Figure 1 shows a block diagram of a system to be identified. Its state-space equation is given by

$\Sigma \left(x\right):\{\begin{array}{l}\stackrel{\dot{}}{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)\\ y\left(t\right)=Cx\left(t\right)+Du\left(t\right)\end{array}$, (4)

where t is the continuous time. $u\in {ℛ}^m$ is the input, $y\in {ℛ}^p$ is the output, and $x\in {ℛ}^n$ is the state vector. The measurable output is given by

$\tilde{y}\left(t\right)=y\left(t\right)+v\left(t\right)$. (5)

where $v\in {ℛ}^p$ is the measurement noise. The input and the output for identification, denoted as ID-input and ID-output, are therefore given by

ID-input: $u\left(t\right)$, ID-output: $\tilde{y}\left(t\right)$

As a matter of fact, in the system identification calculation, discrete-time signals sampled with a constant sampling interval T_s are used. To distinguish the discrete-time signals from the continuous-time signals, the discrete-time input obtained by sampling at t = kT_s is denoted as

$u\left[k\right]\triangleq u\left(k{T}_s\right)\left(k=0,1,2,\cdots \right)$. (6)

Other discrete-time signals are similarly denoted. The objective of the identification in this paper is to estimate the continuous-time LTI system Equation (4) using N paired discrete-time ID-input and ID-output data

$\left\{u\left[k\right],\tilde{y}\left[k\right]\right\}\left(k=0,1,2,\cdots ,N-1\right)$. (7)

The identification procedures are given as follows, where the order of the system n is known.

Step 1: Using N paired discrete-time ID-input and ID-output data (7), the following discrete-time LTI model is estimated using a subspace identification method.

$\{\begin{array}{l}\stackrel{\dot{}}{\xi }\left[k+1\right]=P_d\xi \left[k\right]+Q_{d}u\left[k\right]\\ y\left[k\right]=R_d\xi \left[k\right]+S_{d}u\left[k\right]\end{array}$, (8)

where $\xi \in {ℛ}^n$ is a state vector used in the estimated model.

Step 2: A continuous-time LTI model is obtained by inversely transforming Equation (8) with the zero-th order hold; that is,

$\{\begin{array}{l}\stackrel{\dot{}}{\xi }\left(t\right)=P\xi \left(t\right)+Qu\left(t\right)\\ y\left(t\right)=R\xi \left(t\right)+Su\left(t\right)\end{array}$ (9)

where

$\left[\begin{array}{cc}P& Q\\ 0& 0\end{array}\right]=\frac{1}{T_s}\ln \left[\begin{array}{cc}{P}_d& Q_d\\ 0& I_m\end{array}\right],R=R_d\begin{array}{cc},& S=S_d\end{array}$ (10)

Figure 2 shows a system combined with an actuator system $\Gamma \left(z\right)$. It is assumed that $\Gamma \left(z\right)$ is known in advance. Its state-space equation is given by

$\Gamma \left(z\right):\{\begin{array}{l}\stackrel{\dot{}}{z}\left(t\right)=Fz\left(t\right)+G\delta \left(t\right)\\ u\left(t\right)=Hz\left(t\right)\end{array}$, (11)

where $\delta \in {ℛ}^m$ is the command input to the actuator system, and $z\in {ℛ}^{2m}$ is state vector. In this paper, the actuator is given by the second-order model for each input channel. The state-space equation of the i-th actuator is given by

$\{\begin{array}{l}{\stackrel{\dot{}}{z}}_i\left(t\right)=F_i{z}_i\left(t\right)+G_i{\delta }_i\left(t\right)\\ u_i\left(t\right)=H_i{z}_i\left(t\right)\end{array}\left(i=1,\cdots ,m\right)$, (12)

where $z_i\in {ℛ}^2$ and

$F_i=\left[\begin{array}{cc}0& 1\\ -{\omega }_i^2& -2{\varsigma }_i{\omega }_i\end{array}\right],G_i=\left[\begin{array}{c}0\\ {\omega }_i^2\end{array}\right],H_i=\left[\begin{array}{cc}1& 0\end{array}\right]\left(i=1,\cdots ,m\right)$. (13)

