AIAA 2006-6347
AIAA Guidance, Navigation, and Control Conference and Exhibit
21 - 24 August 2006, Keystone, Colorado
Are Phase and Time-delay Margins Always Adversely
Affected by High-Gain?∗
Chengyu Cao, Vijay V. Patel, C. Konda Reddy, Naira Hovakimyan
Dept. of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24061, USA
Eugene Lavretsky, Kevin Wise
Phantom Works, The Boeing Company
This paper addresses the important issue of robustness for design of feedback control
systems. A well-known fact in robust control is that a high gain in the feedback loop leads
to increased control effort and reduced phase margin. In an adaptive control framework,
a high adaptive gain, implying fast adaptation, often leads to undesirable high-frequency
oscillations in the control signal and sensitivity to time-delays. Since adaptive controllers
are nonlinear, the notion of the phase margin cannot be defined for these architectures.
A more generalized notion is the time-delay margin that can serve as a measure for the
system robustness. In this paper, we consider a linear system under constant disturbance
in the presence of some type of a high-gain in the feedback loop. We explore two different
adaptive control architectures, conventional model reference adaptive control (MRAC)
and an L1 adaptive controller. Since the closed-loop retains the linear structure, one can
explicitly compute the corresponding gain and phase margins. We further consider the
time-delay margin of these feedback structures and reveal that the classical definition of
the time-delay margin does not hold for the L1 adaptive controller. Moreover, while the
phase margin of the L1 adaptive controller is independent of the adaptation rate, its timedelay margin is guaranteed to be bounded away from zero as one increases the speed
of adaptation. The message is twofold: first, the time-delay margin cannot be related
to the phase margin straightforwardly and therefore constitutes a broader concept for
measuring system robustness; secondly, high gain can improve robustness if it is internal
to the controller computation block. For the sake of completeness, we present also two nonadaptive systems and generalize this phenomenon to a different class of feedback controllers.
I.
Introduction
Direct Model Reference Adaptive Control (MRAC) schemes have proven to be extremely useful in a
number of flight tests for recovering nominal performance in the presence of modeling and environmental
uncertainties (see Ref.1, 2 and references therein). A major challenge in analysis of these systems is determining their stability margins, which is directly dependent upon the adaptive control learning rate. Today this
analysis largely relies on the numerical sensitivity provided by Monte-Carlo analysis. It has been observed
that increasing learning rate leads to high-frequency oscillations in the control signal and reduces the system
tolerance to the time-delay in the control and the sensor channels. Recent Refs.3, 4 have introduced a new
paradigm for the design of adaptive controllers, the resulting architecture being named L1 adaptive control.
The L1 adaptive control architectures adapt fast, leading to desired transient performance with analytically
computable performance bounds, and therefore provide the right framework for verification and validation
∗ This material is based upon work supported by the United States Air Force under Contracts No. FA9550-05-1-0157 and
FA9550-04-C-0047.
Copyright © 2006 by Chengyu Cao, Vijay V. Patel, C. Konda Reddy, Naira Hovakimyan, Eugene Lavretsky. Published by the American Institute of Aeronautics and Astronautics, Inc., with
of adaptive systems. Moreover, as demonstrated in Ref.,5 the L1 adaptive control architectures have guaranteed stability margins, as opposed to conventional MRAC controllers. Since both of these architectures are
nonlinear, and as of today there is no systematic design theory available for MRAC, the comparison between
the performance and the stability margins of these two controllers cannot be pursued systematically. Therefore, in this paper, we consider a linear system under constant disturbance such that its closed-loop retains
a linear structure in the presence of both MRAC and L1 type controllers, which reduce to the well-known
PI structure and a filtered version of it, respectively. Consequently, the phase margin and the time-delay
margin can be directly computed using standard tools and definitions. The relationship between these two
margins is given by
phase margin
time delay margin =
,
(1)
crossover frequency
which is a valid definition for a MRAC type of linear PI controller. This relationship implies that a
small/decreasing time-delay margin can be caused by a decreasing phase margin and/or an increase in
the loop gain crossover frequency. These small time-delay margins have been observed in practice in the
development of MRAC systems if the adaption gain is too large.1, 2 Thus, if one attempts to adapt fast, the
MRAC system displays little to no time-delay margin.
