where $A_{i,n}^{\alpha}=B(A_{i}^{\alpha},n)$. We then put a coarse proximity
structure on
$X_{\alpha,n}=\bigcup_{i=1}^{\phi(\alpha)}(A_{i,n}^{\alpha}\times\\{i\\})$ by
setting $\mathcal{B}_{\alpha,n}$ to be the bornology of all finite sets and
define the coarse proximity relation ${\bf b}_{\alpha,n}$ by $A{\bf
b}_{\alpha,n}$ if and only if the projections of
$(A\cap(A_{i,n}^{\alpha}\times\\{i\\}))$ and
$(C\cap(A_{i,n}^{\alpha}\times\\{i\\}))$ into $X$ are close in $X$ with the
metric coarse proximity. It is easily seen that this is a coarse proximity
structure on $X_{\alpha,n}$. For $\beta\geq\alpha$ we see that $X_{\beta,n}$
surjects onto $X_{\alpha,n}$ via the inclusions $A_{i,n}^{\beta}\rightarrow
A_{\rho_{\beta\alpha},n}^{\alpha}$, and that this map is a coarse proximity
map. Denote these maps by $g_{\beta\alpha}$. Similarly, for each $\alpha$ we
see that that there is a natural injection
$h_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}X_{\alpha}\rightarrow
X_{\alpha,n}$ again by the simple inclusion $A_{i}^{\alpha}\rightarrow
A_{i,n}^{\alpha}$. It is easily seen that $h_{\alpha}$ is a coarse proximity
isomorphism for each $\alpha$. In summary, we have the following commutative
diagram.

${{X_{\gamma}}}$${{X_{\beta}}}$${{X_{\alpha}}}$${{X_{\gamma,n}}}$${{X_{\beta,n}}}$${{X_{\alpha,n}}}$$\scriptstyle{F_{\gamma\beta}}$$\scriptstyle{h_{\gamma}}$$\scriptstyle{f_{\beta\alpha}}$$\scriptstyle{h_{\beta}}$$\scriptstyle{h_{\alpha}}$$\scriptstyle{g_{\gamma\beta}}$$\scriptstyle{g_{\beta\alpha}}$

Denoting $\varprojlim_{\alpha}X_{\alpha,n}$ by $X_{\infty,n}$ we have the
following.

######  Proposition 8.6.

The coarse proximity space $X_{\infty}$ coarse embeds into $X_{\infty,n}$.

###### Proof.

Denote the projection maps from $X_{\infty}$ to each $X_{\alpha}$ by
$\pi_{\alpha}$. Similarly, denote the projection maps from $X_{\infty,n}$ to
$X_{\alpha,n}$ by $\rho_{\alpha}$. By construction we have that
$g_{\beta\alpha}\circ h_{\beta}\circ\pi_{\beta}=h_{\alpha}\circ\pi_{\alpha}$
for each $\beta,\alpha\in\Omega$. Therefore, by the universal property of the
inverse limit there is a unique coarse proximity map
$h\mathrel{\mathop{\mathchar 58\relax}}X_{\infty}\rightarrow X_{\infty,n}$
such that $h_{\alpha}\circ\pi_{\alpha}=\rho_{\alpha}\circ h$ for each
$\alpha\in\Omega$. As each $h_{\alpha}$ is an injective coarse equivalence we
have that $h$ is an injective coarse proximity embedding. ∎

######  Corollary 8.7.

For each $n,m\in\mathbb{N}$ with $n\leq m$, the inclusion
$X_{\infty,n}\rightarrow X_{\infty,m}$ is an injective coarse proximity
embedding.

######  Proposition 8.8.

The limit coarse proximity space $X_{\infty}$ is witnessed by uniform covers.

###### Proof.

Let $A,C\subseteq X_{\infty}$ be such that $A{\bf b}_{\infty}C$. Then
$\delta(A)$ and $\delta(C)$ are close via the metric coarse proximity
structure on $X$. Therefore there is an $n\in\mathbb{N}$ such that
$st(\delta(A),\mathcal{V})\cap\delta(C)$ contains some unbounded set $D$,
where $\mathcal{V}$ is the uniformly bounded cover of $X$ by all sets of
diameter at most $n$. We then Consider $X_{\infty,3n}$. As shown above and in
Proposition 8.6 there is an injective coarse proximity embedding
$h\mathrel{\mathop{\mathchar 58\relax}}X_{\infty}\rightarrow X_{\infty,3n}$.
For every $\alpha\in\Omega$ let $\mathcal{V}_{\alpha,3n}$ be the collection of
all subsets of $X_{\alpha,3n}^{\alpha}$ of the form
$(B\times\\{i\\})\cap(A_{i,3n}^{\alpha}\times\\{i\\})$ where $B$ is n element
of $\mathcal{V}$. Then $\mathcal{V}_{\alpha,3n}$ is a uniformly bounded cover
of $X_{\alpha,3n}$ such that $g_{\beta\alpha}(\mathcal{V}_{\beta,3n})$ refines
$\mathcal{V}_{\alpha,3n}$ when $\beta\geq\alpha$. For each $\alpha$ we have
that $(\rho_{\alpha}\circ h)(A){\bf b}_{\alpha,3n}(\rho_{\alpha}\circ h)(C)$
for all $\alpha$. Moreover, for each $\alpha$ this relation is witnessed by
$\mathcal{V}_{\alpha,3n}$. Then, the uniformly bounded cover of
$X_{\infty,3n}$ defined by $\\{B\subseteq X_{\infty,3n}\mid\exists
E\in\mathcal{V}_{\alpha,3n},\,\rho_{\alpha}(B)\subseteq E\\}$ witnesses $h(A)$
being close to $h(C)$. Then by Proposition 3.15 we have that $A$ being close
to $C$ in $X_{\infty}$ is witnessed by a uniformly bounded cover of
$X_{\infty}$.

∎

Notice that in the proof of Proposition 8.8 we in fact show that the coarse
proximity of $X_{\infty}$ is witnessed by a countable family of uniformly
bounded covers. Also, by the construction, the bornologies of $X_{\infty}$ and
$X$ are isomorphic with respect to the order structure given by set inclusion.
Therefore by Theorem 4.2 we have the following.

######  Corollary 8.9.

The coarse proximity space $(X_{\infty},\mathcal{B}_{\infty},{\bf
b}_{\infty})$ is metrizable and any metric inducing the coarse proximity
structure on $X_{\infty}$ makes it into a proper metric space.

######  Proposition 8.10.

The continuous map $\mathcal{U}\delta\mathrel{\mathop{\mathchar
58\relax}}X_{\infty}\rightarrow X$ is $n+1$-to-$1$.

###### Proof.

Let $\\{C_{1},\ldots,C_{k}\\}$ be a collection of gradually disjoint unbounded
subsets of $X_{\infty}$ such that $\\{\delta(C_{1}),\ldots,\delta(C_{k})\\}$
is not divergent. Assume towards a contradiction that $k>n+1$. Because the
$C_{i}$ are gradually disjoint we have that $C_{i}\bar{\bf b}_{\infty}C_{j}$
for $i\neq j$. Then there is some $\alpha$ such that
$\pi_{\alpha}(C_{i})\bar{\bf b}_{\alpha}\pi_{\alpha}(C_{j})$ for $i\neq j$.
Because $\delta$ is a bijective function and the
$\delta(C_{i})=\delta_{\alpha}(\pi_{\alpha}(C_{i}))$ are not divergent we have
that each $\pi_{\alpha}(C_{i})$ must be in a different element of the cover
$\mathcal{F}_{\alpha}$. However, this would imply that there are $k>n+1$
members of the cover that are not divergent, contradicting the fact that the
order of every $\mathcal{F}_{\alpha}$ does not exceed $n+1$. Therefore $k\leq
n+1$, which implies that $\mathcal{U}\delta$ is $n+1$-to-$1$ by Theorem 6.9. ∎

######  Corollary 8.11.

$asdim(X)\leq n$.

###### Proof.

The space $X_{\infty}$ is metrizable by Corollary 8.9 and $\mathcal{U}\delta$
is $n+1$-to-$1$ and surjective by Theorem 6.9. Then by Theorem 6.4 we have
that $\delta$ is coarsely $n+1$-to-$1$ and coarsely surjective, so by Theorem
6.5 we have that $asdim(X)\leq n$. ∎

We then quickly have the following.

