**1. Introduction**

As an intelligent control strategy, iterative learning (IL) control has a simple structure and doesn't require accurate system modeling. According to past control experience, this method can improve the current control performance of the system by operating repetitively over a fixed time interval [1]. In 1978, IL control was first proposed by Uchiyama in Japanese [2], which did not receive much attention. After one critical report published by Arimoto in English [3], IL control made significant progress in both theories and applications [4,5]. Practically, IL control has been applied to a wide range of engineering applications, including flexible structures [6], nonholonomic mobile robots [7], flapping wing micro aerial vehicles [8] and the sheet metal forming process [9]. Considering the repeatability of the structural dynamic response in the vibration process, IL control is expected to provide a feasible solution for the control issue here. Several research groups have applied IL control methods to the active vibration control of piezoelectric smart structures. Zhu et al. [10] and Tavakolpour et al. [11] first applied P-type IL control for the attenuation of vibrations in piezoelectric smart structures for the design of the feedback gain, and the efficiency of P-type IL control was proven in their papers. In addition, Fadil et al. [12] proposed a new intelligent proportional-integral-derivative (PID) controller for vibration suppression by using P-type IL control and PID control, in which the P-type IL control was applied to tune the parameters of the PID controller.

In the studies above, although P-type IL controllers can effectively attenuate structural vibrations at some excitation frequencies, the performances of the controllers for vibration suppression are still not obvious when the piezoelectric smart structure is excited by its first natural frequency. Besides, the control effectiveness of the actuators is obvious at the locations of the sensors, and they are not able to effectively compensate for unwanted vibrations at other locations [13]. Moreover, thousands of iterations in P-type IL control are needed for achieving satisfactory control precision, leading to the slow learning speed [11,14]. In addition, the learning process should be accomplished within a limited period, as overlearning may lead to system instability [11]. Therefore, the iterative number should be limited to a predefined value. Finally, the unreasonable selections of learning gains may directly lead to control spillover or system instability [15].

For accelerating the convergence rate and improving the stability of the learning algorithm, adaptive control is needed to modify the learning process of the P-type IL control. Adaptive control has been successfully incorporated into various learning algorithms for the adjustment of parameters of the learning process [16,17]. Effective adaptive control strategies are necessary for the ability to automatically tune parameters to the desired performance at each sampling period, such as an internal model control method with an adaptive algorithm, implemented to reduce fatigue loads and tower vibrations in wind turbines [18]. A real-time control implementation, based on an auto-tuning finite impulse response filter, was applied to active vibration isolation [19]. An online method that tunes the poles of the controller was proposed to adapt to the errors between a real object and its model [20]. An adaptive voltage and frequency control method was proposed for inverter-based distributed generations in a multi-microgrid structure [21]. A characteristic model-based nonlinear golden section adaptive control method was presented for vibration suppression in a flexible Cartesian smart material robot [22].

As mentioned above, most of the adaptive control methods are model-based, in which the dynamic model of system has been already known before the design of the controller. However, for complex practical systems, the mechanism models of the plants are often difficult to establish, and the parameters are also hard to identify, making the design and application of controllers unpractical. Controlling vibrations in plate and shell structures always brings a challenge because of the complexity and density of the vibration modes. The strategy of using piezoelectric actuator-sensor pairs with discrete locations, glued on both surfaces of the plate, realizes a low weight and effective control for structural vibration [23]. The plate-integrated piezoelectric actuator-sensor pairs thus become a multi-input-multi-output (MIMO) system. If an actuator fails to perform as expected, the performance of its neighboring actuators will be negatively affected. In this system, the interaction among all actuators exists in the whole process of active vibration control, and this kind of interaction is always uncertain. The uncertainty caused by this interaction greatly presents a grea<sup>t</sup> challenge when designing a controller, and the model-based adaptive controller cannot deal with these conditions. Data-driven control methods, which are designed by directly using input and output (I/O) data of the system, can serve as an efficient alternative. The control problems caused by time-varying parameters and uncertainties of the model are challenging for model-based control, but not with data-driven control approaches [24].

Model free adaptive (MFA) control, as an effective data-driven control method, is an attractive technique which has gained a large amount of interest in recent years. It is easily implemented, with small computational burden for its simple structure and strong robustness. Unlike the neural-network-based adaptive control methods and model-based methods, no additional signal testing or training processes are required during the design of data-driven control methods. Instead of identifying the model of the plant, the MFA method builds an equivalent linearization of the data at each operation by introducing a novel concept named the pseudo-partial derivative (PPD), and the time-varying PPD can be estimated by merely using the I/O measurements of the plant.