On the contrary, the phase margin and the time-delay margin of the L1 adaptive controller do not display
this characteristic. By explicitly incorporating a time-delay in the relevant transfer functions, we observe
that the time-delay margin of the L1 adaptive control architecture is guaranteed to stay bounded away from
zero in the presence of fast adaptation. In the absence of any time-delay its phase margin is independent of
the adaptation gain. This fact, first of all, invalidates (1) for general LTI systems, and, with that, reveals
that the phase margin cannot be used to assess the sensitivity of the closed-loop system to time-delays.
As the subsequent analysis proves, the fast adaptation ability of the novel L1 adaptive control architecture
improves robustness, as opposed to conventional MRAC. For a complete picture, we first revisit the nonadaptive control architecture, introduced in Refs.,6, 7 for control of nonaffine systems via fast dynamics. We
analyze the similarities of these two closed-loop systems, and using a scalar linear system that has linear
closed-loop response, we demonstrate the results of our observations.
The rest of the paper is organized as follows. In Section II, we introduce first the non-adaptive control
architecture from Refs6, 7 and compare it to linear high-gain feedback structure in terms of its performance
and robustness. Section III compares the two different adaptive control architectures, MRAC and L1 , and
analyzes their robustness with respect to the speed of adaptation. Conclusions are summarized in Section
IV.
II.
Non-adaptive Feedback Schemes
In this section, we introduce two non-adaptive control schemes leading to the same closed-loop transfer
function, albeit having different robustness properties.
II.A.
High-gain Feedback with Command Shaping Module
Consider a linear system:
x(s) = h(s)u(s) ,
(2)
where x(s) and u(s) are Laplace transformations of system’s scalar input u(t) and scalar output x(t) signals.
We further assume that a tracking controller is defined via the following proportional gain-feedback:
u(s) = K(r(s) − x(s)) ,
(3)
where K ∈ R, and r(s) is the reference input of interest to track.
High-gain feedback controller is the one in (3) with large K that achieves satisfactory tracking of r(t).
To avoid the adverse effects of fast rate and large amplitude due to a large K, a common approach is to
introduce a command shaping module for r(t), which leads to a redefined control law:
u(s) =
K(C(s)r(s) − x(s)) ,
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(4)
d(s)
r(s)
u(s)
+
C(s)
+
x(s)
h(s)
K
xd (s)
+
_
+
+
n(s)
xn(s)
Figure 1. Feedback System
where C(s) is a strictly proper stable system with C(0) = 1. The block diagram of this system is shown in
Figure 1, with possible output disturbance d(t) and measurement noise n(t).
II.B.
Control of Nonaffine Systems via Fast Dynamics
In Ref.,6 we consider a simple single-input system like
ẋ = f (x, u),
x(0) = x0 ,
t ≥ 0,
(5)
where x ∈ R is the system state, u ∈ R is the control input, and f cannot be inverted in terms of elementary
∂f
functions. Assuming that f is Lipschitz continuous function of its arguments, and that ∂u
is bounded away
from zero for (x, u) ∈ Ωx × Ωu ⊂ R × R, where Ωx , Ωu are compact sets, fast dynamics can be introduced to
invert the system in (5) as follows:
µ ¶
∂f
²u̇ = −sign
(f (x, u) + ax), a > 0, ² ¿ 1.
∂u
Refs.6, 7 present the complete framework for single-input and multi-input nonaffine systems and refer to
Tikhonov’s theorem for stability proofs. Decreasing ² in this framework leads to better tracking performance.
II.C.
Phase Margin, Time-Delay Margin and Sensitivity Analysis for A High-Gain
Feedback System
In this section, we analyze the phase and the time-delay margins, as well robustness to measurement
noise and external disturbance, in the presence of static high-gain feedback in the loop. First, we introduce
some notations. Let Ho (s) be the open-loop transfer function for breaking the loop at the plant input. Also,
define
Hc (s)
,
Hd (s)
,
Hn (s)
,
xd (s)
,
r(s)
xd (s)
,
d(s)
xd (s)
,
n(s)
(6)
(7)
(8)
where r(s) is the reference input, x(s) is the linear system’s output, d(s) is some type of deterministic
output disturbance, n(s) models the measurement noise, xd (s) = x(s) + d(s) is the actual system state in
the presence of output disturbance, and xn (s) = xd (s) + n(s) is the state measurement corrupted by the
measurement noise. We note that Hc (s) is the closed-loop transfer function, while Hd (s) and −Hn (s) are
the sensitivity and complementary sensitivity functions, respectively.8 The signals x(s), xs (s), xn (s) are
illustrated in Figure 1.
