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Unformatted text preview: Example of Exact Tradeoffs in Linear Controller Design
Craig Barrall and Stephen Boyd ABSTRACT: The design of a linear time
invariant controller for a given linear time—
invariant plant, like any engineering design,
involves tradeoffs among many desirable
qualities, such as fast response to commands
without excessive overshoot, low actuator
authority, robustness low controller come
plexity, and so on. Only for a few very spe
cial cases are analytic methods known for
ﬁnding the exact form of these tradeeolfs.
Two examples of such analytic methods are
Linear Quadratic Gaussian theory, where the
plant actuator and output variance can be
traded off, and NevanlinnaPick theory,
where, for instance, the achievable distur
bance rejection can be traded off in two dif
ferent bandwidths. In many cases, the limit
of performance achievable with a linear time
invariant controller, and thus the exact form
of the trade—oft, can be computed numeriv
cally. To demonstrate this tradeoff, the pa—
per treats the design of a regulator for a very
typical plant, a double integrator with some
excess phase, The tradeoffs are presented
for two different measures of robustness with
noise sensitivity. The exact form of these
trade—offs is determined numerically by using
the techniques described in the appendixes. Introduction Given a (possibly unstable) linear time—
invariant plant, a basic design problem is to
ﬁnd a linear timeinvariant controller that will
stabilize the plant and meet various design
objectives Beyond simple intuition, little is
known about how the different design objec—
tives or goals trade off. The basic character of some of the design
trade—oils is obvious from simple physical
reasoning, e. g. , it takes more force (actuator
authority) to move a mass from one place to
another more quickly (faster command re
sponse). Other tradeoffs are more subtle, For
example, it is well known to control engi
neers that robustness or stability margins
trade off against closed—loop bandwidth for Craig Barratt and Stephen Boyd are with the In
formation Systems Laboratory in the Electrical
Engineering Department at Stanford University,
Stanford, CA 94305. 46 any plant with delay or other excess phase
[1]. Recently [2], it was observed that the limit
of performance achievable with a linear time—
variant controller can be computed numeri
cally, where perfomtance reflects many
practical constraints and qualities, including
fast response to commands without exces—
sive overshoot or undershoot, small and
quick reactions to disturbances or noises, low
actuator authority, and certain measures of
robustness or insensitivity to unknown or un
modeled plant dynamics. Some design goals,
such as low controller complexity and open—
loop controller stability, cannot be included
using the methods in [2]. Finding the Limit of Performance Recent theoretical results [3] show that the
set of closedloop responses achievable with
stabilizing controllers is aﬁine. This means
that if controllers C1 and C2 each stabilize a
given ﬁxed plant P, and yield closedeloop
transfer functions H. and H2, then, for every
real number )x, there is a controller C x~ which
stabilizes P and yields the closed—loop trans
fer function )xH. + (l — MHZ. Geometri
cally, this means every closed—loop transfer
function on the line through H1 and H2 is
achieved by some stabilizing controller. Note
that, in general, the controller Cy is not given
by )xC. + (1 — )x) C3. These results allow
us to express simply every closed—loop trans
fer function achievable with stabilizing con—
trollers. Now consider the controller design probe
lent, which is the problem of ﬁnding a con—
troller that achieves particular design goals.
