# Data-Enabled Predictive Repetitive Control

Kai Zhang, Riccardo Zuliani, Efe C. Balta, and John Lygeros Research supported
by the Swiss National Science Foundation under NCCR Automation (grant
agreement 51NF40_180545). K. Zhang, R. Zuliani and J. Lygeros are with the
Automatic Control Laboratory (IfA), ETH Zurich, 8092 Zurich, Switzerland
<EMAIL_ADDRESS>E. C. Balta is with Inspire AG,
8092 Zurich, Switzerland<EMAIL_ADDRESS>also with IfA.

###### Abstract

This work introduces the Data-Enabled Predictive Repetitive Control (DeePRC)
algorithm, a direct data-driven approach for repetitive LTI systems. The
DeePRC learns from previous iterations to improve its performance and achieves
the optimal cost. By utilizing a tube-based variation of the DeePRC scheme, we
propose a two-stage approach that enables safe active exploration using a
left-kernel-based input disturbance design. This method generates informative
trajectories to enrich the historical data, which extends the maximum
achievable prediction horizon and leads to faster iteration convergence. In
addition, we present an end-to-end formulation of the two-stage approach,
integrating the disturbance design procedure into the planning phase. We
showcase the effectiveness of the proposed algorithms on a numerical
experiment.

##  I Introduction

_Direct data-driven control_ enables decision-making directly from raw data,
bypassing the need for a parametric model, which received significant
attention in the recent literature. The Fundamental Lemma [1] establishes that
all finite-length trajectories of a linear time-invariant (LTI) system belong
to the span of the Hankel matrix constructed using a suitably persistently
exciting input-output trajectory. The Fundamental Lemma has inspired numerous
novel controller designs in the framework of behavioral systems theory. For
example, [2] focuses on robustly stabilizing data-driven feedback control
design, while [3, 4] develop a predictive control scheme in a receding-horizon
manner. The celebrated data-enabled predictive control (DeePC) scheme of [3]
enables novel applications and theories in the field of direct data-driven
control. Later extensions of DeePC provide also closed-loop stability
guarantees by enforcing the last segment of the predicted trajectory to match
the target [4].

The DeePC scheme uses a Hankel matrix representation of the system dynamics,
which is assumed to satisfy a _persistency of excitation condition_ (PE
condition) [1]. The degree of excitation of the Hankel matrix limits the
maximum prediction horizon of the controller; hence, a lack of sufficiently
rich historical input-output data may produce short prediction horizons,
potentially leading to suboptimal performance. Our approach is to use the
measurement data obtained during system execution to improve the PE condition,
thus allowing for longer prediction horizons. This requires persistently
exciting inputs which need careful design to ensure exploration without
sacrificing performance. We address this problem in the context of repetitive
systems by providing a novel scheme with active exploration and theoretical
guarantees.

Repetitive systems perform iterative tasks, where each single task is referred
to as an iteration. The inherent “repetitive” nature of repetitive processes
enables the possibility of iteratively updating the controller using data
collected from each iteration. In [5], a Learning Model Predictive Control
(LMPC) algorithm for repetitive systems is proposed. Assuming perfect
knowledge of the system model, LMPC uses a safe set constructed from safe
trajectories as the terminal set, where the safe set is updated with the new
(safe) trajectory after each iteration. This strategy ensures asymptotic
stability and a non-increasing cost over iterations, ultimately converging to
the optimal cost under suitable assumptions. To improve robustness, [6]
extends LMPC to systems with additive uncertainties by leveraging tube-MPC
approaches [7]. In this case, robust safe sets are constructed using nominal
trajectories satisfying tightened constraints to ensure the nominal cost is
non-increasing over iterations.

In this paper, we present the Data-Enabled Predictive Repetitive Control
(DeePRC), a direct data-driven control approach for repetitive LTI systems.
The nominal DeePRC uses an input-output convex safe set and a terminal cost
function designed from previous trajectories, similar to the LMPC framework.
We show that the nominal DeePRC enjoys recursive feasibility, asymptotic
stability, non-increasing iteration cost, and convergence to the infinite-
horizon optimum, albeit asymptotically. To ensure global convergence in finite
iterations we develop an active exploration variant of DeePRC, which requires
_only a single initial safe trajectory_ to provide safe convergence guarantees
using a left-kernel-based input design [8] in a tube-based framework. We
further present an end-to-end formulation of the two-stage approach,
integrating the disturbance design procedure into the planning phase.

#### Notation

$[a,b]$ denotes the set of integers within the range from $a\in\mathbb{Z}$ to
$b\in\mathbb{Z}$ with $a\leq b$. For a vector $x$, we use $\|x\|_{p}$ to
denote its $p$-norm, and $\|x\|_{A}^{2}$ to denote $x^{\top}Ax$, where
$\textstyle{A=A^{\top}\succ 0}$ is a matrix. We use ${\rm
col}(x_{0},x_{1},...,x_{\ell})$ and ${\rm col}(A_{0},A_{1},...,A_{\ell})$ to
respectively denote vertically stacked vector
$[x_{0}^{\top}\;x_{1}^{\top}\;\cdots\;x_{\ell}^{\top}]^{\top}$ and matrix
$[A_{0}^{\top}\;A_{1}^{\top}\;\cdots\;A_{\ell}^{\top}]^{\top}$. For a matrix
$A$, $A_{[:m,:]}$ denotes the submatrix of its first $m$ rows, while
$A_{[m:,:]}$ represents the remaining rows and we use $A_{[-m:,:]}$ to denote
its last $m$ rows. ${\rm image}(A)$ and ${\rm leftker}(A)$ denote the image
and left kernel of matrix $A$, respectively. For a generic length-$N$ sequence
$\\{y_{k}\\}_{k=0}^{N-1}$, we write $y_{[a,b]}:={\rm
col}(y_{a},\ldots,y_{b})$.

##  II Preliminaries

###  II-A Problem Setting

Consider the unknown discrete-time LTI system

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bu_{t},$ (1) $\displaystyle
y_{t}$ $\displaystyle=Cx_{t}+Du_{t},$

where $x_{t}\in\mathbb{R}^{n}$ is the state, $u_{t}\in\mathbb{R}^{m}$ is the
input, and $y_{t}\in\mathbb{R}^{p}$ is the output. The state $x_{t}$ is not
directly measurable, and needs to be reconstructed using input-output
information. Denote by $\underline{\ell}\in\mathbb{N}_{>0}$ the lag of the
system, that is, the minimum integer such that the observability matrix
$\mathscr{O}_{\underline{\ell}}(A,C):={\rm
col}(C,CA,...,CA^{\underline{\ell}-1})$ is full rank. We assume that an upper
bound $\ell$ on the lag $\underline{\ell}$ is available.

To represent the state using input/output data, we define the extended state
following [9, Definition 3]:

$\xi_{t}:=\begin{bmatrix}u_{[t-\ell,t-1]}\\\
y_{[t-\ell,t-1]}\end{bmatrix}\in\mathbb{R}^{n_{\xi}},$ (2)

where $n_{\xi}=(m+p)\ell$. Here, $\xi_{t}$ is the state of a non-minimal
representation of (1) with dynamics

$\displaystyle\xi_{t+1}$ $\displaystyle=\tilde{A}\xi_{t}+\tilde{B}u_{t},$ (3)
$\displaystyle y_{t}$ $\displaystyle=\tilde{C}\xi_{t}+\tilde{D}u_{t},$

which is generally not unique. We refer to [9, Equation 6] for one possible
form of (3).

Suppose system (1) performs repetitive tasks, where the task for each
iteration is to steer the system from a starting state $\xi^{\rm S}$ to a
target state $\xi^{\rm F}$. With a slight abuse of notation, we denote the
input/output trajectories of the $j$-th iteration as $u^{j}={\rm
col}(u^{j}_{-\ell},\ldots,u^{j}_{-1},u^{j}_{0},\ldots,u^{j}_{T^{j}-1})$ and
$y^{j}={\rm
col}(y^{j}_{-\ell},\ldots,y^{j}_{-1},y^{j}_{0},\ldots,y^{j}_{T^{j}-1})$, where
$T^{j}\in\mathbb{N}_{>0}$ and $u^{j}_{t}=u^{\rm S}$, $y^{j}_{t}=y^{\rm S}$,
$\forall t\leq 0$ describe the initial state for the associated extended state
sequence $\\{\xi^{j}_{t}\\}_{t=0}^{T^{j}}$, with $\xi^{j}_{0}=\xi^{\rm S}$.

A trajectory is considered safe if it converges to the target state
$\xi^{j}_{T^{j}}=\xi^{\rm F}:=\operatorname{col}(u^{\rm F},\dots,u^{\rm
F},y^{\rm F},\dots,y^{\rm F})$ and satisfies the input and output constraints

$\displaystyle u^{j}_{t}\in\mathcal{U},~{}y^{j}_{t}\in\mathcal{Y},$ (4)

for all $t\in\mathbb{Z}_{[0,T^{j}-1]}$, where
$\mathcal{U}\subseteq\mathbb{R}^{m}$ and $\mathcal{Y}\subseteq\mathbb{R}^{p}$
are convex sets. The control objective is to find an input sequence that
solves the following infinite horizon problem.

