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Proxybased sliding mode control PSMC is an improved version of PID control that combines the features of PID and sliding mode control SMC with continuously dynamic behaviour. However, the stability of the control architecture maybe not well addressed. Consequently, this work is focused on modification of the original version of the proxybased sliding mode control PSMC by adding an adaptive approximation compensator AAC term for vibration control of an EulerBernoulli beam. The role of the AAC term is to compensate for unmodelled dynamics and make the stability proof more easily. The stability of the proposed control algorithm is systematically proved using Lyapunov theory. Multimodal equation of motion is derived using the Galerkin method. The state variables of the multimodal equation are expressed in terms of modal amplitudes that should be regulated via the proposed control system. The proposed control structure is implemented on a simply supported beam with two piezopatches. The simulation experiments are performed using MATLAB/SIMULINK package. The locations of piezotransducers are optimally placed on the beam. A detailed comparison study is implemented including three scenarios. Scenario 1 includes disturbing the smart beam while no feedback loop is established (openloop system). In scenario 2, a PD controller is applied on the vibrating beam. Whereas, scenario 3 includes implementation of the PSMC+AAC. For all previously mentioned scenarios, two types of disturbances are applied separately: 1) an impulse force of 1 N peak and 1 s pulse width, and 2) a sinusoidal disturbance with 0.5 N amplitude and 20 Hz frequency. For impulse disturbance signals, the results show the superiority of the PSMC+AAC in comparison with the conventional PD control. Whereas, both the PSMC+ACC and the PD control work well in the case of a sinusoidal disturbance signal and the superiority of the PSMC is not clear.
proxybased sliding mode control, piezopatches, EulerBernoulli beam, adaptive approximation technique
Much attention has been paid to active vibration control of flexible structures. These structures have been widely used in miscellaneous applications such as space robotics, aircraft structures, gas turbine rotors, skyscrapers, and bridges [14]. They may damage if they undergo unwanted vibrational loads because of possible fatigue and instability. Therefore, they need a suitable control system to suppress the vibrational motion and maintain structural stability. The use of smart materials as actuators and/or sensors witnesses progress in practice and research fields proving a potential solution to reduce the failure of the structures. A smart beamlike structure consists of a regular beam with attached distributed smart materials behaving as actuators and/or sensors. They can actively suppress produced oscillations instead of using passive damping parts [5]. As a result, this work is concerned with modelling and control of slender beams with surface bonded piezopatches (actuators/sensors). Design of a suitable control architecture requires good mathematical modelling for the target dynamic system. In general, three beam theories have been reported in the literature: EulerBernoulli theory that neglects the rotational inertia and shear deformation, Rayleigh theory that considers the shear deformation only while Timoshenko beam theory considers both the rotational inertia and shear deformation [68]. The current work is focused on an EulerBernoulli beam model with attached piezomaterials. The vibrating beam has infinite degreesoffreedom with a large number of vibrational modes. However, the objective of the designed controller is to stabilize the vibration of the first mode shapes since they are dominant in the lowfrequency region of the dynamic response. Therefore, the partial differential equation PDF is transformed into discrete ordinary differential equations ODEs with a definite number of mode shapes.
Miscellaneous control approaches have been used for vibration control of smart flexible beams such as strain rate feedback [9], positive position feedback [10], pulse frequency modulator [11], PID controller [12], Linear quadratic regulator (LQR), state feedback [13], model predictive control [14], and more recent works [1520]. However, the performance and stability of the abovementioned control schemes could be corrupted if undesired external disturbances are applied or unmodeled dynamics are neglected. Above all, the PID family controller has a simple structure but it can work well at the lowfrequency region; the control system performance could be destabilized beyond the cutoff frequency [21]. To overcome PID limitations, a feedforward term is added to the PID term to obtain high bandwidth control. However, if the dynamic modelling includes uncertainties, the adaptive control is integrated such that the parameters associated with the feedforward term are updated based on Lyapunov theory [2233]. In effect, there are two basic strategies for adaptive control: regressorbased control and approximationbased control. In contrast to the regressor approach, the adaptive approximation technique is a modelfree control includes representation of the uncertainties in terms of weighting and basis function vectors/matrices. The weighting coefficients are updates based on Lyapunov theory, see the papers [2233] for more details. However, modelling error should be compensated by using a robust sliding term. The sliding mode control is a powerful robust control strategy for stabilization of motion of dynamic systems with uncertain modelling. It includes a nonlinear discontinuous function, e.g. signumtype function, to enforce the state variables approaching to the sliding surface (modelling and position errors are convergent to zero). However, due to the presence of a discontinuous signum function in the closedloop dynamics, this could lead to chattering problem and hence stability and performance of the sliding mode control can be degraded. Chattering problem can cause highfrequency dynamics with large position error [34]. In general, there are three approaches for attenuation of chattering problem: 1) boundary layer method [35], 2) high order sliding control [36], and 3) proxybased sliding control [37]. Boundary layer method replaces the discontinuous signum function by a continuous function, e.g. a saturated function or a hyperbolic tangent function, etc. However, due to the presence of approximation error, the stability of the control system can be degraded. On the other hand, high order sliding mode control requires a state observer to estimate derivatives of state variables. Above all, proxybased sliding mode control PSMC has brought the attention of most researchers for its simplicity of control architecture. It integrates the features of the PID controller and sliding mode control with ensured continuous closedloop dynamics. In the PSMC, the signum function is rolled out to a saturated function without any approximation and hence the closedloop dynamics have continuous behaviour. PMSC is applied to soft robotics for its superiority to ensure safety and precision in motion [3843]. However, the stability of PSMC is not well treated and further work is required. Thus, the key point of this work is that it integrates three control units (PID+SMC+AAC) to obtain high control performance. First, it reformulates the structure of the PID controller integrating the features of the robust sliding mode control. Secondly, an adaptive approximation compensator is added to the control architecture to compensate for modelling errors if exist.
