Criterion for limit cycle-free state-space digital filters with external disturbances and generalized overflow non-linearities

Abstract

This paper investigates the problem of $H_{\infty}$ elimination of overflow oscillations in fixed-point state-space digital filters using generalized overflow non-linearities and external disturbance. The generalized overflow non-linearities under consideration cover the common types of overflow arithmetic used in practice, for instance zeroing, two’s complement, triangular and saturation. New criteria are established to ensure not only exponential stability, but also reduction in the effect of external disturbance to an $H_{\infty}$ norm constraint. The obtained criteria are in linear matrix inequality (LMI) framework and, hence, are computationally tractable. The presented approach constitutes a generalization over several previously reported approaches for the $H_{\infty}$ elimination of overflow oscillations. For saturation non-linearities, the presented result turns out to be less conservative than several existing criteria. Numerical examples are provided to demonstrate the effectiveness of the presented approach.

Keywords

Digital filters exponential stability finite wordlength effects non-linear systems

Introduction

In the implementation of linear digital filters with fixed-point arithmetic using a digital computer or special-purpose hardware, one encounters several kinds of non-linearities, namely quantization and overflow. The existence of these non-linearities may lead to instability in the designed system. Therefore, the study of stability properties of digital filters with finite wordlength non-linearities is important due to its theoretical interest as well as for application to practical system design. Several researchers have extensively investigated the stability of non-linear systems (Ahn, 2011, 2013a, 2013b, 2013c, 2014a, 2014b, 2015; Ahn and Kim, 2013; Ahn and Lee, 2012; Bose and Chen, 1991; Dey and Kar, 2011a, 2011b; Ebert et al., 1969; Kandanvli and Kar, 2008, 2012; Kar, 2007, 2010; Kar and Singh, 1998, 2000, 2001; Kokil, 2014; Kokil and Kar, 2012; Kokil and Shinde, 2015, 2016; Kokil et al., 2012; Li and Zheng, 2012; Liu and Michel, 1992; Sandberg, 1979; Singh, 1985, 2006). In general, high-order recursive digital filters can be very sensitive to finite wordlength non-linearities. For the implementation of high-order digital filters, several serially cascaded biquad filters need to be implemented. In such a type of implementation, mutual or external interferences between biquad filters that may lead to poor performance and even destruction phenomenon are unavoidable (Monteiro and Leuken, 2010; Tsividis, 2002). There are many results available in literature on the stability analysis of discrete systems, but they are not valid in unfavourable environments with external interference (Bose and Chen, 1991; Dey and Kar, 2011a, 2011b; Ebert et al., 1969; Kandanvli and Kar, 2008, 2012; Kar, 2007, 2010; Kar and Singh, 1998, 2000, 2001; Kokil, 2014; Kokil and Kar, 2012; Li and Zheng, 2012; Liu and Michel, 1992; Sandberg, 1979; Singh, 1985, 2006). In control and filtering problems, energy-to-peak filtering is generally used to deal with the insufficient statistical signal information. Ahn (2013a) has presented a result for $l_{2} - l_{\infty}$ stability of fixed-point state-space digital filters using saturation arithmetic and external interference. Criteria for the realization of limit cycle-free state-space direct form digital filters employing saturation arithmetic have been presented in Ahn and Kim (2013) and Kokil and Shinde (2015). By making use of the induced $l_{\infty}$ approach, stability conditions for the saturation overflow stability of fixed-point state-space interfered digital filters have been derived in Ahn and Lee (2012) and Kokil and Shinde (2016). Recently, a few criteria dealing with $H_{\infty}$ elimination of overflow oscillations in fixed-point state-space digital filters employing saturation arithmetic and external disturbance have been appeared (Ahn, 2011, 2015; Kokil et al., 2012). However, these results (Ahn, 2011, 2015; Kokil et al., 2012) are not applicable for interfered digital filters with generalized overflow non-linearities, which motivated us to carry out the present work.

This paper, therefore, studies the problem of exponential stability and $H_{\infty}$ performance of fixed-point state-space digital filters under the effect of generalized overflow non-linearities and external disturbance. The paper is organized as follows: in the next section, a description of system under consideration is given. New linear matrix inequality (LMI)-based criteria for the exponential stability and $H_{\infty}$ performance of fixed-point state-space digital filters with generalized overflow non-linearities and external disturbance are established. Several numerical examples are provided to demonstrate the effectiveness of the presented results.

