Discrete-time approximation of multivariable continuous-time systems
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Table B: Second simulation £ (y t (kT E ) - 9 t (kT E ) s 0.5- 3 0.1 " 3 K, 0.5- 2 0.1- 2 0.5- 1 0.5- 3 0.1 " 3 y 2 0.5- 2 0.1 " 1 0.5- 1 X(R)U(R) N 3 3 5 5 9 0.14- 2 0.45- 2 0.14 1 0.60 1 0.17 3 0.10- 2 0.35- 2 0.46° 0.19 1 0.57 2 X(R)U(T) N 3 4 5 5 8 0.71- 7 0.35- 6 on- 3 0.17- 2 0.10 1 0.20- 8 0.33- 7 0.36- 4 0.48 " 3 0.29° X(T)U(T) N 3 4 6 7 13 0.60- 7 0.14- 6 0.96- 5 0.11" 3 0.65- 1 0.23- 7 0.21 " 7 0.80 " 5 0.65- 4 0.34"' BPF 0.26- 6 0.18- 6 0.10- 4 0.11 3 0.65- 1 0.30- 8 0.27- 7 0.78- 5 0.65- 4 0.34- 1 BT 0.14 9 0.14 9 0.34 3 0.66 4 0.74- 1 0.52 7 0.24 14 0.56 2 0.42 3 0.36- 1 I I I I S I I I I S value of T E chosen by the authors of paper 2452D in the second simulation. All of the results are different except for those in the first column (X(R)U(R)), which partly overtakes those of Tables 3 and 4 of the paper. In Table B we observe a good internal coherence, in spite of the poor practical interest of the sample interval. The last column (BT) is surprising enough to be commented on. If we determine the stability of the discrete models corresponding to the recursive equations, we note that, for very small values of T E , the discrete models are unstable (/). This shows a big difference between the algorithms, although BT is faster than the others, it leads to bad results, whereas X(T)U(T) and BPF are equivalent in time and precision. For the second simulation, we present in Table C, Tables A and C. This simulation is more informative than the first one. Three reasons, (i) theoretical, (ii) numerical and (iii) pedagogical, lead us to state that our results are more coherent than those of paper 2452D. (i) According to 7^, the discretisation by the bilinear transform gives an unstable discrete model, whereas the continuous model is stable. It should be noted that BT is the only algorithm where the recursive difference equa- tion computes the value y(kT) as a function of u(kT). This is only possible with an instantaneous system. The simi- larity of the results between X(T)U(T) and BPF confirms the good quality of these two algorithms. (ii) In the same simulation from X(T)U(T), we find that the large value of N ( = 30) induces (in simple precision) a convergence problem certainly tied to the dynamics of the system. We note here the similarity of the quality of the results of the two simulations. This is very satisfactory from the points of view of the numerical results and of the choice of algorithm, as no inversion matrix is required. (iii) It is commonly thought that, if the sample period is well chosen, discrete and continuous equations are equivalent; whatever the dynamics of the system. It was absolutely necessary for us to mention this, as shown in the Tables given. At this point, we can observe that the most interesting algorithm is the block pulse function method, which did not present problems in the two examples, and which Table C: Second simulation £ {y t (kT t k-i h 100 K, 1000 100 Y2 1000 T E . s 0.02 0.05 0.10 0.25 0.02 0.05 0.10 0.25 0.02 0.05 0.10 0.25 0.02 0.05 0.10 0.25 X(R)U(R) N 7 9 11 14 X X X X X X X X 0.26 1 0.16 2 0.77 2 0.68 3 0.25 2 0.17 3 0.80 3 0.69 4 0.74° 0.50 1 0.23 2 0.18 3 0.89 1 0.57 2 0.25 3 0.19 4 X(R)U(T) N 1 8 10 16 X X X X X X X X 0.25- 2 0.10° 0.16 1 0.58 2 0.28- 1 0.10 1 0.17 2 0.60 3 0.76- 3 0.29- 1 0.44° 0.14 2 0.78- 2 0.29° 0.49 1 0.15 3 )-UkT E X(T)U(T) N 9 13 19 30 X X X X X X 0.16- 3 0.66- 2 0.10° — 0.17- 2 0.65- 1 0.1 r — 0.86- 4 0.39- 2 0.53- 1 — 0.95- 3 0.34 " 