The $L_1$-induced norm analysis for linear multivariable differential equations

1.   Introduction

Quantitative evaluation of the input/output relationship in differential equations has been regarded as one of the important issues in applied mathematics and control engineering. Depending on the characteristics of the differential equations considered, various systems norms can be taken. For example, the $L_2$-induced (or the $l_2$-induced) norm is used in ^[1,2] to address energy-bounded disturbances, and the $L_\infty$-induced (or the $l_\infty$-induced) norm is taken in ^[3,4,5] to deal with peak-bounded disturbances. Subsequently, the induced norm is considered in a mixed fashion from $L_2$ to $L_\infty$ as in ^[6,7,8].

However, these three norms do not reflect some real-world problems of the maximizing fuel efficiency ^[9], the population management ^[10,11], and so on. These problems can be tackled by taking the $L_1$-induced (or the $l_1$-induced) norm since it corresponds to the ratio of the total sums between the input and the output. In this line, this induced norm is widely used for practical systems such as switched systems ^[12,13], Markov jump systems ^[14], positive systems ^[15,16], and so on. Here, it should be remarked that the $L_1$-induced norm of positive systems is analytically obtained as in ^[16], this induced norm cannot be exactly computed even for general linear time-invariant (LTI) differential equations since it involves integrating the absolute value of a matrix exponential function on the infinite interval $[0, \infty)$.

With respect to computing the $L_1$-induced norm, an adaptive algorithm is introduced in ^[17], but arguments are confined to single variable, strictly proper differential equations, and no clear extension to the case of multivariable proper differential equations is provided in that study. More importantly, the associated convergence order in ^[17] is limited to $1/M^2$ in terms of the approximation parameter $M$. For hybrid continuous/discrete-time differential/difference equations (i.e., sampled-data systems), a method for computing the $L_1$-induced norm is recently developed in ^[18], but the convergence rate is $1/M$.

Motivated by the above facts, we develop methods for computing both the $L_1$-induced norm and the $l_1$-induced norm of linear multivariable differential and difference equations, with the convergence orders higher than $1/M^2$. We first derive a closed-form expression of the $L_1$-induced norm and clarify that the $1$-norm of a matrix exponential function should be integrated on the infinite interval $[0, \infty)$. This interval is then divided into $[0, H)$ and $[H, \infty)$ for a sufficiently large $H$. The $L_1$-induced norm on $[0, H)$ is treated in a relatively rigorous fashion, while that on $[H, \infty)$ is considered in a rough fashion. More precisely, an upper bound and a lower bound on the former are obtained by dividing the interval $[0, H)$ into $M$ subintervals with an equal width together with applying a $p$th order Taylor expansion to the kernel function of the matrix exponential function with $p = 0, 1, 2, 3$, while an upper bound on the latter is only derived. Here, we show that the gap between the upper and lower bounds on the $L_1$-induced norm on $[0, H)$ tends to 0 at the rate of $1/M^{p+1}$ and the upper bound on the $L_1$-induced norm on $[H, \infty)$ converges to 0 in an exponential order of $H$. Combining these bounds leads to an upper bound and a lower bound on the original $L_1$-induced norm, and their gap is ensured to converge to 0 within the order of $1/M^{p+1}$. These arguments are extended to the $l_1$ analysis of discrete-time difference equations in a parallel fashion. The contributions of this paper over the existing studies ^[16,17,18] are also summarized in Table 1.

This paper is organized as follows. In Section 2, we derive a tractable representation, i.e., a closed-form expression, of the $L_1$-induced norm of linear multivariable differential equations, as a preliminary to computing this induced norm. The major results of this paper, i.e., the methods for computing an upper bound and a lower bound on the $L_1$-induced norm and the corresponding convergence proof, are provided in Section 3. The method for computing the $l_1$-induced norm of linear multivariable difference equations is introduced in Section 4. A numerical example is provided in Section 5 to demonstrate the theoretical validity and the practical effectiveness of the overall arguments developed in this paper. Finally, the notations used in this paper are shown in Table 2.