In view of the above, this paper deals with modelling and PMSC with an AAC term for flexible beams with piezopatches. The role of AAC is to compensating for unmodelled dynamics and to ease the task of proof of system stability. Whereas the PMSC unit attempts to capture the features of the PID control and the SMC with continuous closedloop dynamics. The EulerBernoulli beam theory is used for derivation of dynamics of the target smart beam. The partial differential equation PDF of the vibrating beam is transformed into secondorder differential equations ODEs with definite mode shapes using the Galerkin method. A simply supported beam is simulated by MATLAB/SIMULINK package considering the first twomode shapes. Two collocated piezopatches are placed optimally on the vibrating beam working as actuators and sensors. Several experiments are performed to prove verification and precise regulation of PMSC.
The rest of the paper is organized as follows. Dynamic modelling of the EulerBernoulli beam and the proposed control structure are presented in Section 2. Section. Simulation experiments are introduced in Section 3 while Section 4 concludes.
2.1 Dynamic modelling of smart beams
In general, there are three wellknown beam theories for derivation of equation of motion: 1) EulerBernoulli theory that neglects rotational inertia and shear deformation and it is suitable for thin beams, 2) Rayleigh theory in which shear deformation is considered, and 3) Timoshenko beam that considers both rotational inertia and shear deformation and it is suitable for thick beams [8]. This paper is focused on EulerBernoulli beam theory in the modelling of flexible beams. In Figure 1, a thin prismatic beam with constant flexural rigidity $E_{b} I_{b}$. It is excited by a distributed force f (x, t) over the length of the beam and provided with surface bonded piezopatches. The dimension of the beam and piezopatches are shown in the figure. The following assumptions are proposed for modelling of the target beam.
Assumption 1. The beam is thin enough that Euler beam theory is applied.
Assumption 2. The stiffness and inertia of the piezopatches are neglected.
Assumption 3. Sufficient transducers (or piezosensors) are available such that modal amplitudes are measurable.
Figure 1. A flexible beam with piezopatches
Remark 1. In effect, all three assumptions mentioned above are reasonable and necessary for our current work. In Assumption 1, the beam is considered thin such that we can apply EulerBernoulli beam theory. If the beam is considered thick then we should depend on Timoshenko beam theory for modelling purpose [8]. For Assumption 2, the dynamics of the piezotransducers are neglected since the vibration characteristics of the vibrating beam would not be affected by the attached piezotransducers [44]. Assumption 3 means that the modal amplitudes can be observed (measured) depending on installing a sufficient number of sensors [7].
The governing PDF for the transverse bending deflection y of EulerBernoulli beams can be expressed as [7, 45]:
$E_{b} I_{b} \frac{\partial^{4} y(x, t)}{\partial x^{4}}+c \frac{\partial y}{\partial t}+\rho_{b} A_{b} \frac{\partial^{2} y(x, t)}{\partial t^{2}}=f(x, t)\frac{\partial^{2} M_{p}(t)}{\partial x^{2}}$ (1)
where, $E_{b}, I_{b}, \rho_{b}, A_{b}$ and c are modulus of elasticity, moment of inertia, density, crosssectional area of the beam, and damping coefficient respectively. M_{p} (x, t) represents the applied bending moment exerted by a piezoactuator and it can be expressed as [7]:
$M_{p}(x, t)=D v_{a}(t)\left[H\left(xx_{1}\right)H\left(xx_{2}\right)\right]$ (2)
where, $v_{a}(t)$ is an excitation voltage applied to the piezoactuator. The constant D depends on the dimensions and material properties of both beam and piezoactuator [13], $x_{(.)}$ refers to the location of piezoactuator, and H(.) is a Heaviside step function. Using Galerkin method, the transverse displacement of the target beam can be approximated as:
$y(x, t) \cong \sum_{i=1}^{N} \phi_{i}(x) q_{i}(t) \quad i=1,2,3, \cdots N$ （3）
where, q_{i}(t) is a time dependent function (modal amplitudes), N refers to the number of mode shapes, and $\phi_{i}(x)$ is the mode shape of the beam. Thus, Eq. (1) can be expressed as:
$\begin{aligned} E_{b} I_{b} \sum_{j=1}^{N} \phi_{j}^{\prime \prime \prime\prime}(x) q_{j}(t) &+c \sum_{j=1}^{N} \phi_{j}(x) \dot{q}_{j}(t) \\ &+\rho_{b} A_{b} \sum_{j=1}^{N} \phi_{j}(x) \ddot{q}_{j}(t) \\ &=f(x, t)D v_{a}(t) \frac{\partial^{2}}{\partial x^{2}}\left[H\left(xx_{1}\right)\right.\\ &\left.H\left(xx_{2}\right)\right] \end{aligned}$ (4)
Multiplying Eq. (4) by $\phi_{i}(x),(i \neq j)$ and integrating over the length of the beam to obtain the following decoupled system.
