claimClaim remarkRemark hypothesisHypothesis

    Noninvasive Adaptive Control of a Class of Nonlinear Systems Rezaee and Renson

    # Noninvasive Adaptive Control of a Class of Nonlinear Systems With Unknown Parameters††thanks: Submitted to the editors DATE.
    This work was supported by the Engineering and Physical Sciences Research Council (EP/W032236/1).

    Hamed Rezaee
    Department of Mechanical Engineering, Imperial College London, London, UK ().
<EMAIL_ADDRESS>

    Ludovic Renson
    Department of Mechanical Engineering, Imperial College London, London, UK ().
<EMAIL_ADDRESS>

    ###### Abstract

    Control-based continuation (CBC) is a general and systematic method to explore the dynamic response of a physical system and perform bifurcation analysis directly during experimental tests. Although CBC has been successfully demonstrated on a wide range of systems, rigorous and general approaches to designing a noninvasive controller underpinning the methodology are still lacking. In this paper, a noninvasive adaptive control strategy for a wide class of nonlinear systems with unknown parameters is proposed. We prove that the proposed adaptive control methodology is such that the states of the dynamical system track a reference signal in a noninvasive manner if and only if the reference is an open-loop response of the system. Compared to the existing literature, the proposed method does not require any a priori knowledge of some system parameters, does not require a persistent excitation, and is not restricted to linearly-stable systems, facilitating the application of CBC to a much larger class of systems than before. Rigorous mathematical analyses are provided, and the proposed control method is numerically demonstrated on a range of single- and multi-degree-of-freedom nonlinear systems, including a Duffing oscillator with multiple static equilibria. It is shown that both the stable and unstable periodic orbits of the open-loop system can be stabilized and reached, noninvasively, in closed-loop conditions.

    ###### keywords:

    Adaptive control, bi-stability, noninvasive control, nonlinear control, nonlinear dynamics, unstable periodic orbits.

    93C10, 93D21, 34C25, 37G15

    ##
    1 Introduction

    Noninvasive control refers to a class of control strategies for which the control action must asymptotically vanish or become small enough as a desired system behavior or a desired state is approached. Despite vanishing, the control provides a stabilizing effect such that a desired response — for instance, an unstable periodic orbit of the underlying uncontrolled system — can be observed. Noninvasive control is typically employed when it is desirable to preserve (most of) the natural dynamics of the underlying uncontrolled system. For instance, noninvasive control has extensively been used for chaos control where unstable periodic orbits embedded in a chaotic attractor are stabilized [48, 11]. In this context, numerous noninvasive control approaches have been developed such as the so-called OGY method [33, 39], time delay feedback (TDF) [37, 17] and active filters [38, 30]. Chaos control has found applications in electronic circuits [38, 30, 45], chemical oscillators [3, 35, 36], lasers and nonlinear optics [7, 46, 13, 49], secure communication [22, 21] and fluid, mechanical, biological, and biochemical systems [55, 54, 19].

    More recently, noninvasive control has been exploited in the context of control-based continuation (CBC). Introduced in [53], CBC is a method to perform the bifurcation analysis of a physical system directly during experiments. The particularity of the noninvasive control approaches used within CBC is that they rely on the presence of a control reference signal, which contrasts with other noninvasive control methods used in chaos control like the OGY and TDF methods. The significant advantage of using a reference signal is that the reference signal acts as a proxy for the system states. It is therefore straightforward to steer the system dynamics towards regions of interest in the state space. With CBC, the reference signal is _a priori_ unknown and should be iteratively found using Newton-like methods such that by tracking the reference signal, the control input vanishes and becomes noninvasive. This defines the zero-problem whose solutions can be tracked in parameter space using suitable path-following techniques [47]. An apparent challenge of using a reference signal is that a sufficiently accurate approximation of the system dynamics is required to make the controller noninvasive [10]. Provided the controller is properly designed, CBC is insensitive to a change in initial conditions and stability due to bifurcations, which provides a much more systematic and repeatable approach to experimental testing than traditional open-loop approaches. As such, CBC has been used on a wide range of mechanical structures to measure important dynamics characteristics such as frequency response curves [5, 6, 12, 47, 24], backbone curves [41, 1] and fold bifurcation curves [43, 40]. More recently, CBC was used to characterize experimentally the equilibria of the Zeeman catastrophe machine [15], the dynamics of bubbles [18], the buckling of beams [32, 51, 52], the limit cycle oscillations of an airfoil [25, 9], and demonstrated numerically on equilibria and periodic orbits in biological systems [14, 10]. Identified features can then be exploited for model development, calibration, and validation [25, 8, 56].

    Although CBC has been exploited in numerous studies, the majority of the existing literature primarily emphasizes its application rather than the systematic design and theoretical analysis of its underlying noninvasive control method. As such, new applications often require a significant number of trials and errors to obtain a controller that works throughout the parameter range of interest. In this paper, we contribute to filling this void by proposing an adaptive strategy for noninvasive control of a wide class of high-order nonlinear systems with multiple degrees of freedom. The proposed control strategy is designed such that a desired reference can be tracked in a noninvasive manner if and only if it is an open-loop response of the system.

    Recent work in this direction was presented in [34] where the design of noninvasive control using various techniques such as zero-in-equilibrium feedback control and washout filters was addressed. However, due to model nonlinearities and parameter uncertainties, achieving a desired tracking performance based on linear approaches may not be guaranteed. Accounting for model uncertainty, a noninvasive control method based on model-reference adaptive control and pole-placement adaptive control was proposed for linear discrete-time systems with periodic behavior in [27]. The approach was then extended to linear continuous-time systems with periodic behavior in [28] and to a class of nonlinear systems with periodic behavior in [29]. In [28] and [29], the linear terms of the system were assumed to be Hurwitz stable and known. This helps design a model reference for a proper adaptive controller and guarantees stability. However, this assumption prevents the use of the controller in applications exhibiting multiple static equilibria (such as buckled beams and plates) and systems exhibiting self-excited oscillations (as in machine tool vibration, aeroelastic flutter, etc.). Another assumption made in [28] and [29] is that the desired response is persistently exciting, thereby guaranteeing the estimation of the true unknown parameters and the asymptotic tracking of the desired response along with the noninvasiveness of the control action. As shown in Section 5, satisfying this assumption is not always possible or may require a (very) fine discretization of the dynamics (using, for instance, a large number of Fourier modes), which increases the complexity and cost of solving the CBC root problem.

    The control strategy proposed in the present paper lifts those assumptions such that no knowledge of any model parameter is needed, no stability assumptions are made on the linear terms of the system, and the performance of the proposed control strategy does not rely on the persistent excitation of the desired response. This makes our control method applicable to a much larger range of systems than before.

    The structure of this paper is as follows. Notation is provided in the next section. Section 3 defines the noninvasive control problem in more precise terms, and the new noninvasive adaptive control method is presented
    in Section 4 along with rigorous proofs of its convergence and stability. In Section 5, simulation results on three different mechanical structures demonstrate the performance of the proposed control strategy, and Section 6 concludes this paper.

    ##
    2 Notation

    Throughout the paper $\mathbb{R}$ and $\mathbb{R}_{>0}$ denote the set of real and positive real numbers, respectively. $\mathbb{R}^{n}$ and $\mathbb{R}^{n\times m}$ denote the sets of $n\times 1$ real vectors and $n\times m$ real matrices, respectively.
    $v^{(k)}$ denotes the $k^{\text{th}}$ derivative of a variable $v$. $\mathbf{0}_{k}$ is a $k\times 1$ vector of zeros. $|\cdot|$ denotes the absolute value. $\|\cdot\|$ denotes the 2-norm, $|\cdot|_{\infty}$ denotes the supremum value, and ${\rm sat}(\cdot)$ denotes the saturation function which, for a scalar $v$, is defined as

    $\mathrm{sat}(v/\epsilon)=\left\{\begin{array}[]{rl}1&\quad v\geq\epsilon\\
    v/{\epsilon}&\quad-\epsilon\leq v\leq\epsilon\\
    -1&\quad v\leq-\epsilon,\\
    \end{array}\right.$

    where $\epsilon\in\mathbb{R}_{>0}$. Moreover, $\mathrm{pol}(\lambda_{i},k)$ is a polynomial of degree $k$ with parameters $\lambda_{i}\in\mathbb{R},i=1,2,\ldots,k,$ defined as $\mathrm{pol}(\lambda_{i},k)=s^{k}+\lambda_{k}s^{k-1}+\ldots+\lambda_{2}s+\lambda_{1}$.

