OUTLINE: Objectives Motivation Background on Real Sampling Identification Methods

1. Semi-heuristically Obtained DiscreteModels for LTI Systems under Real Sampling with Choice of the Hold DeviceA.J. Garrido, M. De la Sen and R. Bárcenaajgarrid@we.lc.ehu.esmsen@we.lc.ehu.esrbarcena@we.lc.ehu.esInstituto de Investigación y Desarrollo de Procesos (http://www.ehu.es/IIDP). Dpto. de Ingeniería de Sistemas y Automática. Facultad de Ciencias, Universidad del País Vasco, Leioa (Bizkaia), Apdo.644 de Bilbao, 48080. SPAIN.

2. OUTLINE: • Objectives • Motivation • Background on Real Sampling • Identification Methods • Simulated examples and Biestimation • Conclusions

3. Objectives • OBJECTIVES: • To present two different filter-based identification methods for the obtaining of discrete transfer functions of LTI systems from input-output data series, which may be used when the analytical solution becomes very complex or even unfeasible • To apply these methods for the obtaining of discrete-time models for LTI systems under real sampling, that is, using a finite sampling pulse duration rather than an instantaneous ideal one, in the context of a biestimation scheme

4. Motivation Problems: • High complexity or unfeasibility of the analytical solution Real sampling • Continuous transfer function unknown but available input/output data Solution: • Use of parameter estimation techniques to obtain a discrete-time model representing the system MOTIVATION:

5. Background on Real Sampling x[(k+1)T] x[kT] x[(k-1)T] T T x[(k+1)T] x[(k+)T] x[kT] T=p1·T’ T0=T BACKGROUND ON REAL SAMPLING: Given a continuous lineal system: Ideal sampling: Application ofinstantaneoussamplingsampling pulse: Real sampling: Application ofnon-instantaneoussampling of finite sampling pulse duration T0

6. Background on Real Sampling x[(k+1)T] x[(k+)T x[kT] x[(k-1+)T] X[(k-1)T] T0 T T BACKGROUND ON REAL SAMPLING: FROH used for real sampling: State evolution equation obtained (exact hybrid description containing continuous and discrete terms): where:

7. Background on Real Sampling x[(k+)T] =x[(k+p)T] x[(k+p-1)T] x[(k+1)T] x[kT]= x[kT] ..... p times ...... sT T T T0=(p+s)T BACKGROUND ON REAL SAMPLING: Applying a sampling of period T, so that: T=p1T T0= T =(p+s)T, being p=max zZ+ / pTT y s[0,1) and proceeding recursively using a ZOH, an approximate expression for x[(k+)T] is obtained: where:

8. Background on Real Sampling BACKGROUND ON REAL SAMPLING: Then, refering all the description to T and renaming kk, TT, yields to a completely discrete state evolution equation (approximate discrete description), from which the approximate discrete TF may be directly obtained: where: For a complete description of the real sampling technique see: [1] Garrido et al.

9. Identification Methods Continuous system Real sampling Input/output data Hybrid system IDENTIFICATION METHODS: From which, using a filter of adequate order, a discrete model may be identified. In first instance, filters of second, third and fourth order are considered, beholding the relative pole-zero order of the continuous system

10. Identification Methods IDENTIFICATION METHODS: Method 1: Least-Squares minimization Difference equation from input-data series: B, A: numerator and denominator of the discrete transfer function to be estimated d: input-output delay vk: modeling errors, data series truncation errors, noise, etc. Evaluation function to deal with using Least-Squares minimization: N: Total number of input/output pairs considered Identification algorithm:

11. Identification Methods Method 2: Leverrier’s identification algorithm where: Impulse response The method becomes exact when: • Dealing with minimum systems, like absolute minimum or dead-beat systems, the data series describing the impulse response is a polynomial • When the data series may be summed up resulting in a closed expression:

12. Identification Methods Comparison of the methods Method 1: • The accuracy obtained may be arbitrarily increased considering as many input-output pairs as desired for modeling purposes Method 2: • Provides an algebraic (not numerical) result for loworder truncation of the data series • When accounting for more terms of the input/output data series, the method may lead to a incompatible equation system, but even then, it may be applied and provides good results by combining it with a least-squares-type estimation

13. Simulated examples and Biestimation Real sampling SIMULATED EXAMPLES AND BIESTIMATION: Continuous system to consider: Hybrid system Impulse input Impulse response of the real sampled hybrid system

14. Simulated examples and Biestimation Method 1: Choice of the filter order: Mean square performance error: Outputs of the discrete models and the original hybrid system using real sampling with ZOH Discrete approximate third order model for ZOH: yk: Output of the reference system ym: Output of the discrete model m k: Current data being N: Total number of data computed

15.  -1 -0.8 -0.6 -0.4 -0.2 0 Simulated examples and Biestimation 0.197 0.215 0.176 0.163 0.144 0.139 0.2 0.4 0.6 0.8 1 0.109 0.091 0.074 0.068 0.051 Jm,2000 Mean square performance error (defined in the previous slide) of third order discrete models for different  Method 1: Choice of the hold order (): Outputs of best (lower mean square error: Gm2 for =1) and worst (higher mean square error for = -0.8) approximate discrete models and the original continuous system Discrete approximate third order model for =1:

16. Simulated examples and Biestimation Mean square performance error Method 2: Choice of the filter order: Outputs of the discrete models and the original hybrid system using real sampling with FOH Discrete approximate third order model for =1:

17. Simulated examples and Biestimation Biestimation: For a complete description of the internal behaviour of the expert network and the architecture used see: [2] De la Sen et al. Resumed knowledge base

18. Simulated examples and Biestimation Time evolution of the switching map for both methods Evolution of the mean square performance error with respect to the hybrid system obtained under real sampling for the hold order fixed by (=1) Biestimation:

19. Principal References PRINCIPAL REFERENCES: [1] Garrido, A.J., De la Sen, M. and Bárcena, R. Approximate Models to describe Real Sampling and Hold Processes based on Multirate Sampling Techniques. Proceedings of the 2000 American Control Conference. pp. 195-199. 2000. [2] De la Sen, M., Miñambres, J.J. and Garrido A.J. Logical Formal Description of Expert Systems. Informatica Int. Journal. No. 2, Vol. 13, 2002. pp. 177-208. ISSN: 0868-4952. 2002. [3] Ljung, L. System Identification. 2nd Ed. Chapter 4. Prentice-Hall, Upper Saddle River, NJ, 1999. [4] Sévely, Y. Systémes et asservissements linéaires échantillonnés. Chapter 7. Dunod Eds. 1969.

20. Conclusions • CONCLUSIONS: • Two methods for the obtaining of approximated discrete transfer functions for LTI systems under real sampling, using parameter estimation, have been presented • In the application, both methods have been applied to improve the discretization process, in the context of a biestimation scheme that switches to the one that provides more accurate results for different filter and hold orders. This scheme may also be used to evaluate the influence of other sampling parameters not considered here