# Stick-slip and convergence of feedback-controlled systems with Coulomb
friction

Michael Ruderman<EMAIL_ADDRESS>University of Agder, 4604-Norway

###### Abstract

An analysis of stick-slip behavior and trajectory convergence in feedback-
controlled motion systems with discontinuous Coulomb friction is provided. A
closed-form parameter-dependent stiction region, around an invariant
equilibrium set, is proved to be always reachable and globally attractive. It
is shown that only asymptotic convergence can be achieved, with at least one
but mostly an infinite number of consecutive stick-slip cycles, independent of
the initial conditions. Theoretical developments are supported by a number of
numerical results with dedicated convergence examples.

###### keywords:

Coulomb friction , limit cycles , PID control , convergence analysis , sliding
mode , discontinuities

##  1 Introduction

Feedback-controlled motion systems are mostly subject to nonlinear friction,
and the direction-dependent Coulomb friction force plays a crucial role owing
to a (theoretical) discontinuity at velocity zero-crossings. Although the more
complex dynamic friction laws (see, e.g., [1, 2, 3] and references therein)
allow frictional discontinuity to be bypassed during analysis, the basic
Coulomb friction phenomenon continues to present the same challenges in terms
of controller convergence, especially in the presence of an integral control
action. Associated stick-slip behavior and so-called frictional limit cycles
were formerly addressed in [4]. An algebraic prediction of stick-slip, with a
large set of parametric equalities, was compared to the describing function
method, while the Coulomb plus static friction law was assumed for avoiding
discontinuity at the velocity zero-crossing. An explicit solution for
friction-generated limit cycles has also been proposed in [5], necessitating
static friction approximation (to avoid discontinuity) and also requiring the
stiction friction which is larger than the Coulomb friction level. Despite
including explicit analysis of state trajectories for both sticking and
slipping phases, no straightforward conclusions about the appearance and
convergence of stick-slip behavior have been reported. The appearance of
friction-induced (so-called hunting) limit cycles has been briefly addressed
in [6], for the assumed LuGre [7] and so-called switch [8] friction models.
Note that a former analysis of stick-slip behavior and associated friction-
induced limit cycles can be found in [9]. An explanation of how a
proportional-feedback-controlled motion with Coulomb friction comes to
sticking was subsequently shown in [10] by using the invariance principle.
Stick-slip behavior, as an observable phenomenon known in control engineering
practice, was highlighted in [1], and several other control studies have since
attempted to analyze and compensate such processes. For instance, a related
analysis of under-compensation and over-compensation of static friction was
reported in [11]. Issues associated with a slow (creeping-like) convergence of
the feedback-controlled motion in the presence of Coulomb friction have been
addressed and experimentally demonstrated in [12]. More recently, the
convergence problems (of PID feedback control) have been well demonstrated,
with accurate experiments reported, in [13] while attempting to reduce
settling errors by the reset integral control [14]. The related analysis has
been reported before in [15]. Despite a number of experimental observations
and elaborated studies reported in the literature, it appears that no general
consensus has been established in relation to friction-induced stick-slip
cycles in feedback-controlled systems with Coulomb friction. In particular,
questions arise over when and under which conditions the stick-slip cycles
occur, and how a PID-controlled motion will converge to zero equilibrium in
the presence of Coulomb friction (with discontinuity). Note that the problem
of slow convergence in vicinity to a set reference position is of particular
relevance for advanced motion control systems, see, e.g., [16], but in the PID
design and tuning, see, e.g., [17], the associated issues are not widely
accepted and have yet to be formalized, that despite huge demand from
precision engineering control tasks. This gap should not come as a complete
surprise, however, given the nontrivial friction microdynamics, visible in
experimental studies [18, 19, 20], and the uncertain and time-varying friction
behavior, see, e.g., [21].

This paper is intended to contribute to the analysis of feedback-controlled
system convergence with Coulomb friction and, thus, to understanding of stick-
slip cycles occurring in servo mechanisms. In order to keep the analysis as
general as possible and to clarify the principal phenomenon, we have assumed
the classical Coulomb friction law with discontinuity. This led us
(unavoidably) to variable structure system dynamics, distinguishing between
the modes of motion sticking and slipping. At the same time, we show that all
state trajectories always remain continuous and almost always differentiable
(except finite switching between both modes). We provide theorems and identify
the conditions to demonstrate the sticking region around zero equilibrium
reachable and globally attractive. The main contribution of analysis is
further reinforced by several illustrative numerical examples.

###  1.1 Problem statement

Throughout the paper, we will deal with the feedback-controlled systems
described by

$\ddot{x}(t)+K_{d}\dot{x}(t)+K_{p}x(t)+K_{i}\int x(t)dt+F(t)=0,$ (1)

where the derivative, proportional and integral feedback gains are $K_{d}$,
$K_{p}$ and $K_{i}$, respectively. The nonlinear friction (that with
discontinuity) is denoted by $F$, and the set-value control problem is reduced
to the convergence problem for a non-zero initial condition, i.e., $x(0)\neq
0$. Furthermore, we use the following simplifications of the system plant
without loss of generality: the relative motion of an inertial body with unity
mass is considered in the generalized $(x,\dot{x})$ coordinates. The inherent
system damping (including linear viscous friction) and stiffness (of restoring
spring elements) are incorporated (if applicable) into $K_{d}>0$ and
$K_{p}>0$, respectively. There are no actuator constraints, so feedback for
the integral output error is directly applicable via the gain factor
$K_{i}>0$.

