Robust structured controller synthesis for interleaved boost converters using an H ∞ control method

Abstract

The regulation of output voltage and equivalent distribution of phase currents of multi-phase converters which have non-minimum phase characteristic are still challenges, especially in the presence of uncertainties in real parameters, duty cycle, input voltage, and load disturbances. However, in classical third-order integral-lead (Type-III) controller design methodologies, the controller is synthesized considering only the nominal performance conditions. This paper proposes a structured $H_{\infty}$ synthesis framework based on an optimization methodology to the design of a robust Type-III controller for interleaved boost converters. The structured $H_{\infty}$ control approach is adapted for optimization of Type-III feedback and feedforward controllers in two-degree-of-freedom (2-DOF) control system configuration. The robust stability of the closed-loop interleaved boost converter system against model uncertainties is ensured via the classical $μ$ -analysis technique. Numerical comparisons are made among the classical, i.e. unstructured or full order, $H_{\infty}$ -based controller design method, a dual-loop PI controller, and proposed 1-DOF and 2-DOF structured controller synthesis approaches on an interleaved boost converter model. Simulation results verify the effectiveness and advantages of the proposed approach from the viewpoint of the output voltage regulation under different disturbance points.

Keywords

DC/DC interleaved boost converter (IBC)non-minimum phase systems fuel cell applications

Introduction

Polymer electrolyte membrane fuel cells (PEMFCs) are the most promising among the fuel cell stack technologies owing to their distinguishing features including efficient energy conversion, high-power density, and lower temperature/pressure ranges (50–100 ºC). The available terminal voltage range of PEMFC is between 26 and 50 V whereas it has a generally high current capacity (Rahimi et al. 2019). The terminal voltage is negatively affected by operating temperature and load variations although it is desired to keep fuel cell (FC) output voltage at a fixed DC value. On the other hand, maintaining the PEMFC current ripple as small as possible is an essential requirement to expand the lifespan and increase the capacity of FCs. Therefore, non-isolated DC–DC boost converter topologies are widely used in this type of application to step up the terminal voltage at a high DC value and to limit low-frequency current ripple (LFCR) of FC nominal current value within 5% (Zhan et al. 2019). In particular, interleaved boost converters (IBCs), multi-device interleaved boost converters (MDBCs), floating-interleaved boost converters (FIBCs) and inverse coupled interleaved boost converter (IC-IBCs) are viable topologies in such applications (Chakraborty et al. 2019).

To address the model uncertainty, disturbance attenuation and noise suppression requirements of the power electronic converters, the robust control methods can be preferred. A robust constrained predictive fault-tolerant control method was proposed by Han et al. (2020) to decrease online complexity and to improve fault-tolerance in a buck converter. A robust control scheme was proposed for an inverter system with an LCL filter by Chowdhury and Kimball (2020). Stability analysis was discussed in the case of voltage and current sensor failures. Mehida et al. (2018) discussed the mixed sensitivity $H_{\infty}$ controller synthesis for four-phase IBC considering the external disturbances and load variations. A dual-loop control consisting of a combination of voltage and current loops was used to ensure the stability of the converter. The second- and third-order controllers were synthesized for the current and voltage loops, respectively. A fixed-order robust $H_{\infty}$ controller was synthesized for the two-degree-of-freedom (2-DOF) flexure-based motion system concerning the disturbances by Zhu et al. (2019). The complexity of the high-order controllers in practice led researchers to obtain popular structured controllers such as linear quadratic regulator (LQR), lead-lag, PID, or third-order integral-lead (Type-III) controllers. The lead, lag, and lead–lag controllers were synthesized for multiple input–multiple output linear systems using the $H_{\infty}$ control method by Liu et al. (2020). A disturbance estimation-based control structure was presented by Ahmad and Ali (2020) to compensate for external disturbances for a classical boost converter. The controller was rearranged as a 2-DOF PID controller for the simplicity of application. The LQR with integral action (LQI)-type controller was optimized using the genetic algorithm to control a two-phase IBC by Habib et al. (2017). To validate the success of the algorithm, the controller performance was analysed in the case of reference tracking, load, and input disturbances. However, stability analysis, model uncertainties, and noise suppression were not studied in that paper. An LQR controller synthesis was performed by converting the Riccati equation into the linear matrix inequalities (LMIs) formulations by Olalla et al. (2009). However, model uncertainties and disturbances that affect the operation of the converter were ignored. A robust switching strategy was developed by Bayat et al. (2019) by modifying the previous paper, where the model uncertainties, input and load disturbances were also included in the synthesis process. The optimal controller was obtained using LMI formulations based on sophisticated convex optimization. The fact that the LQR needs to measure the inductor currents of the multi-phase converters has canalized researchers to prefer PI/PID/Type-III controllers which use only the voltage mode control (VMC) loop. The Pareto front optimization based robust controller design was presented for a conventional boost converter by Sadek et al. (2016). The main purpose of the design technique was to find a fixed-order robust $H_{\infty}$ -type PI controller. The designed controller by including model uncertainty was compared with a robust loop-shaping controller. Another fixed-order $H_{\infty}$ -based PD controller was synthesized using a data-driven control approach without any parametric model data for position control of an actuation system by Daş and Başlamışlı (2019).

In standard robust controller design techniques, i.e. $H_{\infty}$ , $H_{2}$ and $μ$ synthesis, the controller is computed via Riccati equations or LMI formulations, where the controller order equals the sum of order of the nominal transfer function and the weighting functions. When the $H_{\infty}$ optimization problem is used in an attempt to solve with a predefined controller structure instead of the unstructured controller, the optimization problem becomes non-convex. Then, the problem will be a non-deterministic polynomial-time hard (NP-hard) problem (Apkarian and Noll 2018). Bilinear matrix inequality (Thevenet et al. 2006), metaheuristic (Feyel 2017), convex (Nicoletti et al. 2020), convex–concave (Mercader et al. 2016), and non-smooth optimization techniques (Apkarian and Noll 2006) have been adapted to the fixed-order controller design problem with the $H_{\infty}$ criterion. Among these approaches, the non-smooth optimization is a very effective tool for obtaining structured controllers using a nominal or uncertain model of a real system. This nonlinear approach allows the designers to synthesize the structured controller in the state-space domain or the frequency domain. Therefore, HIFOO and hinfstruct solvers were developed to solve structured robust control problems using specialized non-smooth optimization techniques (Řezáč and Hurák 2013).

