Nonlinear Observer-Based
Nonlinear Observer-Based Control Allocation
123
Control Allocation
9
∂L ∂L
Since
∂λ , ∂ , e exponentially converges to zero as t → +∞, the closed-loop system
u
exponentially converges to
˙ˆx = A d ˆx + B dr
(43)
˙e = Dx f (ˆx + m1, u) − θ(ˆx, u)P − 1[ Dh(ˆx)] TDh(ˆx + m0) e Since A d is a asymptotically stable matrix, we know that ˆx ∈ W is bounded. According to
Assumptions 1 and 4, Dx f (ˆx + m1, u), Dh(ˆx) and Dh(ˆx + m0) are all bounded for m0, m1 ∈ S
and u ∈ Ω. From k 0 > 0, we have 0 < γ 1(ˆx, u) < +∞. According to Assumption 4, we have ker R(ˆx, m0) ⊂ ker Dh(ˆx) which ensures that 0 < νTR(ˆx, m0) ν < +∞ for every ν ∈ ∂S − M, m0 ∈ S and ˆx ∈ W. Thus, we have 0 < γ 2(ˆx) < +∞. As a result, 0 < θ(ˆx, u) < +∞. From (43), we know that ˙e exponentially converges to zero as e exponentially converges to zero.
Moreover, we have
˙x − ˙e = A dx − A de + B dr
(44)
Since ˙e and e exponentially converges to zero, we have the system (1) exponentially converges
to ˙x = A dx + B dr. This completes the proof.
Consider now the issue of solving (26) with respect to ξ 1 and ξ 2. One method to achieve
a well-defined unique solution to the under-determined algebraic equation is to solve a
least-square problem subject to (26). This leads to the Lagrangian
l( ξ 1, ξ 2, ρ) = 1 ( ξTξ
ξ
2
1 1 + ξ T
2 2) + ρ( αT ξ 1 + βT ξ 2 + δ + ωVm)
(45)
where ρ ∈ R is a Lagrange multiplier. The first order optimality conditions
∂l =
∂l =
∂l =
∂ξ
0,
0,
0
(46)
1
∂ξ 2
∂ρ
leads to the following system of linear equations
⎡
⎤ ⎡
⎤
⎡
⎤
Im 0 α
ξ 1
0
⎢
⎢
⎥ ⎢
⎥
⎢
⎥
⎣ 0 I
⎥ ⎢
⎥
⎢
⎥
m β ⎦ ⎣ ξ 2 ⎦ = ⎣
0
⎦
(47)
αT βT 0
ρ
−δ − ωVm
Remark 4. It is noted that Equation (47) always has a unique solution for ξ 1 and ξ 2 if any one of α
and β is nonzero.
4. Example
Consider the pendulum system
˙ x 1
=
x 2
(48)
˙ x
− sin x
2
1 + u 1 cos x 1 + u 2 sin x 1
y = x 1 + x 2
(49)
124
Applications of Nonlinear Control
10
Will-be-set-by-IN-TECH
with x = [ x 1 x 2] T ∈ R2, u = [ u 1 u 2] T ∈ Ω and
Ω = u = [ u 1 u 2] T − 1 ≤ u 1 ≤ 1, − 0.5 ≤ u 2 ≤ 0.5
(50)
As the system is affine in control and its measurement output y is a linear map of its state x,
Assumptions 1, 3 and 4 are satisfied automatically.
