For a linear MIMO system, the discrete state space representation is given by: (1) where , , and are system matrices, is the state, is the input, is the output, and represent the ignored nonlinearity and uncertainties of the piezo-actuator-driven stage and m and q are the number of inputs and outputs, respectively. By iteration, one has: (2) for any , where u_p and y_p are defined as column vectors of the input and output data going p steps towards the future, (3) v_p and w_p are defined as column vectors of the noises and disturbance going p steps towards the future, B_p is the controllability matrix, C_p is the observability matrix, D_p is the Toeplitz matrix for the system Markov parameters and (4)If , there exists an interaction matrix M such that: (5)Substituting Equation (5) into Equation (2) yields: (6)

Combining Equations (2) and (6) leads to the following equation, which is the so-called simplified Hankel-Toeplitz model: (7) where Using the following denotations: (8) one has Equation (7) rewritten as: (9) where represents the combined model noises and can be regarded as the model estimation error.

Define: (10) the square matrix, H₀, can be estimated without knowing M.

Once H₀ is identified, an adjacent can be calculated by using Equation (5) such that: (11) Similarly, (12) Using as building blocks, a Hankel matrix of any size can be constructed. For example: (13)

The Hankel matrix is arranged with Markov parameters of increasing order going from left to right. Let the Hankel matrices be: (14) where . Comparing Equation (14) with Equations (4), (12) and (13); and can then be extracted from H by rearrangement of its elements. The state space matrices are reconstructed from the Hankel matrix by employing the following Lemma 1.

Lemma 1: An s-th order state space model can be reconstructed as: (15) where B is the first m columns of , C is the first q rows of and . The matrix U_s and V_s are made up of s left and right singular vectors of , and the diagonal matrix, , is made up of s corresponding singular values of [24].

Equation (9) can be rewritten as, by ignoring : (16) where

By using the least squares method, θ is identified to be a time-invariant matrix, which might not be able to accurately describe the environment-dependent performance of the piezo-actuator drive stage. In order to apply the state space model in the control of piezo-actuator-driven stage, the model parameters should be updated as new observation data is available. Therefore, MAP online estimation was employed to identify the parameter matrix in Equation (16) instead.

The MAP online estimation method is used to update the parameters as the new observation data points becomes available, which is given by: (17) where X has the same definition as the one given in Equation (16), is the value of identified parameters based on the first i groups of data, P_i is the covariance of identified parameters from the first i groups of data, is the variance matrix of measurement errors and E_i is the estimation error of the i-th group of data. Integration of the prior information regarding the parameters and the information regarding the measurement errors can have the beneficial effect of reducing variances of parameter estimators. As a result, the parameter identification could be improved.

Since the Hankel-Toeplitz model is a regression model given the zero initial condition, E_i was also calculated by using the regression method as: (18) where y_pi is the measurement output of the piezo-actuator-driven stage and is the estimation output of the piezo-actuator-driven stage calculated through i-1 iterations.

A three-input-three-output state space model (1) is employed for the 3-DOF piezo-actuator drive stage. By implementing the singular value decomposition on the Hankel matrix, which is estimated based on the Hankel-Toeplitz model, as shown in Lemma 1, the system matrices of the state space model can be derived.

Since the 3-DOF piezo-actuator-driven stage is previously assumed to be linear, the model identification can be implemented on each input channel individually. For example, when an input signal is only provided in one channel , the three-dimensional output , can be obtained from the identified one-input-three-output model by applying the method mentioned above: (19) where , , and are system matrices of the one-input-three-output system.

The states for all three channels in Equation (19) may be stacked as: (20) According to the definition of the linear system, the output can be expressed as the sum of , such that: (21) As such, the state space model for the three-input-three-output system can be expressed as: (22) where:

To examine the linearity of the 3-DOF piezo-actuator-driven stage, a case study was conducted prior to the system identification. In particular, a 1 Hz 1 μm sinusoidal reference signal with 1 μm offset, a 2 Hz 200 μm sinusoidal reference signal with 2 s time delay and a 100 μm step reference signal with 3 s time delay were provided to the Z, R_x and R_y channel, respectively, and the corresponding outputs were measured. Then, the stage displacement output, as these three signals were applied simultaneously, was measured. The criterion used for the linearity examination is that if the output with three input signals equals or approximately equals the sum of the outputs when the signals is applied individually, the 3-DOF piezo-actuator-driven stage is linear or can be approximately considered to be linear. Figure 2 shows the comparison between the two outputs mentioned above. It can be seen that they overlapped with each other, indicating that the stage can be approximately considered to be a linear system. Differences between the measured output when the three inputs were provided to the different channels simultaneously and the sum of the outputs when the three inputs were provided separately exist. For example, in the R_x direction, the maximum difference is approximately 3 μm, which is only 1.5% of the amplitude of the reference signal. This difference might be due to the nonlinearities of the 3-DOF piezo-actuator-driven stage, which is ignored in the model development presented in this paper.

Figure 3 shows the flow chart of system identification. Since different signals applied in system identification may lead to the difference in the model identified, the effects of applying the random signal and the chirp signal in the parameter estimation were investigated in the signal selection in this study. The two signals were compared, and the one with less model prediction error was employed as the input for order selection, in which state space models with different orders were identified and compared. The one with less model prediction error was employed as the model for the piezo-actuator-driven stage.

