1 Introduction

Quadratic stability and stabilization are important topics in stability analysis and control of dynamical systems. A system is said to be quadratic stable if there exists a Quadratic Lyapunov function (QLF) whose derivative (or first-order difference in discrete time) along the system trajectories is negative [3]. While in the beginning of 1990s, the study of quadratic stability concentrated on systems with uncertainties [5, 15, 20, 25, 28], in the past decade, this topic has been more intensively investigated in switched systems [2, 13, 26, 34] or fuzzy dynamical systems [10, 16, 19, 21].

The main issue in the research of quadratic stability is the sufficient and necessary condition of the existence of a QLF. Nevertheless, the existing results about quadratic stability conditions, such as those mentioned above, are model-based. These approaches rely on the accuracy of the mathematical model describing system dynamics. If a mathematical model is difficult to be established precisely enough or unavailable, then controllers that are designed based on model-based stability conditions may lead to bad control performance or even failure of stabilization.

Consider a system trajectory starting from the initial time to an arbitrary time instant. According to the definition of quadratic stability, a system is quadratic stable if a QLF exists for arbitrary trajectories. If it can be known under which condition a QLF exist for the concerned trajectory, then the system can be controlled to be quadratic stable, as long as at any time instant the trajectory of system under control satisfy this condition of QLF existence. Under this circumstance, the mathematical system model is not necessarily known, and the model-related stabilization problems mentioned above can be avoided.

Motivated by these concerns, this paper concentrates on the data-driven realization of quadratic stability judgment and the corresponding control design of stabilizing nonlinear discrete-time systems. Firstly, this paper uses the measured data to construct two polyhedral cones, and shows that the condition for QLF existence is identical to the emptiness of the intersection between these two polyhedral cones. Secondly, this data-driven stability condition is used to design control for stabilization problems in combination with online system identification.

It should be noticed that the proposed control has no specific requirement for the choice of system identification approaches, as long as they can manage one-step black-box model prediction online. In this paper, the online system identification method using recurrent neural network (RNN) is chosen due to its good performance in dealing with black-box modeling of an arbitrary unknown nonlinear system [9, 27].

Two assumptions have to be made at present stage of this contribution: (i) all system states are measurable, and (ii) the measurements are free of noise. The first assumption is made because the quadratic stability discussed in this paper is in the sense of Lyapunov stability, which is defined on the complete observation of system trajectory (Definition 1 in Sect. 2.1). The second assumption is made because the proposed data-driven stability condition is based on system states, which means that a small variation of true values in system states may probably lead to intolerable bias in the geometrical representation of stability conditions and untrue judgment of system stability.

The paper is organized as follows: at first, the problem definition of quadratic stability and stabilization is introduced in Sect. 2; secondly, the proposed data-driven quadratic stability criterion is presented in Sect. 3; after that, the stabilization using the proposed stability condition is introduced in Sect. 4; in Sect. 5, a numerical example of stabilizing a nonlinear aeroelastic system is presented; finally, this contribution is concluded in the last section of the paper.

2 Problem definition

2.1 Data-driven quadratic stability

The discrete-time nonlinear system considered for stability analysis is described by

$$\begin{aligned} \varvec{x}(k+1) = \varvec{f}(\varvec{x}(k)), \end{aligned}$$
(1)

where \(\varvec{f}(\cdot ):\varOmega \rightarrow \mathbb {R}^n\) is a smooth mapping from a compact set \(\varOmega \subset \mathbb {R}^n\) into \(\mathbb {R}^n\) and \(x\) the state vector defined in \(\varOmega \). According to [3], the definition of quadratic stability for system (1) can be stated as

Definition 1

The system (1) is quadratic stable if there exists a positive definite matrix \(\varvec{P}\) such that the first-order difference of the function \(\varvec{V}(\varvec{x}\,(k)) = \varvec{x}\,(k)^T\,\varvec{P}\,\varvec{x}\,(k)\) along the solution of system (1) satisfies

$$\begin{aligned} \Delta V(\varvec{x}(k))&= V(\varvec{x}(k+1))-V(\varvec{x}(k)) \\&= V(\varvec{f}(\varvec{x}(k)))-V(\varvec{x}(k)) <0. \end{aligned}$$

Correspondingly, the function \(V(\varvec{x}\,(k))\) is called quadratic Lyapunov function (QLF). If in addition \(\varvec{P}\) is diagonal, then \(V(\varvec{x}\,(k))\) is called diagonal quadratic Lyapunov function (DQLF) and the system (1) is diagonally quadratic stable.

In the data-driven context, the existence of a QLF cannot be determined by using the analytical form of \(\varvec{f}(\varvec{x})\) because it is assumed as unknown. Assume that the system (1) be fully observable, and the system states be measured without noise. At the time instant \(t=r\), the data set containing \(r\) consecutive measurements of system states can be denoted as

$$\begin{aligned} {\mathcal {X}}_r = \left\{ \varvec{x}(1),\,...\,, \varvec{x}(r)\right\} . \end{aligned}$$
(2)

As mentioned in the introduction section, the data-driven quadratic stability judgment needs only to judge the stability of the currently running motion of the concerned system, i.e., the measured states of the running trajectory, but not the existence of the QLF for the global state space of the system. Therefore, the objective of this paper is defined as to determine the existence of a QLF for the complete system trajectory under initial condition \(\varvec{x}(1) = \varvec{x}_0\). The system is judged as quadratic stable if and only if a QLF can be found based on the measured data.

2.2 Stabilization using data-driven stability condition

Consider the nonlinear discrete-time system

$$\begin{aligned} \varvec{x}(k+1)= \varvec{f}(\varvec{x}(k),\varvec{u}(k)), \end{aligned}$$
(3)

where \(\varvec{x}\) denotes the state vector and \(\varvec{u}\) the control input. The quadratic stabilization problem related to system (3) is to design a control input

$$\begin{aligned} \varvec{u}(k) = \varvec{u}(\varvec{x}(k),\, \varvec{x}(k-1),\,...\,,\varvec{x}(k-l)), \end{aligned}$$
(4)

where \(l\) is an integer and \(l<k\), such that the origin \(\varvec{x}= \varvec{0}\) in the state space of the system (3) with control input (4) is a uniformly asymptotic stable equilibrium point [14].

In this paper, the known information at an arbitrary instant \(t=r\) is only the measurement of historical states, i.e., the vector set \({\mathcal {X}}_{r}\) defined in Eq. (2). The structural and physical information of the nonlinear function \(\varvec{f}(\cdot )\) described in (3) is assumed unknown. The objective is to determine the suitable control input \(\varvec{u}(k)\) online to stabilize the system (3), with the measured system trajectory being the only utilizable information.

