Aproblem found in several control and identification applications is the reconstruction of the unmeasurable state variables by the measures of the accessible ones [1][9]. This task has been extensively studied for both linear and nonlinear systems. For the formers, a rather standard solution is given by the Luenberger observer and the Kalman filter [10], [11]. On the other hand, when dealing with nonlinear systems, the problem of designing an observer is much more challenging. Many attempts have been made to provide a general framework that allows the structured design of observers. For instance, in [5], [12], [13], the observability problem is addressed by considering observers yielding error dynamics that, possibly after some coordinates transformation, become linear and spectrally assignable. Another technique that is widely used in industrial and manufacturing processes is the extended Kalman filter, whose design is based on a local linearization of the system around a reference trajectory [14]. A remarkable observer design technique has been proposed in [15], where Lyapunovlike conditions have been given for the existence of a nonlinear observer yielding asymptotically stable error dynamics (for more recent procedures allowing the structured design of observers, see [16], [17]).
The observer proposed in this paper belongs to the class of highgain practical observers. Assuming that the system is in observability form and that the time derivatives of the output are bounded, such observers provide estimates of the state of the system yielding arbitrarily small estimation error with arbitrarily fast decay rate. The use of highgains is a classical tool that has been extensively employed to compensate nonlinearities in the system: for instance in [18] a highgain feedback stabilizing control algorithm is proposed for a class of nonlinear systems, in [19][22] it is shown how highgain observers can be exploited to estimate the state of a nonlinear system, while in [23] it is shown how highgain observers can be used in nonlinear feedback control.
The main objective of this paper is to show that, if the highgain practical observer is designed to estimate the time derivatives of the output up to an order that is greater than the dimension of the state of the system (thus leading to the adjective oversized), then the estimation error can be made smaller without increasing the gain. Thanks to their appealing properties (especially the fact that they do not require excessively large values of the observer gain), these observers have been already proved useful in several control and identification applications [24][28]. The performances of oversized and normalsized highgain practical observers are compared by estimating the vertical velocity of an electron beam by measures collected at the Frascati Tokamak upgrade (FTU) facility.
Ⅱ. OBSERVABILITY FOR NONLINEAR SYSTEMConsider the singleoutput, nonlinear system
$ \begin{align} \dot{x}=f(t, x) \end{align} $  (1a) 
$ y = h(x) $  (1b) 
where
In this paper, singleoutput, (possibly, time varying) nonlinear systems that can be written in the following canonical observability form are considered:
$\begin{align} \dot{y}_0 &= y_1\\ &~~\vdots \end{align} $  (2a) 
$ \dot{y}_{N1} = y_N $  (2b) 
$ \dot{y}_N = \bar{p}(t, y_{e, N}) $  (2c) 
$ y = {y_0} $  (2d) 
The goal of this paper is to design an observer for system (2). Such a goal can be pursued by using classical highgain practical observers (see, for instance, [20] and Section Ⅲ where the properties of such a class of observers are recalled). One of the main goals of this paper is to show that the performances of such observers can be improved by estimating, through another highgain observer, more than
In this section, some results about the standard normalsized highgain practical observers introduced in [20] are revised.
Let the polynomial
$\begin{align} \dot{\hat{y}}_0 &= \hat{y}_1+\frac{\bar{\kappa}_1}{\bar{\varepsilon}}(y_0\hat{y}_0)\\&~~\vdots\end{align} $  (3a) 
$ \dot{\hat{y}}_{N1} = \hat{y}_N+\frac{\bar{\kappa}_N}{\bar{\varepsilon}^N}(y_0\hat{y}_0) $  (3b) 
$ \dot{\hat{y}}_N = \frac{\bar{\kappa}_{N+1}}{\bar{\varepsilon}^{N+1}}(y_0\hat{y}_0) $  (3c) 
where
Define the estimation error
$ \dot{\tilde{y}}_{e, N}=A_1\tilde{y}_{e, N}+B_1 \bar{p}(t, y_{e, N}) $  (4) 
where
$ A_1=\left[\begin{array}{cccc} \frac{\bar{\kappa}_1}{\bar{\varepsilon}} & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\bar{\kappa}_N}{\bar{\varepsilon}^N} & 0 & \cdots & 1 \\ \frac{\bar{\kappa}_N+1}{\bar{\varepsilon}^{N+1}} & 0 & \cdots & 0 \end{array}\right], \quad B_1 = \left[\begin{array}{c} 0 \\ \vdots \\ 0 \\ 1 \end{array}\right]. $ 
The following two lemmas and theorem, reported here for completeness, state that the output of the highgain observer given in (3) is a practical estimate of the state of system (2).
