Robust Stability Analysis of Grid-Connected Converters Based on Parameter-Dependent Lyapunov Functions

Abstract

This paper deals with the problem of robust stability analysis of grid-connected converters with LCL filters controlled through a digital signal processor and subject to uncertain grid inductance. To model the uncertain continuous-time plant and the digital control gain, a discretization procedure, described in terms of a Taylor series expansion, is employed to determine an accurate discrete-time model. Then, a linear matrix inequality-based condition is proposed to assess the robust stability of the polynomial discrete-time augmented system that includes the filter state variables, the states of resonant controllers and the delay from the digital control implementation. By means of a parameter-dependent Lyapunov function, the proposed strategy has as main advantage to provide theoretical certification of stability of the uncertain continuous-time closed-loop system, circumventing the main disadvantages of previous approaches that employ approximate discretized models, neglecting the errors. Numerical simulations illustrate the benefits of the discretization technique and experimental results validate the proposed approach.

Introduction

Grid-connected converters have been increasingly used in renewable energy systems for distributed generation (Meza et al. 2012; Chen et al. 2014; Dhar and Dash 2015). One important common feature for grid-connected converters is the grid current control, with total harmonic distortion (THD) within standard limits and guarantee of robust stability in the case of large grid impedance variations. An issue to be considered in the design of the current controller is the damping of the resonance of an output LCL filter, which can be carried out by active damping (Guerrero et al. 2010; Agorreta et al. 2011). In this context, some linear control strategies can be applied to the grid current control, as deadbeat, PI-based and resonant controllers. One of the main problems of the deadbeat controller is related to the parameter mismatches that generate a tracking error and also stability problems (Mattavelli et al. 2005; Kawamura et al. 1988). The PI-based (Dannehl et al. 2009, 2010; Pena-Alzola et al. 2014) and resonant-based controllers (Parker et al. 2014; Liserre et al. 2006) can be used to ensure the tracking of sinusoidal references and the rejection of grid voltage disturbances, being the resonant controllers simpler to implement. To avoid stability problems associated with an infinite gain, one can use nonideal resonant controllers with finite gains at the resonance frequencies, but still high enough to enforce a small steady-state error.

Concerning stability of grid-connected inverters with LCL output filters, some strategies have been proposed in the literature. It has been demonstrated in Liserre et al. (2006) that the use of active damping helps to stabilize the system with respect to many different kinds of resonances. In Dannehl et al. (2009), voltage-oriented PI control is addressed and the system stability is analyzed with respect to different ratios of LCL filter resonance and control frequencies. In Dannehl et al. (2010), various active damping approaches for PI-based current control are provided. Several ratios of LCL resonance frequency and control frequency are considered. It is shown that at high resonance frequencies, only current feedback stabilizes the system. In Mukherjee and De (2013), a hybrid passive-active damping solution with improved system stability margin and enhanced dynamic performance is proposed for high power grid interactive converters. In Yin et al. (2013), the stability problem for a digital single-loop controller based on the grid-side current feedback is analyzed. The stability analysis is performed by means of a Nyquist diagram in the discrete-time domain, providing a stability region in the space of LCL filter and control parameters. In Parker et al. (2014), a theoretical discrete-time analysis framework that identifies three distinct regions of LCL filter resonance is presented, namely a high resonant frequency region where active damping is not required, a critical resonant frequency where a controller cannot stabilize the system, and a low resonant frequency region where active damping is mandatory.

With respect to the study of robust stability, polytopic models can be used to represent the system with uncertain parameters and linear matrix inequalities (LMIs) can be employed to certify the stability (Boyd et al. 1994; Ackermann 1993). This can be found, for instance, for DC–DC converters, in Olalla et al. (2011), where a polytopic model (suitable for continuous-time domain) of the system with a small number of vertices is used for the robust controller synthesis. In Liserre et al. (2006), one has that the possible wide range of grid impedances deteriorates the stability and the effectiveness of current controlled system. Applications for grid-connected converters with state feedback controllers associated with resonant controllers with discrete-time implementation appeared in Gabe et al. (2009) and Maccari et al. (2014). In Gabe et al. (2009), a partial state feedback is used and the gains of the current controller are obtained by means of LMIs described at the vertices of a polytopic model. In Maccari et al. (2014), a more general framework is given for the design of resonant controllers jointly with the gains of full state feedback controllers. In both Gabe et al. (2009) and Maccari et al. (2014), approximate polytopic models for the system with uncertain grid inductance are employed. However, a stability certificate for the closed-loop grid-connected system, based on more accurate models and less conservative robust stability analysis conditions, remains as an important issue to be addressed.

Since most of the current control algorithms applied to power converters are implemented in digital platforms, such as microcontrollers and digital signal processors, a discrete-time controller must be computed to stabilize a continuous-time plant (Hara et al. 1996). Although several discretization techniques have already been applied to different classes of systems, like nonlinear (Katayama and Aoki 2014), or systems with time-varying delays (Moraes et al. 2013), most of the approaches in the literature employ classical methods, conceived for precisely known systems, as in Åström and Wittenmark (1984).

However, a realistic modeling must take into account uncertainties that arise from parameter variations, inaccuracies and external perturbations (Ackermann 1993). The exact discretization of uncertain systems is a difficult task since it requires the computation of the exponential of an uncertain matrix. Several methods use numerical approximations (Su et al. 1998) or a first-order Taylor series expansion approach (Kothare et al. 1996; Lee and Won 2006; Wada et al. 2006; Jungers et al. 2011) to circumvent this difficulty, but this implies that, for large values of the sampling time, the discrete-time model becomes inaccurate and there is no theoretical guarantee of stability of the continuous-time closed-loop system. In the context of electrical converters, Maccari et al. (2012, 2014) use a first-order Taylor series approximation and the stability of the closed-loop system was a posteriori verified by means of exhaustive discretizations for a range of values of the uncertain parameters. Recently, Braga et al. (2013) addressed the problem of constant sampling discretization for networked control of polytopic uncertain continuous-time systems. In Braga et al. (2014), the approach was extended to deal with uncertain sampling rates and network-induced time delays.

This paper presents, as main contribution, a new method for robust stability analysis of grid-connected converters with LCL filters affected by uncertain parameters. The discretization procedure applied to the closed-loop uncertain system follows the lines presented in Braga et al. (2013, 2014). More specifically, the grid inductance is assumed unknown but lying inside an interval and, using a previously designed robust state feedback control gain, the objective is to determine the maximum value of the grid inductance such that the closed-loop grid-connected system remains stable. The uncertain affine parameter-dependent continuous-time model resulting from the union of the converter with the LCL filter and the grid is represented as a polytopic system and, by means of an extension of the Taylor series expansion, it is converted into an equivalent uncertain discrete-time system. The discrete-time matrices are composed by homogeneous polynomials plus norm-bounded terms representing the discretization residual error. The discretized system becomes more accurate with higher degrees of Taylor series expansion at the price of a more complex description. The discretization error is also influenced by the sampling period and the uncertainties associated with the original system.