A Type-III controller also called an integral-double-lead controller, has a pole at the origin and two pole–zero pairs. The controller could ensure the required phase margin by providing reasonable phase contribution at some frequency range, particularly in the non-minimum phase (NMP) systems. Unlike the PI/PID controller synthesis methods widely available in the literature, an advanced design method for a Type-III controller synthesis is not available to the best of the authors’ knowledge. There are a few non-robust design methods described in the literature. The particle swarm optimization (PSO) and gravitational search algorithms are preferred to optimize the Type-III controller for a conventional boost converter (Ghosh et al. 2016). Then, the PSO algorithm was used to optimize the controller, realized by an analogue control circuit, for an improved IBC by Banerjee et al. (2016b). The classical pole placement method was presented to synthesize a Type-III controller by Keskin and Aliskan (2018). The controller which has reasonable performance tracking achievement is not tested in the case of the input voltage and load current disturbances. The K-factor design approach is highly preferred to synthesize the controller in the frequency domain (Anzehaee et al. 2018; Banerjee et al. 2016a; Chan et al. 2015). In Anzehaee et al. (2018), the performance of the controller is quietly acceptable due to the 2-DOF control system structure. However, the designed Type-III controller could be inadequate to suppress the effect of the output voltage oscillations as the controller was designed considering only nominal conditions. As aforementioned methods do not consider the noise and disturbance signals during the synthesis process, they could not ensure the robust stability and performance of the closed-loop converter systems.

The main goal of this paper is to synthesize a robust $H_{\infty}$ structured controller that satisfies the desired reference voltage tracking, model uncertainty compensation, input, and output disturbances suppression requirements. The technical contributions of the paper are summarized as follows.

This paper proposes a systematic way to design a robust Type-III controller in a fixed-order $H_{\infty}$ control framework.

Unlike the above-mentioned nominal Type-III design methods, sensor noise signal, input voltage and load current disturbances are considered during the synthesis process.

A Type-III main feedback and a first-order lead type feedforward controllers in 2-DOF configuration are optimized by using the non-smooth optimization algorithm for the converter, simultaneously.

Thanks to the proposed optimization algorithm-based approach, n-DOF structured $H_{\infty}$ controllers can be synthesized.

For the sake of completeness, the robust stability of the simulated converter against possible uncertainties is investigated utilizing $μ$ -analysis, which is recently popular in the $H_{\infty}$ control applications.

The rest of this paper is organized into five sections. The next section includes the small-signal model of the converter. Then, the unstructured and structured robust $H_{\infty}$ optimization problems are discussed. Next, the comprehensive performance and stability of the designed controllers are investigated under the model uncertainties, input voltage, and load current disturbances of the converter. The final section concludes the paper.

Small-signal model of a two-phase IBC

A two-phase IBC circuit with sets of power semiconductor structures and the 2-DOF control system structure are presented in Figure 1 where $V_{0}$ is the output voltage, $R$ is the load resistance, $L_{1}$ and $L_{2}$ are the inductor values of each phase, $r_{l}$ is the equivalent series resistance (ESR) of the inductors, $C_{o}$ is the capacitance, $D$ is the nominal duty cycle and $V_{IN}$ is the input voltage. The dynamic equations of IBC can be described according to the state of the metal–oxide–semiconductor field-effect transistors (MOSFETs) and diodes denoted by $S W_{1}$ , $S W_{2}$ , $D E_{1}$ and $D E_{2}$ in Figure 1. For the duty cycle of the pulse width modulation signals greater than $50 % (D > 0.5)$ , the state-space equations for four modes are represented as

\overset{\cdot}{x} (t) = {\begin{matrix} A_{1} x (t) + B_{w 1} w (t); Mode I S W_{1} : ON, S W_{2} : OFF \\ A_{2} x (t) + B_{w 2} w (t); Mode II S W_{1} : ON, S W_{2} : ON \\ A_{3} x (t) + B_{w 3} w (t); Mode III S W_{1} : OFF, S W_{2} : ON \\ A_{4} x (t) + B_{w 4} w (t); Mode IV S W_{1} : ON, S W_{2} : ON \end{matrix}

(1)

where $x (t) = [i_{l 1} i_{l 2} v_{C_{o}}]^{T}$ is the state vector, $w (t) = [v_{in} i_{0}]^{T}$ is the disturbance vector, $A_{1, 2, 3, 4}$ are the state matrices and $B_{w 1, w 2, w 3, w 4}$ are the disturbance matrices each number of the converter operating mode. The state vector to output relation is defined as $y (t) = C_{j} [v_{o} i_{l 1} i_{l 2}]^{T}$ , where the index $j$ represents each number of the converter operating mode. The corresponding matrices are given as

\begin{matrix} A_{1} = [\begin{matrix} - r_{l} / L_{1} & 0 & 0 \\ 0 & - r_{l} / L_{2} & - 1 / L_{2} \\ 0 & 1 / C_{o} & - 1 / R C_{o} \end{matrix}] \\ A_{2} = [\begin{matrix} - r_{l} / L_{1} & 0 & 0 \\ 0 & - r_{l} / L_{2} & 0 \\ 0 & 0 & - 1 / R C_{o} \end{matrix}] \\ A_{3} = [\begin{matrix} - r_{l} / L_{1} & 0 & - 1 / L_{1} \\ 0 & - r_{l} / L_{2} & 0 \\ 1 / C_{o} & 0 & - 1 / R C_{o} \end{matrix}] \\ A_{4} = [\begin{matrix} - r_{l} / L_{1} & 0 & 0 \\ 0 & - r_{l} / L_{2} & 0 \\ 0 & 0 & - 1 / R C_{o} \end{matrix}] \\ B_{w 1, w 2, w 3, w 4} = [\begin{matrix} 1 / L_{1} 0; & 1 / L_{2} 0; & 0 - 1 / C_{o} \end{matrix}] \\ C_{1, 2, 3, 4} = [\begin{matrix} 0 & 0 & 1; 1 & 0 & 0; 0 & 1 & 0 \end{matrix}] . \end{matrix}

(2)

Figure 1.

The equivalent circuit diagram of the closed-loop two-phase IBC.