######  Theorem 8.12.

Given a proper metric space $X$ equipped with its metric coarse proximity
structure the following inequality holds:

$\Delta(\nu X)\leq asdim(X)\leq\Delta_{c}(X)$

##  9\. Discussion and Problems

The original problem suggested by Dranishnikov was the problem of whether or
not the asymptotic dimension of an unbounded proper metric space was equal to
the covering dimension of the Higson corona of the space. For general compact
Hausdorff spaces $X$ the inequality $dim(X)\leq\Delta(X)$ holds. There are
compact Hausdorff spaces with finite covering dimension and infinite cofinal
dimension. In fact, for every natural number $n$ there is a compact Hausdorff
space $X_{n}$ such that $dim(X_{n})=n$ and $\Delta(X_{n})=\infty$. Detailed
constructions of such spaces can be found in [15].

######  Question 9.1.

Is there a compact Hausdorff space $Z$ satisfying $dim(Z)<\Delta(Z)$ that can
be realized as the Higson corona of a proper metric space?

We showed that for every unbounded proper metric space that $\Delta(\nu X)\leq
asdim(X)\leq\Delta_{c}(X)$. We can weaken the question of whether or not
$asdim(X)$ and $dim(\nu X)$ agree to the question of whether or not
$\Delta(\nu X)$ and $asdim(X)$ agree.

######  Question 9.2.

Does the equality $\Delta(\nu X)=asdim(X)$ hold for every unbounded proper
metric space?

Stronger questions that imply the previous are the very natural questions:

######  Question 9.3.

Given an unbounded proper metric space $X$, does $\Delta(\nu X)<\infty$ imply
that $\Delta_{c}(X)<\infty$?

######  Question 9.4.

Does $\Delta(\nu X)=\Delta_{c}(X)$ for all unbounded proper metric spaces?

## References

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# Nonlinear integral extension of PID control with improved convergence of
perturbed second-order dynamic systems

Michael Ruderman M. Ruderman is with University of Agder (UiA), Department of
Engineering Sciences. P.B. 422, Kristiansand, 4604 Norway.
Email<EMAIL_ADDRESS>
###### Abstract

Nonlinear extension of the integral part of PID feedback control is proposed
for the perturbed second-order systems. For the matched constant
perturbations, the global asymptotic stability is shown, and for Lipschitz
perturbations an ultimately bounded output error is guaranteed. The second-
order system plants can also be expanded by an additional (parasitic) actuator
dynamics with low-pass characteristics. The proposed nonlinear control is
proven to outperform its linear (PID) benchmarking counterpart during the
settling phase, i.e. at convergence of the residual output error. An
experimental case study of the second-order system with an additional actuator
dynamics and considerable perturbation is demonstrated to confirm and
benchmark the control performance.

##  I INTRODUCTION

Proportional-integral-derivative (PID) controllers, see e.g. [1, 2], as well
as their possible extensions like anti-windup, gain-scheduling, parameters
adaptation, are still mostly seen as a working horse in various industries.
Despite a large number of available and well perceived tutorials and surveys
on PID principles, tuning, extensions, and limitations, e.g. [3, 4, 5] just
for mentioning here the few, numerous applications face (here and again) some
crucial challenges with use of the PID controllers. These lie often in a
trade-off between the performance and robustness, while the metrics associated
with a transient and steady-state response appear to be most significant for
several applications.

Well known, an integral control action is required for asymptotic set-point
regulation under parameter perturbations or (equally usual in practice)
presence of the matched stationary disturbances. For simple first- and second-
order processes, including possibly also time delays, the tuning of the
integral time constant is straightforward and well understood, see e.g. [6].
More challenging becomes an effective integral control action in case of
disturbance processes that vary over time, especially in vicinity of the set-
point, and when the application requires an increased settling accuracy in
shorter time. Not surprising is, hence, that a standard integral control
action cannot, for example, compensate accurately for the nonlinear Coulomb
friction [7, 8] in the set-point tasks. As one of the possible solutions, the
so-called reset integral control (introduced originally in [9]) can be
applied, see e.g. [10]. The reset integral behavior in feedback, and thus a
switching dynamics, restricts however the applicability to a certain class of
the system plants [11]. Worth noting is that despite a relaxation applied to
the underlying base systems and, this way, an attempt to address the stability
by a less stringent circle-criterion conditions [12], the reset integrator
does not entirely satisfy the necessary conditions posed on the feedback
nonlinearity, cf. [13, 14], cf. with preliminaries provided below in section
II. Another approach of the time-varying integral control gain, which assumes
a monotonic decrease of the gain to zero at a sufficiently slow rate, was
presented in [15] for linear system plants subject to a globally Lipschitz and
non-decreasing actuator nonlinearity. Despite the profound theoretical
results, which also form basis for a simple adaptive control strategy, the
approach is not directly applicable to compensate for external disturbances
and to improve the convergence performance. An universal integral controller
for minimum-phase nonlinear systems and the tuning procedures for its key
parameters were proposed in [16]. A single-input-single-output (SISO)
nonlinear system plant that has a well-defined normal form with asymptotically
stable zero dynamics was assumed for the synthesis procedure. Remarkable is
that for relative degree-one systems, this universal controller reduces to the
classical PI controller followed by saturation, and for relative degree-two
systems it reduces to the classical PID controller followed by saturation.
Settling performance, in terms of the convergence time and accuracy were,
however, not explicitly in focus of the proposed methodology. Later, the
problem of robust nonlinear integral control for a class of systems with
dynamic uncertainties was addressed in [17], while a constructive controller
design procedure was presented in the presence of an unknown equilibrium due
to uncertain nonlinearities and dynamic uncertainties. The resulted controller
reduces, notwithstanding, to linear control laws if the nominal system is
linear, and becomes a traditional PI control if, additionally, the relative
degree is one, or the uncertainty falls into the input space, i.e. strict
matching condition.

Motivated by the background provided above, and the steady-state behavior of
motion control systems, see e.g. [18], the present work introduces a possible
nonlinear integral extension of the PID feedback control. To the best of the
author’s knowledge, the proposed control, see eq. (8), is new and was not used
in the previously published works.

The rest of the paper is structured in the following way. Section II provides
the notation in use and the necessary preliminaries of a standard PID feedback
control and circle criterion of absolute stability. The main results are
placed into section III, starting with absolute stability, followed by the
disturbance rejection and extension of the second-order system plant by an
additional actuator dynamics. Also some clarifying numerical examples are
included. An experimental case study used for evaluation of the proposed
control and benchmarking it with the standard PD and PID controllers is
provided in section IV. The paper ends with the short conclusions drawn in
section V.

##  II PRELIMINARIES

### Notations

Unless otherwise said, $x=[x_{1},x_{2},\ldots,x_{n}]^{\top}\in\mathbb{R}^{n}$
is a state vector of real numbers,
$t\in\mathbb{R}_{+}=\\{\tau\in\mathbb{R}\>:\>\tau\geq 0\\}$ is a non-negative
time variable. The time derivative of a variable $x_{n}$ is denoted by
$\dot{x}_{n}\equiv dx_{n}/dt$, respectively the second time derivative is
denoted by $\ddot{x}_{n}\equiv d^{2}x_{n}/dt^{2}$, and so on. An input-output
transfer function is denoted by $H(s)=y(s)/u(s)$, where
$s=(\rho+j\omega)\in\mathbb{C}$ is the complex Laplace variable with the
signed angular frequency $\omega\in\mathbb{R}$, imaginary unit $j^{2}=-1$, and
$\rho\in\mathbb{R}$. An absolute value in $\mathbb{R}$ is denoted by
$|\cdot|$, while $\bigl{|}H(s)\bigr{|}$ and respectively
$\bigl{|}H(j\omega)\bigr{|}$ denote the absolute value of a complex function.
An identity matrix of the appropriate dimension is denoted by $I$.

###  II-A PID controlled feedback loop

We consider the second-order systems of the type

$\ddot{y}(t)=u(t)+\sigma(t),$ (1)

where the output of interest $y(t)$ and its time derivative $\dot{y}(t)$ are
available for a feedback control. The control signal $u(t)\in\mathbb{R}$ is
continuous on $t\in\mathbb{R}_{+}$ and assumed to stay unsaturated111We notice
that a feedback control involving an integral action may be subject to
saturation, which is also quite common practice in control engineering.
However, an explicit measure against the associated wind-up effects is beyond
the scope of the current work. Therefore, neither standard PID control,
assumed as a reference benchmarking control system, nor its nonlinear
extension proposed handle a saturated operation mode. for
$(y,\dot{y})\in\mathbb{R}^{2}$. The matched perturbation function $\sigma(t)$
is assumed to be Lipschitz, i.e. $|\dot{\sigma}|<\Sigma$.

