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Computer determination of low order models for high order systems
Comput. &Elect. Engng, Vol. 2, pp. 117-123.PergamonPress, 1975. Printedin Great Britain.
COMPUTER DETERMINATION OF LOW ORDER MODELS FOR HIGH ORDER SYSTEMS G. J. THALER Department of Electrical Engineering, U.S. Naval Postgraduate School, Monterey, California 93940, U.SA. (Received 22 May 1974)
Abstract--An investigation directed at finding the best low-order model which approximates a given high-order system is presented. New insight is gained into the cost paid for the simplicity of the model and in the accuracy of the transient response of the model related to the magnitude of a cost function. The problem is solved in the time domain by finding the best pole and zero locations of the model which minimize a defined error criterion. The computer is used to estimate these parameters, via a parameter minimization program. A number of examples are included.
INTRODUCTION
There are various situations in which one would like to have a low order model for a high order system. A number of techniques for obtaining the low order model have been proposed [4], many of them relating to rather special problems. Normally one assumes that the high order system is linear, and that the low order model is also to be linear, and the problem then is to choose the "order" of the low order model, and the number of zeros to be used, after which the numerical values of the parameters must be determined. The technique used here answers both questions at the expense of multiple runs! Our results may also provide guidance to help minimize the number of runs required. PHILOSOPHY
The basic method is shown in Fig. 1. The high order system and the proposed model are both disturbed by the same input and the difference between their outputs is the amount by which the model response deviates from the desired response. By definition we would like this difference to be zero at all times, but it is not reasonable to expect this of a low order model, so we choose some functional relationship (operator box) which we would like to minimize. For our studies we use f(E) =
f;
E 2 dt, where T was the approximate settling
time of the high order system. We then feed the available information into a function minimization subroutine which adjusts the model parameter values and repeats the calculations iteratively until it finds a minimum on the cost function surface in parameter space. Referring to Fig. 1, the input signal presumably can be anything one wants. We have used steps[l] and ramps[2], and both work quite satisfactorily, but give slightly different 117
l 18
G.J. THALER
Error, E.
Input
~ ( E
Model States
I Adjustment
q, Function / Minimization" SubroutineF
Fig. 1. Block diagram for model determination.
minima. In our work either model would have been quite satisfactory for either step or ramp inputs. The "high order system" block could conceivably be the actual system, but in practice is more likely to be data obtained from a test, or possibly a sophisticated simulation model which the low order model is to replace. We used data from the high order system and inserted it as a table look-up. The "low order model" block contains the equations of the specific order model you wish to use. The form of the equations depends on the requirements of the function minimization subroutine, and probably the state variable equations will be needed. One must, therefore, choose the order and the initial values of the parameters. The operator block represents any suitable cost function, and this might differ with the application. We used only
E 2 dt. The function minimization subroutine may be any
suitable subroutine available. We used a locally generated one called BOXPLX. We note in passing that this technique can also be used for design--i.e, the "high order system" block can be a table look-up for a desired response defined on the basis of performance specifications; the "model" can be a state description of the hardware to be used with parameter adjustment limited to those parameters which are physically adjustable.
WHICH ORDER MODEL?
The reasons for replacing a high order system with a low order model usually lead to the selection of the lowest order possible; i.e. first or second order. Frequently this is a poor choice because the behavior of the model may differ substantially from that of the high order system. The method indicated in Fig. 1 requires that you first choose the order of the model, and the computer then finds the best parameter values for that order model. If the behavior of this model is not close enough to that of the high order system, then the order of the model must be increased and the optimization repeated. One would like to be able to choose the model order initially with confidence that the optimum parameter values for that order model will provide model response which is acceptable. Unfortunately, there is no known mathematical basis for choosing the "best" order for a low order model. Our studies (see examples) give some results which may be used to guide this choice (of the model order). We find that: (a) A second order model is seldom a good one. (b) A third order model (3 poles, no zeros) is substantially better than a second order model and is frequently "good enough". (c) A fourth order model is sometimes better than a third order model, but orders higher
Computer determination of low order models for high order systems
119