    ##
    3 Problem Statement

    Consider a class of nonlinear systems with $p$ degrees of freedom described as follows:

    (1)

    $\begin{split}\mathbold{x}^{(n)}&=\mathbold{F}(\mathbold{\xi})\mathbold{\theta}+\mathbold{u}\\
    \mathbold{\xi}&=\begin{bmatrix}\mathbold{x}^{(n-1)\top}&\ldots&\dot{\mathbold{x}}^{\top}&\mathbold{x}^{\top}\end{bmatrix}^{\top},\end{split}$

    where $\mathbold{x}^{(k)}\in\mathbb{R}^{p},k\in\{0,1,2,\ldots,n-1\},$ are the state vectors, $\mathbold{F}(\mathbold{\xi})\in\mathbb{R}^{p\times m}$ is a locally Lipschitz nonlinear function of the states which is known, $\mathbold{\theta}\in\mathbb{R}^{m}$ is a vector of unknown constant parameters, and $\mathbold{u}\in\mathbb{R}^{n}$ is the vector of inputs. This system can model a wide range of nonlinear systems that are linear in parameters (e.g., forced Duffing oscillators [2, 26], forced van der Pol oscillators [20, 31], aircraft attitude dynamics [44], and cross-beam structures [42, 16]).

    Assume that the bounded vector $\mathbold{\xi}^{*}=\begin{bmatrix}\mathbold{x}^{*(n-1)\top}&\ldots&\dot{\mathbold{x}}^{*\top}&\mathbold{x}^{*\top}\end{bmatrix}^{\top}$ is a response of the system (1) to a harmonic force $\mathbold{u}=\mathbold{\sigma}$. Such responses also include equilibria and limit cycles when there is no excitation, i.e., when $\mathbold{\sigma}=\mathbf{0}_{n}$.
    In an open-loop setting, various experiments for $\mathbold{u}=\mathbold{\sigma}$ but with different initial states may lead to different established responses and/or significantly different transient behaviors. In a closed-loop setting, it is possible to compensate for the effect of the initial conditions such that $\mathbold{\xi}$ converges to a desired $\mathbold{\xi}^{*}$ in a ‘reasonable’ time, and the control action simultaneously converges to zero, making the controller noninvasive. Since $\mathbold{\theta}$ is unknown, the objective is to propose an adaptive control strategy for the system (1) with input $\mathbold{u}$, defined as the combination of the external excitation $\mathbold{\sigma}$ and a control action $\mathbold{u}^{\prime}$ as $\mathbold{u}=\mathbold{\sigma}+\mathbold{u}^{\prime}$, such that the nonlinear dynamics behavior asymptotically converges to the following open-loop behavior:

    (2)

    $\mathbold{x}^{*(n)}=\mathbold{F}(\mathbold{\xi}^{*})\mathbold{\theta}+\mathbold{\sigma},$

    which implies that

    (3)

    $\lim_{t\rightarrow\infty}(\mathbold{\xi}(t)-\mathbold{\xi}^{*}(t))=\mathbf{0}_{np},$

    while the control action asymptotically converges to zero as follows:

    $\lim_{t\rightarrow\infty}\mathbold{u}^{\prime}=\mathbf{0}_{p}.$

    In the context of CBC, the response $\mathbold{\xi}^{*}$ is _a priori_ unknown and should be iteratively found as a desired reference signal of the control system such that the control action vanishes. This defines a zero problem whose solutions can be found using Newton-like methods and tracked in parameter space by using suitable path-following techniques [47, 43].

    ##
    4 Noninvasive Adaptive Tracking Strategy

    The proposed noninvasive control strategy is presented in this section. The main ideas underpinning the design are first presented before providing a theorem and the associated proof.

    While noninvasive adaptive control of linear and nonlinear systems has already been addressed in the literature, partial knowledge of the system parameters and the persistently exciting nature of the responses were assumed [28, 28, 29]. In this context, the main achievements of the control method proposed here are to consider all the system parameters as unknown and to lift the persistent excitation requirement. To this end, the main idea of the proposed control strategy is to use the adaptive parameters for defining an auxiliary state. This contrasts with the approach followed in [28, 28, 29] where adaptive parameters are directly used to define the control input. The proposed auxiliary state is designed in such a way that if it converges to a desired value (determined based on the desired open-loop response), the real states of the system asymptotically converge to the desired open-loop response. Hence, our objective is to design a noninvasive control strategy such that the auxiliary state converges to its desired value, whereas the adaptive parameters may not converge to the real system parameters. Since the noninvasiveness of the control strategy does not rely on an exact estimation of the unknown parameters, the desired response is not required to be persistently exciting.

    Let us consider the following auxiliary state:

    (4)

    $\mathbold{z}=\mathbold{x}^{(n-1)}-\int_{0}^{t}\Big{(}\mathbold{F}(\mathbold{\xi}(\tau))\hat{\mathbold{\theta}}(\tau)+\mathbold{\sigma}(\tau)+\mathbold{\eta}(\tau)\Big{)}d\tau,$

    where $\hat{\mathbold{\theta}}\in\mathbb{R}^{m}$ are the adaptive parameters and $\mathbold{\eta}\in\mathbb{R}^{p}$ is a control law. We recall that the reference states $\mathbold{\xi}^{*}$ satisfy (2), which implies that

    (5)

    $\mathbold{x}^{*(n-1)}(0)=\mathbold{x}^{*(n-1)}-\int_{0}^{t}\Big{(}\mathbold{F}(\mathbold{\xi}^{*}(\tau))\mathbold{\theta}+\mathbold{\sigma}(\tau)\Big{)}d\tau,$

    where $\mathbold{x}^{*(n-1)}(0)$ is the initial value of the reference state $\mathbold{x}^{*(n-1)}$. Observing the similarity between Equations (4) and (5), we note that if we design the control strategy such that $\mathbold{z}$ converges to $\mathbold{x}^{*(n-1)}(0)$, the dynamics of the state $\mathbold{x}^{(n-1)}$ converges to that of $\mathbold{x}^{*(n-1)}$. Given that $\mathbold{x}^{(n-1)}$ stands for the states with the highest derivative, this does not guarantee the convergence of the other states. Hence, we should also design $\mathbold{\eta}$ such that $\mathbold{x}$ and its derivatives converge to $\mathbold{x}^{*}$ and its derivatives, respectively, and the objective (3) is achieved. Moreover, the parameter adaption law is designed as

    (6)

    $\dot{\hat{\mathbold{\theta}}}=\mathbold{S}\mathbold{F}(\mathbold{\xi})^{\top}(\mathbold{z}-\mathbold{x}^{*(n-1)}(0)),$

    where $\mathbold{S}\in\mathbb{R}^{m\times m}$ is symmetric positive definite. Now, based on the definition of $\mathbold{z}$, the control action $\mathbold{u}^{\prime}$ is designed as

    (7)

    $\mathbold{u}^{\prime}=\mathbold{\eta}-k(\mathbold{z}-\mathbold{x}^{*(n-1)}(0)),$

    where $k\in\mathbb{R}_{>0}$. According to (7), if $\mathbold{z}$ converges to $\mathbold{x}^{*(n-1)}(0)$, $\mathbold{u}^{\prime}$ is noninvasive if $\mathbold{\eta}$ becomes noninvasive as (3) is reached. Accordingly, we design $\mathbold{\eta}$ as follows:

    (8)

    $\mathbold{\eta}=-\lambda_{n-1}\mathbold{e}^{(n-1)}-\cdots-\lambda_{1}\dot{\mathbold{e}}-\phi\mathbold{y}-g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathrm{sat}(\mathbold{y}/\epsilon),$

    where $\mathbold{e}=\mathbold{x}-\mathbold{x}^{*}$ denotes the tracking error, the coefficients $\lambda_{i}\in\mathbb{R}_{>0},\;i\in\{1,2,\ldots,n-1\},$ are such that the polynomial $\mathrm{pol}(\lambda_{i},n-1)$ is Hurwitz stable, and $\mathbold{y}$ is as follows:

    (9)

    $\mathbold{y}=\mathbold{e}^{(n-1)}+\lambda_{n-1}\mathbold{e}^{(n-2)}+\cdots+\lambda_{1}\mathbold{e}.$

    Since $\mathrm{pol}(\lambda_{i},n-1)$ is Hurwitz stable, by convergence of $\mathbold{y}$ to zero, the tracking error and its derivatives, $\mathbold{e}^{(k)},\;k\in\{0,1,\ldots,n-1\},$ converge to zero as well. Indeed, by defining $\mathbold{y}$ as (9), the stabilization of $n$ vectors $\mathbold{e}^{(k)}$ is simplified to the stabilization of just one vector.
    Moreover, $\phi\in\mathbb{R}$ is a time-varying parameter updated as follows:

    (10)

    $\dot{\phi}=\gamma\mathbold{y}^{\top}\mathbold{y},$

    with $\gamma\in\mathbb{R}_{>0},$ and $g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})$ is a control gain considered as

    (11)

    $g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})=\big{\|}(\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*}))\hat{\mathbold{\theta}}\big{\|}+\kappa,$

    with $\kappa\in\mathbb{R}_{>0}$.

