INTRODUCTION
In the control process of practice systems may be appropriately described by
the bilinear model, such as biology, ecology, socioeconomic, nuclear, thermal
and chemical processes etc. (Mohler, 1991; Hsiao
and Wang, 2000; Hsiao et al., 2010). Whereas
the original systems are usually described by high-order difference equations,
it requires large number of computer memory and considerable operation time
to handle the systems. In general, singularly perturbed method can be solved
this problem (Hsiao et al., 2003). Design of
a state feedback controller for singularly perturbed discrete bilinear system
was considered by Chiou et al. (2000) where
design of all states for the controller must be measurable. But this assumption
generally does not hold in practice (Xin et al.,
2009; Asseu et al., 2008; Gholizade-Narm
et al., 2008), i.e., the system’s state may be unavailable to
implement the state feedback. Hence, it is often desirable to design an observer
and an observer-based controller leading to asymptotic stability (Li
and Fang, 2009; Zhang and Li, 2010; Wang
et al., 2009; Luo et al., 2009; Li
et al., 2011; Hassanzadeh and Fallah, 2008;
Asseu et al., 2011) for singularly perturbed
discrete bilinear systems. In this study, the observer and observer-based controller
designs of a system modeled as a singularly perturbed discrete bilinear system
are examined. Two types of observer and observer-based controller are developed,
one is that the controller gain depends on observer gain matrix, the other is
that the controller gain and observer gain matrix can be designed separately
once the so-called quasi-separation property holds.
The following notations will be used throughout the study: The identity matrix
with dimension n is denoted by In, λmin (A) stands
for the minimum eigenvalue of matrix A, the spectral radius of matrix A is denoted
by r (A), ||A|| presents the norm of matrix A, i.e., ||A||
= Max (λ (AT A))1/2 and the symbol “•”
represents the bialternate product in (Fuller, 1968;
Li and Chiou, 2002).
Observer and observer-based controller designs: Consider a singularly
perturbed discrete bilinear system as described by the following difference
equation:
where, X∈Rn, x1∈Rn1, x2∈Rn2,
n (=n1+n2) is the order of the whole system, u is a scalar
control, y ∈ Rm is the output, ε is the singular perturbation
parameter and Aij, Bi, Nij, Ci are
constant matrices with appropriate dimensions for i, j = 1, 2 Let:
For the sake of saving space, we denote B = B (ε) and N = N (ε) in the following derivation.
The objective is to design a full state observer and an observer-based controller for the given system (1) to be stable. Now, it is required to construct an observer for the system (1) as below:
where, 
are the observer states, L1∈Rn1xq and L2∈Rn2xq
are suitable observing gain matrices which will be designed so as to ensure
the estimated state 
and 
as k→∞. It should also be added that the pair (A11, C1)
is detectable.
Here, an observer-based control can be designed as follows:
where, D∈R1xn can be arbitrarily designed, σ is constant.
Hence, the state and error 
equations which can be written as follows:
where,
in which Â0 = A0-LC.
For the stability of the system (4), the eigen values of Â11
and Â22 are designed within the unit circle, i.e., |λ
(Â11)|<1 and |λ(Â22)|<1.
The objective is to find the suitable estimator’s gain matrix L and the
controller gain σ such that the system (4) is stable. In order to prove
the stability of the system (4a) and the error equation (4b),
the Lyapunov function candidates for the system (4a) and the error equation
(4b) can be given by:
in which, P, Pe∈Rnxn may be obtained by solving the Lyapunov equations:
and where, δ is a positive constant satisfying (1+δ) <Min (r―1
(A0), r―1 (Â0)).
For the latter development, the following lemma is helpful.
Lemma 1: (Furnkawa and Shimemura, 1983): Consider
the matrices F, G and H which have the same dimension and let F = G+H. For any
real number δ>0, the following relation holds:
Theorem 1: Suppose A11 is stable and choose L such that Âo is stable in the discrete sense, then the system (1) and observer (2) are guaranteed to be stable under the observer-based controller (3) for all εε (0, ε*p) and σ ε (0, σ*p), if the upper bounds σ*p and ε*p are determined by the following algorithms:
Find:
Proof: It will be useful to divide this proof into two parts, the stability
of the system’s state equation (4a) and the stability
of the error equation (4b).
In fact, this result is the same as Theorem 2 of (Chiou
et al., 2000), i.e., 
(ε*0, q and σ* can be obtained from equations
(11)-(13) of (Chiou et al.,
2000)). Hence, the system (1) can be guaranteed to stable for all 
and 
.
(ii): The stability of the error equation (4b).
From (4b), (5b), (6b) and Lemma 1, the Lyapunov forward difference is given
by:
Inequality (16) is negative if:
Hence, the error system (4b) can be stabilized for all 
and 
.
| • |
It is also obvious that the original system (1) and its observer
(2) are both stable for all 
and  . |
Remark 1: The values ε*0 and σ* are always greater than the values ε*P and σ*P via Theorem 1, that is, the results are conservative by means of the observer-based controller.
Corollary 1: Suppose A11 is stable and choose L1 such that A11-L1C1 is more stable than A11 in the discrete sense, then the system (1) and observer (2) are guaranteed to be robustly stable under the observer-based controller (3), where the upper bounds ε*p and σ*p can be obtained as follows:
Proof: It follows from what has been said that 
for A11- L1C1 is more stable than A11
in the discrete sense. Therefore, reasonably conclude that the system (1) and
observer (2) are guaranteed to be robustly stable for all 
and 
.