The duty cycle of each operation mode of the circuit is $d_{j}, j = 1, 2, 3, 4$ . It can be used to describe the allowed switch states of each mode because the switches are either closed or opened, i.e. $ON / OFF$ states. When (1) is averaged with respect to $d_{j}$ , the averaged signal model of the converter is obtained as

\begin{matrix} \bar{x}' (t) = [A_{1} d_{1} + A_{2} d_{2} + A_{3} d_{3} + A_{4} d_{4}] \bar{x} (t) + B_{wa} \bar{w} (t) \\ \bar{y} (t) = [C_{1} d_{1} + C_{2} d_{2} + C_{3} d_{3} + C_{4} d_{4}] \bar{x} (t) \\ B_{wa} = [B_{w 1} d_{1} + B_{w 2} d_{2} + B_{w 3} d_{3} + B_{w 4} d_{4}] \end{matrix}

(3)

where $\bar{x} (t)$ , $\bar{y} (t)$ , and $\bar{w} (t)$ are average values of the state, output, and disturbance vectors, respectively. The model can be linearized at a given operating point by expanding (3) into a Taylor series about the operating point. Therefore, the small-signal model of IBC is obtained by assuming small-signal perturbations that allow us to consider only first-order conditions. The state variables are composed of DC steady-state magnitudes and AC low-frequency deviations as

\begin{matrix} \bar{y} (t) = Y_{dc} + \tilde{y}, \bar{x} (t) = X_{dc} + \tilde{x}, \bar{w} (t) = W_{dc} + \tilde{w}, d_{1} = d_{3} \\ d_{2} = d_{4}, d_{1} = D_{1} + \tilde{d}, d_{2} = D_{2} - \tilde{d}, D_{1} = 1 - D, D_{2} = D - 0.5 \end{matrix}

(4)

where the DC magnitudes and AC deviations are denoted by capital and small italic letters with a tilde (~), respectively (Rashid 2017). Substituting (4) into (3), neglecting second-order terms ( $\tilde{x} \cdot \tilde{d}, \tilde{w} \cdot \tilde{d} = 0$ ) and taking Laplace transform, the linearized models are obtained. It is assumed that the inductance current per phase is equal. The small-signal model which represents the transfer function from control input to output is obtained as

\begin{matrix} \frac{\tilde{v_{0}} (s)}{\tilde{d} (s)} = C_{av} (sI - A_{av})^{- 1} T + (C_{1} - C_{2} + C_{3} - C_{4}) X_{dc} \\ C_{av} = (C_{1} D_{1} + C_{2} D_{2} + C_{3} D_{3} + C_{4} D_{4}) \\ A_{av} = (A_{1} D_{1} + A_{2} D_{2} + A_{3} D_{3} + A_{4} D_{4}) \\ T = (A_{1} - A_{2} + A_{3} - A_{4}) X_{dc} + (B_{w 1} - B_{w 2} + B_{w 3} - B_{w 4}) W_{dc} \end{matrix}

(5)

by neglecting the AC deviations of the disturbances. Eventually, one obtains the dynamic relationship between the duty cycle and output voltage as

G_{nom} (s) = \frac{\tilde{v_{0}} (s)}{\tilde{d} (s)} = G_{dc} \frac{1 - s / w_{z}}{φ (s)}

(6)

where

G_{dc} = \frac{V_{0}}{1 - D} \frac{(n_{f} R {(1 - D)}^{2} - r_{l})}{(r_{l} + n_{f} R {(1 - D)}^{2})}

(7)

w_{z} = \frac{(n_{f} R {(1 - D)}^{2} - r_{l})}{L}

(8)

φ (s) = \frac{s^{2}}{w_{0}^{2}} + \frac{s}{Q w_{0}} + 1

(9)

w_{0} = \sqrt{\frac{(r_{l} + n_{f} {(1 - D)}^{2} R)}{RL C_{o}}}

(10)

Q = \frac{\sqrt{RL C_{o} (r_{l} + n_{f} {(1 - D)}^{2} R)}}{L + C_{o} R r_{l}}

(11)

D = \frac{V_{0} - V_{IN}}{V_{0}}

(12)

in which $w_{0}$ is the angular corner frequency, $Q^{- 1} / 2$ is the damped ratio and $n_{f}$ is the number of phases. The second-order system in (6) is an NMP system owing to the existence of a right half-plane zero (RHPZ). The dependency of the RHPZ position on the operating point causes complicated converter dynamics and stability issues. The transfer functions from control input to inductance current $(G_{id})$ , input voltage to output voltage $(G d_{in})$ and load current to the output voltage $(G d_{r})$ can be obtained as

\begin{matrix} G_{id} (s) = \frac{\tilde{i_{l}} (s)}{\tilde{d} (s)} = G_{dc} \frac{V_{0} (2 + R C_{o} s)}{(r_{l} + n_{f} R {(1 - D)}^{2}) φ (s)} \\ G d_{in} (s) = \frac{{\tilde{v}}_{0} (s)}{{\tilde{v}}_{in} (s)} = \frac{\frac{1}{L C_{o}}}{s^{2} + (\frac{r_{l}}{L} + \frac{1}{R C_{o}}) s + \frac{r_{l}}{RL C_{o}} + \frac{D^{2} + {(1 - D)}^{2}}{L C_{o}}} \\ G d_{r} (s) = \frac{{\tilde{v}}_{0} (s)}{{\tilde{i}}_{0} (s)} = \frac{- \frac{s + \frac{r_{l}}{L}}{C_{o}}}{s^{2} + (\frac{r_{l}}{L} + \frac{1}{R C_{o}}) s + \frac{r_{l}}{RL C_{o}} + \frac{D^{2} + {(1 - D)}^{2}}{L C_{o}}} \end{matrix}

(13)

by neglecting the other AC deviations.

Structured and unstructured robust control problem

This section includes the unstructured and structured robust $H_{\infty}$ synthesis problems, where the objective function is the infinite norm of transfer functions from external inputs to performance outputs. The 1-DOF control configuration has inherent performance limitations because there exists a trade-off between output disturbance rejection and transient response. To overcome such limitations, a 2-DOF control configuration, consisting of the feedforward and the feedback controller, can be used as given in Figure 2.

Figure 2.

2-DOF control configuration.

