Electricity: motive power systems – Positional servo systems
Reexamination Certificate
2001-08-30
2003-08-26
Ro, Bentsu (Department: 2837)
Electricity: motive power systems
Positional servo systems
C318S568100, C318S569000, C318S632000, C318S635000, C360S061000, C369S044290, C369S044350, C369S044360
Reexamination Certificate
active
06611119
ABSTRACT:
The present invention relates generally to servo control, and more specifically to switching processes for sensor-free servo control.
BACKGROUND
The concept of feedback control has witnessed countless advances and breakthroughs, one after another, not only in theory but in diverse applications far beyond its original engineering horizon. Its richness also has necessitated the creation of new disciplines and finer division of old disciplines, such as adaptive control, optimal control, H
∞
-control, and the like. Amongst the list of challenges and accomplishments from its inception, servomechanisms are one of the most fundamental problems in control theory.
In the simplest setting, a servo can be cast into a standard linear, second order, constant-coefficient ordinary differential equation most often used in modeling a mass-damper-spring system with an external excitation, also referred to in certain cases as an input or a stimulus.
Servomechanisms, or servos, are typically controlled using feedback. However, the use of feedback implies use of sensors to measure observables. Sensors have an associated cost. Sensors take up space which could otherwise be used for payload, and sensors have an associated weight that must be considered for overall performance, specifications, guidance, range, and the like. Further, sensors themselves bring uncertainty and noise into systems. Sensors placed on the servo, or incorporated into the servo, gather and relay information, typically including position, velocity, and acceleration information, to a controller through feedback paths. Such feedback leads to good control of servo operation.
Consider a standard mass-damper-spring system that is severely underdamped. Such systems suffer not only large over- and under-shoots but long settling times. The former poses danger in saturation and excites nonlinear dynamics either unconsidered or ignored, whereas the latter can induce catastrophic resonance among all such systems.
Overdamped and the critically damped systems have zero overshoot but suffer long rising time and so have even longer settling time. A related feature of these systems is their monotony. The ‘steady’ state is reached only asymptotically.
There are a collection of feedback control problems which historically have been solved in this manner: (a) solving the open-loop control as a function of time U=u(t), (b) equating u as an unknown function, or operator F (linear, nonlinear) of the state x, the output y, or the measurement z, and solving for F.
Preliminary Considerations and Previous Results
Consider a basic servomechanism problem to begin with. For single input single output (SISO) linear time invariant (LTI) systems in form of
a
n
x
(n)
+a
n−1
x
(n−1)
+ . . . +a
1
{dot over (x)}+a
0
x=u
(
t
), (2.1)
with a given desired system response x(t) satisfying certain essential Laplace transformability conditions, the pertinent servo control input û(t) can be shown as given by
û=L
−1
{R
n
{circumflex over (X)}−R
n−1
x
0
−R
n−2
{dot over (x)}
0
− . . . −R
1
x
0
(n−2)
−R
0
x
0
(n−1)
} (2.2)
where {circumflex over (X)} (s)=L{{circumflex over (x)}(t)}, {circumflex over (x)}(t) the desired system response, and
R
k
(
s
):=
a
n
s
k
+a
n−1
s
k−1
+ . . . +a
n−k−1
s+a
n−k
=sR
n−1
+a
n−k
, R
0
=a
n
. (2.3)
In particular, if x
0
={dot over (x)}
0
= . . . {dot over (x)}
0
(n−1)
=0, then
û=L
−1
{(
a
n
s
n
+ . . . +a
1
s+a
0
)
{circumflex over (X)}}.
(2.4)
For multiple input multiple output (MIMO) LTI systems {dot over (x)}=Ax+Bu, equation (2.2) changes to
û
=(
B
T
B
)
−1
B
T
[(
sI−A
)
{circumflex over (X)}−x
0
] (2.5)
assuming invertibility of B
T
B. Equations (2.2)-(2.3) are considered as a summary/solution to the most basic linear servomechanism problems. No consideration is given to conditions such as ∥u(t)∥≦1, however.
Note that if {circumflex over (x)}(t)={circumflex over (x)}
f
for t≧t
f
, then
X
^
(
s
)
=
∫
0
t
f
x
^
ⅇ
-
st
ⅆ
t
=
-
ⅇ
-
st
f
s
[
x
^
f
+
∫
0
t
f
x
^
.