The matrices and the vectors of Equation (11) are then constructed as

$\begin{array}{l}F\triangleq \text{block}\cdot \text{diag}\left\{F_1\cdots F_m\right\},G\triangleq \text{block}\cdot \text{diag}\left\{G_1\cdots G_m\right\},\\ H\triangleq \text{block}\cdot \text{diag}\left\{H_1\cdots H_m\right\}.\end{array}$ (14)

$\delta \triangleq {\left[{\delta }_1\cdots {\delta }_m\right]}^{\text{T}},u\triangleq {\left[u_1\cdots u_m\right]}^{\text{T}},z\triangleq {\left[z_1\cdots z_m\right]}^{\text{T}}$. (15)

Table 1 shows the choices of ID-input and ID-output for the system identification with actuator. id-0 indicates the case without actuator system, while the rest are the cases with the actuator system. id-1 is the case in which ID-input is given by δ(t). id-2 is the case in which ID-input is given by u(t) which is the output from the actuator system. A measurement is needed to obtain u(t). id-3 is the case that the measurement noise w(t) is taken into consideration in ID-input; that is,

$\tilde{u}\left(t\right)=u\left(t\right)+w\left(t\right)$. (16)

Maximal length sequence (M-sequence) [13] - [15] is a binary signal having values of {0, 1} or {−1, 1}. The persistent excitation condition holds [1] ; that is, the M-sequence can be used as an exciting signal. In general, the effective frequency range of the discrete-time signal with the sampling interval T_s is given by [0, ω_N], where ω_N is the Nyquist frequency defined as ω_N = π/T_s. However, it may be impossible to realize fast excitation because of the frequency characteristic limits of the actuators for control surfaces of aircraft. In this paper, the required activities of the control surfaces are specified by the bandwidth ratio of the M-sequence, denoted as B_M. Letting T_i be the minimum time length where the value of the M-sequence is not changed, B_M is defined as

$B_M\triangleq \frac{T_s}{T_i}$. (17)

This subsection presents the identification indexes. The two indexes are related to the dynamical modes of aircraft [11] . In the following, the subscript # corresponds to the longitudinal dynamical modes of aircraft: sp: short-period and lp: long-period modes. The first is the bandwidth ratio for the dynamical mode, which is defined as

$B_{\#}\triangleq \frac{{\omega }_N{B}_M}{{\omega }_{\#}}$. (18)

B_# > 1 indicates that the bandwidth of the M-sequence is covered beyond that of the dynamical mode. Since ${\omega }_{sp}>{\omega }_{lp}$ in the longitudinal motion, we have $B_{sp}<B_{lp}$. Then, if B_sp is greater than one, the M-sequence is able to excite both the short- and long-period modes.

The other is the ratio of data for the dynamical mode which is defined as

$R_{\#}\triangleq \frac{{\omega }_{\#}N}{2{\omega }_N}$. (19)

R_# > 1 indicates that the amount of data for identification is greater than the period of the dynamical mode. In longitudinal motion, we have $R_{sp}>R_{lp}$. Then, if R_lp is greater than one, the number of data used for identification is greater than that of data needed for the period of the longitudinal dynamical modes.

When actuators for control surfaces are considered, another index is the bandwidth ratio between the actuator and the dynamical mode. It is defined as

$W_{\#}\triangleq \frac{{\omega }_a}{{\omega }_{\#}}$. (20)

where ω_a is the natural frequency of the actuator. When multiple actuator models are needed as Equations (11)-(15), ω_a is given by

${\omega }_a\triangleq \min \left\{{\omega }_i\right\}\left(i=1,\cdots ,m\right)$. (21)

W_# > 1 indicates that the bandwidth of the actuators is able to excite the dynamical modes of aircraft. Since ${\omega }_{sp}>{\omega }_{lp}$ in the longitudinal motion, we have $W_{sp}<W_{lp}$. Then, W_sp is used as an index to characterize the actuator in this paper.