3 Data-driven quadratic stability criterion

3.1 Geometrical preliminaries

Necessary geometric concepts used in this paper are briefly summarized in this subsection. Most of the geometrical definitions are taken from [6].

The convex hull for a vector set \({\mathcal {C}}\), denoted as \(\mathbf {conv}\, {\mathcal {C}}\), is defined as the minimal convex set containing \({\mathcal {C}}\), i.e., \(\mathbf {conv}\, {\mathcal {C}} = \{\varSigma ^m_{i=1} \theta _i \varvec{x}_i | \varvec{x}_i \in {\mathcal {C}},\, \theta _i \ge 0,\, \varSigma ^m_{i=1}\, \theta _i = 1, i = 1,\,...\,,\,m \}\). A convex conic hull of a set \({\mathcal {C}}\) is the smallest convex cone for a vector set \({\mathcal {C}}\) defined as \(\mathbf {cone}\, {\mathcal {C}} = \{\varSigma ^m_{i=1} \theta _i \varvec{x}_i | \varvec{x}_i \in {\mathcal {C}},\, \theta _i \ge 0,\, i = 1,\,...\,m \}\). The set \(\mathbf {cone}\, {\mathcal {C}}^o \), which is defined as \(\mathbf {cone}\,{\mathcal {C}}^o= \{\varvec{y} | \varvec{x}^T \varvec{y} \le 0, \text{ for } \text{ all }\; \varvec{x} \in \mathbf {cone}\,{\mathcal {C}}\}\), is called the polar cone of \(\mathbf {cone}\,{\mathcal {C}}\).

A set \({\mathcal {C}}\) is said to be a convex polyhedron if it can be written as \(\mathbf {conv}\, {\mathcal {C}} = \{\varvec{x} | \varvec{ A }\varvec{x} \ge \varvec{b}\}\) for some matrix \(\varvec{A}\) and vector \(\varvec{b}\). A set \({\mathcal {C}}\) is a polyhedral cone if it can be represented by the above form of a polyhedron with \(\varvec{b} = \mathbf{0}\), denoted as \(\mathbf {cone}(\varvec{A})\). A typical example of a polyhedral cone is the \({\mathbb R}^n_+\).

A polyhedral cone has two representation methods: the H-representation utilizing the form of the set of inequalities with respect to a matrix \(\varvec{A}\), denoted as \(\mathbf {cone}(\varvec{A})\), and the V-representation utilizing the conic combination of the vectors within a set \({\mathcal {C}}\), denoted as \(\mathbf {cone}\,{\mathcal {C}}\). The Carathéodory theorem [7] shows that if the set \({\mathcal {C}}_{p}\) contains the extreme rays of the cone defined by \({\mathcal {C}}\), then the polyhedral cone \(\mathbf {cone}\, {\mathcal {C}}_{p}\) is identical the polyhedral cone \(\mathbf {cone}\, {\mathcal {C}}\).

3.2 The necessary and sufficient condition for one trajectory

Denote the vector obtained by left-multiplying an orthogonal matrix \(\varvec{\varPhi }\) to the vector \(\varvec{x}(k)\) of the concerned trajectory as \(\tilde{\varvec{x}}(k)\), \(\tilde{\varvec{x}}(k) = \varvec{\varPhi } \varvec{x}(k)\). Define a transformation for every vector \(\tilde{\varvec{x}}(k)\) as

$$\begin{aligned} \tilde{\varvec{v}}(k) = \tilde{\varvec{x}}(k+1) \circ \tilde{\varvec{x}}(k+1) - \tilde{ \varvec{x}}(k) \circ \tilde{\varvec{x}}(k, \end{aligned}$$
(5)

where \(\tilde{\varvec{v}}(k)\) represents the corresponding transformed vector, and \(\circ \) represents the calculation of Hadamard product [12] defined by

$$\begin{aligned} \varvec{a} \circ \varvec{b} = [a_j\,b_j],\; \quad j = 1,\,...\,,\,n. \end{aligned}$$
(6)

In (6), the symbols \(\varvec{a}\) and \(\varvec{b}\) represent two arbitrary \(n\)-dimensional vectors with \(a_j\) and \(b_j\) being their components, respectively. Unlike the inner products between two vectors, the calculation \(\circ \) establish a manipulation from two vectors to a new vector.

Applying the proposed vector manipulation to all the elements within the data set \({\mathcal X}_r\) defined in Eq. (2) for \(r=\infty \), a new vector set of \(\tilde{\varvec{v}}(k)\) can be obtained. Denote the complete vector set of \(\tilde{\varvec{v}}(k)\), \(k=1\,,\,...\,,\,\infty \), as \(\tilde{\mathcal {V}}_\infty \), and the convex conic hull (the smallest convex cone) determined by \(\tilde{\mathcal {V}}_\infty \) as \(\mathbf {cone}\, \tilde{\mathcal {V}}_\infty \). Correspondingly, the polar cone of \(\mathbf {cone}\, \tilde{\mathcal {V}}_\infty \), denoted as \(\mathbf {cone}\, {\tilde{\mathcal V}}_{\infty }^o\), can be represented as

$$\begin{aligned}&\mathbf {cone}\, {\tilde{\mathcal V}}_{\infty }^o\nonumber \\&\quad = \left\{ \varvec{y} | \tilde{\varvec{v}}^T(k) \varvec{y} \le 0, \tilde{\varvec{v}}(k) \in \mathbf {cone}\, {\tilde{\mathcal V}}_{\infty },\, \varvec{y}\in {\mathbb R}^n\right\} . \end{aligned}$$
(7)

Consider the nonlinear discrete-time system (1) with an equilibrium point at the origin of its state space. The necessary and sufficient condition of the existence of a QLF for the considered system trajectory \({\mathcal X}_r\), \(r=\infty \), can be given as the following theorem 1.

Theorem 1

The trajectory \({\mathcal X}_r\), \(r=\infty \), of the nonlinear discrete-time system (1) is quadratic stable, if and only if there exists an orthogonal matrix \(\varvec{\varPhi }\) such that the following two conditions are satisfied:

  1. 1.

    The convex cone \(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }\) is a proper cone.

  2. 2.

    The polar cone \(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o\) defined in (7) satisfies

    $$\begin{aligned} \mathbf {int}\,\left( \mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o \cap \mathbb {R}^n_+\right) \ne \varnothing \,, \end{aligned}$$
    (8)

where \(\mathbf {int}\,(\cdot )\) represents the interior of a set, and \(\varnothing \) represents an empty set.

