Approximation of RCI Set Under p-Norm Ball Constraints for Linear Discrete-Time Systems

Abstract

This paper investigates the approximation of robust control invariant (RCI) sets for linear discrete-time systems subject to p-norm ball constraints ($p\ge 1$). Unlike classical results focusing on specific cases like polytope or ellipsoid constraints, we propose a unified framework for arbitrary p-norm ball constraints. Sufficient conditions for a non-empty set to be contained within a p-norm ball are established, revealing the geometric insight that the set is essentially contained within the inscribed 2-norm ball. Utilizing these results, the approximation problem of the RCI set is formulated as a standard linear programming problem that requires verifying constraints at only n standard basis vectors, significantly reducing computational complexity. Specific optimization models are derived, and numerical experiments demonstrate the effectiveness and competitive accuracy of the proposed method.

1. Introduction

The invariant set of a dynamic system is a class of sets such that once the trajectory of the system enters this set, it will remain within it indefinitely. Various concepts and fundamental theories regarding invariant sets can be referred to in the significant and comprehensive literature [1,2]. The application of invariant sets in controller design can be referred to [3,4,5], and their application in model predictive control can be found in [6,7,8]. In the past few decades, researchers have often focused on the precise calculation, approximation, and determination of conditions of invariant sets. However, computing the exact maximum or minimum invariant set for constrained systems can be challenging. Consequently, most researchers are interested in the approximation of invariant sets, including inner approximation and outer approximation.

In recent decades, there have been numerous significant contributions and achievements in the study of invariant sets. Kolmanovsky and Gilbert [9] provided the set invariance conditions for discrete-time linear systems using Pontryagin’s difference formulation, and it also supplies multiple models and techniques for the computation of invariant sets. An algorithm for computing the inner approximation of the largest control invariant subset for constrained discrete-time linear systems is presented in [10]. The iterative algorithm for approximating the smallest robust positive invariant set of constrained discrete-time linear systems subject to bounded disturbances is provided in [11]. Three types of outer approximation methods for the smallest disturbance invariant set of discrete-time linear systems are investigated in [12], and their strengths and weaknesses are compared. By progressively expanding the initial control invariant set to approximate the target set from within while maintaining invariance, Athanasopoulos and Bitsoris [13] introduced a method for computing invariant sets of constrained discrete-time uncertain systems. Tahir [14] proposed a method to compute the optimal robust invariant set and the corresponding feedback control law for discrete-time linear systems subject to polytopic constraints. This method only needs to use semidefinite programming to calculate the target results in one step, thus effectively avoiding the complexity associated with n-step iterative methods. Raković et al. [15,16] discussed the concept of practical robust invariance for physically interconnected discrete-time linear time-invariant systems by employing set dynamics, which offers a new perspective on characterizing set invariance for large-scale systems. Two novel methods for computing polyhedral invariant sets for both deterministic and uncertain discrete-time linear systems with constraints are investigated in [17]. A class of methods based on linear matrix inequalities for computing polytopic invariant sets of constrained discrete-time linear systems is proposed in [18]. Wang et al. have proposed an algorithm to compute the maximal invariant set for discrete-time linear systems subject to non-convex constraints, and have extended it to linear switched systems and nonlinear systems [19]. Utilizing the duality theory of semidefinite programming, Yang et al. [20] proposed new linear matrix inequality conditions to characterize the robust control invariance for uncertain discrete-time linear systems. Due to the potential limitations of classical iterative algorithms for computing robust control invariant sets with respect to the dimension of the system, Anevlavis et al. have investigated the implicit representation and computational strategies for robust control invariant sets of discrete-time linear systems, which can safely handle high-dimensional systems [21]. Raković and Trodden [22] established the characterization, existence, and uniqueness of the equi-normalized robust positive invariant set for linear difference inclusions. Building on the properties of DC functions, an algorithm for computing robust control invariant sets of nonlinear discrete-time uncertain systems is provided in [23]. Decardi-Nelson et al. transformed the search for invariant sets into a graph-theoretic problem and proposed two approximation algorithms for computing the maximal robust control invariant set of constrained discrete-time nonlinear systems in [24]. Raković and Zhang [25] investigated new characterizations and implicit forms of the maximal positive invariant set for nonlinear discrete-time dynamics subject to closed state constraints. Gao et al. [26] studied the ellipsoidal approximation of the mRPI set for nonlinear dynamic systems by means of the sum of squares (SOS) optimization method. In [27], Comelli et al. proposed an inner-outer approximation method of RCI set for nonlinear discrete-time perturbed systems, and proved that there is an RCI set between the inner set and the outer set. Decardi-Nelson and Liu [28] introduced a distributed architecture founded on graph algorithms to calculate control invariant sets for constrained nonlinear cascade systems. For the approximation of the maximal robust invariant set of polynomial systems with state constraints, refer to [29]. For the computation of the maximal admissible invariant set for discrete-time switched systems subject to semi-algebraic constraints, refer to [30,31].

It can be observed that the approximation of invariant sets for constrained systems has always been a hot topic in research. Raković et al. investigated the computational problem of the optimal robust control invariant set for discrete-time linear systems under $1,\infty$-norm balls constraints [32]. The method is to construct a linear programming problem to calculate the polyhedral RCI sets in one step. Subsequently, Yang and Yang extended this line of research to 2-norm ball and ellipsoidal constraints, studying the minimum-volume outer approximation of RCI sets for discrete-time linear systems [33], as well as the ellipsoidal approximation of minimum robust positive invariant sets under ellipsoidal disturbance constraints [34]. We are now interested in the approximation problem of RCI sets under $p$-norm ball constraints.

Based on the discussions above, the main contributions of this paper are summarized as follows: Unlike classical results that focus on specific cases like polytopes ($p=1,\infty$) or ellipsoids (including $p=2$), this paper proposes a unified framework to approximate RCI sets under arbitrary p-norm ball constraints ($p\ge 1$). The proposed sufficient conditions allow formulating the approximation problem as a standard Linear Programming (LP) problem. This formulation requires verifying constraints at only n standard basis vectors for an $n$-dimensional system, which significantly reduces the problem size compared to classical polytopic approximations (which often require verifying $2^n$ points). In contrast to the one-step LP approach in [35], which requires a sufficiently large number of pre-defined inequalities to achieve a refined approximation (thereby increasing the problem complexity), our method achieves competitive approximation accuracy with a compact formulation involving only a finite number of points. We reveal that the derived conditions essentially imply the set is contained within the inscribed 2-norm ball of the target p-norm ball. Numerical examples are provided to validate the effectiveness of the approach.

The structure of this paper is as follows: Section 2 provides a detailed introduction to the tools and methods required for this study. Section 3 presents the optimization models for calculating the RCI sets under the $p$-norm ball constraints. Section 4 features several numerical examples to validate the obtained results. Section 5 is the discussion. Finally, Section 6 concludes the paper.

