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# Stability guarantees for neural network controllers using Persidskii systems approach

Xubin Ping, Denis Efimov

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Xubin Ping, Denis Efimov. Stability guarantees for neural network controllers using Persidskii systems approach. 2025. hal-05131126v2

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Preprint submitted on 15 Oct 2025

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ARTICLE TYPE

# Stability guarantees for neural network controllers using Persidskii systems approach

Xubin Ping\*<fnref n="1" /> | Denis Efimov<fnref n="2" />

Correspondence

*Xubin Ping, Xidian University, Xi’an, Shaanxi, China. Email: pingxubin@126.com

Funding Information

This research was supported by the Natural Science Basic Research Program of Shaanxi (Program No. 2025JC-YBMS-743

Summary

The problem of establishment of the stability guarantees for closed-loop system with controller approximated by a two-layer feedforward neural network is studied. This problem is reduced to an output stability analysis for a class of generalized Persidskii models. Equivalent conditions of asymptotic stability for this class of systems are formulated, and a sufficient condition is given in terms of linear matrix inequalities. The obtained theoretical results are illustrated by the stability check for a model predictive control approximated by a neural network for a linear plant.

KEYWORDS:

Persidskii system, Feedforward neural network, Stability analysis

# 1 | INTRODUCTION

Nowadays, the artificial neural networks (NNs) find their applications in many different areas of science and technology. Their utilizations are based on the fact that feedforward NNs with sigmoidal activation functions can approximate with any desired accuracy of an arbitrary continuous function on a bounded domain 1, 2. Usually, two layers in NNs are enough to get a desired accuracy (i.e., one hidden layer with nonlinear activations, and the linear output layer), except some special situations which require deeper architectures 3.

Recall that the main approach for the verification of stability in the nonlinear setting is based on the Lyapunov function (LF) method 4 and its extensions 5, 6. The drawback of this approach is that there is no technique assigning proper LFs to generic nonlinear systems, but there are solutions for different canonical forms of nonlinear dynamics. In this line, the class of Persidskii systems has been first studied in 7 with the LFs being a sum of the integrals of the nonlinearities, and next by Persidskii in 8, where he extended the previous design by introducing in LF a combination of the absolute values of the states. This class of systems has been widely investigated in the context of diagonal stability 9, NNs 10, 11, 12 and in other applications 13. In 14, 15, 16, the conditions of input-to-state and input-to-output stability in the form of linear matrix inequalities (LMIs) are obtained for Persidskii systems. A related class of systems with a similar known canonical form of LFs is composed by Lurie models 17. The interval estimation and stabilization for Persidskii systems is investigated in 18, where the efficiency of the designed interval observer and controller is demonstrated for recurrent NNs.

For the control applications, NNs can be used to approximate state feedback controllers or system uncertainties 19. For instance, the idea of using NNs to approximate model predictive control (MPC) has been first presented in 20. In such a case, the optimization problems can be offline solved to obtain solutions, and NNs can be trained to approximate the control inputs (see also a recent result on that topic 21). The main issue in the application of NNs for the control of dynamical systems is to guarantee a certain performance (e.g., stability 22) for the closed-loop systems, which may be lost due to an approximation inaccuracy (or due to linearization of the model used for MPC design). Therefore, a suitable LF has to be found for stability analysis of controlled by NN dynamics. This issue becomes even more important in the MPC setting 23, since frequently in such

<footnote n="1">¹Xidian University, Xi’an, Shaanxi, China</footnote>

<footnote n="2">²Inria, Univ. Lille, CNRS, UMR F-59000, Lille, France</footnote>
a case there is no explicitly defined LF on the whole stabilization domain even for the nominal control before the approximation. A popular and powerful approach for the stability analysis of dynamical systems containing NN is based on the application of integral quadratic constraints for Lurie or Persidskii models 24 (see also 25 for description of the general framework), where the particular features of NNs can be also utilized 26. Thus, there are many results devoted to stability analysis of systems controlled or approximated by NNs. For example, an NN represented system is embedded in a polytope in 27, which introduces a conservatism, but then stability is analyzed using a common quadratic LF. In 28, global stability analysis for Lurie system with ReLU nonlinearities is performed using LF that contains a positive definite quadratic part. There are also methods oriented on a simultaneous search for a LF and a control law 29, 30, which is however a slightly different problem (not applicable in the MPC context).

In this paper, a controller approximated by the trained feedforward neural network is considered, and the problem of establishment of performance guarantees is reduced to the stability analysis of a generalized Persidskii system (for a particular setting). The equivalent conditions of asymptotic stability for the generalized Persidskii system are presented, which decompose the initial problem on the detectability investigation with respect to an auxiliary output and the analysis of stability with respect to this output. Sufficient local conditions in terms of LMIs are formulated (in 14, 15, 16, more restrictive global results are presented). Note that the conditions for the output stability are less confining than ones for the conventional state asymptotic stability considered in 24, 27, 28. Moreover, the existing results are grounded on the assumption that a positive definite state quadratic part is dominating in LF, which we would like to relax for a wider applicability of the result (the stabilizing part is determined by nonlinear NN). The obtained theoretical findings are illustrated through the stability analysis of an NN-based approximation of the MPC optimization problem for a linear system.

The outline of this paper is as follows. Preliminary results are introduced in Section 2. The problem statement is given in Section 3. Equivalent stability conditions are established in Section 4. The sufficient LMIs are formulated in Section 5. Simple illustrating application is considered in Section 6.

## Notation

* $\mathbb{R}$ and $\mathbb{R}_+$ represent the sets of real number and nonnegative real number, respectively. $| \cdot |$ denotes the absolute value in $\mathbb{R}$, $\| \cdot \|$ denotes the Euclidean norm on $\mathbb{R}^n$.