$\begin{aligned} E_{b} I_{b} \int_{0}^{l_{b}} \phi_{i}^{\prime \prime \prime \prime}(x) \phi_{i} &(x) d x q_{j}(t) &+c \int_{0}^{l_{b}} \phi_{i}(x) \phi_{i}(x) d x \dot{q}_{j}(t) &+\rho_{b} A_{b} \int_{0}^{l_{b}} \phi_{i}(x) \phi_{i}(x) d x \ddot{q}_{j}(t) &=\int_{0}^{l_{b}} f(x, t) \phi_{i}(x) d x &\int_{0}^{l_{b}} D v_{a}(t) \phi_{i}(x) \frac{\partial^{2}}{\partial x^{2}}\left[H\left(xx_{1}\right)\right. &\left.H\left(xx_{2}\right)\right] d x \end{aligned}$ （5）
Eq. (5) can systematically be expressed as:
$m_{i} \ddot{q}_{i}(t)+c_{i} \dot{q}_{i}(t)+k_{i} q_{i}(t)=g_{i}(x, t)\mu_{i} v_{a}(t)$
$i=1,2,3, \ldots, N$ (6)
with
$m_{i}=\rho_{b} A_{b} \int_{0}^{l_{b}} \phi_{i}(x) \phi_{i}(x) d x$,
$c_{i}=c \int_{0}^{l_{b}} \phi_{i}(x) \phi_{i}(x) d x$,
$k_{i}=E_{b} I_{b} \int_{0}^{l_{b}} \phi_{i}^{\prime \prime \prime\prime}(x) \phi_{i}(x) d x$,
$w_{n i}^{2}=\frac{k_{i}}{m_{i}}$,
$g_{i}(t)=\int_{0}^{l_{b}} f(x, t) \phi_{i}(x) d x$,
$\mu_{i}=D\left(\phi_{i}^{\prime}\left(x_{2}\right)\phi_{i}^{\prime}\left(x_{1}\right)\right)$ (7)
However, there are $N_{a}$ piezoactuators bonded on the beam and hence by using superposition technique, Eq. (6) can be rewritten as:
$\begin{aligned} m_{i} \ddot{q}_{i}(t)+c_{i} \dot{q}_{i}(t) &+k_{i} q_{i}(t) &=g_{i}(x, t)\sum_{k=1}^{N_{a}} \mu_{i k} v_{a k}(t) , & i=1,2,3, \ldots, N \end{aligned}$ （8）
with $N_{a}$ being the number of piezoactuators and $\mu_{\mathrm{ik}}$ being defined in Eq. (7) with modification of subscripts for x_{1} and x_{2} to $x_{1 k}$ and $x_{2 k}$ respectively to distinguish locations of multipiezoactuators, where k=1,2,3… $N_{a}$.
Eq. (8) can extend for multimodal analysis using matrix representation to get,
$M \ddot{q}+C \dot{q}+K q+g=u$ (9)
where,
$\mathrm{M}=\left(\begin{array}{ccc}m_{1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & m_{N}\end{array}\right) \in R^{N \times N}, C=\left(\begin{array}{ccc}c_{1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & c_{N}\end{array}\right) \in R^{N \times N},$
$K=\left(\begin{array}{ccc}k_{1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & k_{N}\end{array}\right) \in R^{N \times N}, g=\left(\begin{array}{c}g_{1} \\ \vdots \\ g_{N}\end{array}\right) \in R^{N}$
$\mu=\left(\begin{array}{ccc}\mu_{11} & \cdots & \mu_{1 N N_{a}} \\ \vdots & \ddots & \vdots \\ \mu_{N 1} & \cdots & \mu_{N N_{a}}\end{array}\right) R^{N \times N_{a}}, u=\mu v_{a} \in R^{N}$
Eq. (9) is similar to multidegreesoffreedom 2^{nd}order discrete dynamic systems. The following points should be noted:
The matrices M, C and K are diagonal matrices and hence the dynamic system is decoupled on the leftside of Eq. (9). However, the coupling interaction at the input coefficient $\mu$ for multimodal dynamics.
The matrix M has constantvalue elements (assuming beams with constant crosssection) while C is assumed a positive definite matrix.
The state variable q cannot be sensed (measured) directly, its reading depends on sensor voltage readings.