than four show so little improvement over the fourth order model that they probably should not be considered. Thus we conclude that the best low order model, in most cases, will be of third or fourth order. Should the model have zeros? Our results on this are inconclusive. Sometimes the zero helps; i.e. either the order of the model may be reduced ifa zero is used, or the model response is closer to that of the system when a zero is added. Usually, however, the effect of the zero does not seem to be significant. Our best guess is that a zero should be used only if the high order system is known to have a significant zero. In all cases the final mathematical criterion is the value of the cost function at its minimum point. On a comparative basis, the model with the lowest "minimum" cost best reproduces the system response. On an absolute basis the magnitude of the cost indicates the amount by which the model deviates from the system (but this is often hard to interpret !). Since the reasons for wanting a low order model vary With the application, as well as the importance of various features of the model response, the most satisfactory basis for judgement of the model is often just a visual comparison of the two responses. TEST SYSTEMS AND RESULTS In order to define a known "high" order system for our studies we first chose to cascade a pure time delay and a second order system. The transcendental nature of the time delay requires an infinite order polynomial for precise modeling, thus a "high" order system (which is readily simulated) is defined quite accurately. For the second order system we chose roots at - 1 _ j3, and we used two values of time delay: 0-15 and 0.30 sec. The response of this system (to a step and to a ramp) was simulated for 5.2 sec, and a table look-up of 520 points was formed. "Low order" models were formulated using 2-7 poles and 0-3 zeros. Each was optimized using the philosophy of Fig. 1. The results of some of these tests are summarized on Figs. 2 and 3. Figure 2 shows the variation in the minimum value of the cost function as the number of poles in the low order model is increased. Three curves are shown for a model with no zeros and step input; one zero and step input, and with no zeros and ramp input. From the step tests we observe the following: (a) A second order model is quite poor compared with third and higher orders. (b) The cost function minimizes at fourth order for both of the models tested by steps--it is not clear whether or not the model with ramp inputs has a minimum cost function. (c) The difference in cost between the models appears negligible for 3 poles and higher, except for the three pole--one zero model which appears to have relatively high cost. (d) There is only a slight difference in cost between the step and ramp input models for the same high order system. Figure 3 shows the effect of increased time delay on the required order of the model. It is clear that for our "high" order system (which is formed with a time delay) a given order for thelow order model will not be as ?good" when the time delay is increased. The meaning of this for a high order system of finite order is not completely clear, but the implication is that the order of the model required to accurately represent a high order system will increase with the number of significant poles in the high order system. The meaning of the cost values is perhaps clarified by inspection and comparison of the system and model responses. Figure 4 displays typical responses for selected points on Figs. 2 and 3. We note that the second order model is obviously poor, yet any of these
120
G . J . THALER 5.2
b
d=fE=dt 0
to J = . 0 0 3 2
.0008
-
I
t
d = ,014 at p = 2
l .0007
.0006
.0004
I 2
I
I
I
3 4 5 Number of Poles in Low Order Model
I
I
6
7
Fig. 2. Optimization of low order model, high order system has time delay = 0.15 sec
J ~" tO d=.0440 .0070
t --to ~
|
.0060 .0050
.0040 .0030 .0020
-
\\
-
, _ \ \Delay =. IO
_
Deloy=.30
.0010 0
I 2
I
"i
3 4 5 6 Number o f Poles in Low Order Model
i~l
7
Fig. 3. Effect of time delay, model with no zeros.
Computer determinationof low order models for high order systems
I-4
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1'2
1"2
I'0
I'0
~ 0.8 o.
a. 0 ' 8
0
0
0"6
1330
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0'6
0.4
0"4
(o)
(b)
0.2
0"2
, I
, 2
Time
I 3
121
Time
~
4
5
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(c)
Timej 3
5
6
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I-,~ 1330
I'0
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1330
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3
Y
0-4
(e)
0.2 2
4
6
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I
2
I
I
I
3
4
5
I 6
Time Fig. 4. Typical responses. A is the system response, and (a) two poles at ~ = 0'35, o9, = 2 . 7 3 4 ; (b) three poles at ( = 0"388, o9, = 3.145; - 6 . 2 4 ; (c) three poles at ( = 0'25, o9, = 2-99; - 2 . 6 9 and one zero at - 5.0; (d) increased time delay in the system. M o d e l has three poles at ( = 0.228, o9, = 2-995; - 2.93; (e) time delay = 0"15 sec. Model has three poles at ~ = 0.299, o9. = 3.009; - 7.466.
models (including the second order) might be adequate for many problems in which a low order model is desired. To verify these results with a finite order "high order system" we replaced the time delay with five real poles, retaining the second order complex poles. The high order system was then represented by a closed loop transmission function 384 x 107
T(s)
= (s 2 + 2s + 10)(s + 10)(s + 20)(s + 80)(s + 120)(s + 200)
This system was subjected to the same studies with essentially the same results. The variation in cost function for step inputs to a model with no zeros is shown in Fig. 5. These results are similar to those in Fig. 2, though with a lower value for the minimum cost function. Note that for a seven pole model J = 0.00003 rather than J = 0, which is a measure of the numerical accuracy of the computations.