ⅇ
-
st
ⅆ
t
]
=
…
=
-
ⅇ
-
st
f
s
[
x
^
f
+
1
s
x
^
.
f
+
1
s
2
x
^
¨
f
+
…
]
,
(
2.6
)
provided that (2.1) starts from rest. On the other hand, given an input û(t)⇄Û(s),
X
^
=
U
+
∑
k
=
1
n
R
n
-
k
x
0
(
k
-
1
)
R
n
=
∷
N
(
s
)
P
(
s
)
.
(
2.7
)
Smith (Smith, O. J. M.,
Feedback Control Systems
, McGraw-Hill, N.Y., 1958) was the first who proposed Posicast control (for positive-cast) and demonstrated the idea with a very simple mechanism, a pendulum. For time-optimal control systems, Neustadt (Neustadt, L. W., “Time Optimal Control Systems With Position and Integrals Limits,”
J. Math. Anal. and Appl
., Vol. 3,406-427, 1961) included position and integral limits in the Pontryagin Maximum Principle. Wang (Wang, P. K. C., “Analytical Design of Electrohydraulic Servomechanisms with Near Time-Optimal Responses,”
IEEE Trans. Auto. Control
, Vol. 8, No. 1, 15-27, 1963) had a dual-mode closed-loop analytical design for electrohydraulic servomechanisms. Davies (Davies, R. M., “Analytical Design for Time Optimum Transient Response of Hydraulic Servomechanisms,”
J. Mech. Eng. Sci
., Vol. 7, No. 1, 8-14, 1965) had an analytical design for hydraulic servomechanism. Being oriented toward practical application, both Wang and Davies elaborated on modeling and operation of nonlinear electrohydraulic and hydraulic servomechanisms, respectively. Athens (Athens, M., “Minimum-Fuel Control of Second-Order Systems with Real Poles,”
IEEE Trans. Auto. Control
, Vol. 9, No. 5, 148-153, 1964) derived the switching curves for minimum-fuel control of linear second-order systems with real poles. Different from minimum-time problem in which the control is necessarily bang-bang, the minimum-fuel control results in a bang-zero-bang profile. Of related interest, Ellert and Merriam (Ellert, F. J., and Merriam, C. W., “Synthesis of Feedback Controls Using Optimization Theory—An Example,”
IEEE Trans. Auto. Control
, Vol. 8, No. 4, 89-103, 1963) employed the so-called Parametric Expansion Method to vary the weighting factors for synthesis of linear time-varying feedback controls using optimization theory and illustrated by designing an aircraft landing system.
Yastreboff (Yastreboff, M., “Synthesis of Time-Optimal Control by Time Interval Adjustment,”
IEEE Trans. Auto. Control
, Vol. 14, No. 12, 707-710, 1969) seemed to have initiated synthesis of time-optimal control for LTI systems with real modes by time interval adjustment. It has appeared that Goldwyn-Sriram-Graham (Goldwyn, R. M., Sriram, K. P., and Graham, M.,
J. SIAM Control
, Vol. 5, 295, 1967) was the first which explicitly considered the switching times as unknowns to solve. Davison and Monro (Davison, E. J., and Monro, D. M., “A Computational Technique for Finding “Bang-Bang” Controls of Non-Linear Time-Varying Systems,”
Automatica
, Vol. 7, 255-260, 1971) gave a hill climbing-based computational technique finding the bang-bang control switching times of nonlinear time-varying systems. Farlow (Farlow, F. J., “On Finding Switching Times in optimal Control Systems,”
Int. J. Control
, Vol. 17, No. 4, 855-861, 1973) extended Goldwyn-Sriram-Graham and used complex analysis to form and solve the switching times for LTI control systems with real poles. Consider now a unit-ON/OFF input
u
^
=
1
-
2
u
(
t
-
t
1
)
+
2
u
(
t
-
t
2
)
-
…
+
2
(
-
1
)
n
s
-
1
u
(
t
-
t
n
s
-
1
)
+
(
-
1
)
n
s
u
(
t
-
t
n
s
)
↔
U
^
=
1
s
[
1
-
ⅇ
-
st
1
+
2
ⅇ
-
s
Duda Rina I.
Ro Bentsu
Ryan, Laura A. Fogg and Associates, LLC
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