The control problem (1) has long been associated with issues of slow and/or
cyclic convergence of $x(t)$ in the vicinity of steady-state for the set
reference value. This (sometimes called hunting behavior or even hunting limit
cycles) has been addressed in analysis and also observed in several controlled
positioning experiments, e.g., [1, 4, 5, 6, 12, 13]. These phenomena seem to
be associated with integral control action and nonlinear (Coulomb-type)
friction within a vanishing region around equilibria, where the potential
field of proportional feedback weakens and cannot provide $x(t)\rightarrow 0$
at certain (application-required) times $t<\mathrm{const}$. The hunting
behavior is directly cognate with the stick-slip, where a smooth (continuous)
motion alternates with a sticking phase of zero or slowly creeping
displacement. Stick-slip appearance, parametric conditions and convergence in
semi-stable limit cycles are the focus of our study, while we assume the
Coulomb friction force with discontinuity.

##  2 Stiction due to discontinuous Coulomb friction

In this Section, we will analyze the stick-slip behavior of the (1) system,
for which the classical Coulomb friction with discontinuity is represented by
$F(\dot{x})=F_{c}\,\mathrm{sign}(\dot{x})$. Here, the Coulomb friction
coefficient is $F_{c}>0$, and the sign operator is defined by

$\mathrm{sign}(z)=\left\\{\begin{array}[]{ll}1,&\;z>0,\\\
\left[1,1\right],&\;z=0,\\\ -1,&\;z<0.\\\ \end{array}\right.$ (2)

Note that (2) constitutes an ideal relay with instantaneous switching upon
change of the input sign. We also note that for a zero-displacement rate, the
friction equation becomes an inclusion $F(0)\in[-F_{c},F_{c}]$ in the Filippov
sense [22] when seeking the corresponding solution.

We will consider the feedback-controlled system in a minimal state-space
representation, as follows:

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle Ax+Bu,$ (3)
$\displaystyle y$ $\displaystyle=$ $\displaystyle Cx,$ (4) $\displaystyle u$
$\displaystyle=$ $\displaystyle-\mathrm{sign}(y).$ (5)

Note that in this way, we also approach the system notation provided in [23]
for analysis of relay feedback systems (RFSs). Introducing the state vector
$x=(x_{1},x_{2},x_{3})^{T}\in\mathbb{R}^{3}$ of the integral, output and
derivative errors, (1) can be rewritten as (3)-(5), with the system matrix

$A=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ -K_{i}&-K_{p}&-K_{d}\\\
\end{array}\right),$ (6)

and input and output distribution vectors

$B=\left(\begin{array}[]{c}0\\\ 0\\\ F_{c}\\\ \end{array}\right),\quad
C^{T}=\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\ \end{array}\right)$ (7)

correspondingly.

###  2.1 Without integral feedback

Firstly, we consider the (3)-(7) system without an integral feedback action,
meaning $K_{i}=0$. In this case, the phase-plane
$(x_{2},x_{3})\in\mathbb{R}^{2}$ is divided into two regions

$P^{+}=\\{x\in\mathbb{R}^{2}:x_{3}>0\\},\quad
P^{-}=\\{x\in\mathbb{R}^{2}:x_{3}<0\\}$ (8)

by the discontinuity manifold $S=\\{x\in\mathbb{R}^{2}:x_{3}=0\\}$. It can be
seen that in the discontinuity manifold $S$, the vector fields of the state
value $x_{s}$ are given by

$\displaystyle f^{+}(x_{s})$ $\displaystyle=\overset{x\in
P^{+}}{\underset{x\rightarrow
x_{s}}{\lim}}(Ax+Bu)=\left(\begin{array}[]{c}0\\\
-K_{p}x_{2}-F_{c}\end{array}\right),$ (11) $\displaystyle f^{-}(x_{s})$
$\displaystyle=\overset{x\in P^{-}}{\underset{x\rightarrow
x_{s}}{\lim}}(Ax+Bu)=\left(\begin{array}[]{c}0\\\
-K_{p}x_{2}+F_{c}\end{array}\right)$ (14)

and are pointing in opposite directions within $|x_{2}|\leq F_{c}K_{p}^{-1}$.
In contrast, outside of this region (denoted by $S_{0}$ in Figure 1), both
vector fields are pointing in the same direction, towards $P^{+}$ for
$x_{2}<-F_{c}K_{p}^{-1}$ and towards $P^{-}$ for $x_{2}>F_{c}K_{p}^{-1}$.

Figure 1: Phase portrait of $(x_{2},x_{3})$-trajectories of the (3)-(7) system
without integral feedback, attracted to $S_{0}$ from various initial values

Since both vector fields are normal to the manifold $S$, neither smooth motion
nor sliding mode can occur for the $(x_{2},x_{3})\in S_{0}$ trajectories,
which means that any trajectory reaching $S_{0}$ will remain there
$\forall\>t\rightarrow\infty$. Thus, $S_{0}$ constitutes the largest invariant
set of equilibrium points for (3)-(7) without integral control action. Note
that this has also been shown in [10] and is well known when a relative motion
with Coulomb friction is controlled by the proportional-derivative (PD)
feedback only. In this case, the set value error can be reduced by increasing
$K_{p}$ but cannot be driven to zero as long as $F_{c}\neq 0$. The phase
portraits of the trajectories converging to $S_{0}$ are exemplary shown in
Figure 1.