2.   Tractable description of the $L_1$-induced norm

Let us consider the continuous-time (CT) linear time-invariant (LTI) system $\Sigma_C$ with differential and algebraic equations given by

where $x(t)\in {{\mathbb R}}^{n}_1$ is the state, $w(t) \in{{\mathbb R}}^{n_w}_1$ is the input and $y(t) \in {{\mathbb R}}^{n_y}_1$ is the output. With assuming the zero initial condition of $x$, i.e., $x(0) = 0$, the output of $\Sigma_C$ is described by

where ${\mathbf{G}}_C$ corresponds to the operator describing the input/output relation of $\Sigma_C$.

On the basis of the above operator-based representation, we denote the $L_1$-induced norm of $\Sigma_C$ by $\|{\mathbf{G}}_C\|_{L_1/L_1}$, and it is defined as

The matrix $A$ is assumed to be Hurwitz stable (i.e., all the eigenvalues of $A$ are located in the open left half-plane) for the $L_1$-induced norm $\|{\mathbf{G}}_C\|_{L_1/L_1}$ to be bounded and well-defined, throughout the paper.

On the other hand, it is a non-trivial task to compute $\|{\mathbf{G}}\|_{L1/L_1}$ in terms of (2.3) (and (2.2)) because it should be required to consider every $w$ with $\|w\|_{L_1}\leq 1$. To alleviate such a difficulty, we provide the following lemma associated with a more tractable expression of $\|{\mathbf{G}}_C\|_{L_1/L_1}$ by extending the arguments in ^[19] without considering the feed through term $D$.

Lemma 1. The $L_1$-induced norm $\|{\mathbf{G}}_C\|_{L_1 / L_1}$ can be described by

where

and $(\cdot)_{ij}$ means the $(i, j)$th element of $(\cdot)$.

Proof. Let us denote the vector $\infty$-norm by $|\cdot|_\infty$ and denote the $L_\infty$ norm of a function equipped with the vector $\infty$-norm for the spatial space by $\|\cdot\|_{L_\infty}$. We then see from ^[19,20] that

where $\delta(t)$ is the Dirac-delta function and

From ^[21,22], we note that

Combining (2.6) and (2.8) clearly implies that

For $j = 1, \dots, n_w$, let us next take $\hat{w}_j$ by

where $e_j$ is the $j$th standard basis in ${{\mathbb R}}_{1}^{n_w}$. If we denote the corresponding output by $\hat{y}_j$, then we see that

This together with the fact that $\|\hat{w}_j\|_{L_1} = 1$ for $j = 1, \dots, n_w$ leads to that

Combining (2.9) and (2.12) completes the proof. □

It would be worthwhile to note that $f_{ij}$ given by (2.5) coincides with the $L_1$ norm of the impulse response for the single-input/single-output (SISO) system obtained from replacing $C$, $B$ and $D$ in (2.1) with ${\rm row}_i(C)$, ${\rm col}_j(B)$ and $D_{ij}$, respectively. To put it another way, the assertions in Lemma 1 can also be interpreted as showing that the worst input $w^*$ for achieving the $L_1$-induced norm defined as (2.3) corresponds to the impulse signal (i.e., the Dirac-delta function $\delta(t)$). Thus, it is not required to rigorously consider the input $w$ for computing the $L_1$-induced norm $\|{\mathbf{G}}_C\|_{L_1/L_1}$ when we employ the arguments in Lemma 1.

However, it is difficult to directly treat the first term of the right-hand side (RHS) in (2.5), although we can obtain a tractable representation of $\|{\mathbf{G}}_C\|_{L_1/L_1}$ in Lemma 1. In connection with this, we introduce a truncation idea by which the interval $[0, \infty)$ taken in $f_{ij}$ is divided into $[0, H)$ and $[H, \infty)$ with a sufficiently large parameter $H$. More precisely, let us introduce $f_{ij}^{[H-]}$ and $f_{ij}^{[H+]}$ defined respectively as

Then, it readily follows from $f_{ij} = f_{ij}^{[H-]}+f_{ij}^{[H+]}$ that

From the point of view of (2.14), we can lead to the following lemma associated with deriving an upper bound and a lower bound on the $L_1$-induced norm $\|{\mathbf{G}}_C\|_{L_1/L_1}$ given by (2.4).