The 2-DOF control configuration is preferred because the 1-DOF may not ensure both the reference tracking and the disturbance rejection requirements optimally, in the case of RHPZ in the converter transfer function. In Figure 2, the $K_{ff}$ is the feedforward controller which has the first-order transfer function, $K_{fb}$ is the feedback controller, $r$ is the reference signal, $e$ is the error signal, $u$ is the control input, $y_{m}$ is the measured output, $n$ is the sensor noise and $y_{n}$ is the feedback signal. A generalized control system architecture with the unstructured output multiplicative uncertainty is presented in Figure 3, where $W_{r}$ , $W_{n}$ , $W_{1}$ , $W_{2}$ , $W_{3}$ and $W_{m}$ represent the reference, noise, error, control effort, measured output and unstructured multiplicative weighting functions, respectively. In addition, $Δ$ , $G_{nom}$ and $r_{o}$ represent the perturbation block, nominal transfer function and reference output signal of the plant which is necessary only in 2-DOF configuration, respectively. The robust $H_{\infty}$ controller synthesis processes are performed by using the configuration given in Figure 3. The augmented plant ( $P$ ) represents the open-loop transfer function from inputs $[u_{Δ} r n u]^{T}$ to outputs $[y_{Δ} z_{1} z_{2} z_{3} r_{o} e]^{T}$ , where $z_{1}$ , $z_{2}$ and $z_{3}$ are performance outputs. The controllers are synthesized using only the performance channels by neglecting the perturbation channel. The augmented plant can be partitioned as

\begin{matrix} [\begin{matrix} y_{Δ} \\ z \\ r_{o} \\ e \end{matrix}] = [\begin{matrix} P_{11}^{(11)} & P_{11}^{(12)} & P_{12}^{(11)} \\ P_{11}^{(21)} & P_{11}^{(22)} & P_{12}^{(21)} \\ P_{21}^{(11)} & P_{21}^{(12)} & P_{22}^{(11)} \\ P_{21}^{(21)} & P_{21}^{(22)} & P_{22}^{(21)} \end{matrix}] [\begin{matrix} u_{Δ} \\ w \\ u \end{matrix}] \\ P = [\begin{matrix} 0 & 0 & 0 & W_{m} G_{nom} \\ - W_{1} & W_{1} W_{r} & - W_{1} W_{n} & - W_{1} G_{nom} \\ 0 & 0 & 0 & W_{2} \\ W_{3} & 0 & 0 & W_{3} G_{nom} \\ 0 & W_{r} & 0 & 0 \\ - 1 & W_{r} & - W_{n} & - G_{nom} \end{matrix}] \end{matrix}

(14)

Figure 3.

Closed-loop system configuration including output multiplicative uncertainty.

The uncertain P matrix including the perturbation channel is used to investigate the closed-loop systems. The closed-loop transfer function $T_{wz} (s)$ that represents the transfer functions from external inputs to the performance outputs of the system can be calculated using the lower linear fractional transform (LFT) as

\begin{matrix} ‖ T_{wz} (s) ‖_{\infty} = ‖ F_{l} (P (s), K (s)) ‖_{\infty} = \\ ‖ P_{11} + P_{12} K (I - P_{22} K)^{- 1} P_{21} ‖_{\infty} = \\ ‖ \begin{matrix} - J W_{m} / Z & W_{m} UH & - J W_{m} W_{n} / Z \\ W_{1} (J / Z - 1) & H W_{1} (1 - U) & W_{1} W_{n} (J / Z - 1) \\ - (K_{fb} W_{2}) / Z & W_{2} (K_{ff} + K_{fb}) H & - K_{fb} W_{2} W_{n} / Z \\ W_{3} (1 - J / Z) & W_{3} UH & - J W_{3} W_{n} / Z \end{matrix} ‖_{\infty}, \end{matrix}

(15)

where $Z = G_{nom} K_{fb} + 1$ , $U = G_{nom} (K_{fb} + K_{ff})$ , $J = G_{nom} K_{fb}$ , $H = W_{r} / Z$ and $K = [K_{ff} K_{fb}]$ . The 2-DOF Type-III controller is synthesized using the closed-loop transfer function given in (15) whereas the feedforward controller is ignored in the 1-DOF control configuration. Finally, the classical $H_{\infty}$ control problem becomes the problem of finding a sub-optimal admissible controller $K (s)$ which minimizes the effects of external inputs on performance outputs such that $γ < 1$ , where $γ$ is the maximum singular value of the performance channels in (15).

The unstructured 1-DOF feedback control framework

This sub-section considers the synthesis of standard 1-DOF unstructured $H_{\infty}$ controller via two Riccati equations. The problem is finding a full-order $K_{unc} = K_{fb}$ or in the state-space form ( $A_{K_{unc}}$ , $B_{K_{unc}}$ , $C_{K_{unc}}$ , $D_{K_{unc}}$ ) without the rank constraints. The unstructured or the full-order controller synthesis problem, which is solved generally by using two Riccati equations or LMI techniques, is given by

\begin{matrix} min_{Θ} γ \\ s . t . A^{T} X + XA + C_{1}^{T} C_{1} + X (γ^{- 2} B_{1} B_{1}^{T} - B_{2} B_{2}^{T}) X = 0 \\ Y A^{T} + AY + B_{1} B_{1}^{T} + Y (γ^{- 2} C_{1}^{T} C_{1} - C_{2}^{T} C_{2}) Y = 0 \\ ρ (XY) < γ^{2}, \end{matrix}

(16)

where $Θ \overset{Δ}{=} {γ, X, Y}$ . Then the unstructured controller $K_{unc}$ is given by

\begin{matrix} K_{unc} (s) = [\begin{matrix} A_{\infty} & - Z_{\infty} L_{\infty} & Z_{\infty} B_{2} \\ F_{\infty} & 0 & I \\ - C_{2} & I & 0 \end{matrix}] \\ F_{\infty} = - B_{2}^{T} X, L_{\infty} = - {YC}_{2}^{T}, Z_{\infty} = (I - γ^{- 2} YX)^{- 1} \\ A_{\infty} = A + γ^{- 2} B_{1} B_{1}^{T} X + B_{2} F_{\infty} + Z_{\infty} L_{\infty} C_{2}, \end{matrix}

(17)

where $X$ and $Y$ are the real positive matrices and $ρ$ is the magnitude of eigenvalues such that the order of this controller is equal to the order of the plant (Skogestad and Postlethwaite 2007).

