higher order systems. In continuation, this paper is an attempt to propose a computationally efficient approach for obtaining the reduced interval model based on Routh Approximation technique. The proposed approach is a novel method for discrete-time interval system and is discussed in detail in the article content ahead. The provided examples offer the desired explanation for the effectiveness of the proposed algorithm.

Appendix

Intervals $u = \left[ {u^{ - } ,u^{ + } } \right] = \left\{ {u \in :u^{ - } \le u \le u^{ + } } \right\}$

$$ v = \left[ {v^{ - } ,v^{ + } } \right] = \left\{ {v \in :v^{ - } \le v \le v^{ + } } \right\} $$

Lower Limits of interval systems $u^{ - } ,v^{ - }$.

Upper Limits of interval systems $u^{ + } ,v^{ + }$.

Arithmetic operations where $\odot \in \left\{ { + , - , \times , \div } \right\}$ $\left[ u \right] \odot \left[ v \right] = \left\{ {u \odot {v \mathord{\left/ {\vphantom {v {u \in \left[ u \right],v \in \left[ v \right]}}} \right. \kern-0pt} {u \in \left[ u \right],v \in \left[ v \right]}}} \right\}$.

End point formulas for arithmetic operations

$$ u + v = \left[ {u^{ - } + v^{ - } ,u^{ + } + v^{ + } } \right] $$

$$ u - v = \left[ {u^{ - } - v^{ + } ,u^{ + } - v^{ - } } \right] $$

$$ u \times v = \left[ {\min D,\max D} \right],D = \left[ {u^{ - } v^{ - } ,u^{ - } v^{ + } ,u^{ + } v^{ - } ,u^{ + } v^{ + } } \right] $$

$$ {u \mathord{\left/ {\vphantom {u v}} \right. \kern-0pt} v} = u \times \left( {{1 \mathord{\left/ {\vphantom {1 v}} \right. \kern-0pt} v}} \right);{1 \mathord{\left/ {\vphantom {1 v}} \right. \kern-0pt} v} = \left[ {{1 \mathord{\left/ {\vphantom {1 {v^{ + } ,{1 \mathord{\left/ {\vphantom {1 {v^{ - } }}} \right. \kern-0pt} {v^{ - } }}}}} \right. \kern-0pt} {v^{ + } ,{1 \mathord{\left/ {\vphantom {1 {v^{ - } }}} \right. \kern-0pt} {v^{ - } }}}}} \right],0 \notin v $$