# Optimal Active Fault Detection in
Inverter-Based Grids

Mohammad Pirani, Mehdi Hosseinzadeh, , Joshua A. Taylor, and Bruno Sinopoli M.
Pirani and J. A. Taylor are with the Department of Electrical and Computer
Engineering, University of Toronto. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Hosseinzadeh is with the School of Mechanical and
Materials Engineering, Washington State University, Pullman, WA 99164, USA.
E-mail<EMAIL_ADDRESS>Sinopoli is with the Department of
Electrical and Systems Engineering, Washington University in St Louis, St.
Louis, MO 63130, USA. E-mail<EMAIL_ADDRESS>
###### Abstract

Ground faults in converter-based grids can be difficult to detect because,
unlike in grids with synchronous machines, they often do not result in large
currents. One recent strategy is for each converter to inject a perturbation
that makes faults easier to distinguish from normal operation. In this paper,
we construct optimal perturbation sequences for use with the Multiple Model
Kalman Filter. The perturbations maximize the difference between faulty and
fault-free operation while respecting limits on performance degradation.
Simulations show that the optimal input sequence increases the confidence of
fault detection while decreasing detection time. It is shown that there is a
tradeoff between detection and degradation of the control performance, and
that the method is robust to parameter variations.

###### Index Terms:

Fault detection, Inverter-based grids, Multiple model Kalman filter.

##  I Introduction

###  I-A Motivation

Faults are relatively easy to detect in conventional power grids because of
the large currents they draw from synchronous machines. In inverter-based
grids, e.g., microgrids, the converter controllers prevent fault currents from
rising substantially above nominal [1, 2]. As a result, traditional detection-
schemes might fail in inverter-based grids.

One promising approach is for each converter to inject a small perturbation
that makes faults easier to detect without degrading quality. This was done in
[3] to detect unintentional islanding, and more recently in [4] to detect
faults. We build on the latter by optimizing the perturbation sequence. This
maximizes the probability that a multiple model Kalman filter (MMKF) [5] will
be able to distinguish between faulty and fault-free operation.

###  I-B Literature Review

A fault in a power system is an unintentional electrical path to the ground or
between phases. The standard remedy is to de-energize them by opening
protection relays [6]. Detection of faults as fast as possible is vital to
protect the grid. Several passive, e.g., signal processing techniques [7, 8,
9], and active methods, e.g., zero and negative sequence power injection [10,
3], have been proposed in the literature.

The current controllers in inverters prevent faults from drawing large
currents in inverter-based grids, which makes faults harder to detect. To
improve detectability, [4] proposed injecting a harmonic signal to increase
the difference between normal and faulty operation. We build on this by
injecting an optimized signal instead of a harmonic, and by using an MMKF to
detect faults.

The MMKF determines which mode of operation a system is in. In this case, it
determines if the system is faulty or not. The MMKF is inspired from multiple-
model adaptive estimation [11, 12, 13]. More recently the MMKF was used to
detect faults and attacks [14]. We apply this to faults in inverter-based
grids.

###  I-C Contributions

In this paper, we use the MMKF to design an active fault detection scheme for
inverter-based grids. We evaluate the scheme in simulation, and find that the
optimal perturbation sequence dramatically enhances detection, and outperforms
harmonic-based schemes.

We remark that here we only consider three phase, balanced faults and a single
converter. Most real faults, however, are unbalanced, and most grids have
multiple converters and synchronous machines. Extending our approach to more
realistic settings is a topic of future work.

The structure of the paper is as follows. In Section II, model the system. In
Section III, we present the MMKF, show how to solve for the optimal input
sequence, and discuss their application to fault detection. Section IV has the
simulation results and Section V concludes the paper.

##  II Modeling

We consider a single inverter-based distributed generator (DG). We assume that
the system is balanced and only consider balanced, three phase-to-ground
faults. We model the system in the $dq$ frame.

###  II-A Inverter Modeling

The control objective is to keep the voltage of the DG at some nominal value.
This is commonly achieved with an inner controller that keeps the current in a
safe range, and an outer controller that tracks the voltage setpoint and sets
the reference for the inner controller [15].

Figure 1:  Inverter and control loops.

The system is shown in Fig. 1. $V_{i}$ is the inverter output voltage, $V_{\rm
DC}$ the DC voltage, and $I_{g}$ the grid-side line current. $R_{1},L_{1}$ and
$R_{2},L_{2}$ are converter and the transformer’s impedances, respectively.
$C$ and $I_{C}$ are the filter capacitance and its current. The voltage
controller drives $V_{\rm PCC}$ to a reference value, and sends a reference
signal to the current controller. The inverter current must track this
reference value. The fault current limiter (FCL) is a saturation block that
protect the converter from large currents [16].

###  II-B Voltage and Current Control Loops

Let $e_{V}(t)=V_{\rm PCC}^{\rm ref}-V_{\rm PCC}$. The voltage at the PCC is
managed by the PI controller

$u_{V}(t)=k_{p}^{V}e_{V}(t)+k_{i}^{V}\int_{0}^{t}e_{V}(\tau)d\tau,$ (1)

where $k_{p}^{V}$ and $k_{i}^{V}$ are gains. The voltage controller determines
the reference inverter current. The current controller is designed starting
from the inverter’s AC voltage [15], which in the $dq$ frame is given by

$V_{i}^{d}=\frac{1}{2}V_{\rm DC}u_{I}^{d},\quad V_{i}^{q}=\frac{1}{2}V_{\rm
DC}u_{I}^{q},$ (2)

where $u_{I}^{d}$ and $u_{I}^{q}$ are duty cycles. Denoting the capacitor
voltage by $V_{C}$, the dynamics of the inverter line current are

$V_{C}^{j}=V_{i}^{j}-R_{1}I_{i}^{j}-L_{1}\frac{dI_{i}^{j}}{dt},\quad\quad
j\in\\{a,b,c\\}.$ (3)

Transforming (3) into the $dq$ frame and using (2) gives

$\displaystyle V_{C}^{d}$ $\displaystyle=\frac{1}{2}V_{\rm
DC}u_{I}^{d}-R_{1}I_{i}^{d}+\omega L_{1}I_{i}^{q}-L_{1}\frac{dI_{i}^{d}}{dt},$
$\displaystyle V_{C}^{q}$ $\displaystyle=\frac{1}{2}V_{\rm
DC}u_{I}^{q}-R_{1}I_{i}^{q}-\omega L_{1}I_{i}^{d}-L_{1}\frac{dI_{i}^{q}}{dt}.$
(4)

To decouple (II-B), define the new control signals

$\displaystyle u_{I}^{d}=2\frac{\hat{u}_{I}^{d}-L_{1}\omega I_{i}^{q}}{V_{\rm
DC}},\quad u_{I}^{q}=2\frac{\hat{u}_{I}^{q}+L_{1}\omega I_{i}^{d}}{V_{\rm
DC}}.$ (5)

The dynamics are now decoupled and take the form

$\displaystyle V_{C}^{d}$
$\displaystyle=\hat{u}_{I}^{d}-R_{1}I_{i}^{d}-L_{1}\frac{dI_{i}^{d}}{dt},$
$\displaystyle V_{C}^{q}$
$\displaystyle=\hat{u}_{I}^{q}-R_{1}I_{i}^{q}-L_{1}\frac{dI_{i}^{q}}{dt}.$ (6)

The current controller is given by

$\hat{u}_{I}^{j}(t)=k_{p}^{I}e_{I}^{j}(t)+k_{i}^{I}\int_{0}^{t}e_{I}^{j}(\tau)d\tau,\quad\quad
j\in\\{d,q\\}$

where $e_{I}^{d}(t)=I_{i}^{\rm ref,d}-I_{i}^{d}$ and $e_{I}^{q}(t)=I_{i}^{\rm
ref,q}-I_{i}^{q}$.