2. Preliminaries

In this paper, we employ the following notations: ${\mathbb{N}}_k=\{0,1,\dots ,k\}$. The vector $e_j$ in ${\mathbb{R}}^n$ denotes a standard basis vector with a single element of 1 at the jth position, while all other elements are 0. The Minkowski sum of non-empty sets $\mathcal{S}$ and $\mathcal{T}$ is denoted by $\mathcal{S}\oplus \mathcal{T}=\{s+t\mid s\in \mathcal{S},t\in \mathcal{T}\}$. A vector in ${\mathbb{R}}^n$ space whose all elements are 1 is denoted by ${\mathbf{1}}_n.$ The matrix obtained by taking the absolute value of all elements of matrix A is denoted as $abs\left(A\right)$, in particular, the vector obtained by applying this operation to the vector h is denoted as $\left|h\right|$. Polytope stands for a bounded polyhedron. The vectorization of matrix A, denoted as $vec\left(A\right)$, is achieved by concatenating the columns of A into a single column vector. The support function of a non-empty set $\mathcal{X}$ at $\theta \in {\mathbb{R}}^n$ is denoted as $h(\mathcal{X},\theta )={sup}_{x\in \mathcal{X}}{x}^T\theta$. The positive quadrant in ${\mathbb{R}}^n$ represents the quadrant where all coordinate components are positive, denoted by ${\mathbb{R}}_{+}^n$.

Let $1\le p<\infty ,x\in {\mathbb{R}}^n$ be a vector, then the $p$-norm of x is defined as

$${∥x∥}_p={\left({\displaystyle \sum _{i=1}^n}{\left|x_i\right|}^p\right)}^{\frac{1}{p}},$$

in particular, ${∥x∥}_{\infty }={max}_{1\le i\le n}\left|x_i\right|.$

There are two important properties of the vector norm. Let $x,y\in {\mathbb{R}}^n,$ then Hölder’s inequality for the vector norm is as follows:

$${\displaystyle \sum _{i=1}^n}\left|x_i{y}_i\right|\le {∥x∥}_p{∥y∥}_q,{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,p>1,q>1,$$

When $p=q=2$, Hölder’s inequality becomes Cauchy-Schwartz inequality. The $p$-norm of a vector exhibits monotonicity. When $1\le p\le q,$ the following inequality holds for all $x\in {\mathbb{R}}^n$

$${∥x∥}_q\le {∥x∥}_p\le n^{\frac{1}{p}-\frac{1}{q}}{∥x∥}_q.$$

The $p$-norm ball with radius $r>0$ is defined as $B_p\left(r\right)=\left\{x\in {\mathbb{R}}^n\mid {∥x∥}_p\le r\right\},$ and it is a bounded closed convex set. When $p=2$, it corresponds to the Euclidean ball, when $p=1$ or $p=\infty$, it is a polytope.

When the center of the $p$-norm ball is at the origin of the space rectangular coordinate system in ${\mathbb{R}}^n$, any $p$-norm ball is a symmetric set about the origin, any coordinate axis, and coordinate plane. When $p\ge 2$, the inscribed $2$-norm ball of the $p$-norm ball $B_p\left(r\right)$ is $B_2\left(r\right)$. However, the scenario differs when $1\le p\le 2$. Since the $p$-norm ball is a symmetric set, it has a tangent point with its inscribed $2$-ball in each quadrant of the space rectangular coordinate system, and the absolute values of the coordinate components of the tangent point are equal. Therefore, we only need to calculate the radius $\tilde{r}$ of the inscribed $2$-norm ball in the positive quadrant.

In the $2$-dimensional case: Let the tangent point be ${[a,a]}^T\in {\mathbb{R}}_{+}^2$. Because the tangent point lies on the boundary of the $B_p\left(r\right)$, it follows that ${(a^p+a^p)}^{\frac{1}{p}}=r,$ hence $a={\displaystyle \frac{r}{2^{\frac{1}{p}}}},$ thus $\tilde{r}=\sqrt{a^2+a^2}={\displaystyle \frac{r}{2^{\frac{1}{p}-\frac{1}{2}}}}.$

In the $3$-dimensional case: Let the tangent point be ${[a,a,a]}^T\in {\mathbb{R}}_{+}^3.$ Because the tangent point lies on the boundary of the $B_p\left(r\right)$, it follows that ${(a^p+a^p+a^p)}^{\frac{1}{p}}=r,$ hence $a={\displaystyle \frac{r}{3^{\frac{1}{p}}}},$ thus $\tilde{r}=\sqrt{a^2+a^2+a^2}={\displaystyle \frac{r}{3^{\frac{1}{p}-\frac{1}{2}}}}.$

Following the same logic, in the $n$-dimensional case: Let the tangent point be ${[a,a,...,a]}^T\in {\mathbb{R}}_{+}^n.$ Because the tangent point lies on the boundary of the $B_p\left(r\right)$, it follows that ${(a^p+a^p+\dots +a^p)}^{\frac{1}{p}}=r,$ hence $a={\displaystyle \frac{r}{n^{\frac{1}{p}}}},$ thus $\tilde{r}=\sqrt{a^2+a^2+\dots +a^2}={\displaystyle \frac{r}{n^{\frac{1}{p}-\frac{1}{2}}}}.$

The $1.5$-norm ball and its inscribed $2$-norm ball in 2-dimensional and 3-dimensional cases are shown in Figure 1, where A is the tangent point.

Figure 1. Geometric relationship between a 1.5-norm ball and its inscribed 2-norm ball in (a) 2-dimensional and (b) 3-dimensional spaces.

We are currently interested in the conditions for a non-empty set to be contained within a $p$-norm ball. Let us first review the conditions where $p=2.$

Proposition 1 

([33]). A non-empty set $\mathcal{X}$ in ${\mathbb{R}}^n$ is contained within a $2$-norm ball $B_2\left(r\right)=\left\{x\mid {∥x∥}_2\le r\right\}=\{r\theta \mid {∥\theta ∥}_2\le 1\},$ i.e.,

$$\forall x\in \mathcal{X},{∥x∥}_2\le r$$

(1)

if and only if

$$h(\mathcal{X},\tilde{\theta })=\underset{x\in \mathcal{X}}{sup}{x}^T\tilde{\theta }\le r,\mathit{for}\mathit{all}\tilde{\theta }\in \left\{\theta \in {\mathbb{R}}^n\mid {∥\theta ∥}_2=1\right\}.$$

(2)

Proposition 2 

([33]). Let $\mathcal{X}$ is a non-empty set in ${\mathbb{R}}^n,$ if

$$\left|h(\mathcal{X},e_j)\right|\le {\displaystyle \frac{r}{\sqrt{n}}},\mathit{for}\mathit{all}j=1,2,\dots ,n$$

(3)

then $\mathcal{X}\subseteq B_2\left(r\right).$

When $1\le p\le 2,$ the conditions for a non-empty set to be contained in the $p$-norm ball are as follows.

Proposition 3. 

Let $\mathcal{X}$ be a non-empty set in ${\mathbb{R}}^n,$ $1\le p\le 2,$ if

$$h(\mathcal{X},\theta )=\underset{x\in \mathcal{X}}{sup}{x}^T\theta \le {\displaystyle \frac{r}{n^{\frac{2-p}{p}}}},\mathit{for}\mathit{all}\tilde{\theta }\in \left\{\theta \in {\mathbb{R}}^n\mid {∥\theta ∥}_p=1\right\}$$

(4)

then $\mathcal{X}\subseteq B_p\left(r\right).$

Proof. 

