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<h1><a href="index.html" style="text-decoration:none;">Underactuated Robotics</a></h1>
<p data-type="subtitle">Algorithms for Walking, Running, Swimming, Flying, and Manipulation</p>
<p style="font-size: 18px;"><a href="http://people.csail.mit.edu/russt/">Russ Tedrake</a></p>
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© Russ Tedrake, 2020<br/>
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<chapter style="counter-reset: chapter 8"><h1>Lyapunov
Analysis</h1>
<p>Optimal control provides a powerful framework for formulating control
problems using the language of optimization. But solving optimal control
problems for nonlinear systems is hard! In many cases, we don't really care
about finding the <em>optimal</em> controller, but would be satisfied with any
controller that is guaranteed to accomplish the specified task. In many
cases, we still formulate these problems using computational tools from
optimization, and in this chapter we'll learn about tools that can provide
guaranteed control solutions for systems that are beyond the complexity for
which we can find the optimal feedback.</p>
<p>There are many excellent books on Lyapunov analysis; for instance
<elib>Slotine90</elib> is an excellent and very readable reference and
<elib>Khalil01</elib> can provide a rigorous treatment. In this chapter I
will summarize (without proof) some of the key theorems from Lyapunov
analysis, but then will also introduce a number of numerical algorithms...
many of which are new enough that they have not yet appeared in any mainstream
textbooks.</p>
<section><h1>Lyapunov Functions</h1>
<p>Let's start with our favorite simple example. </p>
<example><h1>Stability of the Damped Pendulum</h1>
<center><img width="25%" src="figures/simple_pend.svg"/></center>
<p>Recall that the equations of motion of the damped simple pendulum are
given by \[ ml^2 \ddot{\theta} + mgl\sin\theta = -b\dot{\theta}, \] which
I've written with the damping on the right-hand side to remind us that it
is an external torque that we've modeled.</p>
<p>These equations represent a simple second-order differential equation;
in chapter 2 we discussed at some length what was known about the
solutions to this differential equation--even without damping the equation
for $\theta(t)$ as a function of $\theta(0)$, $\dot{\theta}(0)$, involves
elliptic integrals of the first kind, and with damping we don't even have
that. Since we couldn't provide a solution analytically, in chapter 2 we
resorted to a graphical analysis, and confirmed the intuition that there
are fixed points in the system (at $\theta = k\pi$ for every integer $k$)
and that the fixed points at $\theta = 2\pi k$ are asymptotically stable
with a large basin of attraction. The graphical analysis gave us this
intuition, but can we actually prove this stability property? In a way
that might also work for much more complicated systems?</p>
<p>One route forward was from looking at the total system energy (kinetic
+ potential), which we can write down: \[ E(\theta,\dot{\theta}) =
\frac{1}{2} ml^2\dot{\theta}^2 - mgl \cos\theta. \] Recall that the
contours of this energy function are the orbits of the undamped
pendulum.</p>
<center><img width="50%" src="figures/pend_undamped_phase.svg"/></center>
<p>A natural route to proving the stability of the downward fixed points
is by arguing that energy decreases for the damped pendulum (with $b>0$)
and so the system will eventually come to rest at the minimum energy, $E =
-mgl$, which happens at $\theta=2\pi k$. Let's make that argument
slightly more precise. </p>
<p>Evaluating the time derivative of the energy reveals \[ \frac{d}{dt} E
= - b\dot\theta^2 \le 0. \] This is sufficient to demonstrate that the
energy will never increase, but it doesn't actually prove that the energy
will converge to the minimum when $b>0$ because there are multiple
states(not only the minimum) for which $\dot{E}=0$. To take the last
step, we must observe that set of states with $\dot\theta=0$ is not an
invariant set; that if the system is in, for instance
$\theta=\frac{\pi}{4}, \dot\theta=0$ that it will not stay there, because
$\ddot\theta \neq 0$. And once it leaves that state, energy will decrease
once again. In fact, the fixed points are the only subset the set of
states where $\dot{E}=0$ which do form an invariant set. Therefore we can
conclude that as $t\rightarrow \infty$, the system will indeed come to
rest at a fixed point (though it could be any fixed point with an energy
less than or equal to the initial energy in the system, $E(0)$).</p>
</example>
<p>This is an important example. It demonstrated that we could use a
relatively simple function -- the total system energy -- to describe
something about the long-term dynamics of the pendulum even though the
actual trajectories of the system are (analytically) very complex. It also
demonstrated one of the subtleties of using an energy-like function that is
non-increasing (instead of strictly decreasing) to prove asymptotic
stability.</p>
<p>Lyapunov functions generalize this notion of an energy function to more
general systems, which might not be stable in the sense of some mechanical
energy. If I can find any positive function, call it $V(\bx)$, that gets
smaller over time as the system evolves, then I can potentially use $V$ to
make a statement about the long-term behavior of the system. $V$ is called
a <em>Lyapunov function</em>. </p>
<p>Recall that we defined three separate notions for stability of a
fixed-point of a nonlinear system: stability i.s.L., asymptotic stability,
and exponential stability. We can use Lyapunov functions to demonstrate
each of these, in turn.</p>
<theorem><h1>Lyapunov's Direct Method</h1>
<p>Given a system $\dot{\bx} = f(\bx)$, with $f$ continuous, and for some
region ${\cal B}$ around the origin (specifically an open subset of
$\mathbf{R}^n$ containing the origin), if I can produce a scalar,
continuously-differentiable function $V(\bx)$, such that \begin{gather*}
V(\bx) > 0, \forall \bx \in {\cal B} \setminus \{0\} \quad V(0) = 0,
\text{ and} \\ \dot{V}(\bx) = \pd{V}{\bx} f(\bx) \le 0, \forall \bx \in
{\cal B} \setminus \{0\} \quad \dot{V}(0) = 0, \end{gather*} then the
origin $(\bx = 0)$ is stable in the sense of Lyapunov (i.s.L.). [Note: the notation $A \setminus B$ represents the set $A$ with the elements of $B$ removed.]</p>
<p>If, additionally, we have $$\dot{V}(\bx) = \pd{V}{\bx} f(\bx) < 0,
\forall \bx \in {\cal B} \setminus \{0\},$$ then the origin is (locally)
asymptotically stable. And if we have $$\dot{V}(\bx) = \pd{V}{\bx} f(\bx)
\le -\alpha V(x), \forall \bx \in {\cal B} \setminus \{0\},$$ for some $\alpha>0$,
then the origin is (locally) exponentially stable. </p>
</theorem>
<p>Note that for the sequel we will use the notation $V \succ 0$ to denote a
<em>positive-definite function</em>, meaning that $V(0)=0$ and $V(\bx)>0$
for all $\bx\ne0$ (and also $V \succeq 0$ for positive semi-definite, $V
\prec 0$ for negative-definite functions).</p>
<p>The intuition here is exactly the same as for the energy argument we made
in the pendulum example: since $\dot{V}(x)$ is always zero or negative, the
value of $V(x)$ will only get smaller (or stay the same) as time progresses.
