A robustification of the two degree-of-freedom controller based upon multivariable generalized predictive control law and robust H ∞ control for a doubly-fed induction generator

Abstract

A robustification method of primary two degree-of-freedom (2-DOF) controllers is proposed in this paper to control the wind turbine system equipped with a doubly-fed induction generator DFIG. The proposed robustification method should follow the following three step-procedures. First, the primary 2-DOF controller is designed through the initial form of the multivariable generalized predictive control MGPC law to ensure a good tracking dynamic of reference trajectories. Second, the robust $H_{\infty}$ controller is independently designed for the previous system to ensure good robustness properties of the closed-loop system against model uncertainties, neglecting dynamics and sensor noises. Finally, both above mentioned controllers are combined to design the robustified 2-DOF-MGPC controller using Youla parameterization method. Therefore, the obtained controller conserves the same good tracking dynamic that is provided by the primary 2-DOF-MGPC controller. It ensures the same good robustness properties which are produced by the robust $H_{\infty}$ controller. A wind turbine system equipped with a DFIG is controlled by the robustified 2-DOF-MGPC controller. Its dynamic behaviour is modelled by an unstructured-output multiplicative uncertainty plant. The controller performances are valid by comparison with those given through both controllers, which are primary 2-DOF-MGPC and robust $H_{\infty}$ controllers in time and frequency domains.

Keywords

Multivariable generalized predictive control (MGPC)primary and robustified 2-DOF-MGPC controllers Youla parameterization doubly-fed induction generator (DFIG)

Introduction

In the last few years, renewable energy has received greater attention; wind energy is one of the most promising renewable energy resources due to the fact it is: cost efficient, clean and environmentally friendly (De Battista et al., 2003). One of the main types of wind generators is the DFIG system, which plays an important role among all possible renewable energy resources (De Battista et al., 2003; Han and Shen, 2014). This system consists of an induction wound rotor generator with stator winding connected directly to the three phase grid (Soens et al., 2005). The rotor-voltages have been controlled to ensure constant stator-voltages with a constant frequency in electrical grid, regardless the active and/or reactive energy variation, as well as, the load and speed variation conditions (Vieira et al., 2009; Belfedal et al., 2010). These goals will be attained by a specic synthesis controller choice because the requirement corrections are applied in the electrical part of the DFIG system (Belfedal et al., 2010; Qu and Song, 2011).

Generally, when stringent time-domain specications are set on the output responses, a one degree-of- freedom (1-DOF) controller may not be enough. Therefore, the 2-DOF control conguration based on the MGPC law must be used to ensure a fast tracking response whether DFIG’s dynamic behaviour changed in a short time (Zdešar et al., 2014).

Notice that, the fast tracking dynamic depends heavily on the good choice of the MGPC law tuning parameters, which are a priori given by the user (Scattolini and Schiavoni, 1992; Camacho and Alba, 2013). It also depends on the existence of a perfect linear model, called thr Controlled Auto Regressive Integrated Moving Average CARIMA model (Camacho and Alba, 2013; Zenghui et al., 2006; Clarke et al., 1987). Unfortunately, a linear model is rarely available in practice, due to the inherent complexity and nonlinearity behaviour of the DFIG system (Soens et al., 2005; Vieira et al., 2009). To resolve this problem, all nonlinearities and un-modelled dynamics should be neglected in the modelling step in which an uncertainty plant-model is produced (Camacho and Alba, 2013; Sedraoui et al., 2012). Therefore, no robustness is guaranteed by a primary 2-DOF-MGPC controller where a robustification controller step against uncertainties becomes unavoidable (Stoica et al., 2008; Gossner et al., 1997).

The MGPC robustness problem in the presence of modelling errors and sensor noises is commonly treated by increasing the robustness margin of existing designs where some degrees of freedom based on Youla parameterization are introduced. This parametrization was introduced in Kouvaritakis et al. (1992); Yoon and Clarke (1995); Gossner et al. (1997); Youla et al. (1976); Olaru and Ayerbe (2006); Stoica et al. (2008); Thomsen et al. (2011) by which an optimization problem was solved and then a Q-parameter was derived. Consequently, various robustification methods have been proposed since the 90’s. Kouvaritakis et al. (1992) first applied the Q-parameter for a Single-Input and Single-Output (SISO) system to guarantee stability and to provide a systematic means of optimizing robust stability margins. Yoon’s C-polynomial method was in fact proposed as an alternative for the Q-parametrization method to ease the computation issues (Yoon and Clarke, 1995). Unfortunately, this choice remains complicated for model-plants that are given by a transfer matrix of higher order. Rossiter et al. (1998) proposed the stable GPC approach (SGPC) that improved the robustness margin of the feedback control system. Unfortunately, the dynamic disturbance rejection always failed. Olaru and Ayerbe (2006) suggested an alternative for Q-parametrization to ensure a good trade-off between nominal performance (NP) and robust stability (RS). The trade-off problem was formulated as a weighted-mixed sensitivity design problem, which was solved by an optimization tool. However, the obtained trade-off is difficult in practice for the Multi-Inputs and Multi-Outputs (MIMO) plant-model case. Stoica et al. (2008) proposed the Q-parametrization, in which the Q-parameter is given in a MIMO state space format. Its design problem was formulated as a convex optimization problem and then it was approximated by linear programming with inequality constraints. Afterward, the components of the Q-parameter were determined through the optimal solution of above mentioned optimization problem. Unfortunately, the obtained convergence to the optimal solution becomes very hard since the convergence speed is slows down whenever approaching to the optimal solution because of the higher number of the components to be determined (Sedraoui et al., 2012). Therefore, the computational cost increases exponentially with the optimized variable number, which affects the state-space order of the Q-parameter that leads to numerical ill-conditioning (Sedraoui et al., 2012).

In the literature, the robust $H_{\infty}$ control strategies have been proposed for DFIG control system, which is modelled by a plant-model uncertainty (Belfedal et al., 2010). Moreover, the trade-off problem is formulated as Linear Matrix Inequalities (LMI) which leads to a weighted-mixed sensitivity problem (Apkarian et al., 1995; Gahinet and Apkarian, 1994; Ortega and Rubio, 2004). This problem is resolved by the $H_{\infty}$ method based on LMI’s (Apkarian et al., 1995). The obtained $H_{\infty}$ controller can ensure a good trade-off between NP and RS. However, its tracking dynamic is slower if the secure margin will increase (Apkarian and Tuan, 2000; Apkarian and Noll, 2006). In this work, we will investigate the analysis of the above-mentioned problems. Furthermore, the desired controller should conserve the same good tracking dynamic of the primary MGPC controller, whereas, it should guarantee the same good trade-off robustness that produced through robust $H_{\infty}$ one. The important difference between the robustification method proposed in this paper and those available in the literature, is the way to determine the components of the Q-parameter for MIMO case.