Choose
P = 3
0
0
1
For e = 0 and e ∈ ker[1 1], we have e 1 = −e 2 and
e TP Dx f (x, u)e |e 1= −e 2
= [
e
e
1
1
e 2]
0
3
−
|
cos x
e 1= −e 2
1 − u 1 sin x 1 + u 2 cos x 1
0
e 2
= ( − cos x 1 − u 1 sin x 1 + u 2 cos x 1 + 3) e 1 e 2 |e 1= −e 2
≤ [ − 1.5 cos(arctan 2 ) − sin(arctan 2 ) + 3] e
3
3
1 e 2 |e 1= −e 2
= ( − 1.8028 + 3) e 1 e 2 |e 1= −e 2
= − 0.5986 e 2 |e 1= −e 2
< −k 0 e 2 |e 1= −e 2
with 0 < k 0 < 0.5986. Hence, Assumption 5 is satisfied. Let S be the ball of radius r = 1, centered at zero and ∂S is the boundary of S. Define M ⊂ ∂S and
M = ν = [ ν 1 ν 2] T ∈ R2 : ν = 1, 3 ν 1 ν 2 + 1.8028 |ν 1 ν 2 | < −k 0
Obviously,
∂S − M = ν = [ ν 1 ν 2] T ∈ R2 : ν = 1, 3 ν 1 ν 2 + 1.8028 |ν 1 ν 2 | ≥ −k 0
As γ 1(ˆx, u) = 3 × 1.8028 + k 0 and
γ 2(ˆx) = min ( ν 1 + ν 2)2, ν ∈ ∂S − M = 1 −
2 k 0
3 − 1.8028
γ
choosing k
1( ˆ
x, u)
0 = 0.5, we have
=
γ
35.8699. Let θ(ˆx, u) = 36 > 35.8699 and we have
2( ˆ
x)
Φ(ˆx, u) = −[12 36] T.
Now the nonlinear observer becomes
˙ˆ x 1
˙
=
ˆ x 2
+ 12 ( y − ˆ x
ˆ x
1 − ˆ
x 2)
2
− sin ˆ x 1 + u 1 cos ˆ x 1 + u 2 sin ˆ x 1
36
Choose the reference model (6) where
A d =
0
1
−
,
B
25 − 10
d =
0
25
Nonlinear Observer-Based
Nonlinear Observer-Based Control Allocation
125
Control Allocation
11
and the reference is given by
⎧
⎪
⎪
5
4
3
⎪
t
t
t
⎪
⎪ r
− 15
+ 10
,
0 ≤ t < t
⎪ f 6
1
⎪
⎪
t
t
t
⎪
1
1
1
⎨ rf ,
t 1 ≤ t < t 2
⎡
⎤
r = ⎪
5
4
3
⎪
⎪
t − t
t − t
t − t
⎪
2
2
2
⎪ − ⎣
−
+
⎦ +
⎪ r
6
15
10
r
⎪
f
f , t 2 ≤ t < t f
⎪
t
t
t
⎪
f − t 2
f − t 2
f − t 2
⎩ 0,
t ≥ t f
with t 1 = 10 s, t 2 = 20 s, t f = 30 s and r f = 0.5. Obviously, Assumption 2 is satisfied.
Set H1 = 0, H2 = 10 − 4I2, ω = 1, Γ1 = Γ2 = 2I2, and x 1(0) = 0.3 and x 2(0) = 0.5. Using the proposed approach, we have the simulation result of the pendulum system (48)-(50) shown in
Figures 2-5 where the control u 2 is stuck at − 0.5 from t = 12 s onward.
From Figure 2, it is observed that the estimated states ˆ x 1 and ˆ x 2 converge to the actual
states x 1 and x 2 and match the desired states x 1 d and x 2 d well, respectively, even when u 2 is stuck at − 0.5. This observation is further verified by Figure 3 where both the state
estimation errors e 1(= x 1 − ˆ x 1) and e 2(= x 2 − ˆ x 2) of the nonlinear observer as in (4) and the matching errors τ 1(= 0) and τ 2(= − sin ˆ x 1 + u 1 cos ˆ x 1 + u 2 sin ˆ x 1 + 25 ˆ x 1 + 10 ˆ x 2 − 25r) as in (8) exponentially converge to zero. Moreover, Figure 4 shows that the control u 1 roughly
satisfies the control constraint u 1 ∈ [ − 1, 1] while the control u 2 strictly satisfies the control constraint u 2 ∈ [ − 0.5, 0.5]. This is because, in this example, the Lagrange multiplier λ 1 is first activated by the control u 1 < − 1 at t = 0 (see Figure 5 where λ 1 is no longer zero from t = 0), 0.4
0.2
state variable 1
0
hx
x
x
1
1
1d
0
5
10
15
20
25
30
time(s)
0.5
0
state variable 2 −0.5
hx
x
x
2
2
2d
0
5
10
15
20
25
30
time(s)
Fig. 2. Responses of the desired, estimated and actual states
126
Applications of Nonlinear Control
12
Will-be-set-by-IN-TECH
0.6
e
e
1
2
0.4
0.2
0
estimation error
−0.20
5
10
15
20
25
30
time(s)
10
τ
τ
1
2
5
0
matching error
−50
5
10
15
20
25
30
time(s)
Fig. 3. Responses of estimation error and matching error
1
0.5
1
u
0
−0.5
−1
0
5
10
15
20
25
30
time(s)
0.6
0.4
0.2
2
u
0
−0.2
−0.4
0
5
10
15
20
25
30
time(s)
Fig. 4. Responses of control u
and then the proposed dynamic update law forces the control u 1 to satisfy the constraint
u 1 ∈ [ − 1, 1]. It is also noted from Figure 5 that the Lagrange multiplier λ 2 is not activated in this example as the control u 2 is never beyond the range [ − 0.5, 0.5]. In addition, the output y
and the Lyapunov-like function Vm are shown in Figure 6. From Figure 6, it is observed that
the Lyapunov-like function Vm exponentially converges to zero.
