A Unified Framework for Identification, Estimation, and Control of an Experimental Duffing–Holmes System

Abstract

This paper presents a comprehensive framework for the identification, state estimation, and robust control of a bistable Duffing–Holmes oscillator, validated through an experimental setup. First, to address parametric uncertainty, a Recursive Least Squares Method (RLSM) with a forgetting factor is applied to a filtered model representation, enabling accurate parameter convergence from noisy measurements. Subsequently, a Nonlinear Integral Extended State Observer (NIESO) is designed to reconstruct unmeasured states and estimate total disturbances. A key theoretical contribution is the derivation of explicit gain conditions that guarantee the observer’s stability, overcoming limitations of previous designs. For trajectory tracking, an observer-based backstepping controller is synthesized. Crucially, to bridge the gap between theory and practice, a drift-free integration scheme is implemented to generate feasible position commands for the shake table, preventing actuator saturation. Experimental results confirm the framework’s effectiveness, achieving a 3.7-fold reduction in RMS tracking error compared to open-loop operation, with the tracking error rapidly converging to a small neighborhood within approximately $0.2$ $\mathrm{s}$. Furthermore, the closed-loop system demonstrates superior energy efficiency, requiring significantly lower actuator voltage to sustain stable interwell oscillations.

1. Introduction

Nonlinear oscillators are ubiquitous in physical, mechanical, and engineering systems, representing phenomena where linear models fail to capture essential behaviors such as bifurcations, chaos, and multi-stability [1,2]. Among these, the Duffing oscillator stands out as a fundamental benchmark due to its rich dynamics and versatile mathematical structure [3]. A particularly significant variant is the Duffing–Holmes oscillator, which incorporates specific damping and excitation characteristics that result in a double-well potential energy landscape. This bistable nature makes it an exceptionally powerful model for studying complex dynamics in vibration energy harvesting, structural mechanics, magnetic actuation, and microelectromechanical systems (MEMSs) [4]. However, the practical utilization of these systems is often hindered by the difficulty in accurately characterizing their underlying physics and controlling their sensitive chaotic responses in real-world environments.

Accurate modeling of Duffing-type systems critically depends on the identification of physical parameters, such as damping, linear stiffness, and nonlinear stiffness coefficients, which are often unknown or time-varying. To address this, various advanced identification techniques have been developed. For instance, Erazo [5] evaluated Bayesian filtering methods, highlighting the Unscented Kalman Filter (UKF) for structural parameter estimation. Junker et al. [6] proposed a physical–stochastic framework based on stochastic differential equations to handle model uncertainty in forced Duffing oscillators. However, probabilistic filters like the Extended Kalman Filter (EKF) and the UKF present practical limitations. They require tuning of often-unknown noise covariance matrices, and can suffer from computational overhead and degraded accuracy in highly nonlinear chaotic regimes due to linearization or unscented transform approximation errors. In terms of dealing with complex nonlinearities, Zhu et al. [7] applied nonlinear subspace identification to capture position-dependent friction and stiffness in double-well systems. Furthermore, recent approaches like the one by Aguilar-Ibañez et al. [8] have introduced algebraic methods using Dynamic Regressor Extension and Mixing (DREM) to decouple and estimate parameters using only a single measurable output, reducing the need for persistent excitation.

In addition to parameter uncertainty, a major challenge in practical applications is that the full state vector is rarely available for direct measurement. Consequently, observer design becomes essential for monitoring and control. Zhou et al. [9] addressed this by proposing a dynamic high-gain observer that adjusts gains in real time. Similarly, the Active Disturbance Rejection Control (ADRC) paradigm, which relies on an Extended State Observer (ESO), is widely used to estimate and compensate for uncertainties by treating them as a lumped “total disturbance” [10]. In this context, ESOs have been successfully employed to tackle system uncertainties; for example, Thenozhi et al. [11] developed a Nonlinear Extended State Observer (NESO) for the Duffing–Holmes system, establishing stability via the Gronwall–Bellman inequality. However, standard ADRC frameworks often lack explicit analytical tools to systematically determine the observer gains that guarantee closed-loop stability in such highly nonlinear chaotic systems. Recent trends also incorporate approximation theory and machine learning; for example, Izadbakhsh et al. [12] utilized Szasz–Favard–Mirakyan operators for chaos synchronization without requiring state regressors, while Vargas Álvarez et al. [13] designed Physics-Informed Neural Network (PINN) observers to reconstruct states under incomplete measurements. While achieving high accuracy, data-driven architectures such as PINNs and neural NARX models typically demand substantial offline training data, lack physical interpretability, and impose heavy computational burdens. Fractional-order observers have also been explored to reduce chattering and enhance robustness against uncertainties [14].

Once parameters are identified and states are estimated, the final challenge lies in the precise control of the system trajectory. Given the inherent sensitivity to initial conditions in Duffing systems, linear control strategies are often inadequate. Nonlinear approaches such as backstepping have proven effective [15,16,17]. Hybrid formulations combining backstepping with Sliding Mode Control (SMC) have also been proposed to enhance robustness [18]. Nevertheless, the practical implementation of standard SMC often produces the “chattering” phenomenon. In the context of a Duffing–Holmes oscillator driven by a shake table, this chattering not only results in excessive wear and tear on the actuators but can also excite unmodeled high-frequency structural dynamics. To mitigate this issue, continuous SMC variants such as the Super-Twisting algorithm [19] have been proposed. Alternatively, adaptive schemes based on disturbance observers [20] have emerged to address parametric uncertainties without relying on high-frequency switching. Other modern contributions include fuzzy adaptive control [21], fractional-order synchronization [22], and neural network-based controllers [23,24].

Despite the extensive development of the above-mentioned strategies, their validation has predominantly been limited to numerical simulations. Especially, the experimental investigations that comprehensively address parameter identification, state estimation, and tracking control of the Duffing–Holmes oscillator are still comparatively limited in the existing literature. To address these challenges, this paper proposes a comprehensive framework that unifies parameter identification, state estimation, and robust control for the Duffing–Holmes oscillator, validated through a rigorous experimental setup. The specific contributions of this work are threefold:

• Parameter Identification: This paper proposes a parameter identification approach for the Duffing–Holmes system based on a filtered model representation combined with a Recursive Least Squares Method (RLSM) incorporating a forgetting factor. The effectiveness of the proposed identification strategy is experimentally validated using a Duffing–Holmes prototype.
• State Estimation: A Nonlinear Integral Extended State Observer (NIESO) is developed to estimate the unmeasured velocity and lumped uncertainty from position measurements. In contrast to previous works such as [11], the proposed observer integrates a projection mechanism that guarantees bounded estimation errors. The observer gain design is systematically derived, and its performance is validated experimentally.
• Tracking Control: Using the identified system parameters and the state estimates provided by the proposed schemes, a backstepping-based tracking control law is developed to drive the position of the Duffing–Holmes system toward a desired trajectory. Experimental results demonstrate satisfactory control performance under interwell dynamics. Due to the physical limitations of the shake table used in the experiments, chaotic and intrawell behaviors could not be experimentally validated. Nevertheless, comprehensive simulation studies are conducted to verify the effectiveness of the proposed parameter identification and state estimation methods in achieving control performance under intrawell, interwell, and chaotic regimes.

The remainder of this paper is organized as follows. Section 2 describes the nonlinear dynamics of the Duffing–Holmes oscillator and the experimental setup. Section 3 details the parameter identification methodology. Section 4 presents the design and stability analysis of the NIESO. Section 5 introduces the backstepping control strategy. Section 6 discusses the simulation and experimental results, and Section 7 provides concluding remarks.