$\displaystyle J_{\infty}^{*}(\xi^{\rm S})=\min_{u_{0},u_{1},\ldots}\quad$
$\displaystyle\sum_{k=0}^{\infty}{h(y_{k},u_{k})}$ (5a) $\displaystyle{\rm
s.t.}\quad~{}\,$ $\displaystyle\xi_{k+1}=\tilde{A}\xi_{k}+\tilde{B}u_{k},\
\forall k\geq 0,$ (5b) $\displaystyle y_{k}=\tilde{C}\xi_{k}+\tilde{D}u_{k},\
\forall k\geq 0,$ (5c) $\displaystyle\xi_{0}=\xi^{\rm S},$ (5d) $\displaystyle
u_{k}\in\mathcal{U},\ y_{k}\in\mathcal{Y},\ \forall k\geq 0.$ (5e)

The stage cost is $h(u_{k},y_{k})=\|u_{k}-u^{\rm F}\|_{R}^{2}+\|y_{k}-y^{\rm
F}\|_{Q}^{2}$ with $R\succ 0$ and $Q\succeq 0$.

###### Assumption 1

System (1) is observable and stabilizable, $(Q^{1/2},\tilde{A})$ is
detectable, and (5) admits a finite cost solution. The sets $\mathcal{U}$ and
$\mathcal{Y}$ are convex.

Problem (5) is not tractable due to the infinite horizon and the lack of
knowledge of the system dynamics. We develop the DeePRC algorithm as a
tractable solution to (5). Our scheme uses a data-based non-parametric model
for (5b)-(5c) to perform receding horizon control.

###  II-B Review of DeePC

Here, we provide a brief background on the Data Enabled Predictive Control
(DeePC). Let $T,L\in\mathbb{Z}_{\geq 0}$ with $T\geq L>\ell$. The Hankel
matrix of depth $L$ for a signal $u={\rm col}(u_{0},\ldots,u_{T})$ is defined
as

$\mathscr{H}_{L}(u):=\begin{bmatrix}u_{0}&u_{1}&\cdots&u_{T-L}\\\
u_{1}&u_{2}&\cdots&u_{T-L+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
u_{L-1}&u_{L}&\cdots&u_{T-1}\end{bmatrix}.$

We use the identifiability condition presented in [10] for theoretical
guarantees, which is a slight generalization of the Fundamental Lemma [1].

######  Lemma 1 ([10, Corollary 21])

Consider a length $T$ input/output trajectory $\\{u^{\rm d},y^{\rm d}\\}$ of
system (1). The image of $\mathscr{H}_{L}(u^{\rm d},y^{\rm d}):={\rm
col}(\mathscr{H}_{L}(u^{\rm d}),\mathscr{H}_{L}(y^{\rm d}))$ spans all length
$L$ trajectories of the system if and only if

${\rm rank}(\mathscr{H}_{L}(u^{\rm d},y^{\rm d}))=mL+n.$ (6)

This result can be used for direct data-driven prediction. Given an initial
horizon $T_{\rm ini}\in\mathbb{N}_{>0}$ and a prediction horizon
$N\in\mathbb{N}_{>0}$, let

$\begin{bmatrix}U_{\rm P}\\\ U_{\rm F}\end{bmatrix}:=\mathscr{H}_{T_{\rm
ini}+N}(u^{\rm d}),\quad\begin{bmatrix}Y_{\rm P}\\\ Y_{\rm
F}\end{bmatrix}:=\mathscr{H}_{T_{\rm ini}+N}(y^{\rm d}),$ (7)

where $U_{\rm P}~{}\in~{}\mathbb{R}^{mT_{\rm ini}\times H}$, $U_{\rm
F}~{}\in~{}\mathbb{R}^{mN\times H}$, $Y_{\rm P}~{}\in~{}\mathbb{R}^{pT_{\rm
ini}\times H}$, $Y_{\rm F}~{}\in~{}\mathbb{R}^{pN\times H}$ with $H=T-T_{\rm
ini}-N+1$. Suppose that $(u^{\text{d}},y^{\text{d}})$ satisfy the rank
condition (6) with $L=T_{\rm ini}+N$. According to Lemma 1, $\\{{\rm
col}(u_{\rm ini},u),{\rm col}(y_{\rm ini},y)\\}$ is a valid trajectory of the
system if and only if $\exists g\in\mathbb{R}^{H}$ such that

$\begin{bmatrix}U_{\rm P}\\\ Y_{\rm P}\\\ U_{\rm F}\\\ Y_{\rm
F}\end{bmatrix}g=\begin{bmatrix}u_{\rm ini}\\\ y_{\rm ini}\\\ u\\\
y\end{bmatrix},$ (8)

where $u_{\rm ini}\in\mathbb{R}^{mT_{\rm ini}}$ and $y_{\rm
ini}\in\mathbb{R}^{pT_{\rm ini}}$ are the $T_{\rm ini}$-step past input-output
observations. If $T_{\rm ini}\geq\underline{\ell}$, given any future control
input sequence $u\in\mathbb{R}^{mN}$, the associated output sequence
$y\in\mathbb{R}^{pN}$ is uniquely determined via (8) [11]. In the remainder of
this paper, we assume $T_{\text{ini}}=\ell$.

##  III Nominal DeePRC

In this section, we introduce the nominal DeePRC algorithm and exploration
schemes using past trajectory data.

###  III-A Input-Output Convex Safe Set

The input-output convex safe set ${\rm io}\mathcal{CS}^{j}$ at iteration $j$
is defined as the convex hull of all _extended states_ along previous safe
trajectories:

${\rm io}\mathcal{CS}^{j}:=\left\\{\sum_{i\in
M^{j}}{\sum_{t=0}^{T^{i}}{\gamma_{t}^{i}\xi^{i}_{t}}}\mid\gamma_{t}^{i}\geq
0,\sum_{i\in M^{j}}{\sum_{t=0}^{T^{i}}{\gamma_{t}^{i}}}=1\right\\},$ (9)

where $M^{j}:=\\{i\in\mathbb{Z}_{[0,j-1]}\mid\xi^{i}_{T^{i}}=\xi^{\rm
F},u_{t}^{i}\in\mathcal{U},y_{t}^{i}\in\mathcal{Y}\\}$ indicates past safe
iterations. For all $i\leq j$, $M^{i}\subseteq M^{j}$, and therefore ${\rm
io}\mathcal{CS}^{i}\subseteq{\rm io}\mathcal{CS}^{j}$. Every $\xi\in{\rm
io}\mathcal{CS}^{j}$ is safe, such that it can be steered to $\xi^{\rm F}$
without violating the constraints. We define the cost-to-go of the extended
state $\xi_{t}^{i}$ as

$J^{i}(\xi^{i}_{t})=\left\\{\begin{array}[]{ll}\sum_{k=t}^{T^{i}-1}{h(u_{k}^{i},y_{k}^{i})},&t<T^{i},\\\
0,&t=T^{i},\end{array}\right.$

representing the cost required to steer $\xi_{t}^{i}$ to $\xi^{\rm F}$. Note
that $J^{i}(\xi_{0}^{i})=J^{i}(\xi^{\rm S})$ is the cost of the entire
trajectory $i$. The cost-to-go can be extended to any $\xi\in{\rm
io}\mathcal{CS}^{j}$, by defining

$\displaystyle P^{j}(\xi)=\min_{\gamma\geq 0}$ $\displaystyle\sum_{i\in
M^{j}}{\sum_{t=0}^{T^{i}}{\gamma_{t}^{i}J^{i}(\xi_{t}^{i})}}$ (10)
$\displaystyle{\rm s.t.}$ $\displaystyle\sum_{i\in
M^{j}}{\sum_{t=0}^{T^{i}}{\gamma_{t}^{i}}}=1,\ \sum_{i\in
M^{j}}{\sum_{t=0}^{T^{i}}{\gamma_{t}^{i}\xi^{i}_{t}}}=\xi.$

Note that $P^{j}(\xi)$ represents the cost required to safely steer $\xi$ to
$\xi^{\rm F}$ under the control policy $\pi(\xi)=\sum_{i\in
M^{j}}{\sum_{t=0}^{T^{i}}{\gamma_{t}^{i,*}u_{t}^{i}}}$, where $\gamma^{*}$ is
the minimizer of (10).

###### Theorem 2

The set ${\rm io}\mathcal{CS}^{j}$ is control invariant [12] for all $j$. For
any $\xi\in{\rm io}\mathcal{CS}^{j}$, $P^{j}(\xi^{+})\leq
P^{j}(\xi)-h(\bar{u},\bar{y})$, $\bar{u}\in\mathcal{U}$ and
$\bar{y}\in\mathcal{Y}$, where $\xi^{+}=\tilde{A}\xi+\tilde{B}\bar{u}$,
$\bar{y}=\tilde{C}\xi+\tilde{D}\bar{u}$, and $\textstyle{\bar{u}=\pi(\xi)}$.