###  II-C Fault Modeling

We focus on three phase symmetrical ground faults. The controller during
fault-free operation is shown in Fig. 2 (a).

Figure 2: (a) Fault-free mode, in which the FCL is not active. (b) Faulty mode
in which the FCL is active.

In this case, the FCL is not active. The control loop during a ground fault is
shown in Fig. 2 (b). In this case the current increases, and the upper limit
of the FCL, $\zeta_{U}$, becomes the reference for the current controller. The
capacitor $C$ is designed to compensate the current deficiency in the
converter side during faults.

###### Remark 1

The switching in the FCL is much faster than the dynamics of the controllers.
Due to this time scale separation, we do not account for the transient aspects
of saturation in the fault detection algorithm.

###  II-D State Space Representation

During fault-free operation, the system has the following model.

$\displaystyle\dot{x}$ $\displaystyle=I_{2}\otimes A_{h}x+I_{2}\otimes B_{h}u$
$\displaystyle y$ $\displaystyle=\begin{bmatrix}0&1&0&0&0&0\\\
0&0&0&0&1&0\end{bmatrix}x$ (7)

where $I_{2}$ is the $2\times 2$ identity matrix and $\otimes$ is the
Kronecker product and

$\displaystyle A_{h}$
$\displaystyle=\begin{bmatrix}0&-k_{i}^{I}-k_{i}^{I}k_{p}^{V}&k_{i}^{I}\\\
\frac{V_{\rm DC}}{2L_{1}}&-\left(\frac{k_{p}^{I}V_{\rm
DC}}{2L_{1}}+\frac{R_{1}}{L_{1}}+\frac{k_{p}^{I}k_{p}^{V}V_{\rm
DC}R}{2L_{1}}\right)&\frac{k_{p}^{I}V_{\rm DC}}{2L_{1}}\\\
0&-k_{i}^{V}R&0\end{bmatrix},$ $\displaystyle B_{h}$
$\displaystyle=\begin{bmatrix}k_{i}^{I}k_{p}^{V}&k_{i}^{I}\\\
\frac{k_{p}^{I}k_{i}^{I}V_{\rm DC}}{2L_{1}}&\frac{k_{p}^{I}V_{\rm
DC}}{2L_{1}}\\\ k_{i}^{V}&0\end{bmatrix}.$ (8)

Here, $x=[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}]^{T}$ where $x_{2}=I_{i}^{d}$,
$x_{5}=I_{i}^{q}$, and $x_{1},x_{3},x_{4},x_{6}$ are intermediate states in
the integral paths of PI controllers, i.e., $\dot{x}_{1}=k_{i}^{I}E_{1}^{d}$,
$\dot{x}_{3}=k_{i}^{I}E_{1}^{q}$, $\dot{x}_{4}=k_{i}^{V}E_{2}^{d}$,
$\dot{x}_{6}=k_{i}^{V}E_{2}^{q}$, where $E_{1}$ and $E_{2}$ are the errors
shown in Fig. 1 (a) and (b). The input is $u=\begin{bmatrix}V_{\rm PCC}^{\rm
ref,d}&\Delta u^{d}&V_{\rm PCC}^{\rm ref,q}&\Delta u^{q}\end{bmatrix}^{T}$
where $\Delta u$ is a perturbation (more precisely, $\Delta u$ is a variation
in the reference signal of the current controller) which we describe later.
The output is the inverter current, $I_{i}$, i.e., states $x_{2}$ and $x_{5}$.
In the above dynamics, we have assumed a resistive load with equivalent
resistance $R$.

During a fault, the model is given by

$\displaystyle\dot{x}$ $\displaystyle=I_{2}\otimes A_{f}x+I_{2}\otimes B_{f}u$
$\displaystyle y$ $\displaystyle=\begin{bmatrix}0&1&0&0\\\
0&0&0&1\end{bmatrix}x$ (9)

where

$\displaystyle A_{f}=\begin{bmatrix}0&-k_{i}^{I}\\\ \frac{V_{\rm
DC}}{2L_{1}}&-\left(\frac{k_{p}^{I}V_{\rm
DC}}{2L_{1}}+\frac{R_{1}}{L_{1}}\right)\end{bmatrix},B_{f}=\begin{bmatrix}k_{i}^{I}&k_{i}^{I}\\\
\frac{k_{p}^{I}V_{\rm DC}}{2L_{1}}&\frac{k_{p}^{I}V_{\rm
DC}}{2L_{1}}\end{bmatrix}.$ (10)

Here, $x=[x_{1},x_{2},x_{3},x_{4}]^{T}$ where $x_{2}=I_{i}^{d}$,
$x_{4}=I_{i}^{q}$, and $x_{1},x_{3}$ are intermediate states with
$\dot{x}_{1}=k_{i}^{I}E_{1}^{d}$ and $\dot{x}_{3}=k_{i}^{I}E_{1}^{q}$. The
input in this case is $u=\begin{bmatrix}\zeta_{U}^{d}&\Delta
u^{d}&\zeta_{U}^{q}&\Delta u^{q}\end{bmatrix}^{T}$ and, similar to the fault-
free case, the output is the inverter current. In the rest of the paper,
$\Delta u(t):=[\Delta u^{d}(t)~{}\Delta u^{q}(t)]^{\top}$.

##  III Multiple Model Kalman Filter

We first review the MMKF. Let ${h}$ and $f$ be the indices of fault-free and
faulty modes, respectively. We discretize and combine the fault-free and
faulty modes as

$\displaystyle x_{k+1|i}$ $\displaystyle=A_{i}x_{k|i}+B_{i}u_{k}+w_{k},$
$\displaystyle y_{k|i}$ $\displaystyle=C_{i}x_{k|i}+v_{k},$ (11)

where $i\in\\{h,f\\}$, and $x_{k|h}\in\mathbb{R}^{n_{h}}$ and
$x_{k|f}\in\mathbb{R}^{n_{f}}$ are respectively the state vectors of fault-
free and faulty systems at time $k$, with $n_{h}>0$ and $n_{f}>0$ as the
dimension of the state vectors. Here, $v_{k}$ and $w_{k}$ are measurement and
process noises with covariances $\Sigma_{v}$ and $\Sigma_{w}$, respectively.
Both modes start from the initial condition $x_{0|i}=x_{0}$ with covariance
$\Sigma_{0}$.

Figure 3: Multiple-Model Kalman Filter.