    The main objectives behind the definition of $\mathbold{\eta}$ as (8) are to guarantee the boundedness of the states of the closed-loop control system along with the convergence of $\mathbold{y}$ to zero. In this regard, $-\phi\mathbold{y}$ and the robustifying nonlinear term $-g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathrm{sat}(\mathbold{y}/\epsilon)$ are used to guarantee the boundedness of the states of the closed-loop control system, while $-\lambda_{n-1}\mathbold{e}^{(n-1)}-\cdots-\lambda_{1}\dot{\mathbold{e}}$ and $-\phi\mathbold{y}$ guarantees the convergence of $\mathbold{y}$ to zero. It is worth re-emphasizing that, according to (7) and (8), the estimated parameters are not directly used in the control input. Hence, by proper regulation of $\mathbold{z}$ and the errors $\mathbold{e}^{(k)},k\in\{0,1,\ldots,n-1\},$ without any persistent excitation condition, it is possible to track $\mathbold{\xi}^{*}$ and the control action to be noninvasive. The mathematical analysis of the proposed control strategy is provided in the following theorem.

    ######
    Theorem 4.1.

    Consider the nonlinear system (1) with the desired response $\mathbold{\xi}^{*}$ associated with the excitation force $\mathbold{\sigma}$. Let $\mathbold{u}=\mathbold{\sigma}+\mathbold{u}^{\prime}$ where $\mathbold{u}^{\prime}$ is defined in (7). Under this condition,

    (12)

    $\lim_{t\rightarrow\infty}(\mathbold{\xi}(t)-\mathbold{\xi}^{*}(t))=\mathbf{0}_{np},$

    while $\mathbold{u}^{\prime}$ is bounded and noninvasive.

    ######
    Proof 4.2.

    We prove the theorem in two steps. In the first step, we show that $\mathbold{z}$ remains bounded and converges to $\mathbold{x}^{*(n-1)}(0)$, and $\hat{\mathbold{\theta}}$, $\mathbold{\xi}$, $\mathbold{F}(\mathbold{\xi})$, and $\dot{\mathbold{z}}$ remain bounded. Then, in the next step, the achievement of the objective (12) along with the boundedness of $\phi$ is investigated. Then, we conclude that $\mathbold{u}^{\prime}$ remains bounded and converges to zero.

    Step 1:
    By considering the definition of $\mathbold{z}$ in (4) and since $\mathbold{u}=\mathbold{\sigma}+\mathbold{u}^{\prime}$, along (1) one gets

    (13)

    $\dot{\mathbold{z}}=\mathbold{F}(\mathbold{\xi})\mathbold{\theta}+\mathbold{u}^{\prime}-\mathbold{F}(\mathbold{\xi})\hat{\mathbold{\theta}}-\mathbold{\eta}.$

    To analyze (13) under the control law (7), we consider the following Lyapunov candidate:

    (14)

    $V=\frac{1}{2}\tilde{\mathbold{z}}^{\top}\tilde{\mathbold{z}}+\frac{1}{2}\tilde{\mathbold{\theta}}^{\top}\mathbold{S}^{-1}\tilde{\mathbold{\theta}},$

    where $\tilde{\mathbold{z}}=\mathbold{z}-\mathbold{x}^{*(n-1)}(0)$ and $\tilde{\mathbold{\theta}}=\hat{\mathbold{\theta}}-\mathbold{\theta}$. Since $\mathbold{\theta}$ is constant, the time-derivation of $V$ along (6), (7), and (13), after some simplification yields

    (15)

    $\dot{V}=-k\tilde{\mathbold{z}}^{\top}\tilde{\mathbold{z}}.$

    From (14) and (15), it follows that $\tilde{\mathbold{z}}$, $\mathbold{z}$, $\tilde{\mathbold{\theta}}$, and $\hat{\mathbold{\theta}}$ remain bounded.
    By considering (4) and (5) and according to the definition of $\mathbold{e}$, one gets

    (16)

    $\tilde{\mathbold{z}}=\mathbold{e}^{(n-1)}-\int_{0}^{t}\Big{(}\mathbold{F}(\mathbold{\xi}(\tau))\hat{\mathbold{\theta}}-\mathbold{F}(\mathbold{\xi}^{*}(\tau))\mathbold{\theta}+\mathbold{\eta}(\tau)\Big{)}d\tau.$

    By adding and subtracting $\int_{0}^{t}\mathbold{F}(\mathbold{\xi}^{*}(\tau))\hat{\mathbold{\theta}}d\tau$ to the right hand side of (16) and according to the definition of $\tilde{\mathbold{\theta}}$, one observes that

    (17)

    $\displaystyle\tilde{\mathbold{z}}$
    $\displaystyle=$
    $\displaystyle\mathbold{e}^{(n-1)}-\int_{0}^{t}\Big{(}\mathbold{F}(\mathbold{\xi}^{*}(\tau))\tilde{\mathbold{\theta}}+\big{(}\mathbold{F}(\mathbold{\xi}(\tau))-\mathbold{F}(\mathbold{\xi}^{*}(\tau))\big{)}\hat{\mathbold{\theta}}(\tau)+\mathbold{\eta}(\tau)\Big{)}d\tau.$

    By substituting $\mathbold{\eta}$ defined in (8) into (17), and according to the definition of $\mathbold{y}$ in (9) one gets:

    (18)

    $\displaystyle\tilde{\mathbold{z}}$
    $\displaystyle=$
    $\displaystyle\mathbold{y}-\int_{0}^{t}\Big{(}\mathbold{F}(\mathbold{\xi}^{*}(\tau))\tilde{\mathbold{\theta}}+\big{(}\mathbold{F}(\mathbold{\xi}(\tau))-\mathbold{F}(\mathbold{\xi}^{*}(\tau))\big{)}\hat{\mathbold{\theta}}(\tau)-\phi(\tau)\mathbold{y}(\tau)$

    $\displaystyle-$
    $\displaystyle g(\mathbold{\xi}(\tau),\mathbold{\xi}^{*}(\tau),\hat{\mathbold{\theta}}(\tau))\mathrm{sat}(\mathbold{y}(\tau)/\epsilon)\Big{)}d\tau-\lambda_{n-1}\mathbold{e}^{(n-2)}(0)-\cdots-\lambda_{1}\mathbold{e}(0).$

    Now, by decomposing the $\mathbold{y}$ into its $p$ entries as

    $\mathbold{y}=\begin{bmatrix}y_{1}&y_{2}&\cdots&y_{p}\end{bmatrix}^{\top},$

    we consider two cases. The first is when $|y_{k}|\leq\epsilon,\forall k\in\{1,2,\ldots,p\}$, and the second is when $|y_{k}|>\epsilon$ for some $k\in\{1,2,\ldots,p\}$. These two cases are discussed below:

      *

    Case 1: If $|y_{k}|\leq\epsilon,\forall k\in\{1,2,\ldots,p\},$ according to the definition of $\mathbold{y}$ in (9) and since the polynomial $\mathrm{pol}(\lambda_{i},n-1)$ is Hurwitz stable, $\mathbold{e}^{(k)},k\in\{0,1,\ldots,n-1\},$ are bounded (due to input-to-state stability of Hurwitz linear systems [23]). In this condition, since $\mathbold{\xi}^{*}=\begin{bmatrix}\mathbold{x}^{*(n-1)\top}&\ldots&\dot{\mathbold{x}}^{*\top}&\mathbold{x}^{*\top}\end{bmatrix}^{\top}$ is bounded, $\mathbold{\xi}=\begin{bmatrix}\mathbold{x}^{(n-1)\top}&\ldots&\dot{\mathbold{x}}^{\top}&\mathbold{x}^{\top}\end{bmatrix}^{\top}$ is bounded as well. Then, as $\mathbold{F}(\cdot)$ is a locally Lipschitz function, the boundedness of $\mathbold{\xi}$ implies the boundedness of $\mathbold{F}(\mathbold{\xi})$.
    By considering (7) and (13), we have

    (19)

    $\dot{\mathbold{z}}=\mathbold{F}(\mathbold{\xi})\mathbold{\theta}-k\tilde{\mathbold{z}}-\mathbold{F}(\mathbold{\xi})\hat{\mathbold{\theta}}.$

    Hence, as $\mathbold{F}(\mathbold{\xi})$, $\hat{\mathbold{\theta}}$, and $\tilde{\mathbold{z}}$ are bounded, from (19)
    the boundedness of $\dot{\mathbold{z}}$ can be concluded.