* For a (Lebesgue) measurable function $d: \mathbb{R}_+ \to \mathbb{R}^m$, define the norm $\|d\|_{[t_0, t_1]} = \ess \sup_{t \in [t_0, t_1]} \|d(t)\|$ for $[t_0, t_1) \subseteq \mathbb{R}_+$, then $\|d\|_{\infty} = \|d\|_{[0, +\infty)}$. For the set of $d(t)$ with the property $\|d\|_{\infty} < +\infty$, we further denote as $\mathcal{L}_{\infty}^m$ (the set of essentially bounded measurable functions).

* A continuous function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}$ if $\alpha(0) = 0$, and the function is strictly increasing. The function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}_{\infty}$ if $\alpha \in \mathcal{K}$ and it is increasing to infinity. A continuous function $\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{KL}$ if $\beta(\cdot, t) \in \mathcal{K}_{\infty}$ for each fixed $t \in \mathbb{R}_+$, and $\lim_{t \to +\infty} \beta(s, t) = 0$ for each fixed $s > 0$.

* The notation $\nabla V(x)f(x)$ stands for the directional derivative of a continuously differentiable function $V$ with respect to the vector field $f$ that is evaluated at the point $x$.

* Denote the $n \times n$ identity matrix as $I_n$, and the $n \times m$ dimensional matrix with zeros as $0_{n \times m}$. The notation $\text{diag}\{g\}$ represents a diagonal matrix of dimension $n \times n$ with a vector $g \in \mathbb{R}^n$ on the main diagonal. $\mathbb{D}_+^n \subset \mathbb{R}^{n \times n}$ denotes the subset of the nonnegative definite diagonal matrices.

* For a matrix $A \in \mathbb{R}^{m \times n}$, $A^{(i)} \in \mathbb{R}^{1 \times n}$ denotes its $i$th row for $i = 1, \dots, m$.

* A series of integers $\{1, 2, \dots, n\}$ is denoted as $\overline{1, n}$.

# 2 | PRELIMINARIES

Consider the following nonlinear system:

$$ 
\begin{aligned}
\dot{x}(t) &= f(x(t), d(t)), \forall t \ge 0, \\
y(t) &= h(x(t)),
\end{aligned} 
\tag{1} $$
where $x(t) \in \mathbb{R}^n$ is the system state, and $x(0) = x_0 \in \mathbb{R}^n$; $d(t) \in \mathbb{R}^m$ is the external disturbance, and $d \in \mathcal{L}_\infty^m$; $y(t) \in \mathbb{R}^p$ is the system output. The function $f: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ is locally Lipschitz continuous with $f(0, 0) = 0$, and $h: \mathbb{R}^n \to \mathbb{R}^p$ is a continuously differentiable function. For an initial state $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^m$, the corresponding solution of the system (1) is denoted as $x(t, x_0, d)$ for time $t \ge 0$, and the corresponding output is $y(t, x_0, d) = h(x(t, x_0, d))$.

**Definition 1.** The system (1) is forward complete if for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^m$, the solution $x(t, x_0, d)$ is uniquely defined for all time $t \ge 0$.

**Definition 2.** 31, 32 A forward complete system (1) is said to be:

1) Practical input-to-output stable (pIOS) if there exist $\beta \in \mathcal{KL}$, $\gamma \in \mathcal{K}$ and $c \in \mathbb{R}_+$ such that

$$ \|y(t, x_0, d)\| \le \beta (\|x_0\|, t) + \gamma (\|d\|_\infty) + c, \forall t \ge 0 $$

for any $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^m$. The system is input-to-output stable (IOS) if $c = 0$. When $y = x$, the IOS property is called input-to-state stability (ISS).

2) State-independent input-to-output stable (SIIOS) if there exist $\beta \in \mathcal{KL}$, $\gamma \in \mathcal{K}$ such that

$$ \|y(t, x_0, d)\| \le \beta(\|h(x_0)\|, t) + \gamma(\|d\|_\infty), \forall t \ge 0 $$

for any $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^m$.

The IOS property for $h(x) = x$ and $d \equiv 0$ is equivalent to global asymptotic stability (GAS); if $h(x) \neq x$, then such a property for $d \equiv 0$ is called output global asymptotic stability (oGAS).

**Definition 3.** 31 A forward complete system (1) is said to be uniformly bounded-input-bounded-state stable (UBIBS) if there exists $\sigma \in \mathcal{K}$ such that

$$ \|x(t, x_0, d)\| \le \max\{\sigma(\|x_0\|), \sigma(\|d\|_\infty)\}, \forall t \ge 0 $$

for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^m$.

**Definition 4.** 31, 32 For the system (1), a smooth function $V: \mathbb{R}^n \to \mathbb{R}_+$ is:

1) an IOS-Lyapunov function if there exist $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$, $\chi \in \mathcal{K}$ and $\alpha_3 \in \mathcal{KL}$ such that

$$ \alpha_1(\|h(x)\|) \le V(x) \le \alpha_2(\|x\|), \\ V(x) \ge \chi(\|d\|) \Rightarrow \nabla V(x)f(x, d) \le -\alpha_3(V(x), \|x\|) $$

for all $x \in \mathbb{R}^n$ and $d \in \mathbb{R}^m$.

2) an SIIOS-Lyapunov function if there exist $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\chi, \alpha_3 \in \mathcal{K}$ such that

$$ \alpha_1(\|h(x)\|) \le V(x) \le \alpha_2(\|h(x)\|), \\ V(x) \ge \chi(\|d\|) \Rightarrow \nabla V(x)f(x, d) \le -\alpha_3(V(x)) $$

for all $x \in \mathbb{R}^n$ and $d \in \mathbb{R}^m$.

**Theorem 1.** 31, 32 A UBIBS system (1) is IOS (SIIOS) if and only if it admits an IOS (SIIOS)-LF.

**Remark 1.** Note that for a sufficient condition of IOS/SIIOS, the UBIBS requirement can be discarded provided that the system (1) is forward complete (or it possesses the unboundedness observability property 33) and an IOS/SIIOS-LF satisfies (2) ((3) in SIIOS case) and (4) for all $x \in \mathbb{R}^n$ and $d \in \mathbb{R}^m$, some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\chi, \alpha_3 \in \mathcal{K}$.