The new robust stability analysis conditions proposed in this paper are suitable to handle discrete-time linear systems with polynomially parameter-dependent matrices and additive norm-bounded uncertainties. The conditions are formulated in terms of LMIs and are based on homogeneous polynomially parameter-dependent Lyapunov functions of arbitrary degree that ensure stability for both the discrete and the continuous-time closed-loop system. Therefore, the main appeal of the proposed approach is to provide a theoretical stability certificate for the grid-connected converter with an LCL filter and a digital controller with the grid inductance lying in an interval, circumventing the drawback of the available approaches that only assess the stability for approximate discretized models, that is, without taking into account the discretization error. Numerical simulations and experimental results validate the proposed approach.

Problem Description

Consider the pulse width modulation (PWM) converter connected to an inductive grid at the point of common coupling (PCC), described in Fig. 1.

Fig. 1
figure1

PWM inverter with output LCL grid-connected filter

The inductances of the LCL filter and the grid are given by \(L_{\mathrm{g}1}\) and \(L_{\mathrm{g}2}\), respectively. The grid inductance, defined by \(L_\mathrm{g} = L_{\mathrm{g}1} + L_{\mathrm{g}2}\), is not precisely known, being represented by an uncertain parameter belonging to a given interval \(L_\mathrm{g} \in [L_{\mathrm{g}1},~ L_{\mathrm{g_{max}}}]\), where \(L_{\mathrm{g_{max}}}\) is a parameter to be maximized such that the closed-loop system is stable. The PWM converter, the LCL filter and the grid can be described as the uncertain affine parameter-dependent continuous-time model given by

$$\begin{aligned} \begin{aligned} {\dot{x}}(t)&= A_p(\theta )x(t) + B_p u(t) + F_p(\theta )w(t)\\ z(t)&= C_px(t) \end{aligned} \end{aligned}$$
(1)

with

$$\begin{aligned} A_p(\theta )= & {} A_{p_0} + \theta A_{p_1}\nonumber \\= & {} \begin{bmatrix} \dfrac{-r_c}{L_c}&\quad \dfrac{-1}{L_c}&\quad 0\\ \dfrac{1}{C}&\quad 0&\quad \dfrac{-1}{C}\\ 0&\quad 0&\quad 0 \end{bmatrix} + \theta \begin{bmatrix} 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0\\ 0&\quad 1&\quad -r_g \end{bmatrix}, \\ B_p= & {} \begin{bmatrix} \dfrac{1}{L_c}\\ 0\\ 0 \end{bmatrix}, ~ F_p(\theta ) = F_{p_0} + \theta F_{p_1} = \begin{bmatrix} 0 \\ 0\\ 0 \end{bmatrix} + \theta \begin{bmatrix} 0\\ 0\\ -1 \end{bmatrix},\\ C_p= & {} \begin{bmatrix} 0&0&1 \end{bmatrix} \end{aligned}$$

where \(\theta \) is an uncertain parameter given by \(\theta = {1}/{L_\mathrm{g}} \in [\theta _{\min } ~~ \theta _{\max }]\) with \(\theta _{\min } = 1/L_{\mathrm{g}_{\max }}\) and \(\theta _{\max } = 1/L_{\mathrm{g}1}\).

The state vector is \(x(t)'=[i_\mathrm{c}(t) ~ v_\mathrm{c}(t) ~ i_\mathrm{g}(t) ]\), where \(i_\mathrm{c}(t)\) is the converter current, \(v_\mathrm{c}(t)\) is the capacitor voltage, and \(i_\mathrm{g}(t)\) is the grid current. The signal u(t) is the control input, synthesized by the PWM inverter, the controlled output z(t) is the grid current \(i_\mathrm{g}\) and the disturbance w(t) is the grid voltage \(v_\mathrm{d}\).

Introducing the following change of variables

$$\begin{aligned} \alpha _1 = \dfrac{\theta -\theta _{\min }}{\theta _{\max } - \theta _{\min }}, \quad \alpha _2 = 1 - \alpha _1, \quad \theta \in [\theta _{\min } ~~ \theta _{\max }], \end{aligned}$$

system (1) is rewritten as a model belonging to a new convex parameter space domain (polytopic),

$$\begin{aligned} \begin{aligned} {\dot{x}}(t)&= A_c(\alpha )x(t) + B_c(\alpha ) u(t) + F_c(\alpha )w(t)\\\ z(t)&= C_c(\alpha ) x(t). \end{aligned} \end{aligned}$$
(2)

The system matrices \(A_c(\alpha )\), \(B_c(\alpha )\), \(F_c(\alpha )\), and \(C_c(\alpha )\) can be written as a convex combination of two known vertices as

$$\begin{aligned} M_c(\alpha )=\sum _{i=1}^{2}\alpha _i M_{c_i} \end{aligned}$$

where \(M_c(\alpha )\) represents any uncertain continuous-time system matrix, \(M_{c_i}\) are the vertices computed as follows

$$\begin{aligned} A_{c1}= & {} A_{p_0}+ \theta _{\max }A_{p_1}, \quad A_{c2} = A_{p_0}+\theta _{\min }A_{p_1}, \\ F_{c1}= & {} F_{p_0}+ \theta _{\max }F_{p_1},\quad F_{c2} = F_{p_0}+\theta _{\min }F_{p_1},\\ B_{c1}= & {} B_{c2} = B_p,\quad C_{c1} =C_{c2} = C_p, \end{aligned}$$

and \(\alpha =(\alpha _1,\alpha _2)\) is a vector of time-invariant parameters belonging to the unit simplex, generically defined (arbitrary number of parameters) as

$$\begin{aligned} {\mathcal {U}}=\Big \{\lambda \in \mathbb {R}^N: \sum _{i=1}^{N}\lambda _i = 1, ~\lambda _i \ge 0, ~i=1,\ldots ,N \Big \}. \end{aligned}$$
(3)

The problem to be investigated in this paper is:

Problem 1

Determine the largest value of the parameter \(L_{g_{\max }}\) such that the continuous-time system (1) is robustly stable for a given state feedback discrete-time controller.