###### Proof:

For any $\xi\in{\rm io}\mathcal{CS}^{j}$ it holds that

$\displaystyle\xi^{+}$
$\displaystyle=\tilde{A}\xi+\tilde{B}\bar{u}=\textstyle\sum_{i\in
M^{j}}{\textstyle\sum_{t=0}^{T^{i}}{\gamma_{t}^{i,*}\left(\tilde{A}\xi^{i}_{t}+\tilde{B}u^{i}_{t}\right)}}$
$\displaystyle=\textstyle\sum_{i\in
M^{j}}{\textstyle\sum_{t=0}^{T^{i}-1}{\gamma_{t}^{i,*}\xi^{i}_{t+1}}+\gamma_{T^{i}}^{i,*}\xi^{\rm
F}}\in{\rm io}\mathcal{CS}^{j},$

where $\gamma^{*}$ is the minimizer of (10). This proves that ${\rm
io}\mathcal{CS}^{j}$ is control invariant. Next, from (10) we have

$\displaystyle P^{j}(\xi^{+})$ $\displaystyle\leq\textstyle\sum_{i\in
M^{j}}{\textstyle\sum_{t=0}^{T^{i}-1}{\gamma_{t}^{i,*}J^{i}(\xi_{t+1}^{i})}}$
$\displaystyle=\textstyle\sum_{i\in
M^{j}}{\textstyle\sum_{t=0}^{T^{i}-1}{\gamma_{t}^{i,*}\left(J^{i}(\xi_{t}^{i})-h(u_{t}^{i},y_{t}^{i})\right)}}$
$\displaystyle\leq P^{j}(\xi)-h(\bar{u},\bar{y}).$

Also, due to the convexity of constraints and safety of the trajectories in
${\rm io}\mathcal{CS}^{j}$, we have $\bar{u}\in\mathcal{U}$ and
$\bar{y}\in\mathcal{Y}$. ∎

###  III-B The Nominal DeePRC Algorithm

The nominal DeePRC algorithm utilizes a single safe trajectory
$\\{u^{0},y^{0}\\}$ to construct both the Hankel matrix and the safe set ${\rm
io}\mathcal{CS}^{1}$.

###### Assumption 2

A safe trajectory $\\{u^{0},y^{0}\\}$ satisfying the rank condition (6) with
$\textstyle{L=T_{\rm ini}\\!+\\!N}$ and $\textstyle{N\geq\ell}$, is available.

The nominal DeePRC for time-step $t$ and iteration $j$ is formulated as
follows:

$\displaystyle
J^{j,*}_{N}(\xi_{t}^{j})\\!=\\!\operatorname*{min}_{g(t),u(t),y(t)}J_{N}^{j}(u(t),y(t),\xi_{N}(t))$
(11a) $\displaystyle{\rm s.t.}~{}\begin{bmatrix}U_{\rm P}\\\ Y_{\rm P}\\\
U_{\rm F}\\\ Y_{\rm F}\end{bmatrix}g(t)=\begin{bmatrix}u_{\rm ini}(t)\\\
y_{\rm ini}(t)\\\ u(t)\\\ y(t)\end{bmatrix},~{}\begin{bmatrix}u_{\rm
ini}(t)\\\ y_{\rm ini}(t)\end{bmatrix}=\xi^{j}_{t},$ (11b)
$\displaystyle\hphantom{\rm
s.t.~{}}\xi_{N}(t)=\begin{bmatrix}u_{[N-\ell,N-1]}(t)\\\
y_{[N-\ell,N-1]}(t)\end{bmatrix},~{}\xi_{N}(t)\in{\rm io}\mathcal{CS}^{j},$
(11c) $\displaystyle\hphantom{\rm
s.t.~{}}u_{k}(t)\in\mathcal{U},~{}y_{k}(t)\in\mathcal{Y},~{}\forall
k\in[0,N-1],$ (11d)

with
$\operatorname{col}(U_{P},Y_{P},U_{F},Y_{F})=\mathscr{H}_{T_{\text{ini}}+N}(u^{0},y^{0})$,
$u(t)\in\mathbb{R}^{mN}$, $y(t)\in\mathbb{R}^{pN}$, $g(t)\in\mathbb{R}^{H}$,
and

$\displaystyle
J_{N}^{j}(u(t),y(t),\xi_{N}(t)):=\sum_{k=0}^{N-1}{h(u_{k}(t),y_{k}(t))}+P^{j}(\xi_{N}(t)).$

###  III-C Hankel Matrix Update

The DeePRC (11) utilizes a static Hankel matrix. As new iterations
$\\{u^{j},y^{j}\\}$ are measured, we can augment the prediction horizon by
modifying the Hankel matrix. At iteration $j$, the largest feasible prediction
horizon is

$N^{j}=\max\left\\{N\,\middle|\,{\rm
rank}(\mathcal{H}^{j}_{N})=m(\ell+N)+n\right\\},$ (12)

with
$\mathcal{H}^{j}_{N}=\begin{bmatrix}\mathscr{H}_{\ell+N}(u^{0},y^{0})&\ldots&\mathscr{H}_{\ell+N}(u^{j-1},y^{j-1})\end{bmatrix}$.
Suppose our goal is to reach $\bar{N}\geq\ell$, requiring
$\mathcal{H}_{\bar{N}}^{j}$ to satisfy the rank condition (6) with
$L=\bar{L}:=\ell+\bar{N}$. If $\bar{N}$ has not been achieved at iteration
$j$, i.e., if $\bar{N}>N^{j}$, then we update $\mathcal{H}^{j}_{\bar{N}}$ with
the length-$\bar{L}$ trajectory
$\\{u^{j}_{[t-\bar{L}+1,t]},y^{j}_{[t-\bar{L}+1,t]}\\}$ for all
$t\in\mathbb{Z}_{[\bar{N}-1,T^{j}-1]}$ via

$\mathcal{H}_{\bar{N}}^{j,t+1}=\begin{bmatrix}\mathcal{H}_{\bar{N}}^{j,t}&\begin{bmatrix}u^{j}_{[t-\bar{L}+1,t]}\\\
y^{j}_{[t-\bar{L}+1,t]}\end{bmatrix}\end{bmatrix},$ (13)

with $\mathcal{H}_{\bar{N}}^{j,N^{j}-1}\\!=\mathcal{H}_{\bar{N}}^{j}$,
$\mathcal{H}_{\bar{N}}^{j+1}\\!=\mathcal{H}_{\bar{N}}^{j,T^{j}}$. The process
continues until ${\rm
rank}(\mathcal{H}_{\bar{N}}^{\bar{j},\bar{t}})=m\bar{L}+n$ for some $\bar{j}$
and $\bar{t}$, after which we initialize (11) with $\bar{N}$ and
$\mathcal{H}_{\bar{N}}^{\bar{j},\bar{t}}$. For any $j<\bar{j}$, we solve (11)
with prediction horizon $N^{j}$ and matrix $\mathcal{H}_{N^{j}}^{j}$.

##  IV Safe and Active Exploration

The update (13) can be inefficient, as new trajectories are not guaranteed to
increase the rank condition and extend the prediction horizon. In this
section, we present an _active exploration_ strategy to safely reach the
desired prediction horizon $\bar{N}$. We design the control input as the sum
of two components $u_{t}=\tilde{u}_{t}+d_{t}$, where $\tilde{u}_{t}$ is chosen
to optimize closed-loop performance, and $d_{t}$ is an artificial input
disturbance ensuring active exploration.

###  IV-A Safe Exploration: Tube DeePRC

To ensure safety, we assume $d_{t}$ is chosen from a known, user-designed
bounded set $\mathcal{D}$, and develop a tube-based MPC technique to ensure
robust constraint satisfaction. We consider the nominal system

$\displaystyle\zeta_{t+1}$ $\displaystyle=\tilde{A}\zeta_{t}+\tilde{B}v_{t}$
$\displaystyle z_{t}$ $\displaystyle=\tilde{C}\zeta_{t}+\tilde{D}v_{t}.$

where $z_{t}$, $v_{t}$, and $\zeta_{t}$ are the nominal output, input, and
extended state, respectively. The error is defined as
$e_{t}=\xi_{t}-\zeta_{t}$. Thanks to [11, Lemma 1], we can obtain
$(\tilde{A},\tilde{B})$ from (8) with $N=1$ as

$\xi_{t+1}\\!=\\!\begin{bmatrix}{\rm col}(U_{{\rm P}[m:,:]},U_{\rm F})\\\ {\rm
col}(Y_{{\rm P}[p:,:]},Y_{\rm F})\end{bmatrix}\begin{bmatrix}U_{\rm P}\\\
Y_{\rm P}\\\ U_{\rm F}\end{bmatrix}^{\dagger}\begin{bmatrix}u_{\rm ini}(t)\\\
y_{\rm ini}(t)\\\ u(t)\end{bmatrix}\\!=\\!\tilde{A}\xi_{t}+\tilde{B}u_{t}.$

###### Assumption 3

The pair $(\tilde{A},\tilde{B})$ is stabilizable.