System description

This paper considers the problem of exponential stability and $H_{\infty}$ performance of fixed-point state-space interfered digital filters, which operate under the effect of generalized overflow non-linearities. Specifically, the system under consideration is given by

\begin{matrix} x (r + 1) = f (y (r)) + w (r) \\ = {[f_{1} (y_{1} (r)) f_{2} (y_{2} (r)) \dots f_{n} (y_{n} (r))]}^{T} \\ + {[w_{1} (r) w_{2} (r) \dots w_{n} (r)]}^{T} \end{matrix}

(1a)

y (r) = {[y_{1} (r) y_{2} (r) \dots y_{n} (r)]}^{T} = Ax (r)

(1b)

where $x (r) \in R^{n}$ is a state vector, $w (r) \in R^{n}$ is an external disturbance, $f (.)$ represents generalized overflow non-linearities, $A \in R^{n \times n}$ is the coefficient matrix. The generalized overflow characteristic is given by

\begin{matrix} L \leq f_{i} (y_{i} (r)) \leq 1, & y_{i} (r) > 1 \\ f_{i} (y_{i} (r)) = y_{i} (r), & - 1 \leq y_{i} (r) \leq 1 \\ - 1 \leq f_{i} (y_{i} (r)) \leq - L, & y_{i} (r) < - 1 \end{matrix}} \begin{matrix} i = 1, 2, \dots, n \end{matrix},

(2)

where

- 1 \leq L \leq 1 .

(3)

By suitable selection of L, (2) represents the various overflow non-linearities used in practice, i.e. saturation $(L = 1)$ , zeroing $(L = 0)$ , two’s complement $(L = - 1)$ , triangular $(L = - 1)$ , etc.

Now we recall the following definitions.

Definition 1(Liu and Michel, 1992). The equilibrium $x_{e} = 0$ of system (1) without external interference is said to be asymptotically stable if (i) it is stable in the sense of Lyapunov, i.e. for every $ε > 0$ there is a $δ = δ (ε)$ such that $‖ x (r) ‖ < ε$ for all $r = 0, 1, 2, \dots,$ whenever $‖ x (0) ‖ < δ$ and (ii) every solution of (1) tends to the origin as $r \to \infty$ , i.e. $lim_{r \to \infty} x (r) = 0$ .

Definition 2(Yeganefar et al., 2013). The equilibrium $x_{e}$ of system (1) without external interference is called exponentially stable if there exist constants $q \in (0, 1)$ and $C > 0$ such that for every trajectory x and all $r \geq 0$ , we have $‖ x (r) ‖ \leq C q^{r} ‖ x (0) ‖$ .

For a known interference attenuation level $γ > 0$ , the paper aims to establish criteria such that the system (1)–(3) with $w (r) = 0$ is exponentially stable and

\sum_{r = 0}^{\infty} x^{T} (r) S x (r) < γ^{2} \sum_{r = 0}^{\infty} w^{T} (r) w (r)

(4)

under zero-initial conditions for all non-zero $w (r)$ , where S is a positive definite symmetric matrix. In this case, the system (1)–(3) is said to be exponentially stable with $H_{\infty}$ performance $γ$ .

Equations (1)–(3) can be used to describe a broader class of realistic interfered discrete-time dynamical systems, which include digital filters implemented with finite wordlength register and external disturbance, digital control systems in presence of external disturbance and overflow non-linearities, neural networks defined on hypercubes, material rolling process, metal cutting process and many other engineering problems.

The problem of exponential stability and $H_{\infty}$ performance of fixed-point state-space digital filters subject to the saturation non-linearities (L=1) has been studied in Ahn (2011) and Kokil et al. (2012). The main result of Kokil et al. (2012) may be stated as follows.

Theorem 1(Kokil et al., 2012). For a given level $γ > 0$ , suppose there exist positive definite symmetric matrices $P$ , $S$ and a positive scalar $δ$ such that

[\begin{matrix} δ A^{T} A + S - P & A^{T} N & 0 \\ N^{T} A & P - δ I - (N + N^{T}) & P \\ 0 & P & P - γ^{2} I \end{matrix}] < 0

(5)

where N is a row diagonally dominant matrix with positive diagonal elements, then the system (1) employing saturation arithmetic is exponentially stable with $H_{\infty}$ performance $γ$ .

