Volume 2013, Issue 1 528695

Research Article

Open Access

H_∞ Control of Discrete-Time Singularly Perturbed Systems via Static Output Feedback

Dan Liu

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China pku.edu.cn

Search for more papers by this author

Lei Liu,

Lei Liu

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China pku.edu.cn

Search for more papers by this author

Ying Yang,

Corresponding Author

Ying Yang

[email protected]

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China pku.edu.cn

Search for more papers by this author

Dan Liu,

Dan Liu

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China pku.edu.cn

Search for more papers by this author

Lei Liu,

Lei Liu

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China pku.edu.cn

Search for more papers by this author

Ying Yang,

Corresponding Author

Ying Yang

[email protected]

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China pku.edu.cn

Search for more papers by this author

First published: 31 March 2013

https://doi.org/10.1155/2013/528695

Citations: 1

Academic Editor: Guanglu Zhou

Share a link

Email
Wechat
Bluesky

Abstract

This paper concentrates on H_∞ control problems of discrete-time singularly perturbed systems via static output feedback. Two methods of designing an H_∞ controller, which ensures that the resulting closed-loop system is asymptotically stable and meets a prescribed H_∞ norm bound, are presented in terms of LMIs. Though based on the same matrix transformation, the two approaches are turned into different optimal problems. The first result is given by an ϵ-independent LMI, while the second result is related to ϵ. Furthermore, a stability upper bound of the singular perturbation parameter is obtained. The validity of the proposed two results is demonstrated by a numerical example.

1. Introduction

Singularly perturbed systems widely exist in industrial processes, such as aircraft and racket systems, power systems, and nuclear reactor systems. These kind of systems usually embrace complicated dynamic phenomena which are characterized by slow and fast modes with multiple time-scales. This property causes high dimensionality and ill-conditioning problems. In control theory, a parameter-related state-space model is frequently used to describe a singularly perturbed system. With important practical meaning, the stability bounds of the singular perturbation parameter have been extensively studied by many researchers. In early times, a traditional method of decomposing the original system into fast and slow subsystems was frequently used, see [1, 2]. In [3], a method to testify the stability of singularly perturbed systems without the fast-slow decomposition was established. The stability bound was proved to have close relationship with the system matrix, which contributes to analyse some robust control problems. Furthermore, two algorithms to compute and improve the stability bound were developed in [4]. More stability problems are discussed in [5–10] and the references therein.

In recent years, computer science is increasingly applied to industrial processes. Therefore, discrete-time singularly perturbed systems have attracted much attention. An H_∞ control problem for uncertain discrete-time singularly perturbed systems via state feedback was studied in [11], where two methods of designing H_∞ controllers were given in terms of LMIs. Fast sampling discrete-time singularly perturbed systems were taken into consideration in [12], and the obtained results were generalized to robust controller design. In [13], a new sufficient condition which guaranteed the existence of state feedback controllers and made the closed-loop system asymptotically stable while satisfying a prescribed H_∞ norm requirement was proposed. This condition was also given in the form of LMIs but was proved to be less conservative than that in [11]. It will be more perfect if a theoretical proof was given. Interested readers can refer to [14–16] for more information of discrete-time singularly perturbed systems.

Though state feedback can achieve desired properties, it requires the availability of all state variables, which cannot be satisfied in most of the practical systems. On the other hand, dynamic output feedback usually increases the dimension of the original system. Therefore, static output feedback plays an important role in control theory considering that it is the simplest control technique in a closed-loop sense and can be easily realized with a cost not as high as that in state feedback case [17]. The primal point involved in static output feedback is the decoupling problem. A variety of approaches are developed to solve such issues. In [18], special inequality and some tuning scalars were used to transfer nonlinear matrix inequality to a linear one. A stabilizing state feedback controller gain and some matrix transformations were introduced to deal with the nonlinear inequalities in [19]. This method is effective and easy to implement. Thus, the same decoupling technique is adopted in this paper.