122
G . J . THALER
d .0005 .0004 .0003 .0002 .0001
m
B
B
m
I
0
2
3
4
I
I
5
6
--'t. 7
Number of Poles in Model
Fig. 5. Optimization of low order model of seventh order system.
To further verify the results we choseE3] a pitch rate control for a supersonic aircraft for which 3-75,000 (s + 0.08333) R ( s ) - s 7 + 0.64s 6 + 4,092s 5 + 70,342s 4 + 85,370s 3 + 2,814,271s 2 + . . . Y(s)
• .. 3,310,875s + 281,250. F o r this system the characteristic polynomial has roots of - 0 . 0 9 2 ; - 2 . 0 2 4 _+ j0-965; - 7 - 6 7 5 _+ j13-445; - 3 2 . 0 6 5 _ j38.863. Using the same technique we obtained the cost function curves of Fig. 6. The step response of the three p o l e - - o n e zero model is shown on Fig. 7.
d .0006 .0005 --
~
NO Zeros
.0004 .0005 .0002 .0001 0
2
~
4
5
6
7
Number of Poles in Model
Fig. 6. Optimization of low order model for pitch rate control system.
Computer determination of low order models for high order systems
:::F ,.,
123
,3.
o
0.0
°°Zll 0
I
5
I
I
I0
I
15
20
I
25
Time Fig. 7.
CONCLUSIONS
This paper describes a simple technique by which the computer can be used to define the best parameters for a low order model. This technique uses a function minimization subroutine, and the recommended cost function is
f:
E 2 dr, where E is defined in Fig. 1.
We also conclude that the low order model should be third order (or possibly fourth order) and that the model contain no zeros unless the system is known to have significant zeros.
REFERENCES 1. J. Cantalapiedra, Low-order models for dynamic systems, M. S. Thesis, Naval Postgraduate School (1972). 2. N. V. Lam, Development of low order models for high order systems, M. S. Thesis, Naval Postgraduate School (1973). 3. N. K. Sinha and G. T. Bereznai, Optimum approximation of high-order systems by low order models, Intern. J. Control (1971). 4. L. Meier, Approximation of linear constant systems with linear constant systems of lower order, Ph.D. Thesis, Stanford University (1965).
COMPUTER DETERMINATION OF LOW ORDER MODELS FOR HIGH ORDER SYSTEMS G. J. THALER Department of Electrical Engineering, U.S. Naval Postgraduate School, Monterey, California 93940, U.SA. (Received 22 May 1974)
Abstract--An investigation directed at finding the best low-order model which approximates a given high-order system is presented. New insight is gained into the cost paid for the simplicity of the model and in the accuracy of the transient response of the model related to the magnitude of a cost function. The problem is solved in the time domain by finding the best pole and zero locations of the model which minimize a defined error criterion. The computer is used to estimate these parameters, via a parameter minimization program. A number of examples are included.
INTRODUCTION
There are various situations in which one would like to have a low order model for a high order system. A number of techniques for obtaining the low order model have been proposed [4], many of them relating to rather special problems. Normally one assumes that the high order system is linear, and that the low order model is also to be linear, and the problem then is to choose the "order" of the low order model, and the number of zeros to be used, after which the numerical values of the parameters must be determined. The technique used here answers both questions at the expense of multiple runs! Our results may also provide guidance to help minimize the number of runs required. PHILOSOPHY
The basic method is shown in Fig. 1. The high order system and the proposed model are both disturbed by the same input and the difference between their outputs is the amount by which the model response deviates from the desired response. By definition we would like this difference to be zero at all times, but it is not reasonable to expect this of a low order model, so we choose some functional relationship (operator box) which we would like to minimize. For our studies we use f(E) =
f;
E 2 dt, where T was the approximate settling
time of the high order system. We then feed the available information into a function minimization subroutine which adjusts the model parameter values and repeats the calculations iteratively until it finds a minimum on the cost function surface in parameter space. Referring to Fig. 1, the input signal presumably can be anything one wants. We have used steps[l] and ramps[2], and both work quite satisfactorily, but give slightly different 117
l 18
G.J. THALER
Error, E.