Main results

Consider the following two ranges for L:

0 \leq L \leq 1

(6)

- 1 \leq L < 0

(7)

These two ranges together constitute (3). In the following, a criterion applicable to (6) and a different criterion applicable to (7) for the exponential stability of (1) and (2) with $H_{\infty}$ performance $γ > 0$ employing generalized overflow arithmetic are presented.

Several useful lemmas, which are needed in the proof of our main results are given in Appendix A. Pertaining to (1), (2) and (6), our main result may be stated as follows.

Theorem 2. For a known interference attenuation level $γ > 0$ , if there exist positive definite symmetric matrices $P \in R^{n \times n}$ and $S \in R^{n \times n}$ such that

ζ = [\begin{matrix} A^{T} GA + S - P & A^{T} C & 0 \\ C^{T} A & P - C - C^{T} - G & P \\ 0 & P & P - γ^{2} I \end{matrix}] < 0

(8)

where $G = [g_{i j}] \in R^{n \times n}$ and $C = [c_{i j}] \in R^{n \times n}$ are characterized by (A.1) and (A.4), respectively, then the system (1), (2) and (6) is exponentially stable with $H_{\infty}$ performance $γ$ .

Proof. Consider the following Lyapunov function

V (x (r)) = x^{T} (r) P x (r)

(9)

which satisfies the following Rayleigh inequality (Strang, 1986)

λ_{\min} (P) ‖ x (r) ‖^{2} \leq V (x (r)) \leq λ_{\max} (P) ‖ x (r) ‖^{2}

(10)

where $λ_{\min} (P)$ and $λ_{\max} (P)$ represent minimum and maximum eigenvalue of matrix P , respectively.

Application of (9) to (1a) leads to

\begin{matrix} Δ V (x (r)) = V (x (r + 1)) - V (x (r)) \\ = [f (Ax (r)) + w (r)]^{T} P [f (Ax (r) + w (r))] - x^{T} (r) P x (r) \\ = f^{T} (Ax (r)) P f (Ax (r)) + f^{T} (Ax (r)) P w (r) + w^{T} (r) P f (Ax (r)) \\ + w^{T} (r) P w (r) - x^{T} (r) P x (r) \end{matrix}

(11)

By using Lemmas A.1 and A.2, it follows from (11) that

\begin{matrix} Δ V (x (r)) \leq f^{T} (Ax (r)) P f (Ax (r)) + f^{T} (Ax (r)) P w (r) \\ + w^{T} (r) P f (Ax (r)) + w^{T} (r) P w (r) \\ - x^{T} (r) P x (r) + x^{T} (r) A^{T} C f (Ax (r)) \\ + f^{T} (Ax (r)) C^{T} A x (r) - f^{T} (Ax (r)) (C + C^{T}) f (Ax (r)) \\ + x^{T} (r) A^{T} GA x (r) - f^{T} (Ax (r)) G f (Ax (r)) \\ = μ^{T} (r) ζ μ (r) - x^{T} (r) S x (r) + γ^{2} w^{T} (r) w (r) \end{matrix}

(12)

where

μ^{T} (r) = [\begin{matrix} x^{T} (r) & f^{T} (Ax (r)) & w^{T} (r) \end{matrix}]

(13)

Thus, if (8) is satisfied then

Δ V (x (r)) < - x^{T} (r) S x (r) + γ^{2} w^{T} (r) w (r)

(14)

Summation both sides of (14) from 0 to $\infty$ yields

V (x (\infty)) - V (x (0)) < - \sum_{r = 0}^{\infty} x^{T} (r) S x (r) + γ^{2} \sum_{r = 0}^{\infty} w^{T} (r) w (r)

(15)

As $V (x (\infty)) \geq 0$ and $V (x (0)) = 0$ , the relation (4) follows trivially from (15).

Now, it remains to show that, under the condition (8), the system (1), (2) and (6) with $w (r) = 0$ is exponentially stable. When $w (r) = 0$ , (14) implies

Δ V (x (r)) < - x^{T} (r) S x (r) \leq - λ_{\min} (S) ‖ x (r) ‖^{2}

(16)

In the light of Theorem 3.1 in Lee (2002), (10) and (16) guarantee the exponential stability of the system under consideration. This completes the proof of Theorem 2.