First, we recall briefly a standard PID feedback control

$u(t)=-a\,\dot{y}(t)-b\,y(t)-c\int y(t)dt,$ (2)

with the design parameters $a,b,c>0$, while focusing below on the most
relevant (for us) properties and features. Control parameters are assumed to
be appropriately determined.

One can easily show, either in time or frequency domain, that for a constant
disturbance, i.e. $\dot{\sigma}=0$, the PID feedback control (2) provides a
global asymptotic stabilization of (1). For a Lipschitz disturbance, mapped
into frequency domain by $\Sigma\propto\omega$, the PID feedback control (2)
provides an ultimately bounded output error $|y(t)|<\varepsilon(\omega)$ for
$t\rightarrow\infty$. Regarding this statement, which is rather self-
understood in the linear feedback control theory, cf. e.g. [19, 2], we will
have a closer look below in section III-B.

Substituting (2) into (1) and analyzing the characteristic polynomial of the
unperturbed system dynamics, i.e.

$s^{3}+as^{2}+bs+c=0,$ (3)

one can show by, for example Routh-Hurwitz stability criterion, see e.g. [19],
that the closed-loop control system is asymptotically (and exponentially)
stable if and only if

$ab-c>0.$ (4)

That means every solution of the state trajectories is uniformly bounded and
satisfies $y(t)\rightarrow 0$, $\dot{y}(t)\rightarrow 0$ as
$t\rightarrow\infty$.

###  II-B Circle criterion

For a linear time invariant (LTI) system in the minimal state-space
realization, see e.g. [20] for detail, the nonlinearity $\phi(\cdot,\cdot)$
that is acting in feedback and satisfying

$\displaystyle\phi(0,t)$ $\displaystyle=$ $\displaystyle
0\quad\forall\;t\in\mathbb{R}_{+},$ $\displaystyle z\,\phi(z,t)$
$\displaystyle\geq$ $\displaystyle
0\quad\forall\;z\in\mathbb{R},\>t\in\mathbb{R}_{+}$

results in the so called Lur’e problem [21]. The latter is also refereed to as
the _absolute stability problem_ and is formulated for a generic class of
systems given by

$\displaystyle\dot{x}(t)$ $\displaystyle=$ $\displaystyle Ax(t)+Bu(t),$
$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle Cx(t)+Du(t),$ (5)
$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle-\phi\bigl{(}z(t),t\bigr{)}.$

Note that the single-input-single-output (SISO) systems are considered, so
that $u,\,z\in\mathbb{R}$, while $A,\,B,\,C,\,D$ are the matrices
(correspondingly vectors) of an appropriate dimension and contain the system
parameters. Here and for the rest of the section, we will closely follow the
notations and developments of _circle criterion_ [22, 23] as these are
provided in [14]; for additional explanations see also [24].

For applying the circle criterion of absolute stability to (II-B), the
following necessary assumptions are placed.

  1. i)

$A$, $B$, and $C$ are the minimal realization of a linear subsystem in (II-B),
i.e. the pair $(A,B)$ is fully controllable and the pair $(A,C)$ is fully
observable.

  2. ii)

The system matrix $A$ has all eigenvalues $\lambda_{i}$ lying in
$\mathbb{C}_{-}^{\circ}$, i.e. $\mathrm{Re}\bigl{[}\lambda_{i}(A)\bigr{]}<0$
for $i=1,\ldots,n$.

  3. iii)

The nonlinearity $\phi(\cdot,t)$ belongs to the sector $[k_{1},k_{2}]$ for
$k_{2}\geq k_{1}\geq 0$. That means

$k_{1}z\leq\phi(z,t)\leq k_{2}z.$

Moreover, it is assumed that

  1. iv)

$\phi(z,t)$ is memoryless, possibly time-varying, and piecewise continuous in
$t$ and locally Lipschitz in $z$.

For the input-output transfer function

$H(s)=C(sI-A)^{-1}+D$ (6)

of the linear subsystem in (II-B), the absolute stability of the closed-loop
(II-B) with $\phi(\cdot,t)$ belonging to the sector $[0,k]$, is guaranteed by
the circle criterion stated in the below theorem.

###### Theorem 1

Given a feedback system described by the equations (II-B). Let the assumptions
(i)-(iv) be effective. Then, the origin of (II-B) is globally asymptotically
stable if

$\mathrm{Re}\bigl{[}1+kH(j\omega)\bigr{]}>0,\quad\forall\>\omega.$ (7)

The proof of Theorem 1 can be found in [14, page 241].

It is worth recalling that the circle criterion given by (7) for the sector
$[0,k]$ is equivalent to, correspondingly can be transformed into (and also
from), the circle criterion formulated for the sector $[k_{1},k_{2}]$ in terms
of a disc $\mathcal{D}(k_{1},k_{2})$ in the polar coordinates of
$H(j\omega)$-locus, see e.g. [24, 14].

##  III MAIN RESULTS

The proposed feedback control law with nonlinear extension of the integral
part is given by, cf. (2),

$u(t)=-a\,\dot{y}(t)-b\,y(t)-c\,\Bigl{(}1+d\exp\bigl{(}e|y(t)|\bigr{)}\Bigr{)}\int
y(t)dt.$ (8)

The linear control parameters $a,b,c>0$ are in the same sense as for (2). The
nonlinear parameters $d\geq 0$ and $e<0$ provide a smooth variation of the
overall integral gain

$\Omega\equiv c\,\Bigl{(}1+d\exp\bigl{(}e|y(t)|\bigr{)}\Bigr{)},$ (9)

that is monotonically increasing with decrease of the magnitude of the control
error. Worth emphasizing is that

  1. (i)

$c\leq\Omega\leq c+cd$;

  2. (ii)

$\Omega\rightarrow c$ for a continuously growing $|y|$;

  3. (iii)

the control (8) recovers completely to the benchmarking reference controller
(2) if $d=0$.

In the following, we are proving the absolute stability of an unperturbed
system (1), (8) by using the circle criterion, cf. with section II-B.
Afterwards, the closed-loop system (1), (8) with an exogenous input $\sigma$
is analyzed in terms of the convergence performance of $y(t)$, and that in
comparison to the control law (2), i.e. for $d=0$ in (8).

###  III-A Absolute stability

The unperturbed closed-loop system (1), (8) is transferable into the Lur’e
problem (II-B) with $D=0$ and

$A=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ -c&-b&-a\\\
\end{array}\right),\,B=\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
\end{array}\right),\,C^{\top}=\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
\end{array}\right).$ (10)

The necessary assumptions i) and ii), cf. with section II-B, are fulfilled
since the state-space realization (10) is both fully controllable and
observable, and one can guarantee that the system matrix $A$ is Hurwitz under
the condition that inequality (4) is satisfied. The feedback nonlinearity

$\phi(z,t)=c\,d\,\exp\bigl{(}e|x_{2}|\bigr{)}\,z$ (11)

proves to satisfy the necessary assumption iii), since it belongs to the
sector $[0,k]$ with $k=cd>0$ for all $|x_{2}(t)|\in\mathbb{R}_{+}$.
Fulfillment of the necessary assumption iv) follows from the fact that (11) is
single-valued with respect to the $(z,|x_{2}|)$ pair, i.e. it is memoryless,
and continuous in $t$ and Lipschitz in $z$. Two of the last statements are due
to the exponential term, and since $x_{2}(t)\in\mathcal{C}^{3}$ for all $u(t)$
and $\sigma(t)$ to be Lipschitz. Now, we are in position to show the absolute
stability of the unperturbed control system (1), (8).