2. Dynamics of a Duffing–Holmes System

The bistable Duffing–Holmes oscillator comprises a ferromagnetic elastic beam clamped at one end to a rigid support, as depicted in Figure 1. Under external harmonic excitation of the base, the oscillation of the beam with a magnetic tip interacts with the magnetic field generated by two symmetrically positioned permanent magnets, leading to a strongly nonlinear bistable dynamic response.

The dynamics of the Duffing–Holmes system, illustrated in Figure 1, is governed by the following differential equation:

$$m\ddot{q}\left(t\right)+c\dot{q}\left(t\right)-k_{l}q\left(t\right)+k_n{q}^3\left(t\right)=-m\ddot{p}\left(t\right),$$

(1)

where m denotes the effective beam mass, c the viscous damping coefficient, and $k_l$ and $k_n$ the linear and nonlinear stiffness coefficients, respectively; $q\left(t\right)$ represents the beam displacement, and $\ddot{p}\left(t\right)$ denotes the base acceleration. In particular, the interaction between the negative linear stiffness and the cubic nonlinear stiffness in (1) gives rise to rich nonlinear dynamics, attracting considerable attention in the field of nonlinear vibration analysis [25].

The Duffing–Holmes system (1) exhibits three equilibrium points located at ${\mathbf{q}}_1^{*}={[0,0]}^{\mathrm{T}}$ and ${\mathbf{q}}_{2,3}^{*}={[\pm \sqrt{k_l/k_n},0]}^{\mathrm{T}}$. Among these equilibrium points, the equilibrium point ${\mathbf{q}}_1^{*}$ is unstable, whereas the equilibrium points ${\mathbf{q}}_{2,3}^{*}$ are stable, thereby characterizing the system as bistable.

Figure 2a illustrates the phase portrait of the system, highlighting its complex behavior, which strongly depends on the system parameters, initial conditions, as well as the amplitude and frequency of the excitation force. Depending on these factors, the Duffing–Holmes system can exhibit three distinct dynamical behaviors [26,27,28]:

• Interwell motion: The trajectories oscillate between the two potential wells, crossing the potential barrier.
• Intrawell motion: The system trajectories are confined within a single potential well, oscillating around one stable equilibrium point without crossing the potential barrier.
• Chaotic motion: The system produces unpredictable and aperiodic motion due to extreme sensitivity to initial conditions.

Another way to demonstrate the complexity of the Duffing–Holmes system is by analyzing its potential energy. The elastic potential energy $U\left(q\right)$ of the system is depicted in Figure 2b and is obtained as

$$U\left(q\right)={\int }_0^q\left(k_n{\xi }^3-k_l\xi \right)d\xi ={\displaystyle \frac{1}{4}}{k}_n{q}^4-{\displaystyle \frac{1}{2}}{k}_l{q}^2.$$

(2)

On the other hand, the gravitational potential energy is neglected by maintaining the Duffing–Holmes system in a horizontal configuration, as shown in Figure 1. In this way, the beam motion is confined to the x–y plane, while gravity acts along the negative direction of the z-axis. Since the resulting vertical displacement of the beam tip is negligible, the effect of gravity can be ignored.

Problem Formulation

By dividing (1) by the mass m, the dynamics of the Duffing–Holmes system can be reformulated as:

$$\ddot{q}\left(t\right)+\alpha \dot{q}\left(t\right)-\beta q\left(t\right)+\gamma q^3\left(t\right)=u\left(t\right),$$

(3)

where $\alpha =c/m$, $\beta =k_l/m$, and $\gamma =k_n/m$ represent the unknown constant system parameters. The term $u\left(t\right)=-\ddot{p}\left(t\right)$ denotes the control input, corresponding to the base acceleration. The primary objectives of this study are defined as follows:

• Parameter Identification: To estimate the unknown parameters $\alpha$, $\beta$, and $\gamma$ in real time.
• State and Uncertainity Estimation: To design an observer capable of estimating the unmeasured velocity $\dot{q}\left(t\right)$ and lumped system uncertainties using only the position measurement $q\left(t\right)$.
• Trajectory Tracking: To design an observer-based control law $u\left(t\right)$ that ensures the system position $q\left(t\right)$ asymptotically tracks a desired trajectory $q_d\left(t\right)$ generated by a reference model.

3. Parameter Identification of the System

This section addresses the online identification of the unknown parameters of the Duffing–Holmes system: damping ($\alpha$), linear stiffness ($\beta$), and nonlinear stiffness ($\gamma$). To this end, we propose an approach based on state variable filtering and RLSM with a forgetting factor. The filtered formulation of the model transforms the nonlinear dynamic equation into a linear parametric form, thereby facilitating the estimation process using available measurements of beam and base displacements without requiring direct velocity measurements.

Applying the Laplace transform to (3) yields:

$$s^{2}Q\left(s\right)=-s\alpha Q\left(s\right)+\beta Q\left(s\right)-\gamma \mathcal{L}\left[q^3\left(t\right)\right]-s^{2}P\left(s\right),$$

(4)

where $\mathcal{L}$ denotes the Laplace operator, $Q\left(s\right)=\mathcal{L}\left[q\right(t\left)\right]$, and $P\left(s\right)=\mathcal{L}\left[p\right(t\left)\right]$. To estimate the unknown parameters $\alpha ,\beta ,$ and $\gamma$ utilizing the beam position measurement $q\left(t\right)$ and the base position $p\left(t\right)$, both sides of (4) are multiplied by a second-order low-pass filter $F\left(s\right)$:

$$F\left(s\right)={\displaystyle \frac{{\omega }_c^2}{{(s+{\omega }_c)}^2}},$$

(5)

where ${\omega }_c$ is the filter cut-off frequency. This operation results in:

$$s^{2}F\left(s\right)Q\left(s\right)=-\alpha sF\left(s\right)Q\left(s\right)+\beta F\left(s\right)Q\left(s\right)-\gamma F\left(s\right)\mathcal{L}\left[q^3\left(t\right)\right]-s^{2}F\left(s\right)P\left(s\right).$$

(6)

By defining the filtered variables $q_f\left(t\right)={\mathcal{L}}^{-1}\left[F\left(s\right)Q\left(s\right)\right]$ and a filtered nonlinear term ${\psi }_f\left(t\right)={\mathcal{L}}^{-1}\left[F\left(s\right)\mathcal{L}\left[q^3\left(t\right)\right]\right]$, (6) can be represented in the time domain as:

$${\ddot{q}}_f\left(t\right)=-\alpha {\dot{q}}_f\left(t\right)+\beta q_f\left(t\right)-\gamma {\psi }_f\left(t\right)-{\ddot{p}}_f\left(t\right).$$

(7)

Rearranging terms to isolate the known signals from the unknown parameters, we obtain the linear parametric model:

$$\chi \left(t\right)={\mathit{\varphi }}^{\mathrm{T}}\left(t\right)\mathit{\theta },$$

(8)

where the measured signal $\chi \left(t\right)$, regression vector $\mathit{\varphi }\left(t\right)$, and the parameter vector $\mathit{\theta }$ are defined as:

$$\chi \left(t\right)={\ddot{q}}_f\left(t\right)+{\ddot{p}}_f\left(t\right),{\mathit{\varphi }}^{\mathrm{T}}\left(t\right)=\left[-{\dot{q}}_f\left(t\right),q_f\left(t\right),-{\psi }_f\left(t\right)\right],\mathit{\theta }={\left[\alpha ,\beta ,\gamma \right]}^{\mathrm{T}}.$$

(9)