###  2.2 With integral feedback

When allowing for $K_{i}\neq 0$, it is intuitively apparent that having
reached point $x(t_{s})\in S_{0}$ at $t=t_{s}$, the trajectory cannot remain
there for all times $t_{s}<t<\infty$. While for the motion states
$x_{3}(t_{s})=0$ and $x_{2}(t_{s})=\mathrm{const}\neq 0$, the integral control
effort $K_{i}x_{2}(t_{s})\int^{t}_{t_{s}}dt$ grows continuously and, at some
finite time ($t_{c}>t_{s}$), should lead to the breakaway [24] and a new onset
of continuous motion. This alternating phase, upon system sticking, is often
referred to as _slipping_ , cf., e.g., [5], so that a stick-slip motion [1, 3]
appears, also in the form of limit cycles. In order to analyze the friction-
induced limit cycles, sometimes denoted as hunting-limit cycles, cf. [6], we
firstly need to look into the system dynamics during stiction, i.e., for
$t_{s}<t<t_{c}$. Recall that in this phase, the system (3)-(7) yields as
continuously switching, owing to $x_{3}=0$ and to the discontinuous relay
nonlinearity in the feedback loop. The developments given below are motivated
by analysis of the existence of fast switches, provided in [23] for RFS, while
the obtained results rely on the sliding-mode principles, see, e.g., [25, 26].

Consider the switching variable (or, more generally, surface) $S=Cx(t)=0$, for
which the sliding mode should occur on the manifold $S$. This requires that
the existence and reachability condition, cf. [25],

$\dot{S}S\leq-\eta|S|$ (15)

is fulfilled, where $\eta$ is a small positive constant.

######  Theorem 1.

Given is the control system (3)-(7) with Coulomb friction. The system is
sticking at $x_{3}=0$ iff

$|K_{i}x_{1}|+|K_{p}x_{2}|\leq F_{c}.$ (16)

###### Proof.

The system remains sticking as long as it is in the sliding mode for which
(15) is fulfilled. The sliding-mode condition (15) can be rewritten as

$\dot{S}\,\mathrm{sign}(S)\leq-\eta,$ (17)

while the time derivative of the sliding surface is

$\dot{S}=(CAx\pm CB)=\bigl{(}CAx-\mathrm{sign}(S)CB\bigr{)},$ (18)

depending on the sign of $Cx$. Substituting (18) into (17) results in

$\displaystyle CAx$ $\displaystyle\leq$ $\displaystyle
CB-\eta\qquad\hbox{for}\quad\mathrm{sign}(S)>0,$ (19) $\displaystyle-CAx$
$\displaystyle\leq$ $\displaystyle
CB-\eta\qquad\hbox{for}\quad\mathrm{sign}(S)<0.$ (20)

Since $CB,\eta>0$, the inequalities (19) and (20) can be summarized in

$|CAx|\leq CB-\eta.$ (21)

Evaluating (21) with $x_{3}=0$ and $0\neq\eta\rightarrow 0^{+}$ results in
(16) and completes the proof. ∎

######  Remark 1.

The condition obtained using the Theorem 1 is equivalent to the set of
attraction $\\{x\in S:|CAx|<|CB|\\}$ for $CB>0$ demonstrated in [23, Section
4].

Now, we are interested in the state dynamics during stiction, which means
within the sliding mode. Since staying in the sliding mode (corresponding to
$S\equiv 0$ on the switching surface) requires

$\dot{S}=C\dot{x}=CA\dot{x}+CBu=0\qquad\hbox{for}\quad t_{s}<t<t_{c},$ (22)

one obtains the so-called equivalent control as

$u_{e}=-(CB)^{-1}CAx.$ (23)

Recall that an equivalent control, [26], is the linear one (i.e., with no
relay action) which is required to maintain the system in an ideal sliding
mode without fast-switching. Consequently, substituting (23) into (3) results
in the equivalent system dynamics

$\dot{x}_{e}=\bigl{[}I-B(CB)^{-1}C\bigr{]}Ax=OAx,$ (24)

which governs the state trajectories as long as the system remains in the
sliding mode. $O$ is the so-called projection operator of the original system
dynamics, satisfying the properties $CO=0$ and $OB=0$. Evaluating (24) with
(6) and (7) yields the system dynamics during stiction as

$\left(\begin{array}[]{c}\dot{x}_{1}\\\ \dot{x}_{2}\\\ \dot{x}_{3}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 0&0&0\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}(t_{s})\\\ x_{2}(t_{s})\\\ 0\\\
\end{array}\right).$ (25)

It can be seen that neither relative displacement nor its rate will change
when the system is sticking, although the integral error grows according to

$x_{1}(t)=x_{1}(t_{s})+x_{2}(t_{s})\int\limits^{t_{c}}_{t_{s}}dt.$ (26)

It can be noted that if $K_{i}=0$ then the condition (16), correspondingly the
inequality $|CAx|\leq|CB|$, reduces to $|x_{2}|\leq F_{c}K_{p}^{-1}$, while
the sliding mode (25) reduces to the zero dynamics of the system in stiction
(the results we obtained in Section 2.1).

###  2.3 Region of attraction

The given Theorem 1 provides the necessary (and sufficient) condition as the
(3)-(7) system remains sticking. Yet it is also necessary to demonstrate
global attraction of state trajectories to the stiction region. Recall that
the latter corresponds to the subset

$S_{0}=\\{x\in\mathbb{R}^{3}\,:\,x_{3}=0,\,|K_{i}x_{1}|+|K_{p}x_{2}|\leq
F_{c}\\}$ (27)

where the sliding mode occurs (cf. (16) and (25)).