In many cases, the design goals can be ex—
pressed as convex constraints on various
closedloop transfer functions. A constraint
on a closed—loop transfer function is convex
if the average value of two closedloop trans
fer functions (Hl + H2)/2 satisﬁes the con
straint whenever HI and H2 do (here H1 and
112 are the closedloop transfer functions
yielded by controllers Cl and C2). Examples
of convex constraints are design goals in
volving envelope bounds on closed—loop time
responses to any input (e.g., step responses
and impulse responses), sensitivity to noises, 027271708/89/010070046 $01.00 © 1989 IEEE closed—loop frequency responses, and some
robustness requirements. Examples of de
sign goals that cannot be expressed as convex
constraints on closedloop transfer functions
are restrictions on the structure or order of
the controller, e.g., the number of controller
poles. In the case where every design goal can
be expressed as a convex constraint on some
closedloop transfer function, the controller
design problem is equivalent to ﬁnding a
closedloop transfer matrix (a collection of
the appropriate closedloop transfer func—
tions) that satisﬁes the design goals. The
simple representation of the achievable
closedloop transfer functions means that this
new problem can be solved numerically to
arbitrary accuracy [2]. A point on a trade—off curve can be found
by constraining all but one of the design goals
and minimizing the remaining one (it is as—
sumed that all design goals are expressed so
that “smaller” is “better"). The corre
sponding optimization problem is convex.
This means there are no local minima. Such
convex optimization problems can be solved
numerically to arbitrary accuracy. The Plant We consider the design of a regulator for
a single—input/singleoutput plant that con—
sists of a double integrator with some excess
phase, Pm = (1/52) (4 — s)/(4 + s) The all—pass term (4 — s)/(4 + s) approxi—
mates a 0.5sec delay (exp (is/2)) at low
frequencies. We may think of the allpass
term as accounting for any and all of a va—
riety of sources of excess phase in a real
control loop, e.g., small delays. antialias ﬁl
ters, equivalent excess phase contributed by
a sampleandzero—orderhold plant input,
and so on. The idea of using a double inte—
grator plant with some excess phase as a
simple but realistic typical plant with which
to explore control design tradeoffs is taken
from a study presented by Gunter Stein in
[41‘ The plant is discretized using a zero—order lEEE Control Systems Magazine hold at 10 Hz, giving the following transfer
function: P (z) —0.00379(z2 — 0.7241z — 1.1457)
_ Z3 — 2.67032,2 + 2.3406z — 0.6703 _ —0.00379(z ~ 1.492) (2 + 0.7679)
A (z , 1f (2 — 0.6703) The objective is to investigate some trade
oﬁs in the design of a regulator C for this
plant, where the closed—loop system is shown
in Fig. 1. The discretetime inputs w and v
represent actuator (input—referred process
noise) and sensor noise, which are taken to
be independent white noise. The discrete—
time outputs u and y are the actuator and
plant outputs. LQG: An Analytically
Computable Tradeoff Consider the steady—state noise at u and y
in Fig. 1 due to the injection of noises at v
and w. As usual, assume that v and w are
zero mean, white, independent, with root—
meansquare (nns) values of 1 for U and 10
for w. The trade—oilP between the steady—state out
put variance limb a, Eyf and the steady—state
actuator variance lim,Hm Euﬁ is analytically
computable from Linear Quadratic Gaussian
(LQG) theory [5]. Given a ﬁxed positive p,
the LQG optimal regulator minimizes (over
all stabilizing regulators) the weighted cost
function J, which is a linear combination of
the actuator and output variance, J = lim E{yi + pug}
k—> on
Appendix B gives an example where the dis
crete—time LQG optimal regulator is found
for the plant with p equal to 0.0001. Note that for each p, the LQG optimal
regulator gives the best noise sensitivity tie.
the minimum J) that can be achieved by any
linear timeinvariant regulator (of any com
plexity or structure) that stabilizes the plant
P. Thus, the rms values of u and )1 achieved
by the LQG regulator give a point on the
trade—off curve between these quantities. Let
us justify this assertion. If some other linear
timeinvariant regulator stabilized P and
achieved better rtns values of both it and y,
then it would achieve a cost J smaller than
the LQG regulator (recall that p is greater
than 0). This is not possible, since the LQG
optimal regulator gives the smallest J
achievable by any linear timeinvariant reg—
ulator. As the number p varies. the nns values of
u and y achieved by the corresponding LQG January 7989 Fig. l. optimal regulator sweep out the tradeoff
curve. For our plant. the tradeoff between
the rrns values of u and y is shOWn in Fig.
