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Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems



Ming-Yuan Shieh and Juing-Shian Chiou
 
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ABSTRACT

This study has presented two types of observer and observer-based controller for stabilization of singularly perturbed discrete bilinear systems. The first one is an ε-dependent observer-based controller that stabilizes the closed-loop system for all εε (0, ε*p) where, ε*p is the prespecified upper bound of the singular perturbation parameter. The other one is an ε-independent observer-based controller, which is able to stabilize the closed-loop system for all εε (0, εmax) where, εmax is the exact upper ε-bound. The designs of observer and observer-based controller are suitable for the quasi-separation property. An example is exploited to illustrate the proposed schemes.

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  How to cite this article:

Ming-Yuan Shieh and Juing-Shian Chiou, 2012. Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems. Information Technology Journal, 11: 141-147.

DOI: 10.3923/itj.2012.141.147

URL: https://scialert.net/abstract/?doi=itj.2012.141.147
 
Received: July 03, 2011; Accepted: September 13, 2011; Published: November 22, 2011



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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(1a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(1b)

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(2)

where, Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems as k→∞. It should also be added that the pair (A11, C1) is detectable.

Here, an observer-based control can be designed as follows:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(3)

where, D∈R1xn can be arbitrarily designed, σ is constant. Hence, the state and error Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems equations which can be written as follows:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(4a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(4b)

where,

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(5a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(5b)

in which, P, Pe∈Rnxn may be obtained by solving the Lyapunov equations:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(6a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(6b)

and where, δ is a positive constant satisfying (1+δ) <Min (r―1 (A0), r―10)).

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(7)

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(8)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(9)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(10)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(11)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(12)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(13)

Find:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(14)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(15)

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).

The stability of the system’s state equation (4a)

In fact, this result is the same as Theorem 2 of (Chiou et al., 2000), i.e., Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems*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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems. (ii): The stability of the error equation (4b).

From (4b), (5b), (6b) and Lemma 1, the Lyapunov forward difference is given by:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Inequality (16) is negative if:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(16)

Hence, the error system (4b) can be stabilized for all Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems.

It is also obvious that the original system (1) and its observer (2) are both stable for all Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems.

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(17)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(18)

Proof: It follows from what has been said that Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems. 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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(19a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(19b)

where, Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems are the observer states and L1∈Rn1xq is a suitable observing gain matrix that will be designed so as to ensure the estimated state Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems as . Here, an observer-based control of the form can be designed as follows:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(20)

to stabilize the system (1), where D1 ∈R1xn1 can be arbitrarily designed, σ is constant:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Hence, the state and error equations which can be written as follows:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(21a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(21b)

where,

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

in which Â11 = A11-L1C1 and Â12 = A12-L1C1.

Before deriving the main result, a preliminary lemma is presented.

For the nominal discrete system:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(22)

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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(23a)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(23b)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(23c)

where,

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

For convenience, the following notations are introduced:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

By using the three conditions of stability criterion (23) in order to test the stability of the system (21), it is needed that Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems are both stable, hence, the value σ must be designed such that Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems, the result is written as the following corollary.

Corollary 2: Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(24)

Where:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(25)

in which

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Calculate to satisfy stability criteria by:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(26)

where,

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(27)

in which:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Find:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems to satisfy stability criteria by:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(29)

where,

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(30)

in which:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Calculate Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems to satisfy stability criteria by:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(31)

Where:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(32)

in which:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Find:
Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(33)

Find:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(34)

Proof: The proof can be also divided into two parts, the stability of the system’s state equation (21a) for all Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems and the stability of the error equation (21b) for all Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems.

The stability of the system’s state equation (21a) for all Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems .

In fact, this result is the same as Theorem 3 of Chiou et al. (2000). The proof of this part is benign neglect.

The stability of the error equation (21b) for all Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems.

From condition (a) of Lemma 2 and (21.a):

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

According to Corollary 2, Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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):

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Since , then the above inequality holds if ε<εe2.

The third condition of Lemma 2 can be operated as follows:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

Since, Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems is assumed to be stable, i.e., Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems then the critical stability for the above inequality is:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(35)

or, equivalently:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

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, Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems) 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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(36)

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

where:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems

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 Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems =0.1758 can be obtained from (17). The observer-based controller to stabilize the state system and the observer is then given by:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(37)

with ε = 0.1, D = [1 1 1 0 0 0] and qnc-1 = 0.6764.