Firstly, we will explore the persistence of the sliding mode, meaning we will
prove whether the system can stay incessantly inside of
$S_{0}^{s}=\\{S_{0}\,:\,x_{2}\neq 0\\}$, i.e., in all times
$t_{s}<t\rightarrow\infty$. By making $(x_{1},x_{2})$-projection of
$x\in\mathbb{R}^{3}$, one can show that (16) results in a rhombus, as
schematically illustrated in Figure 2.

Figure 2: Rhombus-shape, in $(x_{1},x_{2})$-projection, of the $S_{0}$-region
of attraction, with vector-field during the stiction mode and example of an
entering and leaving trajectory at $t_{s}$ and $t_{c}$, respectively

The indicated vector field is unambiguous due to the integral control action
(cf. the sliding-mode dynamics (25)). This means that after reaching
$S_{0}^{s}$ at $t_{s}$, any trajectory will leave it at $t_{c}$ having hit the
boundary of $S_{0}$. Denoting the point of reaching $S_{0}$ by
$x(t_{s})\equiv(x^{s}_{1},x^{s}_{2},0)$, one can calculate the new point of
leaving $S_{0}^{s}$ as

$x_{1}(t_{c})=:x^{c}_{1}=x^{s}_{2}\bigl{(}F_{c}|x^{s}_{2}|^{-1}-K_{p}\bigr{)}K^{-1}_{i}.$
(28)

Correspondingly, from (26) and (28) one obtains the time of leaving
$S_{0}^{s}$ as

$t_{c}=\Bigl{[}x^{s}_{2}\bigl{(}F_{c}|x^{s}_{2}|^{-1}-K_{p}\bigr{)}K^{-1}_{i}-x^{s}_{1}+x^{s}_{2}t_{s}\Bigr{]}(x^{s}_{2})^{-1}.$
(29)

From the above, it can be recognized that if $K_{i}\rightarrow 0$, the
stiction region $S_{0}$ blows to the entire $(x_{1},x_{2})$-subspace and,
consequently, $t_{c}\rightarrow\infty$. This means that a system trajectory
will never leave $S_{0}$ having reached it (a result fully in line with what
was demonstrated in Section 2.1). On other hand, if allowing for
$K_{i}\rightarrow\infty$ the time instant $t_{c}\rightarrow t^{+}_{s}$,
according to (29) with $x^{s}_{1}\rightarrow 0$, and this is due to $S_{0}$
collapsing to $\mathrm{proj}_{x_{2}}S_{0}$.

Let us now demonstrate that $S_{0}$ is globally attractive for all initial
values outside of $S_{0}$, meaning
$\forall\>x(t_{0})\in\mathbb{R}^{3}\backslash S_{0}$. As with the eigen-
dynamics in (3) and (6), which is linear, one can ensure the global
exponential stability by analyzing the characteristic polynomial

$s^{3}+K_{d}s^{2}+K_{p}s+K_{i}=0,$ (30)

and applying the standard Routh-Hurwitz stability criterion. The control
parameters condition

$K_{d}K_{p}>K_{i}$ (31)

should then be satisfied, guaranteeing that all eigenvalues $\lambda_{i}$ of
the (6) system matrix have $\mathrm{Re}\\{\lambda_{i}\\}<0$ with
$i=1,\ldots,3$. Then, the resulting (switched) $\dot{x}=Ax\mp B$ subsystems
behave as asymptotically stable in both subspaces
$\\{x\in\mathbb{R}^{3}\backslash S_{0}:x_{3}\gtrless 0\\}$ correspondingly. It
should be noted that the condition of the above parameters is conservative,
since the Coulomb friction itself is always dissipative, independently of
whether $x_{3}>0$ or $x_{3}<0$. This can be shown by considering the
dissipated energy

$\bar{V}(t)=-F(t)\int\dot{x}(t)dt=-F(t)\bar{x},$ (32)

which is equivalent to a mechanical work provided by the constant friction
force $F$ along a unidirectional displacement $\bar{x}$. Taking the time
derivative of (32) and substituting the Coulomb friction law results in

$\displaystyle\frac{d}{dt}\bar{V}(t)$ $\displaystyle=$
$\displaystyle-\frac{d}{dt}F(t)\bar{x}-F(t)\frac{d}{dt}\bar{x}$ (33)
$\displaystyle=$ $\displaystyle
0-F_{c}\mathrm{sign}\bigl{(}\dot{x}(t)\bigr{)}\dot{x}(t)=-F_{c}\bigl{|}\dot{x}(t)\bigr{|}.$

Therefore, $\dot{\bar{V}}(t)<0$ for all $x_{3}(t)\neq 0$. This quite intuitive
yet relevant condition reveals the relay feedback (5) as an additional (rate-
independent) damping, which contributes to stabilization of the closed-loop
dynamics (3)-(7). This result will be further used for proof of the Corollary
1. Notwithstanding this, we will keep the conservative stability condition
(31) as the sufficient (but not necessary) one. This appears reasonable due to
the usually uncertain Coulomb friction parameter and, hence, in order for
increasing overall robustness of the feedback control system. The following
example should, however, exemplify the additionally stabilizing behavior of
the Coulomb friction, even when (31) is violated.

