CDS 212, Homework 9, Fall 2010 - Revision history

Sojoudi at 23:51, 25 November 2010

2010-11-25T23:51:23Z

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	~~{{CDS 212 draft HW}}~~
	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle

Sojoudi: /* Problems */

2010-11-23T09:38:56Z

Problems

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	<ol type="a">		<ol type="a">
	<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>		<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
	<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)		<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath> (wherever you need them) of different degrees.)
	</ol>		</ol>
	</li>		</li>
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	V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi		V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
	</amsmath></center>		</amsmath></center>
	as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La ~~Salla~~'s invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)		as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salle's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
	</li>		</li>

Sojoudi: /* Problems */

2010-11-23T09:31:07Z

Problems

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	</li>		</li>

	<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>		<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath><br>
	Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>		Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
	<ol type="a">		<ol type="a">

Sojoudi: /* Problems */

2010-11-23T09:28:49Z

Problems

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	<li> Is <amsmath>p</amsmath> sum-of-squares? </li>		<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
	<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>		<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
	<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use ~~{\tt~~ issos}.)</li>		<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use issos.)</li>
	<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>		<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
	</ol>		</ol>

Sojoudi: /* Problems */

2010-11-23T09:05:13Z

Problems

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	<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>		<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
	<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>		<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
	<li> How can we use the results of the first two parts of this question to conclude that $p$ is positive semidefinite (even though it is not sum-of-squares)? </li>		<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
	</ol>		</ol>
	</li>		</li>

Sojoudi: /* Problems */

2010-11-23T09:04:32Z

Problems

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	V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi		V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
	</amsmath></center>		</amsmath></center>
	as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)		as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
	</li>		</li>

Sojoudi: /* Problems */

2010-11-23T09:03:00Z

Problems

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	</amsmath></center>		</amsmath></center>
	<ol type="a">		<ol type="a">
	<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~~~{}~~~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>		<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
	<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)		<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
	</li>		</li>
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	</li>		</li>

	<li> Use the data in http://www.cds.caltech.edu/\~{}utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.		<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
	<center><amsmath>		<center><amsmath>
	\begin{array}{c}		\begin{array}{c}

Sojoudi: /* Problems */

2010-11-23T09:01:47Z

Problems

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	</amsmath></center>		</amsmath></center>
	<ol type="a">		<ol type="a">
	<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~{}~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)		<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~{}~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
	~~\item~~ If you cannot find a quadratic Lyapunov function, try a 4th		<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
	degree one. If you cannot find a 4th degree Lyapunov function, then		</li>
	increase the degree of the candidate Lyapunov functions until you		</ol>
	find one. (Hint: 4th degree should work.)		</li>
	~~\end{itemize}~~
			<li> Use the data in http://www.cds.caltech.edu/\~{}utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
	~~\item~~ Use the data in http://www.cds.caltech.edu/\~{}utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables $V$ and $f.$ If you care, $V$ is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by $\dot{x} = f(x).$ Compute a lower bound on the optimal value of the following optimization problem.		<center><amsmath>

	~~\begin{equation}~~
	\begin{array}{c}		\begin{array}{c}
	\displaystyle{\max_{\mu > 0}} ~~[[User:Sojoudi\|Sojoudi]] 08:45, 23 November 2010 (UTC)~~\mu \\		\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
	\text{subject to}~~08:45, 23 November 2010 (UTC)~~ \{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.		\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
	\end{array}		\end{array}
	~~\end{equation}~~ (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)		</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
			</li>
	~~\item~~ Consider the system \[		<li> Consider the system
	\begin{~~array}{l~~}		<center><amsmath>
	\dot{x}_1 = -x_2\\		\begin{aligned}
	\dot{x}_2 = -f(x_2) - g(x_1),		&\dot{x}_1 = -x_2\\
	\end{~~array~~}		&\dot{x}_2 = -f(x_2) - g(x_1),
	\] where the functions $f$ and $g$ satisfy the following conditions:		\end{aligned}
	~~\begin{itemize}~~		</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
	~~\item $~~f$ and $g$ are continuous.		<ol>
	~~\item $~~f(0)=g(0)=0.~~$ $~~\sigma f(\sigma) >0$ and $\sigma g(\sigma)>0$ whenever $\sigma\neq 0.$		<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
	~~\item $~~\int_0^\sigma g(\xi) d\xi \rightarrow \infty$ as $\|\sigma\| \rightarrow \infty.$		<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
	~~\end{itemize}~~ Using		<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>\|\sigma\| \rightarrow \infty.</amsmath>
	\[		</ol>
			Using
			<center><amsmath>
	V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi		V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
	\] as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)		</amsmath></center>
			as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)
			</li>

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #9 | issued = 23 Nov 2010 | due = 2 Dec 2010 }} …'

2010-11-23T08:45:17Z

Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #9 | issued = 23 Nov 2010 | due = 2 Dec 2010 }} …'

New page

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
<li> How can we use the results of the first two parts of this question to conclude that $p$ is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~{}~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)
\item If you cannot find a quadratic Lyapunov function, try a 4th
degree one. If you cannot find a 4th degree Lyapunov function, then
increase the degree of the candidate Lyapunov functions until you
find one. (Hint: 4th degree should work.)
\end{itemize}

\item Use the data in http://www.cds.caltech.edu/\~{}utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables $V$ and $f.$ If you care, $V$ is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by $\dot{x} = f(x).$ Compute a lower bound on the optimal value of the following optimization problem.

\begin{equation}
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} [[User:Sojoudi|Sojoudi]] 08:45, 23 November 2010 (UTC)\mu \\
\text{subject to}08:45, 23 November 2010 (UTC) \{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
\end{equation} (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)

\item Consider the system \[
\begin{array}{l}
\dot{x}_1 = -x_2\\
\dot{x}_2 = -f(x_2) - g(x_1),
\end{array}
\] where the functions $f$ and $g$ satisfy the following conditions:
\begin{itemize}
\item $f$ and $g$ are continuous.
\item $f(0)=g(0)=0.$ $\sigma f(\sigma) >0$ and $\sigma g(\sigma)>0$ whenever $\sigma\neq 0.$
\item $\int_0^\sigma g(\xi) d\xi \rightarrow \infty$ as $|\sigma| \rightarrow \infty.$
\end{itemize} Using
\[
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
\] as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)