The MMKF is shown in Fig. 3. The procedure starts from an initial guess of the
probabilities of the system being faulty or fault-free, denoted by $p_{0|h}$
and $p_{0|f}$. Then, we estimate the output of the system for each mode and
update the probability based on the residual signals, i.e., the difference
between the measured output and the estimated output. More formally, using
measurements at each time step $k$, the Kalman filter for each mode
$i\in\\{h,f\\}$ is given by

$\displaystyle\hat{x}_{k+1|i}$
$\displaystyle=A_{i}\hat{x}_{k|i}+B_{i}u_{k}+H_{i}\left(y_{k}-C_{i}\hat{x}_{k|i}\right)$
$\displaystyle H_{i}$
$\displaystyle=\Sigma_{i}C_{i}^{T}\left[C_{i}\Sigma_{i}C_{i}^{T}+\Sigma_{v}\right]^{-1}$
(12)

where $\hat{x}_{k|h}\in\mathbb{R}^{n_{h}}$ and
$\hat{x}_{k|f}\in\mathbb{R}^{n_{f}}$ are the estimated states at time $k$,
respectively. $\Sigma_{i}$ is the posterior error covariance matrix in steady
state, which is the solution of the discrete Riccati equation,

$\displaystyle 0$ $\displaystyle=-\Sigma_{i}+A_{i}\Sigma_{i}A_{i}^{T}$
$\displaystyle+\Sigma_{w}-A_{i}^{T}\Sigma_{i}C_{i}^{T}[C_{i}\Sigma_{i}C_{i}^{T}+\Sigma_{v}]^{-1}C_{i}\Sigma_{i}A_{i}.$
(13)

After estimating the states of each mode, we use the measurement $y_{k}$ to
generate the residual functions $\alpha_{k|f}=\|y_{k}-\hat{y}_{k|f}\|$ and
$\alpha_{k|h}=\|y_{k}-\hat{y}_{k|h}\|$. These residuals are used to update the
posterior probability of each mode

$p_{k+1|i}=\frac{\beta_{i}e^{-\alpha_{k|i}}}{p_{k|f}\beta_{f}e^{-\alpha_{k|f}}+p_{k|h}\beta_{h}e^{-\alpha_{k|h}}}p_{k|i},$
(14)

where $\beta_{i}=\frac{1}{\sqrt{|\Sigma_{y|i}|}}$ and $\Sigma_{y|i}$ is the
covariance of the residual $y_{k}-\hat{y}_{k|i}$. The residual is updated as

$\displaystyle\Sigma_{y|i}=C_{i}\Sigma_{i}C_{i}^{T}+\Sigma_{v}$ (15)

with $\Sigma_{i}$ as in (13).

The detected mode is the one for which the corresponding posterior probability
is maximized, i.e.,

$p^{*}=\arg\max_{i\in\\{h,f\\}}p\left(i|y_{0:N},\Delta u_{0:N-1}\right)$ (16)

In the following example, we see that without triggering the system with some
additional perturbation, this method can fail to accurately detect faults.

###### Example 1

Suppose that system (11) is faulty. For a horizon of eight time steps, we
calculate the the posterior probabilities (14) for the faulty and fault-free
cases, starting from $p_{0|f}=p_{0|h}=0.5$. In a second instance, we add a
perturbation signal with a bounded norm, $\|\Delta u\|_{\infty}\leq 0.5$, to
help distinguish faulty from fault-free modes (the design of the input will be
discussed in the next section). Fig. 4 shows the evolution of $p_{k+1|i}$ over
time. The passive method, i.e., with no perturbation, is far less sensitive to
faults than the active method.

Figure 4: (Top) Posterior probabilities with and without a perturbation.
(Bottom) Perturbations over time in $d$ and $q$ frames.

###  III-A Designing the Input Sequence

As shown in Example 1, the passive MMKF can lack adequate sensitivity on its
own. We now design the perturbation, $\Delta u$, as shown in Fig. 2, which
makes the faulty and fault-free modes easily distinguishable from the system’s
output. More precisely, we consider the following problem.

###### Problem 1

Consider system (11) with $N>0$ as the detection horizon. Find an input
sequence $\Delta u_{k}$, $k=0,1,2,...,N-1$, such that (i) $\|\Delta
u_{k}\|_{\infty}\leq\gamma$, where $\|\cdot\|_{\infty}$ is the infinity norm
of a vector, and $\gamma$ is a design parameter, and (ii) controller
performance is not degraded, and (iii) the system’s true mode is almost surely
identified. The detection horizon $N$ and the bound on the input signal
$\gamma$ are chosen by the designer.

Note that the control input $u_{k}$ in (11) is augmented with the perturbation
$\Delta u_{k}$. We use the objective function

$J_{d}(\Delta u_{0:N-1})=\mathbb{E}\big{[}\sigma(\hat{\kappa})|\Delta
u_{0:N-1}\big{]},$ (17)

where $\Delta u_{0:N-1}=[\Delta u_{0}~{}\cdots~{}\Delta
u_{N-1}]\in\mathbb{R}^{N}$, $\hat{\kappa}$ is the identified mode of the
system, and $\sigma(\hat{\kappa})$ is zero when the identified mode is the
actual mode of the system, and is non-zero (e.g., equal to one) otherwise. In
order to find the optimal input sequence for (17), $J_{d}$ must be written in
terms of $\Delta u_{0:N-1}$. We will show that (17) can be upper bounded by an
explicit function of the input sequence $\Delta u_{0:N-1}$. To find that, we
first need to calculate the average measurement $\bar{y}_{0:N|i}$ and its
covariance matrix $\Sigma_{y|i}$ within the detection horizon. The mean output
$\bar{y}_{0:N|i}$ is

$\bar{y}_{0:N|i}=C_{i}\bar{x}_{0:N|i},$ (18)

where $\bar{x}_{0:N|i}=[\bar{x}_{0|i}^{T};\hskip
2.84526pt\bar{x}_{1|i}^{T};\hskip 2.84526pt...\hskip
2.84526pt;\bar{x}_{N|i}^{T}]^{T}$, and $\bar{x}_{k|i}$ is the mean value of
the state at time $k$,

$\bar{x}_{k|i}=A_{i}^{k}\bar{x}_{0|i}+A_{i}^{k-1}B_{i}u_{0}+...+B_{i}u_{k-1}.$
(19)

The covariance matrix of the state at times $k$ and $l$ ($k\geq l$) is

$\displaystyle\Sigma_{x,(k,l)|i}$
$\displaystyle=\mathbb{E}\left\\{(x_{k|i}-\bar{x}_{k|i})(x_{k|i}-\bar{x}_{k|i})^{T}\right\\}$
$\displaystyle=A_{i}^{k}\Sigma_{0}(A_{i}^{l})^{T}+A_{i}^{k-1}\Sigma_{w}(A_{i}^{l-1})^{T}$
$\displaystyle+A_{i}^{k-2}\Sigma_{w}(A_{i}^{l-2})^{T}+...+A_{i}^{k-l}\Sigma_{w},$
(20)

where $\Sigma_{0}$ and $\Sigma_{w}$ are covariance matrices of the initial
state vector and the process noise, respectively. Due to the symmetry of
covariance matrix, we have $\Sigma_{x,(k,l)|i}=\Sigma_{x,(l,k)|i}^{T}$. The
covariance matrix of the output is

$\displaystyle\Sigma_{y,(k,l)|i}=\begin{cases}C_{i}\Sigma_{x,(k-1,l-1)|i}C_{i}^{T}\quad\quad\quad\hskip
4.2679ptk>l\\\ C_{i}\Sigma_{x,(k-1,l-1)|i}C_{i}^{T}+\Sigma_{v}\quad
k=l\end{cases},$ (21)

where $\Sigma_{v}$ is the covariance matrix of the measurement noise.