System Discretization

To analyze the continuous-time system controlled by a digital controller, a discrete-time representation of (2) is necessary. Using a zero-order holder with constant sampling period T, one has (Åström and Wittenmark 1984)

$$\begin{aligned} \begin{aligned} x(k+1)&= A(\alpha ) x(k) + {\hat{B}}(\alpha ) v(k)\\ z(k)&= C(\alpha ) x(k), \end{aligned} \end{aligned}$$
(4)

where \(v(k)'=[u(k)' ~~ w(k)']\) and the uncertain matrices \(A(\alpha )\), \({\hat{B}}(\alpha )\) and \(C(\alpha )\) are, respectively,

$$\begin{aligned} \begin{aligned} A(\alpha )&= \exp (A_c(\alpha )T), ~~ B(\alpha ) = \int \limits _{0}^{T} \exp (A_c(\alpha )s)ds B_c(\alpha ) \\ F(\alpha )&= \int \limits _{0}^{T} \exp (A_c(\alpha )s)ds F_c(\alpha ), ~ {\hat{B}}(\alpha ) = [B(\alpha ) ~ F(\alpha )], \\ C(\alpha )&= C_c(\alpha ). \end{aligned} \end{aligned}$$

Since it is a challenging problem to compute exponentials of uncertain matrices, the discretization method presented in Braga et al. (2014), based on a Taylor series expansion, is employed. Therefore, the system matrices of (4) can be written as

$$\begin{aligned} \begin{aligned} A(\alpha )&= A^{\ell }(\alpha ) + \varDelta A^{\ell }(\alpha ), \\ {\hat{B}}(\alpha )&= \left[ B^{\ell }(\alpha ) + \varDelta B^{\ell }(\alpha ) ~~ F^{\ell }(\alpha ) + \varDelta F^{\ell }(\alpha ) \right] \end{aligned} \end{aligned}$$
(5)

with

$$\begin{aligned} A^{\ell }(\alpha )&= \sum \limits _{j=0}^{\ell }\dfrac{A_c(\alpha )^j}{j!}T^j \end{aligned}$$
(6)
$$\begin{aligned} \left[ B^{\ell }(\alpha ) ~~ F^{\ell }(\alpha )\right]&= \sum \limits _{j=1}^{\ell }\dfrac{A_c(\alpha )^{j-1}}{j!}T^j \left[ B_c(\alpha ) ~~ F_c(\alpha ) \right] \end{aligned}$$
(7)

and

$$\begin{aligned} \begin{aligned} \varDelta A^{\ell }(\alpha )&= \exp (A_c(\alpha )T) -A^{\ell }(\alpha )\\ \begin{bmatrix} \varDelta B^{\ell }(\alpha )' \\ \varDelta F^{\ell }(\alpha )' \end{bmatrix}'&= \int \limits _{0}^{T} \exp (A_c(\alpha )s)ds \begin{bmatrix} B_c(\alpha )' \\ F_c(\alpha )' \end{bmatrix}' - \begin{bmatrix} B^{\ell }(\alpha )' \\ F^{\ell }(\alpha )' \end{bmatrix}. \end{aligned} \end{aligned}$$
(8)

From the definitions above and using the notations and results described in Appendix “Multi-nomial Series Development”, matrices (6) and (7) are written as

$$\begin{aligned} A^{\ell }(\alpha )&= I + A_c(\alpha )T + A_c(\alpha )^2 \dfrac{T^2}{2} + \cdots + A_c(\alpha )^{\ell }\dfrac{T^{\ell }}{\ell !} \nonumber \\&= \sum \limits _{j = 0}^{\ell } \left( \sum \limits _{i=1}^{N} \alpha _i \right) ^{\ell -j} \dfrac{T^j}{j!} A_c(\alpha )^j \nonumber \\&= \sum \limits _{k\in {\mathcal {K}}(\ell )} \alpha ^k \sum \limits _{j = 0}^{\ell } \dfrac{T^{j}}{j!} \sum \limits _{\mathop {k \succeq {\hat{k}}}\limits ^{{\hat{k}}\in {\mathcal {K}}(\ell -j)}} \sum \limits _{p \in {\mathcal {R}}(k - {\hat{k}})} \dfrac{(\ell -j)!}{{\hat{k}}!} A_{c_p} \nonumber \\&\triangleq \sum \limits _{k\in {\mathcal {K}}(\ell )} \alpha ^k A_k \end{aligned}$$
(9)
$$\begin{aligned} B^{\ell }(\alpha )&= \left( TI + A_c(\alpha )\dfrac{T^2}{2} + \cdots + A_c(\alpha )^{\ell -1}\dfrac{T^{\ell }}{\ell !} \right) B_c(\alpha ) \nonumber \\&= \sum \limits _{j = 1}^{\ell } \left( \sum \limits _{i=1}^{N} \alpha _i \right) ^{\ell -j} \dfrac{T^j}{j!} A_c(\alpha )^{j-1} B_c(\alpha ) \nonumber \\&= \sum \limits _{k\in {\mathcal {K}}(\ell )} \alpha ^k \sum \limits _{j = 1}^{\ell } \dfrac{T^{j}}{j!} \sum \limits _{\mathop {k \succeq {\hat{k}}}\limits ^{{\hat{k}}\in {\mathcal {K}}(\ell -j)} } \sum _{\mathop { k_{i} - {\hat{k}}_{i} \succ 0 }\limits ^{i \in \{1,\ldots ,N \}} } \sum \limits _{p \in {\mathcal {R}}(k - {\hat{k}} -e_i)} \dfrac{(\ell -j)!}{{\hat{k}}!} A_{c_p} B_{c_i} \nonumber \\&\triangleq \sum \limits _{k\in {\mathcal {K}}(\ell )} \alpha ^k B_k, \end{aligned}$$
(10)

where k! stands for \(k_{1}! k_{2}! \cdots k_{N}! \), the N-tuple \(e_{i}\) is defined as a null vector composed by N elements with ith component equal to one, and \(A_{k}\) and \(B_{k}\) are coefficients of the discretized system polynomial matrices \(A_c(\alpha )^{\ell }\) and \(B_c(\alpha )^{\ell }\). Analogously, \( F^{\ell }(\alpha ) \triangleq \sum _{k\in {\mathcal {K}}(\ell )} \alpha ^k F_k,~ \) with the coefficients \(F_k\), is obtained in the same way of \(B_k\).

As can be seen, the matrices of the discrete-time system (4) are described by (5) in terms of homogeneous polynomial matrices depending on an uncertain parameter belonging to the unit simplex domain plus norm-bounded terms \( \varDelta A^{\ell } (\alpha )\), \(\varDelta B^{\ell } (\alpha )\), and \(\varDelta F^{\ell } (\alpha ) \). These additional terms, related to the residue of the approximation, depend on the degree \(\ell \) of the Taylor series expansion, the sampling period and the original continuous-time system uncertain domain.