###### Theorem 2

Given an unperturbed feedback system (1), (8). Then, its origin is globally
asymptotically stable if

$\frac{c^{2}d-acd\,\omega^{2}}{(b\omega-\omega^{3})^{2}+(c-a\,\omega^{2})^{2}}>-1,\quad\forall\>\omega.$
(12)

###### Proof:

Substituting (10) into (6) and evaluating the loop transfer function for
$s=j\omega$ results in

$H(j\omega)=-\frac{1}{a\,\omega^{2}-c+j\omega\,(\omega^{2}-b)}.$ (13)

Evaluating the real part of (13) and substituting it, together with $k=cd$,
into (7) results in

$1+\frac{c^{2}d-acd\,\omega^{2}}{(b\omega-\omega^{3})^{2}+(c-a\,\omega^{2})^{2}}>0.$

Comparing it to (12), this completes the proof. ∎

###### Remark 1

For both boundary frequencies $\omega=0$ and $\omega=\infty$ the condition
(12) is always fulfilled. Indeed, for $\omega=\infty$ one obtains
$-1/\infty>-1$. Also, substituting $\omega=0$ into (12) results in $d>-1$,
that is always true for the control (8).

Figure 1: Visualization of the circle condition for $H(j\omega)$ and sector
$[0,cd]$.

A graphical visualization of the circle stability criterion applied to
$H(j\omega)$, with an exemplary stable characteristic polynomial (3)
satisfying (12) and $k=cd$, is depicted in Fig. 1 for the sake of
explainability.

###  III-B Disturbance rejection

Consider a standard feedback control loop as shown in Fig. 2. Note that if a
reference $r(t)$ is also applied, then both controllers (2) and (8) will
incorporate the control error $\epsilon=r-y$ instead of $y$ and, respectively,
its derivative and integral. Consequently, they will change to additive
inverse, i.e. $-a\dot{y}$ changes to $a\dot{\epsilon}$ and so on. In the
following, for the sake of simplicity and due to the focus on disturbance
rejection, we assume $r=0$. Same time, $\sigma\neq 0$ is entering the loop.
Further, the control (8) is denoted by nl-PID while its linear counterpart (2)
by PID, respectively, cf. Fig. 2.

Figure 2: Closed loop system with (nl-)PID control.

While the system plant transfer function is given by

$G(s)=\frac{1}{s^{2}},$ (14)

one can temporary assume (9) as a ’frozen’ (i.e. constant) parameter and write
the control transfer function as

$\\{\hbox{PID},\,\hbox{nl-PID}\\}\equiv C(s)=\frac{as^{2}+bs+\Omega}{s}.$ (15)

The corresponding perturbation-to-output transfer function is

$S_{\sigma y}(s)=\frac{y(s)}{\sigma(s)}=\frac{G(s)}{1+C(s)G(s)}.$ (16)

Substituting (14), (15) and evaluating (16), one can recognize that for any
$\sigma=\mathrm{const}$, the resulting $y(s)\rightarrow 0$ as $s\rightarrow
0$, i.e. at steady-state. This is irrespectively of the $\Omega$-value for
which the characteristic polynomial (3) with $c=\Omega$ is guaranteed to
remain stable. Evaluating the magnitude of the frequency response function
$S_{\sigma y}(j\omega)$ one obtains

$\bigl{|}S_{\sigma
y}(j\omega)\bigr{|}=\frac{\omega}{\sqrt{\bigl{(}b\omega-\omega^{3}\bigr{)}^{2}+\bigl{(}\Omega-a\omega^{2}\bigr{)}^{2}}}\,.$
(17)

One can show that for an increasing $\Omega$, while satisfying $\Omega<ab$,
the peak value of (17) also increases, i.e.
$\Omega\nearrow\>\Rightarrow\>\max\,\bigl{|}S_{\sigma
y}(j\omega)\bigr{|}\nearrow$. A qualitative behavior of (17) is shown in Fig.
3 for the exemplary assigned $\Omega\in\\{1c,2c,3c,4c\\}$.

Figure 3: Magnitude of frequency response function $S_{\sigma y}$ depending on
$\Omega$.

Same time, one can also show that for frequencies lower than the peak
frequency $\omega_{\mathrm{peak}}$, an increasing $\Omega$ leads to a
decreasing magnitude value, i.e.
$\Omega\nearrow\>\Rightarrow\>\bigl{|}S_{\sigma y}(j\omega)\bigr{|}\searrow$
for $\omega\ll\omega_{\mathrm{peak}}$, cf. Fig. 3. Here we recall that higher
peak values give rise to higher overshoots during the transients, like for
example in case when all frequencies are excited by a stepwise $\sigma$. On
the contrary, lower $|S_{\sigma y}|$ values reduce the ultimately bounded
output error, correspondingly, improve the convergence performance. Thus, an
appropriately varying $\Omega$ value, cf. (9), can improve both the transient
and steady-state behavior of the control system. We should recall, however,
that the behavior (17), cf. also Fig. 3, discloses only a steady-state
response of the feedback control system.

In order to assess the transient response of the control system to a matched
perturbation $\sigma$, and that depending on the $\Omega$ parameter, we apply
the unit step $\sigma(s)$ so that

$y(s)=\frac{G(s)}{1+C(s)G(s)}\cdot\frac{1}{s}.$ (18)

In the following, for the sake of simplicity when developing and analyzing the
time domain solution of (18) we assume the real eigenvalues
$\lambda_{1},\lambda_{2}>0$ so that the stable transfer function (16) has a
double-pole pair at $-\lambda_{1}$ and, respectively, one pole at
$-\lambda_{2}$. One can show that such poles configuration corresponds to the
control gains equal to

$a=2\lambda_{1}+\lambda_{2},\quad
b=\lambda_{1}^{2}+2\lambda_{1}\lambda_{2},\quad\Omega=\lambda_{1}^{2}\lambda_{2}.$
(19)

Solving (18) in time domain and putting outside the brackets the common term
$\Gamma=(\lambda_{1}-\lambda_{2})^{-2}$ results in

$y(t)=\Gamma\Bigl{(}\exp(-\lambda_{2}t)-\bigl{(}1-(\lambda_{2}-\lambda_{1})\,t\bigr{)}\exp(-\lambda_{1}t)\Bigr{)}.$
(20)

Expressing $\lambda_{2}$ in terms of $\Omega$, cf. (19), one obtains from (20)

$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\hat{\Gamma}\biggl{(}\exp\Bigl{(}-\frac{\Omega}{\lambda_{1}^{2}}t\Bigr{)}-\Bigl{(}1-\frac{\Omega-\lambda_{1}^{3}}{\lambda_{1}^{2}}\,t\Bigr{)}\exp\bigl{(}-\lambda_{1}t\bigr{)}\biggr{)},$
$\displaystyle\hat{\Gamma}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}^{4}}{(\lambda_{1}^{3}-\Omega)^{2}}.$ (21)

Analyzing (21), one can show that an increasing $\Omega$ (for a fixed
$\lambda_{1}$ value) leads to a decreasing peak in the transient response of
$y(t)$, cf. Fig. 4 for exemplary $\Omega\in\\{0.5c,1c,2c\\}$.

Figure 4: Exemplary transient response of (21) for varying $\Omega$.

### Numerical example

The following numerical example visualizes convergence of the control system
(1), (8) for various parameter settings. The assigned (most simple)
disturbance is a constant value $\sigma=-100$. The set linear control
parameters are $a=60$, $b=1100$, $c=3000$, resulting in a poles configuration
$\lambda\in\\{-3.28,\,-28.36\pm j10.47\\}$, i.e. when the nonlinear integral
gain is excluded, i.e. $d=0$. Note that for an integral gain equal to
$2c=6000$, that corresponds to $d=1$ when $y\rightarrow 0$, the poles
configuration results in $\lambda^{*}\in\\{-10,\,-20,\ -30\\}$. The initial
value $y(0)=-1$ is assigned for a simultaneous set-point and disturbance
rejection control task.

For a fixed parameter $e=-10$, the output convergence on the logarithmic scale
is compared for the set parameter values $d=\\{1,2,3\\}$ in Fig. 5. The linear
PID case, i.e. for $d=0$, is also included for the sake of comparison.

Figure 5: Output convergence on logarithmic scale for varying $d$.

Next, for a fixed parameter $d=2$, the output convergence on the logarithmic
scale is compared for the set parameter values $e=\\{-10,-100,-1000\\}$ in
Fig. 6. Also here, the linear PID case of $d=0$ is shown for comparison.

Figure 6: Output convergence on logarithmic scale for varying $e$.