######  Example 1.

Consider the (3)-(7) system with $K_{d}=0$, $K_{p}=100$ and $K_{i}=1$. The
eigenvalues of the $A$ system matrix are $\lambda_{1}=-0.01$,
$\lambda_{2,3}=0.0005\pm 10j$, which implies the linear subsystem is
asymptotically unstable. It should also be noted that (31) is not fulfilled.
To evaluate the trajectories of the (3)-(7) system, one can use the particular
solution

$x(t)=e^{At}x(\tau)+A^{-1}\bigl{(}e^{At}-I\bigr{)}Bu,$ (34)

for the constant control $u=\mp 1$, which corresponds to the (5) relay,
switched in the $x_{3}>0$ and $x_{3}<0$ subspaces. The initial values
$x(\tau)=[x_{1}(t),x_{2}(t),0]^{T}$ at $t=\tau$ should be reassigned each time
the trajectory crosses $(x_{1},x_{2})$-plane, meaning the relay switches at
$x_{3}=0$ outside of $S_{0}$. The $x_{3}$-state trajectory, with an initial
value $x_{3}(0)=10$, is shown in Figure 3, once for $F_{c}=0$ (solid red line)
and once for $F_{c}=1$ (blue dash-dot line).

Figure 3: Velocity trajectories (from $x_{3}(0)=10$) of the Example 1 system
with $F_{c}=0$ (solid red line) and $F_{c}=1$ (blue dash-dot line)

It is easy to recognize, here, that even low-amplitude Coulomb friction
($F_{c}=1$ compared to the proportional feedback gain $K_{p}=100$) leads to
stabilization of the, otherwise, unstable closed-loop control system.

######  Corollary 1.

Consider the (3)-(7) system with the control parameters satisfying (31). The
stiction region (27), given by Theorem 1, is globally attractive for all
initial values outside of this region: $x(t_{0})\in\mathbb{R}^{3}\backslash
S_{0}$.

###### Proof.

By virtue of the passivity theorem, e.g., [27, 28], the feedback
interconnection of energy dissipating systems is also energy dissipating.
Since (3), (4) is dissipative when (31) is fulfilled, and (5) is also
dissipative for $x_{3}\neq 0$, their feedback interconnection yields
dissipative almost everywhere (except $x_{3}=0$) outside of $S_{0}$. This
implies that any $x(t)$ system trajectory, starting from the outside of
$S_{0}$, converges continuously, and some ball $\mathcal{B}\equiv\|x\|$ around
the origin shrinks over time:

$\|x(t_{2})\|<\|x(t_{1})\|\quad\forall\quad
t_{2}>t_{1},\>x\in\mathbb{R}^{3}\backslash S_{0}.$ (35)

For some $t_{3}>t_{2}$, the shrinking circle
$\mathrm{proj}_{(x_{1},x_{2})}\mathcal{B}\subseteq S_{0}$, and for $t_{4}\geq
t_{3}$, zero velocity $x_{3}(t_{4})=0$ will be reached. This implies
$x(t_{4})\in S_{0}$, which completes the proof.

∎

######  Remark 2.

The sliding-mode condition (17), which results in $|CAx|\leq CB$ and,
correspondingly, proves Theorem 1, constitutes the existence and reachability
condition for $S_{0}$, which is necessary but not sufficient. This is because
(16) does not contain any requirements imposed on the $K_{d}$-parameter value.
Theorem 1 and Corollary 1 constitute the necessary and sufficient conditions
for $S_{0}$ to be both globally reachable and attractive from outside of
$S_{0}$.

##  3 Analysis of stick-slip convergence

In this Section, we will analyze the convergence behavior of stick-slip
trajectories in the (3)-(7) system. Recall that having reached $S_{0}^{s}$ at
$t_{s}$, the $x(t)$ trajectory will leave it at $t_{c}$, given by (29), which
is due to the progressing $|x_{1}(t)|$ value, which will (unavoidably) violate
the stiction condition (16). To show (qualitatively) how state trajectories
evolve during a stick-slip cycle, consider the triple-integrator chain (see
Figure 4(a)), which arises out of the closed-loop dynamics (1).

(a)
(b)  (c)
(d)

Figure 4: Phase portrait at stick-slip; (a) triple-integrator chain, (b)
$(x_{1},x_{2})$-projection during sticking, (c) $(x_{2},x_{3})$-projection
during slipping, (d) typical trajectory during one stick-slip cycle

Eliminating the time argument, which is a standard procedure for a phase-plane
construction, one can write in general terms

$x_{n}dx_{n}=\dot{x}_{n}dx_{n-1},\quad n=\\{3,2\\},$ (36)

for the first and second (from the left) integrator. For a unidirectional
motion (here, $\mathrm{sign}(x_{3})=1$, for instance, is assumed) and
piecewise constant approximation $\dot{x}_{n}=\mathrm{const}$, one obtains

