# Risk-Aware Stability, Ultimate Boundedness, and Positive Invariance

Masako Kishida M. Kishida is with the National Institute of Informatics,
Tokyo, Japan<EMAIL_ADDRESS>
###### Abstract

This paper introduces the notions of stability, ultimate boundedness, and
positive invariance for stochastic systems in the view of risk. More
specifically, those notions are defined in terms of the worst-case Conditional
Value-at-Risk (CVaR), which quantifies the worst-case conditional expectation
of losses exceeding a certain threshold over a set of possible uncertainties.
Those notions allow us to focus our attention on the tail behavior of
stochastic systems in the analysis of dynamical systems and the design of
controllers. Furthermore, some event-triggered control strategies that
guarantee ultimate boundedness and positive invariance with specified bounds
are derived using the obtained results and illustrated using numerical
examples.

##  I Introduction

Whether it is a financial portfolio or engineering system, we often need to
accept a certain level of risk to balance the pros and cons of spending costs
in the decision-making processes under uncertainties. This is particularly
true when the uncertainty distributions have an unbounded support because a
guarantee of 100% ideal satisfaction is impossible or requires an infinite
amount of cost. Therefore, risk has been studied for a long time, not only in
the financial industry [1, 2, 3, 4], but also in the wide areas of engineering
[5, 6, 7] using many different approaches.

The notions of stability, ultimate boundedness, and positive invariance are
fundamental in the analysis of dynamical systems and in the design of
controllers [8]. To deal with stochastic uncertainties in dynamical systems,
the concept of probabilistic stability was introduced in [9]. Later, the
concepts of probabilistic set invariance and ultimate boundedness were
introduced for discrete-time linear systems in [10] and extended to
continuous-time linear systems in [11]. Those probabilistic notions are
defined using chance constraints. Another popular approach to dealing with
stochastic uncertainties is to consider mean square and $p$-stability [12]. In
this direction, ultimate boundedness [13] and positive invariance [14, 15]
have been also investigated. However, those notions are not applicable to
guarantee that the expected value of the constraint-violating cases is small.

To take into account this additional consideration of risk, this paper
introduces the definitions of stability, ultimate boundedness, and positive
invariance in terms of the worst-case Conditional Value-at-Risk (CVaR) for
stochastic systems. CVaR is a relatively new risk measure that is defined as
the conditional expectation of losses exceeding a certain threshold [16]. The
worst-case CVaR is the supremum of CVaR over a set of possible disturbances
[4, 17]. Using the (worst-case) CVaR, the tail behavior of the stochastic
systems can be quantified; it quantifies the risk which has a low probability
of occurring, but if it does occur, it will result in a large loss. Assessing
the tail risk is especially beneficial when the uncertainty distribution has a
fat tail. In such a case, the use of the chance constraints or standard
expected value for analysis may result in a significant loss. Because the
(worst-case) CVaR is a coherent risk measure [4], it enjoys nice mathematical
properties. In particular, we see the worst-case CVaR on the squared norm of
the states using the first two moments of the disturbance allows us to obtain
elegant results for discrete-time linear stochastic systems. Such a problem
setup can be easily applied to cases wherein the disturbance is non-Gaussian
or the probability density function is not available.

The rest of the paper is organized as follows. After introducing notation,
definitions, and properties of the worst-case CVaR as well as some basic
results in Section II, Section III and Section IV present the definitions of
stability, ultimate boundedness, and positive invariance using the worst-case
CVaR, without inputs and with bounded inputs, respectively. Based on the
results in Section IV, approaches to risk-aware event-triggered control are
developed in Section V, which is followed by conclusion in Section VI.

##  II Preliminaries

###  II-A Notation

The sets of real numbers, real vectors of length $n$, and real matrices of
size $n\times m$ are denoted by $\mathbb{R}$, $\mathbb{R}^{n}$, and
$\mathbb{R}^{n\times m}$, respectively. The sets of nonnegative numbers,
nonnegative integers, and positive integers are denoted by $\mathbb{R}_{\geq
0}$, $\mathbb{Z}_{\geq 0}$ and $\mathbb{Z}_{>0}$, respectively. For
$M\in\mathbb{R}^{n\times n}$, $M\succ 0$ indicates $M$ is positive definite.
$M^{\top}$ denotes the transpose of a real matrix $M$ and $\text{Tr}(M)$
denotes the trace of $M$. $I_{n}$ denotes the identity matrix of size $n$. The
Kronecker product of two matrices $X$ and $Y$ is denoted as $X\otimes Y$. For
$x\in\mathbb{R}$, $x^{+}=\max\\{x,0\\}$. For a vector $v\in\mathbb{R}^{n}$,
$\|v\|$ denotes the Euclidean norm. For a matrix $M$, $\|M\|$ denotes the
maximum singular value norm. Recall that a function $\gamma:\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$ is a $\mathcal{K}$ function if it is
continuous, strictly increasing and $\gamma(0)=0$. A function
$\beta:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}$ is a $\mathcal{KL}$ function if for each fixed $t\geq 0$,
$\beta(s,t)\in\mathcal{K}$ with respect to $s$ and for each fixed $s\geq 0$,
$\beta(s,t)$ is decreasing with respect to $t$ and
$\lim_{t\rightarrow\infty}\beta(s,t)=0$.

###  II-B Conditional Value-at-Risk

Let $\mu\in\mathbb{R}^{n}$ be the mean and $\Sigma\in\mathbb{R}^{n\times n}$
be the covariance matrix of the random vector $\xi\in\mathbb{R}^{n}$ under the
true distribution $\mathbb{P}$, which is the probability law of $\xi$. Thus,
it is implicitly assumed that the random vector $\xi$ has finite second-order
moments. Let $\mathcal{P}$ denote the set of all probability distributions on
$\mathbb{R}^{n}$ that have the same first- and second-order moments as
$\mathbb{P}$, i.e.,

$\displaystyle\mathcal{P}=\left\\{\mathbb{P}:\mathbb{E}_{\mathbb{P}}\left[\begin{bmatrix}\xi_{i}\\\
1\end{bmatrix}\begin{bmatrix}\xi_{j}\\\
1\end{bmatrix}^{\\!\top}\right]=\begin{bmatrix}\Sigma\delta_{ij}&0\\\
0^{\top}&1\end{bmatrix},\forall i,j\right\\}.$

Here $\delta_{ij}$ denotes the Kronecker delta and
$\mathbb{E}_{\mathbb{P}}[\cdot]$ denotes the expectation with respect to
$\mathbb{P}$. The true underlying probability measure $\mathbb{P}$ is not
known exactly, but it is known that $\mathbb{P}\in\mathcal{P}$.