Remark 1. The matrix inequality (8) is linear in the unknown parameters $α_{i j}$ , $β_{i j}$ , $λ_{i j}$ , $η_{i j}$ $(i, j = 1, 2, \dots, n (i \neq j))$ , $k_{i}$ $(i = 1, 2, \dots, n)$ , S and P . Therefore, (8) can be easily solved using MATLAB LMI Toolbox (Boyd et al., 1994; Gahinet et al., 1995; Sturm, 1999).

Remark 2. It may be observed that, for saturation non-linearities (i.e. L = 1), C (A.4) in Theorem 2 is identified as a general row diagonally dominant matrix N with positive diagonal elements and G (A.1) becomes a general symmetric positive definite row diagonally dominant matrix (Dey and Kar, 2011a). Consequently, with $G = δ I$ and $L = 1$ , (8) reduces to (5). Thus, Theorem 1 is recovered from Theorem 2 as a special case.

Remark 3. By selecting $α_{i j}$ = $β_{i j}$ $(i, j = 1, 2, \dots, n (i \neq j))$ and $L = 1$ (i.e. saturation non-linearities), the matrix $C$ given by (A.4) reduces to a positive definite diagonal matrix. Therefore, Theorem 1 in Kokil et al. (2012) is a special case of Theorem 2, i.e. corresponding to $α_{i j}$ = $β_{i j}$ $(i, j = 1, 2, \dots, n (i \neq j))$ , $L = 1$ and $G = δ I$ . Referring to the saturation non-linearities, the superiority of Theorem 2 will be illustrated in Example 1.

The stability result for the system described by (1), (2) and (7) may be stated as follows.

Theorem 3. For a given interference attenuation level $γ > 0$ , suppose there exist positive definite symmetric matrices $P \in R^{n \times n}$ , $S \in R^{n \times n}$ and a positive definite diagonal matrix $D \in R^{n \times n}$ satisfying

Γ = [\begin{matrix} A^{T} G A - 2 L A^{T} D A + S - P & (1 + L) A^{T} D & 0 \\ (1 + L) D A & P - G - 2 D & P \\ 0 & P & P - γ^{2} I \end{matrix}] < 0

(17)

where $G = [g_{ij}] \in R^{n \times n}$ is characterized by (A.1), then the system (1), (2) and (7) is exponentially stable with $H_{\infty}$ performance $γ$ .

Proof. By employing Lemmas A.1 and A.3, (11) yields

\begin{matrix} Δ V (x (r)) \leq f^{T} (Ax (r)) P f (Ax (r)) + f^{T} (Ax (r)) P w (r) \\ + w^{T} (r) P f (Ax (r)) + w^{T} (r) P w (r) - x^{T} (r) P x (r) \\ + x^{T} (r) A^{T} GA x (r) - f^{T} (Ax (r)) G f (Ax (r)) \\ + (1 + L) y^{T} (r) D f (y (r)) \\ + (1 + L) f^{T} (y (r)) D y (r) - 2 f^{T} (y (r)) D f (y (r)) - 2 L y^{T} (r) D y (r) \\ = μ^{T} (r) Γ μ (r) - x^{T} (r) S x (r) + γ^{2} w^{T} (r) w (r) \end{matrix}

(18)

where $μ^{T} (r) = [\begin{matrix} x^{T} (r) f^{T} (Ax (r)) w^{T} (r) \end{matrix}]$ . Equation (18) implies

Δ V (x (r)) < - x^{T} (r) S x (r) + γ^{2} w^{T} (r) w (r)

(19)

if (17) holds true. By summing both sides of (19) over $r = 0 \to$ $\infty$ and making use of the relations $V (x (\infty)) \geq 0$ , $V (x (0)) = 0$ , one can obtain (4). Furthermore, it follows from (19) that

Δ V (x (r)) < - x^{T} (r) S x (r) \leq - λ_{\min} (S) ‖ x (r) ‖^{2}

(20)

if $w (r) = 0$ . According to Theorem 3.1 in Lee (2002), (10) and (20) ensure the exponential stability of the system under consideration. This completes the proof of Theorem 3.□

Remark 4. The condition (17) is in the form of LMI and therefore is computationally tractable (Boyd et al., 1994; Gahinet et al., 1995; Sturm, 1999).

Remark 5. For triangular and two’s complement overflow non-linearities, one is required to choose L = −1 in Theorem 3, in which case G characterized by (A.1) reduces to a positive definite diagonal matrix. Observe that (2) together with (7) fails to make any distinction between triangular and two’s complement overflow non-linearities. Therefore, the characterization of triangular non-linearities with the help of (2) and (7) appears to be overly restrictive.

