λ
U PU)
2 2
+ γ β
0
0
max
Theorem 2: Consider the system (1) and the cost function (7), for the given index Φ( q, r)
and H∞ norm-bound index γ , if there exists symmetric positive matrix X , matrix Y and
scalars ε > 0( = 4 ~ 9)
i
i
such that the following linear matrix inequality
⎡ X
0
(AX BY) T
⎤
−
+
21
⎢
Σ ⎥
2
T
⎢
γ
− I
D
⎥
*
0 < 0
⎢
⎥
(13)
⎢ *
*
T
−X + ε a I + ε b I + ε BJB
0
4
5
6
⎥
⎢ *
*
*
⎥
⎣
22
Σ ⎦
holds, where
T
T
T
T
T
T
T
Σ = CX +
T
−
Σ =
−I + EJE −Q −
21
[(
EY) ,X,X,Y ,Y ,Y ,Y ,Y ,Y J] ,
1
22
dia [
g
ε
ε
7
,
, 4 ,
I
1
−
1
−
1
−
1
ε I ε J ε J
ε J
ε J ε
−
−
+
−
−
−
I − R
ε
−
5
8 ,
6
, 7
, 8
, 9
, 9 ]
I . Then for all admissible uncertainties and
possible faults M , the faulty closed-loop system (6) with satisfactory fault-tolerant
controller u( k) = Kx( k) =
1
−
1
−
M0 YX (
x k) is asymptotically stable with an H∞ norm-bound γ ,
and the corresponding closed-loop cost function (7) is with
T
1
−
2 2
J ≤ λ
U X U + γ β
max(
)
.
According to Theorem 1 and 2, the consistency of the quadratic D stabilizability constraint,
H∞ performance and cost function indices for fault-tolerant control is deduced as the
following optimization problem.
Theorem 3: Given quadratic D stabilizability index Φ( q, r) , suppose the system (1) is robust
fault-tolerant state feedback assignable for actuator faults case, then LMIs (10), (13) have a
feasible solution. Thus, the following minimization problem is meaningful.
min(γ ) : (
X, Y,γ ,ε i ) S.t. LMIs (10), (13)
(14)
Proof: Based on Theorem 1, if the system (1) is robust fault-tolerant state feedback
assignable for actuator faults case, then inequality
T
A PA − P < 0
C
C
has a feasible solution P , K . And existing λ > 0 , 0
δ > , the following inequality holds
T
T
T
λ ⎡A PA − P⎤ + C C + Q + K MRMK + δI < 0
⎣ C
C
⎦
C
C
(15)
Then existing a scalar γ
γ > γ
0 , when
0 , it can be obtained that
T
A P D(
T
−
γ I − D P D) 1
2
T
D P A < δ
C 1
1
1 C
I
where P = λ
1
P . Furthermore, it follows that
T
T
T
T
T
−
T
A P A − P + C C + Q + K MRMK + A P D γ I − D P D
D P A <
C 1 C
1
C
C
C 1
(
1
) 1
2
0
1 C
Using Schur complement and Theorem 2, it is easy to show that the above inequality is
equivalent to linear matrix inequality (13), namely, 1
P , K , γ is a feasible solution of LMIs
200
Discrete Time Systems
(10), (13). So if the system (1) is robust fault-tolerant state feedback assignable for actuator
faults case, the LMIs (10), (13) have a feasible solution and the minimization problem (14) is
meaningful. The proof is completed.
Suppose the above minimization problem has a solution X
ε
γ
L ,
L
Y , iL , L , and then any
index γ > γ L , LMIs (10), (13) have a feasible solution. Thus, the following optimization
problem is meaningful.
Theorem 4: Consider the system (1) and the cost function (7), for the given quadratic D
stabilizability index Φ( q, r) and H∞ norm-bound index γ > γ L , if there exists symmetric
positive matrix X , matrix Y and scalars ε > 0( = 1 ~ 9)
i
i
such that the following
minimization
2 2
min λ
+ γ β (16)
S.t. (i) (10), (13)
T
⎡− I U ⎤
(ii) λ
⎢
⎥ < 0
⎢ U
−
⎣
X⎥⎦
has a solution X
Y
ε
λ
min , min ,
min ,
i
min , then for all admissible uncertainties and possible faults
M ,
1
−
1
(
u k) K (
x k)
−
=
= M
is an optimal guaranteed cost satisfactory fault-tolerant
0
m
Y inXminx( k)
controller, so that the faulty closed-loop system (6) is quadratically D stabilizable with an H∞
norm-bound γ , and the corresponding closed-loop cost function (7) satisfies
2 2
J ≤ λ
+ γ β .
min
According to Theorem 1~4, the following satisfactory fault-tolerant controller design
method is concluded for the actuator faults case.
Theorem 5: Given consistent quadratic D stabilizability index Φ( q, r) , H∞ norm index
γ > γ
>
+
L and cost function index *
2 2
J
λ
γ β
min
, suppose that the system (1) is robust fault-
tolerant state feedback assignable for actuator faults case. If LMIs (10), (13) have a feasible
solution X , Y , then for all admissible uncertainties and possible faults M ,
1
−
1
(
u k) K (
x k)
−
=
= M0 YX (
x k) is satisfactory fault-tolerant controller making the faulty closed-
loop system (6) satisfying the constraints (a), (b) and (c) simultaneously.
In a similar manner to the Theorem 5, as for the system (1) with quadratic D stabilizability,
H∞ norm and cost function requirements in normal case, i.e., M = I , we can get the
satisfactory normal controller without fault tolerance.