######  Definition II.1 (Conditional Value-at-Risk [16, 18])

For a given measurable loss function $L:\mathbb{R}^{n}\rightarrow\mathbb{R}$,
a probability distribution $\mathbb{P}$ on $\mathbb{R}^{n}$ and a level
$\varepsilon\in(0,1)$, the CVaR at $\varepsilon$ with respect to $\mathbb{P}$
is defined as

$\displaystyle\mathbb{P}\text{-CVaR}_{\varepsilon}[L({\xi})]=\inf_{\beta\in\mathbb{R}}\left\\{\beta+\frac{1}{\varepsilon}\mathbb{E}_{\mathbb{P}}[(L({\xi})-\beta)^{+}]\right\\}.$

CVaR is the conditional expectation of loss above the
($1-\varepsilon$)-quantile of the loss function [18] and quantifies the tail
risk (see Figure 1).

The worst-case CVaR is the supremum of CVaR over a given set of probability
distributions as defined below:

######  Definition II.2 (Worst-case CVaR [18])

The worst-case CVaR over $\mathcal{P}$ is given by

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L({\xi})]=\inf_{\beta\in\mathbb{R}}\left\\{\beta+\frac{1}{\varepsilon}\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{E}_{\mathbb{P}}[(L({\xi})-\beta)^{+}]\right\\}.$

Here, the exchange between the supremum and infimum is justified by the
stochastic saddle point theorem [19].

Figure 1: Illustration of mean and CVaR

If $L(\xi)$ is quadratic with respect to $\xi$, the worst-case CVaR can be
computed by a semidefinite program [18], [20]. Furthermore, if the mean of the
random vector $\xi$ is zero, the following easy-to-compute bounds are
obtained.

######  Lemma II.3 (Bounds for worst-case CVaR [21])

Suppose $\mu=0$ and $L(\xi)=\|A\xi+b\|^{2}+c$ with some
$A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^{n}$ and $c\in\mathbb{R}$, then

$\displaystyle c+b^{\top}b+\frac{1}{\varepsilon}\left(\text{Tr}(\Sigma
A^{\top}A)\right)\leq$ $\displaystyle\
\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L({\xi})]$
$\displaystyle\leq$ $\displaystyle\
c+\frac{1}{\varepsilon}\left(\text{Tr}(\Sigma A^{\top}A)+b^{\top}b\right).$

If $b=0$ and $c=0$, then it follows that

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L({\xi})]=\frac{1}{\varepsilon}\text{Tr}(\Sigma
A^{\top}A).$

To deal with dynamical systems, the following is a simple, but useful result.

###### Lemma II.4

Suppose $\xi_{1},\xi_{2},...,\xi_{m}\in\mathbb{R}^{n}$ are independent and
identically distributed random vectors under the true distribution
$\mathbb{P}\in\mathcal{P}$. Let
$\bar{\xi}=[\xi_{1}^{\top},\xi_{2}^{\top},...,\xi_{m}^{\top}]^{\top}$. Then,
$\bar{\xi}\in\mathbb{R}^{nm}$ is a random vector with the mean zero and
covariance $I_{m}\otimes\Sigma$. Thus, the true underlying probability measure
$\mathbb{P}$ for $\xi$ satisfies $\mathbb{P}\in\mathcal{P}_{\text{aug}}$,
where

$\displaystyle\mathcal{P}_{\text{aug}}=\left\\{\mathbb{P}\\!:\\!\mathbb{E}_{\mathbb{P}}\left[\begin{bmatrix}\bar{\xi}_{i}\\\
1\end{bmatrix}\begin{bmatrix}\bar{\xi}_{j}\\\
1\end{bmatrix}^{\\!\top}\right]=\begin{bmatrix}\\!(I_{m}\otimes\Sigma)\delta_{ij}&0\\\
0&1\end{bmatrix},\forall i,j\right\\}.$

Moreover, for a matrix $A\in\mathbb{R}^{\ell\times nm}$, it holds that

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}_{\text{aug}}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|A\xi\|^{2}]=\frac{1}{\varepsilon}\text{Tr}((I_{m}\otimes\Sigma)A^{\top}A).$

###### Proof:

Follows from the assumption that
$\xi_{1},\xi_{2},...,\xi_{m}\in\mathbb{R}^{n}$ are independent and identically
distributed. The second part follows from Lemma II.3. ∎

One reason that the (worst-case) CVaR is popular for risk assessment is its
mathematically attractive properties of coherency.

######  Proposition II.5 (Coherence properties [4, 22])

The worst-case CVaR is a coherent risk measure, i.e., it satisfies the
following properties: Let $L_{1}=L_{1}(\xi)$ and $L_{2}=L_{2}(\xi)$ be two
measurable loss functions.

  * •

Sub-additivity: For all $L_{1}$ and $L_{2}$,

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{1}+L_{2}]$
$\displaystyle\leq\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{1}]+\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{2}];$

  * •

Positive homogeneity: For a positive constant $a>0$,

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[aL_{1}]=a\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{1}];$

  * •

Monotonicity: If $L_{1}\leq L_{2}$ almost surely,

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{1}]\leq\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{2}];$

  * •

Translation invariance: For a constant $c$,

$\displaystyle\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{1}+c]=\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[L_{1}]+c.$

###  II-C Useful Result

This paper repeatedly utilizes the following inequality.

######  Lemma II.6 (Cauchy-Schwarz inequality)

For two vectors $x,y\in\mathbb{R}^{n}$, it holds that

$\displaystyle
2x^{\top}y\leq\alpha^{2}\|x\|^{2}+\frac{1}{\alpha^{2}}\|y\|^{2},\
\forall\alpha>0.$

##  III Linear System without Inputs

This section considers linear systems subject to stochastic disturbances.
After introducing a system model, stability, ultimate boundedness, and
positive invariance are investigated.