Structured control framework

This sub-section considers that the controller has a predefined structure and parameters of the controller are tunable. The problem given in (16) becomes non-convex due to $n_{K}$ < $n_{P}$ constraint, where $n_{P}, n_{K}$ are the order of the plant and controller, respectively. As the problem can no longer be solved with the Riccati equations, another solution algorithm based on using the non-smooth optimization will be used.

The $N - Δ$ and $M - Δ$ structures are presented in Figure 4 to synthesize the controllers and analyse the robust stability of the systems, respectively. A Type-III controller denoted by $K_{ty}$ in steady-state or transfer function form is given by

\begin{matrix} K_{ty} (s) = \frac{a_{1} s^{2} + a_{2} s + a_{3}}{b_{1} s^{3} + b_{2} s^{2} + b_{3} s} = C_{K_{ty}} (sI - A_{K_{ty}})^{- 1} B_{K_{ty}} + D_{K_{ty}} \\ [\begin{matrix} A_{K_{ty}} & B_{K_{ty}} \\ C_{K_{ty}} & D_{K_{ty}} \end{matrix}] = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & - b_{3} / b_{1} & - b_{2} / b_{1} & 1 / b_{1} \\ a_{3} & a_{2} & a_{1} & 0 \end{matrix}], \end{matrix}

(18)

where $a_{1}$ , $a_{2}$ , $a_{3}$ , $b_{1}$ , $b_{2}$ and $b_{3}$ are the tunable parameters of the controller. The goal is to optimize the parameters that satisfying mentioned various performance specifications. A powerful way to ensure this goal is to formulate the objective function as $‖ T_{wz} (s) ‖_{\infty}$ in the robust $H_{\infty}$ control framework. Then, the optimization problem for 2-DOF control system design becomes

\begin{matrix} \underset{C_{K_{ff}}, D_{K_{ff}}}{\underset{D_{K_{ty}}, A_{K_{ff}}, B_{K_{ff}},}{\min_{γ, A_{K_{ty}}, B_{K_{ty}}, C_{K_{ty}},}}} γ \\ s . t . ‖ T_{wz} (s) ‖_{\infty} \leq γ \\ Re (λ_{i} (A_{cl} (κ))) \leq λ_{\max} < 0 \\ κ \in D, \end{matrix}

(19)

where $κ \in R^{n_{K}}$ is a vector of the decision variables procured by parameters of the controller, $D$ is the search space and $λ_{i}$ is the ith eigenvalue of the closed-loop transfer function. The $H_{\infty}$ Type-III and the feedforward controllers in 2-DOF configuration are obtained from the solution of the optimization problem (19) in the 2-DOF configuration (Toscano 2013). A 1-DOF Type-III controller $(K_{1 ty})$ can be synthesized by ignoring the feedforward controller. A first-order feedforward controller is given by

K_{ff} (s) = \frac{m_{1} s + m_{2}}{n_{1} s + n_{2}} = [\begin{matrix} A_{K_{ff}} & B_{K_{ff}} \\ C_{K_{ff}} & D_{K_{ff}} \end{matrix}] .

(20)

Figure 4.

General $N - Δ$ and $M - Δ$ structures.

This controller is used in the feedforward path to obtain smoother transitions in the case of reference changes.

Selection of the weighting functions

The closed-loop IBC system which has high bandwidth could easily reject the fast dynamical disturbances. However, in cases where the closed-loop system bandwidth of the converter exceeds the RHPZ frequency, the system generates a zero-output response which means that the converter becomes a short circuit (Skogestad and Postlethwaite 2007). Taking this information into consideration the performance weighting functions are selected under this constraint. Consider the first-order error weighting function, i.e.,

W_{1} (s) = \frac{s / M_{m} + w_{be}}{s + w_{be} A_{e}},

(21)

where $M_{m}$ , $w_{be}$ and $A_{e}$ are the modulus margin, i.e. the maximum infinity norm, bandwidth and DC value of the $W_{1}^{- 1} (s)$ function, respectively. The $W_{1} (s)$ shapes the sensitivity function $(S = (1 + G_{nom} K_{fb})^{- 1})$ of the system to get the upper bandwidth, minimum steady-state error and reasonable overshoot, particularly to reject the disturbances. The restriction on selection of the function is given by

w_{be} = w_{z} \frac{1 - 1 / M_{m}}{1 - A_{e}} .

(22)

Considering that $A_{e}$ , which represents the steady-state error, is approximately zero, the modulus margin and bandwidth of the system are severely restricted. In addition, the main constraint imposed by RHPZ is that the maximum bandwidth of the closed-loop system ( $w_{bc}$ ) could be half of RHPZ frequency ( $2 w_{bc}$ < $w_{z}$ ). As the $w_{be}$ is closely related to $w_{bc}$ , this constraint also affects the $w_{be}$ . The appropriate performance and noise weighting functions are chosen under these constraints as

\begin{matrix} W_{1} (s) = \frac{s + 500.1}{1.3 s + 0.00005}, W_{2} (s) = \frac{s + 4369}{0.04 s + 2621} \\ W_{3} (s) = \frac{s + 1285}{0.99 s + 2493}, W_{n} (s) = \frac{s + 1481}{2 s + 148120} . \end{matrix}

(23)

Uncertainty modelling

This sub-section covers the small-signal stability of closed-loop IBC system by using the $μ$ -analysis method, which is recently popular analysis method in the power electronic area, to determine the reliable operation of the uncertain system (Sumsurooah et al. 2017; Rosso et al. 2018). The robust stability of the system is investigated by using the structure in Figure 3. The uncertainty bounds are obtained by variating the parameters of the IBC circuit in a certain range which is given in Table 1.

Table 1.

The IBC parameters.

Parameter	Nominal Value	Uncertainty range
$R$	50 ∈ $[15, 85]$ $Ω$	$\pm 60 %$
$C_{o}$	1 mF	$\pm 20 %$
$L_{1} = L_{2}$	5 mH	$\pm 10 %$
$V_{in}$	46 V	$\pm 25 %$
$V_{0}$	100 V	−
$f$	10 kHz	−

The parametric uncertain model of the IBC is transformed into an unstructured uncertain model considering the output multiplicative uncertainty as given in Figure 3. The frequency response characteristics of the possible models are shown in Figure 5(b), where $W_{m}$ represents the upper bound of the magnitude response of relative error, i.e. multiplicative uncertainty weighting function. Note that the ESRs of the inductors and capacitor are included in the uncertain model definition algorithm, where the ESRs of the inductance and capacitor are assumed as 1 and 0.1 Ω, respectively. To determine the accurate uncertainty weighting transfer function, four weighting functions that have different orders are constructed. Then, the second-order function is preferred to represent the worst multiplicative uncertainty model of the IBC.

