Articles of control theory

Strict Inequality in Homogenous LMI

I’m studying Stephen Boyd’s notes for EE 363, here. In particular, I’m working through lecture 15, slide 9 on strict linear matrix inequalities. An LMI is an expression of the form $G(x) = G_0 + x_1G_1 + \cdots + x_nG_n \geq 0$, where $G_i$ are symmetric $m \times m$ matrices and $x \in \mathbf{R}^n$. The […]

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ \lambda_1^{p_3} & \lambda_2^{p_3} & \cdots & \lambda_n^{p_3} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_1^{p_n} & \lambda_2^{p_n} & \cdots & \lambda_n^{p_n} \end{bmatrix}$$ where $p_1=0$ and $p_k > p_i$ for $i < k […]

Inverse of State-space representation (control)

Ask two questions from a paper (2012 ACC): Consider the plant: Let X be the stabilizing solution of the Riccati equation: where . Define the LQR gain by . The transfer matrix has a left spectral factorization , where WL is given by Questions: If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom […]

Robust Control VS Optimal Control

What’s the diffrents between Optimal Control and Robust Control? I know that Optimal Control have the controllers: LQR – State feedback controller LQG – State feedback observer controller LQGI – State feedback observer integrator controller LQGI/LTR – State feedback observer integrator loop transfer recovery controller (for increase robustness) And Robust Control have: $H_{2}$ controller $H_{\infty}$ […]

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 &= x_2 – 314.2 \\ \dot{x}_2 &= 122\sin{x_1}\cos{x_1} – 154.8x_3\sin{x_1} – 0.5x_2 + 201.2 \\ \dot{x}_3 &= 0.64\cos{x_1} – 0.8x_3 + 0.2u \end{split} \end{equation} where $\mathbf{x} = \mathbf{x}(t) = [x_1(t), x_2(t), x_3(t)]^T \in X […]

Existence of global solution of Riccati equation

Consider a Riccati differential equation $$ \dot P + A(t)^{T}P + PA(t) -PB(t)R(t)B(t)^{T}P + Q(t) = 0,\;\;\; P(t_0) = P_0 = P_0^{T} \geqslant 0 $$ where $Q(t) = Q(t)^{T} \geqslant 0$, $R(t) = R(t)^{T} > 0$, all matrices are real and continuous. How to show that for any $t_0$ there exists (and unique) a solution […]

Optimal control

Consider the growth equation: $ \dot{x} = tu $, with $x(0)=0$ and $x(1)=1$, and with the cost function: $ J= \int_0^1 u^2 dt $. Show that $u^*=3t$ is a successful control, with $x^*=t^3$ and $J^*=3$ the corresponding trajectory and cost. If $u=u^* + v $ is another successful control, show that $\int_0^1 vt dt = […]

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by Allen Tannenbaum. I do not have the access of the paper or even if I had, without any basic knowledge on control theory, I would, perhaps, not […]

Saturated damped harmonic oscillator

Consider a critically damped harmonic oscillator: $$x’ = v$$ $$v’ = -2v – x$$ Such a system has the property that if $x(0) < 0$ and $x(0)+v(0)\leq 0$, then $x(t) < 0$ for all $t$. Now suppose the system is modified so that the acceleration is subject to some minimum value, i.e. $v’ \geq a_{min}$. […]

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ $x \in \Re^n$, $u \in \Re^m$, $Q \in \Re^{n\times n}$, $R \in \Re^{m\times m}$, $A \in \Re^{n\times n}$, $B \in \Re^{n\times m}$. It can be […]