Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller ...

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Relation between Riccati Algebraic Equation and optimization problem

Reading this page: http://www.mathworks.com/help/robust/ug/minimizing-linear-objectives-under-lmi-constraints.html I got stuck in the result that says it can be show that minimizing Trace of X (a ...
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22 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
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21 views

State space system, with a state space system as feedback

I have two state space systems. Now I want to compute the state space system where the first state space system is the input of the other... $$M_1 = \begin{cases}\dot{x}_1 = A_1 x_1 + B_1 u_1 \\ y_1 ...
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15 views

Use linear quadratic regulator to minimize output error

I would like to create an Infinite-horizon, continuous-time LQR with a cost functional defined as $$J = \int_{0}^\infty \left( e^T Q e + u^T R u \right) dt$$ where e is the states' error $x-x_d$, ...
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38 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
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53 views

Reachable Space by an ODE

Let $\dot{x}(t) = Ax(t) + Bu(t)$ be an $n$-dimensional first order ODE where $u(t) \in \mathcal{P}$ for some convex polytope $\mathcal{P}$, for every $t \in \mathbb{R}$. Assume $x(0) = 0$. Is there a ...
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25 views

writing differential equation into state space

i have 2 equations of second order that model the same system and i have to model with state variables $$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+3y(t)+2\frac{dz}{dt}+z(t)=U_1(t)$$ ...
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41 views

Is an Invariant set Connected?

Let an autonomous dynamical system is characterized by the state equation $$ \dot x(t) = f(x(t)),\quad x(0)=x_0 $$ with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, ...
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19 views

Poles and zeros of a system matrx

While I am reading lecture notes on Poles and Zeros of MIMO systems, I find the following example, which is not clear for me. $$ H(s) =\pmatrix{1 & \dfrac{1}{s-3} \\ 0 & 1 } $$ The ...
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34 views

What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
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34 views

Euler equation for $\int_0^{\infty}e^{-rt}(x^2+2x+\dot x^2) \ \mathrm dt$? Is $\infty$ in the boundary open or closed?

I am pondering this problem here, the course Mat-2.3148 Dynamic Optimization in Aalto University, i.e. Find the function $x(t)$ such that $\int_0^{\infty}e^{-rt}(x^2+2x+\dot x^2)\ \mathrm dt$ has ...
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11 views

Backstepping analysis of multi-input system

Suppose I have a simple system that's like following: $\dot{x}_1 = A x_2 + Bx_3 \\ \dot{x}_2 = u_1 \\ \dot{x_3} = u_2$ I am familiar with a standard method of backstepping if there was only one ...
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110 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
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1answer
43 views

differential equation into state space

I have this dynamic system $$ J \ddot{\theta} + F\dot{\theta} = u $$ I would like to acquire the state space of the system. This is what I've done $$ x_{1} = \theta, \\ x_{2} = \dot{\theta}, \\ ...
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45 views

How to draw the Bode diagram for a given transfer function?

With this transfer function: $$G(s)=\displaystyle\frac{10(s+1)}{s(0.1s+1)}$$ I need to do operations to draw the Bode diagram manually I have this: $G(jw)=\displaystyle\frac{10jw+10}{-0.1w^2+jw}$ ...
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1answer
28 views

Explanation of notation $f(t)\in L_{\infty}$ in a control theory textbook

In a control theory textbook I saw the following notation : $$f(t)\in L_{\infty}$$ Since I am not familiar with this kind of notation could someone explain What does it mean?
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45 views

Conversion of continuous, linear stochastic system to discrete, LQR/LQG

I have the standard stochastic, linear time varying system $dx(t) = (A(t)x(t) + B(t)u(t))dt + G(t)dw(t) $ with $x(t_0) = x_0$ with quadratic cost $J = x(t_F)^TQ_Fx(t_F) + \int_{t_0}^{t_F}\left( ...
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28 views

Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
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36 views

Inverse of State-space representation

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 ...
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1answer
18 views

Does controller K have to be Hurwitz?

Ask a few simple questions but confused me. "A" (plant) is unstable. From ARE (or DRE), we find "K", and obtain Ac = A - BK, which is Hurwitz(stable) Must K be PD(>0), PSD, ND, or NSD? or no such ...
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29 views

Multiplication of state space transfer function (state-space form)

If I know the following transfer function (ss-form) How to obtain the following efficiently: Thanks
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32 views

Weird derivative computation

I found the following formulas in a control theory textbook : $$s(x,t)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)}\varepsilon $$ where $\varepsilon(t)=T\left(\frac{e(t)}{p(t)}\right)$ and ...
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Rigorous mathematical treatments of engineering topics

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I ...
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20 views

how to prove unobservable subspace ($\text{null}(C, A)$) is $A$-invariant

Given $$ \begin{align*} \dot{x} &= Ax + Bu \\ y &= Cx \end{align*} $$ where $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{p \times n}$. How to prove the ...
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1answer
30 views

In what cases are the eigenvalue equal to the pole points?