Since these signals are sampled with a sampling period $T_s$, the continuous model in (8) is discretized. Let k denote the discrete time step such that $t=k{T}_s$. The discrete parametric equation is given by:

$$\chi \left(k\right)={\mathit{\varphi }}^{\mathrm{T}}\left(k\right)\mathit{\theta }.$$

(10)

Finally, the estimation of the parameter vector $\widehat{\mathit{\theta }}\left(k\right)={[\widehat{\alpha }\left(k\right),\widehat{\beta }\left(k\right),\widehat{\gamma }\left(k\right)]}^{\mathrm{T}}$ is performed using the RLSM with a forgetting factor $\lambda$, formulated as follows:

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(k\right)& =\widehat{\mathit{\theta }}(k-1)+\mathbf{L}\left(k\right)\tilde{\chi }\left(k\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathbf{L}\left(k\right)& ={\displaystyle \frac{\mathbf{P}(k-1)\mathit{\varphi }\left(k\right)}{\lambda +{\mathit{\varphi }}^{\mathrm{T}}\left(k\right)\mathbf{P}(k-1)\mathit{\varphi }\left(k\right)}},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \mathbf{P}\left(k\right)& ={\displaystyle \frac{1}{\lambda }}\left[\mathbf{P}(k-1)-{\displaystyle \frac{\mathbf{P}(k-1)\mathit{\varphi }\left(k\right){\mathit{\varphi }}^{\mathrm{T}}\left(k\right)\mathbf{P}(k-1)}{\lambda +{\mathit{\varphi }}^{\mathrm{T}}\left(k\right)\mathbf{P}(k-1)\mathit{\varphi }\left(k\right)}}\right],\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \tilde{\chi }\left(k\right)& =\chi \left(k\right)-{\mathit{\varphi }}^{\mathrm{T}}\left(k\right)\widehat{\mathit{\theta }}(k-1),\hfill \end{array}$$

(14)

where $\mathbf{P}\left(k\right)$ is the covariance matrix, $\mathbf{L}\left(k\right)$ is the estimator gain vector, and $\tilde{\chi }\left(k\right)$ represents the a priori prediction error. The forgetting factor $\lambda$ ($0<\lambda \le 1$) determines the algorithm’s memory horizon; a value close to 1 retains information for longer periods, whereas a smaller value (e.g., $\lambda =0.98$) allows the algorithm to track time-varying parameters more rapidly by discarding older data.

4. Nonlinear Integral Extended State Observer

Using the identified parameters, this section proposes a new NIESO to estimate the unmeasured velocity states and external disturbances relying solely on position measurements. The observer employs a high-gain structure combined with an integral action on the system position, thereby enhancing its robustness against measurement noise and parametric uncertainties.

For that purpose, the Duffing–Holmes model in (3) can be rewritten in state-space form by defining the state variables $x_1\left(t\right)=q\left(t\right)$ and $x_2\left(t\right)=\dot{q}\left(t\right)$. This leads to:

$$\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =x_2\left(t\right)\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =\beta x_1\left(t\right)-\gamma x_1^3\left(t\right)-\alpha x_2\left(t\right)+\ddot{p}\left(t\right)\hfill \\ \hfill y\left(t\right)& =x_1\left(t\right)\hfill \end{array}$$

(15)

yielding the following equivalent representation:

$$\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)=& x_2\left(t\right)\hfill \\ \hfill {\dot{x}}_2\left(t\right)=& \widehat{\beta }{x}_1\left(t\right)-\widehat{\gamma }{x}_1^3\left(t\right)-\widehat{\alpha }{x}_2\left(t\right)+\tilde{\beta }{x}_1\left(t\right)-\tilde{\alpha }{x}_2\left(t\right)-\tilde{\gamma }{x}_1^3\left(t\right)+\ddot{p}\left(t\right)\hfill \end{array}$$

(16)

where $\tilde{\beta }=\beta -\widehat{\beta }$, $\tilde{\alpha }=\alpha -\widehat{\alpha }$, and $\tilde{\gamma }=\gamma -\widehat{\gamma }$ represent the bounded parameter estimation errors.

By introducing the following two new states:

$$\begin{array}{cc}\hfill x_0\left(t\right)& ={\int }_0^{t}y\left(\tau \right)\mathrm{d}\tau ,\hfill \\ \hfill x_3\left(t\right)& =\tilde{\beta }{x}_1\left(t\right)-\tilde{\alpha }{x}_2\left(t\right)-\tilde{\gamma }{x}_1^3\left(t\right),\hfill \end{array}$$

(17)

where the state $x_3\left(t\right)$ captures the structured uncertainty due to parameter estimation errors, the dynamics of the Duffing–Holmes system (15) can be expressed as the following extended system:

$$\begin{array}{cc}\hfill {\dot{x}}_0\left(t\right)& =x_1\left(t\right),\hfill \\ \hfill {\dot{x}}_1\left(t\right)& =x_2\left(t\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =\widehat{\beta }{x}_1\left(t\right)-\widehat{\alpha }{x}_2\left(t\right)-\widehat{\gamma }{x}_1^3\left(t\right)+x_3\left(t\right)+\ddot{p}\left(t\right),\hfill \\ \hfill {\dot{x}}_3\left(t\right)& =\dot{g}\left(t\right),\hfill \end{array}$$

(18)

where the derivative of the uncertainty term is given by

$$\dot{g}\left(t\right)=\tilde{\beta }{\dot{x}}_1\left(t\right)-\tilde{\alpha }{\dot{x}}_2\left(t\right)-3\tilde{\gamma }{x}_1^2\left(t\right)x_2\left(t\right),$$

(19)

which is bounded since the system parameters and the physical signals $x_1\left(t\right)$ and $x_2\left(t\right)$ are bounded.

The following NIESO is proposed to estimate the states of the system (18):

$$\begin{array}{cc}\hfill {\dot{\widehat{x}}}_0\left(t\right)& ={\widehat{x}}_1\left(t\right)+4{\omega }_o[x_0\left(t\right)-{\widehat{x}}_0\left(t\right)],\hfill \\ \hfill {\dot{\widehat{x}}}_1\left(t\right)& ={\widehat{x}}_2\left(t\right)+6{\omega }_o^2[x_0\left(t\right)-{\widehat{x}}_0\left(t\right)],\hfill \\ \hfill {\dot{\widehat{x}}}_2\left(t\right)& =\widehat{\beta }{\widehat{x}}_1\left(t\right)-\widehat{\alpha }{\widehat{x}}_2\left(t\right)-\widehat{\gamma }{\widehat{x}}_{1*}^3\left(t\right)+{\widehat{x}}_3\left(t\right)+4{\omega }_o^3[x_0\left(t\right)-{\widehat{x}}_0\left(t\right)]+\ddot{p}\left(t\right),\hfill \\ \hfill {\dot{\widehat{x}}}_3\left(t\right)& ={\omega }_o^4[x_0\left(t\right)-{\widehat{x}}_0\left(t\right)],\hfill \end{array}$$

(20)

where ${\omega }_o$ is the observer gain, which determines the observer’s bandwidth. Since the cubic stiffness nonlinearity $x_1^3$ is not globally Lipschitz, large estimation errors could destabilize the observer. To guarantee the boundedness of the observer’s nonlinear term and ensure stability, we define the state ${\widehat{x}}_{1*}\left(t\right)$ as a saturation projection of the estimated position ${\widehat{x}}_1\left(t\right)$:

$${\widehat{x}}_{1*}\left(t\right)=\mathrm{sat}\left\{{\widehat{x}}_1\left(t\right)\right\}=\left\{\begin{array}{cc}{x}_{1,min},\hfill & \mathrm{if}{\widehat{x}}_1\left(t\right)<x_{1,min}\hfill \\ {\widehat{x}}_1\left(t\right),\hfill & \mathrm{if}{x}_{1,min}\le {\widehat{x}}_1\left(t\right)\le x_{1,max}\hfill \\ x_{1,max},\hfill & \mathrm{if}{\widehat{x}}_1\left(t\right)>x_{1,max}\hfill \end{array}\right.$$

(21)

where $x_{1,min}$ and $x_{1,max}$ denote the physical limits of the beam displacement, which are $-0.09 \mathrm{m}$ and $0.09 \mathrm{m}$, respectively.