Lemma 1 [20]: Let system (4) be given. There exists an
$ A_1=\frac{1}{{\bar{\varepsilon}}}\bar{E}_{\bar{\varepsilon}}^{1}\Delta \bar{E}_{\bar{\varepsilon}}, \qquad B_1=\frac{1}{{\bar{\varepsilon}}}\bar{E}_{\bar{\varepsilon}}^{1}\Gamma $ 
where
$ \Delta=\left[\begin{array}{ccccc} \bar{\kappa}_1 & 1 & 0 & \cdots & 0\\ \bar{\kappa}_2 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \bar{\kappa}_N & 0 & 0 & \cdots & 1\\ \bar{\kappa}_{N+1} & 0 & 0 & \cdots & 0\\ \end{array}\right], \quad\Gamma = \left[\begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ {\bar{\varepsilon}}^{N+1} \end{array}\right] $ 
additionally, one has that
$ \begin{array}{rcl} &&\exp\left(A_1\tau\right) =\bar{E}_{\bar{\varepsilon}}^{1}\exp\left(\dfrac{1}{{\bar{\varepsilon}}}\Delta\, \tau\right)\bar{E}_{\bar{\varepsilon}} \quad\quad\forall\tau>0\\ &&\exp\left(A_1\tau\right)B_1 = \dfrac{1}{{\bar{\varepsilon}}}\bar{E}_{\bar{\varepsilon}}^{1}\exp\left(\dfrac{1}{{\bar{\varepsilon}}}\Delta\, \tau\right)\Gamma \quad\quad\forall\tau>0. \end{array} $ 
Lemma 2 [20]: Let
$ \Delta^{T} \bar{P}+\bar{P} \Delta = I. $  (5) 
Then, setting
$ \begin{array}{rcl} &&A_1^{T} \bar{P}_{\bar{\varepsilon}}+\bar{P}_{\bar{\varepsilon}} A_1 = \dfrac{1}{{\bar{\varepsilon}}}\bar{E}_{\bar{\varepsilon}}^{T} \bar{E}_{\bar{\varepsilon}}\\ &&B_1^{T} \bar{P}_{\bar{\varepsilon}} =\dfrac{1}{{\bar{\varepsilon}}} \Gamma^{T} \bar{P} \bar{E}_{\bar{\varepsilon}} \end{array} $ 
where
Theorem 1 [20]: Consider the error dynamics given in (4). If there exists
$ \tilde{y}_{\bar{\varepsilon}}(t)\in\{\tilde{y}_{\bar{\varepsilon}}:\, \tilde{y}_{\bar{\varepsilon}}^{T} \bar{P} \tilde{y}_{\bar{\varepsilon}}\leq 4\mu^2{\bar{\varepsilon}}^{2N+2}{\bar{P}}^3\}\quad\quad \forall t\geq \bar{T} $  (6) 
where
Consider now the following system:
$\begin{align} \dot{{\xi}}_0 &= {\xi}_1+\frac{{\kappa}_1}{\varepsilon}(y_0\xi_0)\\ &~~\vdots \end{align} $  (7a) 
$ \dot{{\xi}}_{N1} = {\xi}_N+\frac{{\kappa}_N}{\varepsilon^N}(y_0\xi_0) $  (7b) 
$\begin{align} \dot{{\xi}}_N& = {\xi}_{N+1}+\frac{{\kappa}_{N+1}}{\varepsilon^{N+1}}(y_0\xi_0)\\ &~~\vdots \end{align} $  (7c) 
$ \dot{{\xi}}_{N+h} = \frac{{\kappa}_{N+h+1}}{\varepsilon^{N+h+1}}(y_0\xi_0) $  (7d) 
$ \breve{y}_{e, N} = [\begin{array}{cc} I_{N+1} ~~~~ 0_{h}\end{array}] \xi $  (7e) 
where
The goal of this section is to show that the signal
$ \dot{\check{y}}_{e, N}=A_2\check{y}_{e, N}+B_2(\bar{p}(t, y_{e, N1})\xi_{N+1}) $ 
where
$ A_2=\left[\begin{array}{cccc} \frac{{\kappa}_1}{\varepsilon} & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ \frac{{\kappa}_N}{\varepsilon^N} & 0 & \cdots & 1\\ \frac{{\kappa}_N+1}{\varepsilon^{N+1}} & 0 & \cdots & 0 \end{array}\right], \quad B_2 = \left[\begin{array}{c} 0 \\ \vdots \\ 1 \end{array}\right] $ 
and
$ \begin{array}{rcl} &&\dot{\zeta} = A_3\zeta + B_3\check{y}_{0} \\ &&\xi_{N+1} = C_3\zeta \end{array} $ 
where