Lemma 2. The inequality

holds, where

From the stability assumption on $A$, it could be expected that $F^{[H+]}$ converges to 0 as the truncation parameter $H$ becomes larger. Hence, it would be reasonable to take a sufficiently large $H$ for computing $\|{\mathbf{G}}_C\|_{L_1/L_1}$ if we can explicitly compute $F^{[H-]}$. In this sense, the following section provides two methods for computing an upper bound on each entry of $F^{[H+]}$ and an upper bound and a lower bound on each entry of $F^{[H-]}$, respectively, and derives relevant convergence rates.

3.   Computing upper and lower bounds on $\|{\mathbf{G}}_C\|_{L_1/L_1}$

As mentioned at the end of the preceding section, $F^{[H+]}$ converges to 0 by taking $H$ larger. Thus, it might be useful to compute $F^{[H-]}$ as accurately as possible but $F^{[H+]}$ is treated in a relatively simple fashion when we take a sufficiently large $H$. With respect to this, we derive an upper bound and a lower bound on each entry of $F^{[H-]}$ while an upper bound on each entry of $F^{[H+]}$ is only obtained.

3.1.   Upper bound on each entry of $F^{[H+]}$

Even though $F^{[H+]}$ is very close to 0 when $H$ is large enough, its direct computation is also a non-trivial task. Thus, this subsection aims at obtaining an upper bound on $f_{ij}^{[H+]}$ and showing a relevant convergence property. To this end, we note from the stability assumption on $A$ that there should exist an $q \in {{\mathbb N}}_0$ ensuring $\left|e^{Aq}\right|_{1} < 1$ since $e^{At} \to 0\ (t\to \infty)$. Using the $q$, we can derive the following Lemma.

Lemma 3. For an $q$ with $|e^{Aq}|_1 < 1$, the following inequality holds.

Furthermore, $f_{ij, {\rm U}}^{[H+]}$ converges to 0 regardless of the choice of $q$ with an exponential order of $H$.

Proof. Note that

The second assertion follows by the fact that $|{\rm row}_i(C)e^{AH}|_1$ tends to 0 with an exponential order of $H$. This completes the proof. □

From Lemma 3, an upper bound on $f_{ij}^{[H+]}$ can be computed when $H$ is fixed and an appropriate $q$ with $|e^{Aq}|_1 < 1$ is selected. We next consider $F_{{\rm U}}^{[H+]} \in {{\mathbb R}}_{1}^{n_y \times n_w}$ defined as

where the subscript 'U' stands for the upper bound, and this will be used for computing $\|{\mathbf{G}}_C\|_{L_1/L_1}$ instead of $F^{[H+]}$ in Subsection 3.3.

3.2.   Upper and lower bounds on each entry of $F^{[H-]}$

To determine each entry of $F^{[H-]}$ as accurately as possible through its upper and lower bounds, we consider dividing the interval $[0, H)$ into $M$ subintervals with an equal width. To put it another way, we note that

with

Because this allows us to deal with the interval $[0, h)( = [0, H/M)$ smaller than the original interval $[0, H)$, an approximate scheme on the former interval could be expected to lead to an approximation error smaller than that on the latter interval.