The GPC robustification problem

In MIMO-GPC control, the CARIMA model with $m \times n$ inputs-outputs is defined by (Scattolini and Schiavoni, 1992; Camacho and Alba, 2013)

Δ (q^{- 1}) A (q^{- 1}) y (t) = Δ (q^{- 1}) B (q^{- 1}) u (t - 1) + C (q^{- 1}) ζ (t)

(1)

where $y = [y_{1}, y_{2}, \dots, y_{m}]^{T}$ , $u = [u_{1}, u_{2}, \dots, u_{n}]^{T}$ and $ζ = [ζ_{1}, ζ_{2}, \dots, ζ_{m}]^{T}$ , are respectively, the input and output vectors of the nominal plant-model and the white Gaussian noise vector given with a constant variance and mean zero in each channel output i, for $i = 1, 2, \dots, m$ . In addition, $q^{- 1}$ is the backward-shift operator. However, $Δ = (1 - q^{- 1}) I_{m \times m}$ is a diagonal matrix, which represents the integrator part in the CARIMA model and its objective is to cancel the constant disturbance in the steady-state. Moreover, A, B and C are polynomials in backward-shift operator $q^{- 1}$ , which are respectively defined as

{\begin{matrix} A = I_{m \times m} + \sum_{j = 1}^{j = n_{a}} A_{j} q^{- j} \\ B = \sum_{j = 0}^{j = n_{b}} B_{j} q^{- j} \\ C = \sum_{j = 0}^{j = n_{c}} C_{j} q^{- j} \end{matrix}

(2)

In this paper, A will be assumed diagonal matrix and C will be equal to the identity (i.e. $C = I_{m \times m}$ ). The MIMO-GPC algorithm consists of applying a control sequence that minimizes a multistage cost function of the form

J = \sum_{k = N_{1}}^{N_{2} + N_{1} - 1} ‖ \hat{y} (t + k / t) - w (t + k) ‖_{ψ}^{2} + \sum_{k = 1}^{N_{u}} ‖ Δ u (t + k - 1) ‖_{λ}^{2}

(3)

where $N_{1}$ and $N_{2}$ are respectively, the minimum and the maximum costing horizons (Scattolini and Schiavoni, 1992; Camacho and Alba, 2013). The adjustable parameters $ψ$ and $λ$ are positive denite weighting matrices that penalize the error and control signals respectively, $N_{u}$ is the control horizon, $w (t + k)$ is the future set-point or reference sequence and $\hat{y} (t + k / t)$ is the optimum k-step ahead prediction of the plant-model outputs on data-up to a time t. The weighting norm is computed as $‖ X ‖_{F}^{2} = X^{T} FX$ . Moreover, the horizons $N_{1}$ , $N_{2}$ and $N_{u}$ can be taken independently for each input and output. Thus, $N_{1}^{(i)}$ , $N_{2}^{(i)}$ are used for the channel output i and $N_{u}^{(j)}$ is used for the channel input j.

In this work, the costing horizons $N_{1}^{(i)}$ and $N_{2}^{(i)}$ are respectively assumed equal to $N_{1}^{(i)} = 1$ and, $N_{2} = max (N_{2}^{(i)})$ for each channel output i. The control horizon $N_{u}^{(j)}$ (with respect to $N_{u}^{(j)} \leq N_{2}^{(i)}$ ) is assumed equal to $N_{u} = N_{u}^{(j)}$ for each channel input j.

The optimum k-step ahead prediction over the costing horizon $1 \leq k \leq N_{2}$ is given as

\hat{y} (t + k / t) = H_{k} Δ u (t - 1) + F_{k} y (t) + G_{k} Δ u (t + k - 1)

(4)

where, $H_{k}$ , $F_{k}$ and $G_{k}$ and are polynomial matrices in the backward-shift operator $q^{- 1}$ . Their components are determined through resolving iteratively of two Diophantine equations. For more details, see Stoica et al. (2008).

In the MIMO-GPC algorithm, the receding horizon principle assumes that only the n first rows of the optimal control sequences, which performed through minimizing of equation (3), are applied where the same procedure is repeated at the next sampling time. Afterward, the obtained control law can be transformed to the 2-DOF-RST controller-structure that given as (Camacho and Alba, 2013; Ansay et al., 1998; Morari and Lee, 1999)

S_{0} (q^{- 1}) Δ (q^{- 1}) u (t) = T_{0} (q^{- 1}) w (t) - R_{0} (q^{- 1}) y (t)

(5)

where $S_{0}$ , $R_{0}$ and $T_{0}$ are respectively defined by

{\begin{matrix} S_{0} = I_{n \times n} + q^{- 1} M_{1} (q^{- 1}) H (q^{- 1}) \\ R_{0} = M_{1} (q^{- 1}) F (q^{- 1}) \\ T_{0} = M_{1} (q^{- 1}) [q . I_{m \times m}, \dots, q^{N_{2}} . I_{m \times m}]^{T} \end{matrix}

(6)

with $F = [F_{1}, \dots, F_{N_{2}}]^{T}$ and $H = [H_{1}, \dots, H_{N_{2}}]^{T}$ are the polynomial vectors in the backward-shift operator $q^{- 1}$ and $M_{1}$ is the $n -$ first rows of the polynomial matrix M, where

\begin{matrix} M = {(G_{v}^{T} Ψ_{v} G_{v} + λ_{v})}^{- 1} G_{v}, Ψ_{v} = diag (\underset{N_{u} times}{\underset{︸}{ψ, \dots, ψ}}) \\ λ_{v} = diag (\underset{N_{u} times}{\underset{︸}{λ, \dots, λ}}), G_{v} = (\begin{matrix} G_{1} & 0 & \dots & 0 \\ G_{2} & G_{1} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ G_{N_{2}} & G_{N_{2} - 1} & \dots & G_{1} \end{matrix}) \end{matrix}

Recall that, the good reference tracking performances that are provided by the primary 2-DOF-MGPC controller depend heavily on an adequate choice of MGPC law tuning parameters, which are $N_{2}$ , $N_{u}$ , $Ψ$ and $λ$ (Stoica et al., 2008; Thomsen et al., 2011).

Now, let’s consider the feedback-control scheme based upon the primary 2-DOF-MGPC controller, which is shown in Figure 1 (Stoica et al., 2008; Gossner et al., 1997; Olaru and Ayerbe, 2006) where $d_{y} (t)$ , $η (t)$ and $w_{f} (t)$ are respectively, load disturbance vector, sensor noise vector and filtered set-point reference where $w_{f} (t) = T_{0} w (t)$ . Here, $w (t)$ and $e (t)$ are respectively, the set-point reference vector and the control error vector.