Inside the subset ${\cal B}$, for every $\epsilon$-ball, I can choose a
$\delta$ such that $|x(0)|^2 < \delta \Rightarrow |x(t)|^2 < \epsilon,
\forall t$ by choosing $\delta$ sufficiently small so that the sublevel-set
of $V(x)$ for the largest value that $V(x)$ takes in the $\delta$ ball is
completely contained in the $\epsilon$ ball. Since the value of $V$ can only
get smaller (or stay constant) in time, this gives stability i.s.L.. If
$\dot{V}$ is strictly negative away from the origin, then it must eventually
get to the origin (asymptotic stability). The exponential condition is
implied by the fact that $\forall t>0, V(\bx(t)) \le V(\bx(0)) e^{-\alpha
t}$.</p>
<p> Notice that the system analyzed above, $\dot{\bx}=f(\bx)$, did not have
any control inputs. Therefore, Lyapunov analysis is used to study either the
passive dynamics of a system or the dynamics of a closed-loop system (system
+ control in feedback). We will see generalizations of the Lyapunov
functions to input-output systems later in the text. </p>
<subsection><h1>Global Stability</h1>
<p>The notion of a fixed point being stable i.s.L. is inherently a local
notion of stability (defined with $\epsilon$- and $\delta$- balls around
the origin), but the notions of asymptotic and exponential stability can
be applied globally. The Lyapunov theorems work for this case, too, with
only minor modification.</p>
<theorem><h1>Lyapunov analysis for global stability</h1>
<p>Given a system $\dot{\bx} = f(\bx)$, with $f$ continuous, if I can
produce a scalar, continuously-differentiable function $V(\bx)$, such
that \begin{gather*} V(\bx) \succ 0, \\ \dot{V}(\bx) = \pd{V}{\bx}
f(\bx) \prec 0, \text{ and} \\ V(\bx) \rightarrow \infty \text{ whenever
} ||x||\rightarrow \infty,\end{gather*} then the origin $(\bx = 0)$ is
globally asymptotically stable (G.A.S.).</p>
<p>If additionally we have that $$\dot{V}(\bx) \preceq -\alpha V(\bx),$$
for some $\alpha>0$, then the origin is globally exponentially
stable.</p>
</theorem>
<p>The new condition, on the behavior as $||\bx|| \rightarrow \infty$ is
known as "radially unbounded", and is required to make sure that
trajectories cannot diverge to infinity even as $V$ decreases; it is only
required for global stability analysis.</p>
</subsection> <!-- global stability -->
<subsection id="lasalle"><h1>LaSalle's Invariance Principle</h1>
<p>Perhaps you noticed the disconnect between the statement above and the
argument that we made for the stability of the pendulum. In the pendulum
example, using the mechanical energy resulted in a Lyapunov function that
was only negative semi-definite, but we eventually argued that the fixed
points were asymptotically stable. That took a little extra work,
involving an argument about the fact that the fixed points were the only
place that the system could stay with $\dot{E}=0$; every other state with
$\dot{E}=0$ was only transient. We can formalize this idea for the more
general Lyapunov function statements--it is known as LaSalle's
Theorem.</p>
<theorem><h1>LaSalle's Theorem</h1>
<p>Given a system $\dot{\bx} = f(\bx)$ with $f$ continuous. If we can
produce a scalar function $V(\bx)$ with continuous derivatives for
which we have $$V(\bx) \succ 0,\quad \dot{V}(\bx) \preceq 0,$$ and
$V(\bx)\rightarrow \infty$ as $||\bx||\rightarrow \infty$, then $\bx$
will converge to the largest <i>invariant set</i> where $\dot{V}(\bx) =
0$.</p>
</theorem>
<p>To be clear, an <em>invariant set</em>, ${\cal G}$, of the dynamical
system is a set for which $\bx(0)\in{\cal G} \Rightarrow \forall t>0,
\bx(t) \in {\cal G}$. In other words, once you enter the set you never
leave. The "largest invariant set" need not be connected; in fact for the
pendulum example each fixed point is an invariant set, so the largest
invariant set is the <em>union</em> of all the fixed points of the
system. There are also variants of LaSalle's Theorem which work over
a region.</p>
<p>Finding a Lyapunov function which $\dot{V} \prec 0$ is more difficult
than finding one that has $\dot{V} \preceq 0$. LaSalle's theorem gives us
the ability to make a statement about <i>asymptotic</i> stability even in
this case. In the pendulum example, every state with $\dot\theta=0$ had
$\dot{E}=0$, but only the fixed points are in the largest invariant
set.</p>
</subsection> <!-- lasalle -->
<subsection><h1>Relationship to the Hamilton-Jacobi-Bellman
equations</h1>
<p> At this point, you might be wondering if there is any relationship
between Lyapunov functions and the cost-to-go functions that we discussed
in the context of dynamic programming. After all, the cost-to-go
functions also captured a great deal about the long-term dynamics of the
system in a scalar function. We can see the connection if we re-examine
the HJB equation \[ 0 = \min_\bu \left[ \ell(\bx,\bu) +
\pd{J^*}{\bx}f(\bx,\bu). \right] \]Let's imagine that we can solve for the
optimizing $\bu^*(\bx)$, then we are left with $ 0 = \ell(\bx,\bu^*) +
\pd{J^*}{\bx}f(\bx,\bu^*) $ or simply \[ \dot{J}^*(\bx) = -\ell(\bx,\bu^*)
\qquad \text{vs} \qquad \dot{V}(\bx) \preceq 0. \] In other words, in
optimal control we must find a cost-to-go function which matches this
gradient for every $\bx$; that's very difficult and involves solving a
potentially high-dimensional partial differential equation. By contrast,
Lyapunov analysis is asking for much less - any function which is going
downhill (at any rate) for all states. This can be much easier, for
theoretical work, but also for our numerical algorithms. Also note that
if we do manage to find the optimal cost-to-go, $J^*(\bx)$, then it can
also serve as a Lyapunov function so long as $\ell(\bx,\bu^*(\bx)) \succeq
0$.</p>
</subsection> <!-- relationship to HJB -->
<todo>Include instability results, as in Briat15 Theorem 2.2.5</todo>
</section> <!-- Lyapunov definitions -->
<section><h1>Lyapunov analysis with convex optimization</h1>
<p>One of the primary limitations in Lyapunov analysis as I have presented
it so far is that it is potentially very difficult to come up with suitable
Lyapunov function candidates for interesting systems, especially for
underactuated systems. ("Underactuated" is almost synonymous with
"interesting" in my vocabulary.) Even if somebody were to give me a
Lyapunov candidate for a general nonlinear system, the Lyapunov conditions
can be difficult to check -- for instance, how would I check that $\dot{V}$
is strictly negative for all $\bx$ except the origin if $\dot{V}$ is some
arbitrarily complicated nonlinear function over a vector $\bx$?</p>
<p>In this section, we'll look at some computational approaches to verifying
the Lyapunov conditions, and even to searching for (the coefficients of) the
Lyapunov functions themselves.</p>
<p>If you're imagining numerical algorithms to check the Lyapunov conditions
for complicated Lyapunov functions and complicated dynamics, the first
thought is probably that we can evaluate $V$ and $\dot{V}$ at a large number
of sample points and check whether $V$ is positive and $\dot{V}$ is
negative. <a
href="https://github.com/RobotLocomotion/drake/blob/master/systems/analysis/test/lyapunov_test.cc">This
does work</a>, and could potentially be combined with some smoothness or
regularity assumptions to generalize beyond the sample points. <todo>Add
python bindings for the pendulum energy lp lyapunov example.</todo> But in
many cases we will be able to do better -- providing optimization algorithms
that will rigorously check these conditions <em>for all $\bx$</em> without
dense sampling; these will also give us additional leverage in formulating
the search for Lyapunov functions. </p>
<subsection><h1>Lyapunov analysis for linear systems</h1>
<p>Let's take a moment to see how things play out for linear systems.</p>
<theorem><h1>Lyapunov analysis for stable linear systems</h1>
<p>Imagine you have a linear system, $\dot\bx = {\bf A}\bx$, and can
find a Lyapunov function $$V(\bx) = \bx^T {\bf P} \bx, \quad {\bf P} =
{\bf P^T} \succ 0,$$ which also satisfies $$\dot{V}(\bx) = \bx^T {\bf
PA} \bx + \bx^T {\bf A}^T {\bf P}\bx \prec 0.$$ Then the origin is
globally asymptotically stable.</p>
</theorem>
<p>Note that the radially-unbounded condition is satisfied by ${\bf P}
\succ 0$, and that the derivative condition is equivalent to the matrix
condition $${\bf PA} + {\bf A}^T {\bf P} \prec 0.$$</p>
<p>For stable linear systems the existence of a quadratic Lyapunov
function is actually a necessary (as well as sufficient) condition.