Based on those reasons and motivated by the above studies, we aim to design an H_∞ controller via static output feedback to stabilize a discrete-time singularly perturbed system and guarantee that the transfer function of the resulting closed-loop system satisfies a prescribed H_∞ norm bound.

The rest of this paper is organized as follows. Section 2 states the system description and some useful lemmas. In Section 3, two LMI-based methods are proposed to design a static output feedback controller for the system presented in Section 2. A numerical example is given to demonstrate the effectiveness of the proposed results in Section 4. Finally, conclusions are given in Section 5.

The following notation will be adopted throughout this paper. I denotes an identity matrix with appropriate dimension. A^T denotes the transpose of matrix A. Sym{A} denotes A + A^T. For a symmetric block matrix, ( ^*) stands for the blocks induced by symmetry.

2. Problem Formulation

In this paper, we consider a class of linear fast sampling discrete-time singularly perturbed systems of the following form:

()

where

, x_k ∈ Rⁿ is the state vector, in which

, and n₁ + n₂ = n,

is the control input,

is the disturbance input which belongs to L₂[0, ∞),

is the measurement output,

is the controlled output. The scalar ϵ > 0 denotes the singular perturbation parameter.

In the rest of this paper, we will assume that system (1) is completely controllable and observable. We will also assume that not all of its state variables are available. Therefore, the following static output feedback control law is taken into consideration:

()

then the resulting closed-loop system can be obtained as follows:

()

where

()

The H_∞ problem studied in this paper can be described as follows: given a scalar γ > 0 and a discrete-time singularly perturbed system (1), design a static output feedback controller in the form of (2) such that the closed-loop system (3) is asymptotically stable and its transfer function from ω to z,

()

satisfies ∥G∥_∞ < γ.

The following lemmas will be used in establishing our main results.

Lemma 1 (see [19].)Consider a discrete-time transfer function G(z) = C(zI − A) ⁻¹B + D. The following statements are equivalent:

(i) ∥G(z)∥_∞ < 1 and A are stable in the discrete-time sense (|λ_i(A)| < 1);
(ii) there exists X = X^T > 0 such that
()
holds.

Lemma 2 (see [20].)The following two statements are equivalent.

(i) Let A, B, and C be given such that the LMI
()
is feasible in X and Y.
(ii) Let A, B, and C be given such that the following LMI
()
holds.

Lemma 3 (see [20].)The following two statements are equivalent.

(i) Let A, B, and C be given such that the LMI
()
is feasible in G.
(ii) Let A, B, and C be given such that the LMIs
()
hold.

3. Main Results

In this section, two LMI-based methods of designing a static output feedback controller are proposed to ensure asymptotical stability of a closed-loop discrete-time singularly perturbed system (3). The first result is given in the form of an ϵ-independent LMI, while the second result is presented by two LMIs which are related to the singular perturbation parameter ϵ. Furthermore, we can obtain the stability upper bound of ϵ.

Before giving the results, let F₀ be a stabilizing state feedback controller gain of the system (1). In other words, the matrix F₀ is chosen to make matrix A_ϵ0 = A_ϵ + B_2ϵF₀ stable. Then, we introduce the following transformations which will be used in the rest of this paper:

()

where

()

Theorem 4. For a discrete-time singularly perturbed system in the form of (1), given a stabilizing state feedback controller with gain F₀, if there exist matrices P₁₁ > 0, P₂₂ > 0, and G > 0 and matrices P₁₂, L, X_ij, Y_kj, and i = 1, …, 5, k, j = 1,2 with appropriate dimensions, such that the following LMI holds:

()

where

()

then there exists a scalar ϵ^* > 0 such that for every ϵ∈(0, ϵ^*], the system (3) is asymptotically stable, and its transfer function satisfies ∥G∥_∞ < γ with the static output feedback controller in the form of (2) whose control gain is given by F = LG^−T.