According to Figure 1, the corresponding standard feedback control configuration can be given by Figure 2 (Alamo et al., 2007). where $G (q^{- 1}) \in ℜ H^{m \times m}$ is the plant model, $H_{0} (q^{- 1}) \in ℜ H^{m \times m}$ and $K_{0} (q^{- 1}) \in ℜ H^{m \times m}$ are respectively, the pre-compensator and the post-compensator of primary 2-DOF-MGPC controller. Therefore, the process output vector can be expressed as follows

y (t) = G_{c l_{0}} w (t) + S_{d_{0}} d_{y} (t) - S_{c_{0}} η (t)

(7)

where $G_{c l_{0}} (q^{- 1})$ denotes the closed-loop system that gives information on the tracking dynamic of the set-point references, $S_{d_{0}}$ denotes the direct sensitivity matrix that provides information on NP against the load disturbance input vector in low frequency range (Skogestad et al., 1988; Zames, 1981), $S_{c_{0}}$ denotes the complementary sensitivity matrix that gives information on RS toward the neglected dynamics and the sensor noises suppression dynamic in high frequency range (Skogestad et al., 1988; Zames, 1981). The above matrices are given by

{\begin{matrix} G_{c l_{0}} = {(I_{m \times m} + G K_{0})}^{- 1} G K_{0} H_{0} \\ S_{d_{0}} = {(I_{m \times m} + G K_{0})}^{- 1} \\ S_{c_{0}} = {(I_{m \times m} + G K_{0})}^{- 1} G K_{0} \end{matrix}

(8)

Noticing that the desired form of the direct sensitivity plot is achieved when their maximal singular values, called also $σ_{max} [S_{d_{0}} (j ω)]$ , are vanishing as much as possible in low-frequencies and approaching to the unity in high-frequencies (Sedraoui et al., 2012; Ortega and Rubio, 2004). On the other hand, the desired form of the complementary sensitivity plot is achieved when their maximal singular values, called also $σ_{max} [S_{c_{0}} (j ω)]$ , are vanishing as much as possible in high-frequencies and approaching to the unity in low-frequencies (Sedraoui et al., 2012; Ortega and Rubio, 2004).

Knowing that, when modelling errors, sensor noises and un-modelled dynamics are taken into account. Therefore, no optimal trade-off between NP and RS is guaranteed by the primary 2-DOF-MGPC controller where a GPC robustification step based upon Youla parametrization should be included.

Youla parametrization for GPG controllers

In this section, a parametrization method is introduced that can help simplify the GPC robustification problem. This parametrization is formulated as follows:

According to Figure 2, if there exist, $N_{r}$ , $D_{r}$ , $Z_{r}$ , $X_{r_{0}}$ , $Y_{r_{0}}$ and $N_{ℓ}$ , $D_{ℓ}$ , $Z_{ℓ}$ , $X_{ℓ_{0}}$ , $Y_{ℓ_{0}}$ such that $G = N_{r} D_{r}^{- 1} = D_{ℓ}^{- 1} N_{ℓ}$ , $K_{0} = Y_{r_{0}}^{- 1} X_{r_{0}} = X_{ℓ_{0}} Y_{ℓ_{0}}^{- 1}$ and $H_{0} = X_{r_{0}}^{- 1} Z_{r} = Z_{ℓ} X_{ℓ_{0}}^{- 1}$ , Thus, by parametrizing (Youla et al., 1976; Susanu and Dumur, 2006) $X_{r_{1}}$ , $Y_{r_{1}}$ , $X_{ℓ_{1}}$ , and $Y_{ℓ_{1}}$ as

{\begin{matrix} X_{r_{1}} = X_{r_{0}} + Q D_{ℓ} \\ Y_{r_{1}} = Y_{r_{0}} - Q N_{ℓ} \\ X_{ℓ_{1}} = X_{ℓ_{0}} + D_{r} Q \\ Y_{ℓ_{1}} = Y_{ℓ_{0}} - N_{r} Q \end{matrix}

(9)

Therefore, the post-compensator and pre-compensator of the robustified 2-DOF-MGPC controller are respectively defined as $K_{1} = Y_{r_{1}}^{- 1} X_{r_{1}} = X_{ℓ_{1}} Y_{ℓ_{1}}^{- 1}$ and $H_{1} = X_{r_{1}}^{- 1} Z_{r} = Z_{ℓ} X_{ℓ_{1}}^{- 1}$ .

Furthermore, according to the Bezout’s theorem (Maciejowski, 1989), the transfer matrices mentioned above are interconnected by

{\begin{matrix} D_{ℓ} Y_{ℓ_{0}} + N_{ℓ} X_{ℓ_{0}} = I_{m \times m} \\ Y_{r_{0}} D_{r} + X_{r_{0}} N_{r} = I_{m \times m} \\ D_{ℓ} Y_{ℓ_{1}} + N_{ℓ} X_{ℓ_{1}} = I_{m \times m} \\ Y_{r_{1}} D_{r} + X_{r_{1}} N_{r} = I_{m \times m} \\ N_{ℓ} D_{r} - D_{ℓ} N_{r} {= 0}_{m \times m} \\ D_{r} N_{r} - N_{r} D_{r} {= 0}_{m \times m} \end{matrix}

(10)

where Q is a free stable polynomial matrix that determined later.

Now, according to equation (9) and Figure 2, the standard feedback control system based upon the robustified 2-DOF-MGPC controller is presented by Figure 3.

According to Figure 3, the obtained robustness properties (initial RS and initial NP) of the primary 2-DOF-MGPC controller can be enhanced by an optimal choice of Q-parameter, whereas its tracking dynamic becomes always conserved. These above-mentioned statements are proved as follows:

Proof 1. Enhancement of the obtained initial RS by an optimal choice of Q-parameter.

Noticing that the desired enhancement is interpreted through the singular values plot of $S_{c_{1}} (q^{- 1})$ , which is given by $S_{c_{1}} (q^{- 1}) = [I_{m \times m} + G K_{1}]^{- 1} G K_{1}$ . It is also equivalent to

S_{c_{1}} = {(I_{m \times m} + G K_{1})}^{- 1} G K_{1} = G {(I_{m \times m} + K_{1} G)}^{- 1} K_{1}

(11)

By substituting both matrices $G = N_{r} D_{r}^{- 1}$ and $K_{1} = Y_{r_{1}}^{- 1} X_{r_{1}}$ into equation (11) we can obtain $S_{c_{1}} = N_{r} {D_{r}}^{- 1} {(I_{m \times m} + {Y_{r_{1}}}^{- 1} X_{r_{1}} . N_{r} {D_{r}}^{- 1})}^{- 1} {Y_{r_{1}}}^{- 1} X_{r_{1}}$ , or equivalently, $S_{c_{1}} = N_{r} X_{r_{1}}$ .

From equations (9) and (10) we obtain

S_{c_{1}} = N_{r} (X_{r_{0}} + Q D_{ℓ})

(12)

and a simple manipulation of equation (12) leads to

S_{c_{1}} = S_{c_{0}} (I_{m \times m} + {X_{r_{0}}}^{- 1} Q D_{ℓ})

(13)

According to equation (13), it is easy to see that $σ_{\max} [S_{c_{1}} (ω)]$ plot in the log-log axis is obtained through the sum magnitude of both matrices $σ_{max} [S_{c_{0}} (ω)]$ and $σ_{max} [I_{m \times m} + {X_{r_{0}}}^{- 1} Q D_{ℓ}]$ in each frequency point. As a result, the curve of $σ_{max} [S_{c_{1}} (ω)]$ has the steepest slope when it compares with the slope of the curve of $σ_{max} [S_{c_{0}} (ω)]$ in low frequency. This confirms that the initial RS can be improved if an optimal Q-parameter is well chosen.

Proof 2. Enhancement of the obtained initial NP by an optimal choice of Q-parameter.

