From a2a1fa35042ea8c97a61d8b327d6b62e97bac557 Mon Sep 17 00:00:00 2001
From: PaoloPazzaglia <paolopazzaglia.pp@gmail.com>
Date: Fri, 26 Apr 2019 19:00:33 +0200
Subject: [PATCH] Fixed many typos and errors up to section 6
---
paper/figures/fig_simple_example.tex | 2 +-
paper/sec/analysis.tex | 19 ++++++++++---------
paper/sec/behavior.tex | 8 ++++----
paper/sec/design.tex | 6 +++---
paper/sec/intro.tex | 8 ++++----
paper/sec/method.tex | 6 +++---
paper/sec/model.tex | 14 ++++++++------
7 files changed, 33 insertions(+), 30 deletions(-)
diff --git a/paper/figures/fig_simple_example.tex b/paper/figures/fig_simple_example.tex
index e04d637..6bad29a 100644
--- a/paper/figures/fig_simple_example.tex
+++ b/paper/figures/fig_simple_example.tex
@@ -23,7 +23,7 @@ xlabel near ticks,
\nextgroupplot[
legend columns=4,
- legend style={draw=none,fill=none,at={(2.6,1.3)}},
+ legend style={draw=none,at={(2.6,1.3)}},
xlabel = {$T_d$ -- Kill},
]
\addplot[thick, purple, mark=pentagon*] table[x index=0, y index=1, col sep=comma] {\dataaa};
diff --git a/paper/sec/analysis.tex b/paper/sec/analysis.tex
index 6e13068..3e85df5 100755
--- a/paper/sec/analysis.tex
+++ b/paper/sec/analysis.tex
@@ -49,8 +49,8 @@ process $1-\beta$. \textcolor{red}{Scenario theory has for example found use in
\subsection{Scenario Theory}
\label{sec:analysis:scenario}
-The Scenario Theory has been developed in the field of robust
-control~\cite{calafiore2006scenario} to provide robustness guarantees
+The scenario theory has been developed in the field of robust
+control to provide robustness guarantees
for convex optimization problems in presence of probabilistic
uncertainty.
In these problems,
@@ -58,7 +58,7 @@ In these problems,
accounting
for all the possible uncertainty realization might be achieved
analytically, but is computationally too heavy or results in
-pessimistic bounds. The Scenario Theory proposes an empirical method
+pessimistic bounds. The scenario theory proposes an empirical method
in which samples are drawn from the possible realizations of
uncertainty, \textcolor{red}{finding a lower bound on the number of
samples}. It provides statistical
@@ -70,14 +70,14 @@ One of the advantages of this approach is that there is no need to
enumerate the uncertainty sources, the only requirement being the
possibility to draw representative samples. This eliminates the need
to make assumptions on the correlation between the probability of
-deadline miss in subsequent jobs. If interference is happening
+deadline misses in subsequent jobs. If interference is happening
between the jobs, this interference empirically appears when the
system behavior is sampled. While there is no requirement on
subsequent jobs interfering with one another, there is a requirement
that different sequences are independent (i.e., each sequence
represents an execution of the entire taskset of a given length, in
the same or possibly different conditions). Taking the worst observed
-case in a set of experiments, the Scenario Theory allows us to
+case in a set of experiments, the scenario theory allows us to
estimate the probability that something worse than what is observed
can happen during the execution of the system.
@@ -88,7 +88,8 @@ Denoting with $\mu_{\text{tot}}(\omega)$ the total number of job
skips and deadline misses that the control task experienced in
$\omega$, and with $\mu_{\text{seq}}(\omega)$ the
maximum number of consecutive deadline misses or
-skipped jobs in $\omega$, we use as a cost function
+skipped jobs in $\omega$, we chose to use as a cost function the following
+expression:
\begin{equation}\label{eq:Jseq}
J_{seq}(\omega) = \mu_{\text{tot}}(\omega)\,\mu_{\text{seq}}(\omega)
\end{equation}
@@ -110,7 +111,7 @@ deadline misses would be other viable choices.
\subsection{Formal Guarantees}
\label{sec:analysis:guarantees}
-The Scenario Theory allows us to compute the number $n_{\text{sim}}$
+The scenario theory allows us to compute the number $n_{\text{sim}}$
of simulations that we need to conduct to reach the required
robustness $\varepsilon$ and confidence $1-\beta$. The parameter
$\varepsilon$ is a bound on the probability of the obtained result
@@ -141,7 +142,7 @@ controller with high confidence.
\label{sec:analysis:application}
Similarly to any other empirical approach, the validity of the
-Scenario Theory depends on the representativeness of the sampling
+scenario theory depends on the representativeness of the sampling
set. In our case, for example the validity of our results depends on
the significance of the probabilistic execution time distributions
for all the tasks.