Proof. Based on Lemma 1, the closed-loop system (3) is asymptotically stable, and its transfer function satisfies ∥G∥_∞ < γ if there exists a positive definite matrix P such that the following LMI holds:

()

then for all ϵ ∈ (0, ϵ^*], let

()

satisfy (15), and we have

()

where

()

Using the Schur complement to (17), we conclude that

()

where

()

Pre- and post-multiplying (19) by diag(I, (1/ϵ)I, I, I, I), we have

()

where

()

because ϵ is small enough, we can simplify (21) as in the following form:

()

To turn (23) into an LMI, we will use the same idea proposed in [17]. By substituting the transformation defined before, we get another form of (23):

()

where Ξ, Φ, S, and T are given as before.

Applying Lemma 2 to (24), we get

()

It is obvious that (25) can be rewritten as follows:

()

Taking the expression of S into account, we have

()

Using Lemma 3 to (27), we have

()

By letting L = FG^T, we can finally get (13). This completes the proof.

Both the static output feedback gain matrix F and the minimum H_∞ norm γ can be obtained by solving the following optimization problem:

where i = 1, …, 5, m ≤ j, m, k, j = 1,2, subject to γ > 0, P₁₁ > 0, P₂₂ > 0, and G > 0 and the LMI in (13).

Note that the result in Theorem 4 is finally obtained by solving a γ-related optimal problem. While in the following part, a different approach of designing a static output feedback controller is given by solving an optimal problem in which ϵ is involved.

Theorem 5. Given a scalar γ > 0 and a stabilizing state feedback controller with gain F₀, if there exist matrices P₁₁ > 0, P₂₂ > 0, G > 0, and Q_mm > 0, m = 1, …, 4 and matrices P₁₂, L, Y, , , , X_i, Q_mn, n < m, i, m, n = 1, …, 4, with appropriate dimensions, such that the following set of LMIs hold:

()

where

()

then for any singular perturbation parameter ϵ ∈ (0, 1/ϵ^*], by introducing a static output feedback controller in the form of (2), the closed-loop system (3) is asymptotically stable, and its transfer function satisfies ∥G∥_∞ < γ with the controller gain F = G⁻¹L.

Proof. According to Lemma 1, we know that the closed-loop system (3) is asymptotically stable, and its transfer function is less than γ if there exists a positive definite matrix P such that the following LMI holds:

()

For every singular perturbation parameter ϵ ∈ (0, 1/ϵ^*], let

()

satisfy (31), and we get

()

Pre- and post-multiplying (33) by

()

and its transpose, respectively, then for every ϵ ∈ (0, 1/ϵ^*], we have

()

where

()

If there exists a positive definite matrix Q = [Q_ij], i, j = 1, …, 4 satisfies

()

and for all ϵ ∈ (0, 1/ϵ^*],

()

then we can easily conclude that

()

Applying the Schur complement to (39), we get

()

Taking (37) into consideration, we have (35).

To turn (37) and (40) into LMIs, we still adopt the same technique used in [17]. By introducing the transformations defined before, we can rewrite (37) and (40) into the following form:

()

Applying Lemma 2 to (41), we have

()

It is obvious that (42) and (43) can be rewritten as

()

where Ψ is given as before.

Taking Lemma 3 into account, we know that (43) and (44) are equivalent to

()

respectively.

From the above analysis, we learn that if there exists G₁ = G₂ = G which satisfies both (46) and (47), then (29) can be finally concluded by considering the expression of S. This completes the proof.

We can obtain the static output feedback controller gain matrix F, as well as the stability upper bound of the singular perturbation parameter which we record as 1/ϵ^* by solving the following optimal problem:

where i ≤ j, i, j, l = 1,2, m, n, k = 1, …, 4, subject to ϵ^* > 0, P_ii > 0, Q_nn > 0, and G > 0 and the LMIs in (29).

Remark 6. The result in Theorem 4 finally turns into (OP1), which is solved by optimizing γ, the minimum H_∞ norm of the closed-loop system (3). While results in Theorem 5 are obtained by solving a different optimal problem (OP2), which optimizes the stability upper bound 1/ϵ^* with a fixed performance index γ.

4. A Numerical Example

In this section, a numerical example is presented to illustrate the effectiveness of the proposed results.