Defining the estimation errors as $e_i\left(t\right)=x_i\left(t\right)-{\widehat{x}}_i\left(t\right)$ for $i=0,\dots ,3$, leads to the following error dynamics:

$$\begin{array}{cc}\hfill {\dot{e}}_0\left(t\right)& =e_1\left(t\right)-4{\omega }_o{e}_0\left(t\right),\hfill \\ \hfill {\dot{e}}_1\left(t\right)& =e_2\left(t\right)-6{\omega }_o^2{e}_0\left(t\right),\hfill \\ \hfill {\dot{e}}_2\left(t\right)& =\widehat{\beta }{e}_1\left(t\right)-\widehat{\alpha }{e}_2\left(t\right)-\widehat{\gamma }\left[x_1^3\left(t\right)-{\widehat{x}}_{1*}^3\left(t\right)\right]+e_3\left(t\right)-4{\omega }_o^3{e}_0\left(t\right),\hfill \\ \hfill {\dot{e}}_3\left(t\right)& =-{\omega }_o^4{e}_0\left(t\right)+\dot{g}\left(t\right).\hfill \end{array}$$

(22)

The above dynamics can be represented in matrix form as:

$$\dot{\mathbf{e}}\left(t\right)=\mathbf{Ae}\left(t\right)+\mathbf{\Delta }\left(t\right),$$

(23)

where

$$\mathbf{A}=\left[\begin{array}{cccc}-4{\omega }_o& 1& 0& 0\\ -6{\omega }_o^2& 0& 1& 0\\ -4{\omega }_o^3& \widehat{\beta }& -\widehat{\alpha }& 1\\ -{\omega }_o^4& 0& 0& 0\end{array}\right],\mathbf{e}\left(t\right)=\left[\begin{array}{c}{e}_0\left(t\right)\\ e_1\left(t\right)\\ e_2\left(t\right)\\ e_3\left(t\right)\end{array}\right],\mathbf{\Delta }\left(t\right)=\left[\begin{array}{c}0\\ 0\\ -\widehat{\gamma }\left[x_1^3\left(t\right)-{\widehat{x}}_{1*}^3\left(t\right)\right]\\ \dot{g}\left(t\right)\end{array}\right].$$

It is worth noting that the disturbance term $\mathbf{\Delta }\left(t\right)$ in (22) is bounded because $x_1\left(t\right)$, ${\widehat{x}}_{1*}\left(t\right)$, and $\dot{g}\left(t\right)$ are all bounded signals. Therefore, there exists a constant $\delta >0$ such that:

$$\Vert \mathbf{\Delta }\left(t\right)\Vert \le \delta ,\forall t\ge 0.$$

(24)

The characteristic polynomial of the matrix $\mathbf{A}$ is given by:

$$\Lambda (s,{\omega }_o)=s^4+(4{\omega }_o+\widehat{\alpha })s^3+(6{\omega }_o^2+4\widehat{\alpha }{\omega }_o-\widehat{\beta })s^2+(4{\omega }_o^3+6\widehat{\alpha }{\omega }_o^2-4\widehat{\beta }{\omega }_o)s+{\omega }_o^4,$$

(25)

where the observer gain ${\omega }_o$ must be chosen sufficiently large such that $\mathbf{A}$ is Hurwitz. In practice, the tuning of ${\omega }_o$ involves a fundamental trade-off: it should be selected as large as possible to ensure that the estimated states rapidly converge to their true values, yet small enough to prevent the significant amplification of high-frequency measurement noise. A practical guideline for this system is to set ${\omega }_o$ between 20 and 30 times its estimated natural frequency, given by ${\widehat{\omega }}_n=\sqrt{\widehat{k_l/m}}=\sqrt{\widehat{\beta }}$. Let ${\lambda }_{max}\left(\mathbf{A}\right)$ denote the eigenvalue of $\mathbf{A}$ with the largest real part, and define $\rho =-\Re \left\{{\lambda }_{max}\left(\mathbf{A}\right)\right\}>0$. The following theorem establishes the stability of the observer.

Theorem 1.

The norm of the estimation error $\mathbf{e}\left(t\right)$ satisfies the following inequality:

$$\Vert \mathbf{e}\left(t\right)\Vert \le \mu e^{-\rho t}\left(\Vert \mathbf{e}\left(0\right)\Vert -{\displaystyle \frac{\delta }{\rho }}\right)+{\displaystyle \frac{\mu \delta }{\rho }},$$

(26)

which implies that $\mathbf{e}\left(t\right)$ is uniformly ultimately bounded, where μ is a positive constant. Furthermore, the asymptotic error bound $\mu \delta /\rho$ decreases as the observer gain ${\omega }_o$ increases.

Proof of Theorem 1.

The general solution of (23) is:

$$\mathbf{e}\left(t\right)=e^{\mathbf{A}t}\mathbf{e}\left(0\right)+{\int }_0^t{e}^{\mathbf{A}(t-\tau )}\mathbf{\Delta }\left(\tau \right)\mathrm{d}\tau .$$

(27)

where $e^{\mathbf{A}t}$ is the exponential matrix. Applying the norm and using the triangle inequality yields:

$$\Vert \mathbf{e}\left(t\right)\Vert \le \Vert e^{\mathbf{A}t}\Vert \Vert \mathbf{e}\left(0\right)\Vert +{\int }_0^t\Vert e^{\mathbf{A}(t-\tau )}\Vert \Vert \mathbf{\Delta }\left(\tau \right)\Vert \mathrm{d}\tau .$$

(28)

Since $\mathbf{A}$ is Hurwitz, the norm of the exponential matrix fulfills:

$$\Vert e^{\mathbf{A}t}\Vert \le \mu e^{-\rho t},\forall t\ge 0,$$

(29)

which gives

$$\begin{array}{cc}\hfill \Vert \mathbf{e}\left(t\right)\Vert & \le \mu e^{-\rho t}\Vert \mathbf{e}\left(0\right)\Vert +\mu \delta {\int }_0^t{e}^{-\rho (t-\tau )}\mathrm{d}\tau \hfill \\ \hfill & =\mu e^{-\rho t}\Vert \mathbf{e}\left(0\right)\Vert +\mu \delta e^{-\rho t}{\int }_0^t{e}^{\rho \tau }\mathrm{d}\tau \hfill \\ \hfill & =\mu e^{-\rho t}\Vert \mathbf{e}\left(0\right)\Vert +\mu \delta e^{-\rho t}\left[{\displaystyle \frac{e^{\rho t}-1}{\rho }}\right]\hfill \\ \hfill & =\mu e^{-\rho t}\Vert \mathbf{e}\left(0\right)\Vert +{\displaystyle \frac{\mu \delta }{\rho }}(1-e^{-\rho t}).\hfill \end{array}$$

(30)

Rearranging the terms, we obtain:

$$\Vert \mathbf{e}\left(t\right)\Vert \le \mu e^{-\rho t}\left(\Vert \mathbf{e}\left(0\right)\Vert -{\displaystyle \frac{\delta }{\rho }}\right)+{\displaystyle \frac{\mu \delta }{\rho }},$$

(31)

which concludes the proof that the estimation error $\mathbf{e}\left(t\right)$ is ultimately bounded. □