###  III-A System Model

Consider the discrete-time linear stochastic system

$\displaystyle x_{t+1}=Ax_{t}+Ew_{t},$ (1)

where $x_{t}\in\mathbb{R}^{n_{x}}$ is the state and
$w_{t}\in\mathbb{R}^{n_{w}}$ is the disturbance, respectively, at discrete
time instant $t\in\mathbb{Z}_{\geq 0}$. $A\in\mathbb{R}^{n_{x}\times n_{x}}$
and $E\in\mathbb{R}^{n_{x}\times n_{w}}$ are constant matrices. It is assumsed
that the initial condition $x_{0}\in\mathbb{R}^{n_{x}}$ is given, and that
$w_{t}$ are independent and identically distributed random vectors with the
mean zero and covariance $\Sigma_{w}\succ 0$ for all $t\in\mathbb{Z}_{\geq
0}$. The true underlying probability measure $\mathbb{P}$ is not known
exactly, but it is known that $\mathbb{P}\in\mathcal{P}$, where

$\displaystyle\mathcal{P}\\!=\\!\left\\{\mathbb{P}\\!:\\!\mathbb{E}_{\mathbb{P}}\left[\begin{bmatrix}w_{i}\\\
1\end{bmatrix}\begin{bmatrix}w_{j}\\\
1\end{bmatrix}^{\\!\top}\right]\\!\\!=\begin{bmatrix}\Sigma_{w}\delta_{ij}&0\\!\\\
\\!0^{\top}&1\end{bmatrix},\forall i,j\right\\}.$ (2)

For $t\in\mathbb{Z}_{>0}$, the state evolution of (1) can be expressed as

$\displaystyle x_{t}=F_{t}x_{0}+H_{t}\bar{w}_{t},$ (3)

using

$\displaystyle\begin{aligned} \bar{w}_{t}&=[w_{0}^{\top},\
w_{1}^{\top},\cdots,w_{t-1}^{\top}]^{\top},\\\ F_{t}&=A^{t},\\\
H_{t}&=\begin{bmatrix}A^{t-1}E&A^{t-2}E&\cdots&E\end{bmatrix}.\end{aligned}$
(4)

With this notation, from Lemma II.4, $\bar{w}_{t}\in\mathbb{R}^{n_{w}t}$ is a
random vector with the mean zero and covariance $I_{t}\otimes\Sigma_{w}$.
Thus, the true underlying probability measure $\mathbb{P}$ for $\bar{w}_{t}$
satisfies $\mathbb{P}\in\mathcal{P}_{t}$, where

$\displaystyle\mathcal{P}_{t}\\!=\\!\left\\{\mathbb{P}\\!:\\!\mathbb{E}_{\mathbb{P}}\left[\begin{bmatrix}\bar{w}_{t,i}\\\
1\end{bmatrix}\begin{bmatrix}\bar{w}_{t,j}\\\
1\end{bmatrix}^{\top}\right]\\!\\!=\begin{bmatrix}(I_{t}\otimes\Sigma_{w})\delta_{ij}&0\\!\\\
\\!0&1\end{bmatrix},\forall i,j\right\\}.$ (5)

###  III-B Practical Stability and Ultimate Boundedness

Here, the stability notion in terms of the worst-case CVaR is introduced.

######  Definition III.1 (Practical stability)

The system (1) is practically asymptotically worst-case CVaR stable if there
exist $\beta\in\mathcal{KL}$ and a constant $c\geq 0$ such that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}]\leq\beta(\|x_{0}\|^{2},t)+c,\
\forall t\in\mathbb{Z}_{\geq 0}.\end{aligned}$ (6)

The following lemma provides a sufficient condition for a system to be
practically asymptotically worst-case CVaR stable.

######  Lemma III.2 (Sufficient condition for practical stability)

If $\|A\|<1$ and $(A,E)$ is reachable, then the linear system (1) is
practically asymptotically worst-case CVaR stable.

###### Proof:

Using Lemma II.6 and norm sub-multiplicativity,

$\displaystyle\begin{aligned}
\|x_{t}\|^{2}&=\|F_{t}x_{0}+H_{t}\bar{w}_{t}\|^{2}\\\
&\leq(1+\alpha^{2})\|F_{t}\|^{2}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\|H_{t}\bar{w}_{t}\|^{2}\\\
&\leq(1+\alpha^{2})\lambda^{t}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\|H_{t}\bar{w}_{t}\|^{2}\end{aligned}$
(7)

for any $\alpha>0$, where $\lambda=\|A\|^{2}\in[0,1)$. Using Proposition II.5
along with Lemma II.4, it follows that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}]\\\
&\leq(1+\alpha^{2})\lambda^{t}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\frac{1}{\varepsilon}\text{Tr}((I_{t}\otimes\Sigma_{w})H_{t}^{\top}H_{t})\\\
&\leq(1+\alpha^{2})\lambda^{t}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\frac{1}{\varepsilon}\text{Tr}(P).\end{aligned}$
(8)

Here, we used

$\displaystyle\begin{aligned}
\text{Tr}((I_{t}\otimes\Sigma_{w})H_{t}^{\top}H_{t})&=\text{Tr}\left(\sum_{k=0}^{t-1}(A^{k})E\Sigma_{w}E^{\top}(A^{\top})^{k}\right)\\\
&\leq\text{Tr}\left(\sum_{k=0}^{\infty}(A^{k})E\Sigma_{w}E^{\top}(A^{\top})^{k}\right)\\\
&=\text{Tr}(P),\end{aligned}$ (9)

where $P\succ 0$ is the solution to the Lyapunov equation

$\displaystyle APA^{\top}-P+E\Sigma_{w}E^{\top}=0.$ (10)

The last equality in (9) is due to $A$ is Schur and $(A,E)$ is reachable.
Thus, choosing

$\displaystyle\beta(s,t)=(1+\alpha^{2})s\lambda^{t},\
c=\left(1+\frac{1}{\alpha^{2}}\right)\frac{1}{\varepsilon}\text{Tr}(P)$ (11)

satisfies Definition III.1. ∎

###### Remark III.3

For linear systems, assuming that $A$ is Schur is sufficient for practically
asymptotically worst-case CVaR stability as for other kinds of stability.
However, we impose the assumption $\|A\|<1$ in this paper, and Lemma III.2 is
focused on such a case.

Here, we introduce a notion of worst-case CVaR ultimate bound, which is an
extension of the probabilistic ultimate bound [10]. The notion of ultimate
bound is closely related to practical asymptotic stability.

######  Definition III.4 (Ultimate bound)

Let $f:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}$. A domain
$\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:f(x)\leq 0\\}$ is a worst-case CVaR
ultimate bound for the system (1) if for every initial state $x_{0}$, there
exists $T=T(x_{0})>0$ such that
$\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[f(x_{t})]\leq
0$ for all $t\geq T$.