######  Theorem 1 ([14])

The objective, (17), is bounded by

$J_{d}(\Delta
u_{0:N-1})\leq\underbrace{\sqrt{p_{0|h}p_{0|f}}e^{-\phi}}_{\triangleq\hat{J}_{d}(\Delta
u_{0:N-1})}$ (22)

where $p_{0|h}$ and $p_{0|f}$ and $\phi\geq 0$ are prior probabilities, and
$\phi\geq 0$ is:

$\displaystyle\phi$
$\displaystyle=\frac{1}{4}\left(\bar{{y}}_{0:N|h}-\bar{y}_{0:N|f}\right)^{T}\left(\Sigma_{y|h}+\Sigma_{y|f}\right)^{-1}$
$\displaystyle\left(\bar{y}_{0:N|h}-\bar{y}_{0:N|f}\right)+\frac{1}{2}\ln\bigg{(}\frac{\det\left(\frac{\Sigma_{y|h}+\Sigma_{y|f}}{2}\right)}{\sqrt{\det\Sigma_{y|h}\det\Sigma_{y|f}}}\bigg{)}.$
(23)

In order to limit the degradation of the control performance due to the
addition of the perturbation $\Delta u_{k}$, we add a constraint on the norm
of the input signal over the horizon. The final optimization takes the
following form

$\displaystyle\underset{\Delta u_{0:N-1}}{\text{min}}$
$\displaystyle\hat{J}_{d}(\Delta u_{0:N-1})$ (24) subject to
$\displaystyle\|\Delta u_{0:N-1}\|_{\infty}\leq\gamma.$

###### Remark 2

On the one hand, since $p_{0|h}$ and $p_{0|f}$ are constant, minimizing
$\hat{J}_{d}(\Delta u_{0:N-1})$ is equivalent to maximizing $\phi$. On the
other hand, as seen in (1), $\phi$ depends on the distance between expected
outputs of fault-free and faulty systems. Therefore, minimizing
$\hat{J}_{d}(\Delta u_{0:N-1})$ maximizes the distance between expected
outputs of the fault-free and faulty systems. This means that the perturbation
signal should be designed such that the distance between expected outputs of
the fault-free and faulty systems is maximized.

###### Remark 3

In order to avoid degrading the system’s performance in fault-free times, the
perturbation $\Delta u$ is applied only when the reference inverter current
$I_{i}^{\rm ref}$ approaches the upper limit of the FCL. Thus, the measure
$m=|I_{i}^{\rm ref}-\zeta_{U}|$ is used as a passive indicator of fault
detection. Note that relying on the difference between the reference current
$I_{i}^{\rm ref}$ and the upper limit $\zeta_{U}$ may lead to a false alarm,
as there are several scenarios in which $I_{i}^{\rm ref}$ can approach
$\zeta_{U}$ without a fault in the system. For instance, an abrupt change in
the load can lead to a large overshoot in the current reference generated by
the voltage controller, and consequently saturate the FLC.

##  IV Simulations

We now evaluate the performance of the MMKF algorithm in simulation. The
parameters used for the voltage and current controllers are $k_{p}^{I}=170$,
$k_{i}^{I}=100$ and $k_{p}^{V}=0.1$, $k_{i}^{V}=8$. The system parameters are
$R=10\Omega$, $R_{1}=1.5m\Omega$, $L_{1}=300mH$, and $V_{\rm DC}=150V$. These
physical parameters were taken from [16].

###  IV-A Detection-Performance Tradeoff

As shown in Example 1, perturbing the input improves fault detection. However,
large perturbations may degrade performance. The design parameter $\gamma$ in
(24) plays an important role in balancing this tradeoff. Figs. 5 and 6 show
the effect of the perturbation $\Delta u_{0:N-1}$ on the deviation of the
output voltage from its reference for the faulty and fault-free systems,
respectively. As seen in these figures, faster and more accurate fault
detection, quantified in terms of the separation of the posterior
probabilities, comes at the price of performance degradation. We also observe
that the separation of the posterior probabilities increases with time. More
specifically, in both faulty and fault-free systems, the actual mode of the
system can be detected within a single 60 Hz cycle.

Figure 5: The effect of the magnitude of the input signal $\Delta u$ on the
detection time and tracking error for the faulty system

.

Figure 6: The effect of the magnitude of the input signal $\Delta u$ on the
detection time and tracking error for the fault-free system.

###  IV-B Effect of perturbation on the Output Voltage

The influence of the perturbation on the output voltage can also be used as an
indicator of faults. In the fault-free case, the controller consists of two
cascaded PID blocks, while there is only one block in the faulty case. As a
result, the perturbation will be more attenuated during fault-free than faulty
operation. This is shown in Fig. 7, in which a perturbation $\Delta u$ with
$\|\Delta u_{0:N-1}\|_{\infty}\leq 1$ is applied to both the faulty and fault-
free systems. The impact on the output voltage in faulty mode is significantly
higher than in the fault-free case.

Figure 7: The deviation from reference voltage for faulty and fault-free
cases.

###  IV-C Comparison with Harmonic Injection

We now compare our approach to [4], in which the perturbation is constructed
from a few harmonics. We expect our perturbation to lead to better performance
because it is optimized over a larger set of signals. This is shown in Fig. 8.
The dashed lines correspond to the case where the input sequence $\Delta
u_{0:N-1}$ is restricted to contain $3^{\rm rd}$, $5^{\rm th}$, and $7^{\rm
th}$ harmonics of the fundamental frequency, 60 Hz. As expected, the
probability of fault detection converges faster to 1 when the perturbation is
not restricted to harmonics.

Figure 8: The posterior probabilities for MMKF without the harmonics
constraint (bold lines) and with the harmonic constraint (dashed lines).

###  IV-D Robustness to Parameter Uncertainty

An advantage of the MMKF is its robustness to model uncertainty. This is due
to the fact that uncertainties and disturbances are identically modeled in
both the faulty and fault-free modes, and hence have a similar effect on the
residual functions (and consequently on the posterior probabilities). This
feature is beneficial, because identifying the exact parameters, and
particularly the load $R$, is unrealistic in practice.

As an example, we changed the parameters of the inner current controller by
10%. As shown in Fig. 9, this causes no significant change in the posterior
probabilities.

Figure 9: The posterior probabilities for MMKF for nominal control parameters
(bold line) and when they are perturbed by $10\%$ (dashed line).

To analyze the robustness of the proposed method against parameter
uncertainty, we consider an identification error of $20\%$ in the load $R$
(i.e., $R\pm 0.2R$). Simulation results are shown in Fig. 10. Based on this
figure, the algorithm can detect faults despite the existence of large
uncertainties in the load.

Figure 10: (Top) The posterior probabilities for MMKF for a nominal load (bold
line) and when it is perturbed by $20\%$ (dashed line). (Bottom) perturbation
signal $\Delta u$ for nominal and perturbed load.

The perturbation inputs are shown in this figure. Based on this, when the load
is increased ($R+0.2R$), the magnitude of the perturbations decreases. This is
because when the load is larger, the difference between faulty (open loop,
$R=0$) and fault-free cases becomes more apparent and less detection effort is
needed.

###  IV-E Sensitivity Analysis—Impact of Detection Horizon $N$

The simulations are run on an Intel(R) Core(TM) i7-7500U CPU 2.70 GHz with
16.00 GB of RAM, and we use the YALMIP toolbox to solve the associated
optimization problems. Setting $N=8$ (which is the length of the detection
horizon in the results of the previous subsections), the mean computation time
for 1000 runs is 0.612 ms. This is within the time available for real-time
implementation, as the discretizing step is 1 ms.