Consider the following linear system
x(s)
1
=
(9)
u(s)
s(s + 1)
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and let the control objective be to achieve the following closed-loop system response
Wm (s) =
s2
k0
,
+ s + k0
where k0 = 2. We apply the controller from (4), taking into consideration the disturbance and the noise:
u(s) =
−K(xn (s) − C(s)r(s)) ,
k0
.
s2 + s + k0
C(s) =
(10)
(11)
For the system in (9) and the high-gain feedback controller with command shaping module in (10)-(11), we
can easily derive:
Ho (s) =
Hc (s) =
Hd (s) =
Hn (s) =
K
,
s(s + 1)
(12)
k0 K
,
(s2 + s + k0 )(s2 + s + K)
s2 + s
,
2
s +s+K
K
− 2
.
s +s+K
(13)
(14)
(15)
In the presence of large K in the feedback path, we have the following approximation of the closed-loop
transfer function:
k0
Hc (s) ≈ 2
.
s + s + k0
Equation (12) is used to calculate the phase margin. Bode plots for various values of K are presented in
Figure 2. We see that the phase margin decreases while the cross-over frequency increases as the feedback
gain K is increased.
Magnitude (dB)
Bode PK=1(P
lot
m=51.80,
0.786 rad/sec)
K=2(Pm=38.70, 1.25 rad/sec)
100
K=100(Pm=5.720, 9.97rad/sec)
50
0
_ 50
Phase (deg)
_ 100
_ 90
_ 135
_ 180
10-
2
10-
1
0
1
10
Frequency
10
2
10
(rad/sec)
Figure 2. Bode Plot for Ho (s)
In the presence of a time delay τ > 0 in the plant input, the open-loop transfer function is given by
Ho (s) =
K
e−sτ
s(s + 1)
(16)
From Equation (16), it can be seen that the time-delay margin is the phase margin divided by the cross-over
frequency as given by (1). It is evident that the time-delay margin decreases as K increases.
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The frequency responses for sensitivity and complementary sensitivity functions are plotted in Fig. 3. It
can be seen from Fig. 3 that as K increases, the bandwidth of the complementary sensitivity function Hn (s)
increases along with a resonance peak. Similar increase in the resonance peak is observed in the sensitivity
function Hd (s). Recalling that the performance of the sensitivity function Hd (s) is related to the closedloop system’s external disturbance rejection ability, while the performance of the complementary sensitivity
function Hn (s) is related to the closed-loop system’s measurement noise rejection ability, we conclude that
all these properties in the system are adversely affected by the high-gain K.
a (High Gain)
b
30
20
20
0
10
−20
0
−10
−40
−20
−60
−30
−40
−80
−50
K=1
2
100
−100
−60
−70 −2
10
0
10
Frequency (rad/s)
2
10
−120 −2
10
0
10
Frequency (rad/s)
2
10
Figure 3. (a) Sensitivity Hd (s) and (b) Complimentary Sensitivity Hn (s) for high gain feedback system.
II.D.
Phase Margin, Time-Delay Margin and Sensitivity Analysis for A System
with Fast Dynamics
We now apply fast dynamics to invert the same system in (9). Let the desired reference system be:
xr (s) = k0 (r(s) − xn (s)) .
(17)
²u̇(t) = −(u(t) − xr (t)).
(18)
Fast dynamics can be chosen as
For the system in (9) and the controller in (17)-(18), we have the following four transfer functions
2
,
s(s + 1)(²s + 1)
2
Hc (s) =
,
3
²s + (² + 1)s2 + s + 2
Hd (s) = 1 − Hc (s) ,
Hn (s) = −Hc (s) .
Ho (s)
=
(19)
(20)
(21)
(22)
The block-diagram of this system is shown in Figure 4. We notice that fast dynamics is in fact a high-gain
design from xr (s) to u(s), as illustrated in Figure 4. When ² is small, we have the following approximation
of the closed-loop transfer function:
k0
.
