Fixed many typos and errors up to section 6

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};

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.

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 $.

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.

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

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).

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