Sufficient conditions for the stabilizability of $(\tilde{A},\tilde{B})$ have
been studied e.g. in [2, Theorem 3]. Under Assumption 3 we can design a
stabilizing feedback controller $K\in\mathbb{R}^{m\times n_{\xi}}$ and an
associated robust positive invariant set $\mathcal{E}$ (either polytopic [13]
or ellipsoidal [14]) such that

$e_{t+1}=(\tilde{A}+\tilde{B}K)e_{t}+\tilde{B}d_{t}\in\mathcal{E},\ \forall
e_{t}\in\mathcal{E},\ \forall d_{t}\in\mathcal{D}.$ (14)

We use the set $\mathcal{E}$ to tighten the constraints
$\bar{\mathcal{U}}=\mathcal{U}\ominus\mathcal{E}_{u}$ and
$\bar{\mathcal{Y}}=\mathcal{Y}\ominus\mathcal{E}_{y}$, where
$\mathcal{E}_{u}=\\{e_{u}\in\mathbb{R}^{m}\mid\exists e\in\mathcal{E},\ {\rm
s.t.}\ e_{u}=T_{u}e\\}$ and
$\mathcal{E}_{y}=\\{e_{y}\in\mathbb{R}^{p}\mid\exists e\in\mathcal{E},\ {\rm
s.t.}\ e_{y}=T_{y}e\\}$, with $T_{u}$ and $T_{y}$ selecting the entries
associated to the last input and the last output in $e$, respectively.

The Tube DeePRC problem can be obtained by solving (11) with nominal dynamics
and tightened constraints.

$\displaystyle\min_{g(t),v(t),z(t)}$
$\displaystyle~{}J^{j}_{N}(v(t),z(t),\zeta_{N}(t))$ (15a) $\displaystyle{\rm
s.t.}\quad~{}$
$\displaystyle\left(g(t),v(t),z(t),\zeta_{N}(t)\right)\in\mathcal{C},$ (15b)
$\displaystyle v_{k}(t)\in\bar{\mathcal{U}},\ z_{k}(t)\in\bar{\mathcal{Y}},\
\forall k\in[0,N-1],$ (15c)

where $\mathcal{C}$ denotes the constraints (11b)-(11c) with $(u,y,\xi)$
replaced with $(v,z,\zeta)$. We then choose

$\tilde{u}^{j}_{t}=K(\xi^{j}_{t}-\zeta^{j}_{t})+v_{0}^{*}(t).$ (16)

For safe planning, any element in the ${\rm io}\mathcal{CS}$ of problem (1) is
required to meet the tightened constraints
$\bar{\mathcal{U}},\bar{\mathcal{Y}}$. As in [6], we need to strengthen
Assumption 2 as follows.

###### Assumption 4

A robustly safe trajectory $\\{v^{0},z^{0}\\}$ (i.e., with
$v^{0}_{t}\in\bar{\mathcal{U}},z^{0}_{t}\in\bar{\mathcal{Y}},\forall
t\in\mathbb{Z}_{\geq 0}$) satisfying the rank condition (6) with $L=T_{\rm
ini}+N$ and $N\geq\ell$, is available.

To ensure robust constraint satisfaction, we construct the ${\rm
io}\mathcal{CS}$ in (15) using the nominal trajectories $\\{v^{j},z^{j}\\}$.

###  IV-B Active Exploration: LKB Input Disturbance Design

Since $\tilde{u}_{t}^{j}$ in (16) satisfies the nominal constraints for all
$d_{t}^{j}\in\mathcal{D}$, we can design $d_{t}^{j}$ to ensure the Hankel
matrix update (13) always produces an increment in the rank of the matrix.

Choosing $d_{t}^{j}$ independently and uniformly from $\mathcal{D}$ at each
time step may be inefficient, since the control input $\tilde{u}_{t}^{j}$ may
already contain sufficient information to excite the system. We propose a
left-kernel-based (LKB) disturbance design inspired by [8] as a sample-
efficient exploration strategy.

For the first $N^{j}-2$ time-steps of iteration $j$ we set $d_{t}^{j}=0$.
Next, for $t\geq N^{j}-1$, let
$\operatorname{col}(\bar{U}^{j,t},\bar{Y}^{j,t}):=\mathcal{H}_{\bar{N}}^{j,t}$,
where $\bar{U}^{j,t}\in\mathbb{R}^{m\bar{L}\times H_{c}}$,
$\bar{Y}^{j,t}\in\mathbb{R}^{p\bar{L}\times H_{c}}$ with
$H_{c}=t-\bar{N}+1+\sum_{i=0}^{j-1}{\left(T^{i}-\bar{N}+1\right)}$, and let
$\bar{Y}^{j,t}_{\rm up}=\bar{Y}^{j,t}_{[:p(\bar{L}-1),:]}$. If

$\begin{bmatrix}y^{j}_{[t-\bar{L}+1,t-1]}\\\ u^{j}_{[t-\bar{L}+1,t-1]}\\\
\tilde{u}^{j}_{t}\end{bmatrix}\notin{\rm
image}\begin{bmatrix}\bar{Y}^{j,t}_{\rm up}\\\ \bar{U}^{j,t}\end{bmatrix},$
(17)

then the candidate control input $\tilde{u}^{j}_{t}$ will excite the
trajectory and thus we set $d^{j}_{t}=0$. Otherwise, there exists some
$\kappa\in\mathbb{R}^{p(\bar{L}-1)+m\bar{L}}$ in the left kernel of
$\operatorname{col}(\bar{Y}^{j,t}_{\rm up},\bar{U}^{j,t})$ with
$\kappa_{[-m:]}\neq 0$ satisfying

$\exists
d\in\mathcal{D},\quad\kappa^{\top}\begin{bmatrix}y^{j}_{[t-\bar{L}+1,t-1]}\\\
u^{j}_{[t-\bar{L}+1,t-1]}\\\ \tilde{u}^{j}_{t}+d\end{bmatrix}\neq 0,$ (18)

where $\kappa_{[-m:]}$ denotes the last $m$ elements of $\kappa$ [8]. If
$\mathcal{D}$ is absorbing (e.g. any norm ball), there always exists some
$d_{t}^{j}$ satisfying (18). We then apply the input
$u_{t}^{j}=\tilde{u}_{t}^{j}+d_{t}^{j}$ with $\tilde{u}_{t}^{j}$ as in (16).
This ensures that ${\rm rank}(\mathcal{H}_{\bar{N}}^{j,t+1})={\rm
rank}(\mathcal{H}_{\bar{N}}^{j,t})+1$ for all $t\geq N^{j}-1$, meaning that
the desired horizon $\bar{N}$ can be reached in exactly $m\bar{L}+n-{\rm
rank}(\mathcal{H}^{1}_{\bar{N}})$ time steps. Afterwards, $d_{t}^{j}\equiv 0$
and the nominal DeePRC (11) can be used instead of (15). The DeePRC scheme
with LKB active exploration is summarized in Algorithm 1.

Algorithm 1 DeePRC with LKB Active Exploration

1: Init: $\\{u^{0},y^{0}\\}$, $\xi^{\rm S}$, $\xi^{\rm F}$, $\bar{N}\geq\ell$,
$\mathcal{D}$, ${\rm io}\mathcal{CS}^{1}$, $j=1$.

2: while $y^{j}\neq y^{j-1}$ do

3: $t=0$, $\xi^{j}_{0}=\xi^{\rm S}$, $\zeta^{j}_{0}=\xi^{\rm S}$.

4: if $\bar{N}$ has been reached then

5: Setup (11) using $\mathcal{H}_{\bar{N}}^{\bar{j},\bar{t}}$.

6: while not converged to $\xi^{\rm F}$ do

7: Solve (11) and apply $u_{t}^{j}$. Get $y_{t}^{j}$ and $\xi_{t+1}^{j}$.

8: end while

9: ${\rm io}\mathcal{CS}^{j+1}={\rm io}\mathcal{CS}^{j}\cup\\{u^{j},y^{j}\\}$.

10: else

11: Compute $N^{j}$ using (12).

12: Setup (15) using $\mathcal{H}_{N_{j}}^{j}$.

13: while not converged to $\xi^{\rm F}$ do

14: Solve (15) and get $\tilde{u}_{t}^{j}$ from (16).

15: if $t\leq\bar{N}\\!-\\!2$ or (17) holds then

16: $d^{j}_{t}=0$.

17: else

18: Select any $d^{j}_{t}$ satisfying (18).

19: Update $\mathcal{H}_{\bar{N}}^{j,t}$ with (13).

20: end if

21: Apply $u^{j}_{t}=\tilde{u}^{j}_{t}+d^{j}_{t}$. Get $y_{t}^{j}$ and
$\xi_{t+1}^{j}$.

22: end while

23: ${\rm io}\mathcal{CS}^{j+1}={\rm
io}\mathcal{CS}^{j}\cup\\{v^{j},z^{j}\\}$.

24: end if

25: $j\leftarrow j+1$.