$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle
x^{2}_{2}(2x_{3})^{-1}-x_{1}^{min},$ (37) $\displaystyle x_{2}$
$\displaystyle=$ $\displaystyle x^{2}_{3}(2\dot{x}_{3})^{-1}-x_{2}^{c}$ (38)

after integrating the left- and right-hand sides of (36). Obviously, for
$t\geq t_{c}$, the $x_{1}(t)$-trajectory evolves parabolically with a
dependency of $x_{2}(t)$, while $x_{3}(t)$ is square-root-dependent on
$x_{2}(t)$ (see Figures 4(b) and (c) correspondingly). Note that the
increasing and decreasing segments of the corresponding trajectories are both
asymmetric, effectively due to non-constant $\dot{x}_{n}$-value as the motion
evolves. At the same time, one can stress that the extremum $x_{1}^{min}$
(here, minimal due to the assumed positive sign of velocity) always lies on
the $x_{1}$-axis (cf. Figure 4(b)) owing to $x_{1}=\int x_{2}dt$ and
$\mathrm{sign}(x_{3})=\mathrm{const}$. Differently, the
$(x_{2},x_{3})$-projection of the $x(t)$-trajectory can be shifted along the
$x_{2}$-axis, while it always ends in $x_{3}(t_{s})=0$ for $x(t)\in S_{0}$
(cf. Figure 4(c)). The resulting alternation of the stick-slip phases is
schematically shown in Figure 4(d).

######  Proposition 1.

Having reached $S_{0}$, the (3)-(7) system does not leave
$\Omega\in\mathbb{R}^{3}$ with $\mathrm{proj}_{(x_{1},x_{2})}\Omega\subseteq
S_{0}$ and converges _asymptotically_ to
$x(t)\underset{t\rightarrow\infty}{=}\bigl{\\{}(x_{1},0,0):\,|x_{1}|\leq
F_{c}K_{i}^{-1}\bigr{\\}}$ with multiple (or at least one) stick-slip cycles.
The stick-slip cycles can occur with either zero-crossing of $x_{2}$ or
keeping the same $\mathrm{sign}(x_{2}(t_{s,1}))$, where $t_{s,1}$ is the time
instant when $x(t)$ reaches $S_{0}$ for the first time.

When first disregarding the friction side effect, i.e., $F_{c}=0$, it is well
understood that non-overshoot of the set reference value cannot be reached
independently of the assigned control parameters, provided $K_{i}\neq 0$ and
the initial conditions are such that either $x_{1}(0)=0$ or
$\mathrm{sign}(x_{1}(0))=\mathrm{sign}(x_{2}(0))$. This becomes evident since
the integral error state $x_{1}(t)$ accumulates the output error over time.
That is, in order for $|x_{1}(t)|$ to start decreasing, at least one change of
the $x_{2}(t)$-sign is required. An exception is when
$\mathrm{sign}(x_{1}(0))\neq\mathrm{sign}(x_{2}(0))$, which allows both
$x_{1}(t)$ and $x_{2}(t)$ to converge to zero from the opposite directions.
Thus, at least one overshoot should appear, even if all control gains are
assigned to have the real poles only; cf. later Example 4.

When the Coulomb friction becomes effective, i.e., $F_{c}\neq 0$, the system
can change to stiction again, and also without overshoot of $x_{2}=0$ after
starting to slip at $x(t_{c})$. This means that a motion trajectory lands
again onto $S_{0}$ at time $t_{s,2}>t_{c}>t_{s,1}$ and with
$\mathrm{sign}(x_{2}(t_{s,2}))=\mathrm{sign}(x_{2}(t_{s,1}))$. Since the
system with $K_{d},F_{c}>0$ is dissipative, the energy level is
$V(t_{s,2})<V(t_{c})$, meaning the motion trajectory $x(t)$ always lands onto
$S_{0}$ closer to the origin than it was leaving $S_{0}$ in $x(t_{c})$. Note
that the system energy within $S_{0}$ can be expressed by the potential field
of the proportional and integral control errors, yielding

$V(t)=\frac{1}{2}K_{i}x_{1}^{2}+\frac{1}{2}K_{p}x_{2}^{2}\quad\hbox{for}\quad
t\in[t_{s},\ldots,t_{c}].$ (39)

One can recognize that the energy level (39), of the system in stiction, is an
ellipse

$\frac{x_{2}^{2}}{a^{2}}+\frac{x_{1}^{2}}{b^{2}}=1\quad\hbox{with}\quad
a^{2}=2VK_{p}^{-1},\;b^{2}=2VK_{i}^{-1}.$ (40)

Since the energy $V(t_{c})$ is bounded by $S_{0}$, cf. Figure 2, one can show
that the semi-major axis $a\leq F_{c}K_{p}^{-1}$ and semi-minor axis $b\leq
F_{c}K_{i}^{-1}$. From the system dissipativity and global attractiveness of
$S_{0}$, cf. Corollary 1, it follows that the trajectory becomes sticking
again, meaning $x(t)\in S_{0}$ for $t\geq t_{s,2}>t_{c}$. Since
$V(t_{s,2})<V(t_{c})$, the ellipse (40) shrinks as $a$ and $b$ become smaller
(both being proportional to $V(t)$). It is important to notice that during
system sticking, on the contrary, the energy level increases as
$V(t_{c})>V(t_{s,1})$. This is logical since the integral control action
pushes additional energy into the control loop when the system is at a
standstill, thus leading to a breakaway and allowing for the motion to restart
again once the sticking trajectory reaches the $S_{0}$-boundary.

######  Remark 3.

When the state trajectory reattains the stiction region $x(t_{s,2})\in S_{0}$
without overshoot, meaning
$\mathrm{sign}(x_{2}(t_{s,2}))=\mathrm{sign}(x_{2}(t_{c}))$, the system is
over-damped by the Coulomb friction. Otherwise,
$\mathrm{sign}(x_{2}(t_{s,2}))\neq\mathrm{sign}(x_{2}(t_{c}))$ means the
system is said to be under-damped by the Coulomb friction. A special (but not
feasible, as will be shown) case of $x_{2}(t_{s,2})=0$, meaning the system
reaches equilibrium $S_{0}\backslash S_{0}^{s}$ and remains there
$\forall\>t\geq t_{s,2}$, is analyzed below by proving Theorem 2.

