REDUCED ORDER MODELLING AND ITS APPLICATION IN AN INDUSTRIAL PROCESS

Abstract

Reduced order modelling of linear time invariant systems had been a topic of interest to a large number of research workers all over the world. It is interesting to note that different methodologies developed for order reduction can be put broadly under two groups, like, time and frequency domain, being obviously the two accepted methods of system modelling. The reasons behind interest in reduced order modelling are many, but can be precisely put as (i) To have less computation time for analysis and design of system and (ii) To have economy in hardware requirements for on line simulation of a system. i Present attempt is towards development of some new methods in frequency domain using transfer function model with functional criterion defined 'a priori1. Methods developed are in two groups, according to functional criterion chosen as unit step response and frequency response. Four methods have been developed using step response matching where as two methods are proposed using frequency response matching. These have been tried both on single input single output and multivariate continuous systems. One of the methods has been extended to Discrete Systems also. Examples chosen to demons trate the methods are of wide variety and comparisons with other methods as well as between the methods developed have been done. There are some common philosophies among the methods using step response matching and these can be put as : s (i) Retention of dominant poles of the original system in the reduced order system. (ii) Exact forced matching of the steady state parts of the unit step responses of the original and reduced order systems. (iii) Minimization of the error between the transient parts of the unit step responses of the original and reduced order systems. Only in one of the methods, not all of the above have been taken care of, but instead, poles are synthesized using some other criteria. In three of the methods in this category difference is due to the method of. minimising the transient error and the number of zeroes are also different in each case. The first method consists of minimising the transient error with respect to the difference of residues of the original and the reduced order system and the number of zeroes are equal to the number of pole.s. In the second method, the transient error is minimised with respect to the coefficients of the-'numerator polynomial of the reduced order transfer function and the number of zeroes in the reduced order system is one less than the number of poles. The third method includes the idea of having only one zero for all the order of the reduced system and consequently the ' transient error is minimised with respect to only one coefficient of the numerator polynomial. In the fourth method instead of retaining the dominant poles these are synthesized using some conditions defined 'a priori' like equality of system stiffness and mean eigenvalue of the original and reduced order systems. Zeroes are determined here • also by exact forced matching of steady state parts and minimization • of the transient error as done in earlier methods. * v» In one of the methods using frequency response matching, dominant poles are retained according to the order to be reduced to and the zeroes are obtained by exact matching of the steady state parts of the frequency response where the transient error is minimum. The other method consists of retention of dominant poles but the zeroes are synthesized by minimising the transient error at a frequency where the steady state error is minimum. One of the methods using step response matching has been extended to- discrete systems. The system is presumed to be linear and time invariant. Sampling rate is taken as uniform. The method consists of dominant pole retention followed by zero synthesis by exact forced matching of steady state response and minimization of the transient error with respect to the-difference of the residues of the original and the reduced order systems. Another method is propos ed for discrete systems using impulse response error minimization. A design procedure for system compensators (lead, lag, lag-lead and PID) is proposed using low order models. The desired closed loopsystem performance is decided using a second order model including both the plant and the compensator. The original system is reduced to second order using its closed loop unity feedback transfer func tion. Resulting transfer function including the compensator (lead, lag and PID) is of third order having unknown terms as the elements of the compensator which are determined by mathcing the steady state parts of the unit step responses of the second order model transfer > function and this third order transfer function followed by transient error minimization of the same. In case of lag-lead type, the resulting transfer function is of fourth order but the design procedure is exactly the same. Two examples have been successfully tried here. The method developed for discrete systems has been tried on the model of paper making system. A model of the system is identified using input/output data and the order of the same is ascertained. Finally the work ends with design of a controller for a multivariable industrial process. The system is first reduced to a suitable low order using techniques developed and then controller of PID type is designed after decoupling the coupled modes. This gives an oppor tunity to apply both low order modelling techniques and the controller, design method on a multivariable process in existance. The order reduction methods developed are extremely simple computationally and will always give stable models from a stable system. Steady state error is zero in all the methods except one using frequency response matching. Compensators successfully designed using low order models improves the system performance as desired. Modelling and order reduction of an industrial process further justifies the techniques developed. Handling of a multivariable industrial process using its reduced order model gives encouraging results. Apart from trying some new ideas in order reduction the work opens up a number of avenues for further research in this area.