Hc (s) ≈ 2
s + s + k0
Equation (19) for Ho (s) serves to find the phase margin. Figure 5 shows the Bode plots for various ²’s. It
can be seen from the Bode plots that the phase margin is almost invariant with the variation in ².
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d(s)
r(s) +
+
_
k0
1
1
²
_
+
x(s)
u(s)
h(s)
s
xd (s)
+
+
+
n(s)
xn (s)
Figure 4. Feedback System with Fast Dynamics
Magnitude (dB)
100
0
_ 100
_ 200
_ 300
_ 90
²
²
²
Phase (deg)
_ 135
_ 180
=0.01(PM=380)
=0.001(PM=38.60)
=0.0001 ((PM=38.70)
_ 225
_ 270
_2
10
0
10
2
10
Frequency (rad/sec)
4
10
6
10
Figure 5. Bode Plot of the Feedback System with Fast Dynamics
The frequency responses for sensitivity and complementary sensitivity functions are plotted in Figure
6. It can be seen from Fig. 6 that both sensitivity and complementary sensitivity functions are almost
independent of ².
a (Fast Dynamics)
b
10
20
5
0
0
−20
−5
−40
−10
−60
−15
−80
−20
−100
−25
−120
−30
−35 −2
10
0
10
Frequency (rad/s)
2
10
−140 −2
10
ε=0.01
0.001
0.0001
0
10
Frequency (rad/s)
2
10
Figure 6. (a) Sensitivity Hd (s) and (b) Complimentary Sensitivity Hn (s) for feedback system with fast dynamics.
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For this system, the open-loop transfer function in the presence of a delay in the plant input is given by
Ho (s) =
2
e−sτ
s(s + 1)(²s + 1)
(23)
From Equation (23), we see that the usual relation between the phase margin and the time-delay margin
given by (1) holds. Therefore, as the phase margin and the cross-over frequency are invariant with ², so is
the time-delay margin.
III.
Adaptive Schemes
In this section, we introduce the conventional MRAC and the novel L1 adaptive control architectures from
Refs.3, 4 In section III.C, we apply both controllers to a first order linear system with constant disturbance
so that both closed-loop MRAC and L1 degenerate into linear systems. First, using classical frequency
domain tools, we analyze the phase margin of both systems. For the MRAC architecture, the time-delay
margin can be computed from its open-loop transfer function, using the classical definition in (1). For the L1
architecture, the phase margin appears to be independent of the adaptation rate. However, in the presence
of time-delay the relationship in (1) no longer holds, and therefore one cannot conclude that its time-delay
margin is independent of the adaptation rate. Next, in analysis of the time-delay margin of L1 adaptive
control architecture, we reveal that the time-delay margin is guaranteed to stay bounded away from zero
while increasing the adaptation rate, as opposed to MRAC. The general nonlinear theoretical result for the
L1 adaptive control architecture is summarized in Ref.5
III.A.
MRAC
Consider the following single-input single-output system dynamics:
ẋ(t) = Am x(t) − bθ> x(t) + bu(t),
x(0) = x0
>
y(t) = c x(t) ,
(24)
where x ∈ Rn is the system state vector (measurable), u ∈ R is the control signal, b, c ∈ Rn are known
constant vectors, Am is a given n×n Hurwitz matrix, y ∈ R is the regulated output, and unknown parameter
θ belongs to a given compact convex set θ ∈ Θ. Let
ẋm (t) = Am xm (t) + bkg r(t),
xm (0) = x0 ,
>
ym (t) = c xm (t)
(25)
be the desired reference system, where xm ∈ Rn , Am is the same as in (24), kg is a design gain
kg = lim
s→0
and
1
1
= >
,
c> Ho (s)
c Ho (0)
Ho (s) = (sI − Am )−1 b .