Augmented System

In the implementation of the controller in a digital signal processor (DSP), the control signal is delayed by one sample. To take this delay into account, an augmented state is introduced (Åström and Wittenmark 1984), yielding the following state-space representation of (4)

$$\begin{aligned} \begin{aligned} {\hat{x}}(k+1)&= G(\alpha ){\hat{x}}(k)+H{\hat{u}}(k)+H_{\mathrm{dist}}(\alpha ){\hat{w}}(k)\\ {\hat{z}}(k)&= C_{\mathrm{dist}}(\alpha ){\hat{x}}(k) \end{aligned} \end{aligned}$$
(11)

where

$$\begin{aligned} G(\alpha )= & {} \begin{bmatrix} A(\alpha )&\quad B(\alpha )\\ 0&\quad 0 \end{bmatrix},~~ H = \begin{bmatrix} 0\\1 \end{bmatrix}, \\ H_{\mathrm{dist}}(\alpha )= & {} \begin{bmatrix} F(\alpha ) \\ 0 \end{bmatrix}, ~~ C_{\mathrm{dist}}(\alpha ) = \begin{bmatrix} C(\alpha )&0 \end{bmatrix}, \\ {\hat{x}}(k)'= & {} \begin{bmatrix} x(k)'&u(k-1)' \end{bmatrix}, ~~ {\hat{w}}(k)= w(k), \\ {\hat{u}}(k)= & {} u(k), ~~ {\hat{z}}(k) = z(k). \end{aligned}$$

To ensure that the current injected into the grid tracks sinusoidal references and also rejects harmonic disturbances, resonant controllers, based on the internal model principle (Francis and Wonham 1976), can be used. The inclusion of the resonant controller

$$\begin{aligned} \xi (k+1)=R\xi (k) -M C_{\mathrm{dist}} {\hat{x}}(k), \end{aligned}$$

in (11) yields

$$\begin{aligned} {\tilde{x}}(k+1) = ({\tilde{A}}(\alpha )+\varDelta {\tilde{A}}(\alpha )){\tilde{x}}(k)+{\tilde{B}}{\tilde{u}}(k) + {\tilde{F}} {\tilde{w}}(k) \end{aligned}$$
(12)

where R and M are the gains of the resonant controller,Footnote 1 \({\tilde{B}} = \begin{bmatrix} H'&0 \end{bmatrix}'\), \({\tilde{F}} = \begin{bmatrix} H_{\mathrm{dist}}(\alpha )'&0 \end{bmatrix}\),

$$\begin{aligned} {\tilde{A}}(\alpha )+\varDelta {\tilde{A}}(\alpha ) = \begin{bmatrix} G(\alpha )&0\\ -M C_{\mathrm{dist}}&R \end{bmatrix}, \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} {\tilde{A}}(\alpha ) = \begin{bmatrix} A^{\ell }(\alpha )&\quad B^{\ell }(\alpha )&\quad 0\\ 0&\quad 0&\quad 0\\ -M C(\alpha )&\quad 0&\quad R \end{bmatrix}, \\ \varDelta {\tilde{A}}(\alpha ) = \begin{bmatrix} \varDelta A^{\ell }(\alpha )&\quad \varDelta B^{\ell }(\alpha )&\quad 0\\ 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0 \end{bmatrix}, \end{aligned} \end{aligned}$$
(13)

and

$$\begin{aligned} {\tilde{x}}(k)'= \begin{bmatrix} {\hat{x}}(k)'&\xi (k)' \end{bmatrix}, ~~ {\tilde{u}}(k) = {\hat{u}}(k), ~~ {\tilde{w}}(k) = {\hat{w}}(k). \end{aligned}$$

The new augmented state vector in (12) includes the filter state variables, the states of the resonant controller and the delay from the digital control implementation.

In the next section, appropriate conditions, which employ the proposed discretized model, are developed to verify the robust stability of the closed-loop system. To prove the LMI analysis condition, the following lemma (Boyd et al. 1994) is used.

Lemma 1

Given a scalar \(\lambda > 0\) and matrices U and V of compatible dimensions, then \(UV + V'U' \le \lambda UU' + \lambda ^{-1} V'V\).

Robust Stability Analysis

Assuming that the state vector x(t) is available for feedback, that the synchronization with the PCC voltage is already ensured (\(w(t) = 0\)) and that \(v_{cc}\) voltage is stabilized at a constant value, consider the state feedback control law

$$\begin{aligned} {\tilde{u}}(k) = K {\tilde{x}}(k), \end{aligned}$$
(14)

with known stabilizing gain K designed as in Maccari et al. (2014) and reported in Appendix “System Parameters and Control Gains”, assigning the closed-loop eigenvalues in a circular region inside the unit circle to ensure good transient response. Neglecting the disturbance input, the closed-loop system asymptotic stability can be assessed through the representation

$$\begin{aligned} {\tilde{x}}(k+1) = A_{cl}(\alpha ){\tilde{x}}(k). \end{aligned}$$
(15)

with

$$\begin{aligned} A_{cl}(\alpha ) = A_{cl}^{\ell }(\alpha ) + \varDelta A_{cl}^{\ell }(\alpha ), ~~ A_{cl}^{\ell }(\alpha ) = {\tilde{A}}(\alpha )+{\tilde{B}}K \end{aligned}$$
(16)

where \(\varDelta A_{cl}^{\ell }(\alpha ) = \varDelta {\tilde{A}}(\alpha )\).

The discretization residual error \(\varDelta {\tilde{A}}(\alpha )\) given in (13) can be bounded by

$$\begin{aligned} \delta _{{\tilde{A}}} = \sup _{\alpha \in {\mathcal {U}}} \left\| \varDelta {\tilde{A}}(\alpha ) \right\| . \end{aligned}$$
(17)

Estimates for \(\delta _{{\tilde{A}}}\) can be calculated, for instance, using interval analysis methods (Oppenheimer and Michel 1988; Althoff et al. 2007), that are in general very conservative. In this paper, inner approximations of \(\delta _{{\tilde{A}}}\) will be obtained by evaluating the values of \(\alpha \in {\mathcal {U}}\) through an exhaustive grid search.

Problem 1 can be restated in terms of robust LMI conditions based on a homogeneous polynomially parameter-dependent Lyapunov function that ensures the stability of (15), as presented in Theorem 1.