2. The interpretation of the curve in Fig. 2
is as follows. No linear timeinvariant reg—
ulator C that stabilizes P can achieve rms
values ofu and y, which lie below the curve.
This is true for regulators of any order, de—
signed by any method. We can restate this
as: every regulator C that stabilizes P
achieves rrns values of u and y, which lie in
the shaded region, on or above the tradeoff
curve. For example, the following simple
leadlag regulator stabilizes P and achieves
n'ns values of u and y of 27.55 and 5.98,
respectively. C(z) = 10(z * 0.98)/(z  0.56) This property is shown in Fig. 3. This reg
ulator achieves closedloop performance,
which lies in the shaded region of Fig. 2, as
it must. Any regulator C that gives closed—loop
performance in the shaded region in Fig. 3
will perform the same or better (in terms of
rrns values of u and y) than the simple lead—
lag regulator C. This means that C will
achieve the same or better output regulation
(rms y no worse than 5.98) and use the same
or less actuator effort (rms u no worse than
27.55). One family of such regulators is the
LQG optimal regulators with p between
000135 and 2.31. These regulators lie on
the boundary of the shaded region. There are
infinitely many other regulators that give
closed—loop performance in the shaded re—
gion. This idea gives us a second interpretation
of the tradeoff curve in Fig. 2 in terms of
achievable design goals. Any design goals
(i.e., upper bounds on the rrns values of u
and y) that lie in the shaded region in Fig. 2
can be met or exceeded by some linear time
invariant regulator C, which stabilizes P.
Design goals outside of the shaded region
cannot be achieved by any linear timein—
variant regulator C, which stabilizes the plant Regulator system. 10 rms value of y 1 ’ 10 100
rms value of u Fig. 2. Tradeoff between achievable in»
put and output nns noise sensitivities, on a
log—log scale. p 20.00135 rms value of y
.t + _. 10 100
rms value of 11
Fig. 3. Simple regulator C, which
achieves performance above the tradeoilr
curve. P. For example, consider the design goal
that the rms value of u should be less than
10 and the rms value of y should be less than
4. This design goal can be achieved by a
linear timevinvariant regulator that stabilizes
the plant P, whereas the design goal that the
rrns value of u should be less than 3 and the
mis value of y should be less than 5 cannot
be achieved by any linear timeinvariant reg—
ulator that stabilizes the plant P. For p equal to 10—4, the details of the LQG
design are given in Appendix B. The LQG
optimal (current estimator) regulator is
shown to be 45.9748 — 72.79z2 + 28.138z C(z) — z3 — 0.8061:2 + 0.7107z — 0.1071 45.974z(z — 0.91305) (z — 0.6703)
(z — 0.17897) (z — 0.3136 + 0.7072j) (z  0.3136 — 07072)) 47 For this regulator. the rms values of u and y are 63.73 and 2.0184, respectively. The
value of the cost function J is 4.48. Two Measures of Robustness
Tolerance of Additive Loop Perturbations One measure of robustness of a control
system, which combines the gain and phase
margins, is the Mcircle radius, defined as
the minimum distance from the Nyquist plot
of the loop gain PC (exp 10) to the critical
point — 1. Mathematically, this can be writ
ten as M l l dist (PC(exp jﬂ), — 1)
min 1 + PC(expj9) Osnsr The M—circle radius gives a measure of the
sensitivity to additive loop perturbations. If
the distance M is small, then slight variations
in the loop gain PC could change the number
of net clockwise encirclements of _ 1 by the
loop gain, resulting in an unstable closed