We prove that for all $x\in \mathcal{X},$ there is ${∥x∥}_p\le r.$

If $\tilde{x}=0$, then ${∥\tilde{x}∥}_p=0<r,$ and the conclusion holds trivially.

For $\forall \tilde{x}\in \mathcal{X}$ and $\tilde{x}\ne 0,$ ${∥{\displaystyle \frac{\tilde{x}}{{∥\tilde{x}∥}_p}}∥}_p=1,$ then

$${\displaystyle \frac{{∥\tilde{x}∥}_2^2}{{∥\tilde{x}∥}_p}}={\displaystyle \frac{{\tilde{x}}^T\tilde{x}}{{∥\tilde{x}∥}_p}}\le \underset{x\in \mathcal{X}}{sup}{x}^T{\displaystyle \frac{\tilde{x}}{{∥\tilde{x}∥}_p}}\le {\displaystyle \frac{r}{n^{\frac{2-p}{p}}}},$$

(5)

$1\le p\le 2,$ it follows that ${∥\tilde{x}∥}_p\le n^{\frac{2-p}{2p}}{∥\tilde{x}∥}_2,$ that is ${\displaystyle \frac{{∥\tilde{x}∥}_p}{n^{\frac{2-p}{2p}}}}\le {∥\tilde{x}∥}_2.$ Thus

$${\displaystyle \frac{{∥\tilde{x}∥}_p}{n^{\frac{2-p}{p}}}}={\displaystyle \frac{{∥\tilde{x}∥}_p^2}{n^{\frac{2-p}{p}}{∥\tilde{x}∥}_p}}\le {\displaystyle \frac{{∥\tilde{x}∥}_2^2}{{∥\tilde{x}∥}_p}}\le {\displaystyle \frac{r}{n^{\frac{2-p}{p}}}}$$

that is ${∥\tilde{x}∥}_p\le r,$ therefore $\mathcal{X}\subseteq B_p\left(r\right).$ □

Proposition 4. 

Let $\mathcal{X}$ be a non-empty set in ${\mathbb{R}}^n,$ $1\le p\le 2,$ if

$$\left|h(\mathcal{X},e_j)\right|=\underset{x\in \mathcal{X}}{sup}{x}^T{e}_j\le {\displaystyle \frac{r}{n^{\frac{1}{p}}}},j=1,2,\dots ,n$$

(6)

then $\mathcal{X}\subseteq B_p\left(r\right).$

Proof. 

For $\forall \tilde{\theta }\in \left\{\theta \in {\mathbb{R}}^n\mid {∥\theta ∥}_p=1\right\},$ let $\tilde{\theta }={\theta }_1{e}_1+{\theta }_2{e}_2+\dots +{\theta }_n{e}_n,$ where ${\theta }_j\in \mathbb{R}.$ Since ${∥\tilde{\theta }∥}_p=1,$ then ${\left|{\theta }_1\right|}^p+{\left|{\theta }_2\right|}^p+\dots +{\left|{\theta }_n\right|}^p=1.$

When $p>1,$ by Hölder inequality, we have that

$$\begin{array}{cc}\hfill & |{\theta }_1|·1+|{\theta }_2|·1+\dots +|{\theta }_n|·1\hfill \\ & \le \sqrt[p]{{\left|{\theta }_1\right|}^p+{\left|{\theta }_2\right|}^p+\dots +{\left|{\theta }_n\right|}^p}·\sqrt[q]{1^q+\dots +1^q}\hfill \\ \hfill & =\sqrt[q]{n},\hfill \end{array}$$

where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$ So for all $\tilde{\theta }\in \left\{\theta \in {\mathbb{R}}^n\mid {∥\theta ∥}_p=1\right\},$

$$\begin{array}{cc}\hfill h(\mathcal{X},\tilde{\theta })& =\underset{x\in \mathcal{X}}{sup}{x}^T({\theta }_1{e}_1+{\theta }_2{e}_2+\dots +{\theta }_n{e}_n)\hfill \\ \hfill & \le |{\theta }_1|·\left|h(\mathcal{X},e_1)\right|+|{\theta }_2|·\left|h(\mathcal{X},e_2)\right|+\dots +\left|{\theta }_n\right|·\left|h(\mathcal{X},e_n)\right|\hfill \\ \hfill & \le \left(\right|{\theta }_1|+|{\theta }_2|+\dots +|{\theta }_n\left|\right)·{\displaystyle \frac{r}{n^{\frac{1}{p}}}}\hfill \\ \hfill & ={\displaystyle \frac{r}{n^{\frac{1}{p}-\frac{1}{q}}}}\hfill \\ \hfill & ={\displaystyle \frac{r}{n^{\frac{1}{p}-(1-\frac{1}{p})}}}\hfill \\ \hfill & ={\displaystyle \frac{r}{n^{\frac{2-p}{p}}}}.\hfill \end{array}$$

By Proposition 3, it follows that $\mathcal{X}\subseteq B_p\left(r\right).$

When $p=1,$ $\left(6\right)\iff \left|h(\mathcal{X},e_j)\right|\le {\displaystyle \frac{r}{n}},j=1,2,\dots ,n.$ So for all $\tilde{\theta }\in \left\{\theta \in {\mathbb{R}}^n\mid {∥\theta ∥}_1=1\right\},h(\mathcal{X},\tilde{\theta })\le \left(\right|{\theta }_1|+|{\theta }_2|+\dots +|{\theta }_n\left|\right)·\frac{r}{n}=\frac{r}{n}$, according to Proposition 3, $\mathcal{X}\subseteq B_1\left(r\right).$ □

We have found that Propositions 3 and 4 correspond exactly to the facts described in Propositions 1 and 2 when $p=2$. However, Proposition 3 is no longer a necessary and sufficient condition.

Remark 1. 

From Equation (5), if we shrink the denominator to ${∥\tilde{x}∥}_p\le n^{\frac{1}{p}-\frac{1}{2}}{∥\tilde{x}∥}_2$, then for all $\tilde{x}\in \mathcal{X},{∥\tilde{x}∥}_2\le {\displaystyle \frac{r}{n^{\frac{1}{p}-\frac{1}{2}}}}.$ This essentially indicates that the set $\mathcal{X}$ is contained within a 2-norm ball $B_2\left(\tilde{r}\right)$, where $\tilde{r}={\displaystyle \frac{r}{n^{\frac{1}{p}-\frac{1}{2}}}}.$ Geometrically, for $1\le p\le 2,$ $B_2\left(\tilde{r}\right)$ is the inscribed $2$-norm ball of $B_p\left(r\right)$. Equation (6) is equivalent to $\left|h(\mathcal{X},e_j)\right|\le \tilde{r}/\sqrt{n}=r/\sqrt[p]{n},$ by Proposition 2 there is $\mathcal{X}\subseteq B_2\left(\tilde{r}\right)\subseteq B_p\left(r\right).$ Therefore, Propositions 3 and 4 essentially characterize the conditions for a non-empty set to be contained within the inscribed $2$-norm ball of the $p$-norm ball.

When $p\ge 2,$ the following results are obtained.

Proposition 5. 