Furthermore, a Lyapunov function can always be found by finding the
positive-definite solution to the matrix Lyapunov equation
\begin{equation}{\bf PA} + {\bf A}^T{\bf P} = - {\bf
Q},\label{eq:algebraic_lyapunov} \end{equation} for any ${\bf Q}={\bf
Q}^T\succ 0$.</p>
<todo> add an example here. double integrator? re-analyze the LQR output?
</todo>
<p>This is a very powerful result - for nonlinear systems it will be
potentially difficult to find a Lyapunov function, but for linear systems
it is straight-forward. In fact, this result is often used to propose
candidates for non-linear systems, e.g., by linearizing the equations and
solving a local linear Lyapunov function which should be valid in the
vicinity of a fixed point.</p>
</subsection> <!-- lyapunov for linear -->
<subsection><h1>Lyapunov analysis as a semi-definite program
(SDP)</h1>
<p>Lyapunov analysis for linear systems has an extremely important
connection to convex optimization. In particular, we could have also
formulated the Lyapunov conditions for linear systems above using
<em>semi-definite programming</em> (SDP). Semidefinite programming is a
subset of convex optimization -- an extremely important class of problems
for which we can produce efficient algorithms that are guaranteed find the
global optima solution (up to a numerical tolerance and barring any
numerical difficulties).</p>
<p>If you don't know much about convex optimization or want a quick
refresher, please take a few minutes to read the optimization
preliminaries in the appendix. The main requirement for this section is
to appreciate that it is possible to formulate efficient optimization
problems where the constraints include specifying that one or more
matrices are positive semi-definite (PSD). These matrices must be formed
from a linear combination of the decision variables. For a trivial
example, the optimization $$\min_a a,\quad \subjto \begin{bmatrix} a & 0
\\ 0 & 1 \end{bmatrix} \succeq 0,$$ returns $a = 0$ (up to numerical
tolerances).</p>
<p>The value in this is immediate for linear systems. For example, we can
formulate the search for a Lyapunov function for the linear system
$\dot\bx = {\bf A} \bx$ by using the parameters ${\bf p}$ to populate a
symmetric matrix ${\bf P}$ and then write the SDP: \[ \find_{\bf p} \quad
\subjto \quad {\bf P} \succeq 0, \quad {\bf PA} + {\bf A}^T {\bf P}
\preceq 0.\] Note that you would probably never use that particular
formulation, since there specialized algorithms for solving the simple
Lyapunov equation which are more efficient and more numerically stable.
But the SDP formulation does provide something new -- we can now easily
formulate the search for a <i>"common Lyapunov function"</i> for uncertain
linear systems.</p>
<example><h1>Common Lyapunov analysis for
linear systems</h1>
<p>Suppose you have a system governed by the equations $\dot\bx = {\bf
A}\bx$, where the matrix ${\bf A}$ is unknown, but its uncertain
elements can be bounded. There are a number of ways to write down this
uncertainty set; let us choose to write this by describing ${\bf A}$ as
the convex combination of a number of known matrices, $${\bf A} =
\sum_{i} \beta_i {\bf A}_i, \quad \sum_i \beta_i = 1, \quad \forall i,
\beta_i > 0.$$ This is just one way to specify the uncertainty;
geometrically it is describing a polygon of uncertain parameters (in the
space of elements of ${\bf A}$ with each ${\bf A}_i$ as one of the
vertices in the polygon.</p>
<figure> <img width="50%"
src="figures/affine_combination_of_matrices.svg"/> </figure>
<p>Now we can formulate the search for a common Lyapunov function using
\[ \find_{\bf p} \quad \subjto \quad {\bf P} \succeq 0, \quad
\forall_i, {\bf PA}_i + {\bf A}_i^T {\bf P} \preceq 0.\] The solver
will then return a matrix ${\bf P}$ which satisfies all of the
constraints, or return saying "problem is infeasible". It can easily be
verified that if ${\bf P}$ satisfies the Lyapunov condition at all of
the vertices, then it satisfies the condition for every ${\bf A}$ in the
set: \[ {\bf P}(\sum_i \beta_i {\bf A}_i) + (\sum_i \beta_i {\bf
A}_i)^T {\bf P} = \sum_i \beta_i ({\bf P A}_i + {\bf A}_i^T {\bf P})
\preceq 0, \] since $\forall i$, $\beta_i > 0$. Note that, unlike the
simple Lyapunov equation for a known linear system, this condition being
satisfied is a sufficient but not a necessary condition -- it is
possible that the set of uncertain matrices ${\bf A}$ is robustly
stable, but that this stability cannot be demonstrated with a common
quadratic Lyapunov function.</p>
<p> You can try this example for yourself in <drake></drake>.</p>
<jupyter>examples/lyapunov.ipynb</jupyter>
<p>As always, make sure that you open up the code and take a look.</p>
<p>There are many small variants of this result that are potentially of interest. For instance, a very similar set of conditions can certify "mean-square stability" for linear systems with multiplicative noise (see e.g. <elib>Boyd94</elib>, § 9.1.1).</p>
</example>
<todo>Add example or exercise based on e.g. Briat15 (LPV) example 1.3.1, showing off how powerful this can be. Also perhaps the parameter-dependent robust stability of his Def 2.3.5.</todo>
<p>This example is very important because it establishes a connection