      *

    Case 2: If there exist $k\in\{1,2,\ldots,p\}$ such that $|y_{k}|>\epsilon$, according to the definition of the saturation function, $\mathrm{sat}(y_{k}/\epsilon)=y_{k}/|y_{k}|$. Let us decompose $\big{(}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{)}\hat{\mathbold{\theta}}$ to $p$ entries as follows:

    $\big{(}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{)}\hat{\mathbold{\theta}}=\begin{bmatrix}\mathfrak{F}_{1}&\mathfrak{F}_{2}&\ldots&\mathfrak{F}_{p}\end{bmatrix}^{\top}.$

    By considering (11) and according to the definition of $\mathfrak{F}_{k}$, $\mathrm{sat}(y_{k}/\epsilon)=y_{k}/|y_{k}|$ implies that there exists $\beta_{k}\in\mathbb{R}_{>0}$, such that

    (20)

    $-\mathfrak{F}_{k}+g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathrm{sat}(y_{k}/\epsilon)=\beta_{k}y_{k}.$

    Now, let us decompose $\tilde{\mathbold{z}}$, $\mathbold{F}(\mathbold{\xi}^{*})\tilde{\mathbold{\theta}}$, and $\mathbold{e}$ as follows:

    $\begin{split}\tilde{\mathbold{z}}&=\begin{bmatrix}\tilde{z}_{1}&\tilde{z}_{2}&\cdots&\tilde{z}_{p}\end{bmatrix}^{\top}\\
    \mathbold{F}(\mathbold{\xi}^{*})\tilde{\mathbold{\theta}}&=\begin{bmatrix}\mathfrak{F}^{\prime}_{1}&\mathfrak{F}^{\prime}_{2}&\ldots&\mathfrak{F}^{\prime}_{p}\end{bmatrix}^{\top}\\
    \mathbold{e}&=\begin{bmatrix}e_{1}&e_{2}&\cdots&e_{p}\end{bmatrix}^{\top}.\end{split}$

    By considering (20), from (18) one gets

    (21)

    $\displaystyle y_{k}$
    $\displaystyle=$
    $\displaystyle\int_{0}^{t}\Big{(}\mathfrak{F}^{\prime}_{k}(\tau)-\beta_{k}(\tau)y_{k}(\tau)-\phi(\tau)y_{k}(\tau)\Big{)}d\tau$

    $\displaystyle+$
    $\displaystyle\lambda_{n-1}e_{k}^{(n-2)}(0)+\cdots+\lambda_{1}e_{k}(0)+\tilde{z}_{k}.$

    Since $\mathfrak{F}^{\prime}_{k}$ and $\tilde{z}_{k}$ are bounded and $\beta_{k}+\phi$ is positive, by considering (21), it follows that $y_{k}$ remains bounded. Hence, based on the arguments given for Case 1, the boundedness of $\mathbold{\xi}$ and $\mathbold{F}(\mathbold{\xi})$ and then the boundedness of $\dot{\mathbold{z}}$ can be concluded.

    According to the aforementioned two cases, $\mathbold{\xi}$, $\mathbold{F}(\mathbold{\xi})$, and $\dot{\mathbold{z}}$ are bounded.
    By considering (15),
    $\dot{V}$ is negative semidefinite. Hence, the limit of $V$ exists, that is, for any $s\in\mathbb{R}_{>0}$,

    (22)

    $\lim_{t\rightarrow\infty}\int_{t}^{t+s}\dot{V}(\tau)d\tau=0.$

    Since $\dot{V}$ is nonpositive, according to (22), $\dot{V}$ is nonzero in the period $[t,t+s],t\to\infty,$ only when $\ddot{V}$ is unbounded for bounded $\dot{V}$, which according to (19) is not possible (it can be also concluded by the Barbalat lemma [57] and the invariance principle for nonautonomous systems [4]).
    Therefore, $\dot{V}$ converges to zero which implies that $\tilde{\mathbold{z}}$ converges to zero.

    Step 2:
    From (18) one gets

    (23)

    $\displaystyle\dot{\mathbold{y}}=\big{(}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{)}\hat{\mathbold{\theta}}-\phi\mathbold{y}-g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathrm{sat}(\mathbold{y}/\epsilon)+\dot{\mathbold{z}}+\mathbold{F}(\mathbold{\xi}^{*})\tilde{\mathbold{\theta}}.$

    Since $\tilde{\mathbold{z}}$ converges to zero,
    for any $s\in\mathbb{R}_{>0}$,

    (24)

    $\lim_{t\rightarrow\infty}\int_{t}^{t+s}\dot{\mathbold{z}}(\tau)d\tau=0.$

    Equation (24) can imply a nonzero $\dot{\mathbold{z}}$ in the period $[t,t+s],t\rightarrow\infty,$ only when $\ddot{\mathbold{z}}$ is unbounded for bounded $\dot{\mathbold{z}}$ which is not possible according to (19).
    Hence, the convergence of $\dot{\mathbold{z}}$ to zero can be concluded.
    Since $\dot{\mathbold{z}}$ and $\tilde{\mathbold{z}}$ remain bounded and converge to zero, from (19) it follows that $\mathbold{F}(\mathbold{\xi})\tilde{\mathbold{\theta}}$ remains bounded and converges to zero. By adding and subtracting $\mathbold{F}(\mathbold{\xi})\tilde{\mathbold{\theta}}$ to the right side of (23) and since $\tilde{\mathbold{\theta}}=\hat{\mathbold{\theta}}-\mathbold{\theta}$, after some manipulation one gets

    (25)

    $\displaystyle\dot{\mathbold{y}}=\big{(}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{)}\mathbold{\theta}-\phi\mathbold{y}-g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathrm{sat}(\mathbold{y}/\epsilon)+\mathbold{w},$

    where $\mathbold{w}=\dot{\mathbold{z}}+\mathbold{F}(\mathbold{\xi})\tilde{\mathbold{\theta}}$ is bounded and converges to zero.
    As $\mathbold{\xi}$ and $\mathbold{\xi}^{*}$ are bounded and $\mathbold{F}(\cdot)$ is locally Lipschitz, there exists a finite constant $\ell_{1}\in\mathbb{R}_{>0}$ such that

    (26)

    $\big{\|}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{\|}\leq\ell_{1}\|\mathbold{\xi}-\mathbold{\xi}^{*}\|.$

    Moreover, according to (9), since the polynomial $\mathrm{pol}(\lambda_{i},n-1)$ is Hurwitz stable, there exists a vanishing bounded function $h\left(t,\mathbold{e}(0),\ldots,\mathbold{e}^{(n-2)}(0)\right)$ and a finite constant
    $\ell_{2}\in\mathbb{R}_{>0}$
    such that [23]

    (27)

    $\|\mathbold{\xi}-\mathbold{\xi}^{*}\|\leq h+\ell_{2}\|\mathbold{y}\|.$

    Hence, by considering (26) and (27), one gets

    (28)

    $\big{\|}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{\|}\leq\ell_{1}h+\ell_{1}\ell_{2}\|\mathbold{y}\|.$

    Now, we consider the following Lyapunov candidate:

    $V^{\prime}=\frac{1}{2}\mathbold{y}^{\top}\mathbold{y}+\frac{1}{2}\gamma^{-1}(\phi-\phi^{*})^{2},$

    where
    $\phi^{*}\in\mathbb{R}_{>0}$ is a constant that satisfies the following inequality for an arbitrary $\varrho\in\mathbb{R}_{>0}$,

    (29)

    $\phi^{*}>\frac{\varrho}{2}+\ell_{1}\ell_{2}\|\mathbold{\theta}\|.$

    The time derivation of $V^{\prime}$ along (10) and (25) yields

    (30)

    $\displaystyle\dot{V}^{\prime}$
    $\displaystyle=$
    $\displaystyle\mathbold{y}^{\top}\big{(}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{)}\mathbold{\theta}-\phi\mathbold{y}^{\top}\mathbold{y}-g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathbold{y}^{\top}\mathrm{sat}(\mathbold{y}/\epsilon)$

    $\displaystyle+$
    $\displaystyle\mathbold{y}^{\top}\mathbold{w}+(\phi-\phi^{*})\mathbold{y}^{\top}\mathbold{y}.$

    According to the Young inequality, one observes that

    (31)

    $\begin{split}\mathbold{y}^{\top}\mathbold{w}&\leq\frac{\varrho}{2}\mathbold{y}^{\top}\mathbold{y}+\frac{1}{2\varrho}\mathbold{w}^{\top}\mathbold{w}.\end{split}$

    By considering (28) and (31), (30) can be simplified as follows:

    (32)

    $\dot{V}^{\prime}\leq\ell_{1}h\|\mathbold{\theta}\|\|\mathbold{y}\|+\ell_{1}\ell_{2}\|\mathbold{\theta}\|\mathbold{y}^{\top}\mathbold{y}+\frac{\varrho}{2}\mathbold{y}^{\top}\mathbold{y}+\frac{1}{2\varrho}\mathbold{w}^{\top}\mathbold{w}-\phi^{*}\mathbold{y}^{\top}\mathbold{y},$

    where $\ell_{1}h\|\mathbold{\theta}\|\|\mathbold{y}\|$ is bounded and converges to zero. Note that as $\mathbold{\xi}$ and $\mathbold{\xi}^{*}$ are bounded, $\mathbold{y}$ is also bounded.
    From (29) and (32), it follows that there exists $\chi\in\mathbb{R}_{>0}$ such that

    (33)

    $\dot{V}^{\prime}\leq-\chi\mathbold{y}^{\top}\mathbold{y}+\ell_{1}h\|\mathbold{\theta}\|\|\mathbold{y}\|+\frac{1}{2\varrho}\mathbold{w}^{\top}\mathbold{w}.$

    Since $\ell_{1}h\|\mathbold{\theta}\|\|\mathbold{y}\|+1/(2\varrho)\mathbold{w}^{\top}\mathbold{w}$ is bounded and converges to zero, according to (33), for any nonzero $\mathbold{y}$ there exists a finite time $t_{f}$ such that for $t\geq t_{f}$, $\dot{V}^{\prime}$ is negative semidefinite. Note that according to (10) and (25), $V^{\prime}$ cannot become unbounded in finite time. Therefore, as $\dot{V}^{\prime}$ is negative semidefinite for $t\geq t_{f}$, $V^{\prime}$ is bounded in all time.
    Hence, the limit of $V^{\prime}$ exists, that is, for any $s\in\mathbb{R}_{>0}$,

    (34)

    $\lim_{t\rightarrow\infty}\int_{t}^{t+s}\dot{V}^{\prime}(\tau)d\tau=0.$

    According to (34), $\dot{V}^{\prime}$ is nonzero in the period $[t,t+s],t\to\infty,$ only when $\ddot{V}^{\prime}$ is unbounded for bounded $\dot{V}^{\prime}$, which is not possible according to (25).
    Hence, $\dot{V}^{\prime}$ converges to zero which implies that $\mathbold{y}$ converges to zero.
    According to (9), by convergence of $\mathbold{y}$ to zero, since the polynomial $\mathrm{pol}(\lambda_{i},n-1)$ is Hurwitz stable, $\mathbold{e}^{(k)},k\in\{0,1,\ldots,n-1\},$ converge to zero. Therefore,
    the objective (12) is satisfied. Moreover, since $V^{\prime}$ is bounded, $\phi$ remains bounded. Hence, as $\mathbold{y}$ and $\mathbold{\xi}$ are bounded, $\mathbold{u}^{\prime}$ is bounded. To complete the proof, we need to show that $\mathbold{u}^{\prime}$ converges to zero. We have shown that $\mathbold{e}^{(k)},k\in\{0,1,\ldots,n-1\},$ and $\mathbold{y}$ converge to zero.
    It was also shown that $\tilde{\mathbold{z}}$ converges to zero.
    Therefore, according to (7) and (8), $\mathbold{u}^{\prime}$ converges to zero.

    According to the results of Theorem 4.1, if the reference signal $\mathbold{\xi}^{*}$ is a response of the system (1) to the excitation force $\mathbold{\sigma}$, $\mathbold{\xi}$ converges to $\mathbold{\xi}^{*}$, while $\mathbold{u}^{\prime}$ is noninvasive. In the context of CBC, the open-loop behavior of the system is _a priori_ unknown. Hence, to discover such open-loop behavior, the reference signal $\mathbold{\xi}^{*}$ will be found iteratively by checking whether it makes the control action noninvasive or not.
    Hence, it is desirable to guarantee that the reference signal results in noninvasive control “only if” it is an open-loop response of the nonlinear system (1) to $\mathbold{\sigma}$. In other words, in cases when the reference $\mathbold{\xi}^{*}$ is not an open-loop response of the system (1) to $\mathbold{\sigma}$, $\mathbold{\xi}$ should not converge to $\mathbold{\xi}^{*}$ and/or $\mathbold{u}^{\prime}$ should be invasive. This is addressed by the following corollary and its associated proof.

    ######
    Corollary 4.3.

    Consider the nonlinear system (1) with $\mathbold{u}=\mathbold{\sigma}+\mathbold{u}^{\prime}$, where $\mathbold{u}^{\prime}$ is defined in (7). If a bounded smooth reference $\mathbold{\xi}^{*}$ is not a response of the system to $\mathbold{\sigma}$; then, while $\mathbold{\xi}$ and $\mathbold{u}$ remain bounded, $\mathbold{e}$ does not converges to zero, and a stable equilibrium of the tracking error system implies nonzero control input.

    ######
    Proof 4.4.

    If $\mathbold{\xi}^{*}$ is not a response of the system to $\mathbold{\sigma}$, from (2) it follows that

    (35)

    $\mathbold{x}^{*(n)}=\mathbold{F}(\mathbold{\xi}^{*})\mathbold{\theta}+\mathbold{\sigma}+\mathbold{\Delta},$

    where $\mathbold{\Delta}\in\mathbb{R}^{p}$ is bounded and smooth and not identical to zero.
    According to (35), (5) becomes

    (36)

    $\mathbold{x}^{*(n-1)}(0)=\mathbold{x}^{*(n-1)}-\int_{0}^{t}\Big{(}\mathbold{F}(\mathbold{\xi}^{*}(\tau))\mathbold{\theta}+\mathbold{\Delta}(\tau)+\mathbold{\sigma}(\tau)\Big{)}d\tau.$

    In the proof of Theorem 4.1, it was show that if $|y_{k}|\leq\epsilon,\forall k\in\{1,2,\ldots,p\}$, $\mathbold{\xi}$ remains bounded. Let us now consider the case when $|y_{k}|>\epsilon$ for some $k\in\{1,2,\ldots,p\}$.
    By decomposing $\mathbold{\Delta}$ as

    $\mathbold{\Delta}=\begin{bmatrix}\Delta_{1}&\Delta_{2}&\cdots&\Delta_{p}\end{bmatrix}^{\top},$

    and by considering (36), (21) becomes

    (37)

    $\displaystyle\tilde{z}_{k}$
    $\displaystyle=$
    $\displaystyle y_{k}-\int_{0}^{t}\Big{(}\mathfrak{F}^{\prime}_{k}(\tau)-\beta_{k}(\tau)y_{k}(\tau)-\phi(\tau)y_{k}(\tau)-\Delta_{k}(\tau)\Big{)}d\tau$