(26)
(27)
Following standard Lyapunov arguments, one can prove that the following controller
u(t)
˙
θ̂(t)
= θ̂> (t)x(t) + kg r(t),
=
>
ΓProj(θ̂(t), x(t)e (t)P b) ,
(28)
θ̂(0) = θ̂0 ,
(29)
where θ̂(t) ∈ Rn is the vector of the adaptive parameters, Proj(·, ·) denotes the projection operator,9 e(t) =
xm (t)−x(t) is the tracking error, Γ ∈ Rn×n is a positive definite matrix of adaptation gains, and P = P > > 0
is the solution of the algebraic equation A>
m P + P Am = −Q for arbitrary Q > 0, ensures that lim e(t) = 0.
t→∞
The MRAC architecture is shown in Fig. 7.
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r
Adaptive Control
Reference
Model
u
xm
x
System
+
- e
µ^
Adaptive Law
Figure 7. MRAC architecture
III.B.
L1 Adaptive Controller
For the linearly parameterized system in (24), we consider the following companion model
˙
x̂(t)
= Am x̂(t) + b(u(t) − θ̂> (t)x(t)) ,
x̂(0) = x0
>
ŷ(t) = c x̂(t) .
(30)
The adaptive law for θ̂(t) is given by
˙
θ̂(t) = ΓProj(θ̂(t), x(t)x̃> (t)P b),
θ̂(0) = θ̂0 ,
(31)
where x̃(t) = x̂(t) − x(t) is the tracking error, Γ ∈ Rn×n = Γc In×n is the matrix of adaptation gains, and P
is the solution of the algebraic equation A>
m P + P Am = −Q, Q > 0. Letting
r̄(t) = θ̂> (t)x(t),
(32)
we consider the following filtered adaptive controller:
¡
¢
u(s) = C(s) r̄(s) + kg r(s) ,
(33)
where u(s), r̄(s), r(s) are the Laplace transformations of u(t), r̄(t), r(t), respectively, C(s) is a stable and
strictly proper system with low-pass gain C(0) = 1, and kg is the same as in (26).
Letting
n
X
θmax = max
|θi | ,
θ∈Ω
i=1
th
where θi is the i element of θ, the complete L1 adaptive controller consists of (30), (31), (33) subject to
the following L1 stability criterion:
kHo (s)(1 − C(s))kL1 θmax < 1 ,
where kHo (s)(1 − C(s))kL1 is the L1 gain of stable transfer function Ho (s)(1 − C(s)).3, 4
We define a linear time-invariant reference system using the non-adaptive version of (33)
¡
¢
xref (s) = Ho (s) kg C(s)r(s) + (C(s) − 1)θ> xref (s)
¡
¢
uref (s) = C(s) kg r(s) + θ> xref (s)
The main result in Refs.3, 4 is given by
lim (x(t) − xref (t)) =
0,
∀t ≥ 0 ,
lim (u(t) − uref (t)) =
0,
∀t ≥ 0 .
Γ→∞
Γ→∞
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(34)
(35)
We notice that when C(s) = 1, uref (t) reduces to the following ideal controller
uideal (t) = kg r(t) + θ> xref (t)
(36)
and (34) reduces to (25) by cancelling the uncertainties exactly. We note that control law uref (t) is not
implementable since its definition involves the unknown parameters. However, the L1 adaptive controller
ensures x(t) and u(t) track the state xref (t) and the control signal uref (t) of this reference system both
in transient and steady-state, if the adaptation rate is sufficiently fast. In Ref.,4 we further provide design
guidelines for selection of C(s) to ensure that the reference system in (34) can track a desired reference
system arbitrarily closely both in transient and steady state, and these performance bounds are true for
system’s both signals, input and output, simultaneously.
The L1 adaptive control architecture is illustrated in Fig. 8, and its complete design and analysis
framework is developed in Refs.3, 4 The L1 adaptive controller and its variants have been used for control of
wingrock,10 aerial refueling,11 and also flight tested on a miniature unmanned air vehicle.12
Companion
Model
u
x
^
Optional
Feedback K
Low-pass
Filter
C(s)
System
r
x
+
- e
Adaptive Control
µ^
Adaptive Law
Figure 8. Closed-loop system with L1 adaptive controller
III.C.