Input
~ ( E
Model States
I Adjustment
q, Function / Minimization" SubroutineF
Fig. 1. Block diagram for model determination.
minima. In our work either model would have been quite satisfactory for either step or ramp inputs. The "high order system" block could conceivably be the actual system, but in practice is more likely to be data obtained from a test, or possibly a sophisticated simulation model which the low order model is to replace. We used data from the high order system and inserted it as a table look-up. The "low order model" block contains the equations of the specific order model you wish to use. The form of the equations depends on the requirements of the function minimization subroutine, and probably the state variable equations will be needed. One must, therefore, choose the order and the initial values of the parameters. The operator block represents any suitable cost function, and this might differ with the application. We used only
E 2 dt. The function minimization subroutine may be any
suitable subroutine available. We used a locally generated one called BOXPLX. We note in passing that this technique can also be used for design--i.e, the "high order system" block can be a table look-up for a desired response defined on the basis of performance specifications; the "model" can be a state description of the hardware to be used with parameter adjustment limited to those parameters which are physically adjustable.
WHICH ORDER MODEL?
The reasons for replacing a high order system with a low order model usually lead to the selection of the lowest order possible; i.e. first or second order. Frequently this is a poor choice because the behavior of the model may differ substantially from that of the high order system. The method indicated in Fig. 1 requires that you first choose the order of the model, and the computer then finds the best parameter values for that order model. If the behavior of this model is not close enough to that of the high order system, then the order of the model must be increased and the optimization repeated. One would like to be able to choose the model order initially with confidence that the optimum parameter values for that order model will provide model response which is acceptable. Unfortunately, there is no known mathematical basis for choosing the "best" order for a low order model. Our studies (see examples) give some results which may be used to guide this choice (of the model order). We find that: (a) A second order model is seldom a good one. (b) A third order model (3 poles, no zeros) is substantially better than a second order model and is frequently "good enough". (c) A fourth order model is sometimes better than a third order model, but orders higher
Computer determination of low order models for high order systems
119
than four show so little improvement over the fourth order model that they probably should not be considered. Thus we conclude that the best low order model, in most cases, will be of third or fourth order. Should the model have zeros? Our results on this are inconclusive. Sometimes the zero helps; i.e. either the order of the model may be reduced ifa zero is used, or the model response is closer to that of the system when a zero is added. Usually, however, the effect of the zero does not seem to be significant. Our best guess is that a zero should be used only if the high order system is known to have a significant zero. In all cases the final mathematical criterion is the value of the cost function at its minimum point. On a comparative basis, the model with the lowest "minimum" cost best reproduces the system response. On an absolute basis the magnitude of the cost indicates the amount by which the model deviates from the system (but this is often hard to interpret !). Since the reasons for wanting a low order model vary With the application, as well as the importance of various features of the model response, the most satisfactory basis for judgement of the model is often just a visual comparison of the two responses. TEST SYSTEMS AND RESULTS In order to define a known "high" order system for our studies we first chose to cascade a pure time delay and a second order system. The transcendental nature of the time delay requires an infinite order polynomial for precise modeling, thus a "high" order system (which is readily simulated) is defined quite accurately. For the second order system we chose roots at - 1 _ j3, and we used two values of time delay: 0-15 and 0.30 sec. The response of this system (to a step and to a ramp) was simulated for 5.2 sec, and a table look-up of 520 points was formed. "Low order" models were formulated using 2-7 poles and 0-3 zeros. Each was optimized using the philosophy of Fig. 1. The results of some of these tests are summarized on Figs. 2 and 3. Figure 2 shows the variation in the minimum value of the cost function as the number of poles in the low order model is increased. Three curves are shown for a model with no zeros and step input; one zero and step input, and with no zeros and ramp input. From the step tests we observe the following: (a) A second order model is quite poor compared with third and higher orders. (b) The cost function minimizes at fourth order for both of the models tested by steps--it is not clear whether or not the model with ramp inputs has a minimum cost function. (c) The difference in cost between the models appears negligible for 3 poles and higher, except for the three pole--one zero model which appears to have relatively high cost. (d) There is only a slight difference in cost between the step and ramp input models for the same high order system. Figure 3 shows the effect of increased time delay on the required order of the model. It is clear that for our "high" order system (which is formed with a time delay) a given order for thelow order model will not be as ?good" when the time delay is increased. The meaning of this for a high order system of finite order is not completely clear, but the implication is that the order of the model required to accurately represent a high order system will increase with the number of significant poles in the high order system. The meaning of the cost values is perhaps clarified by inspection and comparison of the system and model responses. Figure 4 displays typical responses for selected points on Figs. 2 and 3. We note that the second order model is obviously poor, yet any of these
120
G . J . THALER 5.2
b
d=fE=dt 0
to J = . 0 0 3 2
.0008
-
I
t
d = ,014 at p = 2
l .0007
.0006
.0004
I 2
I
I
I
3 4 5 Number of Poles in Low Order Model
I
I
6
7
Fig. 2. Optimization of low order model, high order system has time delay = 0.15 sec
J ~" tO d=.0440 .0070
t --to ~
|
.0060 .0050
.0040 .0030 .0020
-
\\
-
, _ \ \Delay =. IO
_
Deloy=.30
.0010 0
I 2
I
"i
3 4 5 6 Number o f Poles in Low Order Model
i~l
7
Fig. 3. Effect of time delay, model with no zeros.