Theorem 1

If there exist degrees g and \(f \in \mathbb {N}\), a symmetric positive definite matrix \(W(\alpha ) = \sum \limits _{k \in {\mathcal {K}}(g)} \alpha ^{k}W_{k},~\) matrices

$$\begin{aligned} X(\alpha ) = \sum _{k \in {\mathcal {K}}(f)} \alpha ^{k} X_{k} \quad \text{ and } \quad Y(\alpha )=\sum _{k \in {\mathcal {K}}(f)}\alpha ^{k} Y_{k} \end{aligned}$$

of suitable dimensions, and a positive scalar \(\lambda _{{\tilde{A}}}\), such that the following parameter-dependent LMIFootnote 2

$$\begin{aligned} \begin{bmatrix} W(\alpha ) - \varPsi (\alpha ) - \lambda _{{\tilde{A}}} \delta _{{\tilde{A}}}^{2} I&\quad \star&\quad \star \\ -X(\alpha )' + Y(\alpha ) A_{cl}^{\ell }(\alpha )&\quad \Omega (\alpha ) - W(\alpha )&\quad \star \\ X(\alpha )'&\quad Y(\alpha )'&\quad \lambda _{{\tilde{A}}} I \end{bmatrix} > 0 \end{aligned}$$
(18)

holds for all \(\alpha \in {\mathcal {U}}\), with \( \Omega (\alpha ) = Y(\alpha ) + Y(\alpha )'\), \( \varPsi (\alpha ) = A_{cl}^{\ell }(\alpha )' X(\alpha )'+ X(\alpha ) A_{cl}^{\ell }(\alpha )\), and \(A_{cl}^{\ell }(\alpha )\) and \(\delta _{{\tilde{A}}}\) given, respectively, by (16) and (17), then the closed-loop system (15) is asymptotically stable.

Proof

See Appendix “Proof of Theorem 1\(\square \)

Remark 1

Differently from the results presented in Maccari et al. (2014), where a fine grid on the system parameters is evaluated to provide a necessary condition for stability, and most of the approaches available in the literature, Theorem 1 guarantees the stability of both continuous and discrete-time closed-loop system. The demonstration of the previous statement follows the lines presented in Braga et al. (2014). The appropriate choice of the degree \(\ell \) of the Taylor series expansion yields sufficiently small values for the upper bound of discretization residual error \(\delta _{{\tilde{A}}}\) such that Theorem 1 is feasible. Higher values of \(\ell \), f, and g improve the results, at the expense of increasing the computational burden.

Numerical Implementation

All routines are implemented in MATLAB, version 7.10 (R2010a), using YALMIP (Löfberg 2004) and SeDuMi (Sturm 1999), on an AMD Phenon II X6 1090T (3.2 GHz), 4GB RAM Windows PC. The parameter-dependent LMI condition (18) is implemented by means of a set of finite-dimensional LMI conditions based on a homogeneous polynomially parameter-dependent Lyapunov matrix of arbitrary degree. Details of the numerical implementation and about the procedure to obtain finite-dimensional LMIs from (18), described in terms of matrix-value coefficients derived from monomials with the same degree, can be found in Oliveira and Peres (2007).

Robust Stability Analysis

To verify the efficiency of the proposed conditions for robust stability analysis of system (1) with the discrete-time control law (14), consider the parameters of the converter and the state feedback gain given, respectively, in Table 2 and (24) of Appendix “System Parameters and Control Gains”. More details on the experimental results are given in Sect. 6.2. In Maccari et al. (2014), the controller was designed based on a simplified polytopic model, derived from a discretization method using a first-order Taylor approximation, providing good experimental results, but with no theoretical certification of stability for the continuous-time closed-loop system. The stability was a posteriori verified by means of an exhaustive discretization for a range of values of \(L_g\), implying in high computational effort and even in this case without robust stability guarantees.

Now, employing the discretization procedure presented in Sect. 3 to generate the matrix coefficients \(A_k\) and \(B_k\) that appear in (9) and (10), Theorem 1 can be used to determine the maximum values of \(L_{\mathrm{g}}\) for which the closed-loop stability of (15) is assured. The results for a sampling period \(T = 1/f_{s}\), with \(f_{s} = 20040\) Hz, and filter resistances sufficiently small to be neglected, as in Maccari et al. (2014), are presented in Table 1, where \(\ell \) is the level of the Taylor series expansion, g is the degree of Lyapunov matrix \(W(\alpha )\), f is the degree of the slack variables \(X(\alpha )\) and \(Y(\alpha )\), and \(L_{\mathrm{g}_{\max }}\) is the maximum allowable value of \(L_g\) such that the closed-loop system is stable. The numerical complexity is also presented in Table 1 based on the number of LMI rows (\({\mathcal {L}}\)) and the number of scalar variables (\({\mathcal {V}}\)). Theorem 1 was not feasible for degrees \(\ell \le 4\), since the upper bound of discretization residual error was not sufficiently small. Observe that less conservative results are obtained for higher values of \(\ell \), g, and f, until Theorem 1 reaches \(L_{\mathrm{g}_{\max }}=2.8348\) mH. It was not possible to obtain less conservative values of \(L_{\mathrm{g}_{\max }}\) for levels \(\ell \ge 7\), or degrees g and \(f \ge 1\), once the results do not present significant changes within the adopted precision. Table 1 also shows the bounds on the discretization errors (\(\delta _{{\tilde{A}}}\)) that are directly related to the degree \(\ell \) of the Taylor series expansion. For \(\ell >7\) the reduction of \(\delta _{{\tilde{A}}}\) is no more significant and, therefore, has no effect on the feasibility of Theorem 1.

Table 1 Maximum allowable value of \(L_g\) (mH) such that the closed-loop system is stable, bounds on the discretization errors (\(\delta _{{\tilde{A}}}\)), and numerical complexity described by the number of scalar variables (\({\mathcal {V}}\)) and LMI rows (\({\mathcal {L}}\)) with different parameters \(\ell \), g and f in Theorem 1

To measure the quality of the results provided by Theorem 1, which uses a polynomially parameter-dependent Lyapunov matrix of degree g and slack variable matrices of degree f, it is performed a comparison with the condition in (Gabe et al. 2009, Theorem 2), which is based on quadratic stability (parameter-independent Lyapunov matrix). For that, consider the discretization strategy presented in Gabe et al. (2009), where only the vertices of the polytopic model were discretized by using a zero-order hold (ZOH). Applying (Gabe et al. 2009, Theorem 2) with \(d=0\), \(r=1\), and the same control gains used here, the maximum value obtained for \(L_g\) is 2.1169 mH. If the discretization procedure proposed in (9) and (10), with \(\ell \ge 5\), is used jointly with (Gabe et al. 2009 Theorem 2), one gets the same result, that is, \(L_{g_{\max }}=2.1169\) mH. In both cases, it represents a difference of 0.7179 mH (\(25.32\%\)) with respect to the maximum allowed value of \(L_g\) given by Theorem 1. Another common strategy adopted in the literature is to perform a first-order Taylor expansion in each vertex of the polytopic system, neglecting the discretization error in the stability analysis condition. By employing this procedure, both (Gabe et al. 2009, Theorem 2) and Theorem 1 proposed here do not provide feasible solutions. Note that the proposed approach is always less conservative than (Gabe et al. 2009, Theorem 2), once neglecting the discretization error (\(\delta _{{\tilde{A}}} = 0\)) and for particular choices \(X(\alpha ) = 0\), \(Y(\alpha ) = W(\alpha ) = P\), Theorem 1 recovers the conditions of (Gabe et al. 2009 Theorem 2).