One can recognize that for some lower threshold $\varepsilon=\mathrm{const}$
of the residual error, the convergence of the nl-PID control outperforms that
one of the PID, in terms of $|y(t)|<\varepsilon$ for $t>t_{0}$ with $t_{0}$ to
be a metric of the settling time. Simultaneously, for higher $\varepsilon$
values, for instance with $t_{0}<1$ sec cf. Figs. 5, 6, it is a combination of
$(d,e)$ parameters that allows shaping the output convergence to be superior
comparing to the linear case. Note that in addition to $y(t)$ convergence, the
transient over- or undershoots and the control effort can appear as essential
in a particular application.

###  III-C Extension by an actuator dynamics

When the second-order system plant (1) is extended by a (parasitic) actuator
dynamics

$v(s)=F(s)u(s),$ (22)

where $F(s)$ is a stable and strictly proper transfer function with
$F(0)\rightarrow\mathrm{const}$, the system (1) expands to

$\ddot{y}(t)=v(t)+\sigma(t).$ (23)

Then, the extended plant (23), (22) is controlled in the same way by (8),
provided the resulted Lur’e system equally fulfills the necessary assumptions
i)-iv), cf. with section II-B.

For the extended plant (23), (22), the corresponding input-output loop
transfer function results then in

$H(s)=\frac{F(s)}{s^{3}+as^{2}+bs+c}.$ (24)

Since neither the structure of a feedback system nor the nonlinearity (11)
change by (22), the same circle criterion as given in section II-B is
applicable to (24). Respectively, the parametric inequality (12) of Theorem 2
should modify, so that to take into account also the polynomial coefficients
of $F(j\omega)$. Since both the absolute stability and disturbance rejection
for (24) follow the same lines of argumentation as in sections III-A, III-B,
and a particular actuator dynamics $F(s)$ is rather application specific, we
omit a detailed development here. At the same time, it is worst emphasizing
that the experimental case study provided below in section IV has the second-
order system plant additionally affected by such first-order actuator dynamics
$v(s)=\kappa(s\mu+1)^{-1}u(s)$, with a gain factor $\kappa$ and (not
negligible) time constant $\mu>0$.

##  IV EXPERIMENTAL EVALUATION

The proposed extended nonlinear PID control given by (8) is experimentally
evaluated and additionally benchmarked with a standard PID control (2). In
addition, the corresponding PD control (i.e. $c=0$) is also evaluated for
highlighting the level of perturbations in the system at hand and, this way,
disclosing necessity of the integral control action. The common control
parameters are assumed the same, while all parameter values are summarized in
Table I.

TABLE I: Control parameters control \ params. |  $a$ |  $b$ |  $c$ |  $d$ |  $e$
---|---|---|---|---|---
PD |  $10$ |  $1000$ |  |  |
PID |  $10$ |  $1000$ |  $5000$ |  |
nl-PID |  $10$ |  $1000$ |  $5000$ |  $1$ |  $-100$

The experimental laboratory setup used for evaluation of all three controllers
is depicted in Fig. 7.

Figure 7: Experimental setup of 1DOF actuator system (laboratory view).

The motion system consists of a voice-coil driven actuator with 1DOF and
contactless measured output position $y(t)$. The latter contains a relatively
hight level of the sensing noise. The robust finite-time second-order sliding-
mode differentiator [25, 26] is used for obtaining $\dot{y}(t)$. The sampling
frequency of the implemented real-time control is set to 10 kHz. A more
detailed description of the experimental system, including the identified
system parameters, can be found in [18].

Further it is worth noting that: i) an additional first-order actuator
dynamics (due to electro-magnetic behavior of the voice-coil motor) is in
place, and ii) the unknown perturbation $\sigma(t)$ is mainly due to the
gravity, nonlinear friction, and force ripples of the anchor-stator
interaction.

The measured output response to the step reference $r=0.008$ m, that is
applied at time $t=1$ sec, is shown in Fig. 8 for the nl-PID, PID, and PD
controllers.

Figure 8: Measured output response of the nl-PID, PID, and PD controls.

The absolute output error $|\epsilon|\equiv|r-y|$ of the nl-PID and PID
controllers is additionally visualized on the logarithmic scale in Fig. 9. One
can recognize that for $|\epsilon|<0.0001$ m, both controllers have to deal
with a sensing noise so that the residual control errors are similar. At the
same time, the convergence performance is well distinguishable for the time
range between 1.18 and 1.7 sec. The transient behavior (including over- and
undershoots) at the time between 1 and 1.18 sec is well comparable.

Figure 9: Absolute output error (logarithmic) of the nl-PID, PID controls.

##  V CONCLUSIONS

In this paper, a novel nonlinear extension of the PID feedback control
(denoted by nl-PID) is introduced. The nonlinear extension is proposed for
integral control action that allows improving convergence performance in
presence of the matched unknown perturbations in the second-order systems. The
nonlinear integral control part does not change the system structure and is
continuous in time and Lipschitz in the system output variable, so that the
absolute stability by means of the circle criterion turned out applicable and
sufficient for analysis. The nl-PID control has five design parameters in
total, while keeping the same meaning and tuning rules for the proportional,
integral, and derivative feedback gains. The global asymptotic stability of
the nl-PID control was shown for the case of constant perturbations. For the
case of Lipschitz perturbations, an ultimately bounded output error can be
guaranteed by consideration in frequency domain. Beyond the analysis and
numerical examples, an experimental case study is demonstrated for
benchmarking the standard PD, PID, and proposed nl-PID controls.

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# Light-matter quantum dynamics of complex laser-driven systems

Ivan Gonoskov<EMAIL_ADDRESS>Stefanie Gräfe<EMAIL_ADDRESS>Institute for Physical Chemistry, Friedrich-Schiller-University Jena, Max-
Wien-Platz 1, 07743 Jena, Germany

(13 June 2021)

###### Abstract

We propose a novel general approximation to transform and simplify the
description of a complex fully-quantized system describing the interacting
light and matter. The method has some similarities to the time-dependent Born-
Oppenheimer approach: we consider a quantum description of light rather than
of nuclei and follow a similar separation procedure. Our approximation allows
to obtaing a decoupled system for the light-excited matter and ”dressed” light
connected parametrically. With these equations at hand, we study how intense
light as a quantum state is affected due to the back-action of the interacting
matter. We discuss and demonstrate the possibility of the light-mode
entanglement and nonclassical light generation during the interaction.

††preprint: / to the Journal of Chemical Physics, Special Topic on Quantum
Light

Motivation.– Light is a unique tool to control and probe the world of
matter/particles because of its fascinating properties: nowadays, technologies
allow to generate, measure, and control light radiation at extremely precise
level, both on spatial and temporal scales. This opens extraordinary
possibilities to explore the quantum nature of our world at the
micro/nanometer spatial scales, attosecond time scales, and possibly will
bring some new quantum phenomena to macroscopic level. The quantum-mechanical
description of both, the matter (charged particles) and the electromagnetic
radiation is universal: within conventional non-relativistic quantum theory,
everything can be described by the total wave function which is a solution of
the corresponding time-dependent Scrödinger equation (TDSE) with a complete
Hamiltonian including all the interactions. Since light is a quantum object
Compton , the complete and consistent treatment requires a fully-quantum
description for the whole interacting system including both, the charged
particles and the light modes. It is known that the calculation of the complex
quantum problems is a very complicated issue, especially for strongly
interacting, multidimensional systems. Here, we present a novel theoretical
method which simplifies the problem significantly, while still taking various
purely quantum properties into account.

One of the most general and rigorous approaches which combines classical
electrodynamics (non-relativistic for the particles) and quantum principles
within the time-dependent Schrödinger equation (TDSE) employs the so-called
Pauli-Fierz Hamiltonian (PFH), see PFH  and equation (1), below. This Pauli-
Fierz Hamiltonian contains the description of the dynamics of the charged
particles coupled with the electromagnetic fields in a fully quantum way. This
unifies Maxwell’s equations and Newton’s equations at the fully quantum
description. Within this theory, it is possible also to include the spin of
the particles and several semi-relativistic effects PFH2 ; PFH3 . The possible
solutions of this TDSE could open various opportunities for applications
ranging from new sources of light or states of quantum matter to unique pure
quantum collective effects. Recent experiments demonstrate some surprising
quantum collective phenomena beyond classical
or even semiclassical pictures wP0 ; quantExp .

The paper is organized as follows: we start with the TDSE description with the
PFH. Next, we derive the related simplified system of the parametrically-
connected equations. Finally, we consider selected examples/solutions and
discuss the consequences and the possible applications.