Let $\mathcal{X}$ be a non-empty set in ${\mathbb{R}}^n,$ $p\ge 2,$ if

$$h(\mathcal{X},\theta )=\underset{x\in \mathcal{X}}{sup}{x}^T\theta \le r,\mathit{for}\mathit{all}\tilde{\theta }\in \left\{\theta \in {\mathbb{R}}^n\mid {∥\theta ∥}_p=1\right\},$$

(7)

then $\mathcal{X}\subseteq B_p\left(r\right).$

Proof. 

We prove that for all $x\in \mathcal{X},$ there is ${∥x∥}_p\le r.$

If $\tilde{x}=0$, then ${∥\tilde{x}∥}_p=0<r,$ and the conclusion holds trivially.

For all $\tilde{x}\in \mathcal{X}$ and $\tilde{x}\ne 0,$ ${∥{\displaystyle \frac{\tilde{x}}{{∥\tilde{x}∥}_p}}∥}_p=1,$ then

$${\displaystyle \frac{{∥\tilde{x}∥}_2^2}{{∥\tilde{x}∥}_p}}={\displaystyle \frac{{\tilde{x}}^T\tilde{x}}{{∥\tilde{x}∥}_p}}\le \underset{x\in \mathcal{X}}{sup}{x}^T{\displaystyle \frac{\tilde{x}}{{∥\tilde{x}∥}_p}}\le r.$$

(8)

Since $p\ge 2,$ so ${∥\tilde{x}∥}_p\le {∥\tilde{x}∥}_2,$ thus

$${∥\tilde{x}∥}_p={\displaystyle \frac{{∥\tilde{x}∥}_p^2}{{∥\tilde{x}∥}_p}}\le {\displaystyle \frac{{∥\tilde{x}∥}_2^2}{{∥\tilde{x}∥}_p}}\le r,$$

$\mathcal{X}\subseteq B_p\left(r\right).$□

Proposition 6. 

Let $\mathcal{X}$ be a non-empty set in ${\mathbb{R}}^n,$ $p\ge 2,$ if

$$\left|h(\mathcal{X},e_j)\right|=\underset{x\in \mathcal{X}}{sup}{x}^T{e}_j\le {\displaystyle \frac{r}{n^{\frac{1}{q}}}},j=1,2,\dots ,n$$

(9)

where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,$ then $\mathcal{X}\subseteq B_p\left(r\right).$

Proof. 

The proof process is similar to the proposition 4. □

Remark 2. 

From Equation (8), if we shrink the denominator to ${∥\tilde{x}∥}_p\le {∥\tilde{x}∥}_2$, then for all $\tilde{x}\in \mathcal{X},{∥\tilde{x}∥}_2\le r$. This essentially indicates that the set $\mathcal{X}$ is contained within the inscribed $2$-norm ball of the $p$-norm ball. By defining $\overline{r}=r/n^{\frac{1}{q}-\frac{1}{2}},$ Equation (9) is equivalent to $\left|h(\mathcal{X},e_j)\right|\le \overline{r}/\sqrt{n},j=1,2,\dots ,n.$ According to Proposition 2, there is $\mathcal{X}\subseteq B_2\left(\overline{r}\right)\subseteq B_p\left(r\right).$ Since $\overline{r}\le r$, this implies, from a geometric perspective, that applying Proposition 6 would restrict the set $\mathcal{X}$ to a $2$-norm ball smaller than $B_2\left(r\right)$. Consequently, the condition in Proposition 6 is valid but conservative. Therefore, for $p\ge 2$, it is preferable to use the condition that a non-empty set is contained within the inscribed 2-norm ball of $B_p\left(r\right)$ to obtain a less conservative approximation of the invariant set.

Remark 3. 

(Source of Conservatism) Conservatism fundamentally arises from two aspects. First, instead of verifying set containment over the continuous boundary of the p-norm ball, our method only checks constraints at n standard basis vectors. Second, as revealed by the algebraic derivations in Propositions 3–6, the proposed conditions essentially bound the target set X within a 2-norm ball. For $1\le p\le 2$, the set is restricted to the inscribed 2-norm ball $B_2\left(\tilde{r}\right)$ of $B_p\left(r\right)$, rather than the full $B_p\left(r\right)$. For $p\ge 2$, if the basis-vector condition in Proposition 6 is applied, it restricts the set to a 2-norm ball $B_2\left(\overline{r}\right)$ where $\overline{r}\le r$.

The above discussion also offers us new insights. When the constraint set of a dynamical system is a general symmetric bounded closed convex set, we can calculate the target set within its inscribed $2$-norm ball.

In the subsequent sections of this paper, the following tools will be required.

Lemma 1 

([1]). Let $\mathcal{X}$ be a non-empty set in ${\mathbb{R}}^n,$ $\Pi =\left\{x\in {\mathbb{R}}^n\mid Gx\le g\right\}$ is a polytope

$$\mathcal{X}\subseteq \Pi ⟺h(\mathcal{X},G_i)\le g_i,i=1,2,\dots ,q$$

where $G\in {\mathbb{R}}^{q\times n},g\in {\mathbb{R}}^q,$ $G_i$ is the i-th row vector of G, and $g_i$ is the i-th component of g.

To evaluate the support function of the Minkowski sum, the following lemma is required:

Lemma 2 

([32]). Let $D_i\in {\mathbb{R}}^{n\times m}$ be matrices, and ${\mathcal{W}}_i$ be non-empty compact sets in ${\mathbb{R}}^m,i\in {\mathbb{N}}_k.$ If $\mathcal{X}={⨁}_{i=0}^k{D}_i{\mathcal{W}}_i,$ then $h\left(\mathcal{X},\theta \right)={\sum }_{i=0}^k{max}_{w\in {\mathcal{W}}_i}\left({\theta }^T{D}_i\right)w.$

Lemma 3 

([32]). Let $A\in {\mathbb{R}}^{n\times n}, D\in {\mathbb{R}}^{n\times m}, M\in {\mathbb{R}}^{m\times n}, W=\{Ed+f\mid {∥d∥}_{\infty }\le \eta \}, E\in {\mathbb{R}}^{n\times t}, f\in {\mathbb{R}}^n, w\in W.$ Then

$$\begin{array}{c}\hfill \underset{w\in W}{max}(A+DM)w=\eta ·abs\left((A+DM)E\right)·{\mathbf{1}}_t+(A+DM)f;\end{array}$$

(10)

Moreover, There exists a matrix $L\in {\mathbb{R}}^{n\times t}$ such that $-L\le (A+DM)E\le L,$ where the inequality is carried out element by element, and the formula (10) satisfies

$$\begin{array}{c}\hfill \underset{w\in W}{max}(A+DM)w=\eta ·L·{\mathbf{1}}_t+(A+DM)f.\end{array}$$

(11)

3. Robust Control Invariant Set Under $\mathit{p}$-Norm Ball Constraints

We consider the discrete-time linear time-invariant systems

$$x^{+}=Ax+Bu+w,$$

(12)

where x is the current state and $x^{+}$ is the next state, $x\in {\mathbb{R}}^n,u\in {\mathbb{R}}^m,w\in {\mathbb{R}}^n.$

The system is subject to state constraint set X and control constraint set U. In this paper, it is assumed that the disturbance constraint set $W=\left\{Ed+f\mid {∥d∥}_{\infty }\le \eta \right\},$ where $E\in {\mathbb{R}}^{n\times t},d\in {\mathbb{R}}^t,f\in {\mathbb{R}}^n.$