Finally, consider a class of non-linearities confined to the sector [L, 1], i.e.

f_{i} (0) = 0, {Ly}_{i}^{2} (r) \leq f_{i} (y_{i} (r)) y_{i} (r) \leq y_{i}^{2} (r), i = 1, 2, \dots, n,

(21a)

where

- 1 \leq L < 0

(21b)

In (21a), one is required to choose L = −1/3 for triangular overflow arithmetic and L = −1 for two’s complement overflow arithmetic.

Pertaining to the system (1) and (21), we have the following result.

Theorem 4. For a given interference attenuation level $γ > 0$ , if there exist positive definite symmetric matrices $P \in R^{n \times n}$ , $S \in R^{n \times n}$ and positive definite diagonal matrices $D \in R^{n \times n}$ and $\hat{G} \in R^{n \times n}$ satisfying

Θ = [\begin{matrix} A^{T} \hat{G} A - 2 L A^{T} D A + S - P & (1 + L) A^{T} D & 0 \\ (1 + L) D A & P - \hat{G} - 2 D & P \\ 0 & P & P - γ^{2} I \end{matrix}] < 0

(22)

then the system (1) and (21) is exponentially stable with $H_{\infty}$ performance $γ$ .

Proof. For a positive diagonal matrix $\hat{G}$ , we have

x^{T} (r) A^{T} \hat{G} A x (r) - f^{T} (Ax (r)) \hat{G} f (Ax (r)) \geq 0

(23)

in view of (21). Moreover, (A.7) holds true for the non-linearities given by (21). Therefore, from (11), the upper bound of $Δ V (x (r))$ can be obtained as

\begin{matrix} Δ V (x (r)) \leq f^{T} (Ax (r)) P f (Ax (r)) + f^{T} (Ax (r)) P w (r) \\ + w^{T} (r) P f (Ax (r)) + w^{T} (r) P w (r) - x^{T} (r) P x (r) \\ + x^{T} (r) A^{T} \hat{G} A x (r) - f^{T} (Ax (r)) \hat{G} f (Ax (r)) \\ + (1 + L) y^{T} (r) D f (y (r)) + (1 + L) f^{T} (y (r)) D y (r) \\ - 2 f^{T} (y (r)) D f (y (r)) - 2 L y^{T} (r) D y (r) \\ = μ^{T} (r) Θ μ (r) - x^{T} (r) S x (r) + γ^{2} w^{T} (r) w (r) \end{matrix}

(24)

where $μ^{T} (r) = [\begin{matrix} x^{T} (r) f^{T} (Ax (r)) w^{T} (r) \end{matrix}]$ .

If (22) is satisfied, then

Δ V (x (r)) < - x^{T} (r) S x (r) + γ^{2} w^{T} (r) w (r)

(25)

Summing both sides of (25) over $r = 0 \to$ $\infty$ and using $V (x (\infty)) \geq 0$ , $V (x (0)) = 0$ , one arrives at (4). With $w (r) = 0$ , (25) leads to

Δ V (x (r)) < - x^{T} (r) S x (r) \leq - λ_{\min} (S) ‖ x (r) ‖^{2}

(26)

In view of Theorem 3.1 in Lee (2002), the exponential stability of the system is assured by (10) and (26). This completes the proof of Theorem 4.□

Remark 6. It may be noted that Theorem 4, unlike Theorem 3, makes distinction between triangular and two’s complement overflow non-linearities. It has been observed that, for triangular overflow arithmetic, Theorem 4 always leads to more relaxed stability conditions than Theorem 3.

Remark 7. It is obvious that the presented criteria (Theorems 2–4) for exponential stability also imply asymptotic stability. This ensures the absence of overflow oscillations in the realized digital filter.

Remark 8. Recently, Ahn (2013c, 2014b) has proposed novel $H_{\infty}$ stability criteria for two-dimensional (2-D) digital filters with external interference and saturation arithmetic. The criteria in those studies have been derived by making use of the sector information of the saturation non-linearities. On the other hand, the presented approach utilizes a more precise characterization of the saturation non-linearities (namely (A.3) and (A.6)) than the approaches in Ahn (2013c, 2014b). Moreover, the criteria in those articles are not applicable to 2-D systems involving triangular or two’s complement overflow arithmetic. The possible extension of the presented approach to the study of $H_{\infty}$ stability of 2-D interfered digital filters using generalized overflow arithmetic appears to be an interesting problem for future investigation.

