Consider the 2-D system expressed by Fornasini-Marchesini ...



Robust stability and robust [pic] control of 2-D discrete

systems with polytopic uncertainty

Lixin Gao

Department of Mathematics

Wenzhou Normal College

Wenzhou, Zhejiang, 325000

P. R. CHINA

1. Abstract: In this note, a new sufficient LMI condition for the robust stability is first established with respect to polytopic uncertainty. Then we establish Bounded Real Lemma and extended [pic] control condition for 2-D systems, from which guaranteed [pic] cost condition for uncertain 2-D systems in convex polytopic domains is given in terms of a set of linear matrix inequalities defined at the vertices of uncertainty domain. This approach is less conservative by using a parameter-dependent Lyapunov function and extends the results of the related references. Finally, some numerical examples are given to illustrate the results.

Key-Words: 2-D Systems, Robust Stability, [pic] Control, LMI, Bounded Real Lemma, Polytopic Uncertainty

2. Introduction

Two-dimensional (2-D) dynamical systems exist in many practical applications, such as systems for image data processing and transmission, thermal processes, gas absorption, water stream heating, etc. Thus the area of analysis and synthesis of 2-D control systems is an interesting and challenging problem, and it has received increasing interest in recent literature [1,2,3,4,5,6]. In recent years, the extension of the LMI-based methodology for 1-D systems to 2-D systems has also receiving wide attention, and many results have been established [1,2,3,4]. Unfortunately, many established results are sufficient condition, which are conservative results and not the full extension of the results of 1-D systems, due to analytical and structure complexity of 2-D systems. On the other hand, various types of necessary and sufficient condition for asymptotic stability for 2-D systems have already been established. All these conditions are not computationally tractable since they are based on frequency-dependent conditions.

The LMI-condition are computationally tractable, because there are effective and powerful algorithms such as interior-point method for the solution of LMI problems and a number of software packages are also available for solving LMI problems [10].

The basic approach is termed quadratic stability, which has been widely used for robust analysis, control and filter design for uncertain linear systems, but the quadratic stability may be very conservative due to the use of a single Lyapunov function for the entire uncertainty set. For reducing the conservatism, there have been also a few attempts to imply parameter-dependent Lyapunov matrices for some particular structure of uncertainties [8,9]. For real convex polytopic uncertainty, several LMI conditions have been established for robust stability of continuous-time case and for robust stability of discrete-time case [8,9]. These LMI conditions are sufficient conditions for robust stability of uncertain systems in convex polytopic domains and given in terms of a set of linear matrix inequalities defined at the vertices of uncertainty domain.

[12] establishes extended LMI characterizations for [pic] and [pic] norm computations, these extended analysis conditions feature a symmetric variable coming from Lyapunov theory as well as an extra instrumental variable. Consequently, they inherit all the nice properties from the extended stability and provide less conservative guaranteed [pic] and [pic] cost estimates for time-invariant uncertain systems. In section 3, We first establish Bounded realness lemma to 2-D systems. Then, using the similar ideas of [12], we establish extended stability condition and extended [pic] control condition for 2-D systems in this note. The robust stability condition and robust [pic] control condition uncertain 2-D systems in convex polytopic domains is given in terms of a set of linear matrix inequalities defined at the vertices of uncertainty domain.

The notation of this paper is standard. [pic] ([pic]) is denoted as transpose (conjugate transpose) of a matrix [pic]. For symmetric matrices [pic], [pic] means [pic] is positive (semi-) definite.

2.Stability and robust stability of 2-D discrete systems

Consider the 2-D system expressed by Fornasini-Marchesini second model:

[pic] (1)

where [pic].

Lemma 1[1]. 2-D system is asymptotically stable if there exist positive symmetric matrices [pic] such that:

[pic]. (2)

The condition of lemma 1 can easily be checked since this is an LMI condition. Unfortunately, there exits an asymptotically stable 2-D system which does not satisfy the condition of lemma, but it is less conservative LMI condition by now.