With this in mind, we develop a kernel approximation approach to ${{\mathbf B}}_j(t)$ in (3.4) by introducing ${{\mathbf{B}}}_{j, p}^{[\alpha]}(t)$ defined as

for $p \in \{0, 1, 2, 3\}$ and $\alpha \in [0, 1]$. This ${{{\mathbf{B}}}}^{[\alpha]}_{j, p}(t)$ corresponds to the $p$th order Taylor expansion of $e^{At}$ in ${{\mathbf{B}}}_j(t)$ around $t = \alpha h$. Replacing the ${{\mathbf{B}}}_j(t)$ with ${{\mathbf{B}}}_{j, p}^{[\alpha]}$ in (3.4) derives the approximate treatment of $f_{ij}^{[H-]}$ given by

Regarding an approximation error in terms of $f_{ij, M, p}^{[H-, \alpha]}$ defined as (3.7), we provide the following lemma.

Lemma 4. For $p \in \{ 0, 1, 2, 3\}$, the inequality

holds, where

with $c_\alpha$ given by

Furthermore, $k_{ij, M, p}^{[H-, \alpha]}$ has a uniform upper bound $\hat{k}_{ij, p}^{[H-, \alpha]}$ with respect to $M$ given by

Proof. In terms of the triangular inequality, we first note that

Next, it readily follows from the $p$th order Taylor expansion of $\exp(At)$ around $t = \alpha h$ that

Combining (3.12) and (3.13) establishes the first assertion. The second assertion follows by

This completes the proof. □

Remark 1. The rationale behind taking $p$ as $p\in\{0, 1, 2, 3\}$ is related with explictly computing $f_{ij, M, p}^{[H-, \alpha]}$ in (3.7), although the arguments in Lemma 4 are equivalently extended for the case of $p\geq 4$. To put it another way, it is well-known that no exact solution can be obtained for a general $p$th order polynomial with $p\geq 4$, and thus it is quite difficult to exactly compute $f_{ij, M, p}^{[H-, \alpha]}$ for $p\geq 4$. In contrast, the exact solution formulae exist for general $p$th order polynomials with $p = 0, 1, 2, 3$, and thus $f_{ij, M, p}^{[H-, \alpha]}$ considered in (3.7) can be exactly obtained.

We can easily obtain from Lemma 4 an upper bound and a lower bound on each entry of $F^{[H-]}$. With these values in mind, let us consider the matrices $F_{M, p, {\rm U}}^{[H-, \alpha]}$ and $F_{M, p, {\rm L}}^{[H-, \alpha]}$ whose $(i, j)$the elements are given by $f_{ij, M, p}^{[H-, \alpha]}+\dfrac{k_{ij, M, p}^{[H-, \alpha]}}{M^{p+1}}$ and $f_{ij, M, p}^{[H-, \alpha]}-\dfrac{k_{ij, M, p}^{[H-, \alpha]}}{M^{p+1}}$, respectively, i.e.,

Here, the subscripts 'U' and 'L' stand for the upper bound and the lower bound, respectively, and $F_{M, p, {\rm U}}^{[H-, \alpha]}$ and $F_{M, p, {\rm L}}^{[H-, \alpha]}$ will be used for computing $\|{\mathbf{G}}_C\|_{L_1/L_1}$ instead of $F^{[H-]}$ in Subsection 3.3.

3.3.   Upper and lower bounds on $\|{\mathbf{G}}_C\|_{L_1/L_1}$

The preceding subsections are devoted to providing approximate but asymptotically exact values of $F^{[H+]}$ and $F^{[H-]}$.In terms of $F_{\rm U}^{[H+]}$, $F_{M, p, {\rm U}}^{[H-, \alpha]}$ and $F_{M, p, {\rm L}}^{[H-, \alpha]}$ given by (3.3), (3.15) and (3.16), respectively, combining Lemmas 2–4 leads to the following theorem.

Theorem 1. For a $p\in \{0, 1, 2, 3\}$ and an $\alpha \in [0, 1]$, assume that we take sufficiently large $q \in {{\mathbb N}}$ and $M$ such that $\left|e^{Aq}\right|_{1} < 1$ and every element of $F_{M, p, {\rm L}}^{[H-, \alpha]}$ is not smaller than 0. Then, the following inequality holds.

Furthermore, the gap between the upper and lower bounds in (3.17) tends to 0 as $M$ and $H$ become larger, with the convergence rate proportional to $1/M^{p+1}$ regardless of $q$ and $\alpha$.