26:end while

###  IV-C An End-to-End Formulation

Algorithm 1 presents a two-stage approach for designing the control input. We
also provide an end-to-end formulation, which integrates the online
disturbance design procedure into the optimization problem

$\displaystyle\min_{g(t),v(t),z(t),d(t)}J_{N}^{j}(v(t),z(t),\zeta_{N}(t))\\!+\\!\lambda\|d(t)\|_{1}$
(19a) $\displaystyle\qquad~{}~{}{\rm
s.t.}~{}~{}\left(g(t),v(t),z(t),\zeta_{N}(t)\right)\in\mathcal{C},$ (19b)
$\displaystyle\hphantom{\qquad~{}~{}{\rm
s.t.}~{}~{}}v_{k}(t)\in\bar{\mathcal{U}},\ z_{k}(t)\in\bar{\mathcal{Y}},\
\forall k\in[0,N-1],$ (19c) $\displaystyle\hphantom{\qquad~{}~{}{\rm
s.t.}~{}~{}}u_{0}(t)=v_{0}(t)+K(\xi^{j}_{t}-\zeta^{j}_{t}),$ (19d)
$\displaystyle\hphantom{\qquad~{}~{}{\rm
s.t.}~{}~{}}\|\begin{bmatrix}y^{j}_{[t-\bar{L}+1,t-1]}\\\
u^{j}_{[t-\bar{L}+1,t-1]}\\\
u_{0}(t)+d(t)\end{bmatrix}^{\top}\\!\\!\\!\mathcal{K}^{j,t}\|_{\infty}\geq\epsilon,d(t)\in\mathcal{D}.$
(19e)

Condition (17) is here enforced through the excitation constraint (19e), where
$\epsilon\in\mathbb{R}_{>0}$ and $\mathcal{K}^{j,t}$ is an orthogonal basis of
$\operatorname{col}(\bar{Y}^{j,t}_{\rm up},\bar{U}^{j,t})$. We further add
$1$-norm penalty on the disturbance to ensure that its magnitude is kept to a
minimum. Unlike (15), which is a QP, (19) can be formulated as a mixed-integer
quadratic program (MIQP). The one-stage formulation should generally perform
better than the two-stage, as $d(t)$ is part of the optimization problem.
Similar to the two-stage approach, we only solve (19) until $\bar{N}$ is
reached, then we utilize (11).

##  V Properties and Guarantees

###  V-A Closed-Loop Properties

We first prove that the nominal DeePRC scheme (11) enjoys recursive
feasibility and stability.

###### Proposition 3

Under Assumption 2, the following hold for all $j\geq 1$: 1) Problem (11) is
feasible for all $t\in\mathbb{Z}_{\geq 0}$; 2) $(u^{j},y^{j})$ satisfy the
constraints $\mathcal{U}$, $\mathcal{Y}$; 3) $\xi_{t}^{j}$ asymptotically
converges to $\xi^{\rm F}$.

###### Proof:

Assumption 2 ensures $\xi^{0}_{\tau}\in{\rm
io}\mathcal{CS}^{j},\forall\tau\in[0,T^{0}-1]$. At $t=0$, as
$\xi^{j}_{0}=\xi^{\rm S}=\xi^{0}_{0}$, the length-$N$ trajectory
$\hat{u}(0)=u^{0}_{[0,N-1]}$, $\hat{y}(0)=y^{0}_{[0,N-1]}$ taken from
$\\{u^{0},y^{0}\\}$ is a feasible solution. Next, suppose
$u^{*}(t),y^{*}(t),g^{*}(t)$ is the solution of (11) at time step $t$. At time
step $t+1$, let $\hat{u}(t+1)={\rm col}(u^{*}_{[1,N-1](t)},\bar{u})$,
$\hat{y}(t+1)={\rm col}(y^{*}_{[1,N-1](t)},\bar{y})$, where
$\bar{u}=\pi(\xi^{*}_{N}(t))\in\mathcal{U}$. According to Theorem 2,
$(\hat{u}(t+1),\hat{y}(t+1))$ satisfies all constraints and there exists some
$\hat{g}(t+1)$ satisfying (11b) since $(\hat{u}(t+1),\hat{y}(t+1))$ is a valid
trajectory of the system. This proves recursive feasibility.

Using the candidate solution ${\hat{u}(t+1),\hat{y}(t+1)}$ and Theorem 2,
following the standard MPC arguments, we can prove asymptotic stability by
showing that the optimal cost $J^{j,*}_{N}(\cdot)$ is a Lyapunov function,
similar to [4, 5]. ∎

As the Tube DeePRC (15) solves a similar problem to the nominal DeePRC, it
inherits similar closed-loop properties.

###### Corollary 4

Under Assumption 4, the following holds for all $j\geq 1$: 1) Problem (15) is
feasible for all $t\in\mathbb{Z}_{\geq 0}$; 2) $(v^{j},z^{j})$ satisfy the
tightened constraints $\bar{\mathcal{U}}$, $\bar{\mathcal{Y}}$, and
$\zeta_{t}^{j}$ asymptotically converges to $\xi^{\rm F}$; 3) $(u^{j},y^{j})$
satisfy the nominal constraints $\mathcal{U}$, $\mathcal{Y}$, and
$\xi_{t}^{j}$ converges to the RPI set $\mathcal{E}$ for all
$d^{j}_{t}\in\mathcal{D}$.

###  V-B Recursive Feasibility of the End-to-End Formulation

Let $\mathcal{D}=\\{d\in\mathbb{R}^{m}\mid\|d\|_{\infty}\leq\bar{d}\\}$ for
some $\bar{d}\in\mathbb{R}_{>0}$, and let $\mathcal{K}^{j,t}_{\rm
u}=\mathcal{K}^{j,t}_{[-m:,:]}$.

###### Proposition 5

Under Assumption 4, if $\bar{d}\geq\epsilon$ and the $1$-norm of each non-zero
column of $\mathcal{K}^{j,t}_{\rm u}$ is $1$, then (19) is recursively
feasible.

###### Proof:

Let ${\mathcal{K}}^{j,t}_{{\rm u}[:,i]}$ denote the $i$-th column of
$\mathcal{K}^{j,t}_{\rm u}$, and suppose it is non-zero. Given any
$\hat{u}_{0}(t)$, we have

$\displaystyle\|\begin{bmatrix}y^{j}_{[t-\bar{L}+1,t-1]}\\\
u^{j}_{[t-\bar{L}+1,t-1]}\\\
\hat{u}_{0}(t)+\hat{d}(t)\end{bmatrix}^{\top}{\mathcal{K}}^{j,t}\|_{\infty}$
$\displaystyle=\|c+\hat{d}(t)^{\top}{\mathcal{K}}^{j,t}_{{\rm u}}\|_{\infty}$
$\displaystyle\geq|c_{i}+\hat{d}(t)^{\top}{\mathcal{K}}^{j,t}_{{\rm u}[:,i]}|$
(20)

where $c$ is a constant row vector and $c_{i}$ denotes its $i$-th element. By
choosing

$\hat{d}(t)=\left\\{\begin{array}[]{lc}\bar{d}\cdot{\rm
sign}({\mathcal{K}}^{j,t}_{{\rm u}[:,i]}),&c_{i}\geq 0\\\ -\bar{d}\cdot{\rm
sign}({\mathcal{K}}^{j,t}_{{\rm u}[:,i]}),&c_{i}<0\end{array}\right.$ (21)

where ${\rm sign}(\cdot)$ is applied element-wise, it holds that
$\eqref{eq:deeprc_1s_rfeas_step1}\geq\bar{d}\cdot\|\mathcal{K}^{j,t}_{{\rm
u}[:,i]}\|_{1}\geq\epsilon$. In other words, for any $\hat{u}_{0}(t)$, we can
construct the disturbance $\hat{d}(t)$ such that the excitation constraint
(19e) is satisfied. The rest of the proof constructs a candidate input/output
sequence similar to that of Proposition 3, such that constraints (19b)-(19c)
are satisfied. ∎

###  V-C Convergence Properties

From the discussion in Subsection IV-B, we immediately have the following.

###### Proposition 6

Both the DeePRC with LKB active exploration in Algorithm 1 and the End-to-End
formulation (19) reach the desired prediction horizon $\bar{N}$ in exactly
$m\bar{L}+n-\operatorname{rank}(\mathcal{H}_{\bar{N}}^{1})$ time steps.

Proposition 6 implies we can obtain finite horizon approximations of (5) in
finitely many time steps and a direct data-driven fashion. Using the arguments
in [15], we further have the following.

###### Theorem 7

Let Assumption 2 hold, and suppose the desired prediction horizon $\bar{N}$ is
achieved at iteration $\bar{j}$. Then $J^{j}(\xi^{\rm S})\geq J^{j+1}(\xi^{\rm
S})$ for all $j>\bar{j}$. Moreover, if $u^{j+1}=u^{j}$ in Algorithm 1 for some
finite $j$ and the LICQ condition in [15, Assumption 3] holds, then $u^{j}$
solves (5).

###### Proof:

The first point follows from [5, Theorem 2]. The second point follows from
[15, Theorem 2]. ∎

Furthermore, for $j\leq\bar{j}$, we can similarly prove that the nominal cost
of the Tube DeePRC (15) is non-increasing over iterations. Theorem 7
additionally implies that if $\xi_{t}$ in (5) reaches the origin in finitely
many time-steps $c$, with $c\leq\bar{N}$, then $(u^{j},y^{j})$ solves (5) for
all $j>\bar{j}$.

