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High Accuracy Control of a Proton Synchrotron Magnet Power Supply
T . lnoue et al.
3138
(1 - e-sL)/(l - (1 - G).e- sL ).
(3)
Steady-state Characteristics Let the input to the system in Fig. 3, we t)
=
(4)
r(t) + d(t)
be periodic for t
~
0 and given by
wet) = u(t) 'v(t),
(5)
o
and
vet - L) (- 00 < t < 00).
et < 0) l(t)O),
=
2nk/L
(k
=
IRe s I
0, ±l, ±2,···).
(6)
Because of the poles in the transfer func ti on of the error compensator, transfer zeroes appear in GE(S) at j Wk (k
=
0, ±l, ±2,"') i f I G(j wk) I > O.
Stability and Transient Characteristics The system in Fig. 3 is stab l e if the loop that r elates E to EO is stab le because Es represe nts the control error in the stable basic control system. The loop is stable independent of L if G(s) is stable, and if th e loop gain does not exceed unity: for all w.
(7)
The characte ristic equation for the loop is given by:
where H(s)
H(s) ~
( 8)
1 - G(s) .
=
IH( R) I.
+
n)/L,
IOms)-wl < (l+ n )/L.
(ll)
Let US restrict our attention to the case that exp(-n ) < IH(j w) 1 < e xp( n ),
(12)
e-
1
< Il+h(s-j w)1
(13)
in (10). Then we see that there is at least one root of (8) in D and that the root is a simple root (see appendix). Further, it can be easily shown that H(s) is approximated by a constant in a domain located sufficiently far f r om poles and zeroes in G(s).
if the poles and zeroes in G(s) lie far from the imaginary axis and L is large so that H(s) is well approximated by a cons tant in D, and if AT(W) does not take on an extreme value, as is required from (12). Response to Random Disturbance The case of a periodic input has been discussed in the preceding sec ti ons . However, disturbances may co ntain nonperiodic compo nents . As an example, random disturbance is considered below. Assume that the input to the system in Fig. 3 is random, and let Es(t) be a sample from a stationary random process the power spectrum of which is given by a continuous and slowly varying function of w, Ps(w). The power spectrum PECw) for E(t) is then given by (15)
Let R = a + j w, where a and w are real numbers, be a root of (8) . Then R satisfies exp(aL)
< (1
(14)
If the settling time for Gc(s) in (2) is much smaller than L, the response of Eo to wet) in (5) decays very rapidl y at t > L, and the behavior of E(t) thereafter can be discussed in terms of the zero-input response depending on the characteris ti c roots of the l oop shown in Fig. 3.
=
(10)
From these resul ts we may conclude that, for an a rb itrary w, a simple root exists near j w and the right hand side of (9) is approximated for this root by
Though this is a sufficient cond iti on for stability, it is nearly equivalent to the Nyquist's criterion when L is very large. Because of the rapid phase shift in the dead time e lement , the Nyquist's c riter ion tends to be so tight as (7) with increasing L.
exp(sL)
(1 + h(s - j w» 'H(jw)
and assume that, for any s in D,
Then
the control error E(t ) co nverges to zero with t ~ 00 under the input given by (5) if the system is stable. The location of the zeroes is determined by the dead time L, and is not affected by parameter perturbations in G(s).
11 - G(jw) 1 ( 1
=
for an arbitrary w at which !H(j w) I # O. Consider a domain D defined by such s that
The frequency components of vet) are wk
The right hand side of (8) can be written H(s)
where u(t)
vet)
The left hand side of (9) shows that the peak value of converging or diverging oscillation related to R changes by a factor of !H(R)! every L unit of time provided R is a simple root of (8). The oscillation decays when IH(R)! is less than unity. An approximate but convenient expression for the transient response is obtained from this relation, without solving (8) for infinitel y large number of character istic roots.