In the following, we will attempt to reformulate Eq. (9) such that the output state variables are piezosensor voltages v_{s}. Thus, v_{s }can be represented as [7]:
$v_{s j}=\sum_{k=1}^{N} \alpha_{j k} q_{k}(t)=\left[\begin{array}{ccc}\alpha_{j 1} & \ldots & \alpha_{j N}\end{array}\right]\left[\begin{array}{c}q_{1} \\ \vdots \\ q_{N}\end{array}\right],$
$j=1,2,3, \ldots, N_{s}$ (10)
where, $\alpha_{(.)}$ is a constant that depends on the location of piezosensors and N_{s }is the number of piezosensors that are equal to $N_{a}$ in case of collocated patches. In matrix/vector representation, Eq. (10) can be rewritten as:
$\mathrm{v}_{\mathrm{s}}=\alpha q, \quad \alpha=\left(\begin{array}{ccc}\alpha_{11} & \cdots & \alpha_{1 N} \\ \vdots & \ddots & \vdots \\ \alpha_{N_{S} 1} & \cdots & \alpha_{N_{S} N}\end{array}\right) \in R^{N_{s} \times N}$ （11）
Therefore, substituting Eq. (11) into Eq. (9) results in
$M \alpha^{+} \ddot{v}_{s}+C \alpha^{+} \dot{v}_{s}+K \alpha^{+} v_{s}+g=u$ (12)
where, $\alpha^{+} \in R^{N \times N_{S}}$ is the MoorePenrose inverse matrix of sensor gain matrix ( $\alpha$), see the paper [46] for more details.
2.2 Control structure
This subsection deals with integration of the PSMC and the AAC for vibration suppression of smart beams. The PSMC was well applied to different applications especially soft robotics. It is an improved and continuous version of sliding mode control that integrates the features of PID control family and the robust sliding mode control. The key idea of this strategy is to impose a null virtual mass called a proxy in between PID controller on the side of output target system (the dynamic system that is required to be controlled) and a sliding mode control on the end of the virtual proxy object. It attempts to control (regulate) local and global dynamics associated with the PID control and the SMC respectively [38]. As a result, the PSMC has superior features of accurate tracking and safe recovery while the disturbed system is under external disturbances.
Figure 2. A simplified sketch showing the concept of the PSMC
Figure 2 describes concept of PSMC for the target smart system (equivalent system according to Eq. (9)). The stability of the PSMC is not well addressed. Besides, most previous work was focused on proving that the closed loop dynamics is passive in order to compensate for disturbances if exist. To this end, this section proposes the PSMC combined with an adaptive approximation term to compensate for disturbances if exist. This technique simplifies proof of stability of the dynamic system and makes the derivation straightforward.
According to the principle of PSMC described in Figure 2, consider the following position errors and sliding surfaces:
$e=q_{d}q, \sigma=\left(q_{d}q\right)+\Lambda\left(\dot{q}_{d}\dot{q}\right)=e+\Lambda \dot{e}$ (13a)
$e_{\chi}=\chiq$
$\sigma_{\chi}=(\chiq)+\Lambda(\dot{\chi}\dot{q})=e_{\chi}+\Lambda \dot{e}_{\chi}$ (13b)
$e_{s}=q_{d}\chi$
$\sigma_{s}=\left(q_{d}\chi\right)+\Lambda\left(\dot{q}_{d}\dot{\chi}\right)=e_{s}+\Lambda \dot{e}_{s}$ (13c)
where, $e, e_{x}$ and $e_{s} \in \mathrm{R}^{\mathrm{N}}$ are actual position error, position error associated with PD, and position error associated with sliding mode control respectively with corresponding sliding surfaces $\sigma$, $\sigma_{\chi}$ and $\sigma_{s} \in \mathrm{R}^{\mathrm{N}}$. $q_{d} \in \mathrm{R}^{\mathrm{N}}$ is the desired trajectory, $\chi \in \mathrm{R}^{\mathrm{N}}$ is the proxy position while $\Lambda \in \mathrm{R}^{\mathrm{N} \times \mathrm{N}}$ is a diagonal positive definite feedback gain matrix. According to Figure 2, the intuitive control law for can be designed as:
$u=u_{1}=\underbrace{\hat{\xi}_{\omega}}_{A A C}+\underbrace{K_{\chi} e_{\chi}+K_{d} \dot{e}_{\chi}}_{P D}$ (14a)
with
$\hat{\xi}=\widehat{M} \ddot{q}_{d}+\hat{C} \dot{q}_{d}+\widehat{K} q_{d}+\hat{g}=\widehat{M} \ddot{q}_{d}+\hat{C} \dot{q}_{d}+\hat{\eta}$
$\hat{\eta}=\widehat{K} q_{d}+\hat{g}$
where, $\xi \in \mathrm{R}^{\mathrm{N}}$ is a lumped uncertain vector that can be approximated by a linear combination of orthogonal functions, and the symbol $(\hat{.})$ refers to estimation. Thus, the estimated term $\hat{\xi}$ can be expressed as:
$\hat{\xi}=\widehat{W}^{T} \varphi$ (14b)
where, $\widehat{W} \in \mathrm{R}^{\mathrm{N} \beta \times \mathrm{N}}$ is the weighting coefficient matrix that should be updated and $\varphi \in \mathrm{R}^{\mathrm{N} \beta}$ is the orthogonal basis vector, with $\beta$ referring to the number of basis function terms. The following update law is selected with ensured stability:
$\dot{\widehat{W}}=Q^{1} \varphi \dot{e}^{T}$ (14c)
where, $Q \in \mathrm{R}^{\mathrm{N} \beta \times \mathrm{N} \beta}$ a diagonal positivedefinite adaptation gain matrix. Now the role of PSMC comes. Using the principle of proxybased sliding mode control (see Figure 2) with modification of original version proposed by Kikuuwe et al. [37], the dynamics of proxy can be modelled as
$m_{\chi} \ddot{\chi}=0=u_{1}u_{2}$ (15)
with $m_{\chi} \in \mathrm{R}^{\mathrm{N} \times \mathrm{N}}$ being the virtual proxy mass, and
$u_{2}=\hat{\xi}+U \operatorname{sgn}\left(\sigma_{s}\right)$ (16)
where, $\mathrm{U} \in \mathrm{R}^{\mathrm{N}}$ is constant coefficient vector that limits the values of control input. Alternatively, Eq. (15) can be expressed as:
$\hat{\xi}+K_{\chi} e_{\chi}+K_{d} \dot{e}_{\chi}=\hat{\xi}+U \operatorname{sgn}\left(\sigma_{s}\right)$ (17)
Eq. (17) can be rewritten as:
$K_{\chi} e_{\chi}+K_{d} \dot{e}_{\chi}=\operatorname{Usgn}\left(\psi\Lambda \dot{e}_{\chi}\right)$ (18)
with $\psi=\sigmae_{\chi}$. Using the following mathematical expression [37, 38],
$\begin{aligned} a_{1} &=a_{2}+a_{3} \operatorname{sgn}\left(a_{4}a_{5} a_{1}\right) \\ &=a_{2}+a_{3} \operatorname{sat}\left(\frac{a_{4}}{a_{5} a_{3}}\frac{a_{2}}{a_{3}}\right) \end{aligned}$ (19)
According to the above equation, Eq. (18) can be reformulated as:
$\begin{aligned} \dot{e}_{\chi}=K_{d}^{1} K_{\chi} e_{\chi} &+K_{d}^{1} U_{S a t}\left(U^{1} K_{d} \Lambda^{1} \psi\right.\\ &\left.+U^{1} K_{\chi} e_{\chi}\right) \end{aligned}$ (20)
Thus, the desired control voltage presented in Eq. (14a) can be designed as:
$u=u_{1}=\hat{\xi}+U \operatorname{sat}\left(U^{1} K_{d} \Lambda^{1} \psi+U^{1} K_{\chi} e_{\chi}\right)$ (21)
Equating Eq. (9) to Eq. (14a) leads to the following closedloop dynamics,
$M \ddot{q}+C \dot{q}+\underbrace{K q+g}_{\eta}=\widehat{M} \ddot{q}_{d}+\hat{C} \dot{q}_{d}+\underbrace{\widehat{K} q_{d}+\hat{g}}_{\bar{\eta}}+$
$K_{\chi} e_{\chi}+K_{d} \dot{e}_{\chi}$ (22)
By adding $\left(M \ddot{q}_{d}C \dot{q}_{d}\eta\right)$ to both sides of above equation to obtain,
$M \ddot{e}C \dot{e}=\tilde{M} \ddot{q}_{d}+\tilde{C} \dot{q}_{d}+\tilde{\eta}+K_{\chi} e_{\chi}+K_{d} \dot{e}_{\chi}+\epsilon$ (23)
with $(\hat{.})(.)=(\tilde{.})$. Eq. (23) can be represented as:
$\begin{aligned} M \ddot{e}+C \dot{e} &=\tilde{\xi}K_{\chi} e_{\chi}K_{d} \dot{e}_{\chi}+\epsilon \\ &=\widetilde{W}^{T} \varphiK_{\chi} e_{\chi}K_{d} \dot{e}_{\chi}+\epsilon \end{aligned}$ (24)
where, $\epsilon \in \mathrm{R}^{\mathrm{N}}$ is the approximation modelling error.
Theorem. The dynamics of the target smart beam modelled in Eq. (9) with the control law presented in Eq. (21), the adaptive law of Eq. (14c), and the closedloop dynamics given in Eq. (24) is stable in the sense of Lyapunov theory.
Proof.