Noticing that the desired enhancement is interpreted through the singular values plot of $S_{d_{1}}$ , which is given as

S_{d_{1}} = [I_{m \times m} + G K_{1}]^{- 1}

(14)

By substituting both matrices $G = D_{ℓ}^{- 1} N_{ℓ}$ and $K_{1} = X_{ℓ_{1}} Y_{ℓ_{1}}^{- 1}$ into equation (14) we can obtain $S_{d_{1}} = {(I_{m \times m} + {D_{ℓ}}^{- 1} N_{ℓ} . X_{ℓ_{1}} Y_{ℓ_{1}}^{- 1})}^{- 1}$ , or equivalently

S_{d_{1}} = Y_{ℓ_{1}} D_{ℓ}

(15)

From equations (9) and (10) we obtain

S_{d_{1}} = (Y_{ℓ_{0}} - N_{r} Q) D_{ℓ}

(16)

and a simple manipulation of equation (16) leads to

S_{d_{1}} = (I_{m \times m} - N_{r} Q {Y_{ℓ_{0}}}^{- 1}) S_{d_{0}}

(17)

According to equation (17), it is easy to see that $σ_{max} [S_{d_{1}} (ω)]$ plot in the log-log axis is obtained through the sum magnitude of both matrices $σ_{max} [S_{d_{0}} (ω)]$ and $σ_{max} [I_{m \times m} - N_{r} Q {Y_{ℓ_{0}}}^{- 1}]$ in each frequency point. As a result, the $σ_{max} [S_{d_{1}} (ω)]$ plot has the steepest slope when it compares with the slope of the curve of $σ_{max} [S_{c_{0}} (ω)]$ in low frequency. This confirms that the initial NP can be improved if an optimal Q-parameter is well chosen.

Proof 3. The obtained initial tracking proprieties are conserved by the robustified 2-DOF-MGPC controller i.e. $G_{c l_{1}} = G_{c l_{0}}$ .

Noticing that the obtained tracking reference trajectories by the robustified 2-DOF-MGPC controller is defined as $G_{c l_{1}} = G [I_{m \times m} + K_{1} G]^{- 1} K_{1} H_{1}$ . This implies

G_{c l_{1}} = S_{c_{1}} H_{1}

(18)

However, the obtained tracking reference trajectories by the primary 2-DOF-MGPC controller is defined as $G_{c l_{0}} = G [I_{m \times m} + K_{0} G]^{- 1} K_{0} H_{0} = S_{c_{0}} H_{0}$ , which implies $G_{c l_{0}} = N_{r} Z_{r}$ . By substituting both matrices $H_{1} = X_{r_{1}}^{- 1} Z_{r}$ and $S_{c_{1}} = N_{r} X_{r_{1}}$ into equation (18) we can obtain

G_{c l_{1}} = N_{r} X_{r_{1}} . X_{r_{1}}^{- 1} Z_{r} = N_{r} Z_{r}

(19)

or equivalently

G_{c l_{1}} = G_{c l_{0}}

(20)

Now, according to proofs 1 to 3, the GPC robustification problem is simplified to finding a proper Q-parameter with reduced order which is detailed in the next section.

GPC robustification using approximated Q matrix

The feedback control part of the 2-DOF-MGPC structure controller is designed to meet both RS and NP requirements in a similar manner to that one of 1-DOF design procedure. Thereby, the $H_{\infty}$ control design is used to achieve the good robustness properties in which the corresponding generalized feedback control system can be shown in Figure 4 (Zhou and Ren, 2001).

From Figure 4, the exogenous input vector and exogenous output vector of the generalized plant $Φ (q^{- 1})$ is respectively assumed by $z_{in}$ and $z_{out}$ . Therefore, the design problem of the Q-parameter is formulated by the lower Linear Fractional Transformation LFT, which is determined through the interconnection system of $Φ (q^{- 1})$ by $Q (q^{- 1})$ . It yields also

min_{Q \in R H_{\infty}} ‖ LF T_{l} {Φ (q^{- 1}), Q (q^{- 1})} ‖_{\infty}

(21)

Notice that the direct solving of the optimization problem in equation (21) leads to very high order of the Q-parameter. Therefore, the implementation step of the robustified 2-DOF-MGPC controller leads to high cost, difficult commissioning, poor reliability and potential problems in maintenance.

In this work, the Q-parameter with a reduced order is designed using the following step-procedures, which are:

First, the new weighted-mixed sensitivity problem based upon the adequate performance and stability weights is a priory formulated using the standard $H_{\infty}$ framework given by Figure 5. Its optimal solution, which is obtained with less restriction, determines the state space representation of the proposed robust $H_{\infty}$ controller. Second, an ideal Q-parameter with a full order is derived through the complementary sensitivity matrix, which is provided by the robust $H_{\infty}$ controller. Finally, the above Q-parameter is fitted to the measured transfer matrix in which the Q-parameter with a reduced order is determined by the frequency domain identification method.

In $H_{\infty}$ synthesis controller, the proposed optimization problem is formulated by the LFT, which is determined through the interconnection system of $P (q^{- 1})$ by $K_{H} (q^{- 1})$ (see Figure 5). It yields also

min_{K_{H} \in {R H}_{\infty}} ‖ LF T_{l} {P (q^{- 1}), K_{H} (q^{- 1})} ‖_{\infty}

(22)

According to equation (22), the new weighted-mixed sensitivity problem is given by equation (23) as (Glover and Doyle, 1988; Zames, 1981; Rao and Rao, 2009; Oloomi and Shafai, 2003):

\min_{K_{H} \in ℜ H_{\infty}} {‖ \begin{array}{l} W_{T} (q^{- 1}) S_{c_{H}} (q^{- 1}) \\ W_{S} (q^{- 1}) S_{d_{H}} (q^{- 1}) \end{array} ‖}_{\infty} \Leftrightarrow \min_{K_{H} \in ℜ H_{\infty}} \max_{ω \in [0, \frac{π}{T_{e}}]} {\bar{σ} [\begin{array}{l} W_{T} (ω) S_{c_{H}} (ω) \\ W_{S} (ω) S_{d_{H}} (ω) \end{array}]}

(23)

where equation (23) is numerically equivalent to (Oloomi and Shafai, 2003)

‖ \begin{matrix} W_{T} (ω) S_{c_{H}} (ω) \\ W_{S} (ω) S_{d_{H}} (ω) \end{matrix} ‖_{\infty} \leq γ

(24)

$T_{e}$ and $γ$ denote respectively, the sampling time and the $H_{\infty}$ performance level. $W_{S}$ and $W_{T}$ are the weighting matrices that penalize respectively $S_{d_{H}}$ and $S_{c_{H}}$ .

For the choice of the adequate performance and stability weights, practical considerations such as a rate saturation, a pure time delays and un-modelled dynamics may be included in these weights where this choice is indeed empirical (Oloomi and Shafai, 2003; Nair, 2011). The selection is iterative and may go on until satisfying an acceptable trade-off between RS and NP. Noticing that, the inverse matrices of previous weights provides the perfect forms for direct sensitivity and complementary sensitivity functions. So that, some guidelines suggested by Oloomi and Shafai (2003) are used to select appropriate weights.