(9)
where GE(j w) is defined in (3). Suppose that L is very large. Then Ps( w) and G(j w) are nearly cons tant over an interval with its center at an arbitrary w:
T. Inoue et al.
3140
The freq uency resp onse G(jw), and indices for the transient and noise characteristics AT and Ad 2 are calculated as shown in Fig. 7 using the model given by (20) and (21). The same criterion as in (24) was app lied to the dynamical compensa tion for the other two magnets. The effect of compensation was equally satisfactory.
Experimental Results The control method was tested on the power supply to the three main ring magnets of the KEK 12-GeV proton synchrotron. The referenc e input wavefo rms were as shown in Fig. 1. The input rB(t) to the B magnet exciting current
accuracy tracking of a periodic reference input. The method was applied to the exciting current control for main ring magnets of a proton synchrotron. The 10- 4 accuracy required of the cur rent control was achieved after 16 cycles of a pulsed operation. The contro l can be applied to systems that are subjected to periodic or repeating reference input or disturbance. It may then happen that the stability condition in (7) can not be satisfied because of non linear ities, unknown high-frequency characteristics or other reasons. In such a case, the con troller may induce instability so that the effect of the control would better be eliminated by a filter incorporated in the error compensator. The effect of the band limita tion is discussed by Inoue and others (1981).
control system is plotted also in Fig. 8 (a). The length of a period is 2.37 seconds. REFERENCES The control e rr or was monitored in each digital controller. Several records are shown in B QF QD Figs. 8 (b) through (i). Here Ek ' Ek' and Ek denote the cur rent errors in the B magnet, the QF magnet, and the QD magnet respectively. The subscript k ) 0 specifies the cycle in which the error is measured and indi cates the number of cycles repeated before the measurements. As is seen in Fig. 8, the exciting current in the B magnet is uncontrolled during deceleration, and th e power supply units are switched based solely on the consideration th a t the reactive power on the ac side of the converters should not exceed a given limit. Prior to the operation, the dead time elements in the controllers are loaded with an initial condition determined on an empirical as well as analytical basis. The response to this initial condition is observed during the zero cycle. The control error waveforms in this cycle are shown in Figs. 8 (b) to (d). The control erro r is rapidl y decreased, as is shown in Figs. 8 (e) and (f) for the B magnet, and a specially marked decrease is observed in the low freq uency components. This is con sistent with the dynamical compensator design described in the preceding section.
Inoue, T., M. Nakano , and S. I wai (1981). High accuracy control of servomechanism for repeated contouring. Ppoc . 10th
annual Symp . In cpemen t al Motion Con tpol Systems and Devices . Kubo, T. , A. Kabe, K. Kitagawa, H. Sato, S . Shibata, S. Matsumoto, and H. Baba (1979). Compute r control system of main ring magnet power supply for the KEK 12 GeV P. S . I EEE Tpans . Nuclea p Sci ., NS- 26, 3322-3324 APPENDIX Location of Characteristic Roots First, suppose the case that h(s) _ O. the root s of (8) are given by Rn
r
+ j Wn (n
=
0, ±l, ±2, ··· )
Then
(A-I)
where r wn
{ln IH(j w)I }/L ,
= =
{arg H(j w) + 2n n }/L.
One or two of the roots Rn lie in a closed domain Do defined by s uch s that IRe s i ( TI /L, I (Im s) - w l ( TI /L
The plots in Figs. 8 (g) through (i) show that very high accuracy tracking was achieved in all control systems after sixteen cycles. The 50 Hz ripple dominant in
E~~
and
E~~
is due to
the noise intrinsic in the c urrent detector. Ripples i n the actual mag net exciting current are held in a much lower level by active as well as passive rippl e filters. No indications of instability were observed as the operation was continued.
CONCLUSION This paper has described a method for high
if (12) is satisfied. Second, by taking the logarithm of (8) substi tuting (10) into (8), we have s
=
{Lh(l + h(s - j W» }/ L + Rn'
where Lh is the principal In(l + h('»' From (13), this and the above r esult that (8) has at least one
(A - 2)
value for iLh(·)1 < 1. From for h = 0, we see root in D.
From the derivative for exp(sL) - H(s) we have that the root is simple i f H(s) is sufficient ly approximated by a constant in D so that idH/ds I < L in D.