    $\displaystyle-$
    $\displaystyle\lambda_{n-1}e_{k}^{(n-2)}(0)-\cdots-\lambda_{1}e_{k}(0).$

    Note that whether $\mathbold{\Delta}$ is identical to zero or not, (15) still is satisfied, implying that $\tilde{\mathbold{z}}$ and $\tilde{\mathbold{\theta}}$ are bounded, and hence $\tilde{z}_{k}$ and $\mathfrak{F}^{\prime}_{k}$ are bounded.
    Since $\tilde{z}_{k}$, $\mathfrak{F}^{\prime}_{k}$, and $\Delta_{k}$ are bounded and $\beta_{k}+\phi$ is positive, by considering (37), $y_{k}$ remains bounded. Hence, the boundedness of $\mathbold{\xi}$ can be concluded.
    By considering (36), (25) also becomes

    (38)

    $\displaystyle\dot{\mathbold{y}}=\big{(}\mathbold{F}(\mathbold{\xi})-\mathbold{F}(\mathbold{\xi}^{*})\big{)}\mathbold{\theta}-\phi\mathbold{y}-g(\mathbold{\xi},\mathbold{\xi}^{*},\hat{\mathbold{\theta}})\mathrm{sat}(\mathbold{y}/\epsilon)-\mathbold{\Delta}+\mathbold{w}.$

    Whether $\mathbold{\Delta}$ is identical to zero or not, in a way similar to the proof of Theorem 4.1, one observes that $\tilde{\mathbold{z}}$ and $\dot{\mathbold{z}}$ and then $\mathbold{w}$ converge to zero. Therefore, if $\mathbold{\Delta}$ is not identical to zero, $\mathbold{y}=\mathbf{0}_{p}$ is not the equilibrium point of the system (38). Hence, $\mathbold{e}$ does not converge to zero. Moreover, since $\mathbold{\xi}^{*}$ is not a response of the system to $\mathbold{\sigma}$, (2) cannot be satisfied for the system (1). Therefore, as $\mathbold{u}=\mathbold{\sigma}+\mathbold{u}^{\prime}$, $\mathbold{u}^{\prime}=\mathbf{0}_{p}$ does not imply a stable equilibrium for the tracking error system, that is, a stable equilibrium of the tracking error system implies nonzero control input.
    Note that since $\mathbold{F}(\cdot)$ is locally Lipschitz, according to (1) and (7), $\mathbold{u}$ cannot be unbounded in finite time. In this condition, by considering (1), the boundedness of $\mathbold{\xi}$ implies that $\mathbold{u}$ is bounded.

    Based on the results of Theorem 4.1 and Corollary 4.3, under the proposed control strategy, $\mathbold{\xi}$ converges to a reference signal if and only if the reference signal is an open-loop response of the nonlinear dynamics to the excitation force $\mathbold{\sigma}$. The obtained results are validated by numerical simulation in the next section.

    ######
    Remark 4.5.

    According to the proof of Theorem 4.1, while the proposed control strategy relies on the adaptive vector $\hat{\mathbold{\theta}}$, the convergence of $\hat{\mathbold{\theta}}$ to the real system parameters $\mathbold{\theta}$ is not required. Accordingly, the persistent excitation of $\mathbold{\xi}^{*}$ which is typically required for $\hat{\mathbold{\theta}}$ to converge to $\mathbold{\theta}$, is not a condition for the accurate performance of the proposed control strategy. However, if $\mathbold{\xi}^{*}$ is persistently exciting, it is straightforward to show that under the proposed control strategy, $\hat{\mathbold{\theta}}$ asymptotically converges to $\mathbold{\theta}$.

    ##
    5 Simulation Results

    In this section, the accuracy of the proposed control strategy is numerically demonstrated on three systems. First, a bi-stable Duffing oscillator and then two beam structures exhibiting 1:1 and 3:1 modal interactions, respectively, are considered.

    ###
    5.1 Duffing oscillator

    Consider a forced Duffing oscillator with the following dynamics:

    $\ddot{x}=\theta_{1}\dot{x}+\theta_{2}x+\theta_{3}x^{3}+\sigma+u^{\prime},$

    where $x$ denotes the displacement of the oscillator, $\sigma$ is the applied excitation, $u^{\prime}$ is the control input, and $\theta_{1}=-0.1$, $\theta_{2}=4$, and $\theta_{3}=-2$ are unknown damping, linear stiffness, and nonlinear stiffness parameters, respectively. For illustration, the response of the oscillator under harmonic excitation $\sigma=0.15\cos(\omega t)$, is computed using numerical continuation and shown in Fig. 1-(a).

    |

    ---|---

    (a)

    (b)

    Figure 1: (a) Frequency response of the oscillator around the positive equilibrium in which the specific point associated with $\mathbold{\xi}^{*}$ is shown by a circle (${\color[rgb]{0.00,0.45,0.74}\definecolor[named]{pgfstrokecolor}{rgb}{0.00,0.45,0.74}\circ}$). The dashed line represents unstable responses, and a dot (${\color[rgb]{1.00,0.41,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.41,0.16}\bullet}$) shows a limit-point bifurcation. (b) Desired responses of the oscillator for the initial states $(\dot{x}(0),x(0))=(1.314,1.483)$ and the open-loop response for the initial states $(\dot{x}(0),x(0))=(0,-1)$.

    To illustrate the performance of the proposed control method, a control reference corresponding to an unstable periodic response of the oscillator at $\omega=2.515$ is chosen (see circle marker in Fig. 1-(a)). This reference trajectory is given by $\mathbold{\xi}^{*}=\begin{bmatrix}\dot{x}^{*}&x^{*}\end{bmatrix}^{\top}$ where

    $x^{*}\approx 1.271+0.244\cos(\omega t)-0.026\cos(2\omega t)-0.005\cos(3\omega t)+0.436\sin(\omega t)+0.045\sin(2\omega t),$

    and $\dot{x}^{*}$ is obtained by differentiating $x^{*}$. Moreover, the initial states associated with this reference are $(\dot{x}^{*}(0),x^{*}(0))=(1.314,1.483)$. The response of the oscillator is very sensitive to the initial states as the oscillator has two equilibria with two stable steady-state oscillations around each. The desired open-loop response of the oscillator along with the response of the oscillator to $\sigma=0.15\cos(\omega t)$ for the different initial states $(\dot{x}(0),x(0))=(0,-1)$ is shown in Fig. 1-(b). Therefore, without closed-loop correction of the oscillator response, it is apparent that the oscillator response can be significantly different from the desired one. Note that $x^{*}$ is an unstable open-loop trajectory of the oscillator. Therefore, it may not be precisely revealed even if the initial states of the oscillator are close to the initial states of the desired trajectory.

    |

    ---|---

    |

    (a)

    (b)

    (c)

    (d)

    Figure 2: (a) Desired response of the oscillator for the initial states $(\dot{x}(0),x(0))=(1.314,1.483)$ and the
    closed-loop response of the oscillator for the initial states $(\dot{x}(0),x(0))=(0,-1)$. (b) Noninvasive control input of the oscillator.
    (c) Desired reference $x^{*}=\sin(t)$ (which is not an open-loop response to $\sigma$)
    and closed-loop response of the oscillator. (d) Invasive control input of the oscillator when $\mathbold{\xi}^{*}$ is not an open-loop response to $\sigma$.

    By employing the proposed adaptive control strategy, it is guaranteed that the system response asymptotically converges to the desired open-loop response (see Fig. 2-(a)) while the control action asymptotically converges to zero, i.e., it becomes noninvasive (see Fig. 2-(b)). The control parameters used to obtain these results are $\mathbold{S}=2\mathbold{I}_{3}$, $k=1$, $\kappa=1$, $\lambda_{1}=1$, $\gamma=0.1$, and $\epsilon=1$. Note that, depending on the application, the control gains can be adjusted to reduce or increase control efforts. Smaller control efforts are usually associated with longer times for convergence of the states to the reference signal.

    Under the proposed control strategy, Section 4 also demonstrated that $\mathbold{\xi}$ does not converge to $\mathbold{\xi}^{*}$ and $u^{\prime}$ is invasive if $\mathbold{\xi}^{*}$ is not an open-loop response of the oscillator to $\sigma$. To illustrate this, a closed-loop simulation is performed for an arbitrary reference signal defined as $x^{*}=\sin(t)$. The obtained system response and control input are shown in Fig. 2-(c) and Fig. 2-(d), respectively. As expected, $\mathbold{\xi}$ does not converge to $\mathbold{\xi}^{*}$ and $u^{\prime}$ is invasive.