Proof

To prove sufficiency, denote an arbitrary vector whose entries are real positive numbers as \(\varvec{d}\), \(\varvec{d} \in \mathbb {R}^n_+\), belongs to the set \(\mathbf {int}\,\left( \mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o \cap \mathbb {R}^n_+\right) \). If the convex cone \(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }\) is a proper cone, the polar cone \(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o\) is also proper and has non-empty interior, which indicates that \(\varvec{d}\) also belongs to \(\mathbf {int}\,(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o)\).

According to the definition of the polar cone \(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o\) in (7), it can be obtained that for all the transformed vectors in the data set \(\tilde{\mathcal V}_\infty \), the following condition holds

$$\begin{aligned} \langle \varvec{d},\, \tilde{\varvec{v}}(k)\rangle \le 0, \quad \;\varvec{d} \in {\mathbb R}^n_+, \quad \;k= 1,\,...\,,\,\infty , \end{aligned}$$
(9)

where \(\langle \cdot \rangle \) represents the inner product. Because \(\varvec{d} \in \mathbf {int}\,(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o)\), where \( \mathbf {int}\,(\mathbf {cone}\, \tilde{\mathcal {V}}_{\infty }^o)\) is a convex set, the above inequality can be specified as

$$\begin{aligned} \langle \varvec{d},\, \tilde{\varvec{v}}(k)\rangle < 0, \quad \;\varvec{d} \in {\mathbb R}^n_+, \quad \;k= 1,\,...\,,\,\infty , \end{aligned}$$
(10)

Using the definition of \(\tilde{\varvec{v}}(k)\) in (5), the inner product between \(\tilde{\varvec{v}}(k)\) and \(\varvec{d}\) in (10) can be represented as

$$\begin{aligned} \langle \tilde{ \varvec{v}}(k), \varvec{d} \rangle = \varvec{d}^T \, \left( \tilde{\varvec{x}}(k+1) \circ \tilde{ \varvec{x}}(k+1) - \tilde{ \varvec{x}}(k) \circ \tilde{ \varvec{x}}(k) \right) . \end{aligned}$$
(11)

Define a diagonal matrix \(\varvec{D}\) as \(\varvec{D} = \mathbf {diag}\,[\varvec{d}]\). Obviously \(\varvec{D}\) is positive definite because it is diagonal and its diagonal elements vector \(\varvec{d}\) belongs to \({\mathbb R}^n_+\). With notation that

$$\begin{aligned}&\tilde{\varvec{x}}(k+1) \circ \tilde{\varvec{x}}(k+1) = \mathbf {diag}\,[\tilde{\varvec{x}}(k+1)] \tilde{\varvec{x}}(k+1)\,,\nonumber \\&\varvec{d}^T \mathbf {diag}[\tilde{\varvec{x}}(k+1)] = \tilde{\varvec{x}}^T(k+1)\,\mathbf {diag}[\varvec{d}]\,, \end{aligned}$$
(12)

and the similar relations for \(\varvec{d}\) and \(\tilde{\varvec{x}}(k)\), one can obtain the following equation by substituting (11) and (12) into the inequality (10), as

$$\begin{aligned} \langle \tilde{\varvec{v}}(k), \varvec{d} \rangle&= \tilde{\varvec{x}}^T(k+1)\,\mathbf {diag}\,[\varvec{d}] \,\tilde{\varvec{x}}(k+1)\nonumber \\&- \tilde{\varvec{x}}^T(k) \, \mathbf {diag}\,[\varvec{d}] \, \tilde{\varvec{x}}(k) \nonumber \\&= \tilde{\varvec{x}}^T(k+1)\, \varvec{D} \,\tilde{\varvec{x}}(k+1)\nonumber \\&- \tilde{\varvec{x}}^T(k) \, \varvec{D} \, \tilde{\varvec{x}}(k) < 0. \end{aligned}$$
(13)

Because \(\tilde{\varvec{x}}(k)\) is defined as \(\tilde{\varvec{x}}(k) = \varvec{\varPhi } \varvec{x}(k)\), the Eq. (13) can be represented as

$$\begin{aligned} \langle \tilde{\varvec{v}}(k), \varvec{d} \rangle = {\varvec{x}}^T(k+1)\, \varvec{Q} \,{\varvec{x}}(k+1) - {\varvec{x}}^T(k) \, \varvec{Q} \, {\varvec{x}}(k) < 0, \end{aligned}$$
(14)

where the matrix \(\varvec{Q}\) is defined as \(\varvec{Q} = \varvec{\varPhi }^T \varvec{D} \varvec{\varPhi }\). Because \(\varvec{D}\) is a positive definite diagonal matrix and \(\varvec{\varPhi }\) is an orthogonal matrix, the matrix \(\varvec{Q}\) is also positive definite. Therefore, according to the definition of QLF it can be seen that the function \(V(\varvec{x}(k))=\varvec{x}^T(k)\, \varvec{Q} \,\varvec{x}(k)\) is a QLF for the concerned trajectory of the nonlinear discrete-time system (1), because \(\varvec{Q}\) is a positive definite matrix and \(\Delta V(\varvec{x}(k)) = \varvec{x}(k+1)^T\, \varvec{Q} \,\varvec{x}(k+1) - \varvec{x}(k)^T \, \varvec{Q} \, \varvec{x}(k) \le 0\). This proves the sufficiency of the proposed theorem.

To prove the necessity, suppose there exist a QLF for the concerned trajectory, denoted as \(V_d(\varvec{x}(k))= \varvec{x}(k)^T\,\hat{\varvec{Q}}\,\varvec{x}(k)\). The matrix \(\hat{\varvec{Q}}\) can be decomposed as

$$\begin{aligned} \hat{\varvec{Q}} = \hat{\varvec{\varPhi }}^T \mathbf {diag}\,[\hat{\varvec{d}}] \hat{\varvec{\varPhi }}, \end{aligned}$$
(15)

with \(\hat{\varvec{d}} \in {\mathbb R}^n_+\) being the eigenvalues of \(\hat{\varvec{Q}}\) and \(\hat{\varvec{\varPhi }}\) the eigenvector matrix. Clearly \( \mathbf {diag}\,[\hat{\varvec{d}}]\) is a positive diagonal matrix, and \(\hat{\varvec{\varPhi }}\) is an orthogonal matrix, because \(\hat{\varvec{Q}}\) is positive definite.