Consider the following Lyapunovlike candidate along the trajectory of Eq. (24)
$V=\frac{1}{2} \dot{e}^{T} M \dot{e}+\frac{1}{2} \operatorname{tr}\left(\widetilde{W}^{T} Q \widetilde{W}\right)+\frac{1}{2} e_{\chi}^{T} K_{\chi} e_{\chi}$
$+\left\U e_{s}\right\_{1}$ (25)
Taking derivative of above equation and substituting Eq. (23) into Eq. (25) to get,
$\dot{V}=\dot{e}^{T}\left(C \dot{e}\widetilde{W}^{T} \varphiK_{\chi} e_{\chi}K_{d} \dot{e}_{\chi}+\epsilon\right)$
$+\operatorname{tr}\left(\widetilde{W}^{T} Q \dot{W}\right)+\dot{e}_{\chi}^{T} K_{\chi} e_{\chi}$
$+\dot{e}_{s} U \operatorname{sgn}\left(e_{s}\right)$ (26)
Substituting Eq. (17) into above equation to obtain,
$\dot{V}=\dot{e}^{T} C \dot{e}+\dot{e}^{T} \epsilon+\operatorname{tr}\left(\widetilde{W}^{T}\left(\varphi \dot{e}^{T}+Q \dot{\bar{W}}\right)\right)$
$\dot{e}^{T} U \operatorname{sgn}\left(\sigma_{S}\right)+\dot{e}_{\chi}^{T}\left(\operatorname{Usgn}\left(\sigma_{S}\right)\right.$
$\left.K_{d} \dot{e}_{\chi}\right)+\dot{e}_{s} U \operatorname{sgn}\left(e_{s}\right)$ (27)
Applying Eq. (14c) keeping in mind that $C \geq 0$, then the above equation is reduced to,
$\dot{V}=\dot{e}^{T} C \dot{e}+\dot{e}^{T} \epsilon\dot{e}_{\chi}^{T} K_{d} \dot{e}_{\chi}\dot{e}^{T} U \operatorname{sgn}\left(\sigma_{S}\right)$
$+\dot{e}_{\chi}^{T} U \operatorname{sgn}\left(\sigma_{s}\right)+\dot{e}_{s} U \operatorname{sgn}\left(e_{s}\right)$ (28)
From Eq. (13), we can get,
$\dot{e}_{\chi}+\dot{e}_{s}=\dot{e}$ (29)
Substituting Eqns. (29) and (13c) into Eq. (28) leads to,
$\begin{aligned} \dot{V}=\dot{e}^{T} C \dot{e}+\dot{e}^{T} & \epsilon\dot{e}_{\chi}^{T} K_{d} \dot{e}_{\chi} \\ &\dot{e}_{S}^{T} U \operatorname{sgn}\left(e_{s}+\Lambda \dot{e}_{s}\right) \\ &+\dot{e}_{s}^{T} U \operatorname{sgn}\left(e_{s}\right) \end{aligned}$ (30)
Using the following mathematical inequality,
$y^{T} U[\operatorname{sgn}(z+y)\operatorname{sgn}(z)] \geq 0$ （31）
Therefore, Eq. (30) is reduced to:
$\dot{V} \leq\dot{e}^{T} C \dot{e}\dot{e}_{\chi}^{T} K_{d} \dot{e}_{\chi}+\dot{e}^{T} \epsilon$ (32a)
Now let us deal with the last term exploiting the following inequality [47]:
$\dot{e}^{T} \epsilon \leq \dot{e}^{T} \Omega^{1} \dot{e}+\epsilon^{T} \Omega \epsilon \leq \dot{e}^{T} \Omega^{1} \dot{e}+\bar{\epsilon}$ (32b)
Thus, Eq. (32a) becomes,
$\dot{V} \leq\dot{e}^{T} C \dot{e}\dot{e}_{\chi}^{T} K_{d} \dot{e}_{\chi}+\dot{e}^{T} \Omega^{1} \dot{e}+\bar{\epsilon}$ (32c)
where $\bar{\epsilon}$ is the upper bound of $\epsilon$ such that $\epsilon^{T} \Omega \epsilon \leq \bar{\epsilon}$. To ensure stability, it is necessary to have$\left(C \geq \Omega^{1}\right)$ then the tracking error is bounded and converges to $\bar{\epsilon}$ . Thus, the system is stable in sense of Lyapunov stability [2328].
Remark 2. The abovementioned stability proof requires knowledge of lower bounds of the damping matrix C. Other robust approaches are possible to avoid this choice such as deadzone technique, see, e.g. [48] for more details.
This section deals with simulation of a simply supported beam depicting in Figure 3. The physical parameters of the investigated smart beam are given in Tables 1 and 2. Two collocated piezopatches are bonded to the surface of the beam for suppression of the resulted vibration. The dynamics of piezopatches are neglected in comparison with the regular beam dynamics, see the paper [44] for more details. In effect, considering dynamics of piezopatches would complicate the control problem since the equation of motion for the overall system is no longer decoupled and advanced control strategies are required.
Figure 3. A flexible simply supported beam with the surfacebonded piezopatches. The symbols A and S refer to piezoactuator and sensor respectively. The function h(t) can be impulse or sinusoidal force
Table 1. Parameters of the beam
Property 
Value 
Density, $\rho_{b}$ 
8030 kg/m^{3} 
Young’s modulus, E_{b} 
193×10^{9}pa 
Length, l_{b} 
0.3 m 
Section width, w_{b} 
0.03 m 
Section thickness, t_{b} 
$0.5 \times 10^{3} \mathrm{~m}$ 
Damping coefficients 
$c_{1}=00068 \mathrm{~N} \cdot \frac{\mathrm{s}}{\mathrm{m}}$ 
Table 2. Parameters of the piezopatches^{*}
Property 
Value 
Young’s modulus, E_{p} 
$68 \times 10^{9} P a$ 
Length, l_{p} 
0.075m 
Section width, w_{p} 
0.025m 
Section thickness, t_{p} 
$0.35 \times 10^{3} \mathrm{~m}$ 
Location of 1^{st} piezo, x_{11}, x_{21} 
$x_{11}=0.3\left(l_{b}\right)\frac{l_{p}}{2}$ 
Location of 2^{nd} piezo, x_{12}, x_{22} 
$x_{12}=0.7\left(l_{b}\right)\frac{l_{p}}{2}$ 
*For detailed formulae used for evaluation of values of D and $\alpha_{(.)}$, see ref. [8].