    ###
    5.2 Cross-beam structure

    The cross-beam structure studied in [42] and inspired by the physical system tested in [16] is now considered. This structure has the particularity of having its first two vibration modes very close in frequency (0.5 Hz apart), leading to 1:1 mode interactions. As such, the model of the structure is a 2-DOF modal model given by

    $\ddot{\mathbold{x}}+\mathbold{\Xi}\dot{\mathbold{x}}+\mathbold{\Lambda}\mathbold{x}+\mathbold{N}(\mathbold{x})=\mathbold{\sigma}+\mathbold{u}^{\prime},$

    with

    $\begin{split}\mathbold{x}&=\begin{bmatrix}x_{1}\\
    x_{2}\end{bmatrix},\mathbold{\Xi}=\begin{bmatrix}2\zeta_{1}\omega_{1}&0\\
    0&2\zeta_{2}\omega_{2}\end{bmatrix},\mathbold{\Lambda}=\begin{bmatrix}\omega_{1}^{2}&0\\
    0&\omega_{2}^{2}\end{bmatrix},\mathbold{N}(\mathbold{x})=\begin{bmatrix}\mathbold{N}_{1}(\mathbold{x})\\
    \mathbold{N}_{2}(\mathbold{x})\end{bmatrix},\\
    \mathbold{N}_{1}(\mathbold{x})&=\gamma_{11}x_{1}^{2}+(\gamma_{21}+\gamma_{31})x_{1}x_{2}+\gamma_{41}x_{2}^{2}+\gamma_{51}x_{1}^{3}+(\gamma_{61}+\gamma_{71})x_{1}x_{2}^{2}\\
    &\quad+(\gamma_{81}+\gamma_{91})x_{1}^{2}x_{2}+\gamma_{10,1}x_{2}^{3},\\
    \mathbold{N}_{2}(\mathbold{x})&=\gamma_{12}x_{1}^{2}+(\gamma_{22}+\gamma_{32})x_{1}x_{2}+\gamma_{42}x_{2}^{2}+\gamma_{52}x_{1}^{3}+(\gamma_{62}+\gamma_{72})x_{1}x_{2}^{2}\\
    &\quad+(\gamma_{82}+\gamma_{92})x_{1}^{2}x_{2}+\gamma_{10,2}x_{2}^{3},\end{split}$

    where $x_{1}$ and $x_{2}$ denote the displacements of the first and second modes, respectively, $\zeta_{1}$ and $\zeta_{2}$ denote the linear damping ratios, $\omega_{1}$ and $\omega_{2}$ denote the natural frequencies which here are assumed to be unknown, and $\gamma_{ij},i\in\{1,2,\ldots,10\},i\in\{1,2\},$ are the unknown parameters of the nonlinear term $\mathbold{N}(\mathbold{x})$. It is worth mentioning that such a model can describe a wide range of mechanical structures with geometric nonlinearity, and is therefore not limited to the cross-beam example considered here.

    For the cross-beam system, the unknown parameters are set as $\zeta_{1}=0.0076$, $\zeta_{2}=0.0026$, $\omega_{1}=101.6$, $\omega_{2}=104.6$, $\gamma_{11}=113.321$, $\gamma_{21}=-104.755$, $\gamma_{31}=-104.755$, $\gamma_{41}=-29.740$,$\gamma_{51}=3.836\times 10^{8}$,$\gamma_{61}=2.451\times 10^{7}$, $\gamma_{71}=4.902\times 10^{7}$, $\gamma_{81}=1.929\times 10^{8}$, $\gamma_{91}=9.644\times 10^{7}$, $\gamma_{10,1}=6.104\times 10^{6}$, $\gamma_{12}=-104.755$, $\gamma_{22}=-29.740$, $\gamma_{32}=-29.740$, $\gamma_{42}=85.367$,$\gamma_{52}=9.644\times 10^{7}$,$\gamma_{62}=6.104\times 10^{6}$, $\gamma_{72}=1.221\times 10^{7}$, $\gamma_{82}=4.902\times 10^{7}$, $\gamma_{92}=2.451\times 10^{7}$, and $\gamma_{10,2}=2.351\times 10^{6}$. The response of the structure under the excitation $\mathbold{\sigma}=[1.261\cos(\omega t)~{}0.318\cos(\omega t)]^{\top}$ is computed using numerical continuation and shown in Fig. 3 (for the first mode).

    Figure 3: Frequency response curve associated with the first mode of the cross-beam structure. The specific point associated with $\mathbold{\xi}^{*}$ is shown by a circle (${\color[rgb]{0.00,0.45,0.74}\definecolor[named]{pgfstrokecolor}{rgb}{0.00,0.45,0.74}\circ}$), the dashed line represents unstable responses, and dots (${\color[rgb]{1.00,0.41,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.41,0.16}\bullet}$) and diamonds (${\color[rgb]{0.49,0.18,0.56}\definecolor[named]{pgfstrokecolor}{rgb}{0.49,0.18,0.56}\blacklozenge}$) show limit-point and Neimark-Sacker bifurcations, respectively.

    To demonstrate the control method, the control reference $\mathbold{\xi}^{*}=\begin{bmatrix}\dot{x}_{1}^{*}&\dot{x}_{2}^{*}&x_{1}^{*}&x_{2}^{*}\end{bmatrix}^{\top}$ is chosen as an unstable response of the structure at $\omega=118.814$ (circle in Fig. 3). This reference can be approximated as

    $\begin{split}x_{1}^{*}&\approx-35.344\times 10^{-4}\cos(2\omega t)+0.521\times 10^{-4}\cos(4\omega t)+0.002\times 10^{-4}\cos(7\omega t)\\
    &\quad-10.974\times 10^{-4}\sin(2\omega t)+0.132\times 10^{-4}\sin(4\omega t),\\
    x_{2}^{*}&\approx 42.080\times 10^{-4}\cos(2\omega t)+0.303\times 10^{-4}\cos(4\omega t)-0.006\times 10^{-4}\cos(6\omega t)\\
    &\quad+12.358\times 10^{-4}\sin(2\omega t)+0.077\times 10^{-4}\sin(4\omega t)-0.002\times 10^{-4}\sin(6\omega t),\\
    \end{split}$

    where the initial states associated with this response are

    $(\dot{x}^{*}_{1}(0),\dot{x}^{*}_{2}(0),x^{*}_{1}(0),x^{*}_{2}(0))=(0.5104,0.1495,-0.0035,-0.0011).$

    To achieve the desired response, we employ the proposed adaptive control strategy with $\mathbold{S}=2\times 10^{4}\mathbold{I}_{18}$, $k=10^{-4}$, $\kappa=10^{-4}$, $\lambda_{1}=10^{-4}$, $\gamma=10^{3}$, and $\epsilon=1$. The obtained response and control inputs are presented in Fig. 4. According to the figures, the system response asymptotically converges to the desired response and the control action is noninvasive.

    |

    ---|---

    (a)

    (b)

    Figure 4: (a) Response of the cross-beam structure for the initial states $(\dot{x}_{1}(0),\dot{x}_{2}(0),x_{1}(0),x_{2}(0))=(0.5104,0.1495,-0.0035,-0.0011)$ and the
    closed-loop response of the cross-beam structure for the initial states $(\dot{x}_{1}(0),\dot{x}_{2}(0),x_{1}(0),x_{2}(0))=(0,0,0,0)$.
    (b) Noninvasive control inputs of the cross-beam structure.

