function for µ = 0 , 0 . 01 , 0 . 05 , 0 . 1, respectively. Comment on your simulation results.
9.3 In Problem 9.2, the relative degree of the overall plant transfer function is
n∗ = 1 when µ = 0. Assume that the reference model is chosen to be
Wm 1( s) = am . Discuss the consequences of designing an MRAC scheme
s+ am
for the full order plant using Wm 1( s) as the reference model. Simulate the
closed-loop scheme using the values given in part (c) of Problem 9.2.
9.4 For the MRAC problem given in Problem 9.2,
(a) Choose one reference input signal r that is sufficiently rich but not dom-
inantly rich and one that is dominantly rich.
9.10. PROBLEMS
769
(b) Simulate the MRAC scheme developed in Problem 9.2 using the input
signals designed in (a).
(c) Compare the simulation results with those obtained in Problem 9.2.
Comment on your observations.
9.5 Consider the plant
1
yp =
u
s( s + a) p
where a is an unknown constant and
9
ym =
r
( s + 3)2
is the reference model.
(a) Design a modified MRAC scheme using Method 1 given in Section 9.4.2
(b) Simulate the modified MRAC scheme for different values of the design
parameter τ .
9.6 Consider Problem 9.2.
(a) Replace the standard robust MRAC scheme with a modified one from
Section 9.4. Simulate the modified MRAC scheme using the same pa-
rameters as in Problem 9.2 (c) and τ = 0 . 1.
(b) For a fixed µ (for example, µ = 0 . 01), simulate the closed MRAC scheme
for different τ .
(c) Comment on your results and observations.
9.7 Consider the speed control problem described in Problem 6.2 of Chapter 6.
Suppose the system dynamics are described by
b
V =
(1 + ∆
s + a
m( s)) θ + d
where d is a bounded disturbance and ∆ m represents the unmodeled dynam-
ics, and the reference model
0 . 5
Vm =
V
s + 0 . 5 s
is as described in Problem 6.2.
(a) Design a robust MRAC scheme with and without normalization
(b) Simulate the two schemes for a = 0 . 02 sin 0 . 01 t, b = 1 . 3, ∆ m( s) = − 2 µs µs+1
and d = 0 for µ = 0 , 0 . 01 , 0 . 2. Comment on your simulation results.
770
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
9.8 Consider the plant
1
y =
u − µ∆
s − a
a( s) u
where µ > 0 is a small parameter, a is unknown and ∆ a( s) is a strictly proper
stable unknown transfer function perturbation independent of µ. The control
objective is to choose u so that all signals are bounded and y tracks, as close
as possible, the output ym of the reference model
1
ym =
r
s + 1
for any bounded reference input r as close as possible.
(a) Design a robust MRAC to meet the control objective.
(b) Develop bounds for robust stability.
(c) Develop a bound for the tracking error e 1 = y − ym.
Repeat (a), (b), (c) for the plant
e−τs
y =
u
s − a
where τ > 0 is a small constant.
9.9 Consider the following expressions for the plant
yp = G 0( s) up + ∆ a( s) up
(9.10.1)
N
y
0( s) + ∆1( s)
p =
u
D
p
(9.10.2)
0( s) + ∆2( s)
where ∆ a, ∆1 , ∆2 are plant perturbations as defined in Section 8.2 of Chapter
8.
(a) Design a model reference controller based on the dominant part of the
plant given by
yp = G 0( s) up
(9.10.3)
where G
Zp( s)
0( s) = N 0( s) = k
satisfies the MRAC Assumptions P1 to
D
p
0( s)
Rp( s)
P4 and
Z
y
m( s)
m = Wm( s) r = km
r
Rm( s)
is the reference model that satisfies Assumptions M1 and M2 given in
Section 6.3 of Chapter 6.
(b) Apply the MRC law designed using (9.10.3) to the full-order plants given
by (9.10.1), (9.10.2) and obtain bounds for robust stability.
9.10. PROBLEMS
771
(c) Obtain bounds for the tracking error e 1 = yp − ym.
9.10 Consider the plant
yp = G 0( s) up + ∆ a( s) up
where
yp = G 0( s) up
is the plant model and ∆ a( s) is an unknown additive perturbation. Consider
the PPC laws given by (9.5.3), (9.5.5), and (9.5.7) designed based on the
plant model but applied to the plant with ∆ a( s) = 0. Obtain a bound for
∆ a( s) for robust stability.
9.11 Consider the plant
N
y
0 + ∆1
p =
u
D
p
0 + ∆2
where G 0 = N 0 is the modeled part and ∆
D
1 , ∆2 are stable factor perturba-
0
tions. Apply the PPC laws of Problem 9.10 to the above plant and obtain
bounds for robust stability.
9.12 Consider the robust APPC scheme of Example 9.5.1 given by (9.5.18), (9.5.19)
designed for the plant model
b
yp =
u
s + a p
but applied to the following plants
b
(i)
yp =
u
s + a p + ∆ a( s) up
b
+ ∆1( s)
(ii) y
s+ λ 0
p =
u
s+ a + ∆
p
s+ λ
2( s)
0
where ∆ a( s) is an additive perturbation and ∆1( s) , ∆2( s) are stable factor
perturbations and λ 0 > 0.
(a) Obtain bounds and conditions for ∆ a( s) , ∆1( s) , ∆2( s) for robust stabil-
ity.
(b) Obtain a bound for the mean square value of the tracking error.
(c) Simulate the APPC scheme for the plant (i) when b = 2(1+0 . 5 sin 0 . 01 t),
a = − 2 sin 0 . 002 t, ∆ a( s) = − µs
for µ = 0 , 0 . 1 , 0 . 5 , 1.