The above stability properties and LFs can also be defined locally by restricting the sets of admissible values for the system states and outputs.

**Remark 2.** Note also that local asymptotic stability of the system (1) implies local ISS (it is a consequence of Theorem 1).

**Definition 5.** 22 For $d \equiv 0$, the system (1) is called detectable with respect to the output $y$ if $y(t, x_0, 0) \equiv 0$ for all $t \ge 0$ implies that $\lim_{t \to +\infty} x(t, x_0, 0) = 0$.
## 3 | PROBLEM STATEMENT

Consider the following class of extended Persidskii systems:

$$ \dot{x}(t) = A_0 x(t) + \sum_{j=1}^{M} A_j f^j(H_j x(t)) + Bu(t) + d(t), t \ge 0, \quad (5) $$

where $x(t) = [x_1(t) \dots x_n(t)]^\top \in \mathbb{R}^n$ is the system state, $x(0) \in \mathbb{R}^n$; $u(t) \in \mathbb{R}^p$ is the control input; and $d(t) \in \mathbb{R}^n$ is the external disturbance, $d \in \mathcal{L}_\infty^n$; $f^j(\sigma) = [f_1^j(\sigma_1) \dots f_{k_j}^j(\sigma_{k_j})]^\top$ with $\sigma = [\sigma_1 \dots \sigma_{k_j}]^\top \in \mathbb{R}^{k_j}$ and $j = 1, \dots, M$ are the diagonal functions ensuring the existence of solutions to the system (5) at least locally in the forward time, $f^j(0) = 0$; the matrices $A_r \in \mathbb{R}^{n \times k_r}$, $H_r \in \mathbb{R}^{k_r \times n}$ for $r = 0, \dots, M$ (we use the convention $k_0 = n$ with $H_0 = I_n$). Moreover, all nonlinearities belong to a sector and may take zero values at zero only (so-called passivity condition):

**Assumption 1.** For any $i = \overline{1, k_j}$, $j = \overline{1, M}$:

$$ s f_i^j(s) > 0, \quad \forall s \in \mathbb{R} \setminus \{0\}. $$

Many polynomial, sigmoidal or saturation functions verify this hypothesis, and the class of Persidskii systems serves as a natural generalization of linear ones by including a wide branch of nonlinearities, which allow the analysis of stability for these systems to be reduced to solution of LMIs 14, 15, 16 (a rare achievement in nonlinear setting).

### 3.1 | NN control analysis

Assume that the control input $u(t)$ is realized by using an NN with two layers:

$$ u(t) = \sum_{w=1}^{M} W_w^1 g^w(W_w^0 x(t) + b_w^0) + b^1, \quad (6) $$

where $W_w^1 \in \mathbb{R}^{p \times k_w}$ and $W_w^0 \in \mathbb{R}^{k_w \times n}$ are the weights; $\kappa_w$ is the number of neurons in the hidden layer; $b^1 \in \mathbb{R}^p$ and $b_w^0 \in \mathbb{R}^{k_w}$ are the biases of the NN for $w = \overline{1, M}$; $g^w(\sigma) = [g_1^w(\sigma_1) \dots g_{k_w}^w(\sigma_{k_w})]^\top$ with $\sigma \in \mathbb{R}^{k_w}$ and $w = \overline{1, M}$ are the sigmoidal activation functions of the NN; $M \ge 1$ is the number of activation functions used in the NN (selection of different activation functions can significantly improve the quality of approximation 34). For simplicity, we assume that sector condition (Assumption 1) is satisfied for the activation functions, which is frequently the case: i.e., for any $i = \overline{1, k_w}$, $w = \overline{1, M}$,

$$ \sigma g_i^w(\sigma) > 0, \quad \forall \sigma \in \mathbb{R} \setminus \{0\}, $$

and we do not impose restrictions on boundedness of these functions or their integrals. The constant weights $W_w^1, W_w^0$ and biases $b^1, b_w^0$ with $w = \overline{1, M}$ have been trained up some approximation errors to guarantee the regulation of the system (5) to the origin from the set of initial constraint set $\mathcal{X} = \{x \in \mathbb{R}^n : x_i^{\min} \le x_i \le x_i^{\max}; i = \overline{1, n}\}$, where the pre-specified vectors $x_i^{\min} \in \mathbb{R}^n$ and $x_i^{\max} \in \mathbb{R}^n$, $i = \overline{1, n}$, are the lower and upper bounds of the system states at each dimension.

**Problem 1.** Our goal is to verify local asymptotic stability (which is equivalent to local ISS, see Remark 2) of the closed-loop system (5)-(6) into the set $\mathcal{X}$.

Thus, the aim is to verify that the approximation error does not destroy the stabilization ability of the control (6).

**Remark 3.** In 24, a method is proposed to represent a generic deep NN (with many layers) in a block form suitable for analysis using integral quadratic constraints within Lurie or Persidskii system framework. That approach can be utilized to extend the results obtained for the control input in (6) to a more general scenario. Other kinds of quadratic constraints for NNs are derived in 35, where they are applied to check reachable sets of dynamical systems.