Figure 5.

Frequency response characteristics of the uncertain weighting functions, nominal transfer function and uncertainty model of the converter: (a) the nominal and uncertainty models; (b) the uncertainty weighting functions and magnitude response of the relative error.

Simulation results

This section covers the performance comparison and robust stability analysis of the closed-loop systems. Comprehensive simulation studies are performed to investigate the performance of the controllers under possible disturbance conditions. The combination of the PEMFC stack, the parameters of which are presented in Table 2, and the closed-loop IBC circuits are simulated in the MATLAB/Simulink environment. For comparison purposes, a dual-loop PI controller is synthesized using the classical frequency response method. Considering the maximum bandwidth of the voltage/outer loop is nearly half of the RHPZ frequency which equals 120 Hz, the bandwidths of the current/inner loops are set around 480 Hz. The parameters of inner-loop PI controllers ( $k i_{PI}$ ) and outer-loop PI controller ( $k v_{PI}$ ) are obtained as $k p_{i} = 0.0424$ , $k i_{i} = 44.17$ and $k p_{v} = 0.3391$ , $k i_{v} = 39.95$ using the frequency response method. The gain of current ( $G_{id} \cdot k i_{PI}$ ) and voltage ( $k v_{PI} \cdot k i_{PI} \cdot G_{nom} / (1 + k i_{PI} \cdot G_{id})$ ) loops of the dual-loop control system configuration are presented in Figure 6.

Table 2.

Parameters of the PEMFC stack.

Parameter	Symbol	Nominal value
Nominal voltage	$V_{n}$	46 V
Value of the voltage at in 1 A	$V_{1}$	44 V
Nominal current	$I_{n}$	34 A
The resistance of the membrane	$R_{m}$	0.07882 $Ω$
Maximum power	$P_{\max}$	8092 W
The pressure of the hydrogen partial H₂	$P_{H_{2}}$	0.3 bar
Temperature	$T$	60°C
The pressure of the oxygen partial O₂	$P_{O_{2}}$	2.61 bar
Nominal air flow rate	$Ai r_{Fr}$	300 lpm
Nominal fuel flow rate	$Fue l_{Fr}$	6.922 lpm

Figure 6.

The loop gains of current and voltage loops with dual-loop PI controller.

The structured (19) and unstructured (16) optimization problems are solved via $hinfstruct$ , which is a powerful solver, and $hinfsyn$ functions of robust control toolbox in MATLAB, respectively. The obtained parameters of the Type-III controllers are presented in Table 3. The unstructured controller

\begin{matrix} K_{unc} (s) = \\ \frac{1.4 e^{6} s^{5} + 2.3 e^{11} s^{4} + 9.3 e^{15} s^{3} + 2.4 e^{19} s^{2} + 5.7 e^{21} s + 1.9 e^{24}}{s^{6} + 8.2 e^{7} s^{5} + 6.5 e^{12} s^{4} + 7.8 e^{16} s^{3} + 4.8 e^{20} s^{2} + 8.4 e^{23} s^{1} + 3.2 e^{19}} \end{matrix}

(24)

is a sixth-order controller as expected because it is equal to the sum of the second-order IBC nominal transfer function and fourth-order weighting functions.

Table 3.

Parameters of the structured controllers.

Parameter	1-DOF configuration Type-III	2-DOF configuration
Parameter	1-DOF configuration Type-III	Type-III	$K_{ff}$
$a_{1}$	0.0007479	1428	$m_{1}$	0.02061
$a_{2}$	0.1642	7.289 × 10⁴	$m_{2}$	30.97
$a_{3}$	66.23	2.84 × 10⁷	$n_{1}$	1
$b_{1}$	4.692 × 10⁻⁷	1	$n_{2}$	7691
$b_{2}$	0.004726	1.001 × 10⁴
$b_{3}$	28.58	5.993 × 10⁷

Performance comparison results

To verify the performance of the proposed framework, input voltage and load current of the converter, whose nominal transfer functions are given in (13), are utilized for disturbance inputs of the IBC. The PEMFC voltage, output voltage and current of the converter are shown in Figure 7, where the voltage regulation is tested under four different disturbance cases. The load resistance is first increased from 20 to 85 Ω, then decreased to the previous value in Point 1. By changing the oxygen pressure of the PEMFC stack, the PEMFC voltage is decreased by nearly 8%, then increased to the previous value in Point 2. In Point 3, the load resistance is decreased to 15 Ω, hence the maximum current is transmitted to the output of the converter. Here, the ratio of percentage change of the load is kept high (70%) because the power electronic converters are exposed to the load disturbances continuously. Finally, to demonstrate the reference tracking performances in detail, first the reference signal is decreased to 80 V, then increased to its nominal value in Point 4 as can be shown in Figures 7 and 8 (d). The output voltage regulation is rapidly achieved after each one of the load current and input voltage variations. It is evident that the sixth-order $K_{unc}$ and proposed third-order $K_{1 ty}$ have identical performances such as they have 13% maximum overshoot and 41 ms settling time in Figure 7(a). The controllers have also nearly the same $γ$ which is 0.9827 and 0.9864, respectively. Furthermore, the 2-DOF Type-III controller $(K_{2 ty})$ has $γ =$ 0.984, 3.65% maximum overshoot, 41 ms settling time and no oscillation in the case of disturbances. The achieved bandwidths of the closed-loop system are 93.2, 93.3, and 93.43 Hz for $K_{unc}$ , $K_{1 ty}$ , and $K_{2 ty}$ , respectively. The dual-loop PI controller system has 64.78 Hz closed-loop bandwidth, 27% overshoot and 68 ms settling time. The time domain performance results of the controllers are given in Table 4. These results demonstrate that the Type-III controllers have good performance even though decreasing three orders of the controller as compared with the sixth-order controller. In each disturbance point, the $K_{1 ty}$ has equal performance along with the higher-order $K_{unc}$ . It is noted that the dual-loop configuration has better input voltage disturbance rejection performance because it controls the inductance current in the inner loop with a PI controller that has high bandwidth frequency.