To corroborate the limit of stability provided by Theorem 1, an exhaustive grid in \(L_{g} \in [0.5,~2.8348]\) mH was performed to compute the eigenvalues of the augmented discrete-time closed-loop matrix, which are presented in Fig 2a. The maximum absolute values of these eigenvalues, defined by \(\sigma \triangleq \max {\Vert eig(A_{cl})\Vert }\), where eig(M) stands for the eigenvalues of matrix M, are shown in Fig 2b.

Fig. 2
figure2

a Closed-loop eigenvalues for the augmented system for \(L_{g} \in [0.5,~ 2.8348\)] mH and b \(\sigma \) versus \(L_{g}\)

It is very important to stress that, while the exhaustive evaluation of the closed-loop eigenvalues is only a necessary condition for stability, the conditions of Theorem 1 are sufficient to ensure the closed-loop stability under uncertain and time-invariant grid inductance. The results from Theorem 1 rely on the existence of a parameter-dependent Lyapunov function and provide new information on the stability bounds for this application.

To evaluate how the resistance in the grid affects the system stability, Fig. 3a shows the values of \(\sigma \), for \(L_{\mathrm{g}} \in [2.8348,~2.8350]\) mH, and for the grid-side resistance, \(r_\mathrm{g}\), equals to 0, 0.1, 0.25 and 0.5 \(\Omega \). This analysis shows clearly that the increase in the resistance \(r_g\) improves the stability margin, since the curve for \(r_\mathrm{g}=0.5\) \(\Omega \) is always below the curve for \(r_\mathrm{g}=0\) \(\Omega \). Thus, the stability analysis for \(r_\mathrm{g}=0\) \(\Omega \) is the more stringent evaluation. For instance, with \(r_\mathrm{g} = 0.5~\Omega \), the value of \(L_{\mathrm{g}_{\max }}\) obtained by means of exhaustive gridding is 3.0399 mH, while Theorem 1, for the same situation, was able to find \(L_{\mathrm{g}_{\max }}\) equals to 3.0393 mH, which is a tight evaluation of stability.

Fig. 3
figure3

a \(\sigma \) versus \(L_{\mathrm{g}} \in [2.8348,~2.8350]\) mH, and for \(r_g\) equals to 0, 0.1, 0.25 and 0.5 \(\Omega \) and b detailed view of curve for \(r_g=0\) \(\Omega \) in (a), showing limits of stability

Considering \(r_\mathrm{g} = 0\), a detailed view of the stability limit can be seen in Fig. 3b, which shows that instability occurs for a value of \(L_{\mathrm{g}_{\max }}\) slightly lower than 2.8350 mH. Actually, with a fine grid, one has that instability occurs with \(L_{\mathrm{g}_{\max }}=2.83499333\) mH. The accuracy of the proposed method is proven by the small gap between the bound of stability provided by Theorem 1 (\(L_{\mathrm{g}_{\max }}=2.8348\) mH) and the bound of instability (\(L_{\mathrm{g}_{\max }}= 2.83499\) mH), obtained through exhaustive gridding.

It is worth emphasizing that the proposed discretization procedure provides a more accurate discrete-time model (as \(\ell \) increases), allowing to include parasitic resistances in the filter or even in the grid, to cope, for instance, with passive damping. Note that the conditions of Theorem 1 are always applied off-line with acceptable computational cost, providing an increasingly precise and reliable evaluation of stability domains (as f and g increase), becoming an alternative to exhaustive gridding.

Simulation and Experimental Results

The prototype used in the experiments has the following features:

  • digital signal processor (DSP) from Texas Instruments, model TMS320F28335, floating point, clock of 150 MHz, with 16 A/D converters and 12 PWM outputs;

  • LCL filter: inductors from Semikron and capacitor from Epcos;

  • voltage sensors with transformers and current sensors from LEM;

  • Inverter from Semikron, with IGBT transistors.

Figure 4 shows the photographs of the prototype. In this paper, only one of the phases was used to have a single-phase implementation.

Fig. 4
figure4

Experimental setup a power circuits a inverter; b LCL filter; and b control circuits a DSP where the control law is implemented; b circuits for voltage acquisition; c circuits for control acquisition

Time-domain simulations are given in Fig. 5a, where steady-state grid current waveforms for \(L_\mathrm{g}=0.5000\) mH and \(L_\mathrm{g}=2.8348\) mH show that the closed-loop system is stable for these values. Simulations for several values of \(L_\mathrm{g}\) inside the interval also indicate that the closed-loop system remains stable. Thus, the limit of stability from Theorem 1 is also confirmed through time-domain analysis.

Fig. 5
figure5

a Simulation steady-state responses for the closed-loop system. Waveform in gray for \(L_\mathrm{g}=0.5\) mH and waveform in black line for \(L_\mathrm{g}=2.8348\) mH and b experimental results for \(L_\mathrm{g}=1.5\) mH showing the reference (\(i_{{\mathrm{ref}}}\)), in gray line, and the system output (\(i_\mathrm{g}\)), in black line 7

It is not possible to test, experimentally, the stability of the closed-loop polytopic system (15) for the entire domain of uncertainties \(\alpha \in {\mathcal {U}}\) given in (3). Therefore, to validate the proposed approach, an experimental result was obtained choosing \(\alpha = (\alpha _{1},~\alpha _{2}) = (0.19,~0.81)\), which corresponds to an LCL filter with a grid inductance of \(L_\mathrm{g}= 1.5\) mH. The control algorithm was implemented in a 32-bit floating-point DSP TMS320F28335 from Texas Instruments. The switching frequency is \(f_\mathrm{s}/2\), equals to Maccari et al. (2014). Note that the chosen value (\(L_\mathrm{g} = 1.5\) mH) lies inside the range computed by Theorem 1 for which the closed-loop stability is assured. Fig. 5b shows the stable behavior of the closed-loop system. The curves depicted in Fig. 5b are experimental results, obtained from data stored in the DSP, showing simultaneously the output current and its reference, the transient startup and the steady-state response.