### 3.2 | Reduction to analysis of Persidskii system

Note that in order to keep the origin as the equilibrium for the system, the control input in (6) should be zero at the origin, or equivalently

$$ \sum_{w=1}^{M} W_w^1 g^w(b_w^0) + b^1 = 0. $$
Define new shifted activation functions $\tilde{g}^w(\sigma) = g^w(\sigma + b_w^0) - g^w(b_w^0)$ for $w = 1, \overline{\mathbb{M}}$, which also verify Assumption 1 in the case of sigmoidal function $g^w$ that satisfies the sector condition. Then, the control input (6) can be written as

$$u(t) = \sum_{w=1}^{\mathbb{M}} W_w^1 \tilde{g}^w(W_w^0 x(t)),$$

and the closed-loop system (5)-(6) takes the form

$$\dot{x}(t) = A_0 x(t) + \sum_{j=1}^{\bar{M}} \bar{A}_j \bar{f}^j(\bar{H}_j x(t)) + d(t), \quad (7)$$

where

$$\begin{gathered} \bar{A}_j = A_j, \quad \bar{H}_j = H_j, \quad \bar{f}^j(\sigma) = f^j(\sigma), \quad j = 1, \overline{\mathbb{M}}; \\ \bar{A}_j = BW_{j-\mathbb{M}}^1, \quad \bar{H}_j = W_{j-\mathbb{M}}^0, \quad \bar{f}^j(\sigma) = \tilde{g}^{j-\mathbb{M}}(\sigma), \quad j = \overline{\mathbb{M}}+1, \overline{\mathbb{M}}, \quad \bar{M} = \mathbb{M} + \mathbb{M}. \end{gathered}$$

Following the above observations, and to simplify the terminology and notations, we further consider the problem of asymptotic stability analysis for the system (5) with $u \equiv 0$ and $d \equiv 0$, i.e.,

$$\dot{x}(t) = A_0 x(t) + \sum_{j=1}^{M} A_j f^j(H_j x(t)). \quad (8)$$

System (8) under a small ambiguity in symbols represents the perturbation-free closed-loop dynamics of the system (5)–(6) (i.e., system (7) with $d \equiv 0$). To this end, we need to consider also output asymptotic stability with respect to the variable

$$y(t) = Hx(t), \quad H = \begin{bmatrix} H_1 \\ \vdots \\ H_M \end{bmatrix} \in \mathbb{R}^{K \times n}, \quad (9)$$

where $K = \sum_{j=1}^{M} k_j$ (obviously, we assume that $\mathrm{rank}(H) \ge 1$). If (5) is a linear system, then $H_j = W_j^0$ for $j = 1, \overline{\mathbb{M}}$, i.e., the output (9) is the argument of the activation functions.

## 4 | STABILITY CONDITIONS

Following equivalent conditions of absolute stability established by 36 in terms of the stabilization of linear part of Lurie model by a linear feedback and the output asymptotic stability, let us present similar result for the asymptotic stability of the system (8).

**Theorem 2.** Under Assumption 1, the system (8) is GAS if and only if it is detectable (the pair $(A_0, H)$ is detectable) and oGAS with respect to the output $y$ that is given in (9).

*Proof.* Firstly, for the sufficient condition, let the system (8) be GAS. Since $\|y\| \le \|H\|\|x\|$, it implies forward completeness of the system and oGAS (see Definition 2). For a pair of the matrices $(A_0, H)$, there exists a nonsingular matrix $T \in \mathbb{R}^{n \times n}$ that defines a transformation of coordinates to the canonical observability form (see Theorem 13.12 in reference 37). It means that for $Tx = z = \begin{bmatrix} z_o \\ z_u \end{bmatrix}$ with $z_o \in \mathbb{R}^{n_o}, z_u \in \mathbb{R}^{n_u}$ and $n = n_o + n_u$, we obtain that:

$$\begin{align*} \dot{z}_o(t) &= D_{oo}z_o(t) + \sum_{j=1}^{M} D_{oj}f^j(H_j T^{-1}z(t)), \\ \dot{z}_u(t) &= D_{uo}z_o(t) + D_{uu}z_u(t) + \sum_{j=1}^{M} D_{uj}f^j(H_j T^{-1}z(t)), \\ y(t) &= HT^{-1}z(t) = Cz_o(t), \end{align*}$$

where

$$TA_0T^{-1} = \begin{bmatrix} D_{oo} & 0_{n_o \times n_u} \\ D_{uo} & D_{uu} \end{bmatrix}, \quad TA_j = \begin{bmatrix} D_{oj} \\ D_{uj} \end{bmatrix}.$$
Obviously, $H_j T^{-1}z$ depends only on $z_o$ for all $j = 1, \overline{M}$. Hence, $z_o(t) \equiv 0$ for all $t \ge 0$ is an invariant subspace for the system (8) since $f^j(0) = 0$, which implies that for GAS of the system (8) in the coordinates $z$, the matrix $D_{uu}$ should be Hurwitz 37. The latter property implies detectability of the pair $(A_0, H)$ or detectability of the system (8) with respect to the output $y$.

Secondly, for the necessary condition, let the system (8) be oGAS and detectable in the output $y$, where the latter property implies detectability of the linear part since the nonlinearities only depend on the output. Detectability of the pair $(A_0, H)$ implies that there exists $L \in \mathbb{R}^{n \times K}$ such that the matrix $A_0 - LH$ is Hurwitz. Therefore, there exist $\mathbb{P} = \mathbb{P}^T > 0$ and $\mathbb{Q} = \mathbb{Q}^T > 0$ such that

$$ (A_0 - LH)^T \mathbb{P} + \mathbb{P} (A_0 - LH) = -2\mathbb{Q}. $$

Define the Lyapunov function $V(x) = x^T \mathbb{P} x$, and applies Young's inequality (i.e., for any vectors $a$ and $b$ with proper dimensions, and symmetric positive matrix $X$, the inequality $a^T b \le \frac{1}{2} a^T X a + \frac{1}{2} b^T X^{-1} b$ always holds), the time derivative for the system (8) can be written as follows:

$$ \dot{V} = -2x^T \mathbb{Q} x + 2x^T \mathbb{P} \varphi(y) = -2x^T \mathbb{Q} x + 2x^T \mathbb{Q} \frac{1}{2} \mathbb{Q}^{-\frac{1}{2}} \mathbb{P} \varphi(y) \le -x^T \mathbb{Q} x + \varphi^T(y) \mathbb{P} \mathbb{Q} \varphi(y), $$

where $\varphi(y) = Ly + \sum_{j=1}^{M} A_j f^j(H_j x)$. Due to oGAS property, the output $y(t)$ is bounded by a function of the initial condition $x(0)$ for all $t \ge 0$ and asymptotically converges to zero, which according to the features of $V$ and $\dot{V}$, implies the similar properties for the system state $x(t)$. Thus, the system (8) is GAS. $\square$