General description of the light-matter interaction.– The total Hamiltonian
for the charged particles (numbered with indices $k$ and $k^{\prime}$ ) and
the multi-mode electromagnetic field (modes numbered with index $j$) in the
Schrödinger representation with Coulomb gauge reads as follows (known as
Pauli-Fierz Hamiltonian, see overview and details in PFH ):

$\hat{H}=\sum\limits_{k}\frac{1}{2m_{k}}\left(\,\hat{\vec{p}}_{k}-\frac{e_{k}}{c}\sum\limits_{j}\hat{\vec{A}}_{j}\right)^{2}+\sum\limits_{k\,k^{\prime}}U_{k\,k^{\prime}}+\sum\limits_{j}\omega_{j}\hat{N}_{j}\;.$
(1)

Here, $e_{k}$ and $m_{k}$ are the charges and masses of the particles with
index $k$ , $\sum\limits_{k\,k^{\prime}}U_{k\,k^{\prime}}$ includes particle-
particle mutual interactions, as well as interaction of each particle with
external longitudinal fields. Each mode $j$ of the electromagnetic field is
quantized via the term $\omega_{j}\hat{N}_{j}$ in the Hamiltonian. This is the
most general formulation, without including spin (which can be extended
separately within Pauli-Fierz Hamiltonian) within a non-relativistic
description of matter (particles).

The idea of parametrical-connection approximation: Based on the PFH, we first
consider the quantum interaction of two objects of different nature (one
electron and the light field) having the total Hamiltonian of the form:
$\hat{H}=\hat{H}_{e}+\hat{W}_{int}+\hat{H}_{field}$. We also consider that the
initial state of the whole system is of the form:
$\Psi_{0}(x,q)=F_{0}(x)\otimes{}G_{0}(q)$, where $x$ is related to the
electronic coordinates and $q$ is related to the field-modes coordinates,
respectively. In the strong-field case, when
$\left\langle\hat{H}_{e}\right\rangle\sim\left\langle\hat{W}_{int}\right\rangle\ll\left\langle\hat{H}_{field}\right\rangle$,
it is reasonable to assume that the quantum state of the light field changes
relatively weakly during the interaction compare to the initial state
$G_{0}(q)$ since the interacting electron is a small perturbation to the
strong field. This leads to the idea of the following (full first-order)
approximation of the total solution:
$\Psi(x,q,t)\approx{}F_{(G_{0})}(x,q,t)\otimes{}G_{(F)}(q,t)$, with the error
proportional to some degree of the ratio
$\left\langle\hat{W}_{int}\right\rangle/\left\langle\hat{H}_{field}\right\rangle$.
The most signficant simplification in the following is that the solution for
the electronic subsystem $F_{(G_{0})}(x,q,t)$ is obtained from the TDSE with
only local operators on $q$ (there are no derivatives), i.e. it is an
effectively one-object TDSE with parametrical $q$-dependance. The
corresponding $q$-derivatives can be transformed to the simple $q$-dependent
expressions using a non-perturbed $G_{0}(q)$ within this first-order
approximation. Thus, we transform the initial two-object TDSE to a simplified
system of two equations, one for the $F_{(G_{0})}(x,q,t)$, where $q$ is a
parameter, and other for the field-state evolution $G_{(F)}(q,t)$ including a
small electron back-action via $F$. The detailed derivation of the system for
coherent (Gaussian) initial state of the intense light is given below. The
weaker approximation: $\Psi(x,q,t)\approx{}F(x,\\{q\\},t)\otimes{}G_{0}$ leads
immediately after the field-state averaging of the total Hamiltonian in the
high photon number limit to the known (fixed-amplitude) classical field limit
ClassFL .

Detailed derivation of the general approximation: We first consider one
particle - electron interacting with an arbitrary given configuration of
propagating electromagnetic plane waves. A similar problem statement was
recently considered for a cavity-photon Hamiltonian in the dipole
approximationhoffm . We note, our approach can be transformed for more complex
systems with many particles, as well as for more specific ones. (atomic units
$\hbar=m_{e}=|e|=1$ are used in what follows). We use the coordinate
representation of the field operators quant1 ; quant2 ; quant3 , with the
creation and annihilation operators
$\hat{a}^{+}_{j}=\frac{1}{\sqrt{2}}\left(q_{j}-\frac{\partial}{\partial{q_{j}}}\right)$
and
$\hat{a}_{j}=\frac{1}{\sqrt{2}}\left(q_{j}+\frac{\partial}{\partial{q_{j}}}\right)$
and the field dynamical coordinate $q_{j}$. We assume that the initial state
(at the time $t=0$) of each mode $j$ is coherent:
$G_{j}({{\color[rgb]{0,0,0}q_{j},\,}}0)={{\color[rgb]{0,0,0}G_{0\,j}=}}\;C_{0\,j}\cdot\exp\left[-\frac{1}{2}(q_{j}-e^{-i\theta_{j}}\sqrt{2N_{j}})^{2}\right]$
with $N_{j}$ being an initial arbitrary average number of photons, and
$\theta_{j}$ is initial phase (see Appendix). In this case, if we apply the
following transformation
$\prod\limits_{j}\exp[-i(\omega_{j}t+\theta_{j})\hat{N}_{j}]$ (see also quant1
) to the general TDSE with Hamiltonian Eq.(1), we obtain the TDSE in the
interaction picture with phase-dependent field operators and phase-independent
mode states:

$i\dot{\Psi}=\hat{H}\Psi\;,\;\;\;\;\hat{H}=\frac{1}{2}\left[\,\hat{\vec{p}}-\frac{1}{c}\sum\limits_{j}\hat{\vec{A}}_{j}\right]^{2}+U(\vec{r},t)\;.$
(2)

The initial state for each mode $j$ of the light field in this representation
is
$G_{j}({{\color[rgb]{0,0,0}q_{j},\,}}0)=C_{0\,j}\cdot\exp\left[-\frac{1}{2}(q_{j}-\sqrt{2N_{j}})^{2}\right]$,
the TDSE for the field states in vacuum is $i\dot{G}_{j}^{\;vac}=0$, and the
quantized field vector potential operator reads:

$\displaystyle\hat{\vec{A}}_{j}=$
$\displaystyle{\vec{\chi}}_{j}\;\beta_{j}q_{j}\;\cos(\vec{\kappa}_{j}\vec{r}-\omega_{j}t-\theta_{j})$
(3)
$\displaystyle-i{\vec{\chi}}_{j}\;\beta_{j}\frac{\partial}{\partial{}q_{j}}\;\sin(\vec{\kappa}_{j}\vec{r}-\omega_{j}t-\theta_{j})\,,$

where ${\vec{\chi}}_{j}$ are polarization vectros (linear in this case), and
$\beta_{j}=c\sqrt{2\pi/\omega_{j}{V}}$ are parameters (with the same
dimensionality as ${A}_{j}$) which depend on quantization volume $V$,
according to the conventional plane-wave quantization procedure quant1 . In
our case, when the initial mode state is a coherent state, each $\beta_{j}$
parameter naturally connects the field mode amplitude $A_{0\,j}$ and the
initial average photon number in this mode $N_{j}$ as follows
$A_{0\,j}=\beta_{j}\sqrt{2N_{j}}$, and consequently,
$\beta\sim\left[\left\langle\hat{W}_{int}\right\rangle/\left\langle\hat{H}_{field}\right\rangle\right]^{\frac{1}{2}}$.
A typical laser pulse, which is finite in space and time and consists of a
number of modes, can be described in terms of an effective quantization volume
$V_{eff}$. This volume is related to the sizes of the radiation focusing area
and the effective interaction volume, see also quant1 ; nqvol . Based on the
estimations for the number of relevant experiments with strong laser pulses
interacting with gases or solids, where quantum-optical effects are measurable
wP0 ; quantExp , the effective values are $(\beta_{j}/A_{0j})\sim{}10^{-7}$ or
even much less. These parameters are also responsible for the quantum
dispersion of the quantized radiation in the presence of charged particles,
see my1q . We refer $\beta_{j}$ as the fundamental scaling parameters for the
corresponding light-matter interaction problem, when $E_{0j}\ll{}E_{atomic}$
in the optical wavelength range. Due to its very small values, we may conclude
from Eq.(2) that even in a strong-field case, the back-action of one electron
to the light-mode wavefunction is small and depends somehow on $\beta_{j}$.
Therefore, the state of electromagnetic field $G(q_{j},t)$ is affected as
$G\approx{}G_{0}+\sum{}O(\beta_{j})$. Finding the equation for $G$ is a key
point of our study.