Here, if the previous weights are well selected, the optimization problem (equation (23)) can be solved by the Matlab function DHINFLMI. It provides also the state space representation of the robust $H_{\infty}$ controller in which an optimal trade-off between NP and RS is always attained.

Now, for the design problem of the ideal Q-parameter, equations (13) and (17) are used to formulate the following optimization problem, which is given by

min_{Q \in ℜ H_{\infty}} {{‖ \begin{matrix} W_{S} (ω) S_{d_{1}} (ω, Q) \\ W_{T} (ω) S_{c_{1}} (ω, Q) \end{matrix} ‖}_{\infty} = {‖ \begin{matrix} W_{S} (I_{m \times m} - N_{r} Q {Y_{ℓ_{0}}}^{- 1}) S_{d_{0}} \\ W_{T} S_{c_{0}} (I_{m \times m} + {X_{r_{0}}}^{- 1} Q D_{ℓ}) \end{matrix} ‖}_{\infty}}

(25)

According to equation (25), to achieve the same sensitivities of the $H_{\infty}$ controller, the ideal Q-parameter should be determined, in which $S_{c_{1}}$ (with respect to $S_{d_{1}}$ ) is closely matched $S_{c_{H}}$ (with respect to $S_{d_{H}}$ ). Therefore, its transfer matrix is given by

S_{c_{1}} = S_{c_{H}} \Rightarrow Q = X_{r 0} (S_{c_{0}}^{- 1} S_{c_{H}} - I_{m \times m}) D_{ℓ}^{- 1}

(26)

Noticing that, the transfer matrix of the obtained ideal Q-parameter by equation (26) leads also to a higher order and the implementation problem of the robustified 2-DOF-MGPC controller is therefore always persisted. To remedy this problem, the Q-parameter with a reduced order can be determined using the following proposed algorithm, which is given as:

Step 0: For $i = 1, \dots, n$ and $j = 1, \dots, m$ , the transfer matrix with order $n_{Q_{ij}}$ of the ideal Q-parameter is converted to the $n \times m$ frequency response data using the Matlab function frd.

Step 1: Let $n_{Q_{ij}}^{*}$ the order (to be decreased) of each transfer function of the approximated Q-parameter where $n_{Q_{ij}}^{*} \leq n_{Q_{ij}}$ should be satisfied.

Step 2: For each $n \times m$ frequency response data, the $Q_{ij}^{*}$ with the order $n_{Q_{ij}}^{*}$ is determined by the Matlab function fitfrd.

Step 3: The discrepancy magnitude between both transfer functions $Q_{ij}^{*}$ and $Q_{ij}$ is checked, yields also the magnitude error $ζ_{ij}$ .

Step 4: Decreasing gradually the order $n_{Q_{ij}}^{*}$ (i.e. $n_{Q_{ij}}^{*} = n_{Q_{ij}}^{*} - 1$ ) as long as $ζ_{ij}$ is enough small and go to the step 2, otherwise, stop.

For the designing step of the robustified 2-DOF-MGPC controller, according to Figure 2 and previous notations, when the transfer matrices G, $K_{0}$ and $H_{0}$ are chosen as

{\begin{matrix} G = D_{ℓ}^{- 1} N_{ℓ} = {(Δ A)}^{- 1} . q^{- 1} Δ B \\ K_{0} = Y_{r_{0}}^{- 1} X_{r_{0}} = {(Δ S_{0})}^{- 1} . R_{0} \\ H_{0} = X_{r_{0}}^{- 1} Z_{r} = R_{0}^{- 1} . T_{0} \end{matrix}

(27)

where the transfer matrices $N_{ℓ}$ , $D_{ℓ}$ , $X_{r_{0}}$ , $Y_{r_{0}}$ and $Z_{r}$ are respectively deduced as $N_{ℓ} = q^{- 1} Δ B$ , $D_{ℓ} = Δ A$ , $X_{r_{0}} = R_{0}$ , $Y_{r_{0}} = Δ S_{0}$ and $Z_{r} = T_{0}$ . Therefore, the transfer matrix of the ideal Q-parameter is re-written as

Q = R_{0} (S_{c_{0}}^{- 1} S_{c_{H}} - I_{m \times m}) {(Δ A)}^{- 1}

(28)

Based on equation (28), the Q-parameter with a reduced order, called also $Q^{*}$ , is determined using the previous proposed algorithm. Therefore, from equation (9), the Youla parametrization based upon $Q^{*}$ is derived as

{\begin{matrix} R_{1} = R_{0} + Δ Q^{*} A \\ S_{1} = S_{0} - q^{- 1} Q^{*} B \\ T_{1} = T_{0} \end{matrix}

(29)

Afterward, the transfer matrices of both the pre-compensator $H_{1}$ and the post-compensator $K_{1}$ of the robustified 2-DOF-MGPC controller are determined using equation (29).

Simulation results and discussion

The process to be controlled presents a wind turbine system equipped with a doubly fed generator (see Figure 6) (Han and Shen, 2014; Soens et al., 2005; Vieira et al., 2009).

According to Figure 6, the stator winding is connected directly to the grid and the wind turbine drives the rotor. The power captured by the wind turbine is converted into electrical power through an induction generator and is then transmitted to the grid via stator winding. The produced output power must have the same quality when it reaches over the electrical network, i.e. 220 volts amplitude and 50 Hz frequency. Its harmonics must be kept at a low level, despite the changes of both wind speed and the consumed electrical energy, which may be either active or reactive. Han and Shen (2014) and Qu and Song (2011) give the requirement details of wind turbines. Thus, the stator and rotor in $d - q$ coordinates are expressed as

{\begin{matrix} V_{ds} = R_{s} \cdot I_{ds} + \frac{d ϕ_{ds}}{dt} - ω_{s} \cdot ϕ_{qs} \\ V_{qs} = R_{s} \cdot I_{qs} + \frac{d ϕ_{qs}}{dt} - ω_{s} \cdot ϕ_{ds} \\ V_{dr} = R_{r} \cdot I_{dr} + \frac{d ϕ_{dr}}{dt} - ω_{r} \cdot ϕ_{qr} \\ V_{qr} = R_{r} \cdot I_{qr} + \frac{d ϕ_{qr}}{dt} - ω_{r} \cdot ϕ_{dr} \end{matrix}

(30)

The stator and rotor flux are given by

{\begin{matrix} ϕ_{ds} = L_{s} \cdot I_{ds} + M \cdot I_{dr} \\ ϕ_{qs} = L_{s} \cdot I_{qs} + M \cdot I_{qr} \\ ϕ_{dr} = L_{r} \cdot I_{dr} + M \cdot I_{ds} \\ ϕ_{qr} = L_{r} \cdot I_{qr} + M \cdot I_{qs} \end{matrix}

(31)

This electrical model becomes complete with the following electromagnetic and mechanical torques

{\begin{matrix} C_{e} = p \cdot \frac{M}{L_{s}} (ϕ_{ds} \cdot I_{qr} - ϕ_{qs} \cdot I_{dr}) \\ C_{m} = C_{r} + J \cdot \frac{d ψ}{dt} + f \cdot ψ \end{matrix}

(32)