Phase Margin, Time-Delay Margin and Sensitivity Analysis for MRAC and
L1 adaptive control architectures
Since both MRAC and L1 adaptive controllers are nonlinear controllers, a straightforward analysis for
the phase margin is not available, nor can the sensitivity or complementary sensitivity transfer functions be
derived. To investigate the effect of fast adaptation on the stability/robustness margins of the closed-loop
system, we apply both controllers to the following linear system:
ẋ(t) = ax(t) + θ + u(t) ,
(37)
where a > 0, θ ∈ R is an unknown parameter which belongs to a compact set Θ, x(t) is the state and u(t) is
the control signal. We further let r(t) be the continuous bounded reference signal of interest to track, d(t)
be the output disturbance, and n(t) be the measurement noise. We introduce both MRAC and L1 adaptive
controllers for the system in (37). We define the control signal as:
u(t) = kx(t) + uad (t)
where k = am − a, and uad (t) is the adaptive control signal to be defined shortly. The system in (37) can
therefore be rewritten as
ẋ(t) = am x(t) + θ + uad (t) ,
(38)
so that both MRAC and L1 adaptive controllers can be straightforwardly applied.
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III.C.1.
Phase Margin, Time-Delay Margin and Sensitivity Analysis for MRAC
The MRAC controller for this system is given by:
ẋm (t) = am xm (t) + kg r(t) ,
˙
θ̂(t) = Γ(xn (t) − xm (t)) ,
(39)
u(t) = kxn (t) − θ̂(t) + kg r(t) ,
xn (t) = x(t) + d(t) + n(t) ,
where kg = −am is computed from (26), d(t) is the disturbance, while n(t) is the noise. For the system in
(37) and the MRAC controller in (39), we have
Ho (s)
=
Hc (s)
=
Hn (s)
=
Hd (s)
=
−ks + Γ
,
s(s − a)
−am
,
(s − am )
ks − Γ
,
2
s − am s + Γ
s2 − as
.
s2 − am s + Γ
(40)
(41)
(42)
(43)
The Bode plot for different adaptive gains is shown in Fig. 9. It shows that the phase margin decreases and
the cross-over frequency increases as the adaptation rate increases.
Note that in the presence of time-delay at the input or output of the system, the open-loop transfer
function for the time-delay margin analysis is
Ho (s) =
−ks + Γ −sτ
e
s(s − a)
(44)
In this case, the classical definition of the time-delay margin (1) holds. As the phase margin decreases and
the cross-over frequency increases with the increase of Γ, the time-delay margin decreases.
Γ=1 (PM = 45.80 at 1.82 rad/sec)
100
Γ=10 (PM=17.90 at 3.41 rad/sec)
Magnitude (dB)
0
Γ=100 (PM=5.72 at 3.41 rad/sec)
50
0
−50
Phase (deg)
−100
−90
−135
−180
−225
−270
−2
10
−1
10
0
10
1
10
2
10
3
10
Frequency (rad/sec)
Figure 9. Time-delay margin decreases as Γ increases for MRAC.
The frequency responses for sensitivity and complementary sensitivity functions are plotted in Figure
10. The simulation parameters are a = 1, am = −1 and c = 1. It can be seen from Fig. 10 that as Γ
increases, the bandwidth of the complementary sensitivity function Hn (s) increases along with a resonance
peak. Similar increase in resonance peak is observed in the sensitivity function Hd (s). These observations
are consistent with the similar ones in the case of high-gain feedback.
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20
20
10
10
0
0
50
100
10
¡=1
-10
-10
Mag (dB)
¡=1
-20
-20
-30
-30
10
-40
-40
50
-50
-50
100
-60
-60
-70
-70
-80 -2
10
0
10
Frequency(rad/s)
10
2
-80
-2
10
(a)
0
10
Frequency(rad/s)
10
2
(b)
Figure 10. (a) Sensitivity Hd (s) and (b) Complimentary Sensitivity Hn (s) for MRAC when τ = 0.
III.C.2.
Phase Margin, Time-Delay Margin and Sensitivity Analysis for L1
The L1 adaptive controller for the system in (38) is:
˙
x̂(t)
= am x̂(t) + θ̂(t) + uad (t) ,
˙
θ̂(t) = Γ(xn (t) − x̂(t)) ,
(45)
u̇ad (t) = c(−θ̂(t) − uad (t) + kg r(t)) ,
c
where C(s) = s+c
, c > 0 is the low-pass filter, kg = −am is computed from (26). For the L1 adaptive
controller, we have
Ho (s)
Hc (s)
Hn (s)
Hd (s)
−k
,
s−a
C(s)
= −am
,
s − am
k(s2 − am s + Γ) − Γ(s − a)C(s)
=
,
(s − am )(s2 − am s + Γ)
(s − a)(s2 − am s + Γ(1 − C(s)))
.
