离散Roesser模型的2D广义Kalman-Yakubovich-Popov引理分析与设计

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"离散Roesser模型的广义二维Kalman-Yakubovich-Popov引理" 在信号处理和控制系统领域，离散Roesser模型是一种用于描述和分析二维动态系统的数学工具，它以矩阵差分方程的形式表示系统的行为。这种模型在处理图像处理、多变量控制和通信网络等问题时特别有用。而Kalman-Yakubovich-Popov（KYP）引理是线性系统理论中的一个关键结果，它在单变量系统分析和设计中扮演着重要角色，尤其是在稳定性分析和控制器设计方面。然而，对于二维系统，特别是在离散Roesser模型下，传统KYP引理并未被广泛应用于无限频率域。该研究论文填补了这一空白，提出了一种针对离散Roesser模型的广义二维KYP引理。这个新引理扩展了KYP引理的概念，将二维系统的频率域特性，如正实性和有界实性，与线性矩阵不等式（LMI）关联起来。通过LMI，可以更高效地进行系统分析和设计优化。正实性意味着系统的传递函数在所有实频点上都是正的，这对应于系统无负实部的极点，即系统是稳定的。有界实性则涉及系统在复平面上的极点和零点分布，确保了系统输入和输出之间的能量守恒，即输入信号的平方积分与输出信号的平方积分之间存在界。论文中，作者详细探讨了这个广义KYP引理的特殊情形，即2-Dbounded realness（二维有界实性）和2-D positive realness（二维正实性）。这些特性对于理解二维系统的行为至关重要，因为它们直接影响系统的稳定性和性能。为了进一步证明该引理的实际应用价值，论文提供了数字例子，展示了如何利用这个新引理来设计二维数字滤波器。这些示例突出了广义KYP引理在二维系统设计中的相关性和实用性，为工程师和研究人员提供了一个有力的工具，以解决复杂系统的问题。这篇研究论文不仅在理论上扩展了经典KYP引理，而且在实际应用中为二维系统的分析和设计开辟了新的路径，对进一步推动二维系统理论的发展具有重要意义。

YANG et al.: GENERALIZED 2-D KALMAN–YAKUBOVICH–POPOV LEMMA FOR DISCRETE ROESSER MODEL 3225

B. Generalized KYP Lemma for Roesser Model

By applying Lemma 2, we can obtain the generalized 2-D

KYP Lemma for the 2-D system in Roesser form (1), which is

given in the following theorem.

Theorem 1: Consider the Roesser model (1) and suppose that

the matrix

satisﬁes for all ,

where

denotes the determinant of a matrix. For the matrices

and , two scalars and a

Hermitian matrix

with ,if

there exist Hermitian matrices

and in the form of (2) such

that

(5)

holds, where , then the fol-

lowing ﬁnite frequency condition:

(6)

is satisﬁed for all

, ,

where

Proof: It follows from Lemma 2 that for any

the inequality

(7)

is satisﬁed for any

Considering the Roesser model (1), it is clear that if

where denote the -transforms of the state variable and

the input variable

. Introduce a set

Then, (7) implies that, for each , the inequality

(8)

is satisﬁed for any

(9)

Therefore, if there exists a matrix

such that (5) exists,

then, for any

, it holds that . Obviously, for all

, ,

. Then, (6) holds.

Remark 1: Note that the result of Theorem 1 is only sufﬁcient

since there is no guarantee that there exists a matrix

such

that

even if (6) holds and (8) is satisﬁed for any .

This is in contrast with the 1-D KYP lemma [13], [14] which,

based on the exact (nonconservative) generalized

-procedure

theory, provides a sufﬁcient and necessary condition for the ex-

istence of the ﬁnite frequency property. The reason lies in that

the set

in terms of (7) is not necessarily a lossless set (see

Lemma 4, given in the Appendix), and therefore it cannot lead

to an exact

-procedure.

Note that

, and are generally complex matrices; a cri-

terion for the positive deﬁniteness of a complex matrix is given

below.

Lemma 3: Let

with

. Then, if and only if

(10)

Also, let

with .

Then,

if and only if

(11)

Observe that both the matrices in (10) and (11) are symmetric

because of the fact that

is symmetric and ,

which is a property of Hermitian matrices.

Remark 2: By representing the complex matrices

, , and

then (5) is a complex LMI and, once we represent and ,in

turn, as

with and ,

then, according to Lemma 3, the complex LMI (5) is equivalent

to the following real matrix inequality, which is more convenient

for numerical computation:

(12)

where

Authorized licensed use limited to: SUN YAT-SEN UNIVERSITY. Downloaded on January 1, 2009 at 03:46 from IEEE Xplore. Restrictions apply.

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