between Lyapunov functions and (convex) optimization. But so far we've
only demonstrated this connection for linear systems where the PSD
matrices provide a magical recipe for establishing the positivity of the
(quadratic) functions for all $\bx$. Is there any hope of extending this
type of analysis to more general nonlinear systems? Surprisingly, it
turns out that there is.</p>
</subsection>
<subsection><h1>Lyapunov analysis for polynomial systems</h1>
<p><a href="optimization.html#sums_of_squares">Sums of squares
optimization</a> provides a natural generalization of SDP to optimizing
over positive polynomials (if you are not familiar, take a moment to <a
href="optimization.html#sums_of_squares">read the appendix</a>). This
suggests that it may be possible to generalize the optimization approach
using SDP to search for Lyapunov functions for linear systems to searching
for Lyapunov functions for at least the polynomial systems: $\dot\bx =
f(\bx),$ where $f$ is a vector-valued polynomial function. If we
parameterize a <em>fixed-degree</em> Lyapunov candidate as a polynomial
with unknown coefficients, e.g., \[ V_\alpha(\bx) = \alpha_0 + \alpha_1
x_1 + \alpha_2 x_2 + \alpha_3 x_1x_2 + \alpha_4 x_1^2 + ..., \] then the
time-derivative of $V$ is also a polynomial, and I can formulate the
optimization: \begin{align*} \find_\alpha, \quad \subjto \quad&
V_\alpha(\bx) \sos \\ & -\dot{V}_\alpha(\bx) = -\pd{V_\alpha}{\bx} f(\bx)
\sos. \end{align*} Because this is a convex optimization, the solver will
return a solution if one exists.</p>
<example><h1>Verifying a Lyapunov candidate via SOS</h1>
<p>This example is example 7.2 from <elib>Parrilo00</elib>. Consider
the nonlinear system: \begin{align*} \dot{x}_0 =& -x_0 - 2x_1^2 \\
\dot{x}_1 =& -x_1 - x_0 x_1 - 2x_1^3,\end{align*} and the <i>fixed
</i> Lyapunov function $V(x) = x_0^2 + 2x_1^2$, test if $\dot{V}(x)$ is
negative definite.</p>
<p>The numerical solution can be written in a few lines of code, and
is a convex optimization.</p>
<jupyter>examples/lyapunov.ipynb</jupyter>
</example>
<example><h1>Searching for a Lyapunov function via SOS</h1>
<p>Verifying a candidate Lyapunov function is all well and good, but
the real excitement starts when we use optimization to <i>find</i> the
Lyapunov function. In the following code, we parameterize $V(x)$ as
the polynomial containing all monomials up to degree 2, with the
coefficients as decision variables: $$V(x) = c_0 + c_1x_0 + c_2x_1 +
c_3x_0^2 + c_4 x_0x_1 + c_5 x_1^2.$$ We will set the scaling
(arbitrarily) to avoid numerical issues by setting $V(0)=0$, $V([1,0])
= 1$. Then we write: \begin{align*} \find_{\bc} \ \ \subjto \ \ &
V\text{ is sos, } \\ & -\dot{V} \text{ is sos.}\end{align*}</p>
<jupyter>examples/lyapunov.ipynb</jupyter>
<p>Up to numerical convergence tolerance, it discovers the same
coefficients that we chose above (zeroing out the unnecessary
terms).</p>
</example>
<p>It is important to remember that there are a handful of gaps which make
the existence of this solution a sufficient condition (for proving that
every sub-level set of $V$ is an invariant set of $f$) instead of a
necessary one. First, there is no guarantee that a stable polynomial
system can be verified using a polynomial Lyapunov function (of any
degree, and in fact there are known counter-examples
<elib>Ahmadi11a</elib>) and here we are only searching over a fixed-degree
polynomial. Second, even if a polynomial Lyapunov function does exist,
there is a gap between the SOS polynomials and the positive
polynomials.</p>
<p>Despite these caveats, I have found this formulation to be surprisingly
effective in practice. Intuitively, I think that this is because there is
relatively a lot of flexibility in the Lyapunov conditions -- if you can
find one function which is a Lyapunov function for the system, then there
are also many "nearby" functions which will satisfy the same
constraints.</p>
</subsection>
</section> <!-- computing Lyapunov -->
<section><h1>Lyapunov functions for estimating regions of
attraction</h1>
<p>There is another very important connection between Lyapunov functions and
the concept of an invariant set: <em>any sub-level set of a Lyapunov
function is also an invariant set</em>. This gives us the ability to use
sub-level sets of a Lyapunov function as approximations of the region of
attraction for nonlinear systems.</p>
<theorem><h1>Lyapunov invariant set and region of attraction
theorem</h1>
<p>Given a system $\dot{\bx} = f(\bx)$ with $f$ continuous, if we can find
a scalar function $V(\bx) \succ 0$ and a sub-level set $${\cal G}: \{ \bx
| V(\bx) < \rho \}$$ on which $$\forall \bx \in {\cal G}, \dot{V}(\bx)
\preceq 0,$$ then ${\cal G}$ is an invariant set. By LaSalle, $\bx$ will
converge to the largest invariant subset of ${\cal G}$ on which
$\dot{V}=0$.<p>
<p>Furthermore, if $\dot{V}(\bx) \prec 0$ in ${\cal G}$, then the origin
is locally asymptotically stable and the set ${\cal G}$ is inside the
region of attraction of this fixed point. Alternatively, if $\dot{V}(\bx)
\preceq 0$ in ${\cal G}$ and $\bx = 0$ is the only invariant subset of
${\cal G}$ where $\dot{V}=0$, then the origin is asymptotically stable and
the set ${\cal G}$ is inside the region of attraction of this fixed point.