A worst-case CVaR ultimate bound can be found as below:

######  Theorem III.5 (A ultimate bound)

Let define the bounded set

$\displaystyle\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:\|x\|^{2}-r^{2}\leq 0\\}$
(12)

for $r>0$. Consider the system (1) and assume that $\|A\|<1$ and $(A,E)$ is
reachable. If

$\displaystyle r^{2}>\frac{1}{\varepsilon}\text{Tr}(P),$ (13)

then, $\mathcal{D}$ is a worst-case CVaR ultimate bound for (1).

###### Proof:

Under the condition (13), there exist $\delta,\alpha>0$ such that

$\displaystyle\delta=r^{2}-\left(1+\frac{1}{\alpha^{2}}\right)\frac{1}{\varepsilon}\text{Tr}(P)>0.$
(14)

Using the proof of Lemma III.2 with Proposition II.5,

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}-r^{2}]\\\
&\leq(1+\alpha^{2})\lambda^{t}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\frac{1}{\varepsilon}\text{Tr}(P)-r^{2}\\\
&\leq(1+\alpha^{2})\lambda^{t}\|x_{0}\|^{2}-\delta.\end{aligned}$ (15)

Because the first term above approaches to $0$ as $t$ goes to infinity, there
exists $T$ such that
$\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}-r^{2}]\leq
0$ for all $t\geq T$. ∎

###  III-C Positive Invariance

The notion of positive invariance is yet another important concept, which is
intimately related to ultimate boundedness.

######  Definition III.6 (Positively invariant set)

Let $f:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}$ be a continuous function. A
domain $\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:f(x)\leq 0\\}$ is a worst-case
CVaR positively invariant set for the system (1) if for any state
$x_{t}\in\mathcal{D}$
$\sup_{\mathbb{P}\in\mathcal{P}_{k}}\mathbb{P}\text{-CVaR}_{\varepsilon}[f(x_{t+k})]\leq
0$ for all $k\in\mathbb{Z}_{>0}$.

A worst-case CVaR positively invariant set can be found as below:

######  Theorem III.7 (A positively invariant set)

Let define the bounded set

$\displaystyle\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:\|x\|^{2}-r^{2}\leq 0\\},$
(16)

for $r>0$. Consider the system (1) and assume that $\|A\|<1$ and $(A,E)$ is
reachable. If

$\displaystyle
r^{2}\geq\frac{1}{\varepsilon(1-\|A\|)^{2}}\text{Tr}(\Sigma_{w}E^{\top}E),$
(17)

then, $\mathcal{D}$ is a worst-case CVaR positively invariant set for (1).

###### Proof:

First, we show that $x_{t}\in\mathcal{D}$ implies
$\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+1}\|^{2}-r^{2}]\leq
0$. Using Lemma II.6, for any $\alpha>0$,

$\displaystyle\begin{aligned} \|x_{t+1}\|^{2}&=\|Ax_{t}+Ew_{t}\|^{2}\\\
&\leq(1+\alpha^{2})\|Ax_{t}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\|Ew_{t}\|^{2}.\end{aligned}$
(18)

Choose $\alpha>0$ that satisfies

$\displaystyle(1+\alpha^{2})\|A\|=1.$ (19)

Such an $\alpha$ always exists for $\|A\|<1$. Using Proposition II.5 as well
as the norm sub-multiplicativity and Lemma II.3, it follows that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+1}\|^{2}]\leq\|A\|\|x_{t}\|^{2}+\frac{1}{1-\|A\|}\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E).\end{aligned}$
(20)

Hence, if $x_{t}\in\mathcal{D}$, using the condition (17), it follows that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+1}\|^{2}-r^{2}]\\\
&\leq\|A\|r^{2}+\frac{1}{1-\|A\|}\left(1-\|A\|\right)^{2}r^{2}-r^{2}=0.\end{aligned}$
(21)

Next, we show
$\sup_{\mathbb{P}\in\mathcal{P}_{k}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k}\|^{2}-r^{2}]\leq
0$ for some $k>0$ implies
$\sup_{\mathbb{P}\in\mathcal{P}_{k+1}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k+1}\|^{2}-r^{2}]\leq
0$. Again, using Lemma II.6, for any $\alpha>0$,

$\displaystyle\begin{aligned} \|x_{t+k+1}\|^{2}&=\|Ax_{t+k}+Ew_{t}\|^{2}\\\
&\leq(1+\alpha^{2})\|Ax_{t+k}\|^{2}+\left(1+\frac{1}{\alpha^{2}}\right)\|Ew_{t+k}\|^{2}.\end{aligned}$
(22)

Choose $\alpha>0$ that satisfies (19). As before, we have

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k+1}\|^{2}-r^{2}]\\\
&\leq\|A\|\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k}\|^{2}]+\frac{1}{1-\|A\|}\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)-r^{2}\\\
&\leq\|A\|r^{2}+\frac{1}{1-\|A\|}\left(1-\|A\|\right)^{2}r^{2}-r^{2}=0.\end{aligned}$
(23)

This completes the proof. ∎

###### Remark III.8

With $\alpha$ that does not satisfy (19), a similar argument still holds by
replacing the condition (17) with

$\displaystyle\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)\leq\alpha^{2}\left(\frac{1}{1+\alpha^{2}}-\|A\|^{2}\right)r^{2}.$
(24)

This right-hand-side is maximized with $\alpha$ in (19).

##  IV Linear System with Bounded Inputs

In this section, linear systems with bounded inputs subject to stochastic
disturbances are considered. To deal with such a case, the notion of input-to-
state stability and robust positive invariance are developed using the worst-
case CVaR.

###  IV-A System Model

Consider the discrete-time linear stochastic system

$\displaystyle x_{t+1}=Ax_{t}+Dv_{t}+Ew_{t},$ (25)

where $x_{t}\in\mathbb{R}^{n_{x}}$ is the state, $v_{t}\in\mathcal{V}$ is the
bounded input and $w_{t}\in\mathbb{R}^{n_{w}}$ is the disturbance,
respectively, at discrete time instant $t\in\mathbb{Z}_{\geq 0}$.
$A\in\mathbb{R}^{n_{x}\times n_{x}}$, $D\in\mathbb{R}^{n_{x}\times n_{v}}$ and
$E\in\mathbb{R}^{n_{x}\times n_{w}}$ are constant matrices. It is assumsed
that the initial condition $x_{0}\in\mathbb{R}^{n_{x}}$ is given, and that
$w_{t}$ are independent and identically distributed random vectors defined for
(1).