In this paper, in order to accelerate the convergence speed of the feedback gain, the learning gain of the P-type IL control is designed by the MFA control. The MFA control can realize adaptive control

in parametric and structural manners. And it is also suitable for dealing with system uncertainties [25]. This advantage allows MFA control to be successfully employed in various engineering fields, such as use in the sensing and control of piezoelectrically actuated systems [26], blood pump control [27], multivariable industrial processes control [28], and robotic exoskeleton tracking control [29]. However, the convergence speed of the tracking error may be slow if only MFA control is used to adaptively adjust to the learning gain of the P-type IL control, and external noise in the system will increase the difficulty of vibration control. Various control strategies have been provided and continuously developed for the control of plants with unknown uncertainties and dynamic variations, such as sliding mode (SM) control [30], fuzzy logic control [31], neural networks [32], etc. For nonlinear, time-varying and uncertain systems, neural network approaches have an excellent approximation ability, and fuzzy logic control possesses remarkable robustness and adaptability, nevertheless, the tuning of numerous parameters and complex rules may decrease the efficiency and possibility of these methods [33]. Unlike neural networks and fuzzy logic control, SM control has a simple controller structure and can is easily implemented. Moreover, it has other attractive features, including good transient performance, robustness to parameter variations and insensitivity to disturbances [34]. With the aim of achieving a faster response and better robustness, the SM control was integrated with the MFA control to self-adjust the learning gain of the P-type control. Within the proposed method, the SM control is applied to estimate the parameters by tracking the time-varying PPD, such that the state variables can rapidly converge to the desired trajectory. Additionally, SM control can also be used to compensate for the impact of random disturbance, thereby enabling the system to enhance control effectiveness and maintain superior stability. In the P-type IL algorithm, two parts mainly affect the convergence speed of the feedback gain: The system output error and the learning gain. In the application of vibration suppression, the desired output signal is always zero. Therefore, the measured output signal is the main factor that decides the system output error. In this paper, multi-sensors are used to detect the structural deformation of a cantilever plate. Sensors at various locations generate different measured output signals, and the controllers connected to these sensors may have distinct learning speeds. It is unreasonable to define the same iterative number for all controllers as part of the stopping criteria. After obtaining multi-source information from the controlled plant, it is critical to discover the optimal method of fusing this information. In this paper, in order to solve the multicriteria and multiobjective problems in practical applications, evidence theory was adopted to design the stopping criteria. The evidence theory does not require prior knowledge and has outstanding performance for handling uncertain or inexact information, which makes it an indispensable tool for state diagnosis and defect inspection [35,36]. Applying the combination rules, the evidence theory can carry out reasoning, data fusion or decision making [37]. By using information fusion technology, the learning processes of all controllers can be diagnosed in real-time by the real-time feedback gains obtained from the controllers. On this basis, the stopping criteria were designed for overlearning diagnosis of the robust MFA-IL algorithm.

The finite element (FE) method is a widely accepted and powerful tool to deal with piezoelectric smart structures. Some kinds of efficient and accurate electromechanically coupled dynamic FEs of smart structures have already been developed [38–40]. Among commercial FE analysis codes, ANSYS has the ability to model smart structures with piezoelectric materials, and H Karagülle et al. [41] successfully integrated vibration control actions into ANSYS modeling, where the solution was achieved as well. In this paper, ANSYS parametric design language (APDL) is used to integrate the control law into the ANSYS FE model to perform closed-loop simulations.

In this paper, using the complementary features of P-type IL control, MFA control and SM control, a robust MFA-IL control strategy was developed for the suppression of vibrations in smart structures. Due to its ability to cope with uncertainties in the learning control process, MFA control was applied to adaptively adjust the learning gain of the P-type IL control. By inserting the SM control term into the MFA control, the learning gain can be designed properly and the convergence rate of the tracking error and the robustness of the closed-loop system can be improved. A multi-source information fusion diagnosis method for the overlearning evaluation is presented based on the evidence theory, and the stopping criteria are also be designed. The proposed control method was numerically and experimentally investigated for a clamped plate under various external disturbances, and the results are illustrated and extensively discussed at the end of the present work.

The rest of this paper is organized as follows. In Section 2, based on the FE model of piezoelectric smart structures, the state space model of the equivalent linear system is developed for the purpose of control law design. The P-type IL control is employed for establishing the vibration control equations. Section 3 describes the dynamic transformation and linearization for the vibration control system. Section 4 describes the design of the robust MFA-IL control scheme. Theoretical basis of state diagnosis, based on evidence theory and the design of the stopping criteria, is introduced in Section 5. In Section 6, numerical examples are presented to demonstrate the validity of the proposed method. In Section 7, a complete active vibration control system is set up to conduct an experimental investigation. The conclusions and outlooks are drawn in Section 8.

#### **2. FE Model and P-type IL Control**

The linear electro-mechanically coupled dynamic FE equations of the piezoelectric smart structure can be written as [23]:

$$
\begin{bmatrix} \mathbf{M\_{uu}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{Bmatrix} \mathbf{u} \\ \mathbf{0} \end{Bmatrix} + \begin{bmatrix} \mathbf{C\_{uu}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{Bmatrix} \dot{\mathbf{u}} \\ \dot{\boldsymbol{\Phi}} \end{Bmatrix} + \begin{bmatrix} \mathbf{K\_{uu}} & \mathbf{K\_{u\Phi}} \\ \mathbf{K\_{\phi u}} & \mathbf{K\_{\phi\Phi}} \end{bmatrix} \begin{Bmatrix} \mathbf{u} \\ \boldsymbol{\Phi} \end{Bmatrix} = \begin{Bmatrix} \mathbf{F\_{u}} \\ \mathbf{F\_{\Phi}} \end{Bmatrix} \tag{1}
$$

where *u* and *φ* are the vectors of nodal displacements and electric potentials; *Muu*, *Cuu*, *Kuu*, *Kuφ*(*Kφu*) and *<sup>K</sup>φφ* are the structural mass matrix, the damping matrix, the mechanical stiffness matrix, the piezoelectric coupling matrix and the dielectric stiffness matrix, respectively; *Fu* and *<sup>F</sup>φ* are the vectors of mechanical force and electric load, respectively.