</p>
</theorem>
<example><h1>Region of attraction for a one-dimensional system</h1>
<p> Consider the first-order, one-dimensional system $\dot{x} = -x + x^3.$
We can quickly understand this system using our tools for graphical
analysis.</p> <figure> <img width="80%"
src="figures/cubicPolynomialExample.svg"/> </figure> <p>In the vicinity of
the origin, $\dot{x}$ looks like $-x$, and as we move away it looks
increasingly like $x^3$. There is a stable fixed point at the origin and
unstable fixed points at $\pm 1$. In fact, we can deduce visually that
the region of attraction to the stable fixed point at the origin is $\bx
\in (-1,1)$. Let's see if we can demonstrate this with a Lyapunov
argument.</p>
<p>First, let us linearize the dynamics about the origin and use the
Lyapunov equation for linear systems to find a candidate $V(\bx)$. Since
the linearization is $\dot{x} = -x$, if we take ${\bf Q}=1$, then we find
${\bf P}=\frac{1}{2}$ is the positive definite solution to the algebraic
Lyapunov equation (\ref{eq:algebraic_lyapunov}). Proceeding with $$V(\bx)
= \frac{1}{2} x^2,$$ we have $$\dot{V} = x (-x + x^3) = -x^2 + x^4.$$
This function is zero at the origin, negative for $|x| < 1$, and positive
for $|x| > 1$. Therefore we can conclude that the sub-level set $V <
\frac{1}{2}$ is invariant and the set $x \in (-1,1)$ is inside the region
of attraction of the nonlinear system. In fact, this estimate is
tight.</p>
</example>
<subsection><h1>Robustness analysis using "common Lyapunov functions"</h1>
<p>While we will defer most discussions on robustness analysis until later
in the notes, there is a simple and powerful idea that can be presented
now: the idea of a <em>common Lyapunov function</em>. Imagine that you
have a model of a dynamical system but that you are uncertain about some
of the parameters. For example, you have a model of a quadrotor, and are
fairly confident about the mass and lengths (both of which are easy to
measure), but are not confident about the moment of inertia. One approach
to robustness analysis is to define a bounded uncertainty, which could
take the form of $$\dot{\bx} = f_\alpha(\bx), \quad \alpha_{min} \le
\alpha \le \alpha_{max},$$ with $\alpha$ a vector of uncertain parameters
in your model. Richer specifications of the uncertainty bounds are also
possible, but this will serve for our examples.</p>
<p>In standard Lyapunov analysis, we are searching for a function that
goes downhill for all $\bx$ to make statements about the long-term
dynamics of the system. In common Lyapunov analysis, we can make many
similar statements about the long-term dynamics of an uncertain system if
we can find a single Lyapunov function that goes downhill <em>for all
possible values of $\alpha$</em>. If we can find such a function, then we
can use it to make statements with all of the variations we've discussed
(local, regional, or global; in the sense of Lyapunov, asymptotic, or
exponential).</p>
<example><h1>A one-dimensional system with gain uncertainty</h1>
<p>Let's consider the same one-dimensional example used above, but add
an uncertain parameter into the dynamics. In particular, consider the
system: $$\dot{x} = -x + \alpha x^3, \quad \frac{3}{4} < \alpha <
\frac{3}{2}.$$ Plotting the graph of the one-dimensional dynamics for a
few values of $\alpha$, we can see that the fixed point at the origin is
still stable, but the <em>robust region of attraction</em> to this fixed
point (shaded in blue below) is smaller than the region of attraction
for the system with $\alpha=1$.</p>
<figure> <img width="80%"
src="figures/cubicPolynomialExample_gain_uncertainty.svg"/>
</figure>
<p>Taking the same Lyapunov candidate as above, $V = \frac{1}{2} x^2$,
we have $$\dot{V} = -x^2 + \alpha x^4.$$ This function is zero at the
origin, and negative for all $\alpha$ whenever $x^2 > \alpha x^4$, or
$$|x| < \frac{1}{\sqrt{\alpha_{max}}} = \sqrt{\frac{2}{3}}.$$ Therefore,
we can conclude that $|x| < \sqrt{\frac{2}{3}}$ is inside the robust
region of attraction of the uncertain system.</p>
</example>
<p>Not all forms of uncertainty are as simple to deal with as the gain
uncertainty in that example. For many forms of uncertainty, we might not
even know the location of the fixed points of the uncertain system. In
this case, we can often still use common Lyapunov functions to give some
guarantees about the system, such as guarantees of <em>robust set
invariance</em>. For instance, if you have uncertain parameters on a
quadrotor model, you might be ok with the quadrotor stabilizing to a pitch
of $0.01$ radians, but you would like to guarantee that it definitely does
not flip over and crash into the ground.</p>
<example><h1>A one-dimensional system with additive uncertainty</h1>
<p>Now consider the system: $$\dot{x} = -x + x^3 + \alpha, \quad
-\frac{1}{4} < \alpha < \frac{1}{4}.$$ Plotting the graph of the
one-dimensional dynamics for a few values of $\alpha$, this time we can
see that the fixed point is not necessarily at the origin; the location
of the fixed point moves depending on the value of $\alpha$. But we
should be able to guarantee that the uncertain system will stay near the
origin if it starts near the origin, using an invariant set
argument.</p>
<figure>
<img width="80%" src="figures/cubicPolynomialExample_additive_uncertainty.svg"/>
</figure>
<p>Taking the same Lyapunov candidate as above, $V = \frac{1}{2} x^2$,
we have $$\dot{V} = -x^2 + x^4 + \alpha x.$$ This Lyapunov function
allows us to easily verify, for instance, that $V \le \frac{1}{3}$ is a
<em>robust invariant set</em>, because whenever $V = \frac{1}{3}$, we
have $$\forall \alpha \in [\alpha_{min},\alpha_{max}],\quad
\dot{V}(x,\alpha) < 0.$$ Therefore $V$ can never start at less than
one-third and cross over to greater than one-third. To see this, see
that $$ V=\frac{1}{3} \quad \Rightarrow \quad x = \pm \sqrt{\frac{2}{3}}
\quad \Rightarrow \quad \dot{V} = -\frac{2}{9} \pm \alpha
\sqrt{\frac{2}{3}} < 0, \forall \alpha \in
\left[-\frac{1}{4},\frac{1}{4} \right]. $$ Note that not all sub-level
sets of this invariant set are invariant. For instance $V <
\frac{1}{32}$ does not satisfy this condition, and by visual inspection
we can see that it is in fact not robustly invariant.</p>
</example>
<todo>robust quadrotor example</todo>
</subsection> <!-- end common lyap -->
<subsection><h1>Region of attraction estimation for polynomial systems</h1>
<p>Now we have arrived at the tool that I believe can be a work-horse for
many serious robotics applications. Most of our robots are not actually
globally stable (that's not because they are robots -- if you push me hard
enough, I will fall down, too), which means that understanding the regions
where a particular controller can be guaranteed to work can be of critical
importance.</p>
<p>Sums-of-squares optimization effectively gives us an oracle which we
can ask: is this polynomial positive for all $\bx$? To use this for
regional analysis, we have to figure out how to modify our questions to
the oracle so that the oracle will say "yes" or "no" when we ask if a
function is positive over a certain region which is a subset of $\Re^n$.