We treat $v_{t}\in\mathcal{V}$ as a bounded disturbance and define

$\displaystyle\mathcal{V}=\\{v\in\mathbb{R}^{n_{v}}:\|v\|\leq d\\}$ (26)

for a given $d$.

For $t\in\mathbb{Z}_{>0}$, the state evolution of (25) can be expressed by

$\displaystyle x_{t}=F_{t}x_{0}+G_{t}\bar{v}_{t}+H_{t}\bar{w}_{t},$ (27)

using (4) and

$\displaystyle\begin{aligned} \bar{v}_{t}&=[v_{0}^{\top},\
v_{1}^{\top},\cdots,v_{t-1}^{\top}]^{\top},\\\
G_{t}&=\begin{bmatrix}A^{t-1}D&A^{t-2}D&\cdots&D\end{bmatrix}.\end{aligned}$
(28)

###  IV-B Practical Stability and Ultimate Boundedness

As in the previous section, we define the input-to-state stability using the
worst-case CVaR.

######  Definition IV.1 (Practical stability)

The system (25) is practically worst-case CVaR input-to-state stable if there
exist $\beta\in\mathcal{KL}$, $\gamma\in\mathcal{K}$, and $c\geq 0$ such that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}]\\\
&\leq\beta(\|x_{0}\|^{2},t)+\gamma(\max_{\tau\in[0,t-1]}(\|v_{\tau}\|^{2}))+c,\
\forall t\in\mathbb{Z}_{\geq 0}.\end{aligned}$ (29)

Note that if a system is practically worst-case CVaR input-to-state stable,
then it is practically asymptotically worst-case CVaR stable if there is no
input.

The following lemma provides a sufficient condition for a system to be
practically worst-case CVaR input-to-state stable.

######  Lemma IV.2 (Sufficient condition for practical stability)

If $\|A\|<1$ and $(A,E)$ is reachable, then the linear system (1) is
practically worst-case CVaR input-to-state stable.

###### Proof:

From (26), $v_{\tau}$ is bounded. Write
$\bar{v}_{t-1}=\max_{\tau\in[0,t-1]}(\|v_{\tau}\|))(\leq d)$. Using Lemma II.6
twice and norm sub-multiplicativity,

$\displaystyle\begin{aligned}
\|x_{t}\|^{2}&=\|F_{t}x_{0}+G_{t}\bar{v}_{t}+H_{t}\bar{w}_{t}\|^{2}\\\
&\leq(1+\alpha_{1}^{2})\lambda^{t}\|x_{0}\|^{2}\\\
&+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\alpha_{2}^{2}\right)\left(\frac{\|D\|}{1-\|A\|}\right)^{2}\bar{v}_{t-1}^{2}\\\
&+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\frac{1}{\alpha_{2}^{2}}\right)\|H_{t}\bar{w}_{t}\|^{2}\end{aligned}$
(30)

for any $\alpha_{1},\alpha_{2}>0$, where $\lambda=\|A\|^{2}\in[0,1)$. Here, we
used

$\displaystyle\begin{aligned}
\|G_{t}\bar{v}_{t}\|^{2}&=\left\|\sum_{k=0}^{t-1}A^{k}Dv_{t-1-k}\right\|^{2}\\\
&\leq\left\|\sum_{k=0}^{t-1}\|A\|^{k}\|D\|\|v_{t-1-k}\|\right\|^{2}\\\
&\leq\left\|\sum_{k=0}^{t-1}\|A\|^{k}\|D\|\max_{\tau\in[0,t-1]}(\|v_{\tau}\|)\right\|^{2}\\\
&\leq\left(\frac{\|D\|}{1-\|A\|}\right)^{2}\bar{v}_{t-1}^{2}.\end{aligned}$
(31)

Using Proposition II.5 along with Lemma II.4, it follows that:

$\displaystyle\begin{aligned} &\
\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}]\\\
&\
\leq(1+\alpha_{1}^{2})\lambda^{t}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\alpha_{2}^{2}\right)\left(\frac{\|D\|}{1-\|A\|}\right)^{2}\bar{v}_{t-1}^{2}\\\
&\
+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\frac{1}{\alpha_{2}^{2}}\right)\frac{1}{\varepsilon}\text{Tr}(P),\end{aligned}$
(32)

where $P\succ 0$ is the solution to the Lyapunov equation (10). Thus, choosing

$\displaystyle\begin{aligned} &\beta(s,t)=(1+\alpha_{1}^{2})s\lambda^{t},\\\
&\gamma(\bar{v}_{t-1})=\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\alpha_{2}^{2}\right)\left(\frac{\|D\|}{1-\|A\|}\right)^{2}\bar{v}_{t-1}^{2},\\\
&c=\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\frac{1}{\alpha_{2}^{2}}\right)\frac{1}{\varepsilon}\text{Tr}(P)\end{aligned}$
(33)

satisfies Definition IV.1. ∎

Similar to the previous section, we introduce a notion of worst-case CVaR
ultimate bound with bounded input.

######  Definition IV.3 (Ultimate bound)

Let $f:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}$. A domain
$\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:f(x)\leq 0\\}$ is a worst-case CVaR
ultimate bound with bounded input for the system (25) if for every initial
state $x_{0}$, there exists $T=T(x_{0})>0$ such that
$\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[f(x_{t})]\leq
0$ for all $t\geq T$.

A worst-case CVaR ultimate bound can be found as below:

######  Theorem IV.4 (A ultimate bound)

Let define the bounded set

$\displaystyle\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:\|x\|^{2}-r^{2}\leq 0\\},$
(34)

for $r>0$. Consider the system (25) and assume that $\|A\|<1$ and $(A,E)$ is
reachable. If

$\displaystyle
r^{2}>\left(\frac{\|D\|}{1-\|A\|}d+\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}\right)^{2},$
(35)

then, $\mathcal{D}$ is a worst-case CVaR ultimate bound with bounded input for
(25).