$ A_3 = \left[\begin{array}{cccc} 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & 0\\ 0 & 0 & \cdots & 1\\ 0 & 0 & \cdots & 0 \end{array}\right], \quad B_3 = \left[\begin{array}{c} \frac{{\kappa}_{N+2}}{\varepsilon^{N+2}} \\ \frac{{\kappa}_{N+3}}{\varepsilon^{N+3}} \\ \vdots \\ \frac{{\kappa}_{N+h+1}}{\varepsilon^{N+h+1}} \end{array}\right]. $ 
Hence, by defining
$ \dot{\eta} = \Theta\eta+ \Lambda\bar{p}(y_{e, N1}) $  (8a) 
$ \check{y}_{e, N} = [\begin{array}{cc} I_{N+1} ~~~~~ 0_h\end{array}]\eta $  (8b) 
where
The following two lemmas provide some properties of the matrices
Lemma 3: Let system (8) be given. There exists an
$ \Theta=\frac{1}{\varepsilon}E_\varepsilon^{1}\Phi E_\varepsilon, \qquad\Lambda=\frac{1}{\varepsilon}E_\varepsilon^{1}\Psi $  (9) 
where
$ \begin{array}{rcl} \Phi_1 & = & \left[\begin{array}{ccccc} {\kappa}_1 & 1 & 0 & \cdots & 0 \\ {\kappa}_2 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\kappa}_{N} & 0 & 0 & \cdots & 1 \\ {\kappa}_{N+1} & 0 & 0 & \cdots & 0 \\ \end{array}\right]\\ \Phi_2 & = & \left[\begin{array}{cccc} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & \cdots & 0\\ \end{array}\right]\\ \Phi_3 & = & \left[\begin{array}{ccccc} {\kappa}_{N+2} & 0 & 0 & \cdots & 0 \\ {\kappa}_{N+3} & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\kappa}_{N+h} & 0 & 0 & \cdots & 0 \\ \bar{\kappa}_{N+h+1} & 0 & 0 & \cdots & 0 \\ \end{array}\right]\\ \Phi_4 & = & \left[\begin{array}{cccc} 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1\\ 0 & 0 & \cdots & 0\\ \end{array}\right]. \end{array} $ 
Additionally, one has that
$ \exp\left(\Theta\, \tau\right) = \textstyle E_\varepsilon^{1}\exp\left(\dfrac{1}{\varepsilon}\Phi\, \tau\right)E_\varepsilon $  (10a) 
$ \exp\left(\Theta\, \tau\right)\Lambda = \textstyle \dfrac{1}{\varepsilon}E_\varepsilon^{1}\exp\left(\dfrac{1}{\varepsilon}\Phi\, \tau\right)\Psi $  (10b) 
for all
Proof: The expressions in (9) follow directly from the definition of the matrices
Lemma 4: All the eigenvalues of the matrix
Proof: By construction,
Since, by Lemma 4, the matrix
$ \Phi^{T} P+P \Phi = I. $  (11) 
Thus, letting
$ \Theta^{T} P_\varepsilon+P_\varepsilon\Theta = \frac{1}{\varepsilon}E_\varepsilon^{T} E_\varepsilon $  (12a) 
$ \Lambda^{T} P_\varepsilon = \frac{1}{\varepsilon} \Psi^{T} P E_\varepsilon $  (12b) 
where
The following theorem and corollary state that
Theorem 2: Consider the error dynamics given in (8). Let
$ \eta_\varepsilon(t)\in\{\eta_\varepsilon:\, \eta_\varepsilon^{T} P \eta_\varepsilon\leq 4\mu^2\varepsilon^{2N+2}{P}^3\}\quad\quad \forall t\geq T $  (13) 
where
Proof: Let
$ V(\eta)=\eta^{T} P_\varepsilon \eta $ 
which is positivedefinite, because the matrix
$ \begin{array}{lcl} \dot{V}(\eta) & = & \dot{\eta}^{T} P_\varepsilon \eta+\eta^{T} P_\varepsilon \dot{\eta}\\ & = & (\Theta\eta+\Lambda\bar{p})^{T} P_\varepsilon \eta+ \eta^{T} P_\varepsilon (\Theta\eta+\Lambda\bar{p})\\ & = & \eta^{T}(\Theta^{T} P_\varepsilon + P_\varepsilon \Theta)\eta+2\bar{p}\Lambda^{T} P\eta.\\ \end{array} $ 