The transformation from the three-phase stationary coordinate vector $V_{abc} = [V_{a} V_{b} V_{c}]^{T}$ (zero-axis component is assumed 0) to the $d - q$ rotating coordinate vector $V_{dq} = [V_{d} V_{q}]^{T}$ , called also $Park' s$ transformation, is defined as

\begin{matrix} [\begin{matrix} V_{d} \\ V_{q} \end{matrix}] = \frac{2}{3} [\begin{matrix} \cos (2 π F) & \cos (2 π F - \frac{2 π}{3}) & \cos (2 π F + \frac{2 π}{3}) \\ - \sin (2 π F) & - \sin (2 π F - \frac{2 π}{3}) & - \sin (2 π F + \frac{2 π}{3}) \end{matrix} [\begin{matrix} V_{a} \\ V_{b} \\ V_{c} \end{matrix}] \end{matrix}

(33)

However, the inverse $Park' s$ transformation from $V_{dq}$ to $V_{abc}$ may be defined as

\begin{matrix} [\begin{matrix} V_{a} \\ V_{b} \\ V_{c} \end{matrix}] = [\begin{matrix} \cos (2 π F) & - \sin (2 π F) \\ \cos (2 π F - \frac{2 π}{3}) & - \sin (2 π F - \frac{2 π}{3}) \\ \cos (2 π F + \frac{2 π}{3}) & - \sin (2 π F + \frac{2 π}{3}) \end{matrix}] [\begin{matrix} V_{d} \\ V_{q} \end{matrix}] \end{matrix}

(34)

Table 1 summarizes the signification of each DFIG parameter where the plant-model input vector is given by $u = {[I_{r} = {(\begin{matrix} I_{dr} & I_{qr} \end{matrix})}^{T}, V_{r} = {(\begin{matrix} V_{dr} & V_{qr} \end{matrix})}^{T}]}^{T}$ , as well as, the disturbance plant input presents the consumed active and/or reactive energy in electrical network, which implicitly depends on the variation of the consumed stator current vector, called also $I_{s} = {(\begin{matrix} I_{ds} & I_{qs} \end{matrix})}^{T}$ . Noticing that, in the $d - q$ rotation coordinate axis, the components of the consumed stator current $I_{ds}$ and $I_{qs}$ are assumed as constants in the steady-state. Consequently, its derivatives are supposed zeros (Belfedal et al., 2010; Sedraoui et al., 2012). Thus, the rotor voltage vector $V_{r} = {[\begin{matrix} V_{dr} & V_{qr} \end{matrix}]}^{T}$ is assumed as a set-control vector that provided by a controller. Moreover, let us consider the stator voltage vector $y = [\begin{matrix} V_{ds} & V_{qs} \end{matrix}]^{T}$ as the plant-model outputs and the stator flux vector $x_{s} = [\begin{matrix} ϕ_{dr} & ϕ_{qr} \end{matrix}]^{T}$ as the state-variable vector of the DFIG model. By using numerical data given in Table 2 (Belfedal et al., 2010; Sedraoui et al., 2012), the state space representation of DFIG linear model is determined by applying the Matlab function Linmod2 on the Simulink system, which is established from both equation (30) and (31). The transfer matrix of the nominal plant-model is therefore determined through the following formula

G_{N} (s) = C_{s} (sI - A_{s})^{- 1} B_{s} + D_{s}

(35)

Table 1:

Parameters significant to the DFIG.

Parameter	Signification
$V_{ds}$	Stator voltage component in $' d'$ axe
$V_{qs}$	Stator voltage component in $' q'$ axe
$V_{dr}$	Rotor voltage component in $' d'$ axe
$V_{qr}$	Rotor voltage component in $' q'$ axe
$I_{ds}$	Stator current component in $' d'$ axe
$I_{qs}$	Stator current component in $' q'$ axe
$I_{dr}$	Rotor current component in $' d'$ axe
$I_{qr}$	Rotor current component in $' q'$ axe
$ϕ_{ds}$	Stator flux component in $' d'$ axe
$ϕ_{qs}$	Stator flux component in $' q'$ axe
$ϕ_{dr}$	Rotor flux component in $' d'$ axe
$ϕ_{qr}$	Rotor flux component in $' q'$ axe
$R_{s}$	Stator resistance (of one phase)
$R_{r}$	Rotor resistance (of one phase)
$L_{s}$	Stator cyclic self-inductance
$L_{r}$	Rotor cyclic self-inductance
M	Cyclic mutual self-inductance
p	Number of pair of the machine poles
$C_{r}$	Resistant torque
f	Viscous rubbing coefficient
J	Inertia moment
$C_{e}$	Electromagnetic torque
$C_{m}$	Mechanical torque

Table 2:

Numerical data used in the DFIG.

Parameter	Signification	Value
$R_{s}$	Stator resistance	$5.0000$
$R_{r}$	Rotor resistance	$1.0113$
M	Mutual cyclic self-inductance	$0.1346$ H
$L_{s}$	Stator cyclic self-inductance	$0.3409$ H
$L_{r}$	Rotor cyclic self-inductance	$0.0605$ H
$w_{r}$	Rotor pulsation	$146.6$ rad/sec
$w_{s}$	Stator pulsation	$2 π (50)$ rad/sec

From equation (35), the discrete transfer matrix $G_{N} (q^{- 1})$ is given through the bilinear transform, which is also known as Tustins rule where sampling time $T_{e} = 0.01$ seconds is used. The CARIMA of MIMO plant-model is then determined by splitting $G_{N} (q^{- 1})$ into the diagonal part so-called $A (q^{- 1})$ and into the off-diagonal part so-called $B (q^{- 1})$ . Moreover, the polynomial matrix $A (q^{- 1})$ is equal to the least common multiples of the corresponding row denominators of the matrix $G_{N} (q^{- 1})$ . Afterward, in Camacho and Alba (2013) the polynomial matrix $B (q^{- 1})$ is determined by

B (q^{- 1}) = q . A (q^{- 1}) . G_{N} (q^{- 1})

(36)

The proposed robustification procedure is achieved through the following steps:

(a) The primary 2-DOF-MGPC controller under the controller structure $R_{0} S_{0} T_{0}$ representation is given via the MGPC law using the following tuning parameters: $N_{u} = N_{2} = 2, Ψ = diag (1, 1)$ and $λ = 10^{- 3} \times diag (38, 66)$ we get

\begin{array}{l} \begin{matrix} R_{0} = [\begin{matrix} 1.149 & 0.498 \\ - 0.486 & 1.117 \end{matrix}] + [\begin{matrix} - 1.666 & - 0.723 \\ 0.711 & - 1.634 \end{matrix}] q^{- 1} + [\begin{matrix} 1.563 & 0.679 \\ - 0.672 & 1.543 \end{matrix}] q^{- 2} \\ + [\begin{matrix} - 0.857 & - 0.373 \\ 0.369 & - 0.848 \end{matrix}] q^{- 3} + [\begin{matrix} 0.372 & 0.162 \\ - 0.162 & 0.371 \end{matrix}] q^{- 4} \end{matrix} \\ \begin{matrix} S_{0} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + [\begin{matrix} - 0.341 & - 0.033 \\ 0.024 & - 0.318 \end{matrix}] q^{- 1} + [\begin{matrix} 1.371 & 0.620 \\ - 0.613 & 1.362 \end{matrix}] q^{- 2} \\ + [\begin{matrix} - 0.213 & - 0.347 \\ 0.339 & - 0.203 \end{matrix}] q^{- 3} + [\begin{matrix} 0.530 & 0.484 \\ - 0.482 & 0.527 \end{matrix}] q^{- 4} \end{matrix} \\ \begin{array}{l} T_{0} {= 10}^{- 3} [\begin{matrix} 13.52 & 6.667 \\ - 10.3 & 22.66 \end{matrix}] + [\begin{matrix} 0.548 & 0.2371 \\ 0.228 & 0.5268 \end{matrix}] q^{- 1} \end{array} \end{array}