Because \(\varvec{x}(k+1)^T\,\hat{\varvec{Q}}\,\varvec{x}(k+1) = \hat{\varvec{d}}^T \mathbf {diag}\,[\hat{\varvec{\varPhi }} \varvec{x}(k+1)] \hat{\varvec{\varPhi }} \varvec{x}(k+1)\) and \(\varvec{x}(k)^T\,\hat{\varvec{D}}\,\varvec{x}(k)= \hat{\varvec{d}}^T \mathbf {diag}\,[\hat{\varvec{\varPhi }} \varvec{x}(k)] \hat{\varvec{\varPhi }}\varvec{x}(k)\), the difference of \(V_d(\varvec{x}(k))\) can be expressed as

$$\begin{aligned} \Delta V_d(\varvec{x}(k))&= \varvec{x}(k+1)^T\,\tilde{\varvec{D}}\,\varvec{x}(k+1) - \varvec{x}(k)^T\,\tilde{\varvec{D}}\,\varvec{x}(k) \nonumber \\&= \hat{\varvec{d}}^T \, (\mathbf {diag}\,[\hat{\varvec{\varPhi }}\varvec{x}(k+1)]\hat{\varvec{\varPhi }}\varvec{x}(k+1) \nonumber \\&- \mathbf {diag}\,[\hat{\varvec{\varPhi }} \varvec{x}(k)]\hat{\varvec{\varPhi }}\varvec{x}(k))\ \nonumber \\&= \hat{\varvec{d}}^T \varvec{v}(k) < 0, \end{aligned}$$
(16)

where

$$\begin{aligned} \varvec{v}(k)&= \mathbf {diag}\,[\hat{\varvec{\varPhi }}\varvec{x}(k+1)]\hat{\varvec{\varPhi }}\varvec{x}(k+1) \nonumber \\&- \mathbf {diag}\,[\hat{\varvec{\varPhi }} \varvec{x}(k)]\hat{\varvec{\varPhi }}\varvec{x}(k)\nonumber \\&= \tilde{\varvec{x}}(k+1) \circ \tilde{\varvec{x}}(k+1) - \tilde{\varvec{x}}(k) \circ \tilde{\varvec{x}}(k), \end{aligned}$$
(17)

with \(\tilde{\varvec{x}}(k) =\hat{\varvec{\varPhi }}\varvec{x}(k)\). It is shown in Eq. (16) that the inner product of the vector \(\varvec{v}(k)\) with a vector \(\hat{\varvec{d}}\), \(\hat{\varvec{d}} \in \mathbb {R}^n_+\), is always less than zero. Due to this fact, the interior of the polar cone defined in (7) and constructed with the orthogonal matrix \(\hat{\varvec{\varPhi }}\) is not empty, which indicates that the convex cone constructed by \(\varvec{v}(k)\), \(k = 1,\,...\,,\,\infty \) is also proper. Furthermore, because \(\hat{\varvec{d}} \in \mathbb {R}^n_+\), the polar cone has common elements with the real positive space \(\mathbb {R}^n_+\).

In [8], it is shown that the complete set of the matrix \(\varvec{P}\) in QLF can be mapped surjectively from the special orthogonal group \(\textit{SO}(n, {\mathbb R})\) and the conventional topology of \({\mathbb R}^{n}_{+}\). This mapping can be defined as

$$\begin{aligned} ({\varvec{\varPhi }}, {\varvec{d}}) \mapsto \varvec{P} : \varvec{P} = {\varvec{\varPhi }}^T \mathbf {diag}\,[{\varvec{d}}]{\varvec{\varPhi }}, \end{aligned}$$
(18)

where \({\varvec{\varPhi }}\) is an orthogonal matrix in \(SO(n, {\mathbb R})\), and \({\varvec{d}}\) is a real vector in \(\mathbb {R}^n_+\). Because the mapping (18) is surjective, which is proven in [8], it can be concluded that no QLF exists if no element over the complete set \(SO(n, {\mathbb R}) \times {\mathbb R}^{n}_{+}\) can be found to construct a QLF, and vice versa, which is also consistent with the sufficiency and necessity of theorem 1.

Therefore, theorem 1 shows that the existence of a QLF can be determined by searching through the special orthogonal group \(\textit{SO}(n, {\mathbb R})\) and the conventional topology of \({\mathbb R}^{n}_{+}\). It does not require explicitly an analytical form of the nonlinear function \(\varvec{f}(\cdot )\) in system (1), but the complete time history of system state vectors. This fact makes it possible to apply the above theorem in the data-driven context to judge quadratic stability.

3.3 Stability condition for finite time measurements

At the time instant \(t = r\), every system vector \(\varvec{x}(k) \in {\mathbb R}^n\), \(k=1,\,...\,,r-1\), can be transformed with one certain orthogonal matrix \(\mathbf \Phi \) into a new vector \(\tilde{\varvec{v}}(k)\) with use of the mapping defined in Eq. (5). Correspondingly, these transformed vectors determine a new vector set, denoted as \(\tilde{\mathcal V}_{r-1} = \left\{ \tilde{\varvec{v}}(k) \right\} , k = 1,\,...\,,r-1\).

It should be noted that the vector set \(\tilde{\mathcal V}_{r-1}\) is different from the set \({\tilde{\mathcal V}}_\infty \) in theorem 1. The reason is that the set \({\tilde{\mathcal V}}\) is obtained by transforming all the states vectors \(x(k)\) within the measured system states, but due to the finiteness of the measured data, the vector set \(\mathcal {X}_r\) is only a subset of \(\varOmega \) where the nonlinear mapping \(\varvec{f}(\cdot )\) is defined. Thus \(\tilde{\mathcal V}_{r-1} = {\tilde{\mathcal V}}\) is true only if \(r \rightarrow \infty \).

The polar cone \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o\) can be determined by substituting \({\tilde{\mathcal V}}_{\infty }\) with \({\tilde{\mathcal V}}_{r-1}\) into Eq. (7), as

$$\begin{aligned} \mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o = \left\{ \varvec{y} | \tilde{\varvec{v}}^T \varvec{y} \le 0, \tilde{\varvec{v}} \in \mathbf {cone}\, {\tilde{\mathcal V}}_{r-1},\, \varvec{y}\in {\mathbb R}^n\right\} . \end{aligned}$$
(19)

By examining (19), it can be seen that \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o\) is a convex cone by adding an inequality constraint \(-\tilde{\varvec{v}}(r-1)^T\,\varvec{d}>0\) to \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-2}^o\) , which implies that

$$\begin{aligned} \mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o = \bigcap ^{r}_{l=1}\mathbf {cone}\, {\tilde{\mathcal V}}_{l-1}^o \,. \end{aligned}$$
(20)