5. Observer-Based Controller Design

This section presents the design of a nonlinear backstepping control law for trajectory tracking. To this end, the external input $\ddot{p}\left(t\right)$ in the Duffing–Holmes system (15) is considered as the control input $u\left(t\right)$, resulting in the following state-space representation:

$$\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =x_2\left(t\right)\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =\beta x_1\left(t\right)-\gamma x_1^3\left(t\right)-\alpha x_2\left(t\right)+u\left(t\right).\hfill \end{array}$$

(32)

The proposed controller $u\left(t\right)$ utilizes the position, velocity, and uncertainty estimates provided by the NIESO derived in Section 4. The control objective is to design a control law $u\left(t\right)$ such that the estimated position ${\widehat{x}}_1\left(t\right)$ asymptotically tracks a desired smooth trajectory $x_d\left(t\right)$. This reference trajectory is generated by the following nonlinear reference model:

$${\ddot{x}}_d\left(t\right)+{\alpha }_d{\dot{x}}_d\left(t\right)-{\beta }_d{x}_d\left(t\right)+{\gamma }_d{x}_d^3\left(t\right)=\ddot{r}\left(t\right),$$

(33)

where ${\alpha }_d$, ${\beta }_d$, and ${\gamma }_d$ are design parameters chosen to define the desired dynamic behavior, and $\ddot{r}\left(t\right)$ represents the acceleration input.

The backstepping technique employs a recursive design procedure to ensure asymptotic tracking performance [29,30]. Let us define the tracking error coordinates as:

$$\begin{array}{cc}\hfill z_1\left(t\right)& ={\widehat{x}}_1\left(t\right)-x_d\left(t\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill z_2\left(t\right)& ={\widehat{x}}_2\left(t\right)-\nu \left(t\right),\hfill \end{array}$$

(35)

where $z_1\left(t\right)$ represents the position tracking error and $\nu \left(t\right)$ is a virtual control input to be designed. The time derivative of $z_1\left(t\right)$ is given by:

$$\begin{array}{cc}\hfill {\dot{z}}_1\left(t\right)& ={\dot{\widehat{x}}}_1\left(t\right)-{\dot{x}}_d\left(t\right)\hfill \\ \hfill & ={\widehat{x}}_2\left(t\right)+6{\omega }_o^2{e}_0\left(t\right)-{\dot{x}}_d\left(t\right)\hfill \\ \hfill & =z_2\left(t\right)+\nu \left(t\right)+6{\omega }_o^2{e}_0\left(t\right)-{\dot{x}}_d\left(t\right).\hfill \end{array}$$

(36)

By designing the virtual control law $\nu \left(t\right)$ as:

$$\nu \left(t\right)={\dot{x}}_d\left(t\right)-k_1{z}_1\left(t\right)-6{\omega }_o^2{e}_0\left(t\right),$$

(37)

where $k_1>0$ is a control gain, the dynamics of $z_1\left(t\right)$ in (36) stabilizes to:

$${\dot{z}}_1\left(t\right)=-k_1{z}_1\left(t\right)+z_2\left(t\right).$$

(38)

Subsequently, the dynamics of the velocity error coordinate $z_2\left(t\right)$ is derived as:

$$\begin{array}{cc}\hfill {\dot{z}}_2\left(t\right)& ={\dot{\widehat{x}}}_2\left(t\right)-\dot{\nu }\left(t\right)\hfill \\ \hfill & =\left[\widehat{\beta }{\widehat{x}}_1\left(t\right)-\widehat{\alpha }{\widehat{x}}_2\left(t\right)-\widehat{\gamma }{\widehat{x}}_{1*}^3\left(t\right)+{\widehat{x}}_3\left(t\right)+4{\omega }_o^3{e}_0\left(t\right)+u\left(t\right)\right]-\dot{\nu }\left(t\right),\hfill \end{array}$$

(39)

where the time derivative of the virtual control, $\dot{\nu }\left(t\right)$, is obtained from (37) and the error dynamics (22):

$$\begin{array}{cc}\hfill \dot{\nu }\left(t\right)& ={\ddot{x}}_d\left(t\right)-k_1{\dot{z}}_1\left(t\right)-6{\omega }_o^2{\dot{e}}_0\left(t\right)\hfill \\ \hfill & ={\ddot{x}}_d\left(t\right)-k_1{\dot{z}}_1\left(t\right)-6{\omega }_o^2\left[e_1\left(t\right)-4{\omega }_o{e}_0\left(t\right)\right]\hfill \\ \hfill & ={\ddot{x}}_d\left(t\right)-k_1{\dot{z}}_1\left(t\right)+24{\omega }_o^3{e}_0\left(t\right)-6{\omega }_o^2{e}_1\left(t\right).\hfill \end{array}$$

(40)

Finally, substituting the derived terms into (39), the actual control law $u\left(t\right)$ is synthesized as:

$$u\left(t\right)=-k_2{z}_2\left(t\right)-\widehat{\beta }{\widehat{x}}_1\left(t\right)+\widehat{\alpha }{\widehat{x}}_2\left(t\right)+\widehat{\gamma }{\widehat{x}}_{1*}^3\left(t\right)-{\widehat{x}}_3\left(t\right)-4{\omega }_o^3{e}_0\left(t\right)+\dot{\nu }\left(t\right),$$

(41)

where $k_2>0$ is the second control gain. Substituting (41) into (39) yields the closed-loop dynamics for $z_2\left(t\right)$:

$${\dot{z}}_2\left(t\right)=-k_2{z}_2\left(t\right).$$

(42)

The complete closed-loop error dynamics formed by (38) and (42) can be expressed in matrix form as:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]=\underset{{\mathbf{A}}_c}{\underbrace{\left[\begin{array}{cc}-k_1& 1\\ 0& -k_2\end{array}\right]}}\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right].$$

(43)

Since $k_1,k_2>0$, the matrix ${\mathbf{A}}_c$ is Hurwitz, ensuring that the tracking error states $z_1\left(t\right)$ and $z_2\left(t\right)$ converge asymptotically to zero. Consequently, ${\widehat{x}}_1\left(t\right)\to x_d\left(t\right)$ as $t\to \infty$. Analytically, the upper triangular structure of ${\mathbf{A}}_c$ sets the closed-loop eigenvalues directly to $-k_1$ and $-k_2$, dictating the theoretical convergence rate. In practice, these gains are experimentally tuned to balance rapid tracking performance against the physical saturation limits of the shake table actuator.

Regarding the actual position tracking, the total error can be expressed as $x_1\left(t\right)-x_d\left(t\right)=e_1\left(t\right)+z_1\left(t\right)$. Since the observer estimation error $e_1\left(t\right)$ is uniformly ultimately bounded (as proven in Theorem 1) and $z_1\left(t\right)$ converges to zero, the actual system position $x_1\left(t\right)$ is guaranteed to track the desired trajectory $x_d\left(t\right)$ with a bounded error. This residual error is directly proportional to the observer estimation error bound, which can be minimized by properly tuning the observer gain ${\omega }_o$.

6. Results and Discussion

This section presents the experimental validation of the proposed system identification, state estimation, and control strategies, demonstrating their practical efficacy. Additionally, simulation studies are included to further evaluate the control performance under ideal conditions.

6.1. Experimental Setup

Figure 3 illustrates the Duffing–Holmes prototype constructed based on the schematic presented in Figure 1. The elastic beam was fabricated from aluminum with a length of $30.5$ cm, a width of $3.14$ mm, and a thickness of $0.85$ mm. Since aluminum is non-ferromagnetic, two neodymium magnets were attached to the beam’s free end to enable magnetic interaction. Furthermore, two stationary neodymium magnets (diameter: 12 mm, thickness: 5 mm) were positioned symmetrically to induce the bistable potential field, creating stable equilibrium points located at approximately ±$3.3$ cm from the central axis.