Fig. 11 shows how the detection window size impacts the computation time,
where the mean computation time for $N=8$ (i.e., 0.612 ms) is used as a
nominal value for normalization. This figure reports the mean computation time
of 1000 experiments with random initial conditions for every $N$. Fig. 11
shows that as the length of the detection horizon increases, the optimization
problem becomes larger, and the computation time increases.

Figure 11: Impact of detection horizon $N$ on the computation time, where the
mean computation time for $N=8$ is used as normalizing coefficient.

###  IV-F Sensitivity Analysis—Impact of Controller Parameters

We follow the procedure in [17] to maximize the distance between the faulty
and fault-free system, which is assessed by means of the Vinnicombe gap
(a.k.a. $\nu$-gap metric) [18]. This procedure ensures that faulty and fault-
free systems are far apart, which implies that the posterior probabilities
converge fast. This is shown in Fig. 12, where
$\boldsymbol{K}_{c}=[k_{p}^{V}~{}k_{i}^{V}~{}k_{p}^{I}~{}k_{i}^{I}]^{\top}$.
As seen in this figure, even though we detect the actual mode of the system
for all sets of controller parameters, the posterior probabilities for
$\boldsymbol{K}_{c}$ converge faster than those for $\boldsymbol{K}_{c}\pm
0.2\boldsymbol{K}_{c}$.

Figure 12: Impact of PI controller parameters on detection performance.

##  V Summary and Conclusions

We have designed a new active scheme for detecting ground faults in inverter-
based grids. The scheme consists of a multiple model Kalman filter together
with an optimized perturbation, which enhances the sensitivity of the
detector. Simulations confirm that the detector is highly effective, that the
perturbation signal significantly enhances detection, and that it outperforms
harmonic injection-based techniques. Future work will focus on a broader range
of faulty cases, e.g., unbalanced systems and single phase-to-ground fault.

## References

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# On Generators of the Hardy and the Bergman Spaces

Valentin V. Andreev, Miron B. Bekker111Corresponding author, Joseph A. Cima

###### Abstract

A function $\varphi$ which is analytic and bounded in the unit disk
${\mathbb{D}}$ is called a generator for the Hardy space $H^{2}({\mathbb{D}})$
or the Bergman space $A^{2}({\mathbb{D}})$ if polynomials in $\varphi$ are
dense in the corresponding space. We characterize generators in terms of
$\varphi-$invariant subspaces which are also $z-$invariant and study wandering
properties of such subspaces. Density of bounded analytic functions in the
$\varphi-$invariant subspaces of $H^{2}({\mathbb{D}})$ is also investigated.

Keywords: Hardy space, Bergman space, generator, invariant subspace.

MSC2010: 46E20, 30H10, 30H20

##  1 Introduction

It is well known that any (closed) subspace of the Hardy space
$H^{2}({\mathbb{D}})$ or the Bergman space $A^{2}({\mathbb{D}})$ which is
invariant under multiplication by $z$ (the $z-$invariant subspace) is also an
invariant subspace of any analytic and bounded Toeplitz operator. The converse
is not in general true.

In this article we consider analytic Toeplitz operators $M_{\varphi}$ on the
Hardy space $H^{2}({\mathbb{D}})$ and the Bergman space $A^{2}({\mathbb{D}})$
with the property that the function which is a constant (e.g. $f(z)\equiv 1$)
is a cyclic vector for such operators. Such a function $\varphi$ we call a
generator. We characterize generators in terms of $M_{\varphi}-$ invariant
subspaces which are also $z-$invariant.

In [6] and [7] Donald Sarason introduced a notion of a weak-star generator of
$H^{\infty}$. We show that each weak-star generator is a generator in our
sense. If $\psi$ is a weak-star generator of $H^{\infty}$ then
$\psi-$invariant subspaces have some additional property, see Theorem 5 below.

Weak-star generators often occur in the process of studying wandering property
of invariant subspaces of analytic Toeplitz operators. In this article we also
consider this question for the case when the symbol of the Toeplitz operator
is a generator, see Theorem 2.

The condition that $H^{\infty}\cap\EuFrak{M}$ is dense in the invariant
subspace $\EuFrak{M}$ plays the crucial role in our work. In the last section
we study this condition in the case of invariant subspaces of the Hardy space
$H^{2}({\mathbb{D}})$.

##  2 Generators

In what follows ${\mathbb{D}}$ denotes the unit disk and $\EuFrak{H}$ denotes
either Hardy space $H^{2}({\mathbb{D}})$ or Bergman space
$A^{2}({\mathbb{D}})$.

######  Definition 1.

A function $\varphi\in H^{\infty}({\mathbb{D}})$ is called a generator for
$\EuFrak{H}$ if polynomials in $\varphi$ are dense in $\EuFrak{H}$, i.e.

$\overline{l.h.\\{\varphi^{n}:n=0,1,\ldots\\}}=\EuFrak{H}$ (1)

where l.h. means linear hull and bar means closure in the norm of
$\EuFrak{H}$.

Condition (1) implies that a vector ${\mathbf{1}}$ (the constant function $1$)
is a cyclic vector for the analytic Toeplitz operator $M_{\varphi}$ of
multiplication by $\varphi$.

Since in a locally convex space the weak closure of a subspace coincides with
its norm closure, a function $\varphi$ is a generator if polynomials in
$\varphi$ are weakly dense in $\EuFrak{H}$.

Clearly, $\varphi$ is univalent. Indeed, if $\varphi(z_{1})=\varphi(z_{2})$,
$z_{1},z_{2}\in{\mathbb{D}}$, $z_{1}\neq z_{2}$, then
$P(\varphi(z_{1}))=P(\varphi(z_{2}))$ for any polynomial $P$. Consequently,
$f(z_{1})=f(z_{2})$ for any
$f\in\overline{l.h.\\{\varphi^{n}:n=0,1,\ldots\\}}$ which contradicts (1).

A notion of a generator is closely related to the notion of weak-star
generator introduced by D. Sarason in [6], [7]. A function $\psi\in
H^{\infty}$ is called a weak-star generator of $H^{\infty}$ if polynomials in
$\psi$ are dense in the weak-star topology of $H^{\infty}$.

######  Proposition 1.

Every weak-star generator is a generator in the sense of Definition 1.

Let $\psi$ be a weak-star generator. Then for any $h_{0}\in
H^{\infty}({\mathbb{D}})$, for any $\epsilon>0$, and for any $f_{j}\in
H^{1}({\mathbb{D}})$, $j=1,2,\ldots,n$, there is a polynomial $P(\psi)$ in
$\psi$ such that

$|\frac{1}{2\pi}\int\limits_{0}^{2\pi}\left[P(\psi(e^{it}))-h_{0}(e^{it})\right]\overline{f_{j}(e^{it})}dt|<\epsilon,\quad
j=1.2.\ldots,n.$ (2)

Since $H^{2}({\mathbb{D}})\subset H^{1}({\mathbb{D}})$ we take $f_{j}\in
H^{2}({\mathbb{D}})$ and (2) gives

$|<P(\psi)-h_{0},f_{j}>_{H^{2}}|<\epsilon,$

that is the weak closure of polynomials in $\psi$ contains all $H^{\infty}$.
In particular that closure contains reproducing kernels
$K_{z}=(1-\zeta\bar{z})^{-1}$ for each $z\in{\mathbb{D}}.$ Therefore weak
closure of polynomials in $\psi$ coincides with $H^{2}({\mathbb{D}})$.
Consequently any weak-star generator of $H^{\infty}$ is a generator of
$H^{2}$. The same arguments show that any weak-star generator is a generator
of $A^{2}({\mathbb{D}})$. $\Box$

A (closed) subspace of $\EuFrak{H}$ is called $\varphi$-invariant if it is
invariant under the operator $M_{\varphi}$. We also denote by
$Lat_{\EuFrak{H}}(\varphi)$ the lattice of $\varphi$-invariant subspaces of
$\EuFrak{H}$.