Substituting Eq. (20) into the stability condition (8) in theorem (1), it can be obtained that

$$\begin{aligned} \mathbf {int}\,\left( \mathbf {cone}\tilde{\mathcal {V}}_{\infty }^o \cap \mathbb {R}^n_+\right)&= \mathbf {int}\left( \left( \bigcap ^{\infty }_{l=1}\mathbf {cone}\, {\tilde{\mathcal V}}_{l-1}^o\!\right) \cap \mathbb {R}^n_+\!\right) \nonumber \\&= \bigcap ^{\infty }_{l=1}\left( \mathbf {int}(\mathbf {cone} {\tilde{\mathcal V}}_{l-1}^o \cap \mathbb {R}^n_+ )\right) \nonumber \\&\ne \varnothing \,. \end{aligned}$$
(21)

It is shown in Eq. (21) that if \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o \cap {\mathbb R}^n_+ \ne \varnothing \) at \(t=r\), the intersections between \({\mathbb R}^n_+\) and any of the cones \(\mathbf {cone}\, {\tilde{\mathcal V}}_l^o\) formulated at former time instants \(1\le l<r-1\), are inherently nonempty. This fact implies that if the condition

$$\begin{aligned} \mathbf {int}\,\left( \mathbf {cone}\, {\tilde{\mathcal V}}_{l-1}^o \cap \mathbb {R}^n_+\right) \ne \varnothing \, \end{aligned}$$
(22)

holds at every time instant, which guarantees that \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o \ne \varnothing \), the concerned trajectory is quadratic stable, and vice versa.

Therefore, the quadratic stability of the concerned trajectory can be judged by checking the condition (22), the constructed convex cone is proper at every time instant. Hence, theorem 1 can be reformulated for practical implementation as

Theorem 2

The concerned trajectory of nonlinear discrete-time system (1) is quadratic stable if and only if there exists an orthogonal matrix \(\varvec{\varPhi }\) such that at every time instant \(t=r\), \(r =1,\,...\,,\, \infty \), the following two conditions can be satisfied:

  1. 1.

    The convex cone \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}\) constructed by the data set \({\mathcal X}_r\) and the matrix \(\varvec{\varPhi }\) is proper,

  2. 2.

    The polar cone \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o\) of \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}\) follows the relationship

    $$\begin{aligned} \mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o \cap {\mathbb R}^n_+ \ne \varnothing . \end{aligned}$$
    (23)

Theorem 2 states that if the intersection between the set \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o\) and \({\mathbb R}^n_+\) is not empty at every time instant, the system is quadratic stable. Furthermore, letting \(\tilde{\varvec{d}}\) be any vector located within \(\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o \cap {\mathbb R}^n_+\), \(r=\infty \), the QLF for this system can be expressed as

$$\begin{aligned} V(\varvec{x})=\varvec{x^T} \varvec{\varPhi }\, \mathbf {diag}\,[\tilde{\varvec{d}}]\, \varvec{\varPhi }^T \varvec{x}\,. \end{aligned}$$
(24)

The condition (23) must be satisfied at every time instant for a quadratic stable system. Thus, to give a correct stability judgment to the considered trajectory, this criterion has to be checked at every time instant. If at any time instant the constructed cone is not proper or condition (22) cannot be satisfied, then the concerned trajectory can be judged as not quadratic stable.

As for numerical implementation, the stability condition (23) can be judged by solving a max-min problem [33]. If the optimized value of the max-min problem is greater than zero, the interior of the intersection is not empty and shows that there exists at least one QLF for the concerned trajectory of the system (1) at the time instant \(t=r\). If the optimized value of the max-min problem is positive at every time instant, then the concerned trajectory is globally quadratic stable; if at any time instant the optimized value is not positive, then the concerned trajectory is not quadratic stable.

It should be remarked that the proposed algorithmic stability judgment is only necessary and sufficient with respect to the considered trajectory, but not to the whole system. Nevertheless, the proposed algorithm can still be used in control problem, if the control design can be realized online. In this case only the current running trajectory needs to be considered.

For the purpose of control, if a controller can always drive the closed-loop system to satisfy the stability condition under arbitrary initial conditions, all the trajectories of the controlled system starting from this neighborhood converges to the equilibrium. This means all the perturbed motions starting from this neighborhood shall converge to the considered equilibrium. In this case the controlled system shall possess a (both sufficiently and necessarily) stable equilibrium and thereby be stabilized.

In the next section, a preliminary design of control using this stability criterion is discussed in the problem of stabilizing unknown nonlinear systems.

4 Application in stabilizing unknown systems

The control input \(\varvec{u}(k)\) is considered as a function of \(x(k)\), as shown in Eq. (4). Therefore, the system (3) under control of \(\varvec{u}(k)\) can be expressed as the same form of (3). Correspondingly, the stability of controlled system can be discussed by using the proposed data-driven stability criterion.

The stabilization problem defined in Sect. 2 can be reformulated as to find a control input \(\varvec{u}(k)\) such that the complete considered trajectory of closed-loop system

$$\begin{aligned} \mathcal {X}_{\infty }=\{{\varvec{x}}(k), \;\;k = 1,\,...\,,\,\infty \}, \end{aligned}$$
(25)

satisfies the condition (8) in theorem 1.

Specific to an arbitrary time instant \(t=r\), it means that to find the value of \(\varvec{u}(r)\) such that the trajectory \(\mathcal {X}_{r+1}\) satisfies the condition (23) in theorem 2, with \(\mathcal {X}_{r+1}\) being defined as

$$\begin{aligned} \mathcal {X}_{r+1} = \mathcal {X}_{r}\cup \{\varvec{x}(r+1)\} =\left\{ \varvec{x}(1),\,...\,, \varvec{x}(r), \varvec{x}(r+1)\right\} , \end{aligned}$$
(26)

where \(\varvec{x}(k+1)= \varvec{f}(\varvec{x}(k),\varvec{u}(k))\), and \(\mathcal {X}_{r}\) is defined in 2.

Due to the fact the nonlinear function \(\varvec{f}(\cdot )\) in system dynamics is unknown, the states \(\varvec{x}(r+1)\) in (26) cannot be calculated directly with use of system equation at an arbitrary time instant \(r\), even if the value of control input \(u(r)\) is known. Consequentially, the vector set \(\mathcal {X}_{r+1}\) cannot be established and the stability condition (23) cannot be judged. This problem can be solved by using the online system identification techniques using RNN.