Below, we will describe the mode shapes and frequency response for the first two modes shapes neglecting high order mode shapes. Then, a detailed comparison study is implemented including three scenarios. Scenario 1 includes disturbing the smart beam while no feedback loop is established (openloop system). In scenario 2, a PD controller is applied on the vibrating beam to attenuate the produced vibration due to the excitation disturbance force. Whereas, scenario 3 includes application of the proposed control structure (PSMC+AAC) for vibration suppression of the flexible beam. For all previously mentioned scenarios, two types of disturbances are investigated separately: 1) an impulse force of 1 N peak and 1 s pulse width, and 2) sinusoidal disturbance with 0.5 N amplitude and 20 Hz frequency.
3.1 Frequency response and mode shapes
This subsection is focused on frequency response for the vibrating simply supported beam considering multipiezopatches. The piezopatches are located where large sensor voltage readings are obtained. To capture the transfer functions for multiinput multioutput systems, let us recall Eq. (12a) with null excitation force, g=0.
$\alpha^{+} M \ddot{v}_{s}+\alpha^{+} C \dot{v}_{s}+\alpha^{+} K v_{s}=u$ (33)
Multiplying Eq. (33) by $\alpha$, taking Laplace transform, and substituting $u=\mu v_{a}$, it becomes:
$\frac{V_{s}(s)}{V_{a}(s)}=\alpha\left(M s^{2}+C s+K s\right)^{1} \mu$ (34)
Fortunately, $N_{a}=N_{s}=2$ hence four transfer functions are obtained,
$\frac{V_{s}(s)}{V_{a}(s)}=\left[\begin{array}{ll}g_{11} & g_{12} \\ g_{21} & g_{22}\end{array}\right]$ （35）
Eq. (34) can be reformulated to represent the output in terms of the modal amplitudes as follows. Since $V_{s}(s)=\alpha Q(s)$, hence Eq. (34) becomes,
$\frac{Q(s)}{V_{a}(s)}=\left(M s^{2}+C s+K s\right)^{1} \mu$ （36）
However, it is realistic to deal with the total deflection of the plate as an output. By recalling Eq. (8), the following transfer function is obtained,
$G(x, s)=\sum_{i=1}^{N} \frac{\phi_{i}(x)}{m_{i} s^{2}+c_{i} s+k_{i} s}\left[\begin{array}{lll}\mu_{i 1} & \ldots & \mu_{i N a}\end{array}\right]$ (37)
Eq. (37) is used as a basis for determination of frequency response of the vibrating beam considering multipiezopatches with multimode shapes. Here the deflection frequency response depends on the basis function $\phi_{i}(x)$ that is a function of the displacement x. Besides, the first two mode shapes in the lowfrequency region are considered since they have larger amplitudes in comparison to highfrequency amplitudes.
3.2 Impulse disturbance
Now consider the simply supported beam described in Figure 3. The target beam is subjected to an impulse force with 10 N peak and 2 s pulse width. Three experiments are implemented to investigate the validity of the proposed controller. In experiment 1, the target beam is vibrated under the impulse force without using a feedback control (openloop system). Experiment 2 includes implementation of the PD controller on the targetvibrating beam while in experiment 3, the proposed PSMC+AAC is performed. The feedback gains used in experiment 2 are: $K_{p}=400 I_{2}, K_{d}=100 I_{2}$, where I_{i} is i×i identity matrix. Whereas, for experiment 3, the following feedback and adaptation gains are used in simulation: $\Lambda=20 I_{2}, K_{p}=400 I_{2}, K_{d}=100 I_{2}, U=I_{2}, Q=75 I_{22}$.
According to Eq. (8), the imode equation of motion for the simply supported beam can be expressed as:
$m_{i} \ddot{q}_{i}(t)+c_{i} \dot{q}_{i}+k_{i} q_{i}(t)$$=\sum_{k=1}^{N_{a}} \mu_{i k} v_{a k}(t)g_{i}(x, t)$$\quad, i=1,2$ (38)
with $m_{i}, c_{i}, k_{i}$ being defined in Eq. (6), $N=N_{a}=N_{s}=2$ such that the actuator and sensor gain matrices are square that can facilitate the task of controller, and the exciting force is selected as:
$g_{i}(x, t)=\int_{0}^{l_{b}} f(x, t) \phi_{i}(x) d x=h(t) \int_{0}^{l_{b}} \frac{y_{0} x}{l_{b}} \sin \frac{i \pi x}{l_{b}} d x=$
$\frac{l_{b} y_{0}}{i \pi} h(t)(1)^{i}, i=1,2$ (39)
where, y_{0}=1 and h(t) are the excitation impulse force. The objective of the controller is to attenuate (suppress) the vibration resulted due to the applied impulse force. The proposed control law described in Eq. (21) with the update adaptive law presented in Eq. (14c) is implemented using MATLAB/SIMULINK package. As aforementioned, the control law consists of two terms: a PSMC term plus an AAC term for compensation purposes. Orthogonal Chebyshev polynomials with ꞵ=15 is used as approximators. In the tuning process, the gains are gradually increased from zero to a value at which the system is oscillating then the gain values are halved. The output amplitude displacements are depicted in Figure 4 while the control input is plotted in Figure 5. As seen in the plots, it can be concluded that the proposed control architecture damps out the resulted vibration with high performance in comparison with conventional PD. In addition, if the number of piezopatches are not equal to the number of mode shapes, then a pseudoinverse matrix is a powerful tool to determine the input voltages.