It is worth to mention that the effect of  T (sampling period) is important for the control design. As T decreases, the discrete-time polytopic model used in Maccari et al. (2014) can be approximated by a two-vertex polytope, with small discretization error. Concerning the closed-loop stability analysis, the augmentation of T implies in larger discretization errors, thus, it is necessary to employ larger degrees (\(\ell \)) in the Taylor series expansion (Eqs. (9), (10)) to obtain more accurate outcomes which, on the other hand, demands a higher computational burden due to the use of a discrete-time representation of the continuous-time system in terms of polynomial matrices of larger degrees. The sampling period used in this paper is the same used in Maccari et al. (2014), that is, \(f_\mathrm{s} = 20040\) Hz, illustrating that the proposed conditions can provide a more accurate evaluation of the stability domains. Moreover, Theorem 1 can also cope with different values of sampling frequency in the stability analysis, whose precision is related to the variables \(\ell \), g, and f. For instance, considering a sampling frequency of 12 kHz, the upper bound to \(L_\mathrm{g}\) computed by Theorem 1 (with \(\ell = 7\), \(g = f = 1\)) is \(L_{\mathrm{g}_{\max }} = 2.4954\) mH, while the maximum value obtained by exhaustive gridding for this case is \(L_{\mathrm{g}_{\max }} = 2.4960\) mH. It was also observed that Theorem 1 (with \(\ell = 7\), \(g = f = 1\)) ensures that the controller designed for \(f_s = 20040\) Hz can be implemented with rate of 16275 Hz keeping the maximum value of the inductance presented in Table 1, \(L_{\mathrm{g}_{\mathrm{max}}} = 2.8348\) mH. Thus, the proposed approach is able to assure the stability of the closed-loop system for lower sampling frequencies for smaller values of \(L_{\mathrm{g}_{\max }}\). The increase in the sampling rate implicates in a greater interval of feasibility for \(L_\mathrm{g}\). This example illustrates another potential use for Theorem 1: the closed-loop stability analysis of any uncertain system as a function of the sampling period. Thus, Theorem 1 can be applied, for instance, to evaluate the minimum sampling frequency in which the closed-loop stability can be assured for medium power grid-connected converters, which use lower switching and sampling frequencies.

Conclusion

In this paper, the robust stability analysis of a hybrid system, composed by a grid-connected converter and an LCL filter with uncertain grid inductance modeled as an uncertain continuous-time plant controlled by a digital gain, is investigated. For this purpose, the continuous-time plant is converted to the discrete-time domain by means of a new discretization procedure based on a Taylor series expansion of arbitrary degree. The resulting model is a polynomially parameter-dependent discrete-time system plus a norm-bounded term associated with the approximation residue. By means of a parameter-dependent Lyapunov function, sufficient LMI-based conditions provide less conservative robust stability analysis results, in terms of a larger range of values of the grid inductance. As main appeal of the proposed procedure, a theoretical certificate of the closed-loop continuous-time system stability is obtained for the entire range of the grid inductance parameter. The proposed conditions provide new information on stability domains, being an efficient alternative tool to certify stability of uncertain systems, with potential interest in control applied to power electronics. Moreover, it is also possible to use such conditions to investigate the stability of any power converter that has affine or polynomial dependence in one or more physical uncertain parameters. Simulations and experimental results show the efficiency of the approach.

Notes

  1. 1.

    See Appendix “State-Space Description of Resonant Controllers” for a more detailed description of the resonant controller matrices R and M.

  2. 2.

    The symbol \(\star \) represents a symmetric block in the LMI.

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Acknowledgements

The work was supported by Brazilian agencies CAPES, CNPq (Grants 307536/2012-2, 477487/2013-0), and FAPESP (Grants 2011/08312-6, 2014/22881-1).

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Correspondence to Eduardo S. Tognetti.

Appendices

State-Space Description of Resonant Controllers

In order to ensure zero steady-state error and also to reject harmonic disturbances, resonant controllers can be used. In continuous-time, it is possible to consider the following resonant controller

$$\begin{aligned} {\dot{\xi }}_i(t)=R_{ci} \xi _i(t) + M_{ci} e(t) \end{aligned}$$
(19)

where

$$\begin{aligned} R_{ci} = \left[ {\begin{array}{*{20}{c}} 0 &{}1\\ { - {\omega _i}^2}&{}{ - 2{\zeta _i}{\omega _i}} \end{array}} \right] ~,~ M_{ci} = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right] \end{aligned}$$

and the tracking error is given by

$$\begin{aligned} e(t)=i_{ref}(t)-z(t). \end{aligned}$$
(20)

In this representation, \(\omega _i\) is the frequency of resonant controller and \(\zeta _i\) is a damping factor, employed to avoid problems in the discrete-time implementation related with placing controller poles at the border of the unit circle.

A state-space discrete-time representation of this controller is given by

$$\begin{aligned} \xi _i(k+1)= R_i \xi _i(k)+ M_i e(k) \end{aligned}$$
(21)

where \( \xi _i(k) \in \mathbb {R}^{2 \times 1}\), \( R_i \in \mathbb {R}^{2 \times 2}\), \(M_i \in \mathbb {R}^{2 \times 1}\).

Note that (21) also describes a resonant controller, but specialized for state feedback.

In the general case, a set of the above controllers can be used to represent multiple resonant controllers as

$$\begin{aligned} \xi (k+1)= R \xi (k) - M C_{\mathrm{dist}} x(k) + M i_{\mathrm{ref}}(k) \end{aligned}$$
(22)

where

$$\begin{aligned} \xi = \begin{bmatrix} \xi _1\\ \vdots \\ \xi _n \\ \end{bmatrix}, ~~ R = \begin{bmatrix} R_1&\quad&\quad \\&\quad \ddots&\quad \\&\quad&\quad R_n \\ \end{bmatrix}, ~~ M = \begin{bmatrix} M_1\\ \vdots \\ M_n \\ \end{bmatrix}. \end{aligned}$$
(23)

In this representation, the state vector has dimension 2n, where n is the number of resonant controllers to be implemented, and each resonant controller is discretized independently, resulting in the matrices R (block diagonal matrix) and M.

System Parameters and Control Gains

The parameters of the system used for controller design and to obtain the experimental results are given in Table 2.