3138
(1 - e-sL)/(l - (1 - G).e- sL ).
(3)
Steady-state Characteristics Let the input to the system in Fig. 3, we t)
=
(4)
r(t) + d(t)
be periodic for t
~
0 and given by
wet) = u(t) 'v(t),
(5)
o
and
vet - L) (- 00 < t < 00).
et < 0) l(t)O),
=
2nk/L
(k
=
IRe s I
0, ±l, ±2,···).
(6)
Because of the poles in the transfer func ti on of the error compensator, transfer zeroes appear in GE(S) at j Wk (k
=
0, ±l, ±2,"') i f I G(j wk) I > O.
Stability and Transient Characteristics The system in Fig. 3 is stab l e if the loop that r elates E to EO is stab le because Es represe nts the control error in the stable basic control system. The loop is stable independent of L if G(s) is stable, and if th e loop gain does not exceed unity: for all w.
(7)
The characte ristic equation for the loop is given by:
where H(s)
H(s) ~
( 8)
1 - G(s) .
=
IH( R) I.
+
n)/L,
IOms)-wl < (l+ n )/L.
(ll)
Let US restrict our attention to the case that exp(-n ) < IH(j w) 1 < e xp( n ),
(12)
e-
1
< Il+h(s-j w)1
(13)
in (10). Then we see that there is at least one root of (8) in D and that the root is a simple root (see appendix). Further, it can be easily shown that H(s) is approximated by a constant in a domain located sufficiently far f r om poles and zeroes in G(s).
if the poles and zeroes in G(s) lie far from the imaginary axis and L is large so that H(s) is well approximated by a cons tant in D, and if AT(W) does not take on an extreme value, as is required from (12). Response to Random Disturbance The case of a periodic input has been discussed in the preceding sec ti ons . However, disturbances may co ntain nonperiodic compo nents . As an example, random disturbance is considered below. Assume that the input to the system in Fig. 3 is random, and let Es(t) be a sample from a stationary random process the power spectrum of which is given by a continuous and slowly varying function of w, Ps(w). The power spectrum PECw) for E(t) is then given by (15)
Let R = a + j w, where a and w are real numbers, be a root of (8) . Then R satisfies exp(aL)
< (1
(14)
If the settling time for Gc(s) in (2) is much smaller than L, the response of Eo to wet) in (5) decays very rapidl y at t > L, and the behavior of E(t) thereafter can be discussed in terms of the zero-input response depending on the characteris ti c roots of the l oop shown in Fig. 3.
=
(10)
From these resul ts we may conclude that, for an a rb itrary w, a simple root exists near j w and the right hand side of (9) is approximated for this root by
Though this is a sufficient cond iti on for stability, it is nearly equivalent to the Nyquist's criterion when L is very large. Because of the rapid phase shift in the dead time e lement , the Nyquist's c riter ion tends to be so tight as (7) with increasing L.
exp(sL)
(1 + h(s - j w» 'H(jw)
and assume that, for any s in D,
Then
the control error E(t ) co nverges to zero with t ~ 00 under the input given by (5) if the system is stable. The location of the zeroes is determined by the dead time L, and is not affected by parameter perturbations in G(s).
11 - G(jw) 1 ( 1
=
for an arbitrary w at which !H(j w) I # O. Consider a domain D defined by such s that
The frequency components of vet) are wk
The right hand side of (8) can be written H(s)
where u(t)
vet)
The left hand side of (9) shows that the peak value of converging or diverging oscillation related to R changes by a factor of !H(R)! every L unit of time provided R is a simple root of (8). The oscillation decays when IH(R)! is less than unity. An approximate but convenient expression for the transient response is obtained from this relation, without solving (8) for infinitel y large number of character istic roots.
(9)
where GE(j w) is defined in (3). Suppose that L is very large. Then Ps( w) and G(j w) are nearly cons tant over an interval with its center at an arbitrary w:
T. Inoue et al.
3140
The freq uency resp onse G(jw), and indices for the transient and noise characteristics AT and Ad 2 are calculated as shown in Fig. 7 using the model given by (20) and (21). The same criterion as in (24) was app lied to the dynamical compensa tion for the other two magnets. The effect of compensation was equally satisfactory.