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
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:

Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems
(38)

and the relation betweenImage for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems , Image for - Design of Observer and Observer-based Controller for a Class of Singularly Perturbed Discrete Bilinear Systems 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.

REFERENCES

1:  Asseu, O., M. Koffi, Z. Yeo, X. Lin-Shi, M.A. Kouacou and T.J. Zoueu, 2008. Robust feedback linearization and observation approach for control of an induction motor. Asian J. Applied Sci., 1: 59-69.
CrossRef  |  Direct Link  |  

2:  Asseu, O., T.R. Ori, K.E. Ali, Z. Yeo, S. Ouattara and X. Lin-Shi, 2011. Nonlinear feedback linearization and observation algorithm for control of a permanent magnet synchronous machine. Asian J. Applied Sci., 4: 202-210.
CrossRef  |  Direct Link  |  

3:  Chiou, J.S., F.C. Kung and T.H.S. Li, 2000. Robust stabilization of a class of singularly perturbed discrete bilinear systems. IEEE Trans. Automat. Control, 45: 1187-1191.
CrossRef  |  

4:  Fuller, A.T., 1968. Condition for a matrix to have only characteristic roots with negative real parts. J. Math. Anal. Appl, 23: 71-98.
CrossRef  |  

5:  Furnkawa, T. and E. Shimemura, 1983. Stabilizability conditions by memoryless feedback for linear systems with time-delay. Int. J. Contr., 37: 553-565.

6:  Gholizade-Narm, H., A. Azemi, M. Khademi and M. Karimi-Ghartemani, 2008. A state observer and a synchronization method for heart pacemakers. J. Applied Sci., 8: 3175-3182.
CrossRef  |  Direct Link  |  

7:  Hassanzadeh, I. and M.A. Fallah, 2008. Design of augmented extended and unscented kalman filters for differential-drive mobile robots. J. Applied Sci., 8: 2901-2906.
CrossRef  |  Direct Link  |  

8:  Hsiao, C.H. and W.J. Wang, 2000. State analysis and parameter estimation of bilinear systems via haar wavelets. IEEE Trans. Circuits Syst. Fundam. Theory Appl., 47: 246-250.
CrossRef  |  

9:  Hsiao, F.H., J.D. Hwang and S.T. Pan, 2003. D-stability problem of discrete singularly perturbed systems. Int. J. Syst. Sci., 34: 227-236.

10:  Jury, E.I. and S. Gutman, 1975. On the stability of a matrix inside the unit circle. IEEE Trans. Automat. Contr., 20: 533-535.
CrossRef  |  

11:  Li, T.H.S. and J.S. Chiou, 2002. A new d-stability criterion of multiparameter singularly perturbed discrete systems. IEEE Trans. Circuits Syst. I, 49: 1226-1230.
CrossRef  |  

12:  Li, X.S. and H.J. Fang, 2009. Stability of continuous-time vehicles formations with time delays in undirected communication network. Inform. Technol. J., 8: 165-172.
CrossRef  |  Direct Link  |  

13:  Li, X., X. Bei WU and J. Gao, 2011. Observer-based guaranteed cost fault-tolerant controller design for networked control systems. Inform. Technol. J., 10: 394-401.
CrossRef  |  Direct Link  |  

14:  Luo, H., Y. Lv, X. Deng and H. Zhang, 2009. Optimization of adaptation gains of full-order flux observer in sensorless induction motor drives using genetic algorithm. Inform. Technol. J., 8: 577-582.
CrossRef  |  Direct Link  |  

15:  Mohler, R.R., 1991. Nonlinear Systems. Vol. II, Applications to Bilinear Control, Prentice Hall, Englewood Cliffs

16:  Wang, S.J., N.S. Pai and H.T. Yau, 2009. Robust controller design for synchronization of two chaotic circuits. Inform. Technol. J., 8: 743-749.
CrossRef  |  Direct Link  |  

17:  Xin, D., Z. Jin, G. Tao and L. Yang, 2009. Design of full order observer in speed sensorless induction motor drive. Inform. Technol. J., 8: 1150-1159.
CrossRef  |  Direct Link  |  

18:  Zhang, Y. and J. Li, 2010. Membership-dependent stability conditions for takagi-sugeno fuzzy systems. Inform. Technol. J., 9: 968-973.
CrossRef  |  Direct Link  |  

19:  Hsiao, M.Y., C.H. Liu and S.H. Tsai, 2010. Controller design and stabilization and for a class of bilinear system. Inform. Technol. J., 9: 1500-1503.
CrossRef  |  Direct Link  |  

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