For a set $S\subset\EuFrak{H}$ we denote $\left[S\right]_{\varphi}$ the
smallest $\varphi-$ invariant subspace containing $S$. Similarly,
$\left[S\right]_{z}$ is the smallest $z-$invariant subspace containing $S$.

The next theorem gives a characterization of a generator in terms of its
invariant subspaces.

######  Theorem 1.

Let $\varphi$ be a bounded univalent function in the unit disk ${\mathbb{D}}$.
In order for $\varphi$ to be a generator it is necessary that every
$\varphi-$invariant subspace $\EuFrak{M}\subset\EuFrak{H}$ such that
${\overline{\EuFrak{M}\cap H^{\infty}}}=\EuFrak{M}$ is also $z-$invariant, and
sufficient that the invariant subspace $[\mathbf{1}]_{\varphi}$ be
$z-$invariant.

Proof. Suppose $\varphi$ is a generator, $f\in\EuFrak{M}\cap H^{\infty}$ and
$h\in\EuFrak{M}^{\perp}$. Since $\EuFrak{M}$ is $\varphi-$invariant,
$P(\varphi)f\in\EuFrak{M}$ for any polynomial $P$, that is

$<P(\varphi)f,h>=0.$

The left side of the last equality can be written as
$<P(\varphi),\bar{f}h>=<P(\varphi),T_{\bar{f}}h>$, where $T_{\bar{f}}$ is a
Toeplitz operator on $\EuFrak{H}$. Since $f\in H^{\infty}$ the operator
$T_{\bar{f}}$ is bounded and $T_{\bar{f}}h\in\EuFrak{H}$. Thus we have

$0=<P(\varphi)f,h>=<P(\varphi),T_{\bar{f}}h>.$

Now pick $\epsilon>0$. Because $\varphi$ is a generator there is a polynomial
$P$ such that

$|<z-P(\varphi),T_{\bar{f}}h>|<\epsilon.$

Since $\epsilon>0$ is arbitrary it follows that

$0=<z,T_{\bar{f}}h>=<M_{z}f,h>.$

Hence $M_{z}f\in\EuFrak{M}$ for any $f\in\EuFrak{M}\cap H^{\infty}$ and,
because this intersection is dense in $\EuFrak{M}$ one concludes that
$M_{z}\EuFrak{M}\subset\EuFrak{M}$.

To prove the converse statement consider the $\varphi-$invariant subspace
$[\mathbf{1}]_{\varphi}$. Since this subspace is the closure in $\EuFrak{H}$
of polynomials in $\varphi$, functions from $H^{\infty}$ are dense in it.
Consequently, $[\mathbf{1}]_{\varphi}$ is also $z-$invariant. But any
$z-$invariant subspace of $\EuFrak{H}$ which contains the function
${\mathbf{1}}$ coincides with the whole $\EuFrak{H}$. Therefore
$[\mathbf{1}]_{\varphi}=\EuFrak{H}$, that is $\varphi$ is a generator. $\Box$

######  Remark 1.

It was proved earlier that a bounded univalent function $\psi$ is a weak-star
generator of $\EuFrak{H}$ if and only if
$Lat_{\EuFrak{H}}(\psi)=Lat_{\EuFrak{H}}(z)$. For the case
$\EuFrak{H}=H^{2}({\mathbb{D}})$ it was proved by D. Sarason in [6], for
$\EuFrak{H}=A^{2}({\mathbb{D}})$ the statement was proved by P. Bourdon in
[2].

######  Corollary 1.

Let $\varphi$ be a generator of $\EuFrak{H}$ and let $\EuFrak{M}$ be a
$\varphi-$invariant subspace. If ${\rm dim}\;\EuFrak{M}^{\perp}<\infty$ then
the subspace $\EuFrak{M}$ is $z$-invariant.

The corollary follows from the fact that $H^{\infty}$ is dense in $\EuFrak{H}$
and the statement below that was proved for the more general situation in [5],
Lemma 2.1:
Let a Banach space $X$ be decomposed as the direct sum of a subspace $Y$ and a
finite-dimensional subspace $Z$:

$X=Y\dot{+}Z,$

and $L$ is a dense linear subset of $X$. Then $Y\cap L$ is dense in $Y$.

The following statement was proved in [3], Lemma 4.1.

######  Lemma 1.

Let $\varphi$ be a bounded univalent function on the unit disk ${\mathbb{D}}$
with $\varphi(0)=0$. If $\EuFrak{M}$ is a $z-$invariant subspace of
$\EuFrak{H}$, then $\EuFrak{M}\ominus M_{z}\EuFrak{M}=\EuFrak{M}\ominus
M_{\varphi}\EuFrak{M}$.

######  Theorem 2.

Let $\varphi$ be a bounded univalent function on the unit disk ${\mathbb{D}}$
with $\varphi(0)=0$. Assume that $\varphi$ is a generator of $\EuFrak{H}$ and
$\EuFrak{M}$ is a $\varphi-$ invariant subspace of $\EuFrak{H}$ such that:
(a) $\EuFrak{M}\cap H^{\infty}({\mathbb{D}})$ is dense in $\EuFrak{M}$;
(b) $(\EuFrak{M}\ominus\varphi\EuFrak{M})\cap H^{\infty}({\mathbb{D}})$ is
dense in $\EuFrak{M}\ominus\varphi\EuFrak{M}$.
Then

$\EuFrak{M}=\left[\EuFrak{M}\ominus\varphi\EuFrak{M}\right]_{\varphi}.$ (3)

Proof. From Theorem 1 it follows that the subspace $\EuFrak{M}$ is
$M_{z}-$invariant and Lemma 1 gives that $\EuFrak{M}\ominus
M_{z}\EuFrak{M}=\EuFrak{M}\ominus M_{\varphi}\EuFrak{M}$. If
$\EuFrak{H}=H^{2}({\mathbb{D}})$ the fact that
$\EuFrak{M}=\left[\EuFrak{M}\ominus M_{z}\EuFrak{M}\right]_{z}$ is the well
known Wold decomposition. If $\EuFrak{H}=A^{2}({\mathbb{D}})$ then from the
result of Aleman, Richter, and Sundberg(see [1]) it follows that $\EuFrak{M}$
is the smallest $z-$invariant subspace that contains $\EuFrak{M}\ominus
M_{z}\EuFrak{M}$.

Observe that because $\varphi\in H^{\infty}({\mathbb{D}})$ we have
$\left[\EuFrak{M}\ominus M_{\varphi}\EuFrak{M}\right]_{\varphi}\cap
H^{\infty}({\mathbb{D}})$ is dense in $\left[\EuFrak{M}\ominus
M_{\varphi}\EuFrak{M}\right]_{\varphi}$. Since the last subspace is
$\varphi-$invariant we refer again to Theorem 1 and conclude that it is
$z-$invariant, and is a subspace of $\EuFrak{M}$. Therefore
$\left[\EuFrak{M}\ominus M_{\varphi}\EuFrak{M}\right]_{\varphi}=\EuFrak{M}$.
$\Box$

######  Remark 2.

If in Theorem 2 $\dim(\EuFrak{M}\ominus\varphi\EuFrak{M})<\infty$, then
condition (a) implies condition (b).