Proof. The first assertion readily follows from substituting Lemmas 3 and 4 into Lemma 2. The second assertion is easily established by noting from Lemmas 3 and 4 that $f_{ij, {\rm U}}^{[H+]}$ converges to 0 at an exponential order of $H$ regardless of the choice of $q$ and $\hat{k}_{ij, M, p}^{[H-, \alpha]}$ has a uniform upper bound with respect to $\alpha \in [0, 1]$ because $c_\alpha$ achieves its maximum at $\alpha = 1$ and $\left| {\rm row}_i(C)e^{A\alpha h}A^{p+1} \right|_1$ is bounded by $\left| {\rm row}_i(C) A^{p+1} \right|_1\cdot e^{h}$. □

This theorem clearly implies that we can compute an upper bound and a lower bound on the $L_1$-induced norm $\|{\mathbf{G}}_C\|_{L_1/L_1}$, and their gap converges to 0 by taking the approximation parameter $M$ and the truncation parameter $H$ larger, within the order of $1/M^{p+1}$. Here, it would be worthwhile to note that taking $H$ larger to reduce $F_{\rm U}^{[H+]}$ leads to increasing $k_{ij, M, p}^{[H-, \alpha]}$ in (3.15) and (3.16), by which the gap between the corresponding upper and lower bounds becomes larger. Thus, to achieve the desired accuracy of the $L_1$-induced norm based on Theorem 1, it is crucial to select appropriate values of the parameters $H$, $q$, $M$ and $\alpha$. In connection with this, we provide a pseudo-code based guideline for determining those parameters as follows.

4.   Computing the $l_1$-induced norm of LTI system

Stimulated by the success of computing the $L_1$-induced norm of linear multivariable differential equations developed in the preceding section, in this section, we establish parallel results on computing the $l_1$-induced norm of linear multivariable difference equations.

Let us consider the discrete-time (DT) LTI system $\Sigma_D$ with difference and algebraic equations described by

where $x[k]\in {{\mathbb R}}_{1}^{n}$ is the state, $w[k] \in{{\mathbb R}}_{1}^{n_w}$ is the input and $y[k] \in{{\mathbb R}}_{1}^{n_y}$ is the output. The output of $\Sigma_D$ is given by

assuming the zero initial condition of $x$ similar to the continuous-time case. For the $l_1$-induced norm of $\Sigma_D$ to be well-defined and bounded, we assume that $A_d$ is Schur stable; all the eigenvalues of $A_d$ are located in the open unit disc.

In an equivalent fashion to the arguments in Section 2, we denote the $l_1$-induced norm of $\Sigma_D$ by $\|{\mathbf{G}}_D\|_{l_1/l_1}$ and can derive that

where

To alleviate difficulties in treating an infinite number of $g_{ij}$ in (4.3), we propose the truncation idea with a sufficiently large parameter $N$ as follows.

On the basis of (4.5), it immediately follows from (4.3) that

holds, where

From the stability assumption on $A_{d}$, $G^{[N+]}$ is expected to converge to 0 as the truncation parameter $N$ becomes larger, and thus taking a sufficiently large $N$ is reasonable for computing $\|G_D\|_{l_1/l_1}$. In connection with this, we note from the stability assumption on $A_{d}$ that there should exist a $q \in {{\mathbb N}}_0$ ensuring $\left| A^{q}_{d}\right|_1 < 1$, and can derive the following Lemma.

Lemma 5. For a $q\in {{\mathbb N}}_0$ with $\left| A^{q}_{d}\right|_1 < 1$, the following inequality holds.