Nonlinear Observer-Based
Nonlinear Observer-Based Control Allocation
127
Control Allocation
13
0.1
0
−0.1
−0.2
λ 1
−0.3
−0.4
−0.50
5
10
15
20
25
30
time(s)
0.1
0.05
0
λ 2
−0.05
−0.10
5
10
15
20
25
30
time(s)
Fig. 5. Responses of Lagrangian multiplier λ
1
0.5
y
0
−0.50
5
10
15
20
25
30
time(s)
−8
x 10
3
2
m
V
1
00
5
10
15
20
25
30
time(s)
Fig. 6. Responses of output y and Lyapunov-like function Vm
5. Conclusions
Sufficient Lyapunov-like conditions have been proposed for the control allocation design via
output feedback. The proposed approach is applicable to a wide class of nonlinear systems.
As the initial estimation error e(0) need be near zero and the predefined dynamics of the
128
Applications of Nonlinear Control
14
Will-be-set-by-IN-TECH
closed-loop is described by a linear stable reference model, the proposed approach will
present a local nature.
6. References
Ahmed-Ali, T. & Lamnabhi-Lagarrigue, F. (1999).
Sliding observer-controller design
for uncertain triangle nonlinear systems, IEEE Transactions on Automatic Control
44(6): 1244–1249.
Alamir, M. (1999). Optimization-based nonlinear observer revisited, International Journal of
Control 72(13): 1204–1217.
Benosman, M., Liao, F., Lum, K. Y. & Wang, J. L. (2009).
Nonlinear control allocation
for non-minimum phase systems, IEEE Transactions on Control Systems Technology
17(2): 394–404.
Besancon, G. (ed.) (2007). Nonlinear Observers and Applications, Springer.
Besancon, G. & Ticlea, A. (2007). An immersion-based observer design for rank-observable
nonlinear systems, IEEE Transactions on Automatic Control 52(1): 83–88.
Bestle, D. & Zeitz, M. (1983). Canonical form observer design for non-linear time-variable
systems, International Journal of Control 38: 419–431.
Bodson, M. (2002). Evaluation of optimization methods for control allocation, Journal of
Guidance, Control and Dynamics 25(4): 703–711.
Bornard, G. & Hammouri, H. (1991). A high gain observer for a class of uniformly observable
systems, Proceedings of the 30th IEEE Conference on Decision and Control, pp. 1494–1496.
Buffington, J. M., Enns, D. F. & Teel, A. R. (1998). Control allocation and zero dynamics, Journal
of Guidance, Control and Dynamics 21(3): 458–464.
Gauthier, J. P. & Kupka, I. A. K. (1994). Observability and observers for nonlinear systems,
SIAM Journal on Control and Optimization 32: 975–994.
Krener, A. J. & Isidori, A. (1983). Linearization by output injection and nonlinear observers,
Systems & Control Letters 3: 47–52.
Krener, A. J. & Respondek, W. (1985). Nonlinear observers with linearizable error dynamics,
SIAM Journal on Control and Optimization 23: 197–216.
Liao, F., Lum, K. Y., Wang, J. L. & Benosman, M. (2007). Constrained nonlinear finite-time
control allocation, Proc. of 2007 American Control Conference, New York City, NY.
Liao, F., Lum, K. Y., Wang, J. L. & Benosman, M. (2010).
Adaptive Control Allocation
for Non-linear Systems with Internal Dynamics, IET Control Theory & Applications
4(6): 909–922.
Luenberger, D. G. (1964). Observing the state of a linear system,