Let $\EuFrak{M}$ be a $\varphi-$invariant subspace of $\EuFrak{H}$. In what
follows the quantity $\dim(\EuFrak{M}\ominus\varphi\EuFrak{M})$ is called the
$\varphi-$index of $\EuFrak{M}$. The $z-$index of a $z-$invariant subspace is
defined similarly.

######  Theorem 3.

Let $\varphi$ be a generator of $\EuFrak{H}$ and $\varphi(0)=0$. Assume
$f\in(\EuFrak{H}\cap H^{\infty})$ and $\EuFrak{M}=[f]_{\varphi}$. Then the
$\varphi-$index of $\EuFrak{M}$ equals 1 and $\EuFrak{M}=[f]_{z}$.
Consequently, the $z-$index of $\EuFrak{M}$ is also 1.

Proof. The proof of the first statement of the theorem is, in fact, repetition
of the corresponding proof of Theorem 4, Chapter 8, of [4]. We include it for
completeness.

Represent $f$ in the form $f=f_{1}+f_{2}$, where
$f_{1}\in\EuFrak{M}\ominus\varphi\EuFrak{M}$ and $f_{2}\in\varphi\EuFrak{M}$.
Choose an arbitrary $g\in\EuFrak{M}\ominus\varphi\EuFrak{M}$. Then
$g\in[f]_{\varphi}$, consequently there is a sequence of polynomials of
$\varphi$, say $Q_{n}(\varphi)$, such that $\|Q_{n}(\varphi)f-g\|\to 0$.
Define $h_{n}^{(1)}$ and and $h_{n}^{(2)}$ by the formulas

$\displaystyle
h_{n}^{(1)}=Q_{n}(\varphi(0))f_{1}-g\in\EuFrak{M}\ominus\varphi\EuFrak{M},$
$\displaystyle
h_{n}^{(2)}=(Q_{n}(\varphi)-Q_{n}(\varphi(0)))f_{1}+Q_{n}(\varphi)f_{2}\in\varphi\EuFrak{M}.$

The second inclusion follows from the fact that
$Q_{n}(\varphi)-Q_{n}(\varphi(0))=(\varphi-\varphi(0))R_{n}(\varphi)$,
$R_{n}(\phi)$ is a polynomial of $\varphi$ and $\varphi(0)=0$. We have
$h_{n}^{(1)}+h_{n}^{(2)}=Q_{n}(\varphi)f-g$, therefore

$\|h_{n}^{(1)}\|^{2}+\|h_{n}^{(2)}\|^{2}=\|Q_{n}(\varphi)f-g\|^{2}\to 0.$

In particular, the property $\|h_{n}^{(1)}\|=\|Q_{n}(\varphi(0))f_{1}-g\|\to
0$ implies $g=\lambda f_{1}$, and therefore the $\varphi-$index of
$\EuFrak{M}$ is 1.

From our assumptions and Theorem 1 it follows that $\EuFrak{M}$ is
$z-$invariant. Put $\EuFrak{M}^{\prime}=[f]_{z}$. Then
$\EuFrak{M}^{\prime}\subset\EuFrak{M}$ and $\EuFrak{M}^{\prime}$ is
$z-$invariant. Hence $\EuFrak{M}^{\prime}$ is also a $\varphi-$invariant
subspace which contains $f$. But $\EuFrak{M}=[f]_{\varphi}$ is the smallest
$\varphi-$invariant subspace which contains $f$. Consequently,
$\EuFrak{M}\subset\EuFrak{M}^{\prime}$. Hence
$\EuFrak{M}=\EuFrak{M}^{\prime}$.

Finally, any single-generated $z$-invariant subspace has $z-$index 1. $\Box$

##  3 Density of $H^{\infty}$ in the Invariant Subspaces of
$H^{2}({\mathbb{D}})$

The previous considerations raise the following questions:
Let $\varphi$ be a generator of $\EuFrak{H}$ and $\EuFrak{M}$ is a
$\varphi-$invariant subspace of $\EuFrak{H}$ such that
dim$(\EuFrak{H}\ominus\EuFrak{M})=\infty$.
Is it true that $\EuFrak{M}\cap H^{\infty}\neq\\{0\\}$?
Is $\EuFrak{M}\cap H^{\infty}$ dense in $\EuFrak{M}$?
These question for the case $\EuFrak{H}=H^{2}({\mathbb{D}})$ are discussed in
this section. If $\EuFrak{H}=A^{2}({\mathbb{D}})$ the questions remain open.

Let $f\in H^{2}({\mathbb{D}})\setminus H^{\infty}({\mathbb{D}})$. Then a
function $f_{T}\in H^{\infty}$ is called $f-$truncating, if $f_{T}f\in
H^{\infty}$. There are many ways to construct an $f-$truncating function. One
of them is the following. Defined a real valued function $\omega(e^{it})$ as
follows:

$\omega(e^{it})=\begin{cases}1&\text{if}\;|f(e^{it})|\leq 1\\\
1/|f(e^{it})|&\text{if}\;|f(e^{it})|\geq 1.\end{cases}$

We have $|\omega(e^{it})|\leq 1$ and $\log{\omega}\in L^{1}({\mathbb{T}})$.
Therefore there is an outer function $f_{T}\in H^{\infty}\subset H^{2}$ such
that $|f_{T}(e^{it})|=\omega(e^{it})$. It is obvious that $ff_{T}\in
H^{\infty}({\mathbb{D}})$ and $\sup\\{|f(z)f_{T}(z)|:|z|<1\\}\leq 1$.

Assume now that $\varphi$ is a generator. Therefore, there is a sequence of
polynomials of $\varphi$, say $\\{P_{n}(\varphi)\\}$ such that
$\|P_{n}(\phi)-f_{T}\|_{2}\to 0$. Such a sequence $\\{P_{n}(\varphi)\\}$ we
shall call an $f-$truncating sequence. Elements of such sequence are functions
from $H^{\infty}({\mathbb{D}})$. Since

$\int\limits_{0}^{2\pi}|P_{n}(\varphi(e^{it}))-f_{T}(e^{it})|^{2}dt\to 0$

there is a subsequence $P_{n_{k}}(\varphi)$ such that
$P_{n_{k}}(\varphi(e^{it}))\to f_{T}(e^{it})$ for a.a. $t\in[0,2\pi]$. Without
loss of generality we may assume that $\\{P_{n}(\varphi)\\}$ converges to
$f_{T}$ almost everywhere on ${\mathbb{T}}$. In particular
$P_{n}(\varphi(e^{it}))f(e^{it})\to f_{T}(e^{it})f(e^{it})$ on the set of full
measure on ${\mathbb{T}}$.

######  Theorem 4.

Let $\varphi$ be a generator and let $\EuFrak{M}$ be a $\varphi-$invariant
subspace of the Hardy space $H^{2}({\mathbb{D}})$. Suppose $f\in\EuFrak{M}$ is
not in $H^{\infty}({\mathbb{D}})$ and an $f$-truncating sequence
$\\{P_{n}(\varphi)\\}$ is uniformly bounded in $H^{\infty}$-norm, that is
$\sup_{n}{\|P_{n}(\varphi)\|_{\infty}}\leq C<\infty$. Then $\EuFrak{M}\cap
H^{\infty}({\mathbb{D}})\neq\\{0\\}$.

