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8. Parameter Optimisation for Fixed Controllers
8.3. Standard Forms
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Parameter Optimisation for Fixed Controllers
Differentiating to find the optimal value of T yields
2 / 1 2
) 3 3 1 (
1 D D
T
(8.15)and the corresponding value of the ISE is
D D D
1
) 3 3 1
( 2 1/2
ISEmin (8.16)
Differentiation of this expression shows that the absolute minimum value obtainable is 0.866 when D = 1/3 and the value of T = 1.732.
Three simple examples have been taken to illustrate the analytical approach to minimising the ISE ,which corresponds to n = 0 in the general criterion of equation (8.3). This has been done to illustrate the procedure whilst keeping the algebra relatively simple. If, for example, one wished to investigate the last example for the ISTE criterion, that is n = 1, then one has to differentiate E(s) with respect to s. This increases the order of the denominator from 3 to 6, the algebra for the integral becomes ‘horrendous’ and differentiation of the result is then required for the optimum values. Computationally, however, the minimum can be found very quickly, one selects values for T and D, evaluates the ISTE and has an optimisation algorithm built around to adjust T and D to converge to the optimum values, which will of course exist if the compensator is in H.
One reason for having covered the classical optimization approach in some detail is that it leads naturally to the consideration of standard forms. These provide an interesting closed loop direct synthesis approach for obtaining controller parameters.
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Parameter Optimisation for Fixed Controllers
This can be shown to be a minimum for a K and the corresponding closed loop transfer function is
K K s s s K
T
( )
2. (8.18)
Comparing this with the standard form for the second order equation of
2 2
2
) 2 (
o o o
s s s
T
] Z Z
Z
(8.19)shows that K
Z
o2, so the natural frequency increases with K but the damping ratio9 0 . 5 .
This value of 9 gives a step response overshoot of around 16%. The value is less than the unit value required for no overshoot in the step response and the value of 0.707 required for no peak in the frequency response.Please click the advert
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Parameter Optimisation for Fixed Controllers
Time scaling the standard second order equation (8.19), that is replacing s/Zo by sn, gives the transfer function
1 2 ) 1
(
2 nn
n s s
s
T
]
(8.20)This equation is known as the time normalised form and as explained in chapter 3 has exactly the same time response as eqn.(8.19) but eqn. (8.19) is a factor Zo faster. Eqn. (8.20) with ȗ = 0.5 is referred to as the standard form of the second order all pole closed loop transfer function which minimises the ISE, J0, for H = 1. Note also that the forward loop transfer function, GGc, must contain an integrator to ensure zero steady state error to a step input. Standard forms for any order of the denominator polynomial and for various integral performance criteria can be found and written, with the subscript n dropped from s, as
1 ...
) 1 (
1 1
1
d s d s s s
T j
j
j (8.21)
They have been derived in reference [8.3] for the more general performance index ISTnE for different values of n and are denoted as:-
1 ...
) 1 (
1 1
1
0 s s
d sd s
T j
j
j j (8.22)
The required coefficient values as well as the resulting value of the performance index are given in Table 8.2. Note the value of the index increases for larger n because of the time weighting factor as the settling time is greater than unity.
It is interesting to look at the coefficients of these transfer functions. First, purely of academic interest, is the fact that for the ISE, that is n = 0, all the coefficients are integer values. More important, however, is the fact that as n is increased the coefficients increase in value and the step responses have less overshoot with only a small change in settling time. This can be seen for the second order system as the damping ratio 9, equal to d1/2 in Table 8.2, increases as n
increases. These points are further demonstrated by Fig. 8.1 which shows the step responses for j
= 4 for n = 0 to 3.
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Parameter Optimisation for Fixed Controllers
0 2 4 6 8 10 12 14 16 18 20
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time(s)
Magnitude
Step responses of T (s)
ISE ISTE IST E IST E3 2 04
Figure 8.1 Step responses for j = 4
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Parameter Optimisation for Fixed Controllers
)
0
(
sT j n d1 d2 d3 d4 d5 Jn
0 1.000 1
1 1.335 0.8686
2 1.537 3.2823
)
02
(
s T3 1.665 28.1005
0 2.000 1.000 1.5000
1 2.042 1.472 2.1142
2 2.155 1.825 10.400
)
03
(
s T3 2.281 2.082 105.1355
0 2.000 3.000 1.000 2.0000
1 2.372 3.072 1.539 4.2355
2 2.620 3.295 1.990 27.350
)
04
(
s T3 2.809 3.577 2.349 329.4304
0 3.000 3.000 4.000 1.000 2.5000
1 3.052 3.897 4.094 1.576 7.4816
2 3.195 4.572 4.402 2.092 62.0700
)
05
(
s T3 3.360 5.129 4.827 2.527 898.3668
0 3.000 6.000 4.000 5.000 1.000 3.0000
1 3.385 6.145 5.489 5.110 1.597 12.1017
2 3.649 6.570 6.705 5.481 2.157 126.4600
)
06
(
s T3 3.862 7.108 7.756 6.026 2.649 2189.100
Table 8.2 All Pole Standard Forms for T0j (From reference 8.4)
Since adding a compensator often results in a closed loop transfer function having a zero, a suggested use for these standard forms in controller design has been to add a prefilter with a pole to cancel the zero. To avoid the use of a prefilter studies have been done to investigate standard forms with a zero, that is for
1 ...
) 1 (
1 1
1 1
1
s d s
d s
s s c
T j
j
j j (8.23)
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Parameter Optimisation for Fixed Controllers
In early work on this topic results were given to minimise the ITAE criterion but with the requirement that the system should also have zero steady state error to a ramp input, which requires the constraint that c1 d1. Recently [8.4] results have been derived without this constraint but the required d coefficients vary with the choice of c1, as illustrated in Fig 8.2 for T14(s). It is also possible to plot how the poles of the transfer function vary with the zero parameter c1, and some of these can be found in reference [8.4]. These plots show that the optimal pole positions vary appreciably with the value of the zero.
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9
C
d1, d2, d3
d coefficients of T (s) ISE
ISTE IST E IST E
d1 d2
d3 2
3
1 14
Figure 8.2 d coefficients for T14 for 3 values of n.
The concept of trying to design a controller so that the closed loop transfer function has a specific form is useful, as it addresses the closed loop performance directly, and will be illustrated by an example in the next section, as well as being given further consideration in chapter 11. There are many closed loop standard forms that might be chosen apart from those based on the criteria of Table 8.1. For example, the Butterworth filter form could be selected.