@@ -156,5 +157,5 @@ probabilistic nature of the execution times of the tasks. Some samplings
could in fact make the utilization of instances of the taskset greater
than one. For this reason} simulations that include several
hyperperiods should be performed. On top of that significancy with
-respect the controlled of the physical system is required \textcolor{red}{(since the existance of the hyperperiod is not always guaranteed)}, hence
+respect the controlled of the physical system is required \textcolor{red}{(since the existence of the hyperperiod is not always guaranteed)}, hence
the length of the simulated sequences should cover its dynamics.
diff --git a/paper/sec/behavior.tex b/paper/sec/behavior.tex
index 75bf6fb..38f1e88 100644
--- a/paper/sec/behavior.tex
+++ b/paper/sec/behavior.tex
@@ -27,7 +27,7 @@ theory, which is the periodicity of the output
pattern~\cite{pazzaglia2018beyond}. In this work, we exploit the
knowledge of deadline misses directly in the control design step.
For this purpose,
-we need to characterize the effect of deadline misses on the control
+we need to characterize how deadline misses affect the control
performance. We fully describe the effect of deadline
misses of LET-based controllers with two parameters, named
respectively \emph{delay} and \emph{hold} interval.
@@ -56,7 +56,7 @@ $\mathbf{x}(t_k)$ and the control output(s) active in that time span.
Given a control output computed by $J_{d,k}$ and available at the
actuator for the first time at $t_k + \sigma_k$, the \emph{hold
interval} $h_k$ is the time interval between $t_k + \sigma_k$ and
- the first instant where a new control output is written.
+ the first instant where a new control output is made available.
\end{definition}
In other words, the hold interval $h_k$ indicates the lifetime of the
@@ -239,7 +239,7 @@ the job has an overrun. The hold value $h_{k+1}$ is equal to the
delay of the next completed job $J_{d,k+3}$, i.e., $h_{k+1} =
\sigma_{k+3} = T_d$.
The values that $\lambda_{k,\text{Skip-Next}}$ may assume
-are upperbounded by the maximum delay $\bar{\sigma}$.
+are upperbounded by $\lceil R_d^W / T_d \rceil - 1 $.
\subsubsection{Hold Interval with Queue(1) Strategy}
@@ -264,4 +264,4 @@ and the $k+1$-th control jobs complete before during the $k+1$-th
period---then $\sigma_{k+1} - T_d = 0$, and the control signal
produced by $J_{d,k}$ is never actuated.
Finally, values of $\lambda_{k,\text{Queue(1)}}$ are upperbounded by
-$\bar{\sigma} - T_d$.
+$\lceil (R_d^W - T_d) / T_d \rceil - 1 $.
diff --git a/paper/sec/design.tex b/paper/sec/design.tex
index 45a42b3..e88b221 100755
--- a/paper/sec/design.tex
+++ b/paper/sec/design.tex
@@ -185,8 +185,8 @@ ranging between $1$ and $2$, we compare the resulting performance
under the Kill, Skip-Next, and Queue(1) strategies in
Figure~\ref{fig:onetask_results}. Since $J_\text{ctl}$ is defined as
a cost, lower values in the graph mean better performance. For each
-configuration, a standard controller (assuming no missed deadlines),
-a robust controller, and a clairvoyant controller is designed, and
+configuration, a standard controller (designed assuming no missed deadlines),
+a robust controller, and a clairvoyant controller are designed, and
the performance of each controller, measured in terms of the cost
function~\eqref{eq:cost}, is evaluated using
JitterTime~\cite{cervin2019jittertime} in a simulation of 100,000
@@ -196,7 +196,7 @@ clairvoyant control, as expected. This means that designing control
strategies that take into account deadline misses is beneficial in
all cases. The \our\ design does not achieve the optimal cost that
the clairvoyant design is able to achieve, but systematically beats
-classical control design, even when there are no deadline misses, due to its delay and hold compensation.
+classical control design due to its delay and hold compensation.
As the period is decreased from 2 to lower values, the Kill and
Queue(1) strategies initially behave similarly, with decreasing cost.
diff --git a/paper/sec/intro.tex b/paper/sec/intro.tex
index 6c7952d..2b7007c 100755
--- a/paper/sec/intro.tex
+++ b/paper/sec/intro.tex
@@ -18,8 +18,8 @@ structure. In general, adding a new control task to a given taskset
implies combining requirements that come from both control theory and
real-time implementation. These requirements are different and often
conflicting. As an example, selecting a high execution rate for the
-controller improves control performance, but at the same time limits
-the guarantees on the completion of the control task code and forces
+controller improves the control performance, but at the same time limits
+the guarantees on the timely completion of the control task code and forces
the engineers to take into account overruns~\cite{cervin2005analysis,
pazzaglia2018beyond}. Moreover, minimizing the monetary cost of the
final system is an ever-present priority and over-provisioning
@@ -27,7 +27,7 @@ resources is usually not a viable solution.
