# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 2 Robust Estimation of Model Parameters  
The objective of the application-oriented experiment design is to deliver a model of a system using open-loop data, which guarantees an acceptable performance of a control system [5; 29]. We examine the identification of a discrete-time linear time-invariant system within the prediction error framework [30]. The response of the single-input and single-output discrete true system can be expressed as  
\[y(t)=G_{0}(q,\theta_{0})u(t)+H_{0}(q,\theta_{0})e_{0}(t), \tag{1}\]  
where \(u(t)\) is the input signal, \(y(t)\) is the output sequence, \(e_{0}(t)\) is the white Gaussian noise with the zero mean value and the variance \(\lambda_{0}\), \(q\) is the time-shift operator, and \(t\) represents the experiment duration. \(G_{0}\) and \(H_{0}\) are transfer functions, where \(H_{0}\) is the stable, monic, and minimum phase, and \(\theta_{0}\) is a vector of the true parameters of the system. Let us assume the sufficient flexibility of a model; i.e., it is able to capture the dynamics of the true system. Then, the model output can be described as  
\[y(t)=G(q,\theta)u(t)+H(q,\theta)e(t), \tag{2}\]  
where the transfer functions \(G\) and \(H\) are parameterized by a vector \(\theta\in\mathbb{R}^{n}\). The estimates of the unknown parameters of the model (2) are found by minimization of the criterion function based on the prediction error (FEM), with respect to \(\theta\). The PEM is defined as the measure of the difference between the output of the real plant (the true system) and the output signal obtained from the model [30]. The output of model (2), estimated one step ahead, becomes  
\[\hat{y}(t,\theta)=H^{-1}(q,\theta)G(q,\theta)u(t)+\Big{[}1-H^{-1}(q,\theta) \Big{]}y(t). \tag{3}\]  
The closed-loop system setup, assumed for system identification purposes [27], is shown in Figure 1.  
Referring to the above scheme of a control system, the input signal \(u(t)\) is given by  
\[u(t)=-T(q)y(t)+r(t), \tag{4}\]  
where \(r(t)\) is a reference input signal (external and uncorrelated with noise), \(e(t)\) is a noise sequence, and \(T(q)\) is a linear compensator. Furthermore, it has been assumed that all signals have spectral representation. It should be noted that the PEM method is not restricted only to the systems described above. The system may be either continuous or discrete, including nonlinear multivariable systems [30]. To obtain the vector \(\theta\) of unknown parameters, based on \(N\) observations of input-output data and using the one-step-ahead predicted output (3), the following criterion function is minimized:  
\[\hat{\theta}_{N}=\underset{\theta}{\text{argmin}}\frac{1}{2N}\sum_{t=1}^{N} {(y(t)-\hat{y}(t,\theta))^{2}}. \tag{5}\]  
To determine the uncertainty in the parameter estimates, a study of their asymptotic properties is required. The properties of the distribution of the estimates define their precision. The true system is assumed to have the same structure as the model with the parameter vector \(\theta_{0}\) and the noise variance \(\lambda_{0}\), such that: \(G(\theta_{0})=G_{0}\), \(H(\theta_{0})=H_{0}\). Then, using some assumptions, the parameter estimates have the following asymptotic distribution (see [30]):  
\[ \begin{array}{c}\sqrt{N}\big{(}\hat{\theta}_{N}-\theta_{0}\big{)}\to\mathbf{ \mathcal{N}}(0,P),N\to\infty,\\ \lim_{N\to\infty}N\operatorname{E}\!\left[\big{(}\delta_{N}-\theta_{0}\big{)} \big{(}\hat{\theta}_{N}-\theta_{0}\big{)}^{T}\right]=P,\\ P=\lambda_{0}\big{(}\operatorname{E}\!\left[\psi(t,\theta_{0})\psi^{T}(t, \theta_{0})\right]\big{)}^{-1},\\ \psi(t,\theta_{0})=\frac{\partial}{\partial\theta}\hat{y}(t,\theta)\big{|}_{ \theta=\theta_{0}},\end{array}  \tag{6}\]  
where \(\mathbf{\mathcal{N}}\) represents the normal distribution, and \(P\) is the asymptotic parameter covariance matrix. When the model set contains the system under consideration, then the estimate converges to the true value. However, the signal spectrum also influences estimation accuracy.