This paper investigates a class of polyhedral RCI sets in the form of Minkowski sum introduced in [32]

$$R_k\left({\mathbf{M}}_k\right)\triangleq {\displaystyle \underset{i=0}{\overset{k-1}{⨁}}}{D}_i\left({\mathbf{M}}_k\right)W,$$

(13)

where $k\ge n,{\mathbf{M}}_k=(M_0,M_1,\dots ,M_{k-2},M_{k-1}),M_i\in {\mathbb{R}}^{m\times n},$ and

$$\begin{array}{c}\hfill D_0\left({\mathbf{M}}_k\right)=I,D_i\left({\mathbf{M}}_k\right)\triangleq A^i+{\displaystyle \sum _{j=0}^{i-1}}{A}^{i-1-j}B{M}_j,i\ge 1.\end{array}$$

Assume that ${\mathbb{M}}_k$ denote the set of matrices ${\mathbf{M}}_k$ that satisfy the condition $D_k\left({\mathbf{M}}_k\right)=0$:

$${\mathbb{M}}_k=\{{\mathbf{M}}_k\mid D_k\left({\mathbf{M}}_k\right)=0\}.$$

(14)

Lemma 4 

([32]). For any ${\mathbf{M}}_k\in {\mathbb{M}}_k,k\ge n$ and the associated set $R_k\left({\mathbf{M}}_k\right),$ there exists a control law $u:R_k\left({\mathbf{M}}_k\right)\to {\mathbb{R}}^m,$ ensuring that for all $x\in R_k\left({\mathbf{M}}_k\right),$ $Ax+Bu\left(x\right)\oplus W\subseteq R_k\left({\mathbf{M}}_k\right).$ This implies that the set $R_k\left({\mathbf{M}}_k\right)$ is a robust control invariant set for the system (12) and constraint set $({\mathbb{R}}^n,{\mathbb{R}}^m,W).$

For any $x\in R_k\left({\mathbf{M}}_k\right),$ $u\left(x\right)\in U\left({\mathbf{M}}_k\right)={⨁}_{i=0}^{k-1}{M}_{i}W.$ Raković et al. compute the optimal robust control invariant set for system (12) under the $1,\infty$-norm balls by solving linear programming problems, and indicate that the control law u does not need to be pre-selected, it is only necessary to ensure that $U\left({\mathbf{M}}_k\right)\subseteq U$ is sufficient to satisfy the control constraint set. The control law u can be determined by solving a parameterized quadratic programming problem [32]. We are now interested in computing the robust control invariant set for system (12) contained within a p-norm ball where $p\ge 1$.

Using the proposed method, we construct the following optimization problem to calculate the optimal robust control invariant set $R_k\left({\mathbf{M}}_k\right)$ contained within a smaller $p$-norm ball

$$\begin{array}{c}\hfill {\mathbb{O}}_k:\underset{({\mathbf{M}}_k,r)}{min}\left\{r\mid ({\mathbf{M}}_k,r)\in \mathsf{\Psi }\right\}\end{array}$$

(15)

where

$$\begin{array}{c}\hfill \mathsf{\Psi }\triangleq \left\{({\mathbf{M}}_k,r)\mid {\mathbf{M}}_k\in {\mathbb{M}}_k;\left|h(R_k\left({\mathbf{M}}_k\right),e_j)\right|\le {\displaystyle \frac{r}{n^{\xi }}},j=1,2,\dots ,n;r>0.\right\}\end{array}$$

(16)

When $1\le p\le 2$, $\xi$ is set to ${\displaystyle \frac{1}{p}}$, when $2\le p\le \infty ,$ $\xi$ is set to ${\displaystyle \frac{1}{2}}.$

Optimization problem (15) implies that without state constraints and control constraints, the robust control invariant set of system (12) can be nested within how small a $p$-norm ball. It can also be understood as calculating a smaller $p$-norm ball state constraint set for system (12).

It can be seen from Equation (14) that the first condition of (16) is essentially a linear equality.

Using Lemmas 2 and 3, we obtain

$$\begin{array}{cc}\hfill & \left|h(R_k\left({\mathbf{M}}_k\right),e_j)\right|\le {\displaystyle \frac{r}{n^{\xi }}},j=1,2,\dots ,n\hfill \\ \hfill & ⟺\left|{\displaystyle \sum _{i=0}^{k-1}}\underset{w\in W}{max}{e}_j^T{D}_i\left({\mathbf{M}}_k\right)w\right|\le {\displaystyle \frac{r}{n^{\xi }}},j=1,2,\dots ,n\hfill \\ \hfill & ⟺\left|{\displaystyle \sum _{i=0}^{k-1}}\underset{w\in W}{max}{D}_i\left({\mathbf{M}}_k\right)w\right|\le {\displaystyle \frac{r}{n^{\xi }}}·{\mathbf{1}}_n.\hfill \\ \hfill & ⟺\left|{\displaystyle \sum _{i=0}^{k-1}}(\eta ·L_i·{\mathbf{1}}_t+D_i\left({\mathbf{M}}_k\right)f)\right|\le {\displaystyle \frac{r}{n^{\xi }}}·{\mathbf{1}}_n.\hfill \end{array}$$

where $-L_i\le D_i\left({\mathbf{M}}_k\right)E\le L_i,i=0,1,\dots ,k-1.$ The maximum operation is performed element by element.

Therefore, this is a linear programming problem

$${\mathbb{O}}_k:\underset{\psi }{min}\left\{r\mid \psi \in \mathsf{\Psi }\right\}$$

(17)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathsf{\Psi }\triangleq \{\psi \mid -{\displaystyle \frac{r}{n^{\xi }}}·{\mathbf{1}}_n& \le {\displaystyle \sum _{i=0}^{k-1}}(\eta ·L_i·{\mathbf{1}}_t+D_i\left({\mathbf{M}}_k\right)f)\le {\displaystyle \frac{r}{n^{\xi }}}·{\mathbf{1}}_n,\hfill \\ \hfill -L_i& \le D_i\left({\mathbf{M}}_k\right)E\le L_i,i\in {\mathbb{N}}_{k-1},\hfill \\ \hfill A^k& +{\displaystyle \sum _{j=0}^{k-1}}{A}^{k-1-j}B{M}_j=0,r>0.\}\hfill \end{array}\end{array}$$

(18)

and ${\mathbf{L}}_k\triangleq (L_0,L_1,\dots L_{k-1}),\psi \triangleq \left(vec\left({\mathbf{M}}_k\right),vec\left({\mathbf{L}}_k\right),r\right).$

Suppose $X=\{x\in {\mathbb{R}}^n\mid {∥x∥}_p\le r_x\},U=\{u\in {\mathbb{R}}^m\mid {∥u∥}_{\widehat{p}}\le r_u\},e_j\in {\mathbb{R}}^n,$ ${\widehat{e}}_j$ is a vector in ${\mathbb{R}}^m$ with a single element of 1 at the j-th position and 0 elsewhere.