For signal selection, a 20 μm reference chirp signal with 20 μm offset and frequency ranging from 1 to 100 Hz was provided to channel 1 (reference Z channel), and the corresponding output in each channel was measured. Based on the empirical knowledge of our previous study on piezoelectric actuators, the order of the state space model was originally set to be three, and in Equation (17) was set to be a zero matrix. Since the covariance of the parameters is unknown, P₀ is set to be a diagonal matrix with big covariance designated in the diagonal elements. By applying online estimation with the identified Hankel matrix, the system matrices of the state space model (Equation (19), I = 1) were obtained.

The estimation error varied, depending on the values of parameter p in Equation (2). Figure 4 (a–c) shows the estimated error versus the p-value. It can be seen that if p = 8, the estimation errors in all three output directions approached or reached their individual minimum values. Therefore, it is reasonable to set p = 8, as the chirp signal is provided to channel 1.

For other two channels, a 200 μrad reference chirp signal with frequency ranging from 1 to 100 Hz was applied. By employing the aforementioned procedure, p was set to 25 and 27 for channel 2 and 3 respectively. Table 2 shows the prediction error in each direction, as a 1 Hz sinusoidal reference input was applied to the three channels, respectively. The prediction errors are calculated in terms of the 2-norm of the error vector (defined as the difference between the measurement and the model prediction). It is seen that the diagonal prediction error is 0.1944 μm, 4.864 μrad and 4.3387 μrad in the Z, R_x and R_y direction, respectively, which is 0.49%, 2.43% and 2.16% of the desired movement in the individual direction.

Similar to the use of the chirp signal, 40 μm and 200 μrad reference random signals were also applied to each channel, respectively. The order of each sub-model was chosen to be three, and p was set to nine, 14 and 13 for the three channels, respectively. The same 1 Hz sinusoidal inputs were provided to difference channels, and the output was measured and compared with the model prediction. Table 3 illustrates the model prediction error. In contrast to the chirp signal, it can be concluded that the model prediction errors is much bigger when random signals are used in the model identification. For example, when a 1 Hz 200 μrad sinusoidal reference input was provided to channel 2, the model prediction error in the R_x direction reached 57.362 μrad by using the random inputs, which is over 10-times larger than that derived by using the chirp signal. As a result, a chirp signal was employed as the reference input for model identification below.

To determine the order of the state space model, the parameter identification, as described previously, was repeated with varying values of n (Equation (1)) in each channel. Table 4, Table 5, Table 6 show the estimation errors in each channel.

Parameter p was chosen to have different values for varying orders based on the method mentioned above. It can be concluded that if the chirp signal was used in channel 1, the estimation error in the Z direction reached its minimum value of 1.4906 μm with the order of the sub-model being six or seven. For the R_y direction, the optimal choice was to set n = 7. Therefore, the sub-model for channel 1 was considered to be a seventh order state space system. The system matrices were determined as given in Equation (23). Using a similar procedure, the orders of the sub-model for the other two channels were both chosen to be four, and the system matrices were determined, as shown in Equations (24) and (25).

To illustrate the effectiveness of the MAP online estimation method, 1, 5 and 10 Hz sinusoidal reference inputs were provided to different channels, respectively. For comparison, the estimation method introduced in [25] was implemented as well. The parameter p was defined as 21, four and seven, respectively, for the three input channels. Table 7, Table 8 show the prediction error in each direction based on the different identification methods. The prediction errors were calculated in terms of the 2-norm of the error vector, illustrating that the prediction error increases with the frequency.

In contrast to the identification method introduced in [25], the use of posteriori parameter information in MAP online estimation leads to better estimations on the Hankel matrix. For example, the estimation errors for the 5 Hz, 10 μm sinusoidal inputs to channel 1 were 0.3666 μm, 0.3843 μrad and 0.0956 in the Z, R_x and R_y directions, respectively. These results are 7.3%, 40.6% and 24%, respectively, of those derived using the identification method introduced in [25].

Figure 5 shows the output in each direction as a result of a 10 μm 10 Hz sinusoidal reference input with 10 μm offset in the Z direction compared with the model prediction. It can be clearly seen that the identified state space model is able to describe the coupling effect between each axle.

To verify the identified linear state space model with combined inputs, three experiments were implemented. In the first experiment, the reference inputs simultaneously applied to the three channels are a 1 Hz and 20 μm sinusoidal reference with a 20 μm offset, a 2 Hz and 200 μrad sinusoidal reference with a time delay of two seconds and a 100 μrad step input with a time delay of three seconds. The outputs in the three directions were measured, and the predicted outputs were obtained according to the identified state space model of Equations (23–25), respectively. In the second experiment, a 1 Hz and 1 μm sinusoidal input with 1 μm offset and a 2 Hz, 0.5 μrad sinusoidal reference input with a 2 s time delay were provided to the piezo-actuator-driven stage with the third channel kept zero. The outputs were measured and compared to the predicted outputs. To validate the model in high frequency, the input of 1 Hz and 2 μm sinusoid in channel 1 was replaced with a 10 Hz 40 μm one in the third experiment. Also, the corresponding outputs in the three directions were measured and compared to the outputs predicted by the identified state space model. The comparison is shown in Figure 6, from which it can be concluded that the model is able to describe the performance (both dynamics and cross-coupling effect) of the 3-DOF piezo-actuator-driven stage.