Identification results are generally evaluated from several standpoints: visual and numerical methods, global and local viewpoints. The time and frequency responses are visual and global evaluations. As a numerical evaluation, this paper adopts the ν-gap metric and the additive pole error of the dynamical modes. These definitions are given as follows.

Let the transfer functions of the true and estimated continuous-time linear models be $P^{*}\left(s\right)$ and $\hat{P}\left(s\right)$, respectively, where s is the Laplace operator. The ν-gap metric, denoted as δ_ν, is defined as

${\delta }_{\nu }\triangleq \underset{\omega }{\sup }\kappa \left(\hat{P}\left(j\omega \right),P^{*}\left(j\omega \right)\right)$. (22)

where

$\kappa \left(X,Y\right)\triangleq \bar{\sigma }\left[{\left(I+Y\bar{Y}\right)}^{-1/2}\left(Y-X\right){\left(I+X\bar{X}\right)}^{-1/2}\right]$.

$\bar{X}\left(s\right)$ is the conjugate transfer function of $X\left(s\right)$ and $\bar{\sigma }[\cdot ]$ is the maximum singular value. The range of δ_ν is normalized as ${\delta }_{\nu }\in \left[0,1\right]$ [16] . δ_ν represents a measure of the difference between two LTI systems in the frequency response. Therefore, δ_ν is one of the global evaluations. δ_ν is originally derived from the robust stability condition based on the normalized coprime factorization [16] . It is desirable in system identification that δ_ν should be as small as possible.

Letting the true and estimated eigenvalues of the dynamical mode ${\lambda }_{\#}$ be ${\lambda }_{\#}^{*}$ and ${\hat{\lambda }}_{\#}$, respectively, the additive pole error of dynamical mode is defined as

${\epsilon }_a\left({\lambda }_{\#}\right)\triangleq \left|{\hat{\lambda }}_{\#}-{\lambda }_{\#}^{*}\right|$. (23)

This is a local evaluation of the identified model.

The identification results with considering actuator are shown in Table 2 and Figures 3 - 5 . The magnitude of the measurement noises v(t) and w(t) were given by NSR_v = 30% and NSR_w = 20%, respectively. “true” in the figures means the responses and the location of the true longitudinal aircraft model.

In the cases of W_sp = 2 in Table 2 and Figure 5 , compared with id-0, which is the case without an actuator, the identification performance of id-1 and id-2 has deteriorated; in particular, ε_a(λ_sp) of id-1 and id-2 was greater than that of id-0. The influence of ε_a(λ_sp) appeared in δ_ν, the random and frequency responses. In particular, the pitch rate q in id-2 showed vibrational responses, which were close to unstable responses, as shown in Figure 3(a) . The peak gain of q in id-2 was shifted to the higher frequency region, as shown in Figure 4(a) .

In the cases of W_sp = 6, on the other hand, the identification performance of id-1 and id-2 was close to that of id-0. The performance of id-3 was almost the same level as that of id-0 in both W_sp = 2 and 6. The details will be discussed later. The condition number [18] , denoted as “Cond” shown in Table 2 , will also be mentioned. It is seen from Figure 5 that λ_lp of id-1 and id-2 almost coincided with the true location. This means that the long-period mode was not influenced by the actuator.

In the aircraft model identification without an actuator [11] , B_sp was strongly related to ε_a(λ_sp), while R_lp was to ε_a(λ_lp). The reversed pairs were not related so much. Therefore, this subsection discusses the identification performance of id-1 and id-2 with respect to B_sp and W_sp, where R_lp was fixed at R_lp = 1.6.