4.1 Solving the stabilization problem using RNN

The RNN system identification is a well-developed online system identification method [11, 27]. Denote the identified system dynamics by RNN as

$$\begin{aligned} \hat{\varvec{x}}(k+1) = \hat{\varvec{f}}(\varvec{x}(k), \varvec{u}(k)), \end{aligned}$$
(27)

where \(\hat{\varvec{f}}(\cdot )\) represents the identified dynamics of \(\hat{\varvec{f}}(\cdot )\) in  (3), and \(\hat{\varvec{x}}(k+1)\) represents the one-step prediction of the future states of the plant at the time instant \(t=k\). With suitable training algorithms, the RNN can make a one-step prediction of system states online, and the error of one-step prediction by RNN is incrementally precise with time. This means that

$$\begin{aligned} \lim _{k \rightarrow \infty } \parallel \varvec{e}_m(k+1) \parallel \rightarrow \varvec{0}, \end{aligned}$$
(28)

where

$$\begin{aligned} \varvec{e}_m(k+1) = \hat{\varvec{x}}(k+1) - \varvec{x}(k+1), \end{aligned}$$
(29)

with \(\hat{\varvec{x}}(k+1)\) being the one-step prediction at time instant \(t=k\) and \(\varvec{x}(k+1)\) the true system states at \(t=k+1\).

The Eq. (28) indicates that if the complete time series of the predicted states, denoted as

$$\begin{aligned} \hat{\mathcal {X}}_{\infty }=\{\hat{\varvec{x}}(k), \;\;k = 1,\,...\,,\,\infty \}, \end{aligned}$$
(30)

converges to origin, the trajectory of true system states \(\mathcal {X}_{\infty }\) defined in (25) also converges to the origin, because the error between these two time sequences goes to zero with time.

Correspondingly, specific to every arbitrary time instant \(t=r\), if the vector set \(\hat{\mathcal {X}}_{r+1}\), defined as

$$\begin{aligned} \hat{\mathcal {X}}_{r+1} = \hat{\mathcal {X}}_{r}\cup \{\hat{\varvec{x}}(r+1)\} =\left\{ \hat{\varvec{x}}(1),\,...\,, \hat{\varvec{x}}(r), \hat{\varvec{x}}(r+1)\right\} , \end{aligned}$$
(31)

satisfies the condition (23) in theorem 2, the vector set \(\mathcal {X}_{r+1}\) defined in (26) is also stable in the sense of theorem 2.

This fact shows that if the trajectory of predicted system states is stable, the trajectory of true system states is also stable. Therefore, the task of finding \(\varvec{u}(r)\) at \(t=r\) such that the vector set \(\mathcal {X}_{r+1}\) satisfies (23) is identical to find the \(\varvec{u}(r)\) that let vector set \(\hat{\mathcal {X}}_{r+1}\) satisfy condition (23).

The function \(\hat{\varvec{f}}(\cdot )\) in the identified dynamics (27) can be obtained by training the RNN at an arbitrary time instant \(t = r\). As a result, the prediction \(\hat{\varvec{x}}(r+1)\) can be obtained if the control input \(\varvec{u}(r)\) is known. If a suitable control input \(\varvec{u}(r)\) can be found at every time instant \(t=r\), for \(r=1,\,...\,,\,\infty \), such that the data set \(\hat{\mathcal {X}}_{r+1}\), including the predicted states \(\hat{\varvec{x}}(r+1)\), satisfies the geometrical condition (23), the data set \(\hat{\mathcal {X}}_{r+1}\) is stable in the sense of theorem 2 and this input \(\varvec{u}(r)\) can also stabilize the system to be controlled.

4.2 Candidates of feasible control inputs

Denote the vector set obtained by applying the transformation defined in (5) to \(\hat{\varvec{x}}(k)\), \(k = 1,\,...\,,\,r\), as \(\hat{\mathcal {V}}_{r-1}\). The elements of \(\hat{\mathcal {V}}_{r-1}\) are denoted as \(\hat{\varvec{v}}(k)\), \(k = 1,\,...\,,\,r-1\),

Consider the problem of determining the control input \(u(r)\) at an arbitrary time instant \(t=r\) when the historical predicted states \(\hat{\varvec{x}}(k)\), \(k = 1,\,...\,,\,r\), satisfy the proposed stability condition (23). This means that the following fact

$$\begin{aligned} \mathbf int \left( \mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o \cap {\mathbb R}^n_+\right) \ne \varnothing , \end{aligned}$$
(32)

is true, where the superscript \((\cdot )^o\) denotes the polar cone, and the symbol \(\mathbf {cone}\, \mathcal {\hat{V}}_{r-1}\) represents the convex conic hull of \(\mathcal {\hat{V}}_{r-1}\).

The prediction \(\hat{\varvec{x}}(k+1)\) can be obtained by substituting \(\varvec{u}(r)\) into the identified system (27). A new transformed vector \(\hat{\varvec{v}}(r)\) can be obtained by applying the transformation (5) of \(\hat{\varvec{x}}(k)\) and \(\hat{\varvec{x}}(k+1)\). Defining the set \(\hat{\mathcal {V}}_{r}\) as

$$\begin{aligned} \hat{\mathcal {V}}_{r}=\{\hat{\varvec{v}}(r)\} \cup \hat{\mathcal {V}}_{r-1},\end{aligned}$$
(33)

the remaining problem is how to determine the value of \(\varvec{u}(r)\), so that the condition

$$\begin{aligned} \mathbf {cone}\, \mathcal {\hat{V}}_{r}^o \cap {\mathbb R}^n_+ \ne \varnothing , \end{aligned}$$
(34)

can be fulfilled.

Assume the condition (34) is true. It can be obtained from (33) that

$$\begin{aligned} \mathbf {cone}\, \mathcal {\hat{V}}_{r} = \mathbf {cone}\, \mathcal {\hat{V}}_{r-1}\cup \mathbf {cone}\, \{\hat{v}(r)\}, \end{aligned}$$
(35)

and correspondingly

$$\begin{aligned} \mathbf {cone}\, \mathcal {\hat{V}}_{r}^o = \mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o\cap \mathbf {cone}\, \{\hat{v}(r)\}^o, \end{aligned}$$
(36)

where \(\{\hat{\varvec{v}}(r)\}\) represents the vector set containing only the vector \(\hat{\varvec{v}}(r)\).

Substituting the equation above into (34), it can be obtained that

$$\begin{aligned} \mathbf {int}\,\left( \mathbf {cone}\, \{\hat{v}(r)\}^o\cap \mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o\cap {\mathbb R}^n_+ \right) \ne \varnothing . \end{aligned}$$
(37)

The Eq. (32) shows that the polyhedral cone \(\mathcal {\hat{V}}_{r-1}^o\cap {\mathbb R}^n_+ \) is not empty. According to the computational geometrical theory, the Eq. (37) is true if the vector \(\hat{\varvec{v}}(r)\) is not located in the dual cone of \(\mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o\cap {\mathbb R}^n_+\), which contains all the vectors having positive inner products with elements of \(\mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o\cap {\mathbb R}^n_+\).