The result of Theorem 2 allows the analysis of asymptotic stability for the system (8) to be decomposed on verification of the detectability of the pair $(A_0, H)$ and the oGAS property. The former detectability of the pair $(A_0, H)$ can be checked by using respective necessary and sufficient LMIs (if the row rank of $H$ is $n$, then even this verification can be skipped, since detectability is obvious in such a case):

$$ \exists \mathcal{P} = \mathbb{P}^T > 0, Y \in \mathbb{R}^{n \times K} : A_0^T \mathcal{P} + \mathcal{P} A_0 + Y H + H^T Y^T < 0. $$

The latter oGAS property admits simpler equivalent conditions in terms of existence of corresponding LFs (see Definitions 2 and 4, where the conditions for an IOS-LF are weaker than those for an ISS-LF). Sufficient global LF-based conditions of ISS and IOS for the system (5) have been given in 14, 15 in the form of LMIs.

Despite Theorem 2 is formulated in the global setting, its results have the meanings for the local asymptotic stability, which we need to check to obtain the stability guarantees for the control inputs approximated by the trained NNs. Moreover, the detectability conditions on LMIs can be checked even during training of NNs since they are simple and their satisfaction is necessary for stability of the closed-loop system.

## 5 | SUFFICIENT CONDITIONS OF OUTPUT STABILITY

Let us extend the results of 14, 15 by proposing the LMIs to check local output asymptotic stability or local IOS of (8) or (5), respectively. Our motivation is that local analysis can be more efficient since in such a setting, the sector conditions can be refined as follows:

**Assumption 2.** For some $s^* > 0$, and any $i = \overline{1, k_j}$, $j = \overline{1, M}$, suppose that there exist $0 \le \underline{n}_{ij} < \overline{n}_{ij} < +\infty$ such that

$$ (f_i^j(s) - \underline{n}_{ij} s) (\bar{n}_{ij} s - f_i^j(s)) > 0, \quad \forall s \in [-s^*, s^*] \setminus \{0\}. $$

Assumption 2 introduces locally additional restrictions on the sector of admissible values for the nonlinearities, and the constant $s^*$ can be related with the domain $\mathcal{X}$ (i.e., $\max_{1 \le i \le K} |(Hx)_i| \le s^*$). Under Assumption 2, the system (8) can be rewritten
as follows:

$$
\begin{align}
\dot{x}(t) &= A_0 x(t) + \sum_{j=1}^{M} A_j f^j(H_j x(t)) \nonumber \\
&= \left( A_0 + \sum_{j=1}^{M} (A_j + R_j) \underline{N}_j H_j - R_j \overline{N}_j H_j \right) x(t) + \sum_{j=1}^{M} (A_j + R_j) \left( f^j(H_j x(t)) - \underline{N}_j H_j x(t) \right) \nonumber \\
&\quad + \sum_{j=1}^{M} R_j \left( \overline{N}_j H_j x(t) - f^j(H_j x(t)) \right) \nonumber \\
&= A_0 x(t) + \sum_{j=1}^{2M} A_j F^j(H_j x(t)), \tag{10}
\end{align}
$$

where $\overline{N}_j, \underline{N}_j \in \mathbb{R}^{k_j \times k_j}$ are the nonnegative diagonal matrices, and their elements on the main diagonal are bigger or not bigger than $\overline{n}_{ij}$ and $\underline{n}_{ij}$, respectively, for the corresponding elements; the matrices $R_j \in \mathbb{R}^{n \times k_j}$ are arbitrary, and

$$
\begin{align*}
A_0 &= A_0 + \sum_{j=1}^{M} (A_j + R_j) \underline{N}_j H_j - R_j \overline{N}_j H_j; \\
A_j &= A_j + R_j, \quad F^j(\sigma) = f^j(\sigma) - \underline{N}_j \sigma, \quad H_j = H_j; \\
A_{j+M} &= R_j, \quad F^{j+M}(\sigma) = \overline{N}_j \sigma - f^j(\sigma), \quad H_{j+M} = H_j,
\end{align*}
$$

with $F^j : \mathbb{R}^{k_j} \to \mathbb{R}^{k_j}$, $F^{j+M} : \mathbb{R}^{k_{j+M}} \to \mathbb{R}^{k_{j+M}}$ denoting $k_j = k_{j+M} = k_j$ for $j = 1, \overline{M}$. Thus, the system (10) has two times more nonlinearities, which also verifies Assumption 1 under Assumption 2. Therefore, for any $i = 1, \overline{k}_j, j = 1, 2\overline{M}$,

$$s F_i^j(s) > 0 \quad \forall s \in [-s^*, s^*] \setminus \{0\}.$$

Moreover, the matrices $A_r$ for $r = 0, 2\overline{M}$ have additional tuning terms $\overline{N}_j, \underline{N}_j, R_j$ that may be chosen to guarantee verification of the stabiity LMIs that we will formulate below by considering a LF candidate:

$$V(x) = x^\top P x + 2 \sum_{j=1}^{2M} \sum_{i=1}^{k_j} \lambda_i^j \int_0^{(H_j x)_i} F_i^j(s) ds,$$

where a symmetric matrix $P$ and diagonal matrices $\Lambda_j = \text{diag}\{[\lambda_1^j \dots \lambda_{k_j}^j]^\top\}$ belong to the optimization variables. In other words, by a proper selection of $R_j$, we want to provide properties for $A_0, A_j$ and $A_{j+M}$ with $j = 1, \overline{M}$ enforcing existence of a LF $V(x)$ (i.e., a matrix $A_0$ or $A_j$ may not satisfy the needed LMI, but $A_0$ or $A_j$ can be stabilized by a virtual control gain $R_j$):<fnref n="1" />