Hence, by (12a), one has that
$ \dot{V}(\eta)= \frac{1}{\varepsilon}\eta_\varepsilon^{T}\eta_\varepsilon+\frac{2}{\varepsilon}\bar{p}\Psi^{T} P\eta_\varepsilon $ 
where
$ \dot{V}(\eta) \leq \frac{1}{\varepsilon}({\eta_\varepsilon}^22 \mu\varepsilon^{N+1}{P}~{\eta_\varepsilon}). $ 
Hence, for any
Corollary 1: Let the assumptions of Theorem 2 hold. The estimation error
Proof: Let
$ {\eta_\varepsilon}=\Big\Big{E_\varepsilon \left[\begin{array}{c} \check{y}_{e, N}\\ \zeta \end{array}\right]}\Big\Big\geq {\check{E}_\varepsilon \check{y}_{e, N}} $  (14) 
where
$ {\eta_\varepsilon(t)}\leq 4\underline{\lambda}^{1}\mu^2\varepsilon^{2N+2}{P}^3\quad\quad\forall t\geq T $ 
where
In the remainder of this section, the estimates
Assumption 1: Let the coefficients
Assumption 1 is made in order to guarantee that the "gain" of the highgain observer given in (3) is the same "gain" of the highgain observer given in (7). The following proposition and corollary show that the error
Proposition 1: Let Assumption 1 hold, and let the assumptions of Theorems 1 and 2 hold. Let
Proof: Letting
Corollary 2: Let the assumptions of Proposition 1 hold. If, additionally, there does not exist a compact time interval
$ \int_{\mathcal{I}}{\check{y}_{e, N}(\tau)}^2d\tau < M^2\mathcal{I}\delta_1 $ 
for some
Proof: If there does not exist a time interval
$ \int_{\mathcal{I}}{\check{y}_{e, N}}^2(\tau)d\tau < M^2\mathcal{I}  \int_{\mathcal{I}}{\delta(\tau)}d\tau \triangleq M^2\mathcal{I} \delta_1 $  (15) 
for all
Remark 1: Note that if
Remark 2: The main advantage in the use of the oversized observer (7) relies on the fact that, usually, highgain practical observers yield estimates with larger errors in the higher order derivatives. Therefore, if one estimates time derivatives up to an order that is greater than the dimension of the system, the estimation error is gathered on the higher order derivatives (which are neglected for estimation purposes), thus leading to a smaller error in the estimation of the state of system (2), as confirmed theoretically by Proposition 1 and Corollary 2.
Ⅴ. THE LINEAR TIMEINVARIANT CASEIn this section, the filtering properties of the normalsized and of the oversized highgain practical observers given in (3) and in (7), respectively, are discussed.
Consider the error dynamics given in (4) and (8). By considering
Consider the LTI system
$ \dot{x}(t) = Ax(t)+Bu(t), \quad x(0)=x_0\\ $  (16a) 
$ y(t) = C x(t). $  (16b) 
The transfer matrix
$ y(s)=H(s)u(s). $ 
Therefore, letting
$ \tilde{y}_{e, N}(s) = \hat{C}_1(sIA_1)^{1}B_1\bar{p}(s) $  (17a) 
$ \check{y}_{e, N}(s) = \hat{C}_2(sI\Theta)^{1}\Lambda\bar{p}(s) $  (17b) 
respectively. Given
By using such an algorithm, an explicit expression of the transfer matrices of systems (4) and (8) can be obtained.