Example 7. Consider a discrete-time singularly perturbed system described by (1) with

()

First of all, by solving the optimal problem (OP1), we can get the introduced static output feedback controller gain

()

and the minimum H_∞ norm of the closed-loop system

()

Then, referring to (OP2), we can solve the corresponding LMIs with γ = 1. The obtained static output feedback controller gain is

()

while the stability upper bound of the singular perturbation parameter is

()

Remark 8. Though the performance of the closed-loop system is not as good as the state feedback case, static output feedback controller plays more important role in implemental sense with proper performance.

5. Conclusion

In this paper, H_∞ control problems for fast sampling singularly perturbed systems via static output feedback have been discussed. Rather than adopting the traditional design method of decomposing the original system into fast and slow subsystems, two LMI-based sufficient conditions have been given to guarantee the existence of static output feedback controllers and the asymptotical stability of the closed-loop system with a transfer function whose H_∞ norm is less than γ. With LMI toolbox in matlab platform, the obtained LMI results can be solved easily. The proposed methods simplify the controller design procedure.

Acknowledgments

This work is supported by the National 973 Program of China (2012CB821202) and the National Natural Science Foundation of China (61174052, 90916003).

References

1 Sen S. and Datta K. B., Stability bounds of singularity perturbed systems, Transactions on Automatic Control. (1993) 38, no. 2, 302–304, https://doi.org/10.1109/9.250478, MR1206817, ZBL0774.93053.
10.1109/9.250478
Web of Science® Google Scholar
2 Chen B.-S. and Lin C. L., On the stability bounds of singularly perturbed systems, Transactions on Automatic Control. (1990) 35, no. 11, 1265–1270, https://doi.org/10.1109/9.59817, MR1074897, ZBL0721.93059.
10.1109/9.59817
Web of Science® Google Scholar
3 Liu W. Q. and Sreeram V., A new characterization on stability bounds for singularly perturbed systems, Linear Algebra and Its Applications. (1997) 263, 377–388, https://doi.org/10.1016/S0024-3795(96)00597-6, MR1453980.
10.1016/S0024-3795(96)00597-6
Web of Science® Google Scholar
4 Liu W. Q., Paskota M., Sreeram V., and Teo K. L., Improvement on stability bounds for singularly perturbed systems via state feedback, International Journal of Systems Science. (1996) 28, no. 6, 571–578, 2-s2.0-0031166340.
10.1080/00207729708929418
Web of Science® Google Scholar
5 Geromel J. C., de Souza C. C., and Skelton R. E., Static output feedback controllers: stability and convexity, Transactions on Automatic Control. (1998) 43, no. 1, 120–125, https://doi.org/10.1109/9.654912, MR1604277, ZBL0952.93106.
10.1109/9.654912
Web of Science® Google Scholar
6 Shi P. and Dragan V., Asymptotic H_∞ control of singularly perturbed systems with parametric uncertainties, Transactions on Automatic Control. (1999) 44, no. 9, 1738–1742, https://doi.org/10.1109/9.788543, MR1709831, ZBL0958.93066.
10.1109/9.788543
Web of Science® Google Scholar
7 Xu L. G., Exponential p-stability of singularly perturbed impulsive stochastic delay differential systems, International Journal of Control, Automation, and Systems. (2011) 9, 966–972.
10.1007/s12555-011-0518-3
Web of Science® Google Scholar
8 Chen W. H., Yuan G., and Zheng W. X., Robust stability of singularly perturbed impulsive systems under nonlinear perturbation, IEEE Transactions on Automatic Control. (2013) 58, 168–174.
10.1109/TAC.2012.2203029
PubMed Web of Science® Google Scholar
9 Yang C., Sun J., and Ma X., Stabilization bound of singularly perturbed systems subject to actuator saturation, Automatica. (2013) 49, no. 2, 457–462, https://doi.org/10.1016/j.automatica.2012.11.004, MR3004711.