###### Proof:

Under the condition (35), there exist $\delta,\alpha_{1}>0$ such that

$\displaystyle\begin{aligned}
\delta&=r^{2}-\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(\frac{\|D\|}{1-\|A\|}d+\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}\right)^{2}>0,\end{aligned}$
(36)

By using $\bar{v}_{t-1}\leq d$ and substituting

$\displaystyle\begin{aligned}
\alpha_{2}^{2}=\frac{1-\|A\|}{\|D\|d}\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}>0\end{aligned}$
(37)

in (LABEL:eq:sc_ps), it follows that

$\displaystyle\begin{aligned} &\
\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}]\\\
&\
\leq(1+\alpha_{1}^{2})\lambda^{t}\|x_{0}\|^{2}+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(\frac{\|D\|}{1-\|A\|}d+\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}\right)^{2}.\end{aligned}$
(38)

Thus using Proposition II.5 and (36),

$\displaystyle\begin{aligned} &\
\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}-r^{2}]\leq(1+\alpha_{1}^{2})\lambda^{t}\|x_{0}\|^{2}-\delta.\end{aligned}$
(39)

Because the first term approaches to $0$ as $t$ goes to infinity, there exists
$T$ such that
$\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}-r^{2}]\leq
0$ for all $t\geq T$. ∎

###  IV-C Positive Invariance

The notion of robust positively invariant set can be defined for system with
bounded input.

######  Definition IV.5 (Robust positively invariant set)

Let $f:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}$ be a continuous function. A
domain $\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:f(x)\leq 0\\}$ is a worst-case
CVaR robust positively invariant set for the system (25) if for any state
$x_{t}\in\mathcal{D}$
$\sup_{\mathbb{P}\in\mathcal{P}_{k}}\mathbb{P}\text{-CVaR}_{\varepsilon}[f(x_{t+k})]\leq
0$ for all $k\in\mathbb{Z}_{>0}$.

A robust positively invariant set can be found as follows.

######  Theorem IV.6 (A robust positively invariant set)

Let define the bounded set

$\displaystyle\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:\|x\|^{2}-r^{2}\leq 0\\},$
(40)

for $r>0$. Consider the system (25) and assume that $\|A\|<1$ and $(A,E)$ is
reachable. If

$\displaystyle
r^{2}\geq\frac{1}{(1-\|A\|)^{2}}\left(\|D\|d+\sqrt{\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)}\right)^{2},$
(41)

then, $\mathcal{D}$ is a worst-case CVaR robust positively invariant set for
(25). Note $d$ is defined in (26).

###### Proof:

First, we show that $x_{t}\in\mathcal{D}$ implies
$\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+1}\|^{2}-r^{2}]\leq
0$. Using Lemma II.6, for any $\alpha_{1},\alpha_{2}>0$,

$\displaystyle\begin{aligned} \|x_{t+1}\|^{2}&=\|Ax_{t}+Dv_{t}+Ew_{t}\|^{2}\\\
&\leq(1+\alpha_{1}^{2})\|Ax_{t}\|^{2}+\left(1+\frac{1}{\alpha_{1}^{2}}\right)(1+\alpha_{2}^{2})\|D\|^{2}d^{2}\\\
&\
+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\frac{1}{\alpha_{2}^{2}}\right)\|Ew_{t}\|^{2}.\end{aligned}$
(42)

Choose $\alpha_{1}>0$ that satisfies (19) and $\alpha_{2}>0$ that satisfies

$\displaystyle\begin{aligned}
\alpha_{2}^{2}\|D\|d=\sqrt{\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)}.\end{aligned}$
(43)

Using Proposition II.5 as well as the norm sub-multiplicativity and Lemma
II.3, it follows that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}_{t+1}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+1}\|^{2}]\\\
&\leq\|A\|\|x_{t}\|^{2}+\frac{1}{1-\|A\|}\left(\|D\|d+\sqrt{\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)}\right)^{2}.\end{aligned}$
(44)

Hence, if $x_{t}\in\mathcal{D}$, using the condition (41), it follows that

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+1}\|^{2}-r^{2}]\\\
&\leq\|A\|r^{2}+\frac{1}{1-\|A\|}\left(1-\|A\|\right)^{2}r^{2}-r^{2}=0.\end{aligned}$
(45)

Next, we show
$\sup_{\mathbb{P}\in\mathcal{P}_{k}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k}\|^{2}-r^{2}]\leq
0$ for some $k>0$ implies
$\sup_{\mathbb{P}\in\mathcal{P}_{k+1}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k+1}\|^{2}-r^{2}]\leq
0$. Again, using Lemma II.6, for any $\alpha_{1},\alpha_{2}>0$,

$\displaystyle\begin{aligned}
\|x_{t+k+1}\|^{2}&=\|Ax_{t+k}+Dv_{t+k}+Ew_{t+k}\|^{2}\\\
&\leq(1+\alpha_{1}^{2})\|Ax_{t+k}\|^{2}+\left(1+\frac{1}{\alpha_{1}^{2}}\right)(1+\alpha_{2}^{2})\|D\|^{2}d^{2}\\\
&\
+\left(1+\frac{1}{\alpha_{1}^{2}}\right)\left(1+\frac{1}{\alpha_{2}^{2}}\right)\|Ew_{t}\|^{2}.\end{aligned}$
(46)

Choose $\alpha_{1}>0$ that satisfies (19) and $\alpha_{2}>0$ that satisfies
(43). As before, we have

$\displaystyle\begin{aligned}
&\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k+1}\|^{2}-r^{2}]\\\
&\leq\|A\|\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t+k}\|^{2}]+\frac{1}{1-\|A\|}\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)-r^{2}\\\
&\leq\|A\|r^{2}+\frac{1}{1-\|A\|}\left(1-\|A\|\right)^{2}r^{2}-r^{2}=0.\end{aligned}$
(47)

This completes the proof. ∎

##  V Event-triggered Control

In this section, event-triggered control strategies are developed using the
results in Section IV.

###  V-A System Model

Consider the discrete-time linear control system subject to stochastic
disturbance

$\displaystyle x_{t+1}=Ax_{t}+Bu_{t}+Ew_{t},$ (48)

where $x_{t}\in\mathbb{R}^{n_{x}}$ is the state, $u_{t}\in\mathbb{R}^{n_{u}}$
is the control input and $w_{t}\in\mathbb{R}^{n_{w}}$ is the disturbance,
respectively, at discrete time instant $t\in\mathbb{Z}_{\geq 0}$.
$A\in\mathbb{R}^{n_{x}\times n_{x}}$, $B\in\mathbb{R}^{n_{x}\times n_{u}}$ and
$E\in\mathbb{R}^{n_{x}\times n_{w}}$ are constant matrices. It is assumed that
the initial condition $x_{0}\in\mathbb{R}^{n_{x}}$ is given and $w_{t}$ are
independent and identically distributed random vectors defined for (1). It is
also assumed that a linear state feedback control $u_{t}=Kx_{t}$ has been
designed such that $\|A+BK\|<1$ for (48) and $(A+BK,E)$ is reachable.