Figure 6 shows δ_ν and ε_a(λ_sp) with respect to W_sp. When W_sp was increased, δ_ν and ε_a(λ_sp) were monotonously decreased. Furthermore, the identification performance of id-1 and id-2 was improved by increasing B_sp. The performance

was converged to that of id-0 when $B_{sp}\ge 1.4$ and W_sp = 8. This is the lower limit of the identification with the actuator. In the region of small B_sp and W_sp, δ_ν of id-1 was better than that of id-2. In the region of large B_sp and W_sp, on the other hand, this situation was changed as shown in Figure 6(a) .

Figure 7 shows the identification result with respect to the input noise w(t); that is, the case of id-3. When B_sp and W_sp were small, there was the minimum region of δ_ν and ε_a(λ_sp) around NSR_w = 20%. According to the increase of B_sp and W_sp, the minimum region became ambiguous. It disappeared for $B_{sp}\ge 0.6$ and W_sp = 8. NSR_w of id-3 shown in Figure 6 was given by NSR_w = 20%. The identification performance of id-3 was almost the same as that of id-0 for the range of $W_{sp}\in \left[2,8\right]$.

As mentioned above, $\tilde{u}\left(t\right)$ was contributed to improving the performance of aircraft model identification with actuators. The time history of ID-input, the elevator angle δ_e and the throttle angle δ_t for B_sp = 1.0 and W_sp = 2 was shown in Figure 8 , where δ(t), u(t) and $\tilde{u}\left(t\right)$ are ID-input of id-1, id-2 and id-3, respectively. The NSR of w(t) included in $\tilde{u}\left(t\right)$ was given by NSR_w = 20%. The response of u(t) was delayed due to the dynamics of the actuators. Furthermore, to investigate this from the frequency characteristic point of view, Figure 9 shows the spectra of δ_e in δ(t), u(t) and $\tilde{u}\left(t\right)$. Two magenta asterisks indicate the natural frequencies of the short- and the long-period modes. The spectra of the three input signals were large in the frequency region shown by red triangle-line. This means that the bandwidth of ID input covers the range of dynamical modes. In the high-frequency region, the spectrum in δ(t) was decreased, but a specific magnitude was kept until the Nyquist frequency ω_N = 31.4 [rad/s]. On the other hand, the spectrum in u(t) was further reduced, while that in $\tilde{u}\left(t\right)$ was recovered by the noise w(t).

It may be possible to explain the effectiveness of the spectrum recovery in the identification performance as follows. In the N4SID method, the matrices of the estimated discrete-time model in Step 1 shown in Section 2.1 are obtained by applying the least square method to the following linear equation

$\left[\begin{array}{c}{X}_{k+1}\\ Y_k\end{array}\right]=\left[\begin{array}{cc}{P}_d& Q_d\\ R_d& S_d\end{array}\right]\left[\begin{array}{c}{X}_k\\ U_k\end{array}\right]$ (25)

where $U_k\in {ℛ}^{m\times N}$ and $Y_k\in {ℛ}^{p\times N}$ are respectively constructed by ID-input and ID-output, and $X_k\in {ℛ}^{n\times N}$ is obtained by LQ- and SV-decomposition [3] [4] . It is, in general, known in numerical computation that the accuracy of the solution depends on the algorithm and the condition of the problem [19] . Since the N4SID method was used for all identification calculations, the accuracy of the solution depended on the latter. The right-hand side column in Table 2 , denoted as “Cond”, shows the condition number of ${\Phi }_k{\Phi }_k^{\text{T}}$, where ${\Phi }_k\triangleq {\left[\begin{array}{cc}{X}_k^{\text{T}}& U_k^{\text{T}}\end{array}\right]}^{\text{T}}$. In general, a large condition number means ill-conditioned; that is, easily brings a large error in the solution, while a small condition number means well-conditioned [18] [19] . As shown in Table 2 , the Cond of id-2 in W_sp = 2 was larger than that of the others, while it became small in W_sp = 6. As a matter of fact, the identification performance of id-2 in W_sp = 6 was improved. Since the input noise w(t) was included in U_k, it contributed to the reduction of the condition number. This was a reason from the numerical computational point of view why the identification performance of id-3 was superior to that of id-1 and id-2.