10.1016/j.automatica.2012.11.004
Web of Science® Google Scholar
10 Hachemi F., Sigalotti M., and Daafouz J., Stability analysis of singularly perturbed switched linear systems, Transactions on Automatic Control. (2012) 57, no. 8, 2116–2121, https://doi.org/10.1109/TAC.2011.2179876, MR2957186.
10.1109/TAC.2011.2179876
Web of Science® Google Scholar
11 Dong J. and Yang G.-H., Robust H_∞ control for standard discrete-time singularly perturbed systems, IET Control Theory & Applications. (2007) 1, no. 4, 1141–1148, https://doi.org/10.1049/iet-cta:20060234, MR2348638.
10.1049/iet-cta:20060234
Web of Science® Google Scholar
12 Dong J. and Yang G.-H., H_∞ control for fast sampling discrete-time singularly perturbed systems, Automatica. (2008) 44, no. 5, 1385–1393, https://doi.org/10.1016/j.automatica.2007.10.010, MR2531807.
10.1016/j.automatica.2007.10.010
Web of Science® Google Scholar
13 Xu S. and Feng G., New results on H_∞ control of discrete singularly perturbed systems, Automatica. (2009) 45, no. 10, 2339–2343, https://doi.org/10.1016/j.automatica.2009.06.011, MR2890797, ZBL1179.93071.
10.1016/j.automatica.2009.06.011
Web of Science® Google Scholar
14 Ivan M. and Daafouz J., Stabilisation of polytopic singularly perturbed linear systems, International Journal of Control. (2012) 85, no. 2, 135–142, https://doi.org/10.1080/00207179.2011.641128, MR2877333.
10.1080/00207179.2011.641128
Web of Science® Google Scholar
15 Vrabel R., On the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems, Systems & Control Letters. (2012) 61, no. 3, 422–426, https://doi.org/10.1016/j.sysconle.2012.01.002, MR2890401, ZBL1250.93034.
10.1016/j.sysconle.2012.01.002
Web of Science® Google Scholar
16 Chen J., Sun F., Yin Y., and Hu C., State feedback robust stabilisation for discrete-time fuzzy singularly perturbed systems with parameter uncertainty, IET Control Theory & Applications. (2011) 5, no. 10, 1195–1202, https://doi.org/10.1049/iet-cta.2010.0142, MR2850675.
10.1049/iet-cta.2010.0142
Web of Science® Google Scholar
17 Mehdi D., Boukas E. K., and Bachelier O., Static output feedback design for uncertain linear discrete time systems, IMA Journal of Mathematical Control and Information. (2003) 21, no. 1, 1–13, MR2037008, ZBL1049.93069.
10.1093/imamci/21.1.1
Google Scholar
18 Han Q. L., A delay decomposition approach to stability and H_∞ control of linear time-delay systems—part II: H_∞ control, Proceedings of the 7th World Congress on Intelligent Control and Automation (WCICA ′08), June 2008, 289–294, 2-s2.0-52149119103, https://doi.org/10.1109/WCICA.2008.4592939.
10.1109/WCICA.2008.4592939
Google Scholar
19 Gahinet P. and Apkarian P., A linear matrix inequality approach to H_∞ control, International Journal of Robust and Nonlinear Control. (1994) 4, no. 4, 421–448, https://doi.org/10.1002/rnc.4590040403, MR1286148, ZBL0808.93024.
10.1002/rnc.4590040403
Web of Science® Google Scholar
20 Boukas E. K., Static output feedback control for linear descriptor systems: LMI approach, 3, Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA ′05), August 2005, 1230–1234, 2-s2.0-27744437417.
Google Scholar

All articles

H_∞ Control of Discrete-Time Singularly Perturbed Systems via Static Output Feedback

Abstract

1. Introduction

2. Problem Formulation

3. Main Results

4. A Numerical Example

5. Conclusion

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

H∞ Control of Discrete-Time Singularly Perturbed Systems via Static Output Feedback

Abstract

1. Introduction

2. Problem Formulation

3. Main Results

4. A Numerical Example

5. Conclusion

Acknowledgments

References

References

Related

Information

H_∞ Control of Discrete-Time Singularly Perturbed Systems via Static Output Feedback