Proof. Let $f_{T}\in H^{2}\cap H^{\infty}$ be the $f-$ truncating function
constructed in the previous paragraph. We need to show that
$P_{n}(\varphi)f\to f_{T}f$ in $H^{2}$-norm. Since
$P_{n}(\varphi)f\in\EuFrak{M}$ this will prove the statement. We have

$\displaystyle\int_{0}^{2\pi}|P_{n}(\varphi(e^{it}))f(e^{it})-f_{T}(e^{it})f(e^{it})|^{2}dt=$
$\displaystyle\int_{0}^{2\pi}|P_{n}(\varphi(e^{it}))-f_{T}(e^{it})|^{2}|f(e^{it})|^{2}dt.$

The expression under integral sign converges to zero almost everywhere and is
dominated by $C|f|^{2}$ where $C$ is a positive constant. Now Lebesgue’s
dominated convergence theorem gives the desired result. $\Box$

######  Theorem 5.

Let $\psi$ be a weak-star generator of $H^{\infty}$ and let $\EuFrak{M}$ be a
$\psi-$invariant subspace of $H^{2}({\mathbb{D}})$. Then $\EuFrak{M}\cap
H^{\infty}$ is dense in $\EuFrak{M}$ in $H^{2}$ norm.

Proof. Let the functions $f\in\EuFrak{M}$ and $f_{T}$ be as in the previous
theorem. For any $h\in H^{2}({\mathbb{D}})$ and any polynomial $P$ we have

$<P(\psi)f-f_{T}f,h>=\frac{1}{2\pi}\int_{0}^{2\pi}(P(\psi(e^{it}))-f_{T}(e^{it}))f(e^{it})\overline{h(e^{it})}dt.$

Now pick $\epsilon>0$. Since $f\bar{h}\in L^{1}({\mathbb{T}})$ and $\psi$ is a
weak-star generator of $H^{\infty}$, there is a polynomial $P$ in $\psi$ such
that the absolute value of the last integral is less than $\epsilon$. Since
$P(\psi)f\in\EuFrak{M}$ it means that $f_{T}f$ belongs to the weak closure of
$\EuFrak{M}$. Since weak closure of a subspace coincides with its norm
closure, it proves that $\EuFrak{M}\cap H^{\infty}\neq\\{0\\}$. Put
$\EuFrak{M}^{\prime}=c.l.h.\\{P(\psi)f_{T}f:P\text{ is a polynomial}\\}$. Then
the subspace $\EuFrak{M}^{\prime}$ has the following properties:

  1. 1.

$\EuFrak{M}^{\prime}\subset\EuFrak{M};$

  2. 2.

$\EuFrak{M}^{\prime}$ is $\psi-$invariant;

  3. 3.

$\EuFrak{M}^{\prime}\cap H^{\infty}$ is dense in $\EuFrak{M}^{\prime}$ in
$H^{2}$ norm.

Let $\Omega$ is the collection of all subspaces of $\EuFrak{M}$ which have
these three properties. We have shown that $\Omega\neq\emptyset$. Partially
order $\Omega$ by the set inclusion. By the Hausdorff’s maximal principle
there exists a maximal totally ordered subcollection $\Omega^{\prime}$ of
$\Omega$. Denote by $\EuFrak{M}_{0}$ the union of all $\EuFrak{M}^{\prime}$,
where $\EuFrak{M}^{\prime}$ is a member of $\Omega^{\prime}$. Then
$\EuFrak{M}_{0}$ is a maximal subspace of $\EuFrak{M}$ which satisfies three
conditions above.

We claim that $\EuFrak{M}_{0}=\EuFrak{M}$. Observe first that
$\EuFrak{M}\setminus\EuFrak{M}_{0}$ does not contain any $H^{\infty}$ vectors.
Suppose $\EuFrak{M}_{0}\neq\EuFrak{M}$ and pick
$h\in\EuFrak{M}\ominus\EuFrak{M}_{0}$. Then $h\notin H^{\infty}$. Let $h_{T}$
be the $h$-truncating function. Then $h_{T}h\in H^{\infty}$, hence
$h_{T}h\in\EuFrak{M}_{0}$ and $P(\psi)h_{T}h\in\EuFrak{M}_{0}$ for any
polynomial $P$. Consequently,

$\frac{1}{2\pi}\int\limits_{0}^{2\pi}P(\psi(e^{it}))h_{T}(e^{it})|h(e^{it})|^{2}dt=0$

for any polynomial $P$. Since $h_{T}|h|\in L^{1}({\mathbb{T}})$ and $\psi$ is
a weak-star generator of $H^{\infty}$ from the last equality one easily
deduces that

$\frac{1}{2\pi}\int\limits_{0}^{2\pi}e^{int}h_{T}(e^{it})|h(e^{it})|^{2}dt=0,\qquad
n=0,1,2,\ldots$

from which it follows that

$h_{T}|h|^{2}=k,$ (4)

where $k\in H^{1}_{0}$. Therefore

$|h|^{2}=\frac{k}{h_{T}}.$

Since left side is a real-valued function and right side is a function of the
Nevanlina class and $k(0)=0$ one concludes that that equality (3) is possible
only for $h=0$ and $k=0$. It proves our claim and the theorem. $\Box$

Combining the last theorem with Theorem 1 we obtain the following statement.

######  Corollary 2.

Let $\psi$ be a bounded univalent function in the unit disk ${\mathbb{D}}$.
Then $\psi$ is a weak-star generator of $H^{\infty}$ if and only if every
$\psi-$invariant subspace of $H^{2}({\mathbb{D}})$ is also $z-$invariant.

The statement of the Corollary was obtained by D. Sarason [6] using another
method.

## References

  * [1] A. Aleman, S. Richter, C. Sundberg, Beurling’s theorem for the Bergman space. Acta Math. 177 (1996), 275-310.
  * [2] P. Bourdon, Density of the polynomials in Bergman spaces. Pacific. J. Math, 130, no.2 (1987), 215-221.
  * [3] Brent J. Carswell, Univalent mappings and Invariant Subspaces of the Bergman and Hardy Spaces. Proc. of AMS 131, no.4, 1233-1241.
  * [4] P. Duren and A. Schuster, Bergman Spaces. American Mathematical Society, Providence, R.I., 2004.
  * [5] I. C. Gohberg and M. G.Krein, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspehi Mat. Nauk 12 (1957), no. 2 (74), 43-118; English transl. Amer. Math. Soc. Transl. (2) 13 (1960), 185-264.
  * [6] D. Sarason, Invariant subspaces and unstarred operator algebras. Pacific J. Math, 17, no. 3, (1966), 145-162.
  * [7] D. Sarason, Weak-star generators of $H^{\infty}$. Pacific J. Math, 17, no. 3, (1966), 519-528.

Valentin V. Andreev: Department of Mathematics, Lamar University,
Beaumont, TX 77710, USA<EMAIL_ADDRESS>
Miron B. Bekker: Department of Mathematics, The University of Pittsburgh at
Johnstown, 450 Schoolhouse Rd, Johnstown, PA 15904, USA<EMAIL_ADDRESS>
Joseph A. Cima: Department of Mathematics, the University of
North Carolina at Chapel Hill, CB 3250, 329 Phillips Hall, Chapel Hill, NC
27599, USA<EMAIL_ADDRESS>