Figure 7.

Disturbance rejection comparisons of the controlled systems in the time domain: (a) the output voltage regulation; (b) the input voltage variations; (c) the load current variations.

Figure 8.

The reference voltage tracking performance of IBC. (a) Point 1: eliminating the effect on step decreasing of the load current. (b) Point 2: the effect of the input voltage change on the output voltage. (c) Point 3: eliminating the effect on step increasing of the load current. (d) Point 4: the reference voltage tracking.

Table 4.

Performance results of the controllers (l = left, r = right).

Point	Max overshoot (%)			Settling time (ms)			Voltage ripple (mV)
Point	PI	$K_{1 ty}$	$K_{2 ty}$	PI	$K_{1 ty}$	$K_{2 ty}$	PI	$K_{1 ty}$	$K_{2 ty}$
Starting	27	13	3.65	68	41	41	6	9	9
Point 1 (l)	12.5	4.8	5.4	75	40	40	10	4	4
Point 1 (r)	10.5	4.9	5.4	52	44	44	4	4	4
Point 2 (l)	1.9	3.6	5.9	35	37	34	4	6	6
Point 2 (r)	5.3	3.3	1.8	44.5	44.5	44	4	10	10
Point 3 (l)	5.1	2.8	2.9	75	42	42	8	14	14
Point 3 (r)	5.6	2.9	3	50	41	41	6	8	8
Point 4 (l)	0.6	2	0	60	30	60	10	35	35
Point 4 (r)	0.6	2.2	0	55	41	30	6	8	8

The output voltage stability depends on the power level reflected in the open-loop voltage transfer function $(G_{nom})$ via $V_{IN}$ and $R$ values. The open-loop transfer function presented in Figure 5 (a) has under-damped behaviour for the nominal power conditions. The resonance peak values of the voltage and disturbance transfer functions $(G_{nom}, G d_{r}, G d_{in})$ increase in the case of decreasing the output power demand of the converter or vice versa. The amplitude of the output voltage oscillations and overshoots at the reference and disturbance points increase due to the under-damped behaviour of the transfer functions. This problem around resonant peak frequencies is solved by reducing the amplitude of the sensitivity function in the region. In this paper, the problems occurring in these different power demands are solved with the determination of the performance weighting functions especially the error weighting function including the shape $S$ and $K_{fb} \cdot S$ functions. The disturbance rejection properties of the robust controllers for the small disturbances $({\tilde{i}}_{0}, \tilde{v_{in}})$ in the frequency domain are shown in Figure 9. Figure 10 illustrates the steady-state inductor currents and half of the IBC input/PEMFC current. The inductor currents are precisely phase shifted by 180º with equal amplitude and frequency. As a result, the PEMFC current ripple is minimized in order to expand the PEMFC life span.

Figure 9.

Disturbance rejection performance comparisons in the frequency domain for small disturbances on ${\tilde{i}}_{0}$ and $\tilde{v_{in}}$ using $S \cdot G d_{r}$ and $S \cdot G d_{in}$ .

Figure 10.

Inductor currents of each phase (blue and red) and half of the PEMFC current (black).

Robust stability analysis

The $μ$ -analysis tool could be used to determine the upper and lower bounds of structure singular values ( $μ$ ) to guarantee robust stability of the system when parameters of the analysed system are not with accurate values. It is assumed that the parameters of the converter lie within certain ranges as given in Table 1. The sixth-order unstructured controller is reduced to lower-order controllers by using the Hankel singular-value-based model reduction method. Structured singular values of the closed-loop systems are presented in Figure 11 in a sufficiently wide frequency range. The $K_{1 ty}$ , $K_{2 ty}$ and sixth-order $K_{unc}$ guarantee that the system is robustly stable because the maximum $μ$ values are below one. In addition, the fourth- and fifth-order unstructured controllers ensure robust stability of the system whereas the third-order unstructured controller becomes unstable in the uncertainty range. Although the $K_{1 ty}$ has a third-order transfer function, it has identical robust stability with the sixth-order unstructured controller. As can be shown from Figure 11, the proposed $K_{2 ty}$ has better robust stability than the others.

Figure 11.

Structured singular values of the closed-loop systems.

Conclusion and discussion

The IBC is an NMP system because it has an unstable or RHPZ dynamics when operated in continuous conduction mode. The controller synthesis process of this type of system should be conducted carefully because the RHPZ causes additional constraints over the cross-over frequency of the closed-loop system and converter sensitivity function. In this paper, the output voltage stability is addressed in the case of the load and input voltage disturbances. A robust design framework for a Type-III controller synthesis problem does not exist whereas the PI/PID controller synthesis methods are widely available in the literature. The robust $H_{\infty}$ controller design problem has been formulated for the Type-III controller and solved effectively by using the robust control toolbox of MATLAB. The input and output disturbance signals have been considered in the synthesis process of the proposed framework to improve the overall closed-loop stability and performance in comparison with the classic design methods. In addition, a first-order lead-type controller has also been synthesized with the Type-III controller in the 2-DOF configuration, simultaneously. The small-signal stability analysis of synthesized controllers has been performed by using the $μ$ -analysis method for the uncertain system. Finally, the performance of the proposed framework is compared with the robust full-order and dual-loop PI controllers. Here, the weighting functions given in (23) are unchanged during the synthesis of these robust controllers for a fair comparison of the performances. The performance and stability of the obtained 1-DOF Type-III controller are the same as the full-order controller such that they have 13% maximum overshoot, 40 ms settling time in the transient response. In addition, it has noticeably better stability than other reduced order unstructured controllers. The 2-DOF controller, which has 3.65% maximum overshoot, 40 ms settling time and no oscillation, has better robust stability and performance outputs. Obtained results demonstrate that the proposed robust structured controller synthesis framework is useful for optimizing the Type-III controllers to ensure robustness and performance specifications in the power electronics area.

Future work

A funding appeal will be made to realize the experimental application of the closed-loop systems. To increase the output voltage stability, the structured $H_{\infty}$ optimization problem can be extended to the combined design of the input voltage feedforward function and a feedback controller. Furthermore, a convex-optimization-technique-based algorithm can be adapted to the defined optimization problem.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Rıdvan Keskin

Ibrahim Aliskan

Ersin Daş

References

Ahmad

Ali

(2020) Unified disturbance estimation based control and equivalence with IMC, PID: Case study on a DC-DC boost converter. IEEE Transactions on Industrial Electronics. DOI:10.1109/TIE.2020.2987269.