##  VI Numerical Examples

We demonstrate the performance of the proposed methods on a linearized four-
tank system adapted from [16]

$A=\begin{bmatrix}1&0&0.1&0\\\ 0&1&0&0.1\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}\\!,B=\begin{bmatrix}0.1&0\\\ 0&0.1\\\ 0&0.1\\\
0.1&0\end{bmatrix}\\!,C=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\end{bmatrix}$,

with $D=0$, subject to $\mathcal{U}=[-1.5,1.5]^{2}$,
$\mathcal{Y}=[-1.5,1.5]^{2}$. We assume the true lag $\underline{\ell}=2$ is
unknown and set $\ell=4$. The initial and target states are $(u^{\rm S},y^{\rm
S})=([0\ 0]^{\top},[0\ 0]^{\top})$ and $(u^{\rm F},y^{\rm F})=([0\
0]^{\top},[0.4\ -\\!0.4]^{\top})$. The cost matrices are $Q=\mathbf{I}^{4}$
and $R=0.1\cdot\mathbf{I}^{2}$. In this setting, the desired prediction
horizon is $\bar{N}=50$.

The initial safe trajectory $\\{u^{0},y^{0}\\}$ is obtained using the LQR
controller ($\bar{Q}=\mathbf{I}^{4},\bar{R}=50\mathbf{I}^{2}$). To meet
Assumption 4, small random disturbances are added to the LQR control input for
the first $20$ time steps. Using (12), we verify that $\\{u^{0},y^{0}\\}$
produces an initial prediction horizon of $N^{1}=8$.

In Fig. 1, we compare the performance of three methods: passive Hankel update
(Passive), two-stage with LKB design (2s-LKB), one-stage approach (1s). The
figure illustrates the relationship between the rank of the matrix
$\mathcal{H}_{\bar{N}}^{j}$ and its number of columns. Without exploration,
the rank increment is considerably slower, indicating that many added
trajectories to the matrix lack sufficient information. Meanwhile, for the
methods with exploration, the trend is approximately linear, as expected.
Since the rank is computed by singular value thresholding, there are brief
plateaus in the rank increase.

$100$$150$$200$$250$$300$$350$$400$$40$$60$$80$$100$Number of
columnsRankPassive2s-LKB1s$\bar{N}$ max. rank Figure 1: Evolution of the rank
of the Hankel matrix

In Table I we further compare the closed-loop costs of the 2-stage LKB and the
1-stage approaches against a nominal DeePRC scheme with fixed horizon $N=8$
(without exploration). The methods with active exploration achieve worse cost
in $\textstyle{j=1}$ due to constraint tightening, but outperform the nominal
scheme later thanks to the extended prediction horizon.

In the first iteration, the one-stage approach produces a slightly higher
nominal cost due to the excitation constraint. Nevertheless, it attains the
lowest real cost as it incorporates disturbance design in the planning phase.
Starting from $\textstyle{j=2}$, both approaches with exploration converge,
thus achieving identical costs.

| Nominal | 2s-LKB | 1s
---|---|---|---
$j$ | $N$ | $J^{j}=J^{j}_{\rm nom}$ | $N$ | $J^{j}$ | $J^{j}_{\rm nom}$ | $N$ | $J^{j}=J^{j}_{\rm nom}$
0 | \ | 10.7171 | \ | 10.7171 | 10.7171 | \ | 10.7171
1 | 8 | 7.9852 | 8 | 8.0201 | 8.0186 | 8 | 8.0192
2 | 8 | 7.9401 | 50 | 7.7484 | 7.7484 | 50 | 7.7484
3 | 8 | 7.9298 | 50 | 7.7484 | 7.7484 | 50 | 7.7484
4 | 8 | 7.9227 | 50 | 7.7484 | 7.7484 | 50 | 7.7484
TABLE I: Prediction horizon and trajectory costs over iterations

All experiments are implemented in MATLAB. On a Windows 11 64-bit workstation
equipped with a 3.20 GHz AMD Ryzen 7-5800H CPU and 16.0GB RAM, QPs of the two-
stage approaches (261 continuous variables, 0 binary variables, and 274
constraints) have an average solving time of 0.058 seconds. Whereas solving
the MIQP of the end-to-end approach (189 continuous variables, 89 binary
variables, and 464 constraints) avarages 0.757 seconds, By imposing the
excitation constraints, the computation time increases significantly,
highlighting the trade-off between performance and computation.

##  VII Conclusion

In this work, we presented the DeePRC algorithm, a direct data-driven approach
to control LTI systems performing repetitive tasks. The DeePRC requires only
an initial safe trajectory with a low excitation order and it guarantees
sample-efficient rank-increment while maintaining safety in closed-loop. Under
suitable conditions, DeePRC is guaranteed to converge to the best achievable
performance. Extending the formal guarantees for systems with process
disturbances, measurement noise, and nonlinearities is a promising direction
for future extensions.

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# Entropy generating function for past lifetime and its properties

Smitha Sa and Sudheesh K. Kattumannilb
a K E College Mannanam, Kerala, India,
bIndian Statistical Institute, Chennai, India

###### Abstract.

The past entropy is considered as an uncertainty measure for the past lifetime
distribution. Generating function approach to entropy become popular in recent
time as it generate several well-known entropy measures. In this paper, we
introduce the past entropy-generating function. We study certain properties of
this measure. It is shown that the past entropy-generating function uniquely
determines the distribution. Further, we present characterizations for some
lifetime models using the relationship between reliability concepts and the
past entropy-generating function.
keywords: Past entropy generating function, Lifetime models,
Characterizations.

†Corresponding E-mail<EMAIL_ADDRESS>
##  1\. Introduction

In many practical situations, measuring the uncertainty associated with a
random variable is very important. In this connection, the notion of entropy
was originally developed by Shannon (1948). It is of fundamental importance in
different areas such as Physics, Communication theory, Probability, Economics,
etc. Let $X$ be an absolutely continuous non-negative random variable having
cumulative distribution function $F(x)=P(X\leq x)$, probability density
function $f(x)$ and survival function $\bar{F}(x)=P(X>x)$. The Shannon’s
entropy is defined as

$H(F)=-\int\limits_{0}^{\infty}{f(x)\log f(x)dx},$ (1)

where ‘log’ denotes the natural logarithm. Several extensions of (1) are
available in literature each one being suitable for some specific situation.
Some of these measures are; cumulative residual entropy (Rao et al., 2004),
cumulative past entropy (Di Crescenzo and Longobardi, 2009), and the
corresponding weighted and dynamic measures (Chakraborty and Pradhan, 2023).
Sudheesh et al. (2022) proposed general measures of cumulative (residual)
entropy showing that several measures discussed in the literature are special
cases of their proposed measures of entropy.

A convenient tool to generate moments of a probability distribution is its
moment-generating function (m.g.f). It is well known that the successive
derivatives of the m.g.f at point zero give the successive moments of a
probability distribution, provided these moments exist. Golomb (1966)
introduced entropy generating function of a probability distribution and is
given by

$B_{s}(F)=\int_{0}^{\infty}f^{s}(x)dx,\quad s\geq 1.$ (2)

The first order derivative of $B_{s}(F)$ evaluated at value one yields
negative Shannon entropy or negative entropy of a distribution. Guiasu and
Reischer (1985) proposed a relative information-generating function and they
observed that its derivatives provide well-known statistical measures like the
Kullback-Leibler divergence between two probability distributions. Arora and
Nath (1972) proposed an inaccuracy-generating function for two probability
distributions and studied its relationship with the entropy-generating
function proposed by Golomb. The first derivative of this function at point
one gives the inaccuracy measure by Kerridge (1961).

Recently, information-generating functions have received much attention. Clark
(2019) introduced an information-generating function for point processes for
determining statistics related to entropy and relative entropy. In a series of
papers, Kharazmi and Balakrishnan (2021a, 2021b, 2021c) introduced cumulative
residual information generating and relative cumulative residual information
generating measures, cumulative residual Fisher information and relative
cumulative residual Fisher information measures, and studied their properties.
In particular, Kharazmi and Balakrishnan (2021a) proposed two new divergence
measures and showed that some of the well-known information divergences such
as Jensen-Shannon, Jensen-extropy and Jensen-Taneja divergence measures are
all special cases of it. For the information-generating functions associated
with maximum and minimum ranked sets and record values and their properties,
see Zamani et al. (2022) and Kharazmi et al. (2021). Saha and Kayal (2023)
proposed general weighted information and relative information-generating
functions and studied their properties. Recently, Smitha et al. (2023)
proposed a dynamic residual entropy-generating function and studied its
properties. Motivated these recent development, as past life plays an
important role in lifetime data analysis, we consider entropy generating
function associated with past lifetime and study its properties.

The rest of the paper is organised as follows. In section 2, we give the
definition of past entropy generating function and propose the relationship
with reversed hazard rate. Various properties of this measure are also
discussed. In Section 3, we give some characterization results based on the
functional relationship between past entropy-generating functions and
reliability concepts. Finally, in Section 4, we give some concluding remarks.