Remark 9. The $H_{\infty}$ norm (Stoorvogel, 1992) is defined as

‖ T_{x w} ‖_{\infty} = \frac{\sqrt{\sum_{r = 0}^{\infty} x^{T} (r) S x (r)}}{\sqrt{\sum_{r = 0}^{\infty} w^{T} (r) w (r)}}

(27)

where $T_{x w}$ is a transfer function matrix from $w (r)$ to $x (r)$ . As discussed in Ahn (2011), the relation (4) can be represented by $H (\infty) < γ^{2}$ where

H (r) = \frac{\sum_{r = 0}^{\infty} x^{T} (r) S x (r)}{\sum_{r = 0}^{\infty} w^{T} (r) w (r)}

(28)

Numerical examples

To demonstrate the usefulness of the proposed results, let us consider the following examples.

Example 1

Consider a specific example of the system (1) and (2) with

A = [\begin{matrix} 1.1 & - 1 \\ 0.78 & 0 \end{matrix}], w (r) = [\begin{matrix} \cos (r) \\ \sin (r) \end{matrix}], L = 1 .

(29)

Pertaining to the design objective (8), assume the $H_{\infty}$ performance be specified by $γ = 9.9 \times 1 0^{- 5}$ . Using the MATLAB LMI toolbox (Boyd et al., 1994; Gahinet et al., 1995), it can easily be verified that Theorem 1 of Kokil et al. (2012) fails to establish the exponential stability of the present system. Furthermore, in view of Lemma 1 in that study, it turns out that the present example falls outside the application scope of Theorem 1 in that same study. On the other hand, (8) is feasible for the following values of unknown parameters:

\begin{matrix} P = [\begin{matrix} 0.1418 & - 0.0959 \\ - 0.0959 & 0.1909 \end{matrix}] \times 1 0^{- 8}, S = [\begin{matrix} 0.3759 & - 0.1968 \\ - 0.1968 & 0.5288 \end{matrix}] \times 1 0^{- 10}, G = [\begin{matrix} 0.0847 & - 0.0536 \\ - 0.0536 & 0.1092 \end{matrix}] \times 10^{- 8} \\ (k_{1} = 1.7187 \times 1 0^{- 10}, k_{2} = 4.1677 \times 1 0^{- 10}, λ_{12} = 6.9679 \times 1 0^{- 11}, η_{12} = 6.0521 \times 1 0^{- 10}), \\ C = [\begin{matrix} 0.1052 & - 0.0837 \\ - 0.0900 & 0.1439 \end{matrix}] \times 10^{- 8} \\ (α_{12} = 1.0753 \times 1 0^{- 10}, β_{12} = 9.4422 \times 1 0^{- 10}, α_{21} = 2.6942 \times 1 0^{- 10}, β_{21} = 1.1692 \times 1 0^{- 9}) \end{matrix}

(30)

Thus, Theorem 2 succeeds to determine the exponential stability of the system with $H_{\infty}$ performance level $γ = 9.9 \times 1 0^{- 5}$ .

Figure 1 shows the plot of H (r) for the example under consideration. From Figure 1, it is clear that $H (\infty) < γ^{2} = 9.8 \times 1 0^{- 9}$ and consequently the $H_{\infty}$ norm from the external disturbance w (r) to the state vector x (r) is reduced within the $H_{\infty}$ norm bound $γ$ . Hence, according to Theorem 2, the system under consideration is exponentially stable with $H_{\infty}$ performance $γ = 9.9 \times 1 0^{- 5}$ .

Figure 1.

The plot of H (r) for Example 1.