It was rigorously demonstrated that the solution of the two-component TDSE
with an arbitrary Hamiltonian can be written as a product two special
wavefunctions each satisfying the normalization condition, see EF1  for full
details. In our case, it leads to the following exact factorization:
$\Psi=F_{ex}(\vec{r},q_{j},t)\cdot{}G_{ex}({q_{j},t})$, with the related
system of equations for the $F_{ex}$ and $G_{ex}$ of the same complexity as
the original TDSE, see also EF1 . Instead, based on our previous
considerations regarding the dependance on parameters $\beta_{j}$, we consider
the solution of Eq.(2) of the form
$\Psi\approx{}F_{G_{0}}(\vec{r},q_{j},\beta_{j},t)\cdot{}G({q_{j},\beta_{j},t})+\sum{}O(\beta_{j}^{s})$.
In our approximation, we have the same form of the solution with the same
partial normalization conditions: $\left\langle
F|F\right\rangle=\int{}|F(\vec{r},q_{j},t)|^{2}d^{3}r\equiv{}1$, $\left\langle
G|G\right\rangle=\int{}|G(q_{j},t)|^{2}[d\,q_{j}]\equiv{}1$. The
representation is not unique since one can construct others one by choosing
$\Psi=F(\vec{r},q_{j},t)e^{i\chi(q_{j},t)}\,\cdot\,G(q_{j},t)e^{-i\chi(q_{j},t)}$
with an arbitrary function $\chi(q_{j},t)$ matching the initial condition at
$t=0$, see also EF1 . In order to have a uniqueness of the representation for
a given initial state, we require the condition: $\left\langle
F|\dot{F}\right\rangle=0$, which is often referred to as a static gauge of the
factorization ClassFL .

Using the considered form of the solution, we can write:

$i\dot{\Psi}=i\dot{F}G+iF\dot{G}=\hat{H}\,FG\;,$ (4)

and obtain the decoupling in first-order approximation:

$\displaystyle i\dot{F}=$
$\displaystyle\left[G^{-1}\hat{H}G\right]\,F-i\,\frac{\dot{G}}{G}\,F\approx\left[G_{0}^{-1}\hat{H}G_{0}\right]\,F-i\phi\,F\,,$
(5a) $\displaystyle\;\text{since:}\;G=G_{0}+\sum_{j}O(\beta_{j})\;,\;t>0\,;$
$\displaystyle i\dot{G}=$ $\displaystyle\left\langle
F\left|\hat{H}\right|F\right\rangle\,G\,,\;\text{under the static
gauge:}\,\left\langle F|\dot{F}\right\rangle=0\,.$ (5b)

From this it follows that in a first step of a reasonable approximation, for
the electron subsystem, we may use $\beta_{j}$-independent states of light
modes for the approximation of the vector potential operator. Thus, we can
exclude the non-local operator $\frac{\partial}{\partial{}q_{j}}$ in the
equation for the electron subsystem (since
$\beta_{j}\frac{\partial}{\partial{}q_{j}}F\sim{}\beta_{j}^{2}$), having
instead a simple local operator as follows:
$\beta_{j}\frac{\partial}{\partial{}q_{j}}\rightarrow\\{\beta_{j}G_{0\,j}^{-1}\frac{\partial}{\partial{}q_{j}}G_{0\,j}\\}$.
Starting from Eq.(5a), we assume that $G(q,t)$ does not have zeros within a
time interval of consideration. This can be in the case when one starts with
coherent state and considers conditions of relatively small deviations,
leading for example to the various nonclassical/squeezed states. Otherwise the
procedure can be modified in a way that instead of Eq.(5a) we can write
$iG\dot{F}=\hat{H}\;G_{0}F-i\dot{G}F$ with the similar subsequent related
considerations. The described approach allows us to decouple the total system
Eq.(2) and separate the equations for the electronic subsystem and ”dressed”
light subsystem via $\beta_{j}$ parametrical connections. We obtain finally
for the quantum light-electron interaction problem:

$\displaystyle
i\dot{F}=\hat{H}_{P}F\;,\;\;\;\;\hat{H}_{P}=\frac{1}{2}\left[\,\hat{\vec{p}}-\frac{1}{c}\sum\limits_{j}\vec{A}_{j}^{par}\right]^{2}+U(\vec{r},t)\;,$
(6a) $\displaystyle
i\dot{G}=\hat{H}_{B}G\;,\;\;\;\;{{\color[rgb]{0,0,0}\hat{H}_{B}=\left\langle
F\left|\hat{H}\right|F\right\rangle}}\;,$ (6b)

where
$\vec{A}_{j}^{par}={{\color[rgb]{0,0,0}\vec{\bf\chi}_{j}}}\\{\beta_{j}q_{j}\\}\cos(\vec{\kappa}_{j}\vec{r}-\omega_{j}t-\theta_{j})-i{{\color[rgb]{0,0,0}\vec{\bf\chi}_{j}}}\\{\beta_{j}G_{0\,j}^{-1}\frac{\partial}{\partial{}q_{j}}G_{0\,j}\\}\sin(\vec{\kappa}_{j}\vec{r}-\omega_{j}t-\theta_{j})$
is a field vector potential operator which depends on $q_{j}$-expressions and
a set of scaling parameters $\\{\beta_{j}\\}$. In the strong-field case with
coherent initial state of light, when $N_{j}\gg{}1$ (and consequently
$\langle{q_{j}}\rangle=\sqrt{2N_{j}}\gg{}\langle{\frac{\partial}{\partial{}q_{j}}}\rangle=0$),
we can simply write
$\vec{A}_{j}^{par}={{\color[rgb]{0,0,0}\vec{\bf\chi}_{j}}}\\{\beta_{j}q_{j}\\}\cos(\vec{\kappa}_{j}\vec{r}-\omega_{j}t-\theta_{j})$
noting that this is nothing else than the parametric description of classical
vector potential ($\\{\beta_{j}q_{j}\\}$ corresponding to the mode amplitudes
in the classical description). This description however connects naturally
both subsystems: the interacting electron subsystem $F$ with field parameters
$\\{\beta_{j}q_{j}\\}$, and the "dressed" light field which depends on $F$ via
the avereging procedure of $\hat{H}_{B}=\left\langle
F\left|\hat{H}\right|F\right\rangle$. The error of our first-order
approximation in this case is proportional to some combination of
$\beta_{j}^{1}$.

Now, let us apply our approach to some simple model cases of light-electron
interaction. We consider the strong-field case, when $N_{j}\gg{}1$. For
simplicity, we consider also the so-called dipole approximation when the
dependence of the vector potential on the coordinate is neglected, since
$|\vec{\kappa}_{j}(\vec{r}-\langle\vec{r}\rangle_{0})|\ll{}1$, which is common
for the interaction in the optical wavelength range.

First, let us consider a non-bound electron wave packet
($U(\vec{r},t)\equiv{}0$), localized in some region of space with initial
momentum 0. Then, the system of Eqs.(6a-6b) can be solved analytically:

$\displaystyle\Phi(\vec{r},\\{\beta_{j}q_{j}\\},t)=$
$\displaystyle\exp\bigg{[}-\frac{1}{2}it\hat{p}^{2}+\frac{i}{c}\hat{p}\int\limits^{t}\vec{A}_{P}(\tau)d\tau$
(7a)
$\displaystyle-\frac{1}{2c^{2}}i\int\limits^{t}\vec{A}^{2}_{P}(\tau)d\tau\bigg{]}\,\Phi(\vec{r},0)\;,$
$\displaystyle i\dot{G}=$
$\displaystyle\frac{1}{2}\left[-\frac{1}{c}\sum\limits_{j}\hat{\vec{A}}_{j}\right]^{2}G\;.$
(7b)

Eq.(7a) describes the dynamics of the electronic wave packet in a given field
vector potential
$\vec{A}_{P}=\sum\limits_{j}\vec{A}_{j}^{par}(\\{\beta_{j}q_{j}\\},t)$ while
Eq.(7b) gives the correction to the initial field state $G(\\{q_{j}\\},0)$ due
to the interaction with the electron. The solution of Eq.(7b) includes terms
$\exp(d_{ij}(t)q_{i}q_{j})$ (where $d_{ij}(t)$ are some time-dependent
functions), thus it demonstrates entanglement between different modes during
the interaction. In case of a one-mode field, the solution of Eq.(7a-7b)
corresponds to the closed-form solution screp  with a relative error
$\sim{}\beta^{2}$.