The Duffing–Holmes prototype was securely mounted on a Quanser I-40 shake table (Quanser, Markham, ON, Canada) via a 6 mm thick acrylic interface. The shake table provides mechanical base excitation with a bandwidth up to 10 $\mathrm{Hz}$ and a stroke limit of $\pm 2\mathrm{cm}$. It is powered by a Quanser VoltPAQ-X1 linear voltage amplifier and interfaced through a Q2-USB data acquisition (DAQ) board. To monitor the base dynamics, the table is equipped with an accelerometer featuring a measurement range of $\pm 5$ g. The control algorithms were implemented in real time using MATLAB/Simulink, version R2017a, and the QUARC control software, version 2.6.2082. Additionally, safety limit switches were installed to protect the mechanism by deactivating the system upon reaching maximum displacement.

The beam displacement was measured using a Panasonic HG-C1400-P laser displacement sensor (Panasonic Industry, Osaka, Japan), which has a measurement range of 200 mm to 600 mm and a response time of $1.5$ ms. The sensor provides an analog output voltage ranging from 0 V to 5 V. To maximize the resolution of the Q2-USB DAQ board (12-bit resolution over a $\pm 10 \mathrm{V}$ range), the sensor output was conditioned using an AD620 instrumentation amplifier. This amplifier scales and offsets the signal to operate within a $\pm 7 \mathrm{V}$ range, thereby optimizing the dynamic range and ensuring high-resolution displacement measurements across the system’s operating range. Finally, all experimental validations were conducted with a sampling time of 2 ms.

6.2. Parameter Identification

The implementation of the proposed RLSM identification scheme is illustrated in Figure 4. The Quanser I-40 shake table utilizes a closed-loop Proportional–Derivative (PD) controller to regulate the table position $p\left(t\right)$ and track a desired trajectory $p_d\left(t\right)$ by minimizing the error $ϵ\left(t\right)=p_d\left(t\right)-p\left(t\right)$.

To ensure persistent excitation—a necessary condition for the convergence of the parameter estimates—a multi-tonal excitation signal was designed. Consequently, the shake table generated the following acceleration profile $\ddot{p}\left(t\right)$ to drive the Duffing–Holmes oscillator:

$$\ddot{p}\left(t\right)={\displaystyle \frac{d^2}{d{t}^2}}\left[0.005sin\left(6\pi t\right)+0.002sin\left(2\pi t\right)\right]\mathrm{m} {\mathrm{s}}^{-2}.$$

(44)

The identification algorithm employs the second-order low-pass filter $F\left(s\right)$, defined in (5), with a cut-off frequency of ${\omega }_c=20\mathrm{rad} {\mathrm{s}}^{-1}$. This filter processes the measured beam position $q\left(t\right)$ and the base position $p\left(t\right)$ to yield the filtered signals $q_f\left(t\right)$, ${\dot{q}}_f\left(t\right)$, ${\ddot{q}}_f\left(t\right)$, and ${\ddot{p}}_f\left(t\right)$. Subsequently, these signals are utilized to construct the observation scalar $\chi \left(t\right)$ and the regression vector $\mathit{\varphi }\left(t\right)$, as detailed in Section 3. Finally, the RLS algorithm, enhanced with a forgetting factor, recursively updates the estimated system parameter vector $\mathit{\theta }={\left[\alpha ,\beta ,\gamma \right]}^{\mathrm{T}}$.

Figure 5 depicts the time evolution of the parameter estimates $\widehat{\alpha }$, $\widehat{\beta }$, and $\widehat{\gamma }$. Following an initial transient phase, the estimates converge and settle around their average values after approximately 50 $\mathrm{s}$. This clear and stable convergence empirically verifies that the persistent excitation (PE) condition is effectively satisfied during the experimental identification process. The final identified values, listed below, were obtained by averaging the steady-state data over the time interval $t\in [100,300]\mathrm{s}$:

$$\begin{array}{cc}\hfill \widehat{\alpha }& =0.51 {\mathrm{s}}^{-1},\hfill \\ \hfill \widehat{\beta }& =352.2 {\mathrm{s}}^{-2},\hfill \\ \hfill \widehat{\gamma }& =5.15 \times {10}^5{\mathrm{m}}^{-2} {\mathrm{s}}^{-2}.\hfill \end{array}$$

(45)

6.3. State Estimation

The implementation of the NIESO, defined in (20), requires the selection of the observer gain ${\omega }_o$ and the utilization of the identified parameters $\widehat{\alpha },\widehat{\beta }$, and $\widehat{\gamma }$ listed in (45). By substituting these values into the characteristic polynomial (25), the specific stability polynomial is obtained:

$$\begin{array}{cc}\hfill \Lambda (s,{\omega }_o)=& s^4+(4{\omega }_o+0.51)s^3+(6{\omega }_o^2+2.04{\omega }_o-352.2)s^2\hfill \\ \hfill & +(4{\omega }_o^3+3.06{\omega }_o^2-1408.8{\omega }_o)s+{\omega }_o^4.\hfill \end{array}$$

(46)

Applying the Routh–Hurwitz stability criterion to this polynomial reveals that the observer gain must satisfy ${\omega }_o>20.5$ to guarantee stability. To analyze the transient response characteristics, the evolution of the closed-loop poles as a function of ${\omega }_o$ (root locus) is depicted in Figure 6. It can be observed that increasing ${\omega }_o$ shifts the poles further into the left half of the complex plane, thereby accelerating the error convergence. Consequently, the observer gain was set to ${\omega }_o=500$ to ensure fast convergence and small estimation error $\mathbf{e}\left(t\right)$.

To experimentally validate the observer, the Duffing–Holmes system was subjected to a harmonic base excitation designed to induce a dynamic response. The corresponding acceleration input $\ddot{p}\left(t\right)$ is defined as:

$$\ddot{p}\left(t\right)={\displaystyle \frac{d^2}{d{t}^2}}\left[0.0051sin\left(16\pi t\right)\right]\mathrm{m} {\mathrm{s}}^{-2}.$$

Figure 7 presents the experimental results, where Figure 7a compares the measured beam position $x_1\left(t\right)$ obtained via the laser sensor with the estimated position ${\widehat{x}}_1\left(t\right)$ provided by the NIESO. The results demonstrate that the observer converges to the actual state within approximately $0.2$ $\mathrm{s}$ and maintains accurate tracking thereafter. The estimates of the unmeasured velocity ${\widehat{x}}_2\left(t\right)$ and the lumped uncertainty ${\widehat{x}}_3\left(t\right)$ are shown in Figure 7b and Figure 7c, respectively.

Furthermore, to verify that the designed prototype exhibits nonlinear characteristics, the estimated states ${\widehat{x}}_1\left(t\right)$ and ${\widehat{x}}_2\left(t\right)$ were used to construct the Poincaré map shown in Figure 8. The resulting pattern confirms the chaotic behavior of the experimental prototype under the applied excitation.

6.4. Observer-Based Controller Performance

Following the experimental validation of the NIESO, the proposed observer-based backstepping controller (41) is implemented to evaluate the closed-loop tracking performance. The primary objective is to force the system position $x_1\left(t\right)$ to track a desired trajectory $x_d\left(t\right)$ generated by the reference model (33).