The damping matrix *Cuu* is usually defined as a linear combination of the structural mass matrix *Muu* and the mechanical stiffness matrix *Kuu*, which is written as follows:

$$\mathbf{C}\_{\text{uu}} = \alpha \mathbf{M}\_{\text{uu}} + \beta \mathbf{K}\_{\text{uu}} \tag{2}$$

where the constants *α* and *β* are the Rayleigh damping coefficients.

The piezoelectric sensor generates output electric potential as long as the structure is oscillating. The system (1) can be uncoupled into the following independent equations for the sensor output electric potential:

$$\boldsymbol{\phi} = \boldsymbol{K}\_{\boldsymbol{\phi}\boldsymbol{\phi}}^{-1} (\mathbf{F}\_{\boldsymbol{\phi}} - \mathbf{K}\_{\boldsymbol{\phi}\boldsymbol{u}} \boldsymbol{u}) \tag{3}$$

and the structural displacement

$$M\_{uu}\ddot{u} + \mathcal{C}\_{uu}\dot{u} + \mathcal{K}^\*u = F\_u - \mathcal{K}\_{u\phi}\mathcal{K}^{-1}\_{\phi\phi}F\_\phi\tag{4}$$

where *K*∗ = *Kuu* − *KuφK*−<sup>1</sup> *φφ <sup>K</sup>φ<sup>u</sup>*.

Note that *<sup>F</sup>φ* is usually zero in the sensor, thus, Equation (3) can be rewritten as *φ* = − *K*−<sup>1</sup> *φφ <sup>K</sup>φuu*. Then, the rate of change of the sensor output electric potential can be written as:

$$
\dot{\phi} = -\mathcal{K}\_{\phi\phi}^{-1} \mathcal{K}\_{\phi u} \dot{u} \tag{5}
$$

The system output error is defined as:

$$e = y\_d - y\tag{6}$$

where *yd*is the desired output signal and *y* is measured output signal.

In the application of vibration suppression, we require the desired output signal to be zero. The measured output signal in this paper is the rate of change of the sensor output electric potential, namely, *y* = . *φ*. As a discrete-time system, the output error at the *k*th moment can be given as:

$$
\epsilon(k) = \mathbf{0} - \dot{\Phi}(k) = \mathbf{K}\_{\Phi\Phi}^{-1} \mathbf{K}\_{\Phi u} \dot{u}(k) \tag{7}
$$

According to the update rule of the P-type IL control [42], the feedback gain at the *k*th moment can be given as:

$$\mathbf{G}(k) = \mathbf{G}(k-1) + \Phi \mathbf{e}(k-1) \tag{8}$$

where **Φ** and *G*(*k*) are the learning gain matrix and feedback gain matrix, respectively.

The feedback gain *G*(*k*) is stored in memory at the (*k* − 1)th moment and applied for the next iteration when the system operates. The input voltage of the actuator is expressed as:

$$V\_4 = -G\dot{\phi}\tag{9}$$

The electric load vector is defined as:

$$F\_{\Phi} = \mathbb{C}\_{a} V\_{a} \tag{10}$$

where *Ca* is the capacitance constant of the piezoelectric actuator.

The control force can be defined as *Fa* = −*KuφK*−<sup>1</sup> *φφF<sup>φ</sup>* and by combining with Equations (5), (9) and (10), *Fa* can be rewritten as:

$$F\_4 = -\mathcal{K}\_{u\Phi} \mathcal{K}\_{\Phi\phi}^{-1} \mathcal{C}\_{u} \mathcal{G} \mathcal{K}\_{\phi\Phi}^{-1} \mathcal{K}\_{\phi u} \dot{u} \tag{11}$$

The control force generated by actuator is used to suppress structural vibration. Substituting Equations (5), (9) and (10) into Equation (4), the vibration control equation of the piezoelectric smart structure can be expressed as:

$$\mathbf{M}\_{\rm uu}\ddot{u} + (\mathbf{C}\_{\rm uu} + \mathbf{K}\_{\rm u\phi}\mathbf{K}\_{\boldsymbol{\phi}\boldsymbol{\phi}\boldsymbol{\phi}}^{-1}\mathbf{C}\_{\boldsymbol{u}}\mathbf{G}\mathbf{K}\_{\boldsymbol{\phi}\boldsymbol{\phi}\boldsymbol{\phi}}^{-1}\mathbf{K}\_{\boldsymbol{\phi}\boldsymbol{u}})\dot{u} + \mathbf{K}^{\*}u = \mathbf{F}\_{\boldsymbol{u}}\tag{12}$$