Example 2

Consider a second-order system (1) and (2) with

A = [\begin{matrix} 0.95 & - 0.5 \\ 0.2 & 0 \end{matrix}], w (r) = [\begin{matrix} \cos (r) \\ \sin (r) \end{matrix}], L = 0.5

(31)

Let the $H_{\infty}$ performance level be specified as $γ = 0.7$ . It turns out that (8) is feasible for the following values of unknown parameters

\begin{matrix} P = [\begin{matrix} 0.1030 & - 0.0690 \\ - 0.0690 & 0.2794 \end{matrix}], S = [\begin{matrix} 0.0027 & - 0.0060 \\ - 0.0060 & 0.0900 \end{matrix}], G = [\begin{matrix} 0.0393 & - 0.0111 \\ - 0.0111 & 0.5139 \end{matrix}] \\ (k_{1} = 0.0118, k_{2} = 0.4864, λ_{12} = 0.0082, η_{12} = 0.0193), \\ C = [\begin{matrix} 0.0925 & - 0.0432 \\ - 0.9728 & 1.4565 \end{matrix}] \\ (α_{12} = 0.0247, β_{12} = 0.0679, α_{21} = 0.2418, β_{21} = 1.2147) \end{matrix}

(32)

Therefore, according to Theorem 2, the system is exponentially stable with $H_{\infty}$ performance $γ = 0.7$ .

Figure 2 depicts the plot of H (r) for the example under consideration. From Figure 2, it is clear that $H (\infty) < γ^{2}$ = $0.49$ and consequently, the $H_{\infty}$ norm from the external disturbance w (r) to the state vector x (r) is reduced within the $H_{\infty}$ norm bound $γ$ .

Figure 2.

The plot of H (r) for Example 2.

Example 3

Consider the discrete-time system described by (1) and (2) with the following parameters

A = [\begin{matrix} 0.85 & - 0.35 \\ 0.25 & 0 \end{matrix}], w (r) = [\begin{matrix} \cos (r) \\ \sin (r) \end{matrix}], L = - 1

(33)

and $H_{\infty}$ performance $γ = 0.65$ . For the present example, (17) gives the following feasible solutions

\begin{matrix} P = [\begin{matrix} 0.0348 & - 0.0030 \\ - 0.0030 & 0.0442 \end{matrix}], S = [\begin{matrix} 0.0013 & 0.0043 \\ 0.0043 & 0.0197 \end{matrix}], G = [\begin{matrix} 0.0301 & 0 \\ 0 & 0.0463 \end{matrix}] \\ (k_{1} = 0.0123, k_{2} = 0.0284, λ_{12} = η_{12} = 0.0089), \\ D = [\begin{matrix} 0.0044 & 0 \\ 0 & 0.0093 \end{matrix}] \end{matrix}

(34)

Thus, Theorem 2 assures the system is exponentially stable with $H_{\infty}$ performance $γ = 0.65$ .

Figure 3 demonstrates the plot of H (r) and it is clear that $H (\infty) < γ^{2}$ = $0.4225$ . Consequently, the $H_{\infty}$ norm from the external disturbance w (r) to the state vector x (r) is reduced within the $H_{\infty}$ norm bound $γ$ .

Figure 3.

The plot of H (r) for Example 3.

Example 4

Consider a third-order system (1)–(3) with

A = [\begin{matrix} 0.5 & - 0.3 & 0 \\ 0.2 & - 0.5 & 0 \\ 0.3 & 0 & 0.2 \end{matrix}], w (r) = [\begin{matrix} n_{1} (r) \\ n_{2} (r) \\ n_{3} (r) \end{matrix}], L = 1

(35)

where $n_{1} (r)$ , $n_{2} (r)$ and $n_{3} (r)$ are Gaussian noises with mean 0 and variance 1. Let the $H_{\infty}$ performance is specified by $γ = 0.6$ . By using MATLAB LMI toolbox (Boyd et al., 1994; Gahinet et al., 1995), one can verify that Theorem 2 is feasible for the following values of unknown parameters:

\begin{matrix} P = [\begin{matrix} 0.1901 & - 0.0481 & - 0.0177 \\ - 0.0481 & 0.1939 & - 0.0108 \\ - 0.0177 & - 0.0108 & 0.1938 \end{matrix}], S = [\begin{matrix} 0.0393 & 0.0041 & - 0.0179 \\ 0.0041 & 0.0393 & - 0.0023 \\ - 0.0179 & - 0.0023 & 0.0878 \end{matrix}], G = [\begin{matrix} 0.1142 & - 0.0577 & - 0.0371 \\ - 0.0577 & 0.1564 & - 0.0098 \\ - 0.0371 & - 0.0098 & 0.1487 \end{matrix}] \\ (k_{1} = 2.0792 \times 1 0^{3}, k_{2} = 1.8317 \times 1 0^{3}, k_{3} = 164.3347, λ_{12} = - 936.6397, η_{12} = - 936.5820, λ_{13} = - 102.9506, η_{13} = - 102.9135, λ_{23} = 20.8341, η_{23} = 20.8440), \\ C = [\begin{matrix} 0.3995 & - 0.0565 & - 0.0335 \\ - 0.0783 & 0.4357 & - 0.0228 \\ - 0.0649 & - 0.0330 & 0.4657 \end{matrix}] \\ (α_{12} = 0.0715, α_{13} = 0.0832, α_{21} = 0.0751, α_{23} = 0.0922, α_{31} = 0.0850, α_{32} = 0.0989, β_{12} = 0.1280, β_{13} = 0.1167, β_{21} = 0.1533, β_{23} = 0.1150, β_{31} = 0.1499, β_{32} = 0.1319) \end{matrix}