Timing constraints in real-time systems are modeled as
\emph{deadlines}, i.e., a threshold that the execution time of each
-task instance (\emph{job}) must respect. We refer to a job
+task instance (\emph{job}) should respect. We refer to a job
that successfully completes its execution before the corresponding
deadline as a \emph{deadline hit} event. If the job could not
terminate its execution before that deadline instant, we say that it
@@ -72,7 +72,7 @@ overcome this limitation, we obtain an estimate of deadline miss
occurrence simulating the schedule execution, drawing execution times
(for all the tasks) from the corresponding probability distributions.
A robust control tool, the
-\emph{scenario-theory}~\cite{calafiore2006scenario}, provides the
+\emph{scenario theory}~\cite{calafiore2006scenario}, provides the
means to select the worst-case sequence of misses and hits from the
simulations. Leveraging the scenario theory, our approach allows us
to provide probabilistic guarantees for worst-case conditions both in
diff --git a/paper/sec/method.tex b/paper/sec/method.tex
index efb970b..11e7537 100755
--- a/paper/sec/method.tex
+++ b/paper/sec/method.tex
@@ -4,8 +4,8 @@
The aim of \our{} is to provide the first control synthesis method
that is \emph{robust} both with respect to deadline misses and with
respect to the strategy used to handle them. Our control design
-leverages knowledge of the probability of occurrence of different
-sequences of deadline hits and misses and produces a fixed controller
+leverages knowledge of the probability that different
+sequences of deadline hits and misses may occur, and produces a fixed controller
that is (on average) optimal with respect to a defined cost function.
We obtain such knowledge by formulating a chance constrained
optimization problem in a probabilistic framework, and obtaining
@@ -53,7 +53,7 @@ behavior from simulations of a certain number of control jobs.
produced sequences to select the worst-case sequence for the
controller design.
\item $\xi$: The strategy used to handle a deadline miss. We consider
- three different strategies for how to handle a deadline miss:
+ three different strategies:
killing the job that missed the deadline, letting it continue and
skipping the next job, or letting it continue and enqueuing the next
job (up to a maximum of one enqueued job at any point in time).
diff --git a/paper/sec/model.tex b/paper/sec/model.tex
index b579da3..78fcd18 100644
--- a/paper/sec/model.tex
+++ b/paper/sec/model.tex
@@ -72,7 +72,7 @@ task experiences its WCET. Similarly, the Best Case Response Time
considering that every
job executes with its BCET. Finally, in this
work all tasks $\tau_i$ in $\Gamma'$ are \emph{schedulable}, i.e.
-$R_i^W < D_i$ for each $\tau_i$. However, this
+$R_i^W \leq D_i$ for each $\tau_i$. However, this
hypothesis will not be required for $\tau_d$. We will only assume
that at least one job of $\tau_d$ respects its deadline, i.e. $R_d^B
\leq D_d$.
@@ -144,8 +144,9 @@ as shown in Figure~\ref{fig:pandc}. \pp{The behavior of these devices can be mod
%
The plant state is sampled every $T_d$ time units, implying $\mathbf{x}(t_k) = \mathbf{x}(kT_d)$. \pp{The control job $J_{d,k}$ is released at the same instant, i.e. $a_{d,k} = kT_d$, and the sensor data $\mathbf{x}(t_k)$ is immediately available to it.}
Based on the state measurement, the controller computes the feedback control action $\mathbf{u}(t_{k})$.
+
As an hypothesis, our control task $\tau_d$
-executes under the Logical Execution Time (LET) paradigm.
+executes under the Logical Execution Time paradigm.
Indeed, the job $J_{d,k}$
%released at time $a_{d,k}$
% reads the measurement of the plant state available at time
@@ -185,7 +186,7 @@ In this paper, we work under the assumption that $\tau_d$ is the task
with the lowest priority. If other tasks with priority lower
than $\tau_d$ do exist, the design proposed hereafter is still valid
in principle, since those tasks cannot interfere with $\tau_d$.
-However, if this is the case, the choice on the values of $T_d$
+However, if this is the case, the range of possible values of $T_d$
should be tied with schedulability guarantees for the lower
priority tasks.
Due to space constraints, we reserve to analyze
@@ -367,9 +368,10 @@ $\nu$ is a job that successfully completes its execution and whose
generated output is not overwritten before the next deadline instant.
\end{definition}
-For each time interval $[0,t)$, we show that is possible to extract the sequence
-of $v$ valid jobs, defined as $S = \{\nu_1,\nu_2,...,\nu_{v}\}$, where
-the index does not count the passing of time, and the relation $v
+For each time interval $[0,t)$, we show that is possible to extract the
+ordered sequence
+of $v$ valid jobs, defined as $S = \{\nu_1,\nu_2,...,\nu_{v}\}$ (where
+the index does not count the passing of time) and the relation $v
\leq \lceil t/T_d \rceil $ trivially holds. The sequence of valid
jobs depends on the strategy used to handle deadline misses, and will
be described in Section~\ref{sec:behavior}. Our control design should
--
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