Furthermore, $g_{ij, {\rm U}}^{[N+]}$ converges to 0 regardless of the choice of $q$ with an exponential order of $N$

Proof. Note that

On the other hand,

Substituting (4.10) into (4.9) completes the proof of the first assertion. The second assertion follows the fact that the RHS of (4.10) tends to 0 with an exponential order of $N$. □

From Lemma 5, an upper bound on $g_{ij}^{[N+]}$ can be computed when $N$ is fixed and an appropriate $q$ with $|A_d^q|_1 < 1$ is selected. We next consider $G_{{\rm U}}^{[N+]} \in {{\mathbb R}}_{1}^{n_y \times n_w}$ defined as

where the subscript 'U' stands for the upper bound, and this will be used for computing $\|{\mathbf{G}}_D\|_{l_1/l_1}$ instead of $G^{[N+]}$ in the following arguments. The remaining part, determining each entry of $G^{[N-]}$, can be immediately obtained since only finite numbers of summations are required. Substituting $G^{[N-]}$ and $G_{\rm U}^{[N+]}$ given respectively by (4.7) and (4.11) into (4.6) leads to the following result.

Theorem 2. For a sufficiently large $q \in {{\mathbb N}}$ with $|A_d^q|_{1} < 1$, the following inequality holds.

Furthermore, the gap between the upper and lower bounds in (4.12) converges to 0 regardless of the choice of $q$ with an exponential order of $N$.

5.   Numerical example

In this section, we provide a numerical example for verifying the effectiveness of the methods for computing $L_1$-induced and $l_1$-induced norms proposed in this paper. With respect to this, let us consider the twin-rotor MIMO system as shown in Figure 1.

5.1.   $L_1$-induced norm computation

The dynamic behavior of the twin-rotor mimo system is described by the state-space equation ^[23]

where $x: = \begin{bmatrix} \theta_1 & \dot{\theta_1} & \theta_2 & \dot{\theta_2} & \tau_1 & \tau_2 \end{bmatrix}^T$ and $\theta_1$, $\theta_2$, $\tau_1$ and $\tau_2$ denote the pitch angle, the yaw angle, the momentum of the rotor 1 and the momentum of the rotor 2, respectively. The control input $v: = \begin{bmatrix} V_1 & V_2 \end{bmatrix}^T$ contains the input voltages $V_1$ and $V_2$ supplied to the rotor 1 and the rotor 2, respectively, $w$ denotes the disturbance affecting the control input and $y$ is the regulated output. We take the corresponding system parameters as shown in Table 3.

Because the system given by (5.1) is unstable, we consider to employ the full-state stabilizing feedback controller proposed in ^[24], i.e.,

Computing the $L_1$-induced norm of the feedback system consisting of (5.1) and (5.2) is practically meaningful since it corresponds to evaluating the effect of the disturbances occurring in the input voltages on the pitch and yaw angles. In other words, we can clarify how much the angles would rotate by the disturbances in the twin-rotor MIMO system. Based on the fact that $c_\alpha$ in (3.9) attains its minimum at $\alpha = 0.5$, we take $\alpha = 0.5$ for this example. With letting $\epsilon_1 = 10^{-6}$, we can obtain $(H, q) = (150, 6)$ by following Algorithm 1. These parameter values lead to $|F_{\rm{U}}^{[H+]}|_1 = 3.165 \times 10^{-7} < 10^{-6} = \epsilon_1$. After the pair $(H, q) = (150, 6)$ is fixed, we take the approximation parameter $M$ ranging from 800 to 2000 with the approximation order $p = 0, 1, 2, 3$. The computation results are shown in Table 4.

We can observe from Table 4 that both the upper and lower bounds on the $L_1$-induced norm converge to 0.6462 by making $M$ larger, and the gaps between these bounds are decreasing at a rate no smaller than $1/M^{p+1}$ for all the approximation order $p = 0, 1, 2, 3$. To make the practical effectiveness of the arguments in Theorem 1 and Algorithm 1 clearer, the results of computing the $L_1$-induced norm (i.e., an upper bound, a lower bound and their gap) obtained through the conventional arguments in ^[17] are also shown in Table 5. In a comparison between the results in Table 4 for $p = 2, 3$ to those in Table 5, we can observe that the gaps in the former table are quite smaller than that in the latter table under the same parameter $M$. Furthermore, the convergence speeds observed from Table 4 for $p = 2, 3$ are much faster than that from Table 5. From these observations, the arguments developed in this paper (i.e., Theorem 1 and Algorithm 1) are demonstrated in both the theoretical and practical aspects and shown to outperform the conventional arguments in ^[17] for computing the $L_1$-induced norm of linear multivariable differential equations.