I have a transfer function in form of a matrix and want to determine the stability of the whole system. Now I'm wondering if I need to calculate the pole points or the eigenvalue. A friend of mine ...
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60 views

Estimation covariance of the Kalman filter state

I implemented Kalman filtering for a simplest 1D coordinate+velocity model. The prediction worked, but I wanted to estimate the prediction probability distribution. I.e. how likely it is that the ...
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13 views

Condition of RH2 and RH∞

Some notes say: A vector rational function is in RH2, if it is strictly proper and no poles on the closed right-half plane. A matrix rational function is in RH∞, if it is proper and no poles on the ...
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26 views

How to derive H2 transfer function (w -> z)?

Give a general feedback system: The dynamics of G: The dynamics of K: Suppose the A of the closed loop system; My questions is: how to prove the transfer function T(s): w -> z: I know ...
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16 views

Question about Coprime Factorization (CF)

Suppose G(s) = [A B;C D] is a transfer matrix. Suppose M', N' is a left-CF of G(s) a. (V',U') => M'V' + N'U' = I Suppose M, N is a right-CF of G(s) a. (V,U) => UN + VM = I F & H (constant ...
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35 views

Weird differentiation formula explanation

I stumbled upon the following formula in a systems control textbook : $$ s\left(\overline{x}^{(n-1)},t\right)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)} e(t) \in R$$ where $\overline{x}^{(n-1)}=[x\ ...
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1answer
31 views

Controllable & Stabilizable

Followed by the well-know theorem: (A,B) is controllable iff poles of A-BK can be arbitrarily assigned. (A,B) is stabilizable iff poles of A-BK can be arbitrarily assigned on the LHP LHP = left-half ...
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17 views

Describing function of a non linearity with memory

Can anyone help me on finding the correct methodology to compute the describing function of the following NL function? ...
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56 views

Riccati & Lyapunov equations

Hope to ask: Lyapunov Eq: $A'P + PA + Q = 0$ Algebraic Riccati Eq: $A'P + PA + Q + PB*inv(R)*B'P= 0$ It seems that the difference between the two lies in $B = 0$ (zero input) in Lyapunov Eq and both ...
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28 views

Constructing error state kalman filter

I am trying to construct an error state kalman filter for GPS/INS integration using simulated data and I am having problem on a few steps. My error state vector is $\delta x = [\delta\alpha \, ...
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50 views

How does the z transform work in practice?

What I've found I've implemented a PID controller using the equations 7 and 9 of this article, which states that: $$\frac{U(s)}{E(s)}=K_p+\frac{K_i}{s}+K_ds$$ Translates to ...
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25 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
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Prospect of research in some stochastic optimization/approximation field

This question is a not a technical one. Sorry for that. As I am new to the area of stochastic optimization/control, I want to know the active prospect of research in the following areas 1) ...
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25 views

Extension of Schur-Cohn for quadratic matrix equation

Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$: $z^2 + bz+c=0$ and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq ...
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33 views

How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
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41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
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17 views

What about the HUM for a finite-dimensional system?

We consider the finite-dimensional system \begin{equation}\begin{cases}y'(t)=Ay(t)+Bv,\ t\in (0,T)\\ y(0)=y^0\end{cases}\end{equation} Where $v$ is the control, $A\in Mat(N\times N); B\in Mat(N\times ...
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Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb ...
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42 views

how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$
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217 views

deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
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14 views

Controlling a random variable

I've got a system the output of which is a random variable with a certain distribution, which for the purpose of this discussion can be assumed to be normal. The input variable is voltage. The ...
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37 views

designing a controller for an unstable plant?

How do would anyone make a controller for a system which is $G(s) = \frac{(s-2)}{(s-1)(s-6)}$ I do not see how this system can ever become stable, without using pole/zero cancelation. So how ...
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44 views

Bode plot of an unstable system?

I am a bit confused on how to sketch a bode plot for an unstable system? (being a/all pole(s) lies on RHP). I tried plotting it in matlab, but it doesn't resemble the output i was expecting using ...
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23 views

Design feedback control law to make the whole matrix Hurwitz

Suppose $(A_1, B_1)$ and $(A_2, B_2)$ are both stabilizable. Then we know that we can find some $K_1$ and $K_2$ to make $A_1+B_1K_1$ and $A_2+B_2K_2$ Hurwitz, respectively. Now, for non-zero constant ...
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25 views

Control stability problem

Design controllers $u_1$ depending on $x$ and $u_2$ depending on $y$ such that the following system is exponentially stable: $$\dot x = A_1 x + B_1 u_1 + C_1 y \\ \dot y = A_2 y + B_2 u_2 + C_2 x ...
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1answer
24 views

write all functions as rational functions

my question is short and simple : could we write all functions that have the formula : $1 + G(s)H(s)$ as $$\frac{(s+b_0)(s+b_1)\dots(s+b_n)}{(s+p_0)\dots(s+p_m)}$$ if the answer is yes , could you ...