#### **3. Dynamic Transformation and Linearization of Vibration Control Equations**

As a discrete-time system, the system (12) can be approximated by the following form at the *k*th moment:

$$\mathbf{M}\_{\mathbf{m}}\ddot{\boldsymbol{u}}(k) + [\mathbf{C}\_{\mathbf{u}\mathbf{u}} + \mathbf{K}\_{\mathbf{u}\boldsymbol{\Phi}}\mathbf{K}\_{\boldsymbol{\Phi}\boldsymbol{\Phi}}^{-1}\mathbf{C}\_{\mathbf{u}}\mathbf{G}(k)\mathbf{K}\_{\boldsymbol{\Phi}\boldsymbol{\Phi}}^{-1}\mathbf{K}\_{\boldsymbol{\Phi}\mathbf{u}}]\dot{\boldsymbol{u}}(k) + \mathbf{K}^{\*}\mathbf{u}(k) = \mathbf{F}\_{\mathbf{u}}(k) \tag{13}$$

Again, using Equations (5) and (7), the system (13) can be rewritten as:

$$\dot{y}(k) = -(K\_{\text{M}}^{\*}\mathbb{C}\_{\text{uu}}K\_{\text{\textquotedblleft}\text{u}}^{-1}\mathbb{K}\_{\text{\textquotedblleft}\text{\textquotedblright}\text{\textquotedblright}\text{\textquotedblright}})y(k) - [\mathbb{K}\_{\text{M}}^{\*}\mathbb{K}\_{\text{C}}^{\*}\text{G}(k)]y(k) + \mathbb{K}\_{\text{M}}^{\*}\mathbb{F}^{\*}(k) \tag{14}$$

where *<sup>K</sup>*<sup>∗</sup>*M* = *K*−<sup>1</sup> *φφKφuM*−<sup>1</sup> *uu* , *<sup>K</sup>*<sup>∗</sup>*C* = *KuφK*−<sup>1</sup> *φφCa*, *F*∗(*k*) = *<sup>K</sup>*<sup>∗</sup>*u*(*k*) − *Fu*(*k*).

Similar to Equation (8), a time-varying version for P-type IL updating rule is given as [42]:

$$\mathbf{G}(k) = \mathbf{G}(k-1) + \Phi(k-1)\mathbf{e}(k-1) \tag{15}$$

where **Φ**(*k* − 1) is the learning gain matrix, which is now time-varying.

Because .*y*(*k*) = *y*(*k*+<sup>1</sup>)−*y*(*k*) *T* for the sample period *T*, substituting Equation (15) into Equation (14), the discrete-time form of the system (14) can be given as:

$$y(k+1) = -T[\mathbf{K}\_{\mathbf{M}}^{\*}\mathbf{C}\_{\mathbf{M}}\mathbf{K}\_{\mathbf{\Phi}\mathbf{\Phi}}^{-1}\mathbf{K}\_{\mathbf{\Phi}\mathbf{\Phi}} + \mathbf{K}\_{\mathbf{M}}^{\*}\mathbf{K}\_{\mathbf{\mathcal{C}}}^{\*}G(k-1) + \mathbf{K}\_{\mathbf{M}}^{\*}\mathbf{K}\_{\mathbf{\mathcal{C}}}^{\*}\mathbf{\Phi}(k-1)e(k-1) - \frac{1}{T}[y(k) + T\mathbf{K}\_{\mathbf{M}}^{\*}F^{\*}(k)] \tag{16}$$

where **Φ**(*k* − 1) and *y*(*k*) are the system input and the system output, respectively.

It can be known from Equation (16) that the partial derivatives of *y*(*k* + <sup>1</sup>), with respect to output *y*(*k*) and input **Φ**(*k* − <sup>1</sup>), are continuous and the system is generalized *Lipschitz*. For system (16), with [<sup>Δ</sup>*y*(*k*), Δ**Φ**(*k* − 1)]*<sup>T</sup>* = 0 for each fixed *k*, there must exist **<sup>Ψ</sup>**(*k*), named the PPD matrix, such that Equation (16) can be transformed into the following equivalent full form dynamic linearization model:

$$
\Delta y(k+1) = \mathbf{\bar{Y}}(k)[\Delta y(k), \Delta \Phi(k-1)]^T \tag{17}
$$

where **<sup>Ψ</sup>**(*k*)=[*ϕ*1(*k*),*ϕ*2(*k*)], *ϕ*1(*k*), *ϕ*2(*k*) ∈ *RN*×*<sup>N</sup>* and **Ψ**(*k*) < *b*, *b* is a positive constant.

