Yilin Mo

I am an Associate Professor at Tsinghua University, China.

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Notes

Interesting Mathematics, control theory and other scientific notes that I collected. No guarantee for accuracy or that I understand them. You can annotate the page or leave comments below.

Model Reference Adaptive Control

01 Sep 2015

Consider the following first-order linear system:

\begin{equation} \dot{x} = a x + bu, \label{eq:truesystem} \end{equation}

where \(a,\,b\) are unknown parameters. However, the sign of \(b\) is known.

Suppose that we want the linear system \eqref{eq:truesystem} to track the following reference system:

\begin{equation} \dot{x_m} = a_m x_m + b_m r, \label{eq:refsystem} \end{equation}

where \(a_m < 0\) and \(b_m\) are known and \(r(t)\) is a known reference signal.

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How to Get Fair Results from a Biased Coin

05 Jul 2015

This is a test of org-babel.

Problem Formulation

Consider we have a biased coin, such that the probability of getting a head is \(P(H) = p\in(0,1)\) and the probability of getting a tail is \(P(T) = q = 1-p\), where \(p\) is unknown by us. However, different coin tosses are independent from each other.

Our goal is to generate a "fair" binary random variable \(X\), with distribution \(P(X = 0) = P(X=1) = 0.5\).

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Properties of Discrete-time Algebraic Riccati Equation (DARE)

03 Jul 2015

This post serves as a test of emacs org-mode and jekyll static site generator.

Preliminary

We will denote \(\mathbb S_n,\,\mathbb S_n^+,\mathbb S_n^{++}\) to be the set of all real symmetric matrices, the set of all positive semi-definite matrices and the set of all positive definite matrices respectively. For any two matrices \(A,\,B\in \mathbb S_n\), we call \(A \geq B\) (\(A > B\)) if \(A - B\) is positive semi-definite (positive definite).

Let \(A\in \mathbb R^{n\times n}\) and \(C \in \mathbb R^{m\times n}\) be two matrices. Further define \(Q \in \mathbb S_n\) and \(R\in\mathbb S_n\) to be two positive definite matrices. The DARE function \(g: \mathbb S_n^+\rightarrow\mathbb S_n^+\) is defined as

\begin{equation} \label{eq:riccati1} g(X) \triangleq \left[\left(A X A' + Q\right)^{-1} + C'R^{-1}C\right]^{-1}. \end{equation} read more