loop system. The M—circle radius is related to the max
imum sensitivity of the control system. The
Mecircle radius is the inverse of the maxi—
mum sensitivity M=1/lSll°. where S = 1/(1 + PC) is the sensitivity
transfer function, and the maximum of the
sensitivity transfer function over all frequenv
cies is (In this notation, HHIIQ, denotes the maxi—
mum magnitude over real frequencies of a
transfer function H.) Thus. a small M—circle
radius corresponds to peaking of the sensi
tivity function at some frequency. It also fol
lows that constraining the maximum mag
nitude of the closed»loop transfer function S
to be at most 1 over Mm,“ is equivalent to
constraining the M—circle radius to be at least
Mm. For example, the p = 10’4 LQG regulator
gives ilSllm = 3.34, so that M Z 0.30 is
the closest the Nyquist plot comes to the crit
ical point #1. This can be seen from the
Nyquist plot in Fig. 4. The Mcircle of ra
dius 0.30 is also shown in Fig. 4. The mag
nitude of the sensitivity transfer function S
for the p = 10 '4 LQG regulator is shown in
Fig. 5. Note that its maximum is 10.5 dB
(which is equal to —20 log M). Tolerance of Additive Plant Perturbations The second measure of robustness we will
consider concerns the ability of the regulator 48 lm(PC) 15 '2 —1.5 .1 0.5 0 0.5 1 Re(PC) Fig. 4. Nyquist plot of the LQG
regulator. Note: p = 10‘4 and the M 2
0.30 circle is shown. 10
i
{:11 f\//,
s /
9
0.1 §
0.01 ' ‘ ,7 0 0.5 1 1.5 2 2.5 3
Q: (OT Fig. 5. Magnimde of sensitivity transfer
function of LQG regulator. Note: p =
10*. to maintain closed—loop stability in the face
of stable additive plant perturbations AP. as
shown in Fig. 6. The plant perturbation could
represent errors in modeling the plant, com
ponent variations. or deliberately ignored
plant dynamics. Intuitively, if AP is small at all frequen—
cies, then we would expect that if the reg
ulator C stabilizes the plant P, then C should
stabilize P + AP as well. This intuition is
indeed correct, a consequence of the small
gain theorem [6]. [7]. We may ask, what is the smallest (in the
sense of maximum magnitude of frequency
response) stable additive plant perturbation
AP that will destabilize the system in Fig.
6? Let us deﬁne the quantity D as the small
est plant perturbation AP that will destabi
lize the closed»loop system, D = min IIAPH,” APdcstabilizcssystem
It is not difﬁcult to derive that the inverse of
D is equal to the maxinrum of the closed—
loop transfer function from the reference in Fig. 6. Additive plant perturbation. put r to the actuator output u,
l/D = IlC/(l + PC) II... This observation is due to Doyle and Stein
[6]. The positive number D can be interpreted
as the largest size of stable additive plant
perturbation the control system can be guar—
anteed to withstand. A regilator that yields
small values of D corresponds to a system
that can be destabilized by a small stable
additive plant perturbation. It also follows
that constraining the maximum magnitude of
the closed—loop transfer function C/(l + PC)
to he at most l/Dmm is equivalent to con
straining the additive plant perturbation tol
erance D to be at least Dmi“. For the p = 10'4 LQG optimal regulator,
the peak of the closedloop transfer function
C/(l + PC) is 83.02; so D = 0.0121.
Hence, there are stable plant perturbations
AP that destabilize the system in Fig. 6, with
HAPIIm as small as 0.0121. One destabiv
lizing perturbation is shown for which
IIAPIIW = 1/80, which is greater than D. APO _ 0.2152 x 1040 W z)
“ v 22 v 0.985z + 0.9801 This AP destabilizes the system in Fig. 6.