**Proof.** From Equation (16) we have

<sup>Δ</sup>*y*(*k* + 1) = *y*(*k* + 1) − *y*(*k*) = {−*T*[*K*<sup>∗</sup>*MCuuK*−<sup>1</sup> *<sup>φ</sup>uKφφ* + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C<sup>G</sup>*(*<sup>k</sup>* − 1) + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C***<sup>Φ</sup>**(*<sup>k</sup>* − <sup>1</sup>)*e*(*k* − 1) − 1*T* ]*y*(*k*) + *<sup>T</sup>K*<sup>∗</sup>*MF*<sup>∗</sup>(*k*)} −{−*T*[*K*<sup>∗</sup>*MCuuK*−<sup>1</sup> *<sup>φ</sup>uKφφ* + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C<sup>G</sup>*(*<sup>k</sup>* − 1) + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C***<sup>Φ</sup>**(*<sup>k</sup>* − <sup>2</sup>)*e*(*k* − 1) − 1*T* ]*y*(*k* − 1) + *<sup>T</sup>K*<sup>∗</sup>*MF*<sup>∗</sup>(*k*)} <sup>+</sup>{−*T*[*K*<sup>∗</sup>*MCuuK*−<sup>1</sup> *<sup>φ</sup>uKφφ* + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C<sup>G</sup>*(*<sup>k</sup>* − 1) + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C***<sup>Φ</sup>**(*<sup>k</sup>* − <sup>2</sup>)*e*(*k* − 1) − 1*T* ]*y*(*k* − 1) + *<sup>T</sup>K*<sup>∗</sup>*MF*<sup>∗</sup>(*k*)} <sup>+</sup>{−*T*[*K*<sup>∗</sup>*MCuuK*−<sup>1</sup> *<sup>φ</sup>uKφφ* + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C<sup>G</sup>*(*<sup>k</sup>* − 2) + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C***<sup>Φ</sup>**(*<sup>k</sup>* − <sup>1</sup>)*e*(*k* − 2) − 1*T* ]*y*(*k* − 1) + *<sup>T</sup>K*<sup>∗</sup>*MF*<sup>∗</sup>(*<sup>k</sup>* − 1)} −{−*T*[*K*<sup>∗</sup>*MCuuK*−<sup>1</sup> *<sup>φ</sup>uKφφ* + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C<sup>G</sup>*(*<sup>k</sup>* − 2) + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C***<sup>Φ</sup>**(*<sup>k</sup>* − <sup>2</sup>)*e*(*k* − 2) − 1*T* ]*y*(*k* − 1) + *<sup>T</sup>K*<sup>∗</sup>*MF*<sup>∗</sup>(*<sup>k</sup>* − 1)} −{−*T*[*K*<sup>∗</sup>*MCuuK*−<sup>1</sup> *<sup>φ</sup>uKφφ* + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C<sup>G</sup>*(*<sup>k</sup>* − 2) + *<sup>K</sup>*<sup>∗</sup>*MK*<sup>∗</sup>*C***<sup>Φ</sup>**(*<sup>k</sup>* − <sup>1</sup>)*e*(*k* − 2) − 1*T* ]*y*(*k* − 1) + *<sup>T</sup>K*<sup>∗</sup>*MF*<sup>∗</sup>(*<sup>k</sup>* − 1)} (18)

$$\begin{aligned} \theta(k) & \overset{\scriptstyle \Delta}{=} \left\{-T[\mathbf{K}\_{\text{M}}^{\*}\mathbf{C}\_{\text{uu}}\mathbf{K}\_{\text{\textquotedbl}}^{-1}\mathbf{K}\_{\text{\textquotedbl}} + \mathbf{K}\_{\text{M}}^{\*}\mathbf{K}\_{\text{\textquotedbl}}^{\*}\mathbf{G}(k-1) + \mathbf{K}\_{\text{M}}^{\*}\mathbf{K}\_{\text{\textquotedbl}}^{\*}\Phi(k-2)\mathbf{e}(k-1) - \frac{1}{T}\right\} \mathbf{y}(k-1) + T\mathbf{K}\_{\text{M}}^{\*}\mathbf{F}^{\*}(k)\right\} \\ & - \left\{-T[\mathbf{K}\_{\text{M}}^{\*}\mathbf{C}\_{\text{uu}}\mathbf{K}\_{\text{\textquotedbl}}^{-1}\mathbf{K}\_{\text{\textquotedbl}} + \mathbf{K}\_{\text{M}}^{\*}\mathbf{K}\_{\text{\textquotedbl}}^{\*}\mathbf{G}(k-2) + \mathbf{K}\_{\text{M}}^{\*}\mathbf{K}\_{\text{\textquotedbl}}^{\*}\Phi(k-1)\mathbf{e}(k-2) - \frac{1}{T}\right\} \mathbf{y}(k-1) + T\mathbf{K}\_{\text{M}}^{\*}\mathbf{F}^{\*}(k-1) \end{aligned}$$

By virtue of the Cauchy differential mean value theorem, Equation (18) can be rewritten as

$$
\Delta y(k+1) = \frac{\partial y(k+1)}{\partial y(k)} \Delta y(k) + \frac{\partial y(k)}{\partial \Phi(k-2)} \Delta \Phi(k-1) + \theta(k) \tag{19}
$$

where *∂y*(*k*+<sup>1</sup>) *∂y*(*k*) is the partial derivative value of *y*(*k* + <sup>1</sup>), with respect to output *y*(*k*), and *∂y*(*k*) *∂***Φ**(*k*−<sup>2</sup>) represents the partial derivative value of *y*(*k*), with respect to input **Φ**(*k* − <sup>2</sup>).

For each fixed *k*, we consider the following equation with the numerical matrix *H*(*k*) ∈ *RN*×*N*.

$$\theta(k) = H(k)[\Delta y(k), \Delta \Phi(k-1)]^T \tag{20}$$

Since condition [<sup>Δ</sup>*y*(*k*), Δ**Φ**(*k* − 1)]*<sup>T</sup>* = 0, Equation (20) must have at least one solution *<sup>H</sup>*<sup>∗</sup>(*k*). In fact, it must have an infinite number of solutions for each *k*.