1 0.56° — N X X X X X X X )) 2 BPF 0.16- 3 0.67- 2 0.10° 0.59 1 0.17- 2 0.65- 1 0.11 1 0.60 2 0.86- 4 0.33- 2 0.53"' 0.25 1 0.96- 3 0.34- 1 0.56° 0.26 2 N X X X X X X BT 0.15- 1 0.71 ~ 2 0.10° 0.59 1 0.58° 0.74- 1 0.11 1 0.60 2 0.46 " 3 0.35- 2 0.53- 1 0.25 1 0.60- 1 0.36- 1 0.56° 0.26 2 1 S S s 1 s s s results using the same sample interval 7^ as used in Table A. We find again the same coherence as in Table A. The differences between X(R)U(R), X(R)U(T) and X(T)U(T) are more important than in the first simulation. This is certainly tied to differences in the dynamics of the two simulated systems. We have shown that there is, for T E = 0.05 or 0.10, complete coherence between the results for X(T)U(T), BPF and BT. For T E = 0.25, we have noted a divergence of X(T)U(T) probably tied to the number of terms (N — 30) of the F or G development, and to the config- uration of the A and B matrices. For 7^. = 0.02 we have observed an instability from BT. Conclusions Certainly, the peculiarity of the simulation of the second system is the reason for the difference in the analysis of gave good results. We also observe the need to use several different algorithms in numerical problems to improve the quality of the results. L. CARALP G. DEFAYE 6th January 1986 Maitres de Conferences, ENSCPB, UA 503 CNRS, Universite de Bordeaux I, 351 cours de la Liberation, 33405 Talence Cedex, France The authors of paper 2452D wish to thank the corre- spondents for the comments on their paper. As men- 212 1EE PROCEEDINGS, Vol. 134, Pt. D, No. 3, MAY 1987
Transcript of Discrete-time approximation of multivariable continuous-time systems
Table B: Second simulation £ (yt(kTE) - 9t(kTE)
s
0.5-3
0.1 "3
K, 0.5-2
0 .1 - 2
0.5-1
0.5-3
0.1 "3
y2 0.5-2
0.1 "1
0.5-1
X(R)U(R)N
33559
0.14-2
0.45-2
0.141
0.601
0.173
0.10-2
0.35-2
0.46°0.191
0.572
X(R)U(T)N
34558
0.71-7
0.35-6
on-3
0.17-2
0.101
0.20-8
0.33-7
0.36-4
0.48 "3
0.29°
X(T)U(T)N
346713
0.60-7
0.14-6
0.96-5
0.11"3
0.65-1
0.23-7
0.21 "7
0.80 "5
0.65-4
0.34"'
BPF
0.26-6
0.18-6
0.10-4
0.11 3
0.65-1
0.30-8
0.27-7
0.78-5
0.65-4
0.34-1
BT
0.149
0.149
0.343
0.664
0.74-1
0.527
0.2414
0.562
0.423
0.36-1
IIIIS
IIIIS
value of TE chosen by the authors of paper 2452D in thesecond simulation.
All of the results are different except for those in thefirst column (X(R)U(R)), which partly overtakes those ofTables 3 and 4 of the paper. In Table B we observe agood internal coherence, in spite of the poor practicalinterest of the sample interval. The last column (BT) issurprising enough to be commented on. If we determinethe stability of the discrete models corresponding to therecursive equations, we note that, for very small values ofTE, the discrete models are unstable (/). This shows a bigdifference between the algorithms, although BT is fasterthan the others, it leads to bad results, whereas X(T)U(T)and BPF are equivalent in time and precision.
For the second simulation, we present in Table C,
Tables A and C. This simulation is more informativethan the first one.
Three reasons, (i) theoretical, (ii) numerical and (iii)pedagogical, lead us to state that our results are morecoherent than those of paper 2452D.
(i) According to 7^, the discretisation by the bilineartransform gives an unstable discrete model, whereas thecontinuous model is stable. It should be noted that BT isthe only algorithm where the recursive difference equa-tion computes the value y(kT) as a function of u(kT). Thisis only possible with an instantaneous system. The simi-larity of the results between X(T)U(T) and BPF confirmsthe good quality of these two algorithms.