**Theorem 3.** Let Assumptions 1 and 2 be satisfied. A forward complete system (8) is locally asymptotically stable with respect to the output $y$ into the domain $\mathcal{X} = \{x \in \mathbb{R}^n : V(x) \le \varpi\}$ with $\varpi = \arg\sup_{s>0}\{x \in \mathbb{R}^n, V(x) \le s : \max_{1 \le i \le K} |(Hx)_i| \le s^*\}$ if there exist $0 \le P^\top = P \in \mathbb{R}^{n \times n}$, $S \in \mathbb{D}_+^K$, $\Lambda_j \in \mathbb{D}_+^{k_j}$ for $j = 1, 2\overline{M}$, $Y_{g,g} \in \mathbb{D}_+^{k_g}$ for $g = 0, 2\overline{M}$, $Y_{0,j} \in \mathbb{D}_+^{k_j}$ for $j = 1, 2\overline{M}$, $Y_{z,l} \in \mathcal{M}(H_z, H_l)$ for $z = 1, 2\overline{M} - 1$ and $l = z + 1, 2\overline{M}$ such that the following LMIs (11)–(13) are verified:

$$S > 0, \quad \left[ \begin{array}{cc} P + \sum_{j=1}^{2M} H_j^\top \Lambda_j H_j & H^\top \\ H & S \end{array} \right] \ge 0, \tag{11}$$

$$Q = \begin{bmatrix} A_0^\top P + P A_0 + Y_{0,0} & P A_1 + A_0^\top H_1^\top \Lambda_1 + H_1^\top Y_{0,1} & \dots & P A_{2M} + A_0^\top H_{2M}^\top \Lambda_{2M} + H_{2M}^\top Y_{0,2M} \\ * & \Lambda_1 H_1 A_1 + A_1^\top H_1^\top \Lambda_1 + Y_{1,1} & \dots & \Lambda_1 H_1 A_{2M} + A_1^\top H_{2M}^\top \Lambda_{2M} + Y_{1,2M} \\ \vdots & \vdots & \ddots & \vdots \\ * & * & \dots & \Lambda_{2M} H_{2M} A_{2M} + A_2^\top H_{2M}^\top \Lambda_{2M} + Y_{2M,2M} \end{bmatrix} \le 0, \tag{12}$$

$$\rho \left( P + \sum_{j=1}^{2M} H_j^\top \Lambda^j H_j \right) \le \sum_{g=0}^{2M} H_g^\top Y_{g,g} H_g + \sum_{z=1}^{2M} \sum_{l=z+1}^{2M} H_z^\top Y_{z,l} H_l + \sum_{j=1}^{2M} H_j^\top Y_{0,j} H_j, \quad \rho > 0. \tag{13}$$

<footnote n="1">¹Denote by $\mathcal{M}(H_z, H_l) \subset \mathbb{R}^{k_z \times k_l}$ a set of matrices, whose ij-th element is $i$) nonnegative if $H_z^{(i)}$ and $H_l^{(i)}$ are co-directional, ii) nonpositive if $H_z^{(i)}$ and $H_l^{(i)}$ are anti-directional, and iii) zero otherwise, for $i = 1, \overline{k}_z, j = 1, \overline{k}_l, z = 1, 2\overline{M} - 1$ and $l = z + 1, 2\overline{M}$.</footnote>
Proof. According to Definition 4, we should find $\alpha_1, \alpha_2 \in \mathcal{K}$ such that the property in (2) is verified locally for $x \in \mathcal{X}$. Existence of $\alpha_2$ can be substantiated by the continuity of $V$ and Assumptions 1 and 2, the property $V(x) \ge \alpha_1 (\|Hx\|)$ follows from the inequality (the integrals are positive definite with respect to the component of $y$ by Assumptions 1 and 2):

$$P + \sum_{j=1}^{2M} \mathcal{H}_j^\top \Lambda_j \mathcal{H}_j \geq H^\top S^{-1} H, \quad (14)$$

for some $S \in \mathbb{D}_+^K$ with $S > 0$. By using Schur complement, condition (14) is equivalent to LMI (11). Let us calculate the derivative of $V$ along the solutions of the system (8):

$$\begin{aligned} \dot{V} &= x^\top P \dot{x} + \dot{x}^\top P x + 2 \sum_{j=1}^{2M} \dot{x}^\top \mathcal{H}_j^\top \Lambda_j F^j(H_j x) \\ &= \zeta^\top Q \zeta - \sum_{g=0}^{2M} F^g(H_g x)^\top Y_{g,g} F^g(H_g x) - 2 \sum_{j=1}^{2M} x^\top \mathcal{H}_j^\top Y_{0,j} F^j(H_j x) - 2 \sum_{z=1}^{2M-1} \sum_{l=z+1}^{2M} F^z(H_z x)^\top Y_{z,l} F^l(H_l x), \end{aligned}$$

where the vector $F^0(H_0 x)^\top = x$ with $H_0 = I_n$, and the augmented vector $\zeta$ is

$$\zeta^\top = [F^0(H_0 x)^\top \ F^1(H_1 x)^\top \ \dots \ F^{2M}(H_{2M} x)^\top]^\top.$$

Since $Q \le 0$ is given in (12) and by conditions of the theorem,

$$\dot{V} \le - \sum_{g=0}^{2M} F^g(H_g x)^\top Y_{g,g} F^g(H_g x) - 2 \sum_{j=1}^{2M} x^\top \mathcal{H}_j^\top Y_{0,j} F^j(H_j x) - 2 \sum_{z=1}^{2M-1} \sum_{l=z+1}^{2M} F^k(H_z x)^\top Y_{z,l} F^l(H_l x),$$

we need to ensure that

$$\dot{V} \le -\alpha(V)$$

for some $\alpha \in \mathcal{K}$, which requires that the following inequality holds:

$$\alpha(V) \le \sum_{g=0}^{2M} F^g(H_g x)^\top Y_{g,g} F^g(H_g x) + 2 \sum_{j=1}^{2M} x^\top \mathcal{H}_j^\top Y_{0,j} F^j(H_j x) + 2 \sum_{z=1}^{2M-1} \sum_{l=z+1}^{2M} F^z(H_z x)^\top Y_{z,l} F^l(H_l x).$$