In this section, we design event-triggered control strategies that guarantee
ultimate boundedness and positive invariance for the system (48) with the set

$\displaystyle\mathcal{D}=\\{x\in\mathbb{R}^{n_{x}}:\|x\|^{2}-r^{2}\leq 0\\}.$
(49)

To design trigger conditions, let us introduce the following notation: Let the
triggering time sequence $\\{t_{k}\\}_{k\in\mathbb{Z}_{\geq 0}}$, and define
the state used for the control input by

$\displaystyle\hat{x}_{t}$ $\displaystyle=x_{t_{k}},\ \forall
t\in[t_{k},t_{k+1}),$ (50)

and the state error by

$\displaystyle e_{t}$ $\displaystyle=\hat{x}_{t}-x_{t},\ \forall
t\in[t_{k},t_{k+1}).$ (51)

Then the control law can be written as

$\displaystyle u_{t}=K\hat{x}_{t},\ \forall t\in[t_{k},t_{k+1}),$ (52)

and the system (48) can be written as

$\displaystyle x_{t+1}=(A+BK)x_{t}+BKe_{t}+Ew_{t}.$ (53)

The rest of this section considers the event-triggering mechanism in the form
of

$\displaystyle t_{k+1}=\min\\{t>t_{k}:\phi(x_{t},\hat{x}_{t})>\sigma\\},\
t_{0}=0,$ (54)

where the triggering function $\phi$ and the triggering threshold $\sigma$ are
to be designed. Note that such an event-trigger condition guarantees that
$\phi(x_{t},\hat{x}_{t})\leq\sigma$ for all $t\in\mathbb{Z}_{\geq 0}$.

We consider static event-triggered control strategies that use a constant
error threshold $\sigma$.

###  V-B Ultimate Boundedness

Event-triggered control strategies that guarantee ultimate boundedness are
followed from Theorem IV.4.

###### Corollary V.1

Suppose $r>0$ is chosen to satisfy

$\displaystyle r^{2}>\frac{1}{\varepsilon}\text{Tr}(P),$ (55)

where $P\succ 0$ is the solution to the Lyapunov equation

$\displaystyle(A+BK)P(A+BK)^{\top}-P+E\Sigma_{w}E^{\top}=0.$ (56)

Then, the use of the event-triggered condition

$\displaystyle\phi(x_{t},\hat{x}_{t})=\|\hat{x}_{t}-x_{t}\|=\|e_{t}\|>\sigma_{1}$
(57)

in (54) guarantees ultimate boundedness with (49) if $\sigma_{1}$ satisfies

$\displaystyle\sigma_{1}\leq\frac{1-\|A+BK\|}{\|BK\|}\left(r-\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}\right).$
(58)

###### Proof:

By replacing $A$, $D$, $v_{t}$ in (25) and $d$ in (26) by $A+BK$, $BK$,
$e_{t}$ in (48) and $\sigma_{1}$ in (54), respectively, the result follows
from Theorem IV.4. ∎

A similar result can be obtained using the error threshold on the control
input error:

###### Corollary V.2

Suppose $r>0$ is chosen to satisfy the condition (55). The use of the static
event-triggered condition

$\displaystyle\phi(x_{t},\hat{x}_{t})=\|K(\hat{x}_{t}-x_{t})\|=\|Ke_{t}\|>\sigma_{2}$
(59)

with

$\displaystyle\sigma_{2}\leq\frac{1-\|A+BK\|}{\|B\|}\left(r-\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}\right).$
(60)

in (54) guarantees ultimate boundedness with (49).

###### Proof:

Similarly to Corollary V.1, replace $A$, $D$, $v_{t}$ in (25) and $d$ in (26)
by $A+BK$, $B$, $Ke_{t}$ in (48) and $\sigma_{2}$ in (54), respectively, the
result follows from Theorem IV.4. ∎

###  V-C Positive Invariance

Event-triggered control strategies that guarantee positive invariance are
followed from Theorem IV.6.

###### Corollary V.3

Suppose $r>0$ is chosen to satisfy

$\displaystyle
r^{2}>\frac{1}{(1-\|A+BK\|)^{2}}\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E).$
(61)

Then, the use of the event-triggered condition

$\displaystyle\phi(x_{t},\hat{x}_{t})=\|\hat{x}_{t}-x_{t}\|=\|e_{t}\|>\sigma_{3}$
(62)

in (54) guarantees positive invariance with (49) if $\sigma_{3}$ satisfies

$\displaystyle\sigma_{3}\leq\frac{1}{\|BK\|}\left((1-\|A+BK\|)r-\sqrt{\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)}\right).$
(63)

###### Proof:

Follows from Theorem IV.6. ∎

A similar result can be obtained using the error threshold on the control
input error:

###### Corollary V.4

Suppose $r>0$ is chosen to satisfy the condition (61). The use of the static
event-triggered function

$\displaystyle\phi(x_{t},\hat{x}_{t})=\|K(\hat{x}_{t}-x_{t})\|=\|Ke_{t}\|>\sigma_{4}\|x_{t}\|$
(64)

with

$\displaystyle\sigma_{4}\leq\frac{1}{\|B\|}\left((1-\|A+BK\|)r-\sqrt{\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)}\right).$
(65)

in (54) guarantees positive invariance with (49).

###### Proof:

Follows from Theorem IV.6. ∎

###  V-D Numerical Examples

Here we observe the performances of the event-triggered controllers in the
previous subsections using numerical examples.

Consider the system (48) with

$\displaystyle\begin{aligned} A&=\begin{bmatrix}1.2&0.3\\\
0&0.5\end{bmatrix},\ B=\begin{bmatrix}1\\\ 0.5\end{bmatrix},\
E=\begin{bmatrix}1&2\\\ 0.5&-0.5\end{bmatrix},\\\
x_{0}&=\begin{bmatrix}2&3\end{bmatrix}^{\top}\end{aligned}$ (66)

subject to the zero-mean Gaussian disturbance with the covariance

$\displaystyle\Sigma_{w}=\begin{bmatrix}0.5&0\\\ 0&0.25\end{bmatrix},$ (67)

and the state feedback gain

$\displaystyle K=\begin{bmatrix}-0.7&-0.2\end{bmatrix}.$ (68)

Choose $\varepsilon=0.3$ to compute the worst-case CVaR.