    ###
    5.3 Cantilever beam with a nonlinear mechanism at its tip

    A cantilever beam structure with nonlinear springs attached at its free end is now considered. This structure has the particularity to exhibit a 1:3 mode interaction between its first and second bending modes [50]. The system can be described by the following modal equations:

    $\ddot{\mathbold{x}}+\mathbold{\Xi}\dot{\mathbold{x}}+\mathbold{\Lambda}\mathbold{x}+\mathbold{\Phi}^{\top}\mathbold{f}(\mathbold{\Phi}\mathbold{x})=\mathbold{\Phi}^{\top}\mathbold{u},$

    with

    (39)

    $\begin{split}\mathbold{x}&=\begin{bmatrix}x_{1}\\
    x_{2}\end{bmatrix},\mathbold{\Xi}=\begin{bmatrix}2\zeta_{1}\omega_{1}&0\\
    0&2\zeta_{2}\omega_{2}\end{bmatrix},\mathbold{\Lambda}=\begin{bmatrix}\omega_{1}^{2}&0\\
    0&\omega_{2}^{2}\end{bmatrix}\\
    \mathbold{f}(\mathbold{\Phi}\mathbold{x})&=\begin{bmatrix}0&0&0&2k_{0}\ell_{0}x^{\prime}_{4}\left(\frac{1}{a}-\frac{1}{\sqrt{a^{2}+{x^{\prime}_{4}}^{2}}}\right)\end{bmatrix}^{\top},\mathbold{u}=\begin{bmatrix}u_{1}&u_{2}&0&0\end{bmatrix}^{\top}\\
    \mathbold{\Phi}\mathbold{x}&=\begin{bmatrix}x^{\prime}_{1}&x^{\prime}_{2}&x^{\prime}_{3}&x^{\prime}_{4}\end{bmatrix}^{\top}=\mathbold{x}^{\prime},\end{split}$

    where $x_{1}$, $x_{2}$, $\zeta_{1}$, $\zeta_{2}$, $\omega_{1}$, and $\omega_{2}$ have the same meaning as in Section 5.2, $k_{0}$ is the stiffness coefficient of the springs, $\ell_{0}$ is their original length, $a$ is the half span of the mechanism, and $\mathbold{\Phi}\in\mathbb{R}^{4\times 2}$ describes the relation between the modal variables and the physical deflection of the beam at four specific points along the beam. For this system, the parameters are $\zeta_{1}=0.01$, $\zeta_{2}=0.01$, $\omega_{1}=67.395$, $\omega_{2}=235.783$, $k_{0}=910$, $\ell_{0}=0.018$, $a=0.019$, and

    $\mathbold{\Phi}=\begin{bmatrix}-0.1603&-0.6821\\
    -1.7748&-4.4598\\
    -5.9745&6.1940\\
    -6.1389&7.0245\end{bmatrix}.$

    To describe (39) with the model class (1), the mode shape matrix $\mathbold{\Phi}$ is assumed to be known, which in practice can be achieved using standard linear system identification techniques.
    All the other model parameters are considered unknown. Moreover, since the system has two degrees of freedom, we assumed we have two control inputs. The response of the structure under the excitation $\mathbold{\sigma}=[2\cos(\omega t)~{}0~{}0~{}0]^{\top}$ is computed using numerical continuation and shown in Fig. 5 (for the first mode).

    Figure 5: Frequency response associated with the first mode of the beam structure with nonlinear springs. The specific point associated with $\mathbold{\xi}^{*}$ is shown by a circle (${\color[rgb]{0.00,0.45,0.74}\definecolor[named]{pgfstrokecolor}{rgb}{0.00,0.45,0.74}\circ}$), the dashed line represents unstable responses, and dots (${\color[rgb]{1.00,0.41,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.41,0.16}\bullet}$) and diamonds (${\color[rgb]{0.49,0.18,0.56}\definecolor[named]{pgfstrokecolor}{rgb}{0.49,0.18,0.56}\blacklozenge}$) show limit-point and Neimark-Sacker bifurcations, respectively.

    A desired open-loop trajectory
    $\mathbold{\xi}^{*}=\begin{bmatrix}\dot{x}_{1}^{*}&\dot{x}_{2}^{*}&x_{1}^{*}&x_{2}^{*}\end{bmatrix}^{\top}$ is considered to be an unstable response of the beam structure to the harmonic excitation force $\mathbold{\sigma}=[2\cos(\omega t)~{}0~{}0~{}0]^{\top},\omega=83.085$, for the initial states $(\dot{x}_{1}(0),\dot{x}_{2}(0),x_{1}(0),x_{2}(0))=(-0.066,-0.011,-2.613\times 10^{-4},-3.031\times 10^{-4})$, where $x_{1}^{*}$ and $x_{2}^{*}$ are as follows:

    $\begin{split}x_{1}^{*}&\approx-2.834\times 10^{-4}\cos(2\omega t)+0.254\times 10^{-4}\cos(4\omega t)-0.034\times 10^{-4}\cos(6\omega t)\\
    &\quad-0.487\times 10^{-4}\sin(2\omega t)-2.600\times 10^{-4}\sin(4\omega t)+0.055\times 10^{-4}\sin(6\omega t)\\
    x_{2}^{*}&\approx-8.240\times 10^{-4}\cos(2\omega t)+0.066\times 10^{-4}\cos(4\omega t)+0.026\times 10^{-4}\cos(6\omega t)\\
    &\quad-0.469\times 10^{-4}\sin(2\omega t)-0.219\times 10^{-4}\sin(4\omega t)-0.043\times 10^{-4}\sin(6\omega t).\end{split}$

    To achieve the desired open-loop response $\mathbold{\xi}^{*}$, we employ the proposed adaptive control strategy with $\mathbold{S}=2\times 10^{5}\mathbold{I}_{18}$, $k=10^{-4}$, $\kappa=10^{-4}$, $\lambda_{1}=10^{-4}$, $\gamma=10^{5}$, and $\epsilon=1$.
    The simulation results, presented in Fig. 6-(a), show that the system response asymptotically converges to the desired one. As shown in Fig. 6-(b), the control action becomes noninvasive as the system reaches the desired response.

    |

    ---|---

    (a)

    (b)

    Figure 6: (a) Response of the beam structure (with nonlinear springs) for the initial states $(\dot{x}_{1}(0),\dot{x}_{2}(0),x_{1}(0),x_{2}(0))=(-0.066,-0.011,-2.613\times 10^{-4},-3.031\times 10^{-4})$ and the closed-loop response of the cross-beam structure for the initial states $(\dot{x}_{1}(0),\dot{x}_{2}(0),x_{1}(0),x_{2}(0))=(0,0,0,0)$. (c) Noninvasive control inputs of the beam structure.

    One of the contributions of the paper is that contrary to the existing literature, the noninvasiveness of the proposed control strategy does not rely on the persistent excitation of the desired response and thereby does not rely on the true estimation of the system parameters. To show this, the norm of the estimation error $\tilde{\mathbold{\theta}}=\hat{\mathbold{\theta}}-\mathbold{\theta}$ when $\hat{\mathbold{\theta}}$ is initialized to zero and $\|\mathbold{\theta}\|=5.578\times 10^{4}$ is shown in Fig. 7-(a). The estimation error does not converge to zero. In fact, a negligible update in the model parameters $\hat{\mathbold{\theta}}$ occurs. Defining the following excitation matrix:

    $\mathbold{M}_{e}(t)=\int_{0}^{t}\mathbold{F}(\mathbold{\xi}(\tau))^{\top}\mathbold{F}(\mathbold{\xi}(\tau))d\tau,$

    its minimum eigenvalue along the system trajectory is shown in Fig. 7-(b), which is close to zero and verifies that $\mathbold{M}_{e}$ is not positive definite. Hence, the closed-loop system is not persistently excited along the desired reference.

    |

    ---|---

    (a)

    (b)

    Figure 7: (a) Error of estimating the parameters of the beam structure (with nonlinear springs). (b) The smallest eigenvalue of the matrix $\mathbold{M}_{e}(t)$ associated with the beam structure.

    ##
    6 Conclusion and Future Work

    Control-based continuation is an increasingly popular method to characterize the nonlinear dynamics of a physical system directly during tests, without the need for a model. Despite being successfully applied to a wide range of experiments and providing invaluable results for model calibration and validation, the general and systematic design of the noninvasive controller at the core of the method is lacking.

    In this paper, an adaptive strategy for noninvasive control of nonlinear systems that are linear in their unknown parameters is proposed. The method guarantees convergence to the desired response and noninvasive control action if and only if the response is a natural response of the uncontrolled system. Compared to other available methods, the proposed control strategy does not require any knowledge of the system parameters, nor their exact estimation, and it does not assume stable linearized dynamics. This makes the proposed methodology applicable to a much wider range of nonlinear systems, with a wider range of response types. Furthermore, the proposed method does not rely on the persistence of the excitation, which, in the context of CBC, means there is no need to check the persistent nature of the excitation to guarantee convergence to a noninvasive control target. This has the potential to increase CBC accuracy and reduce overall testing time.

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