##  2\. Past Entropy Generating Function and its Properties

In lifetime analysis, on several occasions, one has information about the
current age of the unit under consideration. In such cases, we define a random
variable $X_{t}^{*}=X-t|X\geq t$ to study the properties of the unit. The mean
$E(X_{t})$ is known as the mean residual life and it has many applications.
Measuring the uncertainty about the remaining lifetime of the unit, Shannon’s
entropy associated with $X_{t}$ was developed by Ebrahimi and Pellery (1995)
and has the form

$H(F:t)=-\int\limits_{t}^{\infty}{\frac{f(x)}{\bar{F}(t)}\log\frac{f{(x)}}{\bar{F}(t)}dx}.$

Apart from the residual life of a unit, past life also has important
applications in lifetime analysis. In this case, we are interested in studying
the behavior of the random variable $X_{t}=t-X|X\leq t$. Di Crescenzo and
Longobardi (2004) have shown that in many situations, uncertainty is not
necessarily related to the future but can also refer to the past. By
considering such situations, they proposed the past entropy defined over
$(0,t)$ and is given by

$\bar{H}(F;t)=-\int\limits_{0}^{t}{\frac{f(x)}{F(t)}\log\frac{f{(x)}}{F(t)}dx}.$
(3)

We observed that $\bar{H}(F;t)$ can be viewed as the entropy of inactivity
time $t-X|X\leq t.$ This measure plays an important role in the context of
information theory, forensic science, and other related fields.

The past entropy function can also be expressed in terms of the reversed
hazard rate, $\lambda(x)=\frac{f(x)}{F(x)}$, through the relationship

$\bar{H}(F;t)=1-\frac{1}{F(t)}\int_{0}^{t}f(x)\log\lambda(x)dx.$ (4)

For more works on entropy related to past lifetime, we refer to Di Crescenzo
and Longobardi (2004,2006), Nanda and Paul (2006), Kundu et al. (2010), Mini
(2016), Nair et al. (2021), Sudheesh et al. (2022) and Chakraborty and Pradhan
(2023a). Next, we define the entropy-generating function in the past lifetime.

######  Definition 1.

Let $X$ be an absolutely continuous non-negative random variable with
cumulative distribution function $F(x)=P(X\leq x)$ and probability density
function $f(x)$. Let $X_{t}=t-X|X\leq t$ be the past life associated with $X$.
Then the past entropy generating function is given by

$\bar{B_{s}}(F;t)=\int_{0}^{t}\left(\frac{f(x)}{F(t)}\right)^{s}dx,\quad s\geq
1.$ (5)

In the following example, we show that two distributions can possess the same
entropy generating function, but different past entropy generating function.

######  Example 1.

Consider two components in a system having random lifetimes $X$ and $Y$ with
distribution functions $F$ and $G$ and density functions

$f_{X}(x)=\frac{1}{2},\,0\leq x\leq 2$

and

$f_{Y}(y)=\frac{1}{2},\,2\leq y\leq 4.$

Here

$\bar{B_{s}}(X)=2^{1-s}$

and

$\bar{B_{s}}(Y)=2^{1-s},$

respectively. We can observe that $\bar{B_{s}}(X)=\bar{B_{s}}(Y)=2^{1-s}$.

In this case, the past entropy-generating function is obtained as

$\bar{B_{s}}(X;t)=\frac{2^{s}t^{1-s}}{s+1}$

and

$\bar{B_{s}}(Y;t)=\frac{2^{s}[1-(1-t)^{s+1}]}{(s+1)(2t-t^{2})^{s}}.$

This result implies that the expected uncertainty regarding the predictability
of the outcomes of $X$ and $Y$ in terms of entropy generating functions are
identical for $F$ and $G$. However, the past entropy-generating functions of
$X$ and $Y$ are different.

Next, we study the properties of the past entropy-generating function.
Property 1: The first derivative of $\bar{B_{s}}(F;t)$ at $s=1$ with negative
sign is the past entropy measure proposed by DiCrescenzo and Longobardi
(2004).
Property 2: For any absolutely continuous random variable $X,$ define
$Z=aX+b,$ where $a>0,$ and $b\geq 0,$ are constants. Then, for $t>b$

$\bar{B_{s}}(Z;t)=a^{1-s}.\bar{B}_{s}\left(X;\frac{t-b}{a}\right).$

###### Proof.

$\displaystyle\bar{B_{s}}(Z;t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\left(\frac{f_{Z}(x)}{F_{Z}(t)}\right)^{s}dx$
$\displaystyle=$
$\displaystyle\frac{1}{\left[F\left(\frac{t-b}{a}\right)\right]^{s}}\int_{0}^{t}\left(\frac{1}{a}f\left(\frac{x-b}{a}\right)\right)^{s}dx$
$\displaystyle=$ $\displaystyle
a^{1-s}\int_{0}^{\frac{t-b}{a}}\left(\frac{f(u)}{F\left(\frac{t-b}{a}\right)}\right)^{s}du$
$\displaystyle=$ $\displaystyle
a^{1-s}\bar{B}_{s}\left(X;\frac{t-b}{a}\right).$

Hence the proof. ∎

From Property 2, we have the following identity.

  1. (1)

If $b=0,$ $\bar{B}_{s}(aX;t)=a^{1-s}\bar{B_{s}}(X;\frac{t}{a})$

  2. (2)

If $a=1,$ $\bar{B}_{s}(X+b;t)=\bar{B_{s}}(X;t-b)$

Property 3: The relationship between $\bar{B_{s}}(F;t)$ and reversed hazard
rate is given by

$\bar{B_{s}}^{\prime}(F;t)=\left(\lambda(t)\right)^{s}-s.\bar{B_{s}}(F;t)\lambda(t),$
(6)

where $\bar{B_{s}}^{\prime}(F;t)$ is the derivative of $\bar{B_{s}}(F;t)$ with
respect to $t$.

##  3\. Characterization Results

In this section, we establish several characterization results associated with
past entropy-generating functions. The following result shows that the past
entropy-generating function uniquely determines the underlying distribution
function.

######  Theorem 3.1.

Let $F(x)$ be an absolutely continuous distribution function and assume that
$\bar{B_{s}}(F;t)$ is increasing in $t.$ Then $\bar{B_{s}}(F;t)$ uniquely
determines $F(t).$

###### Proof.

Suppose that $F(x)$ and $G(x)$ are distribution functions such that

$\bar{B_{s}}(F;t)=\bar{B_{s}}(G;t),~{}\forall~{}t\geq 0.$ (7)

That is,

$\frac{1}{\left(F(t)\right)^{s}}\int_{0}^{t}f^{s}(x)dx=\frac{1}{\left(G(t)\right)^{s}}\int_{0}^{t}g^{s}(x)dx,~{}\forall~{}s\geq
1,$ (8)

where $f(x)$ and $g(x)$ are probability generating functions corresponding to
$F(x)$ and $G(x)$, respectively. Differentiating (8) with respect to $t$, we
obtain

$\displaystyle(F(t))^{-s}(f(t))^{s}$ $\displaystyle+$
$\displaystyle(-s)(F(t))^{-s-1}f(t)\int_{0}^{t}f^{s}(x)dx$ $\displaystyle=$
$\displaystyle(G(t))^{-s}(g(t))^{s}+(-s)(G(t))^{-s-1}g(t)\int_{0}^{t}g^{s}(x)dx.$

This implies

$(\lambda_{1}(t))^{s}-s\lambda_{1}(t)\bar{B_{s}}(F;t)=(\lambda_{2}(t))^{s}-s\lambda_{2}(t)\bar{B_{s}}(G;t),$

where $\lambda_{1}(t)$ and $\lambda_{2}(t)$ are the reversed hazard rates of
$f(x)$ and $g(x)$ respectively. Note that the reversed hazard rate determines
the distribution, uniquely. Hence to prove $F(t)=G(t),$ it is enough to prove
$\lambda_{1}(t)=\lambda_{2}(t),~{}\forall~{}t\geq 0.$

Suppose we have the inequality

$\lambda_{1}(t)>\lambda_{2}(t),\quad\lambda_{i}(t)\not=0,~{}i=1,2.$

From (8) we obtain

$\lambda_{1}(t)\left[(\lambda_{1}(t))^{s-1}-s\bar{B_{s}}(F;t)\right]=\lambda_{2}(t)\left[(\lambda_{2}(t))^{s-1}-s\bar{B_{s}}(G;t)\right].$

That is

$\frac{\lambda_{1}(t)}{\lambda_{2}(t)}=\frac{(\lambda_{2}(t))^{s-1}-s\bar{B_{s}}(G;t)}{(\lambda_{1}(t))^{s-1}-s\bar{B_{s}}(F;t)}$
(9)
$\frac{\lambda_{1}(t)}{\lambda_{2}(t)}>1\implies(\lambda_{2}(t))^{s-1}-s\bar{B_{s}}(G;t)>(\lambda_{1}(t))^{s-1}-s\bar{B_{s}}(F;t)$

Using (7), the above equation gives

$(\lambda_{2}(t))^{s-1}>(\lambda_{1}(t))^{s-1}\implies\lambda_{2}(t)>\lambda_{1}(t).$

This is a contradiction. Similarly, we can show that the inequality
$\lambda_{1}(t)<\lambda_{2}(t)$ also leads to a contradiction. Therefore, we
have $\lambda_{1}(t)=\lambda_{2}(t).$ ∎

Now we give the characterization results for uniform and power function
distributions using the functional relationship between the past entropy
generating function and reversed hazard rate.