Figure 4. The modal displacement responseimpulse disturbance signal
Figure 5. The control input voltagesimpulse disturbance signal
Figure 6. The modal displacement responsesinusoidal disturbance signal
Figure 7. The control input voltagessinusoidal disturbance signal
3.3 Sinusoidal disturbance
In a similar manner to Sec. 3.2, three experiments are performed including the openloop system, the PD controller, and the PSMC+AAC respectively. This section differs from the previous subsection in the type of disturbance signal used. A sinusoidal
Excitation force with 0.5 N amplitude and 20 Hz frequency is used for all three experiments. In experiment 1, the vibrating smart beam is disturbed without application of any feedback loop (openloop system) and hence the beam system will vibrate freely. Whereas, in experiments 2 and 3 a PD controller and a PSMC+AAC are applied separately and respectively on the vibrating beam under a harmonic excited force. The same feedback gains mentioned for Sec. 3.2 are used here. In addition, Chebyshev polynomials with ꞵ=15 are used as approximators for adaptive scheme. As expected, the PSMC+AAC works well in vibration suppression of the vibrating beam. The modal responses for the first two mode shapes are depicted in Figure 6 while the response of the input control is described in Figure 7. It should be noted that both the PD and the PSMC work well in a sinusoidal disturbance signal and there could be no superiority for the PSMC in the case of a harmonic sinusoidal disturbance.
This paper is concerned with a modified version of the PSMC by adding AAC term for vibration control of EulerBernoulli beam. The key idea of the PSMC is to combine the features of the PID and the SMC. Whereas, the AAC includes representation of the uncertainty in terms of a weighting coefficient and basis function matrices/vectors. The performance and stability of the proposed control structure is approved using Lyapunov theory. The dynamics of vibrating beam with the attached piezopatches are derived systematically in standard 2^{nd} order differential equations considering definite mode shapes. Despite the current work does not consider a nonlinear model for the smart beam, it can be straightforward used for complex dynamics of smart beams with miscellaneous disturbance signals since the control structure includes a robust feedback term represented by the PSMC and a feedforward term denoted by the adaptive approximation compensator. Future works are required to deal with the following issues:
(1) Spillover phenomenon resulted from the effect of vibration of unmodelled mode shapes.
(2) The nonlinear dynamics are not modelled in the current work; however, our proposed control is a promising control scheme for nonlinear beam models.
(3) Vibration suppression of largescale flexible structures such as plates and shells.
(4) Considering dynamics of the piezotransducers in beam modelling. This can complicate the control problem since the mode shapes are coupled in this case.
A_{b} 
Crosssectional area of beam, $m^{2}$ 
E_{b} 
Young’s modulus of beam, $N / m^{2}$ 
E_{p} 
Young’s modulus of piezomaterial, $N / m^{2}$ 
f(x,t) 
Excitation force, N 
$H_{(.)}$ 
Heaviside step function 
I_{b} 
Moment of inertia of beam, $\mathrm{kg} / \mathrm{m}^{3}$ 
l_{b} 
Length of beam, m 
M_{p} 
Piezomoment per unit length, N. $m / m$ 
N 
Number of mode shapes 
$\mathrm{N}_{a}$ 
Number of piezoactuator 
N_{s} 
Number of piezosensor 
t_{b} 
Thickness of beam crosssection, m 
t_{p} 
Thickness of piezo crosssection, m 
$v_{a}(t)$ 
Voltage of piezoactuator, volt 
w_{b} 
Width of beam crosssection, m 
w_{p} 
Width of piezo crosssection, m 
$x_{(.)}$ 
Location of piezoactuators, m 
y 
Deflection of beam, m 
C 
$\in R^{N \times N}$, damping matrix, N.s/m 
e $_{(.)}$ 
$\in R^{N}$, position error, m 
g 
$\in R^{N}$, excitation force vector, N 
k 
$\in R^{N}$, stiffness matrix, N/m 
$K_{d}$ 
$\in R^{N}$, derivative gain matrix 
$K_{\chi}$ 
$\in R^{N}$, proportional gain matrix 
M 
$\in R^{N}$, mass matrix, kg 
q 
Modal displacement vector, m 
Q 
$\in \mathrm{R}^{\mathrm{N} \beta \times \mathrm{N} \beta}$, adaptation matrix 
W 
$\in \mathrm{R}^{\mathrm{N} \beta \times \mathrm{N}}$w, weighting matrix 
u 
$\in R^{N}$, input vector, N 
v_{s} 
$\in R^{N_{s}}$ the piezosensor voltage vector, volt 
Greek symbols 

$\rho_{b}$ 
Density of beam, $k g / m^{3}$ 
$\beta$ 
Number of basis function terms 
$\phi(x)$ 
Mode shape of beam 
$\sigma_{(.)}$ 
$\in R^{N}$, sliding surface vector, m 
$\alpha$ 
$\in R^{N_{s} \times N}$, sensor gain matrix, m/volt 
$\mu$ 
$\in R^{N \times N_{a}}$, input coefficient matrix, N/volt 
$\xi$ 
$\in R^{N}$, a lumped uncertain vector 
$\varphi$ 
$\in R^{N \beta}$, basis function vector 
$x$ 
$\in R^{N}$, proxy position vector, m 
Subscripts 

b 
beam 
p 
Piezomaterial 
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