(b) A weighted-mixed sensitivity problem is independently formulated using both weighting matrices $W_{T} (s)$ and $W_{S} (s)$ which have been proposed in Belfedal et al. (2010); Sedraoui and Boudjahem (2012) for a same previous nominal plant-model as follows

\begin{matrix} W_{S} (s) = 0.25 \times diag (\frac{0.068 s + 1}{0.068 s + 10^{- 4}}, \frac{0.068 s + 1}{0.068 s + 10^{- 4}}) \\ W_{T} (s) = 0.65 \times diag (\frac{0.065 s + 1}{10^{- 4} s + 1}, \frac{0.065 s + 1}{10^{- 4} s + 1}) \end{matrix}

Whereas, the state-space representation of the robust $K_{H} (q^{- 1})$ controller is given by solving the above-mentioned optimization problem using the Matlab function Hinflmi, which yields also the optimal performance level $γ = 0.697382$ . The obtained continuous transfer matrix of the controller is of order six, which is given by

\begin{matrix} K_{H_{s}} (s) = [\begin{matrix} A_{H_{s}} B_{H_{s}} \\ C_{H_{s}} D_{H_{s}} \end{matrix}] = C_{H_{s}} {(s I_{6 \times 6} - A_{H_{s}})}^{- 1} B_{H_{s}} + D_{H_{s}} \end{matrix}

where

A_{H_{s}} = [\begin{matrix} - 11.222 & 0.197 & - 0.282 & - 9.949 & 520.591 & 45.814 \\ - 0.197 & - 11.222 & 9.951 & - 0.282 & - 45.426 & 525.041 \\ - 0.049 & - 6.233 & - 0.146 \times 10^{- 3} & 313.961 & 16.368 & 1.649 \\ 6.231 & - 0.049 & - 313.961 & - 0.147 \times 10^{- 3} & - 1.634 & 16.511 \\ 335.681 & - 29.286 & 10.663 & - 0.851 & - 15706 & - 0.273 \\ 29.537 & 338.561 & 0.857 & 10.755 & 0.272 & - 15976 \end{matrix}]

$B_{H_{s}} = [\begin{matrix} 0.984 & 0.077 \\ - 0.077 & 0.984 \\ 0.031 & 0.002 \\ - 0.002 & 0.031 \\ 81.001 & - 0.705 \\ 0.711 & 81.694 \end{matrix}]$ , $C_{H_{s}}^{T} = [\begin{matrix} - 1.595 & - 0.154 \\ 0.154 & - 1.596 \\ 167.931 & 12.842 \\ - 12.842 & 167.931 \\ - 0.133 & 0.004 \\ - 0.004 & - 0.133 \end{matrix}]$ , $D_{H_{s}} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ Afterward, the bilinear transform method discretizes the controller transfer matrix where the same previous sampling time $T_{e}$ is used. In which we get $K_{H} (q^{- 1})$ . Next, the polynomial matrix of the ideal Q-parameter is determined by $Q = R_{0} (S_{c_{0}}^{- 1} S_{c_{H}} - I_{m \times m}) (Δ A)^{- 1}$ , which is then fitted to the measured transfer matrix. Finally, the Q-parameter with a reduced order is performed by the Matlab function fitfrd, in which the order of each transfer function $Q_{ij}$ is found after several trials. We obtain

Q = [\begin{matrix} Q_{11} & Q_{12} \\ Q_{21} & Q_{22} \end{matrix}]

where

Q_{11} = \frac{1.031 (1 - 0.067 q^{- 1}) (1 - 1.267 q^{- 1} + 0.575 q^{- 2}) (1 - 0.151 q^{- 1} + 0.489 q^{- 2})}{(1 - 0.856 q^{- 1}) (1 + 0.0168 q^{- 1}) (1 - 0.547 q^{- 1} + 0.809 q^{- 2}) (1 - 0.519 q^{- 1} + 0.802 q^{- 2})}

Q_{12} = \frac{- 0.444 (1 + 0.128 q^{- 1}) (1 - 1.281 q^{- 1} + 0.585 q^{- 2}) (1 - 0.168 q^{- 1} + 0.496 q^{- 2})}{(1 - 0.853 q^{- 1}) (1 + 0.184 q^{- 1}) (1 - 0.560 q^{- 1} + 0.809 q^{- 2}) (1 - 0.504 q^{- 1} + 0.801 q^{- 2})}

Q_{21} = \frac{0.432 (1 + 0.328 q^{- 1}) (1 - 1.301 q^{- 1} + 0.603 q^{- 2}) (1 - 0.181 q^{- 1} + 0.510 q^{- 2})}{(1 - 0.850 q^{- 1}) (1 + 0.359 q^{- 1}) (1 - 0.567 q^{- 1} + 0.816 q^{- 2}) (1 - 0.494 q^{- 1} + 0.799 q^{- 2})}

Q_{22} = \frac{- 1.001 (1 + 0.006 q^{- 1}) (1 - 1.284 q^{- 1} + 0.589 q^{- 2}) (1 - 0.161 q^{- 1} + 0.495 q^{- 2})}{(1 - 0.855 q^{- 1}) (1 + 0.078 q^{- 1}) (1 - 0.554 q^{- 1} + 0.811 q^{- 2}) (1 - 0.512 q^{- 1} + 0.802 q^{- 2})}

Now, by applying equation (29), we calculate the robustified 2-DOF-MGPC controller. Simulation results given by both the primary and the robustified 2-DOF-MGPC controllers are compared with those given by the robust $H_{\infty}$ controller in frequency domains for the range $ω \in [10^{- 2}, 10^{+ 4}]$ radians/second. Therefore, Figure 7 presents the singular value plots of the $S_{d_{0}}$ , $S_{d_{1}}$ and $S_{d_{H}}$ , which are compared with those given by the $W_{T}^{- 1}$ in log-log axis.

Moreover, Figure 8 presents the singular value plots of the $S_{c_{0}}$ , $S_{c_{1}}$ and $S_{c_{H}}$ , which is also compared with those given by $W_{S}^{- 1}$ in log-log axis.

According to Figures 7 and 8, it is easy to observe that the obtained frequency responses of both the robustified 2-DOF-MGPC and the robust $H_{\infty}$ controllers are matched as close as possible in each frequency point. Consequently, the proposed Q-parameter has the ability to improve the obtained robustness properties of the primary 2-DOF-MGPC controller, wherein it becomes similar to those given by the robust $H_{\infty}$ controller.