Prior to the experimental validation, comprehensive simulation studies were carried out to assess the performance of the proposed controller under three distinct dynamical regimes: interwell motion, intrawell motion, and chaotic motion. Because the performance of the proposed controller depends heavily on accurate parameter identification and reliable state estimation, these simulation studies provide a global validation of the complete control framework. Furthermore, experimental validation of the intrawell and chaotic regimes is practically constrained by inherent hardware limitations. These specific behaviors require acceleration control signals with significant low-frequency content, which can easily lead to actuator position drift in the physical shake table. Since practical implementation necessitates protective signal attenuation to safeguard the equipment against these low frequencies, simulations offer an unrestricted and safe platform to evaluate the controller’s true effectiveness across all regimes prior to physical deployment.

6.4.1. Simulation Results

For the simulation study, the reference model (33) was constructed using the identified parameters listed in (45). To generate the different desired trajectories (Interwell, Intrawell, and Chaotic), the reference model was excited with specific acceleration inputs $\ddot{r}\left(t\right)$ and initialized with different conditions.

Table 1. Configuration parameters for the reference model to generate distinct dynamic behaviors.

Parameter	Reference Model Behavior
	Interwell	Intrawell	Chaotic
Input $\ddot{r}\left(t\right)$ amplitude [$\mathrm{m} {\mathrm{s}}^{-2}$]	1.78	1.26	5.76
Input $\ddot{r}\left(t\right)$ frequency [$\mathrm{Hz}$]	3	4	5.4
Initial condition $x_d\left(0\right)$ [$\mathrm{m}$]	0.006	−0.04	0.05
Initial condition ${\dot{x}}_d\left(0\right)$ [$\mathrm{m} {\mathrm{s}}^{-1}$]	0	0	0

summarizes the configuration parameters used to generate these behaviors.

To assess the robustness of the proposed controller against parametric uncertainties, a parameter mismatch was introduced. The Duffing–Holmes system to be controlled was simulated using parameters varied by approximately $\pm 5\%$ with respect to the identified values used in the controller and reference model. The nominal parameters for the system are:

$$\begin{array}{cc}\hfill \alpha & =0.485 {\mathrm{s}}^{-1},\hfill \\ \hfill \beta & =370.7 {\mathrm{s}}^{-2},\hfill \\ \hfill \gamma & =4.9 \times {10}^5 {\mathrm{m}}^{-2} {\mathrm{s}}^{-2},\hfill \end{array}$$

(47)

with system initial conditions set to $x_1\left(0\right)=0.035 \mathrm{m}$ and $x_2\left(0\right)=0 \mathrm{m} {\mathrm{s}}^{-1}$.

Both the NIESO (20) and the control law (41) utilize the identified parameters (45). The observer gain was set to ${\omega }_o=500$, and the backstepping controller gains were tuned to $k_1=600$ and $k_2=250$. Furthermore, to replicate realistic physical constraints and prevent numerical divergence, the control input $u\left(t\right)$ was saturated at $\pm 20 \mathrm{m} {\mathrm{s}}^{-2}$.

Tracking Performance Under Interwell Motion

The reference model is configured according to the parameters listed in Table 1 to generate high-energy interwell oscillations. This motion is characterized by trajectories that repeatedly cross the potential barrier separating the system’s two stable equilibrium points.

Figure 9a depicts the time evolution of the reference trajectory $x_d\left(t\right)$ alongside the measured position $x_1\left(t\right)$ and its estimate ${\widehat{x}}_1\left(t\right)$. The results demonstrate a two-stage convergence process: first, the observer estimate ${\widehat{x}}_1\left(t\right)$ converges almost instantaneously to the actual position $x_1\left(t\right)$; second, the controller forces the actual position $x_1\left(t\right)$ to asymptotically track the desired reference $x_d\left(t\right)$, achieving synchronization within approximately $0.3$ $\mathrm{s}$.

Figure 9b and Figure 9c display the velocity tracking and uncertainty estimation performance, respectively. It is observed that the estimates ${\widehat{x}}_2\left(t\right)$ and ${\widehat{x}}_3\left(t\right)$ converge to their true values within $0.1$ $\mathrm{s}$, effectively reconstructing the unmeasured states even during the transient phase. As shown, all signals settle into a periodic steady state after approximately 15 $\mathrm{s}$. Finally, Figure 9d illustrates the control input $u\left(t\right)$. The controller initially saturates at its limit ($\pm 20 \mathrm{m} {\mathrm{s}}^{-2}$) to overcome the initial error and system inertia, before settling into a smooth periodic trajectory once the tracking error is minimized.

These results confirm that the proposed observer-based controller is capable of robustly driving the Duffing–Holmes system into the interwell motion, inducing and sustaining large-amplitude periodic oscillations. This capability is particularly critical for applications such as vibration energy harvesting, where high-energy orbit crossing is desired [16].

Tracking Performance Under Intrawell Motion

In this scenario, the controller performance is evaluated using an intrawell reference trajectory. Here, the system oscillations remain strictly confined within a single potential well around one of the two stable equilibrium points. This test case is particularly relevant for assessing the controller’s capacity to track small-amplitude motions accurately and to reject disturbances that might otherwise trigger unintended transitions into the neighboring potential well—a critical requirement for applications demanding stable and localized oscillatory behavior.

Figure 10 demonstrates the tracking results. The actual beam position $x_1\left(t\right)$ converges to the desired trajectory $x_d\left(t\right)$ within approximately $0.5$ $\mathrm{s}$. Following this convergence, both signals oscillate around the theoretical equilibrium point of the reference model, calculated as $-\sqrt{\widehat{\beta }/\widehat{\gamma }}\approx -0.0262 \mathrm{m}$.

Furthermore, the beam velocity $x_2\left(t\right)$ and the estimated uncertainty $x_3\left(t\right)$ exhibit periodic responses with amplitudes notably smaller than those observed in the interwell case. Interestingly, the control input $u\left(t\right)$ exhibits a larger peak amplitude during this specific intrawell scenario compared to the interwell case, see Figure 10d. This suggests that maintaining the system in an intrawell orbit might require a higher control effort to counteract the local stiffness and overcome the system’s inherent dynamics.

Tracking Performance Under Chaotic Motion

Finally, this subsection examines the tracking performance when the reference model generates a chaotic trajectory. Given the inherent aperiodicity and extreme sensitivity to initial conditions characteristic of chaotic dynamics, this scenario serves as a rigorous test of the proposed controller’s robustness and its ability to compensate for uncertainties while tracking a complex and aperiodic reference signal.

Figure 11 demonstrates that the beam position $x_1\left(t\right)$ successfully tracks the chaotic reference trajectory $x_d\left(t\right)$ with high precision. Similarly, the state variables $x_2\left(t\right)$ and $x_3\left(t\right)$ exhibit complex aperiodic behavior; nonetheless, their corresponding observer estimates ${\widehat{x}}_2\left(t\right)$ and ${\widehat{x}}_3\left(t\right)$ converge accurately to the true states, validating the NIESO’s performance.

Figure 11d indicates that the control input $u\left(t\right)$ exhibits a larger peak amplitude in this scenario compared to both the interwell and intrawell cases. This increased control effort is attributed to the broad frequency spectrum and unpredictability of the chaotic reference, which demands rapid and forceful control actions to maintain tracking performance. This occurs even though the steady-state displacement amplitude of $x_1\left(t\right)$ remains largest in the interwell case, highlighting that control energy cost is driven more by signal complexity and frequency content than by displacement magnitude alone.

6.4.2. Experimental Control Implementation

The experimental implementation of the proposed control scheme is illustrated in Figure 12. The Duffing–Holmes oscillator is excited by the base acceleration $\ddot{p}\left(t\right)$, which is generated by commanding the shake table to follow a desired position trajectory $p_d\left(t\right)$.