Computer determinationof low order models for high order systems
I-4
1.4
1'2
1"2
I'0
I'0
~ 0.8 o.
a. 0 ' 8
0
0
0"6
1330
J
0'6
0.4
0"4
(o)
(b)
0.2
0"2
, I
, 2
Time
I 3
121
Time
~
4
5
0
I I
2
(c)
Timej 3
5
6
I-4
6
I-,~ 1330
I'0
o
5
1330
O'B
4
0-6
3
Y
0-4
(e)
0.2 2
4
6
0
I
2
I
I
I
3
4
5
I 6
Time Fig. 4. Typical responses. A is the system response, and (a) two poles at ~ = 0'35, o9, = 2 . 7 3 4 ; (b) three poles at ( = 0"388, o9, = 3.145; - 6 . 2 4 ; (c) three poles at ( = 0'25, o9, = 2-99; - 2 . 6 9 and one zero at - 5.0; (d) increased time delay in the system. M o d e l has three poles at ( = 0.228, o9, = 2-995; - 2.93; (e) time delay = 0"15 sec. Model has three poles at ~ = 0.299, o9. = 3.009; - 7.466.
models (including the second order) might be adequate for many problems in which a low order model is desired. To verify these results with a finite order "high order system" we replaced the time delay with five real poles, retaining the second order complex poles. The high order system was then represented by a closed loop transmission function 384 x 107
T(s)
= (s 2 + 2s + 10)(s + 10)(s + 20)(s + 80)(s + 120)(s + 200)
This system was subjected to the same studies with essentially the same results. The variation in cost function for step inputs to a model with no zeros is shown in Fig. 5. These results are similar to those in Fig. 2, though with a lower value for the minimum cost function. Note that for a seven pole model J = 0.00003 rather than J = 0, which is a measure of the numerical accuracy of the computations.
122
G . J . THALER
d .0005 .0004 .0003 .0002 .0001
m
B
B
m
I
0
2
3
4
I
I
5
6
--'t. 7
Number of Poles in Model
Fig. 5. Optimization of low order model of seventh order system.
To further verify the results we choseE3] a pitch rate control for a supersonic aircraft for which 3-75,000 (s + 0.08333) R ( s ) - s 7 + 0.64s 6 + 4,092s 5 + 70,342s 4 + 85,370s 3 + 2,814,271s 2 + . . . Y(s)
• .. 3,310,875s + 281,250. F o r this system the characteristic polynomial has roots of - 0 . 0 9 2 ; - 2 . 0 2 4 _+ j0-965; - 7 - 6 7 5 _+ j13-445; - 3 2 . 0 6 5 _ j38.863. Using the same technique we obtained the cost function curves of Fig. 6. The step response of the three p o l e - - o n e zero model is shown on Fig. 7.
d .0006 .0005 --
~
NO Zeros
.0004 .0005 .0002 .0001 0
2
~
4
5
6
7
Number of Poles in Model
Fig. 6. Optimization of low order model for pitch rate control system.
Computer determination of low order models for high order systems
:::F ,.,
123
,3.
o
0.0
°°Zll 0
I
5
I
I
I0
I
15
20
I
25
Time Fig. 7.
CONCLUSIONS
This paper describes a simple technique by which the computer can be used to define the best parameters for a low order model. This technique uses a function minimization subroutine, and the recommended cost function is
f:
E 2 dr, where E is defined in Fig. 1.
We also conclude that the low order model should be third order (or possibly fourth order) and that the model contain no zeros unless the system is known to have significant zeros.
REFERENCES 1. J. Cantalapiedra, Low-order models for dynamic systems, M. S. Thesis, Naval Postgraduate School (1972). 2. N. V. Lam, Development of low order models for high order systems, M. S. Thesis, Naval Postgraduate School (1973). 3. N. K. Sinha and G. T. Bereznai, Optimum approximation of high-order systems by low order models, Intern. J. Control (1971). 4. L. Meier, Approximation of linear constant systems with linear constant systems of lower order, Ph.D. Thesis, Stanford University (1965).