## 3 Object Identification  
Adaptive control assumes the necessity to identify parameters of an unknown model in a continuous manner so that it is possible to update the parameters (settings) of the controller with the use of an adaptive algorithm. The scheme of such a system is presented in Figure 2, in which the output \(y(t)\) is a function of the object's response to the \(u(t)\) control and the response to the system disturbance \(v(t)\). In the case of the considered plate vibration reduction station (Figure 1), the disturbance \(v(t)\) consists of the vibrations generated from an external source as well as the measurement noise. A stable, time invariant system presented on Figure 2 can be described by the input-output discrete-time model of the form [23]:  
\[ \begin{split} y(t)=&-a_{1}y(t-1)-\dots-a_{n}Ay(t- nA)+\\ &+b_{1}u(t-d-1)+\dots+b_{nB}u(t-d-nB)+v(t),\end{split}  \tag{1}\]  
where \(t\) denotes discrete time: \(t=\frac{\tau}{T_{S}}\), where \(T_{S}\) is the sampling period, \(\tau\) is continuous time and \(d\) is the integer number of sampling periods contained in the time delay of the  
Figure 1: Test stand and MFC actuator. (**a**) Test stand, 1—the object with MFC actuator, 2—a laser head, 3—a vibrometer, 4—PC, 5—computer with real time system and data acquisition cards, 6—high voltage amplifier 7—MPS sensor; (**b**) The Star MFC; (**c**) Illustration of the vibration mode of the plate at 193 Hz. system. In this equation \(y(t)\) represents output signal, \(u(t)\) denotes the input signal and \(v(t)\) is some disturbance signal. These signals are sampled in discrete time \(t=1,2,3,\dots\). Using the backward shift operator \(z^{-1}\) this model (1) can be rewritten as:  
\[A(z^{-1})y(t)=z^{-d}B(z^{-1})u(t)+v(t), \tag{2}\]  
where \(A(z^{-1})\) and \(B(z^{-1})\) are polynomials in the delay operator:  
\[A(z^{-1})=1+a_{1}z^{-1}+\dots+a_{nA}z^{-nA} \tag{3}\]  
\[B(z^{-1})=b_{1}z^{-1}+\dots+b_{nB}z^{-nB}. \tag{4}\]  
The Equation (2) can be converted to the standard linear regression notation:  
\[y(t)=\mathbf{\Theta}^{T}\mathbf{\phi}^{T}(t)+v(t), \tag{5}\]  
where \(\mathbf{\Theta}\) is the parameter vector  
\[\mathbf{\Theta}^{T}=[a_{1},\ \dots,\ a_{nA},\ b_{1},\ \dots,\ b_{nB}], \tag{6}\]  
\(\mathbf{\phi}(t)\) is the vector of lagged input-output data  
\[\mathbf{\phi}^{T}(t)=[-y(t-1),\ \dots,\ -y(t-nA),\ u(t-d-1)\dots,\ u(t-d-nB)]. \tag{7}\]  
This model describes the observed variable \(y(t)\) as an unknown linear combination of the components of the measured vector \(\mathbf{\phi}(t)\) plus disturbance \(v(t)\). To model a disturbance term for the considered object we use the structure commonly called the Auto-Regressive model with exogenous input (ARX) [23]  
\[y(t)=z^{-d}\frac{B(z^{-1})}{A(z^{-1})}u(t)+\frac{1}{A(z^{-1})}v(t). \tag{8}\]  
Using such adopted vectors and the RLS (Recursive Least Squares) algorithm, it is possible to determine the estimated parameters of the model. This leads to a recursive algorithm, the terms of which are defined by [16]  
\[\epsilon(t)=y(t)-\mathbf{\hat{\Theta}}^{T}(t-1)\mathbf{\phi}(t) \tag{9}\]  
\[\mathbf{\hat{\Theta}}(t)=\mathbf{\hat{\Theta}}(t-1)+\mathbf{L}(t)\epsilon(t) \tag{10}\]  
\[A(z^{-1})y(t)=z^{-d}B(z^{-1})u(t)+v(t), \tag{2}\]0  
\[A(z^{-1})y(t)=z^{-d}B(z^{-1})u(t)+v(t), \tag{2}\]1 Arrangement of system identification in an adaptive control system for the object described by ARX model. \[A(z^{-1})y(t)=z^{-d}B(z^{-1})u(t)+v(t), \tag{2}\]2  
The algorithm can be improved by the use of regularization improving its stability and convergence [15; 16]:  
\[A(z^{-1})y(t)=z^{-d}B(z^{-1})u(t)+v(t), \tag{2}\]3  
In the above expression, \(\mathbf{I}\) is the identity matrix, while \(\gamma_{1}\), \(\gamma_{2}\) are the parameters controlling the upper and lower regularization range of the covariance matrix. During the tests, the regularization parameters were set to the values \(\gamma_{1}=2.5e-5\), \(\gamma_{2}=10\), for which the algorithm was stable and adjusted the model parameters relatively quickly to changing conditions.