(ii) In the same simulation from X(T)U(T), we findthat the large value of N ( = 30) induces (in simpleprecision) a convergence problem certainly tied to thedynamics of the system. We note here the similarity ofthe quality of the results of the two simulations. This isvery satisfactory from the points of view of the numericalresults and of the choice of algorithm, as no inversionmatrix is required.
(iii) It is commonly thought that, if the sample periodis well chosen, discrete and continuous equations areequivalent; whatever the dynamics of the system. It wasabsolutely necessary for us to mention this, as shown inthe Tables given.
At this point, we can observe that the most interestingalgorithm is the block pulse function method, which didnot present problems in the two examples, and which
Table C: Second simulation £ {yt(kTtk-i
h
100
K,1000
100
Y21000
TE.s
0.020.050.100.25
0.020.050.100.25
0.020.050.100.25
0.020.050.100.25
X(R)U(R)N
791114
XXXX
XXXX
0.261
0.162
0.772
0.683
0.252
0.173
0.803
0.694
0.74°0.501
0.232
0.183
0.891
0.572
0.253
0.194
X(R)U(T)N
181016
XXXX
XXXX
0.25-2
0.10°0.161
0.582
0.28-1
0.101
0.172
0.603
0.76-3
0.29-1
0.44°0.142
0.78-2
0.29°0.491
0.153
)-UkTE
X(T)U(T)N
9131930
XXX
XXX
0.16-3
0.66-2
0.10°—
0.17-2
0.65-1
0.1 r—
0.86-4
0.39-2
0.53-1
—
0.95-3
0.34 "1
0.56°—
N
XXXX
XXX
))2
BPF
0.16-3
0.67-2
0.10°0.591
0.17-2
0.65-1
0.111
0.602
0.86-4
0.33-2
0.53"'0.251
0.96-3
0.34-1
0.56°0.262
N
XXX
XXX
BT
0.15-1
0.71 ~2
0.10°0.591
0.58°0.74-1
0.111
0.602
0.46 "3
0.35-2
0.53-1
0.251
0.60-1
0.36-1
0.56°0.262
1SSs
1sss
results using the same sample interval 7̂ as used in TableA.
We find again the same coherence as in Table A. Thedifferences between X(R)U(R), X(R)U(T) and X(T)U(T)are more important than in the first simulation. This iscertainly tied to differences in the dynamics of the twosimulated systems.
We have shown that there is, for TE = 0.05 or 0.10,complete coherence between the results for X(T)U(T),BPF and BT. For TE = 0.25, we have noted a divergenceof X(T)U(T) probably tied to the number of terms(N — 30) of the F or G development, and to the config-uration of the A and B matrices. For 7̂ . = 0.02 we haveobserved an instability from BT.
Conclusions
Certainly, the peculiarity of the simulation of the secondsystem is the reason for the difference in the analysis of
gave good results. We also observe the need to useseveral different algorithms in numerical problems toimprove the quality of the results.
L. CARALPG. DEFAYE
6th January 1986
Maitres de Conferences, ENSCPB,UA 503 CNRS,Universite de Bordeaux I,351 cours de la Liberation,33405 Talence Cedex,France
The authors of paper 2452D wish to thank the corre-spondents for the comments on their paper. As men-
212 1EE PROCEEDINGS, Vol. 134, Pt. D, No. 3, MAY 1987
tioned in the paper, the second simulation example is for ments of the correspondents form a suitable complementa stiff system, which has fast as well as slow modes. The to the paper.sampling interval was selected in view of the fast mode ats = -10. It was stated in paper 2452D that the results in 8 t h M a y 1 9 8 6 PROF. N.K. SINHATables 3 and 4 were not very satisfactory and explained Department of Electrical & Computer Engineering,that it might have been caused by the presence of the McMaster Universityzero at s = -0 .1 , which was nearly equivalent to differ- Hamilton, Ontarioentiation. The authors also mentioned that a very small Canada L8S 4L7 'sampling interval may lead to ill-conditioning whichcould result in numerical errors. It seems that the com- 5285D
IEE PROCEEDINGS, Vol. 134, Pt. D, No. 3, MAY 1987 213