The frequency—response magnitude of AP is
plotted in Fig. 7. The transfer function AP
might represent a mechanical resonance,
which was ignored when the plant P was
modeled for the LQG design. In general, the robustness requirements
that M and D be large are independent. A
system may have good margins (i.e., large
M) but be quite sensitive to additive plant
perturbations (i.e., small D) and vice versa. .0
_. tude of A P (expj Q)
o
9 0.001 o Magn '00010 0.5 1 1.5 2 2.5 3 Frequency .0 Fig. 7. Frequency response of a
destabilizing AP. IEEE Control Systems Magozme Tradeoff Curves Involving
Noise Sensitivity and Robustness We wish to minimize the noise sensitivity
of the system and determine how this trades
otf with the two different robustness require—
ments described in the previous section. Spe
ciﬁcally, how does the noise sensitivity .I
trade off against M when we require D 2
D,,,,,,, and how does it trade off against D
when we require M 2 Mmm? The design problem can be expressed
mathematically as Jmin 2 min J
thuimp
l/M= H l/(l + PC) It , s l/Mm,"
l/D:iC/(1+PC)H, 5 no... Unlike LQG, no exact analytical solution
of this minimization problem is known, al
though some work has been done [8]. Never
theless, this minimization problem can be
solved numerically since it can be cast as an
(inﬁnitedimensional) convex optimization
problem [2]. In general, the optimizing con—
trollers are of high order. In general, no
method is known for ﬁnding low»order con~
trollers that achieve close to the optimal per
formance. For three ﬁxed values of Dmm, the trade—
off between 1",,“ and l/Mmin is shown in Fig.
8. The p = 10—4 LQG regulator is also
shown. Since this regulator achieves l/D =
8302, it lies below the I/Dmin = [0 curve
and on the l/Dmin = 83.02 curve. All trade
off curves with l/Dm,n 2 83.02 will pass
through the LQG performance point and be
horizontal to the right of it. Note the interesting fact that by allowing
the Mcircle radius to be less than 0.5, only
modest improvement in the noise response
is gained, with the same tolerance D to ad—
ditive plant perturbations. For four ﬁxed values of Mmm, the trade
ol’f between Jmi“ and l/Dm,n is shown in Fig.
9. The p = 10—4 LQG regulator is also 6 \\
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A” i \ \\1/0510
.3» ,2,
2 \\\1/a:_302
LQG
1
0 _. 1.5 2 2.5 3 3.5
1/M Fig. 8. Tradeoff between rms noise and
M—circle radius. January V989 6 #__,V 2v 2
\
5 \\ \
\\\ \\\ t/M <15
4 \ \\ \\ ’ 22ml
\ \\. 1/M <17
«IN \ \ x ,2 2*, 777 7 7
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2 LQG (
1 . 0O 170 20 30 40 50 60 70 80 90100
1/D Fig. 9. Trade—off between rms noise and
additive plant sensitivities. shown. Since this regulator achieves l/M =
3.34. it lies below the l/Mm,n : 2.0 curve
and on the l/Mmm = 3.34 curve. All trade
off curves with l/Mmin 2 3.34 will pass
through the LQG performance point and be
horizontal to the right of it. From Fig. 9 we can draw some interesting
conclusions. Consider the UM S 3.34
curve. which corresponds to regulators that
yield the same or larger Mcircle radius as
the p = 10—4 LQG regulator. The curve is
relatively ﬂat for [ID 2 20, meaning that D
can be increased to about 0.05 with a rela
tively small increase in ms noise response
and the same or larger Mcircle radius (0.3).
For the p = 10‘4 LQG regulator, this rep—
resents an increase in additive plant perturv
bation tolerance D of a factor of 4. Of course, all regulators that stabilize P
yield a noise sensitivity J 2 JLQG, so that
all curves lie on or above the horizontal
asymptote J 1130 = 2.117. Imposing further
constraints on the two measures of robust
ness naturally will increase the minimum
noise sensitivity J achievable with linear
time»invariant stabilizing regulators. What is
neither intuitively obvious nor analytically
computable is how much the minimum noise
sensitivity J must increase when we impose
various constraints on the two measures of
robustness. Figures 8 and 9 show this trade—
oﬂ" precisely. Let us give two examples of speciﬁc con
clusions we may draw from Figs. 8 and 9.
First, the following design goals can be
achieved with a linear time—invariant regu
lator that stabilizes P (this point is marked
“x" in Fig. 9). 11/2 s 3, M 2 0...
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