When the state constraint set is a $p$-norm ball and the control constraint set is a $\widehat{p}$-norm ball, the following optimization problem can be constructed to calculate the robust control invariant set under the constraint sets

$${\tilde{\mathbb{O}}}_k:\underset{({\mathbf{M}}_k,a,b,\zeta )}{min}\left\{\zeta \mid ({\mathbf{M}}_k,a,b,\zeta )\in \tilde{\mathsf{\Psi }}\right\}$$

(19)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{\mathsf{\Psi }}\triangleq \{({\mathbf{M}}_k,a,b,\zeta )\mid & {\mathbf{M}}_k\in {\mathbb{M}}_k,\hfill \\ \hfill & \left|h(R_k\left({\mathbf{M}}_k\right),e_j)\right|\le {\displaystyle \frac{a{r}_x}{n^{\xi }}},j=1,2,\dots ,n,\hfill \\ \hfill & \left|h(U\left({\mathbf{M}}_k\right),\widehat{e_j})\right|\le {\displaystyle \frac{b{r}_u}{m^{\widehat{\xi }}}},j=1,2,\dots ,m,\hfill \\ \hfill & 0\le a\le 1,0\le b\le 1,\hfill \\ \hfill & q_{a}a+q_{b}b\le \zeta .\}\hfill \end{array}\end{array}$$

(20)

Here, $q_a$ and $q_b$ are weight coefficients, indicating a preference for the relative shrinkage of the constraint sets, respectively. The values of $\xi$ and $\widehat{\xi }$ are chosen as previously described.

Utilizing the previously introduced tools, one can verify that this optimization problem is a linear programming problem

$${\tilde{\mathbb{O}}}_k:\underset{\tilde{\psi }}{min}\left\{\zeta \mid \tilde{\psi }\in \tilde{\mathsf{\Psi }}\right\}$$

(21)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{\mathsf{\Psi }}\triangleq \{\tilde{\psi }\mid & A^k+{\displaystyle \sum _{j=0}^{k-1}}{A}^{k-1-j}B{M}_j=0,\hfill \\ \hfill -{\displaystyle \frac{a{r}_x}{n^{\xi }}}·{\mathbf{1}}_n& \le {\displaystyle \sum _{i=0}^{k-1}}(\eta ·L_i·{\mathbf{1}}_t+D_i\left({\mathbf{M}}_k\right)f)\le {\displaystyle \frac{a{r}_x}{n^{\xi }}}·{\mathbf{1}}_n,\hfill \\ \hfill -L_i& \le D_i\left({\mathbf{M}}_k\right)E\le L_i,i\in {\mathbb{N}}_{k-1},\hfill \\ \hfill -{\displaystyle \frac{b{r}_u}{m^{\widehat{\xi }}}}·{\mathbf{1}}_m& \le {\displaystyle \sum _{i=0}^{k-1}}(\eta ·T_i·{\mathbf{1}}_t+M_{i}f)\le {\displaystyle \frac{b{r}_u}{m^{\widehat{\xi }}}}·{\mathbf{1}}_m,\hfill \\ \hfill -T_i& \le M_{i}E\le T_i,i\in {\mathbb{N}}_{k-1}\hfill \\ \hfill 0& \le a\le 1,0\le b\le 1,q_{a}a+q_{b}b\le \zeta .\}\hfill \end{array}\end{array}$$

(22)

and ${\mathbf{L}}_k\triangleq (L_0,L_1,\dots ,L_{k-1}),{\mathbf{T}}_k\triangleq (T_0,T_1,\dots ,T_{k-1}),\tilde{\psi }\triangleq (vec\left({\mathbf{M}}_k\right),vec\left({\mathbf{L}}_k\right),vec\left({\mathbf{T}}_k\right),a,b,\zeta ).$

4. Numerical Examples

In this section, we use some numerical examples to verify our results. All numerical experiments were implemented in MATLAB R2025b. The linear programming problems were solved using the MATLAB Optimization Toolbox. The computations involving Minkowski sums of polytopic sets and the visualization of the invariant sets were carried out using the Multi-Parametric Toolbox (MPT).

Example 1. 

We consider the system $x^{+}=\left[\begin{array}{cc}1& 1\\ 0& 0.8\end{array}\right]x+\left[\begin{array}{cc}0.6& 0.8\\ 0& 1.2\end{array}\right]u+w.$ And the state constraint set is a $1.5$-norm ball $\{x\in {\mathbb{R}}^2\mid {∥x∥}_{1.5}\le 4\}$, the control constraint set is a $2.5$-norm ball $\{u\in {\mathbb{R}}^2\mid {∥u∥}_{2.5}\le 2.5\}$, the disturbance constraint set $W=\left\{d\mid {∥d∥}_{\infty }\le 1\right\}.$ Take $k=5,q_a=1,q_b=1,$ we construct the following optimization problem to calculate the RCI set for this system under such constraint sets.

$$\underset{\tilde{\psi }}{min}\left\{\zeta \mid \tilde{\psi }\in \tilde{\mathsf{\Psi }}\right\}$$

(23)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{\mathsf{\Psi }}\triangleq \{\tilde{\psi }\mid & A^5+{\displaystyle \sum _{j=0}^4}{A}^{4-j}B{M}_j=0;\hfill \\ \hfill -{\displaystyle \frac{4a}{2^{1.5}}}·{\mathbf{1}}_2& \le {\displaystyle \sum _{i=0}^4}{L}_i·{\mathbf{1}}_2\le {\displaystyle \frac{4a}{2^{1.5}}}·{\mathbf{1}}_2,\hfill \\ \hfill -L_i& \le D_i\left({\mathbf{M}}_5\right)\le L_i,i\in {\mathbb{N}}_4;\hfill \\ \hfill -{\displaystyle \frac{2.5b}{2^{0.5}}}·{\mathbf{1}}_2& \le {\displaystyle \sum _{i=0}^4}{T}_i·{\mathbf{1}}_2\le {\displaystyle \frac{2.5b}{2^{0.5}}}·{\mathbf{1}}_2,\hfill \\ \hfill -T_i& \le M_i\le T_i,i\in {\mathbb{N}}_4;\hfill \\ \hfill 0& \le a\le 1,0\le b\le 1,a+b\le \zeta .\}\hfill \end{array}\end{array}$$

(24)

and $M_i\in {\mathbb{R}}^{2\times 2},L_i\in {\mathbb{R}}^{2\times 2},T_i\in {\mathbb{R}}^{2\times 2},i\in {\mathbb{N}}_4,{\mathbf{L}}_5\triangleq (L_0,L_1,L_2,L_3,L_4),{\mathbf{T}}_5\triangleq (T_0,T_1,T_2,T_3,T_4),\tilde{\psi }\triangleq (vec\left({\mathbf{M}}_5\right),vec\left({\mathbf{L}}_5\right),vec\left({\mathbf{T}}_5\right),a,b,\zeta ).$

The calculation yields

$$\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ M_3\\ M_4\end{array}\right]=\left[\begin{array}{cc}-0.7922& -0.7778\\ -0.3452& -0.6667\\ 0& 0\\ 0.2761& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ -0.0920& 0\\ 0& 0\end{array}\right]$$

The RCI set is visualized in Figure 2, where the red dashed lines represent the inscribed $2$-norm balls of the constraint sets.

Figure 2. Comparison of the computed objective sets (inner coloring area) against the constraint sets (outer boundaries) for Example 1.