This property follows from LMI (13) (in other words, we need to guarantee that the kernel of the quadratic form $\sum_{g=0}^{2M} \mathcal{H}_g^\top Y_{g,g} \mathcal{H}_g + \sum_{j=1}^{2M} \mathcal{H}_j^\top Y_{0,j} \mathcal{H}_j + \sum_{z=1}^{2M} \sum_{l=z+1}^{2M} \mathcal{H}_z^\top Y_{z,l} \mathcal{H}_l$ is the same or inside of the kernel of the quadratic form of $P + \sum_{j=1}^{2M} \mathcal{H}_j^\top \Lambda^j \mathcal{H}_j$). $\square$

For the conditions in Theorem 3, if $V$ is positive definite with respect to the full system state $x$, then the requirement of forward completeness can be skipped.

**Corollary 1.** Let all conditions of Theorem 3 be verified by replacing (11) with $P + \sum_{j=1}^{2M} \mathcal{H}_j^\top \Lambda_j \mathcal{H}_j > 0$, and (13) with $\rho H^\top H \le \sum_{g=0}^{2M} \mathcal{H}_g^\top Y_{g,g} \mathcal{H}_g + \sum_{j=1}^{2M} \mathcal{H}_j^\top Y_{0,j} \mathcal{H}_j + \sum_{z=1}^{2M} \sum_{l=z+1}^{2M} \mathcal{H}_z^\top Y_{z,l} \mathcal{H}_l$. Then, the system (8) is locally asymptotically stable with respect to $y$ in $\mathcal{X}$.

Proof. The proof is a direct consequence of Theorem 3. Notice that if $V$ is positive definite in the system state, then the condition $\dot{V} \le -\alpha(|y|)$, which is verified in such a case for some $\alpha \in \mathcal{K}$, implies boundedness of the system state and convergence of the system output $y$ to zero due to LaSalle invariance principle. $\square$

The presented conditions are rather sophisticated, but they are not restrictive. For example, they allow the quadratic part of LF $V(x)$ determined by the matrix $P$ to be semi-definite, and the stability of the system to be provided by nonlinear terms (by the NN-based control), contrarily to many existing results that treat the nonlinearity as a perturbation by using a purely quadratic LF $x^\top \tilde{P} x$ with a positive definite and symmetric matrix $\tilde{P}$ (see, e.g., 24, 27).

A drawback of the LMIs from Theorem 3 is that they are not solved with respect to the matrix variables $R_j$. By considering $R_j$ as a part of optimization variables, a bilinear matrix inequality problem has to be solved, then handling this issue is an appealing direction of future investigations.
6 | APPLICATION TO MPC OF LINEAR SYSTEMS

For the following linear system (15) (i.e., $M = 0$ in the system (5)), an MPC control is designed to stabilize it with the considerations of constraints on the states and control inputs:

$$\dot{x}(t) = Ax(t) + Bu(t) + d(t), \quad (15)$$

where the system state $x(t) \in \mathcal{X} \subset \mathbb{R}^n$, control input $u(t) \in \mathcal{U} \subset \mathbb{R}^p$, and disturbance $d(t) \in \mathcal{D} \subset \mathbb{R}^n$; $\mathcal{X}, \mathcal{U}$, and $\mathcal{D}$ are the polyhedral sets, and both contain the origin as an interior point. The control input $u(t)$ constrained in the set $\mathcal{U}$ is approximated by the following feedforward NN with one hidden layer and one kind of activation functions ($\mathcal{M} = 1$ in (6)):

$$u(t) = W_1^{-1} g^1(W_1^0 x(t) + b_1^0) + b^1, \quad (16)$$

where all variables have the same meaning as above. Note that the following constraint

$$W_1^{-1} g^1(b_1^0) + b^1 = 0$$

should be verified to keep the origin as an equilibrium of the closed-loop system. Then, for the closed-loop system (15)–(16), the parameters for the system (8) (i.e., disturbance $d \equiv 0$) with $M = 1$ are

$$A_0 = A, \quad A_1 = BW_1^{-1}, \quad H_1 = W_1^0, \\ f^1(\sigma) = g^1(\sigma + b_1^0) - g^1(b_1^0),$$

where the nonlinear function $f^1$ verifies Assumption 1 for a rich variety of activation functions used in NN, e.g., for the following hyperbolic tangent function:

$$g^1(s) = \frac{1 - e^{-2s}}{1 + e^{-2s}}.$$

Obviously, for such a nonlinearity, Assumption 2 is also satisfied. Then for any $s^* > 0$, there exists $0 < \underline{n} < \bar{n} < +\infty$ such that

$$(f^1(s) - \underline{n}s)(\bar{n}s - f^1(s)) > 0 \quad \forall s \in [-s^*, s^*] \setminus \{0\}.$$

Therefore, the results of this work can be employed: we need to check detectability of the pair $(A, W_1^0)$ and LMIs (11)–(13) from Theorem 3 or Corollary 1, whose satisfaction implies the local asymptotic stability of the NN controlled system, and also its local ISS (see Remark 2).

We assume that the MPC problem is formulated for the case $d \equiv 0$, but it also can be solved by taking the presence of uncertainties into account. To demonstrate the robustness of the closed-loop system with the approximated control inputs by the trained NN, we consider disturbances during the simulations (the issues of satisfaction of constraints by the NN-based control signal $u(t)$ and the state vector $x(t)$ do not belong to the scope of this note).