Let

$$\Psi(k) = H^\*(k) + [\frac{\partial y(k+1)}{\partial y(k)}, \frac{\partial y(k)}{\partial \Phi(k-2)}],\tag{21}$$

Then, we have Equation (17).

Based on Equation (17), the system (16) can be rewritten in the following dynamic linearization form:

$$y(k+1) = \mathfrak{p}\_1(k)\Delta y(k) + \mathfrak{p}\_2(k)\Delta \Phi(k-1) + y(k)\tag{22}$$

where the values of *ϕ*1(*k*) and *ϕ*2(*k*) are dynamic changed. 

## **4. Controller Design**

## *4.1. MFA Control*

Consider the following control input criterion function:

$$J(\Phi(k-1)) = \left\| y\_d(k+1) - y(k+1) \right\|^2 + \gamma \left\| \Phi(k-1) - \Phi(k-2) \right\|^2 \tag{23}$$

where *<sup>y</sup>d*(*<sup>k</sup>* + 1) is the desired output signal and *γ* > 0 is a weighting constant. **Φ**(*k* − 1) is the MFA control rate.

Substituting Equation (22) into Equation (23), then differentiating Equation (23) with respect to **Φ**(*k* − <sup>1</sup>), and letting it be equal zero, gives the following:

$$
\Delta\Phi(k-1) = \left(\mathfrak{q}\_2(k)^T\mathfrak{q}\_2(k) + \gamma I\right)^{-1}\mathfrak{q}\_2(k)^T \times \left[y\_d(k+1) - y(k) - \mathfrak{q}\_1(k)\Delta y(k)\right] \tag{24}
$$

Equation (24) includes the calculation of the inverse matrix, which may cause computational burden once the I/O matrixes of the system are of a high dimension. The simplified form of Equation (24) can be expressed as follows:

$$\Phi\_{\rm MEA}(k-1) = \Phi\_{\rm MEA}(k-2) + \frac{\rho \mathfrak{g}\_2(k)^T \left[\mathfrak{y}\_d(k+1) - \mathfrak{y}(k) - \mathfrak{g}\_1(k)\Delta \mathfrak{y}(k)\right]}{\left\|\mathfrak{e}\_2(k)\right\|^2 + \gamma} \tag{25}$$

where *ρ* ∈ (0, 1] is a step-size constant, which is added to make Equation (25) general.

In this paper, a modified projection algorithm is used to estimate the unknown PPD matrix:

$$f(\mathbf{Y}(k)) = \left\| \Delta \mathbf{y}(k) - \mathbf{Y}(k) [\Delta \mathbf{y}(k-1), \,\Delta \Phi(k-2)]^T \right\|^2 + \mu \left\| \mathbf{Y}(k) - \mathbf{Y}(k-1) \right\|^2 \tag{26}$$

where **Ψ**(*k*) = **Ψ**ˆ (*k* − 1) + <sup>Δ</sup>**Ψ**(*k*)=[*ϕ*<sup>ˆ</sup> 1(*k* − <sup>1</sup>),*ϕ*<sup>ˆ</sup> 2(*k* − 1)] + [<sup>Δ</sup>*ϕ*1(*k*), <sup>Δ</sup>*ϕ*2(*k*)], *μ* > 0 is a weighting constant and **Ψ** ˆ (*k* − 1) is an estimated value of **Ψ**(*k* − <sup>1</sup>).

Here, we differentiate Equation (26) with respect to **<sup>Ψ</sup>**(*k*), and letting it be equal to zero. According to the simplified form in Equation (25), we can obtain the parameters of the estimation algorithm as follows:

$$\begin{split} \Phi\_{1}(k) &= \Phi\_{1}(k-1) + \frac{\eta \left[\Delta y(k) - (\Phi\_{1}(k-1)\,\mu\_{2}(k-1))(\Delta y(k-1),\Delta\Phi(k-2))^{\top}\right] \Delta y(k-1)^{\top}}{\mu + \left\|\Delta y(k-1)\right\|^{2} + \left\|\Delta\Phi(k-2)\right\|^{2}} \\ \Phi\_{2}(k) &= \Phi\_{2}(k-1) + \frac{\eta \left[\Delta y(k) - (\Phi\_{1}(k-1)\,\mu\_{2}(k-1))(\Delta y(k-1),\Delta\Phi(k-2))^{\top}\right] \Delta\Phi(k-2)^{\top}}{\mu + \left\|\Delta y(k-1)\right\|^{2} + \left\|\Delta\Phi(k-2)\right\|^{2}} \end{split} \tag{27}$$

where *η* ∈ (0, 1] is a step-size constant, which is added to make Equation (27) general.