In other words, the condition (34) is true if the vector \(\hat{\varvec{v}}(r)\) satisfies

$$\begin{aligned} \hat{\varvec{v}}(r) \in \varLambda , \end{aligned}$$
(38)

where \(\varLambda \) is the complementary set of the dual cone of \(\mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o\cap {\mathbb R}^n_+\), denoted as

$$\begin{aligned} \varLambda = \mathbb {R}^n \backslash (\mathbf {cone}\, \mathcal {\hat{V}}_{r-1}^o \cap \mathbb {R}^n_+)^*. \end{aligned}$$
(39)

If the condition (34) is true, the trajectory of predicted system states \(\hat{\varvec{x}}(k)\), \(k = 1,\,...\,,\,r,\,r+1\) is stable in the sense of theorem 2. As a result, the feasible values of control input \(\varvec{u}(r)\) which can guarantee stability at \(t=r\) are the ones which can make the vector \(\hat{\varvec{v}}(r)\) moved into the set \(\varLambda \) defined in (39).

4.3 Choosing suitable control input from candidates

The discussion about feasible control inputs shows that the control input \(\varvec{u}(k)\) should be chosen from the ones fulfilling (38). In this paper, the problem of seeking \(\varvec{u}(k)\) at one certain time instant is solved by solving an optimization problem, as

$$\begin{aligned} \begin{array}{rl} \underset{\varvec{K}}{\min }. &{} J \\ s.t. &{} \varvec{u}(k) = -\varvec{K}(k) \varvec{x}(k), \\ &{} \tilde{\varvec{v}}(r) \in \mathbb {R}^n \backslash (\mathbf {cone}\, {\tilde{\mathcal V}}_{r-1}^o \cap \mathbb {R}^n_+)^*, \end{array} \end{aligned}$$
(40)

where \(J\) is a suitably chosen performance measure that can be evaluated online with respect to the system states. For example, this measure can be chosen as

$$\begin{aligned} J = \int ^{t_1}_{t_0} \varvec{x}^T \varvec{E} \varvec{x} + \varvec{u}^T \varvec{F} \varvec{u}\, \text{ d }t, \end{aligned}$$
(41)

where \(t_0\) and \(t_1\) are the starting time and the final time of the evaluation, \(\varvec{E}\) and \(\varvec{F}\) are positive definite matrices, which has been used in the classic linear optimal control as one kind of representation of the input and state energy.

figure a

It should be noted here that to minimize the performance measure \(J\) is not the primary goal at the current stage of this research. Instead, it serves as the selecting measure to obtain one certain control input which can be used to achieve the primary goal of stabilization.

One possible algorithm to find the suboptimal solution to the optimization problem (40) is shown in Algorithm 1. In this example algorithm, a finite set of the feedback gain matrix \(\varvec{K}\) is given and denoted as \(\varXi \). The example algorithm calculates the value \(J\) of the cost function defined in the optimization problem (40) for every \(\varvec{K}(k)\) in \(\varXi \), determines the one which possesses the smallest \(J\), and additionally, make the sequence of the predicted states satisfy the stability condition. So a suitable feedback gain \(\varvec{K}\) can be defined to stabilize the plant at the time instant \(t=r\). Although this \(\varvec{K}(k)\) is only optimal for the present time instant, it can be applied to construct the suitable control input \(\varvec{u}(k)\) and applied to the plant to fulfill the goal of control.

5 Numerical example of stabilizing a nonlinear aeroelastic system

In this paper, a nonlinear aeroelastic system [23] is used as a benchmark example to show the effectiveness and flexibility of the proposed control approach. The control of this benchmark system has been widely studied with different control approaches [24, 26, 29–32].

5.1 Configuration of the aeroelastic system

The configuration of the considered 2-D nonlinear aeroelastic wing section is shown in Fig. 1. The two degrees of freedom, the pitching movement and the plunging one, are respectively restrained by a pair of springs attached to the elastic axis (EA) of the airfoil. A single trailing-edge control surface is used to control the air flow, thereby providing more maneuverability to suppress instability. This model is accurate for airfoils at low velocity and has been confirmed by both computational tests [4] and wind tunnel experiments [22].

Fig. 1
figure 1

2-D Wing-flap aeroelastic model

Full size image

According to [23], the equations of motion governing the aerolastic system are given as

$$\begin{aligned} \begin{array}{l} \left[ \begin{array}{cc} m_T &{} m_Wx_{\alpha }b \\ m_Wx_{\alpha }b &{} I_{\alpha } \end{array} \right] \left[ \begin{array}{c} \ddot{h} \\ \ddot{\alpha } \end{array} \right] +\left[ \begin{array}{cc} c_h &{} 0 \\ 0 &{} c_{\alpha } \end{array} \right] \left[ \begin{array}{c} \dot{h} \\ {\dot{\alpha }} \end{array} \right] \\ + \left[ \begin{array}{cc} k_h &{} 0 \\ 0 &{} k_{\alpha } \end{array} \right] \left[ \begin{array}{c} h \\ \alpha \end{array} \right] =\left[ \begin{array}{c} -L \\ M \end{array} \right] , \end{array} \end{aligned}$$
(42)

where plunging and pitching displacement are denoted as \(h\) and \(\alpha \), respectively. In Eq. (42), \(m_W\) denotes the mass of the wing, \(m_T\) represents the total mass of the wing and its support structure, \(b\) the semi-chord of the wing, \(I_{\alpha }\) the moment of inertia, \(x_{\alpha }\) the non-dimensional distance from the center of mass to the elastic axis, \(c_{\alpha }\) and \(c_h\) the pitch and plunge damping coefficients, respectively, \(k_{\alpha }\) and \(k_h\) the pitch and plunge spring constants, respectively, and \(M\) and \(L\) denote the quasi-steady aerodynamic lift and moment. In the case when the quasi-steady aerodynamics is considered, \(M\) and \(L\) should be written as

$$\begin{aligned} \left\{ \begin{array}{l} L = \rho U^2bc_{l_{\alpha }}\left[ \alpha +\frac{\dot{h}}{U}+(\frac{1}{2}-a)b\frac{\dot{\alpha }}{U}\right] +\rho U^2 bc_{l_{\beta }}\beta \; \\ M = \rho U^2b^2c_{m_{\alpha }}\left[ \alpha \!+\!\frac{\dot{h}}{U}\!+\!(\frac{1}{2}-a)b\frac{\dot{\alpha }}{U}\right] +\rho U^2 b^2 c_{m_{\beta }}\beta \end{array}\right. \!, \end{aligned}$$
(43)

where \(\beta \) represents the flap angle, and the other denotations of aerodynamic coefficients are explained in Table 1.