Table 2 Parameters of the converter

Considering the parameters of Table 2 and tunning the resonant controllers at the frequencies of 60, 180, 300 and 420 Hz, the control gains designed in Maccari et al. (2014), used in Sect. 6, are given by (truncated with 4 decimal digits)

$$\begin{aligned} \mathbf{K }= \begin{bmatrix} -13.0046 \\ -0.8727 \\ -3.2444 \\ -0.5887 \\ 87.2641 \\ -86.5638 \\ 43.0993 \\ -41.8932 \\ 38.4751 \\ -37.7920 \\ 37.8061 \\ -36.2425 \end{bmatrix}'. \end{aligned}$$
(24)

Multi-nomial Series Development

In the matrix case, products in multi-nomial series are non-commutative. In this case, one can write \(A_{c}(\alpha )\) as

$$\begin{aligned} A_{c}(\alpha )^q&= \left( \sum \limits _{i=1}^{N} \alpha _i A_{ci} \right) ^q =\sum \limits _{p \in {\mathcal {P}}(q)} \prod \limits _{i=1}^{q} \alpha _{p_i} A_{c_{p_i}} \nonumber \\&= \sum \limits _{p \in {\mathcal {P}}(q)} \alpha _{p_1} A_{c_{p_1}} \cdots \alpha _{p_q} A_{c_{p_q}} \nonumber \\&= \sum \limits _{p \in {\mathcal {P}}(q)} \alpha _{p} A_{c_{p}}, = \sum \limits _{k\in {\mathcal {K}}(q)} \alpha ^{k} \sum \limits _{p \in {\mathcal {R}}(k)} A_{c_{p}} \end{aligned}$$
(25)

where, \( A_{c_{p}} = A_{c_{p_1}} \cdots A_{c_{p_q}}\) \(\alpha ^k=\alpha _1^{k_1} \alpha _2^{k_2} \cdots \alpha _N^{k_N}\), \(k=(k_1 k_2 \cdots k_N)\), \(\alpha _p=(\alpha _{p_1}, \alpha _{p_2}, \ldots , \alpha _{p_q}\)), \(p=(p_1 p_2 \cdots p_q)\),

$$\begin{aligned} {\mathcal {K}}(q) \triangleq \Big \{ k = ( k_1 \cdots k_N ) \in \mathbb {N}^N : \sum _{j=1}^N k_j = q, ~~ k_j \ge 0 \Big \}, \end{aligned}$$

\( {\mathcal {P}}(q) \) is the set of q-tuples obtained as all possible combinations of nonnegative integers \( p_i \), \( i = 1, \ldots , q \), such that \( p_i \in \{ 1,\ldots ,N \} \), that is,

$$\begin{aligned} {\mathcal {P}}(q) \triangleq \Big \{ p \in \mathbb {N}^q : p_i \in \{1,\ldots ,N \}, ~ i = 1, \ldots , q \Big \} \end{aligned}$$

and \( {\mathcal {R}}(k) \), \(k\in {\mathcal {K}}(q)\), is the subset of all q-tuples \(p \in {\mathcal {P}}(q)\) such that the elements j of p have multiplicity \(k_j\), for \(j=1,\ldots ,N\), that is,

$$\begin{aligned} {\mathcal {R}}(k) \triangleq \Big \{ p \in \mathbb {N}^q : m_p(j) = k_j, ~j = 1, \ldots , N \Big \} \end{aligned}$$

where \(m_p(j)\) denotes the multiplicity of the element j in p.

By definition, for N-tuples k and \(k'\), one has that \(k \ge k'\) if \(k_{i} \ge k'_{i}\), \(i = 1, \ldots , N\). Operations of summation \(k + k'\) and subtraction \(k - k'\) (whenever \(k' \le k\)) are defined componentwise.

Proof of Theorem 1

To prove Theorem 1, apply Schur’s complement in (18) to get

$$\begin{aligned} \underbrace{\begin{bmatrix} W(\alpha ) - \varPsi (\alpha )&\star \\ -X(\alpha )' + Y(\alpha ) A_{cl}^{\ell }(\alpha )&\Omega (\alpha ) - W(\alpha ) \end{bmatrix}}_{\varGamma } - \\ \lambda _{{\tilde{A}}} \begin{bmatrix} \delta _{{\tilde{A}}} I \\ 0 \end{bmatrix} \begin{bmatrix} \delta _{{\tilde{A}}} I \\ 0 \end{bmatrix}' - \lambda _{{\tilde{A}}}^{-1} \underbrace{\begin{bmatrix} X(\alpha ) \\ Y(\alpha ) \end{bmatrix}}_{V'} \begin{bmatrix} X(\alpha ) \\ Y(\alpha ) \end{bmatrix}' > 0. \end{aligned}$$

Knowing that \(\varDelta A_{cl}^{\ell }(\alpha )' \varDelta A_{cl}^{\ell }(\alpha ) < \delta _{{\tilde{A}}}^{2}I\) and using Lemma 1, one obtains

$$\begin{aligned} \varGamma - \begin{bmatrix} \varDelta A_{cl}^{\ell }(\alpha ) \\ 0 \end{bmatrix} V - V' \begin{bmatrix} \varDelta A_{cl}^{\ell }(\alpha ) \\ 0 \end{bmatrix}' > 0 \end{aligned}$$
(26)

which is equivalent to

$$\begin{aligned} \begin{bmatrix} \begin{pmatrix} W(\alpha ) -{A_{cl}}(\alpha )' X(\alpha ) ' \\ - X(\alpha ) {A_{cl}}(\alpha ) \end{pmatrix}&\star \\ -X(\alpha )' + Y(\alpha ) {A_{cl}}(\alpha )&\Omega (\alpha ) -W(\alpha ) \end{bmatrix} > 0 \end{aligned}$$
(27)

where \({A_{cl}}(\alpha )\) is given by (16). Finally, multiplying (27) by \({\mathcal {B}}' = \begin{bmatrix} -I&{A_{cl}}(\alpha )'\end{bmatrix}\) on the left and by \({\mathcal {B}}\) on the right, one gets \(W(\alpha ) - (A_{cl}^{\ell }(\alpha ) + \varDelta A_{cl}^{\ell }(\alpha ))' W(\alpha ) (A_{cl}^{\ell }(\alpha ) + \varDelta A_{cl}^{\ell }(\alpha )) > 0,\) that ensures the stability of the closed-loop system (15).

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Braga, M.F., Morais, C.F., Maccari, L.A. et al. Robust Stability Analysis of Grid-Connected Converters Based on Parameter-Dependent Lyapunov Functions. J Control Autom Electr Syst 28, 159–170 (2017). https://doi.org/10.1007/s40313-017-0301-7

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Keywords

  • Robust stability analysis
  • Linear matrix inequalities
  • Parameter-dependent Lyapunov function
  • Grid-connected converters
  • Sampled-data systems