Let us now consider the more complicated case, when the electron is bound in a
quadratic potential $U=u(t)\,{\bf x}^{2}$. This situation can be related to
some model bound potential ($u(t)=\text{const}$) or to the problem of an
electron in magnetic fields (time-independent or time-dependent). Since, in
that case, the total Hamiltonian $\hat{H}_{P}$ for the interacting system is
quadratic, the solution for $\Phi$ can be obtained analytically. The analysis
of the solution is beyond the scope of this article, however the main
exponential part of the solution is evident:

$\displaystyle G$ $\displaystyle(\\{q_{j}\\},t)=$ $\displaystyle
C(t)\cdot\exp\left[\sum\limits_{j}a_{j}(t)q_{j}^{2}+\sum\limits_{j}b_{j}(t)q_{j}+\sum\limits_{i\neq{}j}d_{ij}(t)q_{i}q_{j}\right]\;,$
(8a)

where the coefficients $a_{j}(t)$, $b_{j}(t)$, $d_{i\,j}(t)$ depend on the
initial states of the modes, initial state of the electron, and $u(t)$. We
would like to emphasize that the parameters of the exponential part Eq.(8) can
be controlled by external parameters in various ways. This means that
depending on the conditions and the binding potential, one can generate
nonclassical states of light. In particular, when $(\,a_{j}<-\frac{1}{2}\,)$,
it is possible to observe amplitude-squeezed states. It is possible to obtain
also multi-mode squeezed states of different types.

Finally, we would like to note our approximation in the context of multi-
electron systems. The generalization is straitforward:

$\displaystyle i\dot{F}=$
$\displaystyle\left[\sum_{e}\hat{H}_{P}+\sum_{e\neq{}e^{\prime}}U_{e{}e^{\prime}}\right]F\;,\;\;\;\;$
(9a)
$\displaystyle\hat{H}_{P}=\sum_{e}\frac{1}{2m_{e}}\left[\,\hat{\vec{p}}_{e}-\frac{1}{c}\sum\limits_{j}\vec{A}_{j}^{par}\right]^{2}+\sum_{e}U(\vec{r},t)\;,$
$\displaystyle i\dot{G}=$
$\displaystyle\hat{H}_{B}G\;,\;\;\;\;\hat{H}_{B}=\left\langle
F\left|\hat{H}_{P}\right|F\right\rangle\;,$ (9b)

This system can be further used to unify various multi-electron theories (for
example band structure theory) with quantized light description for the
strong-field interaction problem. The noticeable difference with the one-
electron consideration is that the fundamental scaling parameter here is
$\beta\sim\left[N_{e}\cdot{}W_{e}/W_{field}\right]^{\frac{1}{2}}$, where
$N_{e}\cdot{}W_{e}$ is the total oscillating energy of charged particles.
Although, in many relevant cases it still can be very small, in some other
cases, higher order approximations may be necessary.

——————————–
Conclusions.– We present an approach which opens the possibilities to simplify
and further study some quantum phenomena in interaction of light with complex
quantum systems. This approach is analogous to the BO-approximation for the
multi-system interactions of different nature, including possible non-local
interactions. The general scheme can be further improved by additional
subsequent steps beyond the considered first-order approximation. Our approach
can be applied to the general light-matter interaction problem and the various
related particular problem. We obtain a system of equations connected
parametrically which allows to simplify the problem significantly. We
demonstrate multi-mode and light-electron entanglement during the interaction,
and the possibility to generate nonclassical states of light. Our approach can
explain recent experiments with intense interacting light wP0 ; quantExp  and
shed light to some related phenomena towards strong-field quantum optics and
beyond.

###### Acknowledgements.

I. G. and S.G. highly acknowledge funding by the ERC consolidator grant QUEM-
CHEM (772676).

## Appendix: COHERENT STATE DETAILS

We consider the following function:

$\displaystyle\Psi_{c}(q,t)=$ $\displaystyle
C(t)\cdot{}\exp\left[-\frac{1}{2}q^{2}+q\,e^{-i\omega{t}+i\Theta}\sqrt{2N_{0}}\right]\;,$
(10a) $\displaystyle C(t)=$
$\displaystyle\frac{1}{\sqrt[4]{\pi}}\exp\left[-\frac{1}{2}N_{0}\right]\cdot\exp\left[-\frac{i\omega{t}}{2}-\frac{1}{2}N_{0}\,e^{-2i\omega{t}+2i\Theta}\right]\;,$
(10b)

where $q$ is a parameter of the light wave function (light state internal
"coordinate"); $\Theta$ is an arbitrary given real number (phase of the light
wave, see later); $N_{0}$ is an arbitrary given constant (average photon
number, see later); $\omega$ is an arbitrary given frequency of the light
mode. This function is called quantum coherent state of the plane wave light
mode in coordinate representation because of its following properties:
1) It is an exact solution of the following Schrödinger equation:

$i\partial_{t}\Psi_{c}(q,t)=\omega\hat{N}\,\Psi_{c}(q,t)=\frac{\omega}{2}\left(q^{2}-\frac{\partial^{2}}{\partial{}q^{2}}\right)\Psi_{c}(q,t)\;,$
(11)

which describes the time-dependent states of a light plane wave with frequency
$\omega$ propagating in vacuum. Note, this is a Schödinger representation of
Schrödinger equation.
2) It is normalized:
$\int\limits_{-\infty}^{+\infty}|\Psi_{c}(q,t)|^{2}dq\equiv{1}$.
3) Within the following definitions of the field operators:

$\displaystyle\hat{\vec{A}}={\bf{{x}_{0}}}\,\frac{1}{\sqrt{2}}{\beta}\,\Big{[}\hat{a}\,e^{i\kappa{z}}+\hat{a}^{+}\,e^{-i\kappa{z}}\Big{]}\;,$
(12a)
$\displaystyle\hat{a}=\frac{1}{\sqrt{2}}\Big{(}q+\frac{\partial}{\partial{q}}\Big{)}\;,$
(12b)
$\displaystyle\hat{a}^{+}=\frac{1}{\sqrt{2}}\Big{(}q-\frac{\partial}{\partial{q}}\Big{)}\;,$
(12c)
$\displaystyle\hat{N}=\Big{(}\hat{a}^{+}\hat{a}+\frac{1}{2}\Big{)}=\frac{1}{2}\,\Big{(}q^{2}-\frac{\partial^{2}}{\partial{q}^{2}}\Big{)}\;,$
(12d) $\displaystyle\beta=c\sqrt{2\pi/\omega{V}}\,,$ (12e)

We can obtain the following quantum-mechanically averaged values:

$\displaystyle\langle{q}\rangle_{c}=\sqrt{2N_{0}}\,\cos\left(\omega{t}-\Theta\right)\;,$
(13a)
$\displaystyle\langle\frac{\partial}{\partial{q}}\rangle_{c}=-i\sqrt{2N_{0}}\,\sin\left(\omega{t}-\Theta\right)\;,$
(13b)
$\displaystyle\langle\hat{\vec{A}}\rangle_{c}={\bf{{x}_{0}}}\,\beta\sqrt{2N_{0}}\,\cos\left(\omega{t}-\Theta-\kappa{z}\right)$
$\displaystyle\;\;\;\;\;\;\;\;={\bf{{x}_{0}}}\,A_{0}\,\cos\left(\omega{t}-\Theta-\kappa{z}\right)\;,$
(13c) $\displaystyle\langle\hat{N}\rangle_{c}=\frac{1}{2}+N_{0}\;,$ (13d)

where
$\langle\hat{R}\rangle_{c}=\int\limits_{-\infty}^{+\infty}\Psi_{c}^{*}(q,t)\,\hat{R}\,\Psi_{c}(q,t)\,dq$.
Finally, we note that
$\;\hat{a}\;\Psi_{c}=e^{-i\omega{t}+i\Theta}\sqrt{N_{0}}\;\Psi_{c}$
establishing a connection with the definition of the canonical coherent state
in Schödinger representationquant1 .

## Data Availability Statement

The data that supports the findings of this study are available within the
article.

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