The proof is completed.
Remark 2: The controller constant gain σ depends on the observer gain matrix L in Theorem 1. For bilinear system, it would be possible to argue that the results do not possess the separation property. While, in Corollary 1, the value L cannot affect the value σ directly once A11-L1C1 is more stable than A11. Fundamentally, the property is called the quasi-separation property for Corollary 1.
Furthermore, another type of full state observer and observer-based controller will be introduced for the given system (1) to be stable. Now, a new observer as follows:
where, 
are the observer states and L1∈Rn1xq is a suitable
observing gain matrix that will be designed so as to ensure the estimated state
and 
as . Here, an observer-based control of the form can be designed as follows:
to stabilize the system (1), where D1 ∈R1xn1 can
be arbitrarily designed, σ is constant:
Hence, the state and error equations which can be written as follows:
where,
in which Â11 = A11-L1C1 and Â12 = A12-L1C1.
Before deriving the main result, a preliminary lemma is presented.
For the nominal discrete system:
where, A∈Rnxn, the critical stability criterion is described by the following lemma.
Lemma 2: (Jury and Gutman, 1975): The critical
stability criterion of discrete system (22) is given by the following three
conditions:
where,
For convenience, the following notations are introduced:
By using the three conditions of stability criterion (23) in order to test
the stability of the system (21), it is needed that 
and 
are both stable, hence, the value σ must be designed such that
and
are stable. Hence, the exact σ-bound can be also obtained by means
of the three conditions of stability criterion (23) for the stability of
and ,
the result is written as the following corollary.
Corollary 2: 
and 
are both asymptotically stable matrices for all σ∈(0, σ max),
if σ max is determined by the following algorithm:
| • |
Calculate to satisfy stability criteria by: |
Where:
in which
| • |
Calculate to satisfy stability criteria by: |
where,
in which:
Theorem 2: Consider the singularly perturbed discrete bilinear system (1) and the observer (19), the observer-based controller (20) stabilizes the systems (1) and (19) for all ε∈ (0, εmax), if εmax is determined by the following algorithms:
| • |
Calculate 
to satisfy stability criteria by: |
where,
in which:
| • |
Calculate 
to satisfy stability criteria by: |
Where:
in which:
Proof: The proof can be also divided into two parts, the stability of
the system’s state equation (21a) for all 
and the stability of the error equation (21b) for all 
.
In fact, this result is the same as Theorem 3 of Chiou
et al. (2000). The proof of this part is benign neglect.
From condition (a) of Lemma 2 and (21.a):
According to Corollary 2,
is an asymptotically stable matrix for σ satisfies (28), hence the above
inequality can be satisfied if ε <εe1.
From condition (b) of Lemma 2 and (21.a):
Since , then the above inequality holds if ε<εe2.
The third condition of Lemma 2 can be operated as follows:
Since,
is assumed to be stable, i.e., 
then the critical stability for the above inequality is:
or, equivalently:
Hence, the upper bound εe3, to guarantee the condition (c) of Lemma 2, can be given by ε<εe3.
It is also obvious that the original system (1) and its observer (19) are both
stable for all εε(0, 
)
by each σ which σε (0, σmax). It follows from
what has been proved that the upper bound can be represented by (34) for all
σε (0, σmax). The proof is completed.
Remark 3: From the observer-based controller (20), one can see that
different
can be determined by using different gain σ (< σmax)
in Theorem 2 for a fixed matrix D1. The εmax is the
largest ε-bound for the system (1) and observer (19) under the observer-based
controller (20) for a fixed matrix D1.
AN EXAMPLE
Consider the system (1) as described below:
where:
From Corollary 1, choosing L = [0.163 0.0519 -0.4148 0 0 0]T implies
to place all the eigenvalues of Â0 to be 0.1, 0.2, 0.3, 0,
0 and 0. It is easy to choose δ = 0.5 for solving Lyapunov equation
(6) and 
=0.1758 can be obtained from (17). The observer-based controller to stabilize
the state system and the observer is then given by:
with ε = 0.1, D = [1 1 1 0 0 0] and qnc-1 = 0.6764.
|
| Fig. 1: |
The relation between ε*oc-2, ε*e-2
(Y axis ) and σ (X axis) |
From Corollary 2, let D1 = [1 1 1] and choose L1 = [0.163
0.0514-0.4148]T, then σmax = 0.3199 and σmax
= 1.7278 can be calculated from Eq. 24 and 26,
respectively. Hence, σmax = 0.3199. Furthermore, from Theorem
2, εmax = 0.875 can be obtained. Finally, the closed-loop system
can be stabilized for all εε(0, 0.875) if the observer-based controller
is given by:
and the relation between
, 
and σ are shown in Fig. 1.
CONCLUSION
The ε-dependent and ε-independent observer and observer-based controllers for singularly perturbed discrete systems have been presented. It is illustrated in the simulation results that the ε-bounds of the observer-based controller are more conservative than those of the direct state feedback controller.
ACKNOWLEDGMENT
This study is supported by the National Science Council, Taiwan, Republic of China, under grand number NSC 100-2221-E-218-036 and NSC 100-2632-E-218-001-MY3.
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