That trick is called the S-procedure. It is closely related to the
Lagrange multipliers from constrained optimization, and has deep
connections to "Positivstellensatz" from algebraic geometry.</p>
<subsubsection><h1>The S-procedure</h1>
<p>Consider a scalar polynomial, $p(\bx)$, and a semi-algebraic set
$g(\bx) \preceq 0$, where $g$ is a vector of polynomials. If I can find
a polynomial "multiplier", $\lambda(\bx)$, such that \[ p(\bx) +
\lambda^T(\bx) g(\bx) \sos, \quad \text{and} \quad \lambda(\bx) \sos, \]
then this is sufficient to demonstrate that $$p(\bx)\ge 0 \quad \forall
\bx \in \{ g(\bx) \le 0 \}.$$ To convince yourself, observe that when
$g(\bx) \le 0$, it is only harder to be positive, but when $g(\bx) > 0$,
it is possible for the combined function to be SOS even if $p(\bx)$ is
negative.</p>
<p>We can also handle equality constraints with only a minor
modification -- we no longer require the multiplier to be positive. If
I can find a polynomial "multiplier", $\lambda(\bx)$, such that \[
p(\bx) + \lambda^T(\bx) g(\bx) \sos \] then this is sufficient to
demonstrate that $$p(\bx)\ge 0 \quad \forall \bx \in \{ g(\bx) = 0
\}.$$ Here the intuition is that $\lambda(x)$ can add arbitrary
positive terms to help me be SOS, but those terms contribute nothing
precisely when $g(x)=0$.</p>
</subsubsection>
<subsubsection><h1>Basic region of attraction formulation</h1>
<example id="roa_cubic_system"><h1>Region of attraction for the
one-dimensional cubic system</h1>
<p>Let's return to our example from above: \[ \dot{x} = -x + x^3 \]
and try to use SOS optimization to demonstrate that the region of
attraction of the fixed point at the origin is $x \in (-1,1)$, using
the Lyapunov candidate $V = x^2.$ </p>
<p>First, define the multiplier polynomial, \[ \lambda(x) = c_0 + c_1
x + c_2 x^2. \] Then define the optimization
\begin{align*} \find_{\bf c} \quad & \\ \subjto \quad& - \dot{V}(x) -
\lambda(x) (1-V(x)) \sos \\ & \lambda(x) \sos \end{align*} </p>
<p>You can try this example for yourself in <drake></drake>.</p>
<jupyter>examples/lyapunov.ipynb</jupyter>
</example>
<p>While the example above only verifies that the one-sub-level set of
the pre-specified Lyapunov candidate is negative (certifying the ROA
that we already understood), we can generalize the optimization to allow
us to search for the largest sub-level set (with the objective using a
convex approximation for volume). We can even search for the
coefficients of the Lyapunov function using an iteration of convex
optimizations. There are a number of variations and nuances in the
various formulations, and some basic rescaling tricks that can help make
the numerics of the problem better for the solvers.</p>
<example><h1>Region of Attraction codes in Drake</h1>
<p>In <drake></drake>, we have packaged most of the work in setting up
and solving the sums-of-squares optimization for regions of attraction
into a single method <code>RegionOfAttraction(system, context,
options)</code>. This makes it as simple as, for instance:</p>
<pre><code class="python">x = Variable("x")
sys = SymbolicVectorSystem(state=[x], dynamics=[-x+x**3])
context = sys.CreateDefaultContext()
V = RegionOfAttraction(sys, context)
</code></pre>
<jupyter>examples/lyapunov.ipynb</jupyter>
<p>Remember that although we have tried to make it convenient to call
these functions, they are not a black box. I highly recommend opening
up the <code>RegionOfAttraction</code> method and understanding how it
works. There are lots of different options / formulations, and numerous numerical recipes to improve the numerics of the optimization problem.</p>
</example>
<!--
<example><h1>Region of attraction for the time-reversed Van der Pol oscillator</h1>
<pre><code class="matlab" testfile="testVanDerPol">
cd(fullfile(getDrakePath,'examples'));
VanDerPol.run();
</code></pre>
</example>
-->
</subsubsection>
<subsubsection><h1>Seeding the optimization with linear analysis</h1>
<p>LQR gives the cost-to-go which can be used as the Lyapunov candidate.
Otherwise, use a Lyapunov equation. You may not even need to search for
a better Lyapunov function, but rather just ask the question: what is
the largest region of attraction that can be demonstrated for the
nonlinear system using the Lyapunov function from linear analysis?</p>
</subsubsection>
</subsection> <!-- ROA for polynomial -->
</section>
<section><h1>Rigid-body dynamics are (rational) polynomial</h1>
<p>We've been talking a lot in this chapter about numerical methods for
polynomial systems. But even our simple pendulum has a $\sin\theta$ in the
dynamics. Have I been wasting your time? Must we just resort to polynomial
approximations of the non-polynomial equations? It turns out that our
polynomial tools can perform exact analysis of the manipulation equation for
almost all†<sidenote>†the most notable exception to this is if
your robot has helical/screw joints (see
<elib>Wampler11</elib>).</sidenote>of our robots. We just have to do a
little more work to reveal that structure.</p>
<p>Let us first observe that rigid-body kinematics are polynomial (except
the helical joint). This is fundamental -- the very nature of a "rigid
body" assumption is that Euclidean distance is preserved between points on
the body; if $\bp_1$ and $\bp_2$ are two points on a body, then the
kinematics enforce that $|\bp_1 - \bp_2|_2^2$ is constant -- these are
polynomial constraints. Of course, we commonly write the kinematics in a
minimal coordinates using $\sin\theta$ and $\cos\theta$. But because of
rigid body assumption, these terms only appear in the simplest forms, and
we can simply make new variables $s_i = \sin\theta_i, c_i = \cos\theta_i$,
and add the constraint that $s_i^2 + c_i^2 = 1.$ For a more thorough
discussion see, for instance, <elib>Wampler11</elib> and
<elib>Sommese05</elib>. Since the potential energy of a multi-body system
is simply an accumulation of weight times the vertical position for all of
the points on the body, the potential energy is polynomial.<p>
<p>If configurations (positions) of our robots can be described by
polynomials, then velocities can as well: forward kinematics $\bp_i =
f(\bq)$ implies that $\dot\bp_i = \frac{\partial f}{\partial \bq}\dot\bq,$
which is polynomial in $s, c, \dot\theta$. Since the kinetic energy of
our robot is given by the accumulation of the kinetic energy of all the
mass, $T = \sum_i \frac{1}{2} m_i v_i^Tv_i,$ the kinetic energy is
polynomial, too (even when we write it with inertial matrices and angular
velocities).</p>
<p>Finally, the <a href="multibody.html">equations of motion</a> can be
obtained by taking derivatives of the Lagrangian (kinetic minus
potential). These derivatives are still polynomial!</p>
<example><h1>Global stability of the simple pendulum via SOS</h1>
<p> We opened this chapter using our intuition about energy to discuss
stability on the simple pendulum. Now we'll replace that intuition with
convex optimization (because it will also work for more difficult
systems where our intuition fails).</p>
<p>Let's change coordinates from $[\theta,\dot\theta]^T$ to $\bx =
[s,c,\dot\theta]^t$, where $s \equiv \sin\theta$ and $c \equiv
\cos\theta$. Then we can write the pendulum dynamics as $$\dot\bx =
\begin{bmatrix} c \dot\theta \\ -s \dot\theta \\ -\frac{1}{m l^2}
\left( b \dot\theta + mgls \right) \end{bmatrix}.$$
</p>
<p>Now let's parameterize a Lyapunov candidate $V(s,c,\dot\theta)$ as
the polynomial with unknown coefficients which contains all monomials
up to degree 2: $$V = \alpha_0 + \alpha_1 s + \alpha_2 c + ...