Experimental Results The control method was tested on the power supply to the three main ring magnets of the KEK 12-GeV proton synchrotron. The referenc e input wavefo rms were as shown in Fig. 1. The input rB(t) to the B magnet exciting current
accuracy tracking of a periodic reference input. The method was applied to the exciting current control for main ring magnets of a proton synchrotron. The 10- 4 accuracy required of the cur rent control was achieved after 16 cycles of a pulsed operation. The contro l can be applied to systems that are subjected to periodic or repeating reference input or disturbance. It may then happen that the stability condition in (7) can not be satisfied because of non linear ities, unknown high-frequency characteristics or other reasons. In such a case, the con troller may induce instability so that the effect of the control would better be eliminated by a filter incorporated in the error compensator. The effect of the band limita tion is discussed by Inoue and others (1981).
control system is plotted also in Fig. 8 (a). The length of a period is 2.37 seconds. REFERENCES The control e rr or was monitored in each digital controller. Several records are shown in B QF QD Figs. 8 (b) through (i). Here Ek ' Ek' and Ek denote the cur rent errors in the B magnet, the QF magnet, and the QD magnet respectively. The subscript k ) 0 specifies the cycle in which the error is measured and indi cates the number of cycles repeated before the measurements. As is seen in Fig. 8, the exciting current in the B magnet is uncontrolled during deceleration, and th e power supply units are switched based solely on the consideration th a t the reactive power on the ac side of the converters should not exceed a given limit. Prior to the operation, the dead time elements in the controllers are loaded with an initial condition determined on an empirical as well as analytical basis. The response to this initial condition is observed during the zero cycle. The control error waveforms in this cycle are shown in Figs. 8 (b) to (d). The control erro r is rapidl y decreased, as is shown in Figs. 8 (e) and (f) for the B magnet, and a specially marked decrease is observed in the low freq uency components. This is con sistent with the dynamical compensator design described in the preceding section.
Inoue, T., M. Nakano , and S. I wai (1981). High accuracy control of servomechanism for repeated contouring. Ppoc . 10th
annual Symp . In cpemen t al Motion Con tpol Systems and Devices . Kubo, T. , A. Kabe, K. Kitagawa, H. Sato, S . Shibata, S. Matsumoto, and H. Baba (1979). Compute r control system of main ring magnet power supply for the KEK 12 GeV P. S . I EEE Tpans . Nuclea p Sci ., NS- 26, 3322-3324 APPENDIX Location of Characteristic Roots First, suppose the case that h(s) _ O. the root s of (8) are given by Rn
r
+ j Wn (n
=
0, ±l, ±2, ··· )
Then
(A-I)
where r wn
{ln IH(j w)I }/L ,
= =
{arg H(j w) + 2n n }/L.
One or two of the roots Rn lie in a closed domain Do defined by s uch s that IRe s i ( TI /L, I (Im s) - w l ( TI /L
The plots in Figs. 8 (g) through (i) show that very high accuracy tracking was achieved in all control systems after sixteen cycles. The 50 Hz ripple dominant in
E~~
and
E~~
is due to
the noise intrinsic in the c urrent detector. Ripples i n the actual mag net exciting current are held in a much lower level by active as well as passive rippl e filters. No indications of instability were observed as the operation was continued.
CONCLUSION This paper has described a method for high
if (12) is satisfied. Second, by taking the logarithm of (8) substi tuting (10) into (8), we have s
=
{Lh(l + h(s - j W» }/ L + Rn'
where Lh is the principal In(l + h('»' From (13), this and the above r esult that (8) has at least one
(A - 2)
value for iLh(·)1 < 1. From for h = 0, we see root in D.
From the derivative for exp(sL) - H(s) we have that the root is simple i f H(s) is sufficient ly approximated by a constant in D so that idH/ds I < L in D.