We know that the $1$-norm ball and the $\infty$-norm ball are polyhedra. Previous literature has established the necessary and sufficient condition for a non-empty set to be contained within a polyhedron, that is, Lemma 1. Our conclusion, that is Proposition 4, represents a sufficient condition. Reference [32] utilizes Lemma 1 to construct the linear programming problems (3.18) and (3.24) for calculating the RCI set under polytopic constraints. Subsequently, we will illustrate the distinction between the two methods by examining specific examples. We use the control variable method to set up comparative tests.

Firstly, we employ the above two methods to calculate the smaller $1$-norm ball state constraint set for a given dynamical system.

Example 2. 

We consider the system $x^{+}=\left[\begin{array}{cc}1& 2\\ 0& 1.5\end{array}\right]x+\left[\begin{array}{c}0.6\\ 0.8\end{array}\right]u+w,$ the disturbance constraint set $W=\left\{d\mid {∥d∥}_{\infty }\le 1\right\}.$ Take $k=5,$ ${\mathbf{L}}_5\triangleq (L_0,L_1,\dots ,L_4),\psi \triangleq \left(vec\left({\mathbf{M}}_5\right),vec\left({\mathbf{L}}_5\right),r\right).$ Firstly, the optimization problem is constructed by using the method in this paper

$$\underset{\psi }{min}\left\{r\mid \psi \in \mathsf{\Psi }\right\}$$

(25)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathsf{\Psi }\triangleq \{\psi \mid -{\displaystyle \frac{r}{2}}·{\mathbf{1}}_2& \le {\displaystyle \sum _{i=0}^4}{L}_i·{\mathbf{1}}_2\le {\displaystyle \frac{r}{2}}·{\mathbf{1}}_2,\hfill \\ \hfill -L_i& \le D_i\left({\mathbf{M}}_5\right)\le L_i,i\in {\mathbb{N}}_4,\hfill \\ \hfill A^5& +{\displaystyle \sum _{j=0}^4}{A}^{4-j}B{M}_j=0,r>0.\}\hfill \end{array}\end{array}$$

(26)

and $M_i\in {\mathbb{R}}^{1\times 2},L_i\in {\mathbb{R}}^{2\times 2}.$ The calculation yields $r=4.3077$.

Expressing the $1$-norm ball in the form of a polyhedron: $B_1\left(r\right)=\{x\mid Cx\le rc\},$ where $C={\left[\begin{array}{cccc}1& 1& -1& -1\\ 1& -1& 1& -1\end{array}\right]}^T,c={\left[\begin{array}{cccc}1& 1& 1& 1\end{array}\right]}^T.$ Using the Lemma 1, we construct the following optimization problem (i.e., the optimization problem $\left(3.18\right)$ in [32])

$$\underset{\psi }{min}\left\{r\mid \psi \in \mathsf{\Psi }\right\}$$

(27)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathsf{\Psi }\triangleq & {\displaystyle \{\psi \mid \sum _{i=0}^4}{L}_i·{\mathbf{1}}_2\le r·c,\hfill \\ & -L_i\le C{D}_i\left({\mathbf{M}}_5\right)\le L_i,i\in {\mathbb{N}}_4,\hfill \\ & A^5+{\displaystyle \sum _{j=0}^4}{A}^{4-j}B{M}_j=0,r>0.\}\hfill \end{array}\end{array}$$

(28)

and $M_i\in {\mathbb{R}}^{1\times 2},L_i\in {\mathbb{R}}^{4\times 2}.$ The calculation yields $r=4.1635.$

By calculating the robust control invariant sets in two cases, we find that the RCI sets obtained by the two methods are identical. As shown in Figure 3, our method is based on a sufficient condition, there is an discrepancy of $4.3077-4.1635=0.1442.$ Consequently, treating the $1,\infty$-norm balls as polyhedra will obtain better results.

Figure 3. Comparison of the optimal containment radii for the RCI sets obtained by the proposed method ($r=4.3077$) and the method in [32] ($r=4.1635$) under $1$-norm ball constraints.

Next, we use the above two methods to calculate the RCI set for a dynamic system under the fixed $1$-norm ball state constraint set.

Example 3. 

We consider the system $x^{+}=\left[\begin{array}{cc}1& 2\\ 0& 1.5\end{array}\right]x+\left[\begin{array}{c}1\\ 0.8\end{array}\right]u+w.$ And the state constraint set is a $1$-norm ball $\{x\mid {∥x∥}_1\le 4\},$ the control constraint set is a polytope $\{u\mid Gu\le g\},$ the disturbance constraint set $W=\left\{d\mid {∥d∥}_{\infty }\le 1\right\},$ where $G={\left[\begin{array}{cc}1& -1\end{array}\right]}^T,g={\left[\begin{array}{cc}3.5& 3.5\end{array}\right]}^T.$ We are now interested in the difference between the RCI sets obtained by the two methods under the same constraint sets. Take $k=5,q_a=1,q_b=1.$ We only apply different conditions to the state constraint set for comparison and other conditions are unchanged.

Using the current method, we calculate

$$\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ M_3\\ M_4\end{array}\right]=\left[\begin{array}{cc}-0.3636& -1.9098\\ 0& 0\\ 0.8182& 0.0518\\ 0& 0.0398\\ 0& 0\end{array}\right],$$

Using the previous method (i.e., the optimization problem $\left(3.24\right)$ in [32]), we calculate

$$\left[\begin{array}{c}\overline{M_0}\\ \overline{M_1}\\ \overline{M_2}\\ \overline{M_3}\\ \overline{M_4}\end{array}\right]=\left[\begin{array}{cc}-0.3636& -1.9205\\ 0& 0\\ 0.8182& 0.1023\\ -0.1009e-14& -0.6056e-14\\ 0& 0\end{array}\right].$$

The RCI sets obtained by the two methods are calculated and visualized as shown in Figure 4. The green solid line in the figure represents the RCI set obtained using the proposed method, while the blue dashed line indicates the RCI set obtained using the previous method. Upon separately calculating the volumes, it is found that the polytope enclosed by the solid line has a volume of $11.9753$, whereas the polytope enclosed by the dashed line has a volume of $11.7186$. The results demonstrate that the proposed method yields an approximation quality comparable to the optimal solution derived from the original method.

Figure 4. Comparison of the RCI sets computed by the proposed method (green solid line) and the method in [32] (blue dashed line) under a fixed 1-norm ball state constraint.

It can be observed from Examples 2 and 3 that there is little difference in the approximation effect whether the smaller constraint set is calculated with an identical RCI set, or the optimal RCI set is calculated with a fixed constraint set. Although our method is a sufficient condition and Lemma 1 is a necessary and sufficient condition, both methods only require testing at a finite number of points to ensure the set inclusion relationship holds. However, as can be seen from the examples, our method requires only the verification of n points for an $n$-dimensional system, whereas treating the $1$-norm ball as a polytope would necessitate verifying $2^n$ points. Moreover, the points verified by our method are the standard orthogonal basis vectors $e_j$ in Euclidean space, which simplifies the computational aspect.

Example 4. 