Algorithm 1 [32]: Computation of the Matrix 
Input: A matrix 
Output: The matrix 
1: Compute 
2: Define 
3: for 
4: Compute 
5: Compute 
6: end for 
7: Compute 
8: return 
Lemma 5: Let systems (4) and (8) be given. Letting
$ [H_1(s)]_\ell=\frac{\bar{\varepsilon}^{N+2\ell}\sum\limits_{j=0}^{\ell1}\bar{\kappa}_j(\bar{\varepsilon} s)^{\ell1j}} {(\bar{\varepsilon}s)^{N+1}+\sum\limits_{j=1}^{N+1}{\bar{\kappa}_j}(\bar{\varepsilon}s)^{N+1j}} $  (18) 
whereas the
$ [H_2(s)]_\ell=\frac{\varepsilon^{N+2\ell} (\varepsilon s)^h\sum\limits_{j=0}^{\ell1}{{\kappa}_j}(\varepsilon s)^{\ell1j}} {(\varepsilon s)^{N+h+1}+\sum\limits_{j=1}^{N+h+1}{\bar{\kappa}_j}({\varepsilon}s)^{N+h+1j}}. $  (19) 
Proof: By using Algorithm 1, with
$ \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 1 & 0 & \cdots & 0\\[1mm] \frac{\bar{\kappa}_{i}}{\bar{\varepsilon}^i} & 0 & \cdots & 0 & \frac{\bar{\kappa}_1}{\bar{\varepsilon}} & 1 & \cdots& 0\\[1mm] \frac{\bar{\kappa}_{i+1}}{\bar{\varepsilon}^{i+1}} & \frac{\bar{\kappa}_{i}}{\bar{\varepsilon}^{i}}& \cdots & 0 & \frac{\bar{\kappa}_2}{\bar{\varepsilon}^{2}} & \frac{\bar{\kappa}_1}{\bar{\varepsilon}} & \cdots & 0\\[1mm] \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\[1.6mm] \star & \star & \cdots & 0 & \frac{\bar{\kappa}_{i1}}{\bar{\varepsilon}^{i1}} & \frac{\bar{\kappa}_{i2}}{\bar{\varepsilon}^{i2}} & \cdots & 0\\[2mm] \star & \star & \cdots & 0 & 0 & \frac{\bar{\kappa}_{i1}}{\bar{\varepsilon}^{i1}} & \cdots & 0\\[2mm] \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\[2mm] \frac{\bar{\kappa}_{N}}{\bar{\varepsilon}^N} & \frac{\bar{\kappa}_{N1}}{\bar{\varepsilon}^{N1}} & \cdots & \star & 0 & 0 & \cdots& 0\\[2mm] \frac{\bar{\kappa}_{N+1}}{\bar{\varepsilon}^{N+1}} & \frac{\bar{\kappa}_{N}}{\bar{\varepsilon}^N} & \cdots & \star & 0 & 0 & \cdot& \star\\[2mm] 0 & \frac{\bar{\kappa}_{N+1}}{\bar{\varepsilon}^{N+1}} & \cdots & \star & 0 & 0 & \cdots& \star\\[2mm] \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\[2mm] 0 & 0 & \cdots & \frac{\bar{\kappa}_{N}}{\bar{\varepsilon}^N} & 0 & 0 & \cdots & \frac{\bar{\kappa}_{i2}}{\bar{\varepsilon}^{i2}}\\[2mm] 0 & 0 & \cdots & \frac{\bar{\kappa}_{N+1}}{\bar{\varepsilon}^{N+1}} & 0 & 0 & \cdots & \frac{\bar{\kappa}_{i1}}{\bar{\varepsilon}^{i1}} \end{smallmatrix}\right] $ 
then, by Steps 4 and 5 of Algorithm 1,
$ [H_1(s)]_\ell=\frac{\sum\limits_{j=0}^{\ell1}\frac{\bar{\kappa}_j}{\bar{\varepsilon}^j} s^{\ell1j}} {s^{N+1}+\sum\limits_{j=1}^{N+1}\frac{\bar{\kappa}_j}{\bar{\varepsilon}^j}s^{N+1j}}. $ 
To prove that
$ T = \left[\begin{array}{cc} I_{N+1} & 0\\ 0 &I_h \end{array}\right] $ 
which is trivially nonsingular. Consider now the matrix
$ R_{N+h+1i}= \left[\begin{smallmatrix} 0 & \cdots & 0 & 1 & \cdots & 0\\[2mm] \frac{{\kappa}_{i+1}}{\varepsilon^{i+1}} & \cdots & 0 & \frac{{\kappa}_1}{\varepsilon} & \cdots& 0\\[2mm] \frac{{\kappa}_{i+2}}{\varepsilon^{i+2}} & \cdots & 0 & \frac{{\kappa}_2}{\varepsilon^2} & \cdots & 0\\[2mm] \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\[2mm] \star & \cdots & 0 & \frac{{\kappa}_{i} }{\varepsilon^i} & \cdots & 0\\[2mm] \star & \cdots & 0 & 0 & \cdots & 0\\[2mm] \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\[2mm] \frac{{\kappa}_{N+h}}{\varepsilon^{N+h}} & \cdots & \star & 0 & \cdots& 0\\[2mm] \frac{{\kappa}_{N+h+1}}{\varepsilon^{N+h+1}} & \cdots & \star & 0 & \cdot& \star\\[2mm] 0 & \cdots & \star & 0 & \cdots& \star\\[2mm] \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\[2mm] 0 & \cdots & \frac{{\kappa}_{N+h}}{\varepsilon^{N+h}} & 0 & \cdots & \frac{{\kappa}_{i1}}{\varepsilon^{i1}}\\[2mm] 0 & \cdots & \frac{{\kappa}_{N+h+1}}{\varepsilon^{N+h+1}} & 0 & \cdots & \frac{{\kappa}_{i}}{\varepsilon^i} \end{smallmatrix}\right]. $ 