#### *4.2. Robust MFA-IL Control*

The discrete-time SM is used to compensate the external disturbances and guarantee the fast convergence of feedback gain, which can increase the system robustness and control performance. By assuming the discrete sliding surface is

$$\mathfrak{s}(k) = \mathfrak{e}(k) \tag{28}$$

and combining Equations (6) and (22), the above equation can be rewritten as follows:

$$\mathfrak{s}(k+1) = \mathfrak{e}(k+1) = y\_d(k+1) - \mathfrak{q}\_1(k)\Delta y(k) - \mathfrak{q}\mathfrak{z}(k)\Delta \Phi(k-1) - \mathfrak{y}(k) \tag{29}$$

The sliding reaching law is defined as follows [43]:

$$\begin{cases} \mathbf{s}(k+1) = (1 - qT)\mathbf{s}(k) - \varepsilon T \text{fal}(\mathbf{s}(k), \sigma, \delta) \\\ \delta > \left(\frac{\varepsilon T}{1 - qT}\right)^{\frac{1}{1 - qT}}, \qquad 0 < \frac{\varepsilon T}{1 - qT} < 1 \end{cases} \tag{30}$$

where fal(*s*(*k*), *α*, *δ*) = |*s*(*k*)|*<sup>σ</sup>*sign(*s*(*k*)), |*s*(*k*)| ≥ *δ s*(*k*) *δ*1−*<sup>σ</sup>* , |*s*(*k*)| < *δ* , 0 < *σ* < 1, 0 < *δ* < 1, *ε* > 0, *q* > 0, 1 − *qT* > 0, *T* is the sample period.

Substituting Equation (30) into Equation (29), gives the following:

$$\Delta\Phi(k-1) = \mathfrak{sp}\_2(k)^{-1} \left[ \mathfrak{y}\_d(k+1) - \mathfrak{sp}\_1(k)\Delta\mathfrak{y}(k) - \mathfrak{y}(k) - (1-qT)\mathfrak{s}(k) + \varepsilon T \text{fal}(\mathfrak{s}(k), \mathfrak{e}, \delta) \right] \tag{31}$$

Because the sliding mode reaching law is based on the transformed dynamic linearization, let **<sup>Φ</sup>**SM(*k* − 1) = Δ **Φ**(*k* − <sup>1</sup>). Then, the final learning gain of the IL controller is equal to Equation (25) plus Equation (31), written as follows:

$$
\Phi(k-1) = \Phi\_{\text{MFA}}(k-1) + \Gamma \Phi\_{\text{SM}}(k-1) \tag{32}
$$

where Γ is a weighting factor which is added to make Equation (32) general. **<sup>Φ</sup>**SM(*k* − 1) is used to compensate for the input disturbance and increase the convergence rate.

For convenience, a block diagram of the robust MFA-IL control approach is presented in Figure 1.

**Figure 1.** Block diagram of the robust model-free adaptive-iterative learning (MFA-IL) control.

#### **5. The Design of the Stopping Criteria Based on Evidence Theory**

## *5.1. Evidence Theory*

The evidence theory is a mathematical theory and general framework for reasoning with uncertainty information in systems, which allows one to combine multiple variables from multiple sources, arriving at a degree of belief. The major definitions and concepts of the theory are briefly introduced as follows [44,45].

**Definition 1.** *Let a finite set of elements* Θ = {*<sup>Z</sup>*1, *Z*2, ··· , *ZL*} *be defined as the frame of discernment. An element can be a hypothesis, object or state. The* <sup>2</sup>Θ*, named the power set, is the set of all subsets of* Θ*. It is composed of each element and multi-subset and can be indicated as* 2<sup>Θ</sup> = {<sup>∅</sup>, {*<sup>Z</sup>*1}, {*<sup>Z</sup>*2}, ··· {*<sup>Z</sup>*1, *<sup>Z</sup>*2}, ··· , <sup>Θ</sup>}*, where* ∅ *is an empty set.*

**Definition 2.** *A mass function is a mapping of m from* 2<sup>Θ</sup> *to* [0, 1]*, and be formally defined as:*

$$m: \mathbb{2}^{\Theta} \to [0, 1] \tag{33}$$

*Additionally, the function satisfies the following equation:*

$$m(\mathcal{Q}) = 0, \sum\_{A \subseteq 2^{\Theta}} m(A) = 1 \tag{34}$$

*The function m is named basic probability assignment (BPA). m*(*A*) *expresses the proportion of all relevant and available evidence. It is claimed that a particular element of* Θ *belongs to the set A, but to no particular subset of A. If m*(*A*) > 0*, A is called a focal element of* Θ*. If m*(*A*) = 0*, it means that the proposition totally lacks belief.*

**Definition 3.** *Suppose that two BPAs denoted by m*1 *and m*2 *are obtained from two different information sources in the same frame of discernment* Θ*. The degree of conflict among the evidence is denoted as follows:*

$$K = \sum\_{A \cap B = \mathcal{Q}} m\_1(A)m\_2(B) \tag{35}$$

*where if K* = 0*, it means that the two pieces of evidence are fully compatible with each other. On the contrary, if K* = 1*, it means that the two pieces of evidence totally conflict with each other.*

Dempster's rule of combination is the most basic and widely used rule for the combination of evidence:

$$\begin{cases} \begin{array}{ccc} \frac{1}{1-K} & \sum \\_m(A)m\_2(B) & \forall C \subseteq 2^{\Theta}, \mathcal{C} \neq \mathcal{Q} \\\ 0 & \mathcal{C} = \mathcal{Q} \end{array} \end{cases} \tag{36}$$

Dempster's combination rule is commutative and associates. Thus, the fusion result has nothing to do with the order of the fusion process.