We consider the system $x^{+}=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]x+\left[\begin{array}{c}0.5\\ 1\end{array}\right]u+w,$ the state constraint set is a $\infty$-norm ball $\{x\mid {∥x∥}_{\infty }\le 1\},$ the disturbance constraint set $W=\left\{d\mid {∥d∥}_{\infty }\le 0.1\right\}.$ This is system (15) in Section IV of reference [35]. Both this paper and reference [35] calculate robust invariant sets for discrete-time linear systems by solving linear programming problems. In this example, we provide a comparative analysis of the two methods in terms of computational efficiency and applicability.

First, the linear programming problem (21) is solved to obtain the RCI set where

$$\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ M_3\\ M_4\end{array}\right]=\left[\begin{array}{cc}-0.7500& -1.500\\ 0.7500& 0.5000\\ 0& 0\\ -0.2500& 0\\ 0.2500& 0\end{array}\right].$$

Subsequently, the linear programming method in reference [35] is employed to calculate the robust invariant sets with parameter settings $r=6,20,48.$ The computational results are visualized in Figure 5.

Figure 5. RCI set obtained by the proposed method versus the results in [35] with different parameters.

The volumes of the three robust invariant sets are calculated as $V_{r=6}=0.5688$, $V_{r=20}=0.2646$, $V_{r=48}=0.2518$, $V_{RCI}=0.1975.$ These results indicate that both linear programming approaches are effective in computing robust invariant sets for this system. However, the invariant set derived by the proposed method yields a significantly smaller volume. Furthermore, it is worth noting that with the method in [35], the resulting sets are defined by 20 and 48 non-redundant inequalities for $r=20$ and $r=48$, respectively, which substantially increases the computational complexity of the linear programming problem.

Example 5. 

We consider the system $x^{+}=\left[\begin{array}{ccc}0.5& 0& 0\\ 0& 0.6& 0\\ 0& 0& 0.8\end{array}\right]x+\left[\begin{array}{c}0.4\\ 0.3\\ 0.5\end{array}\right]u+w,$ the state constraint set is a $3$-norm ball $\{x\mid {∥x∥}_3\le 2\},$ the disturbance constraint set $W=\left\{d\in {\mathbb{R}}^3\mid {∥d∥}_{\infty }\le 0.1\right\}.$ Let $k=5,$ using the method in this paper, the RCI set of the system is calculated to be

$$\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ M_3\\ M_4\end{array}\right]=\left[\begin{array}{ccc}-1.2500& 0& -1.6000\\ -0.0128& 11.0076& -6.3952\\ 0.4577& -7.9544& 5.1161\\ 1.1287& -3.8591& 1.2655\\ -0.6769& 2.5422& -1.0124\end{array}\right].$$

The visualization of the state constraint set and RCI set is shown in Figure 6, where the red geometric body represents the 3-norm ball constraint set and the green set represents the RCI set. This demonstrates the applicability of our method in calculating the RCI set for three-dimensional constrained systems.

Figure 6. Visualization of the RCI set under the 3D 3-norm ball.

Remark 4. 

It is theoretically sufficient to select any fixed integer $k\ge n$ to formulate the optimization problems in this paper. However, to achieve the minimal-volume outer approximation of the mRCI set in practical computations, the parameter k should be dynamically determined. We have previously addressed this in [33] by observing that the volume of $R_k\left({\mathbf{M}}_k\right)$ stabilizes as k increases, leading to a practical stopping criterion based on a predefined tolerance.

5. Discussion

5.1. Practical Significance Beyond Computational

Unlike classical results that treat polytopic constraints ($p=1,\infty$) and ellipsoidal constraints ($p=2$) as separate cases requiring distinct mathematical tools, the proposed method establishes a unified LP framework applicable to arbitrary p-norms. From a practical standpoint, this provides system engineers with unprecedented flexibility. For instance, in actuator saturation limits, intermediate norms (e.g., $p=1.5$) often describe the physical constraints more accurately than the diamond-shaped 1-norm or the box-shaped ∞-norm. Prior to this work, computing RCI sets under such non-polytopic, non-ellipsoidal constraints required complex surface discretization or the introduction of a massive number of linear inequalities, rendering LP methods computationally prohibitive. Our formulation bypasses this hurdle seamlessly by adjusting a single parameter $\xi$.

Furthermore, the theoretical insight—that the sufficient condition maps the p-norm constraint to its inscribed 2-norm ball—represents a significant conceptual advancement. This geometric decoupling implies that for any symmetric, bounded, closed convex constraint set, one can potentially compute a valid RCI set by simply evaluating it at its inscribed 2-norm ball. This transforms a highly non-trivial infinite-dimensional boundary verification problem into a trivial finite-point check, opening new theoretical pathways for constrained control.

5.2. Comprehensive Comparative Analysis

To rigorously evaluate the practical significance of the proposed method, we compare it against two mainstream LP-based approaches. Compared with the Optimal Polytopic Method [32], it is acknowledged that for $p=1,\infty$, the method in [32] utilizes necessary and sufficient conditions (verifying $2^n$ extreme points) and yields the optimal polytopic RCI set. Our method uses sufficient conditions (verifying n basis vectors) and thus introduces conservatism. However, as quantitatively demonstrated in Example 3, the volume of the RCI set obtained by our method (11.9753) is only marginally smaller than that of [32] (11.7186)—a volume loss of less than 2%. In exchange for this negligible conservatism, our method cuts the number of verification constraints in half and eliminates the need to pre-calculate the polytopic facets of the constraint set, offering a highly favorable cost-to-performance ratio. The method in [35] approximates invariant sets by solving an LP defined by a finite number of pre-determined normal vectors. To achieve a refined approximation, it requires a large r (number of inequalities). As shown in Example 4, setting $r=20$ and $r=48$ in [35] leads to bloated LP problems that still yield larger (worse) invariant sets (volumes 0.2646 and 0.2518) compared to the set computed by our method (volume 0.1975). This demonstrates that our approach does not merely reduce complexity; it achieves a superior approximation accuracy with a fraction of the variables.

6. Conclusions

This paper addressed the problem of approximating robust control invariant sets for linear discrete-time systems subject to p-norm ball constraints ($p\ge 1$). The main contributions are threefold: (1) We established sufficient conditions for a non-empty set to be contained within a p-norm ball, revealing that this geometrically implies containment within the inscribed 2-norm ball; (2) We proposed a unified framework that formulates the approximation problem as a standard Linear Programming (LP) problem, which requires verifying constraints at only n standard basis vectors, significantly reducing computational complexity compared to classical polytopic methods; (3) We derived specific optimization models for computing RCI sets under both state and control constraints. Numerical results demonstrate the effectiveness of the proposed method, showing that it achieves competitive approximation accuracy with a compact formulation and effectively handles arbitrary p-norm ball constraints.

Despite these contributions, the present study has certain limitations that warrant discussion, including the potential conservatism arising from using sufficient conditions (rather than necessary and sufficient ones) and the current restriction to linear time-invariant systems. Future research will focus on two main directions: (1) investigating necessary and sufficient conditions to mitigate the conservatism inherent in the current sufficient conditions; (2) extending the proposed methodology to handle nonlinear dynamics and Model Predictive Control applications.

This paper was partially funded by the Shandong Provincial Science and Technology SME Innovation Capability Improvement Project, granted by 2024TSGC0376.