By Pedro Balaguer
Dimensional research is an engineering device that's largely utilized to varied engineering difficulties, yet has just recently been utilized to manage thought. program of Dimensional research in structures Modeling and keep watch over layout goals to unravel regulate difficulties similar to identity and version relief, strong keep an eye on, adaptive regulate and PID control.
This new publication introduces the basics of dimensional research to either regulate engineers and theorists with examples of functional applicability to business keep watch over difficulties. via adopting regulate concept study, the writer describes tips to take advantage of the advantages that dimensional research can provide to regulate theoretic and useful difficulties.
• dimensional research and dimensional similarity
• dynamical structures dimensionless representation
• dimensionless platforms id and version order reduction
• homogeneity of PID tuning rules
• dimensionless PID tuning ideas comparison
• dimensional research regulate fundamentals
• keep watch over of dimensionally comparable systems
• strong control
• adaptive regulate within the presence of enter saturation
• time scales keep watch over.
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Additional info for Application of Dimensional Analysis in Systems Modeling and Control Design
The scaling factor associated with any transfer function parameter with time dimension is equal to St := (τp1 )p /(τp1 )m , where (τp1 )p and (τp1 )m are the prototype and model dominant time constant, respectively. Proof. First consider the time delay scaling factor Sh . 2). Finally the dimensionless number involving frequency is given by τp1 s. 8) As a result the time scaling factor St is the same and equal to (τp1 )p /(τp1 )m for every transfer function parameter with time dimension. 11) that provides the relation between dimensionally similar transfer functions in terms of gain and time scaling factors.
An ) B1 , . . , Bm ) C1 , . . , Cp ) D1 , . . , Dq ) Dynamical systems: dimensional similarity 45 where there are n + m + p + q dimensionless numbers related to governing variables. In this case, there are as many scaling factors as governing variables, and they are problem dependent. 2. 22) with √ im k ¯l = l m i g¯ = g c¯ = c 1 ik We have one scaling factor per governing variable, that is a gravity acceleration scaling factor Sg , a length scaling factor Sl , a damping scaling factor Sd , a mass scaling factor Sm , an inertia scaling factor Si , and a stiffness scaling factor Sk .
Proof. 48) y(k) = −a1 y(k − 1) − · · · − an y(k − n) + k(b0 u(k − q) + · · · + bm u(k − m − q)) by dimensional homogeneity considerations , the dimensions of [ai y(k − i)] must be equal to [ y(k)], then [ai y(k − i)] = Y . It then follows that [ai ] = 1. In a similar way [kbj u(k − j − q)] = Y , being Y the output dimension. Note also that [u(k − j − q)] = U , being U the input dimension, then [kbj ] = YU −1 = K. As a result, a possible dimensional decomposition of the parameter kbj into a gain k and a parameter bj is [k] = K and [bj ] = 1.