######  Theorem 3.2.

Let $X$ be a continuous random variable with absolutely continuous
distribution function $F(x)$ and reversed hazard rate $\lambda(t).$ Then the
past entropy generating function $\bar{B_{s}}(F;t)$ has the form

$\bar{B_{s}}(F;t)=(\lambda(t))^{s-1}~{};~{}s>1$ (10)

if and only if $X$ follows uniform distribution with distribution function

$F(x)=\frac{x-a}{b-a}~{};~{}a\leq x\leq b.$ (11)

###### Proof.

Suppose $X\sim U[a,b],$ by direct computation of $\bar{B_{s}}(F;t)$ we have
the necessary part. Conversely, assume (10) holds. On differentiating both
sides of the equation (10) with respect to $t$, we obtain

$\bar{B_{s}}^{\prime}(F;t)=(s-1)(\lambda(t))^{s-2}\lambda^{\prime}(t).$ (12)

Using the relationship between $\bar{B_{s}}(F;t)$ and reversed hazard rate
given in (6), (12) becomes

$(\lambda(t))^{s}-s\bar{B_{s}}(F;t)\lambda(t)=(s-1)(\lambda(t))^{s-2}\lambda^{\prime}(t).$
(13)

Substituting (10) in (13), we obtain

$\displaystyle(\lambda(t))^{s}-s(\lambda(t))^{s}=(s-1)(\lambda(t))^{s-2}\lambda^{\prime}(t).$

Or

$(\lambda(t))^{s}(1-s)=(s-1)(\lambda(t))^{s-2}\lambda^{\prime}(t).$

That is,

$\frac{\lambda^{\prime}(t)}{(\lambda(t))^{2}}=-1,$

which can be written as

$\frac{d}{dt}\left(\frac{1}{\lambda(t)}\right)=1.$

Integrating both sides of the above equation in the interval $(0,t)$, we
obtain the $\lambda(t)$ as the reversed hazard rate of $U(a,b)$ distribution.
Since the reversed hazard rate determines the distribution uniquely, we have
the model in (11). ∎

Next, we characterize power distribution by using the functional relationship
between past entropy generating and reversed hazard rate. The result is stated
as follows.

######  Theorem 3.3.

Let $X$ be a non-negative random variable admitting an absolutely continuous
distribution function $F(x)$. The relationship

$\bar{B_{s}}(F;t)=k(\lambda(t))^{s-1}~{};~{}s>1,\,k\in R,$ (14)

holds for all real $t\geq 0,$ if and only if $X$ has power distribution with
distribution function

$F(x)=x^{c}~{};~{}c>0,~{}0<x<1.$

###### Proof.

By direct computation we get (14). Conversely assume that (14) holds.
Differentiating (14) and using the relationship between $\bar{B_{s}}(F;t)$ and
reversed hazard rate given in (6), we have

$\displaystyle(\lambda(t))^{s-1}-sk(\lambda(t))^{s-1}$ $\displaystyle=$
$\displaystyle k(s-1)(\lambda(t))^{s-3}\lambda^{\prime}(t)$
$\displaystyle(1-sk)(\lambda(t))^{s-1}$ $\displaystyle=$ $\displaystyle
k(s-1)(\lambda(t))^{s-3}\lambda^{\prime}(t)$
$\displaystyle-\frac{\lambda^{\prime}(t)}{(\lambda(t))^{2}}$ $\displaystyle=$
$\displaystyle\frac{sk-1}{k(s-1)}.$

The above equation can be written as

$\frac{d}{dt}\frac{1}{\lambda(t)}=\frac{sk-1}{k(s-1)}.$

Solving the above differential equation we obtain

$\lambda(t)=\frac{c}{t},$

where $c=\frac{k(s-1)}{sk-1}$, which is the reversed hazard rate of the power
distribution. ∎

######  Theorem 3.4.

A non-negative random variable $X$ has an exponential distribution with mean
$\mu$ if and only if we have the following relationship holds:

$\bar{B_{s}}(F;t)=\mu(\lambda(t))^{s}(\frac{e^{st/\mu}-1}{s}),$ (15)

###### Proof.

Let $X$ follows an exponential distribution with mean $\mu$. BY direct
calculation we obtain

$\bar{B_{s}}(F;t)=\frac{\mu}{s}\left(\frac{1}{\mu^{s}(1-e^{-t/\mu})^{s}}-\left(\frac{e^{-t/\mu}}{\mu(1-e^{-t/\mu})}\right)^{s}\right).$

In term of the reversed hazard rate, this reduces to the expression given in
(15).
Now, suppose (15) holds. Using the definition of $\bar{B_{s}}(F;t)$ and
$\lambda(t)$, we obtain the identity

$\int_{0}^{t}f^{s}(x)dx=\frac{\mu}{s}f^{s}(t)(e^{st/\mu}-1).$

Differentiating both sides with respect to $t$ and simplifying we obtain

$\frac{f^{\prime}(t)}{f(t)}=\frac{-1}{\mu}.$

Solving the above differential equation we obtain
$f(x)=\frac{1}{\mu}e^{-t/\mu}$, which proves the result. ∎

Next, we consider the case where the random variable $X$ has support
$(-\infty,\infty).$ Kundu et al. (2010) defined a past entropy measure for the
support $(a,b),$ $-\infty\leq a<b\leq\infty,$ and studied its property. Let
$X$ be a real valued absolutely continuous random variable having support
$(-\infty,\infty)$, then the past entropy generating function is defined as

$\bar{B_{s}}(F;t)=\int\limits_{-\infty}^{t}\left(\frac{f(x)}{F(t)}\right)^{s}dx,~{}s\geq
1.$ (16)

For the general support $(a,b),$ where $a=\inf\\{x:F(x)>0\\}$ and
$b=\sup\\{x:F(x)<1\\},$ $-\infty\leq a<b\leq\infty,$ the past entropy
generating function is defined as

$\bar{B_{s}}(F;t)=\int\limits_{a}^{t}\left(\frac{f(x)}{F(t)}\right)^{s}dx,~{}s\geq
1.$ (17)

If $a=0,$ then (17) is identical to the past entropy generating function
defined in (5).

The next theorem shows the constancy of the past entropy-generating function.

######  Theorem 3.5.

If $X$ is a random variable admitting an absolutely continuous distribution
function $F(x)$, then the past entropy generating function is independent of
$t$ if and only if the distribution $X$ is specified by

$F(x)=\exp{[a(x-b)]};-\infty\leq x\leq b.$ (18)

###### Proof.

Let $\bar{B_{s}}(F;t)=k$, where $k$ is a positive constant. Then

$\bar{B_{s}}^{\prime}(F;t)=0.$

Upon using equation (6), we obtain

$({\lambda(t)})^{s}-sk\lambda(t)=0.$

Hence, $\lambda(t)=({sk})^{\frac{1}{s-1}}$, a constant. By Theorem 2.3 of Nair
et al. (2019), the reversed hazard rate is independent of $t$, if the
distribution function of $X$ is specified by

$F(x)=\exp{[a(x-b)]};-\infty\leq x\leq b.$

Conversely, assume $X$ has a distribution function specified by (18). Then by
direct computation, we have $\bar{B_{s}}(F;t)=\frac{a^{s}}{as},$ which is
independent of $t$. This proves the theorem. ∎

Next, we give a characterization theorem for generalized power function
distribution.

######  Theorem 3.6.

Let $X$ be a continuous random variable with absolutely continuous
distribution function $F(x)$ and mean inactivity time $m(t)=E(t-X|X\leq t).$
Then $X$ follows generalized power function distribution specified by

$F(x)=\left(\frac{cx+d}{cb+d}\right)^{\frac{1-c}{c}}~{};~{}-\frac{d}{c}<t<b~{};~{}0<c<1$
(19)

if and only if

$\bar{B_{s}}(F;t)=k(m(t))^{1-s}.$ (20)

###### Proof.

Suppose $X$ follows generalized power function distribution specified in (19).
Then by direct computation we obtain

$\bar{B_{s}}(F;t)=\frac{\left(c(1-c)\right)^{s}}{(1-2c)^{s+1}}(ct+d)^{1-s}.$

That is,

$\bar{B_{s}}(F;t)=k(m(t))^{1-s}.$

Conversely, assume that (20) holds. On differentiating both sides of the above
equation and using the relationship given in (6), we obtain

$(\lambda(t))^{s}-ks(m(t))^{1-s}\lambda(t)=k(1-s)(m(t))^{-s}m^{\prime}(t).$
(21)

Multiplying by $(m(t))^{s}$ through out the equation (21) and upon using the
relationship $\frac{1-m^{\prime}(t)}{m(t)}=\lambda(t)$, we obtain

$(1-m^{\prime}(t))^{s}-ks(1-m^{\prime}(t))=k(1-s)m^{\prime}(t).$ (22)

On simplifying the above equation, we obtain

$\displaystyle
m^{\prime}(t)=1-\left(\frac{ks-k(1-s)}{s}\right)^{\frac{1}{s-1}},$

a constant. This implies that $m(t)$ is a linear function in $t$. Hence $X$
has generalized power function distribution specified by (19) (Elbatal et al.
2012). ∎

##  4\. Concluding remarks

In this article, we study past entropy using a generating function approach.
Further, we established the relationships between past entropy-generating
functions and reversed hazard rates. Some characterization results are
obtained. One can consider developing a goodness of fit test for power
distribution using the characterization result obtained in this paper.
Different extensions/modifications of measures of entropy and extropy are
studied in the literature. We can study these measures using generating
function approach.

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