Lemma 2. The following statements are equivalent:

1) There exist positive symmetric matrices [pic] such that:

[pic]. (3)

2) There exist positive symmetric matrices [pic], such that

[pic] (4)

3) There exist positive symmetric matrices [pic], and matrix [pic] such that:

[pic] (5)

Proof. 1)[pic]2). If the condition 1) is satisfied, by using applying Schur complement, it is not difficult to get 2).

2)[pic]3). By choosing [pic], hence 2) implies 3).

3)[pic]1). By multiplying (5) to the left by [pic] and to the right by its transpose, we can get

[pic]

[pic]

Assume that the systems matrices [pic] that characterize the 2-D system (1) are uncertain but belong to a convex polytopic set defined as

[pic] (6)

Theorem 3. Uncertain 2-D system (1) is robustly stable in uncertainty domain (6), if there exist positive symmetric matrices [pic] such that:

[pic], (7)

for all [pic].

Proof. By using 2) of lemma 2, the following inequality can be obtained from (7)

[pic]

[pic]

so the uncertain system (1) is robustly stable.

Theorem 4. Uncertain system (1) is robustly stable in uncertainty domain (6), if there exist positive symmetric matrices [pic] and matrix [pic] such that:

[pic] (8)

for all [pic].

Proof. Defining a parameter-dependent matrix

[pic] and [pic], (9)

we multiply inequality (8) by [pic] respectively, and add all terms respectively to obtain

[pic]

Then, by using results of Lemma 1 and Lemma 2, we obtain the uncertain systems (1) is robust stability.

3. Bounded realness of 2-D systems

Consider the 2-D system expressed by Fornasini-Marchesini second model:

[pic] (10)

[pic] (11)

where [pic] represent the state, [pic] is the noise input which belongs to [pic], and [pic] is the controlled output. [pic], [pic], [pic], [pic]. The boundary condition of the system is assumed to be zero. The transfer function of this system is given by

[pic] (12)

where [pic] and [pic] are, respectively, the 1-step back-shift operator along [pic]-axis, and the one along [pic]-axis. The [pic] norm of [pic] is defined by

[pic], (13)

where [pic][11]. If [pic] is asymptotically stable, then

[pic]. (14)

Consider 1-D transfer function given in

[pic] (15)

where [pic], [pic].

Using the fact that

[pic]

we obtain

[pic] (16)

Given a positive scalar [pic], the 2-D system (10)-(11) with zero boundary is said to have an [pic] performance [pic] if it is asymptotically stable and its [pic] induced norm is bounded by [pic], i.e.

[pic]. (17)

Similar to 1-D system case, by using the 2-D Parseval’s Theorem, it is not difficult to show that condition (17) is equivalent to [pic].

The transfer function of 1-D system is given by

[pic]. (18)

The [pic] norm of stable 1-D system is given by

[pic], (19)

where [pic] denote the maximum singular value of a matrix.

To establish the result of this paper, we first recall following complex version discrete-time Bounded Real Lemma, the lemma can be similarly prove as real version discrete-time Bounded Real Lemma by using the results of [10].

Lemma 5. The systems (18) is stable and [pic] if and only if there exists a matrix [pic] satisfying

[pic]. (20)

Lemma 6[3]. Given constant matrices [pic] of appropriate dimensions where [pic] and [pic] are symmetric and [pic], then [pic] if and only if

[pic]or equivalently [pic].

(21)

Theorem 7. Given a positive scalar [pic], the 2-D system (10)-(11) has an [pic] performance [pic], i.e. [pic], if there exist a scale [pic] and positive symmetric matrices [pic] such that

[pic].

(22)

Proof. First, note (22) implies that

[pic],

so the systems is asymptotically stable.

By multiplying (22) to the left by

[pic]

and to the right by its conjugate transpose, then we obtain

[pic]

using lemma 6, the above inequality implies

[pic]

then [pic] from lemma 5, so the 2-D system (10)-(11) has an [pic] performance [pic].

Theorem 8[11]. Given a scalar [pic], the 2-D system (10)-(11) has an [pic] performance, i.e. [pic], if there exist positive symmetric matrices [pic] and matrix [pic] such that

[pic]

(23)

Proof. This lemma can be similarly proved as proving theorem 7 by choosing matrix

[pic].

to multiply the left of (23) and the right by its conjugate transpose.