(36)

Thus, Theorem 2 verifies the exponential stability of the present system with $H_{\infty}$ performance $γ = 0.6$ in this example.

Figure 4 shows the plot of H (r) for the example under consideration, which makes clear that $H (\infty) < γ^{2}$ = 0.36 and thus the $H_{\infty}$ norm from the external disturbance w (r) to the state vector x (r) is reduced within the $H_{\infty}$ norm bound $γ$ .

Figure 4.

The plot of H (r) for Example 4.

Conclusion

LMI-based sufficient conditions are established for the exponential stability and $H_{\infty}$ performance of fixed-point state-space digital filters with generalized overflow non-linearities and external interference. Pertaining to the saturation non-linearities, the proposed approach turns out to be less conservative than those in Ahn (2011) and Kokil et al. (2012). The worthiness of the proposed approach is verified with the help of several numerical examples.

Footnotes

Appendix A

(A.1a)

g_{i i} = k_{i} + \sum_{j = 1, j \neq i}^{n} (λ_{i j} + η_{i j}), i = 1, 2, \dots, n,

(A.1b)

g_{i j} = g_{j i} = \frac{1 + L}{2} (λ_{i j} - η_{i j}), i, j = 1, 2, \dots, n (i \neq j),

(A.1c)

λ_{i j} = λ_{j i} > 0, η_{i j} = η_{j i} > 0, i, j = 1, 2, \dots, n (i \neq j),

(A.1d)

k_{i} > 0, i = 1, 2, \dots, n .

(A.2)

- 1 \leq L \leq 1

When $n = 1$ , G corresponds to a positive scalar. Then

(A.3)

y^{T} (r) G y (r) - f^{T} (y (r)) G f (y (r)) \geq 0,

where $f (y (r))$ is given by (2) and (3).

(A.4a)

c_{i i} = \sum_{j = 1, j \neq i}^{n} (α_{i j} + β_{i j}), i = 1, 2, \dots, n,

(A.4b)

c_{i j} = L (α_{i j} - β_{i j}), i, j = 1, 2, \dots, n (i \neq j),

(A.4c)

α_{i j} > 0, β_{i j} > 0, i, j = 1, 2, \dots, n (i \neq j)

(A.5)

0 \leq L \leq 1

For $n = 1$ , C corresponds to a positive scalar. Then

(A.6)

\begin{matrix} \sum_{i = 1}^{n} 2 [y_{i} (r) - f_{i} (y_{i} (r))] \\ [\sum_{j = 1, j \neq i}^{n} {(α_{i j} + β_{i j}) f_{i} (y_{i} (r)) + L (α_{i j} - β_{i j}) f_{j} (y_{j} (r))}] \\ = y^{T} (r) C f (y (r)) + f^{T} (y (r)) C^{T} y (r) - f^{T} (y (r)) (C + C^{T}) \\ f (y (r)) \geq 0 \end{matrix}

where $f (y (r))$ is specified by (2) and (7).

(A.7)

\begin{matrix} \sum_{i = 1}^{n} 2 d_{i} [y_{i} (r) - f_{i} (y_{i} (r))] [- L (y_{i} (r)) + f_{i} (y_{i} (r))] \\ = (1 + L) y^{T} (r) D f (y (r)) + (1 + L) f^{T} (y (r)) D y (r) \\ - 2 f^{T} (y (r)) D f (y (r)) - 2 L y^{T} (r) D y (r) \geq 0 \end{matrix}

holds true for the non-linearities given by (2) and (8).

Appendix

Acknowledgements

The authors thank the Editors and the reviewers for their constructive comments and suggestions to improve the manuscript.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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