=
(s − am )(s2 − am s + Γ))
=
(46)
(47)
(48)
(49)
Equation (46) defines the open-loop transfer function for determination of the phase margin of the closedloop system. It can be seen that it is independent of Γ. The Bode plot of Ho (s) is plotted in Figure 11.
In the presence of time delay in the input to the plant, the open-loop transfer function is given by
Ho (s) =
−k −sτ
ΓC(s)
e
+ 2
(e−sτ − 1)
s−a
s − am s + Γ
(50)
Equation (50) is the open-loop transfer function for determination of the time-delay margin. We can see
that while the phase margin is independent of Γ, the time-delay margin is not. Since (46) can be derived
from (50) by setting τ = 0, the second term in (50), which is a function of Γ, disappears, rendering the phase
margin independent of Γ. This special structure of the L1 adaptive control architecture is conditioned by
the fact that the control input enters both the plant and the companion model. However, the delay affects
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Bode Diagram
Gm = −6.02 dB (at 0 rad/sec) , Pm = 60 deg (at 1.73 rad/sec)
Magnitude (dB)
10
0
−10
−20
−30
Phase (deg)
−40
−90
−135
−180
−2
10
−1
10
0
10
Frequency (rad/sec)
1
2
10
10
Figure 11. Phase Margin is independent of Γ for L1 Adaptive System
only the input to the plant, while the companion model is internal to the control computation block. Thus,
the relation between the time-delay margin and the phase margin in (1) no longer holds for the L1 system.
The frequency domain responses for sensitivity and complementary sensitivity functions are given in Fig.
12. The simulation parameters are a = 1, am = −1 and c = 1. It can be seen that the bandwidth of Hn (s)
increases with Γ, and the resonance peak decreases as Γ increases. There is no increase in the resonance
peak for Hd (s). This implies that with the L1 adaptive control architecture, the closed-loop system retains
good external disturbance rejection ability in addition to good tracking performance. We also notice that
as we increase Γ, the measurement noise tolerance of this architecture improves. Recall that for MRAC we
had exactly the opposite observations.
10
10
¡=1
10 50
5
0
0
¡=1
-5
Mag (dB)
10
50
5
100
100
-5
-10
-10
-15
-15
-20
-20
-25
-30
-25
-35
-30
-40 -2
10
0
10
Frequency(rad/s)
10
2
-35
10
(a)
-2
0
10
Frequency(rad/s)
10
2
(b)
Figure 12. (a) Sensitivity Hd (s) and (b) Complimentary Sensitivity Hn (s) for L1 when τ = 0.
Using the open-loop transfer function in (50), the following characteristic equation can be obtained for
determination of the time-delay margin
cΓ(s − a)(e−sτ − 1) + (s − a − ke−sτ )(s + c)(s2 − am s + Γ) = 0 ,
(51)
the left-side of which is the numerator of 1 + Ho (s) of (50). To determine the stability boundaries, we set
s = iω, ω ∈ R in (51). Separating the real and imaginary parts and setting them identically equal to zero
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gives
α sin ωτ + β cos ωτ = f
β sin ωτ − α cos ωτ = g,
(52)
where
α = cam Γ + (am − c)kω 2
β = ω(am kc + cΓ − kΓ + kω 2 )
f = −a(am c − Γ)ω − (a + am − c)ω 3
(53)
g = −ω 2 (−aam + (a + am )c − Γ + ω 2 )
Eliminating τ from (52) gives
f 2 + g 2 − α2 − β 2 = p4 + µ3 p3 + µ2 p2 + µ1 p + µ0 = 0
(54)
where p = ω 2 and the coefficients µi = µi (Γ, a, am , c). Suppose that ω ∗ is a positive root of (54). Then,
equation (52) gives the time-delay margin as
τ∗ =
1
αf + βg
.
tan−1
ω∗
βf − αg
(55)
Since eq.(54) cannot be solved in general with respect to ω, we consider the limiting case as Γ → ∞. In that
case eq. (54) reduces to
p2 + (am − c)(2a − am + c)p − a2m Γ2 ≈ 0.