In addition, Figure 7 shows that the singular value plots of three direct sensitivities are bounded by the upper-bound, $1 / σ_{max} [W_{S} (ω)]$ , at all frequencies. Therefore, the NP condition is satisfied. Similarly, a better NP margin is given when the maximum singular values of direct sensitivity matrix become small as much as possible at low frequencies. As a result, a primary 2-DOF-MGPC controller ensures a better margin from others. On the other hand, according to Figure 8, it can be seen that the better RS margin is given when the maximum singular values of complementary sensitivity matrix become small as much as possible at high frequencies. As a result, the $σ_{max} [S_{c_{0}} (ω)]$ plots exceed its upper bounds, $1 / σ_{max} [W_{T} (ω)]$ , at some frequencies except, in frequency range $ω \in [60, 10^{4}]$ radians /seconds. Therefore, the primary 2-DOF-MGPC controller violates the RS condition. This will be explained in time domain by the higher sensitivity of the feedback control system to the sensor noises.

Figure 8 shows also that the singular value plots of $S_{c_{1}} (q^{- 1})$ are reduced at frequencies beyond the system bandwidth in order to secure robustness at high frequency range. Moreover, for frequencies above $ω = 400$ radians/second the curve of $σ_{max} [S_{c_{1}} (ω)]$ is below −20 dB in which the sensor noises are suppressed more than 10 time at the plant output. Consequently, the robustified 2-DOF-MGPC controller provides a better RS margin from others.

Now, to confirm the above mentioned comments in time domain, we use the three exogenous inputs. First, the set-point reference vector $V_{s}^{ref} (t)$ , load disturbance vector $d_{y} (t)$ and the sensor noise vector $η (t)$ . These inputs excite the feedback control system in which the DFIG plant-model is established via the Simulink block system. Therefore, all nonlinearity DFIG dynamics are also taken in consideration. The first set-point reference vector contains the desired stator-voltage that presents in the stationary coordinate by the pure sinusoidal signals with constant amplitude 220 volts and constant frequency $F = 50$ Hz. We get

V_{s}^{ref} (t) = [\begin{matrix} V_{sa}^{ref} (t) = 220 . \sin (2 π Ft) \\ V_{sb}^{ref} (t) = 220 . \sin (2 π Ft - \frac{2 π}{3}) \\ V_{sb}^{ref} (t) = 220 . \sin (2 π Ft + \frac{2 π}{3}) \end{matrix}]

The second input $d_{y} (t)$ presents the consumed stator current in electrical network that assumed in the d-q rotating coordinate by the unit-step function with constant gain 30 amperes at the start-time $t = 0.5$ seconds. The third input vector presents the measurement-noise vector $η (t)$ that is modelled by a random signal Gaussian distributed with zero-mean, i.e. ${\bar{η}}_{y} = 0$ , and variance $VA R_{η_{y}} = 10, 000$ and is applied at the start-time $t = 0.8$ seconds.

Figure 7 shows the curve of the obtained a-phase stator voltage $V_{s_{a}}$ that is provided by the primary 2-DOF-MGPC controller.

Figure 9 contains three different dynamics, which are: First, the obtained tracking dynamic in the transient-state that enlarged in the time range $t \in [0, 0.1]$ seconds. Second, the obtained load disturbance attenuation dynamic that is enlarged in the time range $t \in [0.48, 0.6]$ seconds. Finally, it contains the obtained rejection dynamic of the sensor noise in steady-state, which is enlarged in the time range $t \in [0.8, 1]$ seconds.

Similarly, Figures 10 and 11 give the previous dynamics using respectively, the robust $H_{\infty}$ controller and the robustified 2-DOF-MGPC one.

According to these figures, all controllers ensure a constant stator voltage (i.e. amplitude $= 220$ volts) with a constant frequency (i.e. 50 Hz) in the electrical network. However, the robustified 2-DOF-MGPC controller provides a better tracking dynamic more than the one given by the robust $H_{\infty}$ controller, it can also provide a better trade-off between NP and RS more than the one given by the primary 2-DOF-MGPC controller.

More confirmation of these results: the curves of the stator voltage amplitudes $| V_{s} |$ where $| V_{s} | = \sqrt{V_{s_{a}}^{2} + V_{s_{b}}^{2} + V_{s_{c}}^{2}}$ are compared in Figure 12. Moreover, Table 3 summarizes the obtained time properties of three above controllers where better results are mentioned in bold.

Table 3:

Gives a summary of the time-specifications which are provided by three previous controllers, better results are in bold.

Controller	primary 2-DOF-MGPC
Transient-state:
Rise time (milliseconds)	12.308
Settling time (milliseconds)	71.943
The plant-disturbance $d_{y}$	142.021
Steady-state:
Rejection of measurement-noises ${\| V_{s} \|}_{min} \leq \| V_{s} \| \leq {\| V_{s} \|}_{max}$	153.89 $\leq \| V_{s} \| \leq$ 389.54
Controller	Robust $H_{\infty}$
Transient-state:
Rise time (milliseconds)	143.797
Settling time (milliseconds)	196.002
The plant-disturbance $d_{y}$	216.051
Steady-state:
Rejection of measurement-noises ${\| V_{s} \|}_{min} \leq \| V_{s} \| \leq {\| V_{s} \|}_{max}$	215.45 $\leq \| V_{s} \| \leq$ 222.93
Controller	Robustified 2-DOF-MGPC
Transient-state:
Rise time (milliseconds)	12.308
Settling time (milliseconds)	71.943
The plant-disturbance $d_{y}$	216.051
Steady-state:
Rejection of measurement-noises ${\| V_{s} \|}_{min} \leq \| V_{s} \| \leq {\| V_{s} \|}_{max}$	205.7 $\leq \| V_{s} \| \leq$ 222.77

According to Figure 12, it is easy to confirm that the proposed Q-parameter attains our main goal, which is the enhancement of the obtained robustness properties of the primary 2-DOF-MGPC controller where its good closed-loop tracking dynamic is always conserved.

Conclusion

In this paper we have proposed a robustification method of initial form multivariable GPC law to enhance its trade-off robustness against uncertain plant-models where the neglected dynamic, load disturbances and sensor noise effect are taken in consideration. In this work, we show that the proposed Q-parameter that is designed through the frequency domain identification helps to exploit advantages of the robust $H_{\infty}$ controller, to increase the trade-off margin of the primary 2-DOF-MGPC controller. The proposed robustification method can be also applied for any conventional 2-DOF controller structure and some others type of uncertainties.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers for any comments and suggestions that enhance the technical and scientific quality of this paper. The authors would also like to thank the Pervasive Artificial Intelligence PAI group of the informatics department of Fribourg, Switzerland, for their valuable suggestions and comments which helped us to improve this paper. Special thanks to Prof. Béat Hirsbrunner, Prof. Michèle Courant, Dr. Zanat Kamel and Dr. Belaouar Djamel.

Declaration of Conflicting Interests

The authors declare that there is no conflict of interests.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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