######  Theorem 2.

The (3)-(7) system, with control parameters satisfying (31) and $F_{c}>0$,
converges asymptotically to invariant set $\Lambda=\\{(x_{1},0,0):|x_{1}|\leq
F_{c}K_{i}^{-1}\\}$ by the number of stick-slip cycles $1\leq N<\infty$. There
are no system parameter values and stick-slip initial conditions
$(x_{1}(t_{s,n}),x_{2}(t_{s,n}))$ with $n\in N$ which allow $\Lambda$ to be
reached by the end of the next stick-slip cycle at time
$t_{s,n+1}>t_{c,n}>t_{s,n}$.

###### Proof.

Convergence to $\Lambda$ follows from system dissipativity during the slipping
and shrinking ellipse (40), which implies an ever-decreasing energy level by
the end of one stick-slip cycle, meaning $V(t_{s,n+1})<V(t_{s,n})$. This
implies $|x_{2}(t_{s,n+1})|<|x_{2}(t_{s,n})|$ for $n\in N$ and ensures such
$x(t)$-trajectory, which starts slipping at $t_{c,n}$ and lands closer to the
origin at $t_{s,n+1}$ than before (at $t_{s,n}$).

The proof of the second part of Theorem 2, which says it is impossible to
reach the invariant equilibrium set $\Lambda$ after one particular stick-slip
cycle, follows through contradiction. For this purpose, we should first assume
that there is a particular setting
$\bigl{(}K_{p},K_{i},K_{d},F_{c},x_{1}(t_{s,n}),x_{2}(t_{s,n})\bigr{)}$ for
which the state trajectory $x(t_{s,n+1})\in\Lambda$, i.e. in the next stiction
phase at the finite time $t_{s,n+1}>t_{c,n}>t_{s,n}$. The initial conditions
of a slipping phase are always given, cf. with Section 2.3, by

$\displaystyle x_{2}(t_{c,n})$ $\displaystyle=$ $\displaystyle
x_{2}(t_{s,n}),$ (41) $\displaystyle x_{1}(t_{c,n})$ $\displaystyle=$
$\displaystyle\frac{F_{c}}{K_{i}}-\frac{K_{p}}{K_{i}}x_{2}(t_{c,n})\>\hbox{ in
1st quad.,}$ (42) $\displaystyle x_{1}(t_{c,n})$ $\displaystyle=$
$\displaystyle-\frac{F_{c}}{K_{i}}-\frac{K_{p}}{K_{i}}x_{2}(t_{c,n})\>\hbox{
in 3rd quad.}$ (43)

This becomes apparent when inspecting the stiction phase dynamics (25), (26)
and $S_{0}$-boundary, cf. Figure 2. For reaching $\Lambda$ at a final time
instant $\psi=t_{s,n+1}$, while starting at $\tau=t_{c,n}$, an explicit
particular solution of

$0=C\Bigl{[}e^{A\psi}\left(\begin{array}[]{c}x_{1}(\tau)\\\\[-1.42271pt]
x_{2}(\tau)\\\\[-1.42271pt]
0\end{array}\right)+A^{-1}(e^{A\psi}-I)Bu-\left(\begin{array}[]{c}x_{1}(\psi)\\\\[-1.42271pt]
0\\\\[-1.42271pt] 0\end{array}\right)\Bigr{]}$ (44)

with $u=\pm 1$, should exist, cf. with (34). Due to the symmetry of solutions,
we will consider the 1st quadrant of $S_{0}$ only, i.e., with the above
initial condition (42) and $u=+1$ correspondingly, without loss of generality
when solving (44). Recall that the matrix exponential

$e^{A\psi}=\sum^{\infty}_{k=0}\frac{(A\psi)^{k}}{k!}$ (45)

has to be evaluated to find an explicit solution for (44). Substituting the
initial condition equalities (41) and (42) into (44), we solve (44) with
respect to $x_{2}(\tau)$, and that for an increasing $k=[1,\ldots,40]$. For
all the solutions evaluated with the help of the Symbolic Math ToolboxTM, it
was found that (44) has no initial-value solution other than zero, meaning
$x_{2}(\tau)=x_{2}(t_{s,n})=0$. This means that there are no other initial
conditions than zero for which a stick-slip cycle could lead to
$x(t)\in\Lambda$ at $t=t_{s,n+1}$. This contradicts our initial assumption
that such initial conditions exist and, hence, completes the proof.

∎

######  Remark 4.

Since no relative motion occurs during a stiction phase, cf. Section 2.2, the
trajectories solution (44) represents the single descriptor of the system
dynamics, determining convergence of trajectories during the slipping phases.
One can recognize that discontinuous Coulomb friction contributes as a
constant piecewise-continuous input term $u$, into the solution of
trajectories $x(t)$ at $t_{c,n}<t<t_{s,n+1}$. Thus, it comes as no surprise
that the stick-slip convergence appears only asymptotically, meaning either
within one or an (theoretically) infinite number of stick-slip cycles. We also
note that this is independent of whether $x(t)$ reattains $S_{0}$ with or
without overshooting.