5.2.   $l_1$-induced norm computation of the DT LTI system

With respect to verifying the arguments in Theorem 2 relevant to discrete-time systems, we consider a discretization of (5.1) through the zero-order-hold(ZOH) method ^[25] with the sampling time $T = 0.1$ [s]. Based on such a discretized model, we also consider the discrete-time stabilizing controller given by

and we take the truncation parameter $N$ ranging from 60 to 180 under the condition $q = 28$ that results in $|A^{q}_{d}|_1 = 0.9076(<1)$. The corresponding computation results are shown in Table 6.

We can observe from Table 6 that the upper and lower bounds on the $l_1$-induced norm converge to 1.3951 as the truncation parameter $N$ becomes larger, and the gaps between these bounds tend to 0 in an exponential order of $N$. Both observations demonstrate the theoretical validity and the practical effectiveness of the approximation method developed in this paper for computing the discrete-time $l_1$-induced norm.

6.   Conclusions

In this paper, we developed methods for computing the $L_1$-induced norm of multivariable linear differential equations. We first derived a closed-form representation of the $L_1$-induced norm and clarified that it should be required to integrate the absolute value of a matrix exponential function on the infinite time interval $[0, \infty)$. To alleviate this difficulty, we aimed to compute an upper bound and a lower bound on the $L_1$-induced norm within any degree of accuracy. To this end, the time interval $[0, \infty)$ is divided into $[0, H)$ and $[H, \infty)$, for a sufficiently large but finite $H$. An upper bound on the $L_1$-induced norm relevant to the infinite interval $[H, \infty)$ was derived, and we showed that the upper bound is decreasing with an exponential order of $H$. An upper bound and a lower bound on the $L_1$-induced norm with respect to the finite interval $[0, H)$ were also obtained by dividing the interval into $M$ subintervals with equal width and applying a $p$th order Taylor approximation with $p = 0, 1, 2, 3$. More precisely, the gap between the upper and lower bounds was ensured to converge to 0 at the rate of $1/M^{p+1}$. Combining these bounds for both the intervals $[0, H)$ and $[H, \infty)$ allowed us to compute the $L_1$-induced norm on the original interval $[0, \infty)$, within any degree of accuracy. Parallel results on computing the $l_1$-induced norm of discrete-time differential equations were further established. Some numerical examples were given to demonstrate the theoretical validity and the practical effectiveness of the overall arguments developed in this paper.

Finally, it would also be worthwhile to note that the computation method developed in this paper is straightforward and thus it might be extended to more involved problems. Regarding practical applications of the present study, for example, a new stability criterion for biped walking robots and a new quantitative performance measure could be constructed by modifying the relevant arguments in ^[26,27], respectively. With respect to theoretical improvements, the quasi-finite-rank approximation of compression operators in ^[28] can be reformulated in the $L_1$-induced norm sense. Furthermore, an optimal controller minimizing the $L_1$-induced norm can be also studied for involved systems such as neutral differential-algebraic equations ^[29], piecewise continuous systems ^[30,31,32], and so on. However, the aforementioned extensions are non-trivial tasks because it is unclear how an optimal controller for minimizing the $L_1$-induced norm for multivariable linear differential equations can be obtained, and thus they are left for meaningful but quite difficult future works.

Author contributions

Junghoon Kim: Methodology; software; writing original draft; Jung Hoon Kim: Conceptualization; supervision, writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This work was supported by project for Smart Manufacturing Innovation R&D funded Korea Ministry of SMEs and Startups in 2022 (Project No. RS–202200141122).

Conflict of interest

All authors declare no conflicts of interest in this paper.