Therefore, by (17b) and by Step 7 of Algorithm 1 and by considering that det
$ [H_2(s)]_\ell=\frac{s^h\sum\limits_{j=0}^{\ell1}\frac{{\kappa}_j}{\varepsilon^j} s^{\ell1j}} {s^{N+h+1}+\sum\limits_{j=1}^{N+h+1}\frac{\bar{\kappa}_j}{{\varepsilon}^j}s^{N+h+1j}} $ 
where
The transfer functions given in (18) and (19) can be used to wholly characterize the filtering properties of the observers given in (3) and (7).
Remark 3: The "extra" parameters in
The optimization could be performed either on the
We now present two applications in which it is very important to provide the first derivative of the signals as accurate as possible. The first case is a numerical simulation by which we can show the improved performances of the proposed oversized observer. Consider the second order LTI system
$ P(s)=\frac{k \omega_n^2}{s^2+2\lambda\omega_n+\omega_n^2} $ 
with
$ \begin{align} \left[\begin{array}{cc} \bar{\kappa}_1 & \bar{\kappa}_2 \end{array}\right]^{T} &=\left[\begin{array}{ccc} 7.07 & 49.99 \end{array}\right]^{T}\\ \end{align} $  (20a) 
$ \begin{align} \left[\begin{array}{ccc} \bar{\kappa}_1 & \bar{\kappa}_2 & \bar{\kappa}_3 \end{array}\right]^{T} &= \left[\begin{array}{cccc} 21.38 & 221.81 & 499.99 \end{array}\right]^{T} \end{align} $  (20b) 
that correspond to
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Fig. 1 Bode plots of 
These numerical computations corroborate the theoretical results given in this paper; in fact, they suggest again that, by allowing an "oversizing" of the highgain observer given in [20], improved performances can be achieved.
Fig. 2 depicts the results of such a numerical simulation by showing the difference between the analytical time derivative of the output
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Fig. 2 Results of the numerical simulation. 
The performances of the highgain observer have improved by allowing its "oversizing". As a matter of fact, one has that the estimation error resulting from the use of the oversized highgain practical observer is lower than the error obtained by employing the normalsized one.
The two observers corresponding to the
The measured plasma vertical position is first filtered with a firstorder lowpass filter with cutoff frequency
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Fig. 3 Filtered position and estimated velocities: 
Note that the estimate of the vertical velocity, used by the feedback system to stabilize the runaway electron beam, is of crucial importance in order to avoid damages to the plant [35].
Ⅶ. CONCLUSIONSIn this paper, oversized highgain practical observers have been studied. It has been shown that, if one estimates the time derivatives of the output up to an order that is greater than the dimension of the system and takes into account just the first
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