\alpha_{9} s^2 + \alpha_{10} sc + \alpha_{11} s\dot\theta.$$ Now we'll
formulate the feasibility problem: \[ \find_{\bf \alpha} \quad \subjto
\quad V \sos, \quad -\dot{V} \sos.\] In fact, this is asking too much
-- really $\dot{V}$ only needs to be negative when $s^2+c^2=1$. We can
accomplish this with the S-procedure, and instead write \[ \find_{{\bf
\alpha},\lambda} \quad \subjto \quad V \sos, \quad -\dot{V}
-\lambda(\bx)(s^2+c^2-1) \sos.\] (Recall that $\lambda(\bx)$ is another
polynomial with free coefficients which the optimization can use to
make terms arbitrarily more positive when $s^2+c^2 \neq 1$.) Finally,
for style points, in the code example in <drake></drake> we ask for
exponential stability:</p>
<jupyter>examples/lyapunov.ipynb</jupyter>
<p>As always, make sure that you open up the code and take a look. The
result is a Lyapunov function that looks familiar (visualized as a
contour plot here):</p>
<div style="text-align:center"><img
src="figures/pend_global_sos.svg" width="400px"/></div>
<p>Aha! Not only does the optimization finds us coefficients for the
Lyapunov function which satisfy our Lyapunov conditions, but the result
looks a lot like mechanical energy. In fact, the result is a little
better than energy... there are some small extra terms added which prove
exponential stability without having to invoke LaSalle's Theorem.</p>
</example>
<p>The one-degree-of-freedom pendulum did allow us to gloss over one
important detail: while the manipulator equations $\bM(\bq) \ddot{\bq} +
\bC(\bq, \dot\bq)\dot{\bq} = ...$ are polynomial, in order to solve for
$\ddot{\bq}$ we actually have to multiply both sides by $\bM^{-1}$. This,
unfortunately, is <i>not</i> a polynomial operation, so in fact the final
dynamics of the multibody systems are <i>rational</i> polynomial. Not
only that, but evaluating $\bM^{-1}$ symbolically is not advised -- the
equations get very complicated very fast. But we can actually write the
Lyapunov conditions using the dynamics in implicit form, e.g. by writing
$V(\bq,\dot\bq,\ddot\bq)$ and asking it to satisfy the Lyapunov conditions
everywhere that $\bM(\bq)\ddot\bq + ... = ... + {\bf B}\bu$ is
satisfied, using the S-procedure.</p>
<example id="ex:implicit"><h1>Verifying dynamics in implicit form</h1>
<p>Typically we write our differential equations in the form $\dot\bx =
{\bf f}(\bx, \bu).$ But let us consider for a moment the case where the
dynamics are given in the form $${\bf g}(\bx, \bu, \dot\bx ) = 0.$$ This
form is strictly more general because I can always write ${\bf
g}(\bx,\bu,\dot\bx) = f(\bx,\bu) - \dot\bx$, but importantly here I can
also write the bottom rows of ${\bf g}$ as $\bM(\bq)\ddot\bq +
\bC(\bq,\dot\bq)\dot\bq - \btau_g - \bB \bu$. This form can also
represent <a
href="https://en.wikipedia.org/wiki/Differential-algebraic_system_of_equations">differential
algebraic equations (DAEs)</a> which are more general than ODEs; $\bg$
could even include algebraic constraints like $s_i^2 + c^2 - 1$. Most
importantly, for manipulators, ${\bf g}$ can be polyonimal, even if ${\bf
f}$ would have been rational.</p>
<p>Interestingly, we can check the Lyapunov conditions, $\dot{V}(\bx) \le
0$, directly on a system (with no inputs) in its implicit form,
$\bg(\bx,\dot\bx)=0$. Simply define a new function $Q(\bx, \bz) =
\frac{\partial V(\bx)}{\partial \bx} \bz.$ If we can show $Q(\bx, \bz) \le
0, \forall \bx,\bz \in \{ \bx, \bz | \bg(\bx,\bz) = 0 \}$ using SOS, then
we have verified that $\dot{V}(\bx) \le 0$, albeit at the non-trivial cost
of adding indeterminates $\bz$ and an additional S-procedure.</p>
</example>
<p>There are a few things that <i>do</i> break this clean polynomial view
of the world. Rotary springs, for instance, if modeled as $\tau = k
(\theta_0 - \theta)$ will mean that $\theta$ appears alongside
$\sin\theta$ and $\cos\theta$, and the relationship between $\theta$ and
$\sin\theta$ is sadly <i>not polynomial</i>. Linear feedback from LQR
actually looks like the linear spring, although writing the feedback as $u
= -\bK \sin\theta$ is a viable alternative.</p>
<p>In practice, you can also Taylor approximate any smooth nonlinear
function using polynomials. This can be an effective strategy in
practice, because you can limit the degrees of the polynomial, and because
in many cases it is possible to provide conservative bounds on the errors
due to the approximation.</p>
<p>One final technique that is worth knowing about is a change of
coordinates, often referred to as the <a
href="https://en.wikipedia.org/wiki/Stereographic_projection">
stereographic projection</a>, that provides a coordinate system in which
we can replace $\sin$ and $\cos$ with polynomials:
<figure>
<img width="70%" src="figures/stereographic.svg"/>
<figcaption>The stereographic projection.</figcaption>
</figure>
By projecting onto the line, and using similar triangles, we find that $p
= \frac{\sin\theta}{1 + \cos\theta}.$ Solving for $\sin\theta$ and
$\cos\theta$ reveals $$\cos\theta = \frac{1-p^2}{1+p^2}, \quad \sin\theta
= \frac{2p}{1+p^2}, \quad \text{and} \quad \frac{\partial p}{\partial
\theta} = \frac{1+p^2}{2},$$ where $\frac{\partial p}{\partial \theta}$
can be used in the chain rule to derive the dynamics $\dot{p}$. Although
the equations are rational, they share the denominator $1+p^2$ and can be
treated efficiently in mass-matrix form. Compared to the simple
substitution of $s=\sin\theta$ and $c=\cos\theta$, this is a minimal
representation (scalar to scalar, no $s^2+c^2=1$ required); unfortunately
it does have a singularity at $\theta=\pi$, so likely cannot be used for
global analysis.
</p>
</section> <!-- end rigid bodies are polynomial -->
<todo>convex ROA w/ outer approx.</todo>
<todo>control design w/ alternations.</todo>
<section><h1>Exercises</h1>
<exercise><h1>Valid Lyapunov Function for Global Stability</h1>
<p>For the system \begin{align*} \dot x_1
&=-\frac{6x_1}{(1+x_1^2)^2}+2x_2, \\ \dot x_2
&=-\frac{2(x_1+x_2)}{(1+x_1^2)^2}, \end{align*} you are given the positive
definite function $V(\bx) =\frac{x_1^2}{1 + x_1^2}+ x_2^2$ and told that,
for this system, $\dot V(\bx)$ is negative definite over the entire space.