**Example** Let $n = 2$, and the system model be given by a chain of integrators,

$$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$

The system state constraint set is $\mathcal{X} = \{x \in \mathbb{R}^2 : x^{\min} \le x \le x^{\max}\}$, where $x = [x_1, x_2]^T$; $x^{\min} = [x_1^{\min}, x_2^{\min}]^T = [-1, -1]^T$; $x^{\max} = [x_1^{\max}, x_2^{\max}]^T = [1, 1]^T$. The control input constraint set $\mathcal{U} = \{u(t) \in \mathbb{R} : -1 \le u(t) \le 1\}$. By assuming the sample-and-hold control strategy and discretizing the model with a constant sampling time step $\Delta t = 0.01$ seconds, the following system model is obtained:

$$x_{k+1} = A_d x_k + B_d u_k,$$

where $x_k = x(t_k) = [x_1(t_k), x_2(t_k)]^T$, $u_k = u(t_k)$ for $t_k = \Delta t k$ with the nonnegative integers $k \ge 0$, and

$$A_d = \begin{bmatrix} 1 & \Delta t \\ 0 & 1 \end{bmatrix}, \quad B_d = \begin{bmatrix} \frac{\Delta t^2}{2} \\ \Delta t \end{bmatrix}.$$

For the above system model, define the following optimization performance cost:

$$J_k = x_{k+N}^T W x_{k+N} + \sum_{i=0}^{N-1} (x_{k+i}^T Q x_{k+i} + u_{k+i}^T R u_{k+i}),$$
<figure><img src="image_3.png" /><figcaption>**FIGURE 1** The results of control inputs for the MPC optimization problems. **FIGURE 2** The results of approximation for the control inputs by the trained NN.</figcaption></figure>

for some symmetric positive definite matrices W, Q, R, where the prediction horizon and control horizon $N \ge 1$. Then, the MPC optimization problem is formulated as

$$ \min_{u_{k+i}} J_k, \qquad (17) $$

$$ \text{s.t.} \quad x_{k+i} \in \mathcal{X}, u_{k+i} \in \mathcal{U}, i = \overline{0, N-1}. \qquad (18) $$

In the simulations, select $W = I_2$, $Q = \text{diag}\{[2 \ 1]\}$, $R = 1$ and $N = 25$, then the MPC optimization problem (17)–(18) for this system can be solved by using the toolbox of MATLAB. Let us generate a uniform grid on the state constraint set $\mathcal{X}$:

$$ x^{l,j} = \begin{bmatrix} x_1^{\text{min}} + (x_1^{\text{max}} - x_1^{\text{min}}) \frac{l}{N_g} \\ x_2^{\text{min}} + (x_2^{\text{max}} - x_2^{\text{min}}) \frac{j}{N_g} \end{bmatrix}, l, j = \overline{0, N_g}, $$

where $N_g = 50$. For each pre-specified system state $x^{l,j}, l, j = 0, N_g$, we can generate $u_{\text{MPC}}^{l,j}$ as the respective MPC control input by offline solving the MPC optimization problem (17)–(18). The control inputs $u_{\text{MPC}}^{l,j}$ of the offline solved MPC optimization problems for the defined system states $x^{l,j}, l, j = 0, N_g$, are shown in Figure 1.

Then, an NN with the number of neurons $\kappa_1 = 5$ in the hidden layer and the conventional hyperbolic tangent activation function can be trained to minimize the following errors:

$$ \sum_{l=0}^{N_g} \sum_{j=0}^{N_g} |u_{\text{MPC}}^{l,j} - u_{\text{NN}}^{l,j}|^2 $$

where $u_{\text{NN}}^{l,j}$ are the approximations of the control inputs by the trained NN. The respective toolbox of MATLAB can be applied for this purpose, where the Levenberg-Marquardt method is used for training the NN, which provides us the weights $W_1^0, W_1^1$, and biases $b_1^0, b_1^1$. The results of the approximation of the control inputs $u_{\text{NN}}^{l,j}$ are the outputs of the trained NN and shown in Figure 2.

Now the results of this paper can be applied to get the stability certificates for this control and the continuous-time plant. The detectability in our case is trivially satisfied since $W_1^0$ is a full row rank matrix. Hence, the LMIs (11)–(13) from Theorem 3 or Corollary 1 are checked in MATLAB. The coefficients $n_{i1}, \bar{n}_{i1}$ for $i = 1, 2$ are calculated numerically, and we are looking iteratively for the solutions of $P, \Lambda_1, \Lambda_2$ and $R_1$ to brake the bilinear dependence of the problem on these variables. In the simulations, the responses of the closed-loop system states are shown in Figure 3, where 20 initial system states are randomly selected in the set $\mathcal{X}$, and the control inputs are generated by the outputs of the trained NN. For the verification of robustness of the trained NN, the bounded disturbance $d(t) \in \mathbb{R}^2$ and state measurement noise $v(t) \in \mathbb{R}^2$ (i.e., the control input in (16) uses the signal $x(t) + v(t)$) are respectively generated by $d(t) = 0.002(\text{rand}(t) - 0.5) + 0.001 \sin(\omega_1 t)$ and $v(t) = 0.2(\text{rand}(t) - 0.5) + 0.1 \sin(\omega_2 t)$, where $\text{rand}(t) \in [0, 1]^2$ generates a vector of uniformly distributed random numbers, and the vectors $\omega_1 = [5 \ 6]^T$ and $\omega_2 = [7 \ 8]^T$. It is shown in Figure 3 that due to the presence of disturbances and noises, for the initial conditions close to
<figure><img src="image_4.png" /><figcaption>**FIGURE 3** The responses for the closed-loop system states and control inputs generated by the outputs of the trained NN.</figcaption></figure>

the constraint bounds, the system states may slightly leave the constraint set, while the control inputs generated from the outputs of the trained NN stays bounded.

## 7 | CONCLUSIONS

The study has focused on examining the stability guarantees for the closed-loop system with a controller approximated by a two-layer neural network. This investigation reduces the problem to the output stability analysis for a class of generalized Persidskii models. The paper formulates equivalent conditions for asymptotic stability in this class of systems and provides a sufficient condition expressed in terms of matrix inequalities. The advantages of theoretical findings are demonstrated through checking the guarantees for a model predictive control approximated by a neural network for a linear system.

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<figure><img src="image_5.png" /></figure>