( s+5)2
9.13 Consider the robust APPC scheme based on state feedback of Example 9.5.2
designed for the plant model
b
yp =
u
s + a p
772
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
but applied to the following plants:
b
(i)
yp =
u
s + a p + ∆ a( s) up
b
+ ∆1( s)
(ii)
y
s+ λ 0
p =
u
s+ a + ∆
p
s+ λ
2( s)
0
where ∆ a( s) is an additive perturbation and ∆1( s) , ∆2( s) are stable factor
perturbations and λ 0 > 0.
(a) Obtain bounds and conditions for ∆ a( s) , ∆1( s) , ∆2( s) for robust stabil-
ity.
(b) Obtain a bound for the mean square value of the tracking error.
(c) Simulate the APPC scheme with plant (ii) when
( e−τs − 1)
b = 1 , a = − 2 sin 0 . 01 t, λ 0 = 2 , ∆1( s) =
, ∆
s + 2
2( s) = 0
for τ = 0 , 0 . 1 , 0 . 5 , 1.
9.14 Simulate the ALQC scheme of Example 9.5.3 for the plant
b
yp =
(1 + ∆
s + a
m( s)) up
where ∆ m( s) = − 2 µs and µ ≥ 0.
1+ µs
(a) For simulation purposes, assume b = − 1 , a = − 2(1 + 0 . 02 sin 0 . 1 t). Consider the following values of µ: µ = 0 , µ = 0 . 05 , µ = 0 . 2 , µ = 0 . 5.
(b) Repeat (a) with an adaptive law that employs a dead zone.
9.15 Consider the following MIMO plant
y 1
h
u
=
11( s)
h 12( s)
1
y 2
0
h 22( s)
u 2
where h 11 = b 1 , h
and ∆( s) is strictly proper and
s+ a
12 = µ∆( s) , h 22 =
b 2
1
s+ a 2
stable.
(a) Design a decentralized MRAC scheme so that y 1 , y 2 tracks ym 1 , ym 2, the
outputs of the reference model
y
1
m 1
0
r
=
s+1
1
ym 2
0
2
r
s+2
2
for any bounded reference input signal r 1 , r 2 as close as possible.
(b) Calculate a bound for µ∆( s) for robust stability.
9.10. PROBLEMS
773
9.16 Design a MIMO MRAC scheme using the CGT approach for the plant
˙ x = Ax + Bu
y = C x
where x ∈ R 2 , u ∈ R 2 , y ∈ R 2 and ( A, B, C) corresponds to a minimal state
representation. The reference model is given by
− 2
0
1
˙ xm =
x
r
0
− 2
m +
1
ym = xm
Simulate the MRAC scheme when
− 3
1
1 0
1 0
A =
, B =
, C =
0
0 . 2
0 2
0 1
Appendix
A
Swapping Lemmas
The following lemmas are useful in establishing stability in most of the adap-
tive control schemes presented in this book:
Lemma A.1 (Swapping Lemma A.1) Let ˜
θ, w : R+ → Rn and ˜
θ be
differentiable. Let W ( s) be a proper stable rational transfer function with a
minimal realization ( A, B, C, d) , i.e.,
W ( s) = C ( sI − A) − 1 B + d
Then
W ( s)˜
θ ω = ˜
θ W ( s) ω + Wc( s) ( Wb( s) ω )˙˜
θ
where
Wc( s) = −C ( sI − A) − 1 , Wb( s) = ( sI − A) − 1 B
Proof We have
t
W ( s)˜
θ ω = W ( s) ω ˜
θ = d˜
θ ω + C
eA( t−τ) Bω ˜
θdτ
0
τ
τ = t
= d˜
θ ω + C eAt
e−AσBω ( σ) dσ ˜
θ( τ )
0
τ =0
t
τ
−
e−AσBω ( σ) dσ ˙˜
θ( τ ) dτ
(A.1)
0
0
t
=
˜
θ
dω + C
eA( t−σ) Bω( σ) dσ
0
t
τ
−C
eA( t−τ)
eA( τ−σ) Bω ( σ) dσ ˙˜
θ( τ ) dτ
0
0
774
A. SWAPPING LEMMAS
775
Noting that
t
dω + C
eA( t−σ) Bω( σ) dσ = ( d + C ( sI − A) − 1 B) ω = W ( s) ω, 0
t
eA( t−σ) Bω ( σ) dσ = ( sI − A) − 1 Bω
0
and
t
C
eA( t−τ) f ( τ ) dτ = C ( sI − A) − 1 f
0
we can express (A.1) as
W ( s)˜
θ ω = ˜
θ W ( s) ω − C ( sI − A) − 1
( sI − A) − 1 Bω
˙˜ θ
(A.2)
and the proof is complete.
✷
Other proofs of Lemma A.1 can be found in [201, 221].
Lemma A.2 (Swapping Lemma A.2) Let ˜
θ, ω : R+ → Rn and ˜
θ, ω be
differentiable. Then
˜
θ ω = F 1( s, α 0) ˙˜
θ ω + ˜
θ ˙ ω + F ( s, α 0)[˜
θ ω]
(A.3)
where F ( s, α 0) =
αk 0
, F
, k ≥ 1 and α
( s+ α
1( s, α 0) = 1 −F ( s,α 0)
0 > 0 is an
0) k
s
arbitrary constant. Furthermore, for α 0 > δ where δ ≥ 0 is any given
constant, F 1( s, α 0) satisfies
c
F 1( s, α 0)
≤
∞δ
α 0
for a finite constant c ∈ R+ which is independent of α 0 .
Proof Let us write
˜
θ ω = (1 − F ( s, α 0)) ˜
θ ω + F ( s, α 0)˜
θ ω
Note that
k
k<