##  4 Numerical examples

The following numerical examples serve to illustrate and evaluate the above
analysis. A dedicated numerical simulation of the stick-slip dynamics is
developed by implementing (25), (26) and (34), while the conditions of Theorem
1 provide switching between the piecewise smooth trajectories of the
alternating slipping and sticking phases of relative motion of the (3)-(7)
system.

######  Example 2.

Consider the (3)-(7) system with $K_{d}=20$, $K_{p}=100$, $K_{i}=1000$ and
varying $F_{c}=\\{50,75,100\\}$. The initial values are assigned as
$x(0)=\\{0,-1.1,0\\}$, corresponding to a classical positioning task for the
feedback- controlled system (1). Note that $|x_{2}(0)|>F_{c}K_{p}^{-1}$ so
that the trajectories start outside of $S_{0}$ and are, therefore, inherently
in the slipping phase. The transient and convergence responses for all three
Coulomb friction values are shown opposite to one another in Figure 5, cf.
qualitatively with an experimental convergence pattern reported in [13, Fig
4].

(a)  (b)

Figure 5: Output response of Example 2: transient phase $t=[0,\ldots,12]$ sec
(a), convergence phase $t=[12,\ldots,120]$ sec (b)

######  Example 3.

Consider the (3)-(7) system with $K_{d}=10$, $K_{p}=1040$, $K_{i}=8000$ and
$F_{c}=100$. The initial values $x(0)=(0,-0.15,0)$ are assigned to be close to
but still outside of the $S_{0}$-region. The linear damping $K_{d}$ is
selected, with respect to $F_{c}$, so that the system exhibits only one
initial overshoot; but the stick and slip phases alternate without changing
the sign of $x_{2}$. The output displacement response is shown in Figure 6(a).
The stick-slip convergence without zero-crossing is particularly visible on
the logarithmic scale in Figure 6(b). Note that after the series of stick-slip
cycles, a further evaluation of the alternating dynamics (about $10^{-13}$ in
order of magnitude) is no longer feasible, due to a finite time step and
corresponding numerical accuracy, cf. 1st quadrant of the $S_{0}$-rhombus in
Figure 2.

(a)  (b)

Figure 6: Output (a) and logarithmic (b) response of Example 3

######  Example 4.

Consider the (3)-(7) system with $K_{d}=56$, $K_{p}=1040$, $K_{i}=6400$ and
$F_{c}=100$. Note that the control gains are assigned in such a way that the
linear dynamics (3) and (6) reveal a double real pole at $\lambda_{1,2}=-20$
and one in its vicinity at $\lambda_{3}=-16$. This ensures that all states
converge fairly simultaneously towards zero, once the $\mathrm{sign}(x_{3})$
remains unchanged. For the initial conditions, $x_{1}(0),x_{3}(0)=0$ and the
varying initial displacement is $x_{2}(0)=\\{-0.2,-0.25,-0.3,-0.35\\}$. Note
that all $x(0)$ are outside of $S_{0}$, while the transient overshoot lands
(in all cases) within $S_{0}$, thus leading to the first stiction after
overshoot; see Figure 7. One can recognize that the integral state requires
quite different times before the system then passes again into slipping.
During the slipping phase, all states converge asymptotically towards zero,
provided $F_{c}$ remains constant. Here, it is important to notice that in the
real physical systems, a varying $F_{c}$-value and so-called frictional
adhesion, see, e.g., [29], at extremely low velocities, will both lead to the
system passing into a sticking phase again, therefore, further provoking
multiple stick-slip cycles. Even though it is not the case here, with our
ideal Coulomb friction assumption, the Theorem 2 still holds, since there is
only an asymptotic convergence after at least one stick-slip cycle has
occurred.

Figure 7: Output response of Example 4 for different initial values

######  Example 5.

Consider the (3)-(7) system with $K_{d}=20$, $K_{p}=100$, $K_{i}=1000$ and
$F_{c}=50$. The initial condition $x(0)=(0,-0.5,0)$ is assigned to be on the
boundary of $S_{0}$, thus leading to short initial slipping and, then,
realizing a large number of stick-slip cycles by long-term simulation with
$t=[0,\ldots,100000]$ sec. The output is shown as logarithmic absolute value
(due to the alternating sign) over the logarithmic time argument in Figure 8.
One can recognize that each consequent sticking phase gets closer to the
origin, while the stick-slip period grows exponentially, cf. logarithmic
timescale. This confirms an asymptotic convergence with stick-slip cycles, cf.
Theorem 2.

Figure 8: Output response of Example 5, the logarithmic absolute value over
the logarithmic time argument

##  5 Conclusions

An analysis of stick-slip behavior during convergence of feedback-controlled
motion with Coulomb friction has been developed. The most general case of
frictional discontinuity at velocity zero-crossing has been assumed, and the
parametric conditions for appearance of the stiction region around the
equilibrium (set independently of the initial values) are derived. Theorem 1
and Corollary 1 proved the stiction region to be globally reachable and
attractive. Theorem 2 stated that convergence is only asymptotically possible
and occurs with at least one but mostly an infinite number of stick-slip
cycles in a sequence. In particular, an ’ideal’ convergence of the control
configuration with all real poles in a neighborhood appears with one initial
stick-slip cycle, followed by an asymptotic convergence without new stiction
phases. The number of illustrative numerical examples, with different initial
conditions and parameter settings, argue in favor of the developed analysis
and provide additional insight into stick-slip mechanisms of controlled motion
with Coulomb friction.

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