The above is a quadratic equation. Since a2m Γ2 > 0, there is one positive root
p
−(am − c)(2a − am + c) + (am − c)2 (2a − am + c)2 + 4a2m Γ2
∗
∗2
p =ω ≈
.
2
Also,
(56)
αf + βg
ω ∗ 2 (c − k) + cam a
= ∗
Γ→∞ βf − αg
ω (a(c − k) − cam )
lim
Thus, in the limiting case, the time-delay margin is given by
lim τ ∗ =
Γ→∞
∗2
1
−1 ω (c − k) + cam a
tan
.
ω∗
ω ∗ (a(c − k) − cam )
(57)
The important thing to note is that unlike MRAC where the time-delay margin goes to zero as Γ tends to
infinity, the time-delay margin for L1 adaptive controller approaches a constant value, bounded away from
zero, given by (56) and (57).
Figure 13 compares the time-delay margins for MRAC and L1 . The simulation parameters are a =
1, am = −1 and c = 1. It can be clearly seen that for large values of Γ, the time-delay margin for MRAC
approaches zero whereas the time-delay margin for L1 approaches 0.4. This is indeed the value predicted by
the analytical result given by (56) and (57). This is also consistent with the general theoretical result for
the nonlinear closed-loop system with the L1 adaptive architecture, presented in Ref.5
IV.
Conclusion
This paper addresses robustness of four different control architectures, two of them being adaptive.
Through several comparison studies, we reveal that fast adaptation does not hurt the robustness/stability
margins of the L1 adaptive control architecture, while the same is not true for conventional MRAC. Similar
observations are made for two non-adaptive schemes, namely that the same type of closed-loop transfer
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0.7
0.6
L1
0.5
0.4
¿
0.3
0.2
MRAC
0.1
0
0
50
100
150
200
¡
Figure 13. Comparison of the time-delay margin for MRAC and L1 dependent upon Γ.
function obtained via static high-gain feedback loses its robustness, while if the feedback is generated via
fast dynamics, then the robustness is not affected. Table 1 summarizes these relations for the control
architectures considered in this paper. For MRAC and L1 adaptive controller, we specialize Eqs. (40)-(44)
and Eqs. (46)-(50), respectively, for the case when am = −1, a = 1 and k = (am − a) = −2. We notice that
the fact that the time-delay margin of the closed-loop system with fast dynamics is not adversely affected by
the high-gain is a consequence of the fact that its phase margin (in the absence of time-delay) is not affected
by the high-gain. The time-delay margin and the phase margin in this case have the same type of functional
dependence upon the gain ² of the controller. The situation is different with the adaptive schemes. For the
L1 adaptive controller, the phase margin is independent of Γ, while the time-delay margin is not, and this
change is conditioned by the structure of the open-loop transfer function of this architecture. Thus, while the
two architectures, the closed-loop system with fast dynamics and L1 adaptive control, have good robustness
features, the reasons for this are different. The commonality is that in both cases the controller gain becomes
internal to the controller computational block, and therefore it is not affecting robustness directly, while the
difference is that with the L1 adaptive controller, due to the additional estimation block, the structure of
the open-loop transfer function changes, which renders its phase margin independent of the gain Γ, while
the time-delay margin cannot be concluded from this.
Feedback
High Gain
Fast Dynamics
MRAC
L1
Ho (s) (τ = 0)
K
s(s+1) :
2
s−1 :
Ho (s) (τ 6= 0)
Phase Margin
2
−sτ
: Cross−over
s(s+1)(²s+1) e
Frequency
not affected by ²
2s+Γ −sτ
:
s(s−1) e
affected by Γ
independent of Γ
Time-delay margin
Phase Margin
K
−sτ
: Cross−over
s(s+1) e
Frequency
affected by K
2
s(s+1)(²s+1) :
2s+Γ
s(s−1) :
Phase margin
2
−sτ
s−1 e
+
Phase Margin
Cross−over Frequency
ΓC(s)
−sτ
s2 +s+Γ (e
− 1): 6=
Table 1. Time-delay margin for different control schemes
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Phase Margin
Cross−over Frequency
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