It is important to note that the control law $u\left(t\right)$, derived in (41), represents the required acceleration input. However, the internal low-level controller of the shake table requires a position reference signal $p_d\left(t\right)$. Theoretically, the acceleration signal $u\left(t\right)$ corresponds to the second derivative of the position; thus, $p_d\left(t\right)$ could be obtained by double integration:

$$p_d\left(t\right)={\int }_0^t{\int }_0^{{\tau }_1}u\left({\tau }_2\right)\mathrm{d}{\tau }_2\mathrm{d}{\tau }_1.$$

(48)

However, practical implementation of pure double integration is challenging since it amplifies low-frequency noise and DC offsets present in the signal $u\left(t\right)$. This accumulation of the low-frequency components lead to integral drift, causing the calculated position command $p_d\left(t\right)$ to grow unbounded and potentially exceed the physical stroke limits of the shake table. To mitigate this issue, a high-pass filtering integrator is employed instead of a pure double integrator, which attenuates the low-frequency components while preserving the motion dynamics in the frequency range of interest. By this way, the position command $p_d\left(t\right)$ remains bounded and centered within the workspace of the shake table. The transfer function drift-free integrator $F_d\left(s\right)$ is given by:

$$F_d\left(s\right)={\displaystyle \frac{P_d\left(s\right)}{U\left(s\right)}}={\displaystyle \frac{s}{s^2+10.77s+10.56}}.$$

(49)

Figure 13 compares the Bode magnitude and phase plots of an ideal integrator against the implemented drift-free integrator. Unlike the ideal integrator, which amplifies low-frequency noise, the drift-free filter $F_d\left(s\right)$ effectively attenuates components below its cut-off region. Specifically, for frequencies above $\omega \approx 10\mathrm{rad} {\mathrm{s}}^{-1}$, the magnitude response of $F_d\left(s\right)$ closely matches that of the ideal integrator, thereby yielding a realizable and bounded position command $p_d\left(t\right)$ for the shake table.

Experimental tests were conducted to evaluate the tracking performance of the proposed controller. For these tests, the observer and controller gains were set to the same values used in the simulation study: ${\omega }_o=500$, $k_1=600$, and $k_2=250$. The system was initialized at its stable equilibrium point, with $x_1\left(0\right)=0.033 \mathrm{m}$ and $x_2\left(0\right)=0 \mathrm{m} {\mathrm{s}}^{-1}$. The reference trajectory was generated using the following input acceleration, corresponding to the interwell motion profile:

$$\ddot{p}\left(t\right)=1.78sin\left(6\pi t\right)\mathrm{m} {\mathrm{s}}^{-2}.$$

(50)

It is worth noting that the excitation frequency ($\omega =6\pi \approx 18.85\mathrm{rad} {\mathrm{s}}^{-1}$) lies well within the operating bandwidth of the drift-free integrator (>$10\mathrm{rad} {\mathrm{s}}^{-1}$), ensuring accurate signal integration.

Finally, the effectiveness of the proposed controller was verified through a switching experiment involving both open-loop and closed-loop configurations. The test sequence was defined as follows:

• Open Loop (0–30 s and 120–150 s): The system was driven solely by the external excitation $\ddot{p}\left(t\right)$ from (50), mimicking the reference model’s input but without feedback control.
• Closed Loop (30–120 s): The proposed backstepping control law $u\left(t\right)$ from (41) was activated to force the system to track the reference trajectory.

Figure 14a presents the measured beam position $x_1\left(t\right)$ compared against the desired reference trajectory $x_d\left(t\right)$. Correspondingly, Figure 14b shows the control tracking error, defined as $\tilde{x}\left(t\right)=x_1\left(t\right)-x_d\left(t\right)$. It can be observed that in the open-loop phase, a significant phase lag exists between $x_1\left(t\right)$ and $x_d\left(t\right)$, which is attributed to the inevitable parametric mismatch between the physical system and the theoretical reference model. However, in the closed-loop phase, the proposed controller effectively compensates for these uncertainties. Furthermore, upon closing the control loop, the tracking error converges rapidly to a small neighborhood of the origin, demonstrating a significant reduction within approximately $0.2$ $\mathrm{s}$.

Quantitatively, the Root Mean Square (RMS) value of the tracking error, ${\tilde{x}}_1\left(t\right)$, in the open-loop configuration over the intervals 15–30 $\mathrm{s}$ and 130–150 $\mathrm{s}$ is $0.0224 \mathrm{m}$. In contrast, the closed-loop RMS error over the interval 50–85 $\mathrm{s}$ is reduced to $0.0061 \mathrm{m}$, representing approximately a $3.7$ times improvement in tracking accuracy compared to the open-loop case.

Furthermore, Figure 15 illustrates the control effort signals: the calculated acceleration $u\left(t\right)$ (Figure 15a), the resulting shake table position command $p_d\left(t\right)$ (Figure 15b), and the voltage applied to the shake table actuator (Figure 15c). A supplementary video of the experimental validation is also available in [31]. Both the quantitative data and the visual record clearly demonstrate that the actuator exhibits significantly larger displacements and requires a higher overall control effort during the open-loop phase. Consequently, the noticeable reduction in the applied voltage during closed-loop operation implies that the proposed controller is not only highly accurate but also energy-efficient, demanding considerably less power to sustain the desired interwell oscillations.

Finally, it is important to acknowledge a limitation in the experimental scope. Because the drift-free filter $F_d\left(s\right)$ attenuates low-frequency components to protect the hardware, experimental validation of the intrawell and chaotic regimes could not be carried out, as these behaviors inherently involve significant low-frequency content in the control signals. In future work, we plan to implement a control strategy based on a shake table driven by acceleration input rather than position input. This modification is expected to enable experimental validation across all dynamic regimes, consistent with the results demonstrated in the simulation section.

7. Conclusions

This work presented a comprehensive methodology for the identification, state estimation, and robust control of a bistable Duffing–Holmes oscillator. By combining a filtered model formulation with a Recursive Least Squares Method (RLSM), the system’s nonlinear parameters were accurately estimated despite the presence of noise. Building upon these identified parameters, a Nonlinear Integral Extended State Observer (NIESO) was proposed. Theoretical analysis and numerical validation confirmed that the NIESO effectively estimates unmeasured states and compensates for parametric uncertainties and external disturbances, ensuring the stability of the closed-loop system.

Simulation results demonstrated the versatility of the proposed backstepping controller in tracking diverse reference trajectories, including intrawell, interwell, and chaotic motions. A notable finding from the simulation analysis is that maintaining the system in a tight, high-frequency intrawell orbit required a higher control effort compared to the larger-amplitude interwell motion. This suggests that the controller must actively counteract the local stiffness and overcome the system’s inherent dynamics when operating away from its natural resonance, a critical consideration for energy-efficient control design in bistable systems.

Experimental validation on a shake table setup corroborated the effectiveness of the proposed scheme. The controller achieved a significant performance improvement, registering a 3.7-fold reduction in the RMS tracking error during interwell motion compared to open-loop operation. Furthermore, the experimental results revealed that the closed-loop system requires lower actuator voltage to sustain stable oscillations than the open-loop case, indicating that the proposed control strategy not only improves tracking accuracy but also enhances energy efficiency.

Finally, the practical implementation addressed the challenge of signal drift in the shake table through a drift-free integrator. While this solution enabled reliable long-term testing for interwell motion, its high-pass filtering characteristics limited the experimental validation of low-frequency intrawell and chaotic trajectories. Consequently, future work will focus on implementing a direct acceleration control interface to bypass the limitations of position-commanded tables, thereby allowing experimental verification across all dynamic regimes, including chaos.