To see the performances of the controllers with Corollaries V.1 and V.2,
choose $r=6>\sqrt{\frac{1}{\varepsilon}\text{Tr}(P)}=2.94$ and
$\sigma_{1}=1.36$, $\sigma_{2}=0.99$ which satisfy (58) and (60) with
equalities, respectively. With those parameters, the event-triggered control
performances and control inputs as well as those of a periodic controller
(i.e., standard state feedback controller that updates the control input all
the time) are shown in Figure 2. It is observed that the number of the control
input updates were reduced to 27 and 25 during 60 time-steps, respectively,
while $\|x_{t}\|^{2}$ is always smaller than $15<6^{2}=36$, thus
$\sup_{\mathbb{P}\in\mathcal{P}_{t}}\mathbb{P}\text{-CVaR}_{\varepsilon}[\|x_{t}\|^{2}-r^{2}]\leq
0$ for sufficiently large $t$ is achieved.

Figure 2: Event-triggered controller performances and control inputs (ultimate
boundedness)

To see the performances of the controllers with Corollaries V.3 and V.4,
choose
$r=10>\sqrt{\frac{1}{(1-\|A+BK\|)^{2}}\frac{1}{\varepsilon}\text{Tr}(\Sigma_{w}E^{\top}E)}=6.54$
and $\sigma_{3}=1.52$, $\sigma_{4}=1.11$ which satisfy (63) and (65) with
equalities, respectively. With those parameters, the event-triggered control
performances and control inputs as well as those of a periodic controller are
shown in Figure 3. The number of control input updates were 24 and 23 during
60 time-steps, respectively.

Figure 3: Event-triggered controller performances and control inputs (positive
invariance)

In all cases, it is observed that the number of control inputs achieved a 50%
reduction while achieving the objectives. Those reductions are heavily
dependent on the threshold $\sigma$. Large $r$ and $\varepsilon$, and a small
$\|A+BK\|$ help to reduce the number of updates.

##  VI Conclusion

This paper introduced the novel concepts of stability, ultimate boundedness,
and positive invariance for stochastic systems using the worst-case
Conditional Value-at-Risk (CVaR) to quantify the tail behavior of the
stochastic systems. These notions extend the stochastic correspondences and
allow us to consider risk in the decision-making processes. The introduced
notions are used to design event-triggered controllers and their performances
were illustrated using numerical examples.

## References

  * [1] H. M. Markowitz, _Portfolio selection_. Yale University Press, 1968.
  * [2] R. W. Klein and V. S. Bawa, “The effect of estimation risk on optimal portfolio choice,” _Journal of Financial Economics_ , vol. 3, no. 3, pp. 215–231, 1976.
  * [3] H. M. Markowitz and G. P. Todd, _Mean-variance analysis in portfolio choice and capital markets_. John Wiley & Sons, 2000, vol. 66.
  * [4] S. Zhu and M. Fukushima, “Worst-case conditional value-at-risk with application to robust portfolio management,” _Operations research_ , vol. 57, no. 5, pp. 1155–1168, 2009.
  * [5] C. A. Cornell, “Engineering seismic risk analysis,” _Bulletin of the seismological society of America_ , vol. 58, no. 5, pp. 1583–1606, 1968.
  * [6] P. Whittle, “Risk-sensitive linear/quadratic/Gaussian control,” _Advances in Applied Probability_ , vol. 13, no. 4, pp. 764–777, 1981.
  * [7] R. V. Whitman, “Evaluating calculated risk in geotechnical engineering,” _Journal of Geotechnical Engineering_ , vol. 110, no. 2, pp. 143–188, 1984\.
  * [8] F. Blanchini, “Set invariance in control,” _Automatica_ , vol. 35, no. 11, pp. 1747–1767, 1999.
  * [9] H. J. Kushner, “On the stability of stochastic dynamical systems,” _Proceedings of the National Academy of Sciences of the United States of America_ , vol. 53, no. 1, pp. 8–12, 1965.
  * [10] E. Kofman, J. A. De Doná, and M. M. Seron, “Probabilistic set invariance and ultimate boundedness,” _Automatica_ , vol. 48, no. 10, pp. 2670–2676, 2012.
  * [11] E. Kofman, J. A. De Doná, M. M. Seron, and N. Pizzi, “Continuous-time probabilistic ultimate bounds and invariant sets: Computation and assignment,” _Automatica_ , vol. 71, pp. 98–105, 2016.
  * [12] D. Kannan and V. Lakshmikantham, _Handbook of stochastic analysis and applications_. CRC Press, 2001.
  * [13] M. Zakai, “On the ultimate boundedness of moments associated with solutions of stochastic differential equations,” _SIAM Journal on Control_ , vol. 5, no. 4, pp. 588–593, 1967.
  * [14] A. Benzaouia, E. K. Boukas, and N. Daraoui, “Stability of continuous-time linear systems with Markovian jumping parameters and constrained control,” _IFAC Proceedings Volumes_ , vol. 35, no. 1, pp. 43–48, 2002.
  * [15] E. Boukas and A. Benzaouia, “Stability of discrete-time linear systems with Markovian jumping parameters and constrained control,” _IEEE Transactions on Automatic Control_ , vol. 47, no. 3, pp. 516–521, 2002.
  * [16] R. T. Rockafellar and S. Uryasev, “Optimization of conditional value-at-risk,” _Journal of Risk_ , vol. 2, pp. 21–41, 2000.
  * [17] D. Huang, S.-S. Zhu, F. J. Fabozzi, and M. Fukushima, “Portfolio selection with uncertain exit time: A robust cvar approach,” _Journal of Economic Dynamics and Control_ , vol. 32, no. 2, pp. 594–623, 2008.
  * [18] S. Zymler, D. Kuhn, and B. Rustem, “Distributionally robust joint chance constraints with second-order moment information,” _Mathematical Programming_ , vol. 137, pp. 167–198, 2013.
  * [19] A. Shapiro and A. Kleywegt, “Minimax analysis of stochastic problems,” _Optimization Methods and Software_ , vol. 17, no. 3, pp. 523–542, 2002.
  * [20] S. Zymler, D. Kuhn, and B. Rustem, “Worst-case value at risk of nonlinear portfolios,” _Management Science_ , vol. 59, no. 1, pp. 172–188, 2013.
  * [21] M. Kishida and A. Cetinkaya, “Risk-aware linear quadratic control using conditional value-at-risk,” _IEEE Transactions on Automatic Control_ , 2022\. [Online]. Available: https://doi.org/10.1109/TAC.2022.3142131
  * [22] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent measures of risk,” _Mathematical Finance_ , vol. 9, pp. 203–228., 1999.