It has two characteristics [46]:


#### *5.2. The Design of Stopping Criterions*

Research on stopping criteria based on evidence theory in this paper involves extracting real-time feedback gains from each controller in the vibration control system, constructing the frame of discernment, choosing appropriate feature vectors that describe the learning process of the robust MFA-IL algorithm, calculating the BPAs based on the input signals of actuator, forming the fused BPAs using combination rule and diagnosing the learning states of the control method based on the BPA results.

The real-time feedback gains from each controller can be considered as a piece of evidence for diagnosing the system state, assuming *M* actuators are glued on the plate and that the feedback gains of the corresponding controllers are measured vectors. For the sake of simplicity, suppose that all types of states are independent of each another. Only one state can occur at any given time. Let *Sω* represent the measurement obtained from the *ω*th controller (information source):

$$\mathbf{S}\_{\omega} = [s\_{\omega 1} s\_{\omega 2} \dotsm s\_{\omega m\_{\omega}}] \quad \omega = 1, 2, \dots, M \tag{37}$$

where *sωi* is the *i*th element of *S<sup>ω</sup>*, *i* = 1, 2, ... , *mω*, *mω* is the number of elements provided by the *ω*th controller and *M* ∑ *<sup>ω</sup>*=1*mω* = *n*, *n* is the number of features.

There are *N* types of states. The system states matrix can be described as [47]:

$$H = \begin{bmatrix} \mathbf{X}\_1 \\ \mathbf{X}\_2 \\ \vdots \\ \mathbf{X}\_N \end{bmatrix} = \begin{bmatrix} \mathbf{x}\_{11} & \mathbf{x}\_{12} & \cdots & \mathbf{x}\_{1n} \\ \mathbf{x}\_{21} & \mathbf{x}\_{22} & \cdots & \mathbf{x}\_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{x}\_{N1} & \mathbf{x}\_{N2} & \cdots & \mathbf{x}\_{NN} \end{bmatrix} \tag{38}$$

where *Xj* is a feature vector describing the *j*th state, *xji* is the *i*th feature of the *j*th state, and *i* = 1, 2, . . . , *n*, *j* = 1, 2, . . . , *N*.

*Micromachines* **2019**, *10*, 196

This paper uses the Minkowski distance for quantifying the objective evaluation of the BPAs. The Minkowski distance between *Sω* and *Xj* is reconstructed as:

$$d\_{\omega j} = \begin{cases} \left[ \sum\_{i=1}^{\frac{m\_{\omega l}}{\omega}} \left( \frac{s\_{\omega i} - x\_{ji}}{x\_{ji}} \right)^r \right]^{1/r} & \omega = 1 \\\\ \left[ \sum\_{i=1}^{\frac{m\_{\omega l}}{\omega}} \left( \frac{s\_{\omega i} - x\_{j(i + \sum\_{l=1}^{\omega - 1} m\_l)}}{x\_{j(i + \sum\_{l=1}^{\omega - 1} m\_l)}} \right)^r \right]^{1/r} & \omega = 2, 3, \dots, M \end{cases} \tag{39}$$

where *<sup>d</sup>ω<sup>j</sup>* is the distance between *Sω* and *<sup>X</sup>j*. *r* is a constant, such that if *r* = 2, then the distance converges to the Euclidean distance. On the other hand, if *r* = 1, the distance converges to the corner distance. The distances between all measurement vectors *Sω* and all state vectors *Xj* can be obtained in a matrix form: 

$$D = \begin{bmatrix} d\_{11} & d\_{12} & \cdots & d\_{1N} \\ d\_{21} & d\_{22} & \cdots & d\_{2N} \\ \vdots & \vdots & \vdots & \vdots \\ d\_{M1} & d\_{M2} & \cdots & d\_{MN} \end{bmatrix} \tag{40}$$

The smaller the distance *<sup>d</sup>ωj*, the more probable the *j*th state, based on the feedback gain acquired from the *ω*th controller. By defining *pωj* = 1/*dωj*, a matrix form after normalizing can be expressed as follows: 

$$P = \begin{bmatrix} p\_{11} & p\_{12} & \cdots & p\_{1N} \\ p\_{21} & p\_{22} & \cdots & p\_{2N} \\ \vdots & \vdots & \vdots & \vdots \\ p\_{M1} & p\_{M2} & \cdots & p\_{MN} \end{bmatrix} = \begin{bmatrix} P\_1 \\ P\_2 \\ \vdots \\ \vdots \\ P\_M \end{bmatrix} \tag{41}$$

where *Pω* = *pω*<sup>1</sup> *pω*<sup>2</sup> ··· *pωN* . *Pω* is the BPA assigned by the *ω*th controller to the set of states, satisfying *N* ∑ *j*=1*pωj* = 1.

According to Equation (35), the degree of conflicting evidence among the various controllers can be calculated. Then, the fused BPAs are computed using the combination rule, denoted in Equation (36). The fused BPAs are the pieces of evidence for state diagnosis. The threshold value was predefined as a stopping criterion. Based on the values of BPAs for a certain state, decision making on a system state can be fulfilled by comparing with the threshold value.