Table 1 Denotation list of aerodynamic coefficients
Full size table

The structural nonlinearity is supposed to exist in the pitching spring constant \(k_{\alpha }\), which is assumed as to be a polynomial of \(\alpha \), shown as

$$\begin{aligned} k_{\alpha }=\sum ^4_{i=0}k_{{\alpha }_i}{\alpha }^{i}. \end{aligned}$$
(44)

The control objective is to drive the flap angle \(\beta \) properly so that the instability caused by structural nonlinearities can be suppressed in the vicinity of the nominal system flutter speed with smaller control errors and less input energy.

5.2 Problem settings and specific tasks of control

As pointed in [18], a stable pitch motion of the 2-D wing section leads at the same time to a stable plunging motion, and vice versa. If a control law can stabilize the pitch motion, the plunging motion as well as the whole system is also stable.

Therefore, the control law of stabilizing the aeroelastic wing section can be designed with only consideration of the pitch motion: to stabilize a subsystem with pitching angel \(\alpha \) and pitching velocity \(\dot{\alpha }\) being its states and the flapping angle \(\beta \) as its control input. If this subsystem can be stabilized, then the stability of whole wing section can be guaranteed. This simplification can reduces the problem from four to two dimensions, while the goal of stabilization can be sustained. As mentioned in the introduction, the states \(\alpha \) and \(\dot{\alpha }\) are assumed as fully measurable and free of noise.

The task of control is to stabilize the system following the above assumptions with two different nonlinearities of the pitching spring stiffness according to  [17] and [1], respectively, as

$$\begin{aligned} k_{\alpha 1} = \left[ \begin{array}{ccccc} 6.8&10.0&667.7&26.6&-5087.9 \end{array} \right] [\alpha ^i], \end{aligned}$$
(45)

and

$$\begin{aligned} k_{\alpha 2} = \left[ \begin{array}{ccccc} 2.8&-62.3&3709.7&-24195.6&48756.9 \end{array} \right] [\alpha ^i]. \end{aligned}$$
(46)

The adaptive feedback linearization control in [17] is also applied in the simulation to be compared with the proposed control method. It should be mentioned that the parameter settings of the both control methods, including the feedback gains in adaptive feedback linearization method, are kept as the same in the both simulations when the nonlinearity \(k_{\alpha 1}\) is changed into \(k_{\alpha 2}\), in order to compare the adaptive abilities of the both control approaches.

5.3 Simulation results

The open-loop responses of the concerned aeroelastic system with different nonlinearities \(k_{\alpha 1}\) and \(k_{\alpha 2}\) are shown in Fig. 2a, b. It can be seen from these two figures that the structural nonlinearity in the pitching stiffness leads the system response to the limit cycle oscillation (LCO).

Fig. 2
figure 2

Open-loop system phase portraits with different nonlinearities

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The system functionality measure in this example uses the form of the objective function in linear optimal control introduced in Eq. (41). The optimization problem (40) here is solved by heuristic search (Algorithm 1). The searching space is defined as the varying region of the flapping motion \(\beta \), which is detailed as \((-45\hbox {deg},\;45\hbox {deg})\).

Simulations of the close-loop system are performed with wind speed \(U\) \(=\) 20 m/s and structural parameter \(a=0.8\) (nondimensional distances from midchord to the elastic axis). The initial conditions for the state variables of the system are selected as \(\alpha (0) = 5.75\,\hbox {(deg)}, h(0)=0.01\hbox {m}\), \(\dot{\alpha }(0)=0\,\hbox {(deg/s)}\), and \(\dot{h}\)(0)=0 m/s. The sampling time is set as \(1\times 10^{-4}\) s.

The simulation results of the closed-loop system with the nonlinearity \(k_{\alpha 1}\) and \(k_{\alpha 2}\) are given in Figs. 3 and 4, respectively. The red-dashed curves in Figs. 3 and 4 show the simulation results with the proposed control method, from which it can be seen that in both cases system trajectories converge to the origin of the state space with increasing time. From these simulation results it can be seen that the proposed control method can stabilize the unknown nonlinear aeroelastic system with different nonlinearities. This fact shows that the proposed control strategy cannot only be used to stabilize the unknown nonlinear system, but also adapt the variation of system dynamics and realize the goal of stabilizing unknown plants.

Fig. 3
figure 3

Closed-loop response with nonlinearity \(k_{\alpha 1}\)

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Fig. 4
figure 4

Closed-loop response with nonlinearity \(k_{\alpha 2}\)

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The simulation results of adaptive feedback linearization control with respect to the two different nonlinearities are shown as the blue solid curves in Fig. 3 and green curves in Fig. 4, respectively. It can be seen that in the case of \(k_{\alpha 1}\) the adaptive feedback linearization has better control performance with respect to the error and settling time than the proposed method. However, it cannot stabilize the system to the desired position (origin in the state space) in the second simulation with \(k_{\alpha 2}\), in which case the sytem dynamics are assumed unknown to feedback linearization control, and its parameters are kept as the same as those in the first simulation.

From this comparison, it is shown that the proposed method possesses higher adaptability than the adaptive feedback linearization method when the system dynamics is unknown.

6 Conclusion

This paper firstly proposes the necessary and sufficient criterion for the existence of a QLF only for a trajectory of nonlinear discrete-time systems, without necessity of knowing the mathematical description of the system. The proposed criterion firstly transforms the measured system states in the concerned trajectory into a new data set with use of Hadamard product and orthogonal transformation, and shows that the existence of a QLF for the considered trajectory can be determined by examining the existence of a suitable orthogonal matrix, with which a convex cone can be constructed and have a non-empty intersection with positive real space.

Based on the proposed stability criterion, a new control method is proposed in combination of online black-box system identification using RNN. The recurrent neural network is utilized to identify the unknown plant dynamics. The proposed controller can utilize the identified model and proposed data-driven stability condition to find a suitable control input to realize stabilization, without knowing any precise physical or structural information of the plant.

The proposed control method is applied to stabilize a nonlinear aeroelastic system using only the information of the measured system states. Simulation results show that the proposed control can adaptively fulfill the task of stabilizing unknown systems without changing any parameters.

It should be noticed that the discussions in this paper are based on two strong assumptions: the system states are fully measurable and the measurements are without noise, which cannot be fulfilled in real life applications and will be considered in the further research of this topic.