Anzehaee

Behnam

Hajihosseini

(2018) Augmenting Armarkov-PFC predictive controller with PID-Type III to improve boost converter operation. Control Engineering Practice 79: 65–77.

Apkarian

Noll

(2006) Nonsmooth H∞ synthesis. IEEE Transactions on Automatic Control 51(1): 71–86.

Apkarian

Noll

(2018) Structured H∞-control of infinite-dimensional systems. International Journal of Robust and Nonlinear Control 28(9): 3212–3238.

Banerjee

Ghosh

Rana

(2016a) Design and fabrication of closed loop two-phase interleaved boost converter with Type-III controller. In: IECON 2016 - 42nd Annual Conference of the IEEE Industrial Electronics Society. IEEE, pp. 3331–3336.

Banerjee

Ghosh

Rana

(2016b) An improved interleaved boost converter with PSO-based optimal Type-III controller. IEEE Journal of Emerging and Selected Topics in Power Electronics 5(1): 323–337.

Bayat

Karimi

Taheri

(2019) Robust output regulation of zeta converter with load/input variations: LMI approach. Control Engineering Practice 84: 102–111.

Chakraborty

Hasan

Tran

Baghdadi

Hegazy

(2019) DC-DC converter topologies for electric vehicles, plug-in hybrid electric vehicles and fast charging stations: State of the art and future trends. Energies 12(8): 1569.

Chan

Lam

Zeng

Zheng

Sin

Wong

(2015) Generalized Type III controller design interface for DC-DC converters. In: TENCON 2015-2015 IEEE Region 10 Conference. IEEE, pp. 1–6.

10.

Chowdhury

Kimball

(2020) Robust control scheme for a three phase grid-tied inverter with LCL filter during sensor failures. IEEE Transactions on Industrial Electronics. doi:10.1109/TIE.2020.3013515.

11.

Daş

Başlamışlı

SÇ

(2019) Data-driven fixed-order H controller synthesis in frequency domain: Closed-loop system approach. Transactions of the Institute of Measurement and Control. DOI: 0142331219847741.

12.

Feyel

(2017) Robust Control Optimization with Metaheuristics. New York: John Wiley & Sons.

13.

Ghosh

Banerjee

Sarkar

Dutta

(2016) Design and implementation of Type-II and Type-III controller for DC–DC switched-mode boost converter by using K-factor approach and optimisation techniques. IET Power Electronics 9(5): 938–950.

14.

Habib

Khoucha

Harrag

(2017) GA-based robust LQR controller for interleaved boost DC–DC converter improving fuel cell voltage regulation. Electric Power Systems Research 152: 438–456.

15.

Han

Feng

Zhao

Jiang

Wang

(2020) Robust constrained predictive fault tolerant control with generalized input parameterization and event-triggered regulation: Design and experimental results. IEEE Transactions on Industrial Electronics.

16.

Keskin

Aliskan

(2018) Design of non-inverting buck-boost converter for electronic ballast compatible with LED drivers. Karaelmas Science and Engineering Journal 8(2): 473–481.

17.

Liu

Lam

Xie

Shu

(2020) Generalized lead–lag H∞ compensators for MIMO linear systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems. doi:10.1109/TSMC.2019.2960583

18.

Mehida

Ayad

Saadi

Kraa

Aboubou

(2018) Multi-stack fuel cells and interleaved DC/DC converters interactions for embedded applications. In: 2018 International Conference on Electrical Sciences and Technologies in Maghreb (CISTEM). IEEE, pp. 1–6.

19.

Mercader

Åström

Baños

Hägglund

(2016) Robust PID design based on QFT and convex–concave optimization. IEEE Transactions on Control Systems Technology 25(2): 441–452.

20.

Nicoletti

Martino

Aguglia

(2020) Data-driven approach to iterative learning control via convex optimisation. IET Control Theory and Applications 14(7): 972–981.

21.

Olalla

Leyva

El Aroudi

Queinnec

(2009) Robust LQR control for PWM converters: An LMI approach. IEEE Transactions on Industrial electronics 56(7): 2548–2558.

22.

Rahimi

Hosseini

Sabahi

Gharehpetian

Abapour

(2019) Reliability evaluation of a fault-tolerant three-phase interleaved DC-DC boost converter. Transactions of the Institute of Measurement and Control 41(5): 1278–1289.

23.

Rashid

(2017) Power Electronics Handbook. London: Butterworth-Heinemann.

24.

Řezáč

Hurák

(2013) Structured MIMO H∞ design for dual-stage inertial stabilization: Case study for HIFOO and hinfstruct solvers. Mechatronics 23(8): 1084–1093.

25.

Rosso

Cassoli

Buticchi

Engelken

Liserre

(2018) Robust stability analysis of LCL filter based synchronverter under different grid conditions. IEEE Transactions on Power Electronics 34(6): 5842–5853.

26.

Sadek

Sarjaš

Chowdhury

Svečko

(2016) FPGA-based optimal robust minimal-order controller structure of a DC–DC converter with Pareto front solution. Control Engineering Practice 55: 149–161. doi: 10.1109/TIE.2020.3013521.

27.

Skogestad

Postlethwaite

(2007) Multivariable Feedback Control: Analysis and Design, Vol. 2. New York: Wiley.

28.

Sumsurooah

Odavic

Bozhko

Boroyevich

(2017) Robust stability analysis of a DC/DC buck converter under multiple parametric uncertainties. IEEE Transactions on Power Electronics 33(6): 5426–5441.

29.

Thevenet

Noll

Apkarian

(2006) Non linear spectral SDP method for BMI-constrained problems: Applications to control design. In: Informatics in Control, Automation and Robotics I. Berlin: Springer, pp. 61–72.

30.

Toscano

(2013) Structured Controllers for Uncertain Systems. Berlin: Springer.

31.

Zhan

Guo

Zhu

Liang

Yang

(2019) Comprehensive influences measurement and analysis of power converter low frequency current ripple on PEM fuel cell. International Journal of Hydrogen Energy 44(59): 31352–31359.

32.

Zhu

Pang

Teo

(2019) Dual-axis control of flexure-based motion system for optical fibre transceiver assembly using fixed-order controller. ISA transactions 86: 266–275.