4. Simulative example
Consider a satellite attitude control uncertain discrete-time system (1) with parameters as
follows:
⎡ 2
−
2⎤
⎡1 1⎤
A
B =
C = [
]
⎡0.1⎤
⎡1 0⎤
⎡1 0⎤
=
, ,
0.2 0.3 ,
D =
,
Q =
,
R =
,
a = 0.1,
b =
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
0.2
.
⎣ 2
4⎦
⎣0 1⎦
⎣0.5⎦
⎣0 1⎦
⎣0 1⎦
Suppose the actuator failure parameters M =
{0.4, 0
.6} M =
1.1
l
diag
,
{1.3,
}
u
diag
. Given the
quadratic D stabilizability index Φ(0.5,0.5) , we can obtain state-feedback satisfactory fault-
tolerant controller (SFTC), such that the closed-loop systems will meet given indices
constraints simultaneously based on Theorem 5.
Quadratic D Stabilizable Satisfactory Fault-tolerant Control with
Constraints of Consistent Indices for Satellite Attitude Control Systems
201
⎡ 0.5935
3.0187 ⎤
K
=
SFTC
⎢
6.7827
5.6741⎥
−
−
⎣
⎦
In order to compare, we can obtain the state-feedback satisfactory normal controller (SNC)
without fault-tolerance.
⎡ 0.4632
2.4951 ⎤
K
=
SNC
⎢
5.4682
4.9128⎥
−
−
⎣
⎦
Through simulative calculation, the pole-distribution of the closed-loop system by
satisfactory fault-tolerant controller and normal controller are illustrated in Figure 1, 2 and 3
for normal case and the actuator faults case respectively. It can be concluded that the poles
of closed-loop system driven by normal controller lie in the circular disk Φ(0.5,0.5) for
normal case (see Fig. 1). However, in the actuator failure case, the closed-loop system with
normal controller is unstable; some poles are out of the given circular disk (see Fig. 2). In the
contrast, the performance by satisfactory fault-tolerant controller still satisfies the given pole
index (see Fig. 3). Thus the poles of closed-loop systems lie in the given circular disk by the
proposed method.
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.2
0.4
0.6
0.8
1
Fig. 1. Pole-distribution under satisfactory normal control without faults
5. Conclusion
Taking the guaranteed cost control in practical systems into account, the problem of
satisfactory fault-tolerant controller design with quadratic D stabilizability and H∞ norm-
bound constraints is concerned by LMI approach for a class of satellite attitude systems
subject to actuator failures. Attention has been paid to the design of state-feedback controller
that guarantees, for all admissible value-bounded uncertainties existing in both the state and
control input matrices as well as possible actuator failures, the closed-loop system to satisfy
202
Discrete Time Systems
the pre-specified quadratic D stabilizability index, meanwhile the H∞ index and cost
function are restricted within the chosen upper bounds. So, the resulting closed-loop system
can provide satisfactory stability, transient property, H∞ performance and quadratic cost
performance despite of possible actuator faults. The similar design method can be extended
to sensor failures case.
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Fig. 2. Pole-distribution under satisfactory normal control with faults
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.2
0.4
0.6
0.8
1
Fig. 3. Pole-distribution under satisfactory fault-tolerant control with faults
Quadratic D Stabilizable Satisfactory Fault-tolerant Control with
Constraints of Consistent Indices for Satellite Attitude Control Systems
203
6. Acknowledgement
This work is supported by the National Natural Science Foundation of P. R. China under
grants 60574082, 60804027 and the NUST Research Funding under Grant 2010ZYTS012.
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Part 3
Discrete-Time Adaptive Control
13
Discrete-Time Adaptive Predictive Control
with Asymptotic Output Tracking
Chenguang Yang1 and Hongbin Ma2
1 University of Plymouth
2 Beijing Institute of Technology
1 United Kingdom
2 China
1. Introduction
Nowadays nearly all the control algorithms are implemented digitally and consequently
discrete-time systems have been receiving ever increasing attention. However, as to the
development of nonlinear adaptive control methods, which are generally regarded as smart
ways to deal with system uncertainties, most researches are conducted for continuous-time
systems, such that it is very difficult or even impossible to directly apply many well
developed methods in discrete-time systems, due to the fundamental difference between
differential and difference equations for modeling continuous-time and discrete-time systems,
respectively. Even some concepts for discrete-time systems have very different meaning from
those for continuous-time systems, e.g., the “relative degrees” defined for continuous-time
and discrete-time systems have totally different physical explanations Cabrera & Narendra
(1999). Therefore, nonlinear adaptive control of discrete-time systems needs to be further
investigated.
On the other hand, the early studies on adaptive control were mainly concerning on the
parametric uncertainties, i.e., unknown system parameters, such that the designed control
laws have limited robustness properties, where minute disturbances and the presence of
nonparametric model uncertainties can lead to poor performance and even instability of
the closed-loop systems Egardt (1979); Tao (2003). Subsequently, robustness in adaptive
control has been the subject of much research attention for decades.
However, due to
the difficulties associated with discrete-time uncertain nonlinear system model, there are
only limited researches on robust adaptive control to deal with nonparametric nonlinear
model uncertainties in discrete-time systems. For example, in Zhang et al. (2001), parameter
projection method was adopted to guarantee boundedness of parameter estimates in presence
of small nonparametric uncertainties under certain wild conditions. For another example,
the sliding mode method has been incorporated