Is $V(\bx)$ a valid Lyapunov function to prove global asymptotic stability
of the origin for the system described by the equations above? Motivate
your answer.</p>
</exercise>
<exercise><h1>Invariant Sets and Regions of Attraction</h1>
<p>You are given a dynamical system $\dot \bx = f(\bx)$, with $f$ continuous, which has a fixed point at the origin. Let $B_r$ be a ball of (finite) radius $r > 0$ centered at the origin: $B_r = \{ \bx : \| \bx \| \leq r \}$. Assume you found a continuously-differentiable scalar function $V(\bx)$ such that: $V(0) = 0$, $V(\bx) > 0$ for all $\bx \neq 0$ in $B_r$, and $\dot V(\bx) < 0$ for all $\bx \neq 0$ in $B_r$. Determine whether the following statements are true or false. Briefly justify your answer.</p>
<ol type="a">
<li>$B_r$ is an invariant set for the given system, i.e.: if the initial state $\bx(0)$ lies in $B_r$, then $\bx(t)$ will belong to $B_r$ for all $t \geq 0$.</li>
<li>$B_r$ is a subset of the ROA of the fixed point $\bx = 0$, i.e.: if $\bx(0)$ lies in $B_r$, then $\lim_{t \rightarrow \infty} \bx(t) = 0$.</li>
</ol>
</exercise>
<exercise><h1>Are Lyapunov Functions Unique?</h1>
<p>If $V_1(\bx)$ and $V_2(\bx)$ are valid Lyapunov functions that prove
global asymptotic stability of the origin, does $V_1(\bx)$ necessarily
equal $V_2(\bx)$?</p>
</exercise>
<exercise><h1>Proving Global Asymptotic Stability</h1>
<p>Consider the system given by \begin{align*} \dot x_1 &= x_2 - x_1^3, \\
\dot x_2 &= - x_1 - x_2^3. \end{align*} Show that the Lyapunov function
$V(\bx) = x_1^2 + x_2^2$ proves global asymptotic stability of the origin
for this system.</p>
</exercise>
<exercise><h1>Control-Lyapunov Function</h1>
<p>Consider the problem of synthesizing a stabilizing feedback law $\bu =
\pi (\bx)$ for the dynamical system $\dot \bx = f(\bx, \bu)$. Lyapunov
analysis suggests us a very simple approach to this problem: choose a
candidate Lyapunov function $V(\bx)$, and design a control law $\pi (\bx)$
such that $V(\bx)$ always decreases along the trajectories of the
closed-loop system $\dot \bx = f(\bx, \pi (\bx))$. A function $V(\bx)$
for which such a control law exists is called a <i>control-Lyapunov
function</i>. In this exercise, we use this idea to drive a wheeled
robot, implementing the controller proposed in <elib>Aicardi95</elib>.</p>
<p>Similar to <a href="intro.html#wheeled_robot">this previous
example</a>, we use a kinematic model of the robot. We represent with
$z_1$ and $z_2$ its Cartesian position and with $z_3$ its orientation.
The controls are the linear $u_1$ and angular $u_2$ velocities. The
equations of motion read \begin{align*} \dot z_1 &= u_1 \cos z_3, \\ \dot
z_2 &= u_1 \sin z_3, \\ \dot z_3 &= u_2.\end{align*} The goal is to
design a feedback law $\pi(\bz)$ that drives the robot to the origin
$\bz=0$ from any initial condition. As pointed out in
<elib>Aicardi95</elib>, this problem becomes dramatically easier if we
analyze it in polar coordinates. As depicted below, we let $x_1$ be the
radial and $x_2$ the angular coordinate of the robot, and we define $x_3 =
x_2 - z_3$. Analyzing the figure, basic kinematic considerations lead to
\begin{align*} \dot x_1 &= u_1 \cos x_3, \\ \dot x_2 &= - \frac{u_1 \sin
x_3}{x_1}, \\ \dot x_3 &= - \frac{u_1 \sin x_3}{x_1} -
u_2.\end{align*}</p>
<figure>
<img width="40%" src="figures/exercises/wheeled_robot.svg"/>
<figcaption>Wheeled robot with Cartesian $\bz$ and polar $\bx$
coordinate system.</figcaption>
</figure>
<ol type="a">
<li>For the candidate Lyapunov function $V(\bx) = V_1(x_1) + V_2(x_2,
x_3)$, with $V_1(x_1) = \frac{1}{2} x_1^2$ and $V_2(x_2, x_3) =
\frac{1}{2}(x_2^2 + x_3^2)$, compute the time derivatives $\dot V_1
(\bx, u_1)$ and $\dot V_2(\bx, \bu)$.</li>
<li>Show that the choice \begin{align*} u_1 &= \pi_1(\bx) = - x_1 \cos
x_3, \\ u_2 &= \pi_2(\bx) = x_3 + \frac{(x_2 + x_3) \cos x_3 \sin
x_3}{x_3}, \end{align*} makes $\dot V_1 (\bx, \pi_1(\bx)) \leq 0$ and
$\dot V_2 (\bx, \pi(\bx)) \leq 0$ for all $\bx$. (Technically speaking,
$\pi_2(\bx)$ is not defined for $x_3=0$. In this case, we let
$\pi_2(\bx)$ assume its limiting value $x_2 + 2 x_3$, ensuring
continuity of the feedback law.)</li>
<li>Explain why Lyapunov's direct method does not allow us to establish
asymptotic stability of the closed-loop system.</li>
<li>Substitute the control law $\bu = \pi (\bx)$ in the equations of
motion, and derive the closed-loop dynamics $\dot \bx = f(\bx,
\pi(\bx))$. Use LaSalle's theorem to show (global) asymptotic stability
of the closed-loop system.</li>
<li>In <a
href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/lyapunov/control/control.ipynb"
target="_blank">this python notebook</a> we set up a simulation
environment for you to try the controller we just derived. Type the
control law from point (b) in the dedicated cell, and use the notebook
plot to check your work.</li>
</ol>
</exercise>
<exercise><h1>Limitations of SOS Polynomials in Lyapunov Analysis</h1>
<ol type="a">
<li>Are there positive definite functions that are not representable as sums of squares?</li>
<li>If a fixed point of our dynamical system does not admit a SOS Lyapunov function, what can we conclude about its stability?</li>
</ol>
</exercise>
<exercise><h1>ROA Estimation for the Time-Reversed Van der Pol Oscillator</h1>
<p>In this exercise you will use SOS optimization to approximate the ROA of the time-reversed Van der Pol oscillator (a variation of the <a href="https://en.wikipedia.org/wiki/Van_der_Pol_oscillator">classical Van der Pol oscillator</a> which evolves backwards in time). In <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/lyapunov/van_der_pol/van_der_pol.ipynb" target="_blank">this python notebook</a>, you are asked to test the following SOS formulations.</p>
<ol type="a">
<li>The one from <a href="#roa_cubic_system">the example above</a>, augmented with a line search that maximizes the area of the ROA.</li>
<li>A single-shot SOS program that can directly maximize the area of the ROA, without any line search.</li>
<li>An improved version of the previous, where less SOS constraints are imposed in the optimization problem.</li>
</ol>
</exercise>
</section>
</chapter>
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