Theorem 8 given in [11], but this theorem can be easy to prove as theorem 7. Compare to theorem 8, the LMI condition (22) given in theorem 7 has lower dimension than (23), and it seem as if the theorem 7 is directly extension of Bounded Real of 1-D systems for [pic], [pic]. The result of Theorem 8 has less conservative than the result of Theorem 7 because of the more free parameter [pic].

4. Extended LMI characterization of [pic] performance and guaranteed [pic] cost

Theorem 9. Given a positive scalar [pic], the 2-D system (10)-(11) has an [pic] performance, i.e. [pic], if there exist a scale [pic], positive symmetric matrices [pic] and matrix [pic] such that

[pic].

(24)

Proof. If the inequality (24) is feasible, then [pic], which implies [pic] is nonsingular. Since [pic], we obtain [pic] which yields

[pic]

By multiplying the above inequality to the left by [pic] and to the right by its transpose, then we obtain

[pic].

Assume that the matrices that characterize the dynamic system (10)-(11) are uncertain but belong to a convex and bound set. This set is such that matrix [pic] defined by

[pic] (25)

belongs to a convex bounded polyhedron [pic]. This is, each uncertain matrix in this domain may be written as an unknown convex combination of [pic] given extreme matrices [pic] such that

[pic] (26)

Theorem 10. If there exist scales [pic], positive symmetric matrices [pic] and matrix [pic] such that

[pic]

(27)

where the matrices [pic] and [pic] define the extreme matrices [pic], [pic], then the all 2-D system (10)-(11) characterized by matrix [pic] in the domain [pic] have an [pic] performance [pic], i.e. [pic].

Proof. Consider the parameter dependent Lyapunov matrix defined by

[pic], [pic], [pic] (28)

and [pic]. Multiply each of these inequalities by [pic] and sum them up to obtain

[pic]so theorem 10 can be established by applying result of theorem 9.

Corollary 11. If there exists scales [pic], positive symmetric matrices [pic] such that

[pic] (29)

where the matrices [pic] and [pic] define the extreme matrices [pic], [pic], then the all 2-D system (10)-(11) characterized by matrix [pic] in the domain [pic] have an [pic] performance [pic], i.e. [pic].

Using the results of theorem 8, we can also get similar results as the theorem 9 and theorem 10.

5. Numerical examples

In this section, we give some examples to demonstrate the feasibility of the approach presented in this paper by using LMI solver of MATLAB.

Example 1. Consider the 2-D system expressed by Fornasini-Marchesini second model:

[pic]

and uncertain system matrix [pic] located in polytope [pic] described by the [pic] vertex [pic],

[pic],

[pic]

using theorem 3, we can not get a common matrix [pic] by using LMI solver of MATLAB. But, using theorem 4, we get [pic],[pic] by using LMI solver of MATLAB. So the sufficient condition given by theorem 4 has less conservative than that by theorem 3.

Example 2. Consider the 2-D Fornasini-Marchesini second model with

[pic], [pic],

[pic], [pic], [pic], [pic].

It is easy to get the [pic] norm of this system is 10 by directly computation [11]. By applying theorem 7, (22) is infeasible for [pic] but (22) is feasible for [pic] by using LMI solver of MATLAB, but by applying theorem 8 the optimal admissible [pic] is 10.2 [11]. Thus, the result given by theorem 7 presents a less conservative performance than that by theorem 8 at least in this example.

6. Conclusion

In this note, a new sufficient LMI condition for the robust stability is first established with respect to polytopic uncertainty. Then we establish Bounded Real Lemma and extended [pic] control condition for 2-D systems, form which guaranteed [pic] cost condition for uncertain 2-D systems in convex polytopic domains is given in terms of a set of linear matrix inequalities defined at the vertices of uncertainty domain. This approach is less conservative by using a parameter-dependent Lyapunov function. Several numerical examples have been solved in order to illustrate results of this note. Some LMI condition for feedback synthesis can also be established by similar idea.

References

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9] S. P. Boyd, L. EI Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM, Philadelphia, 1994.

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