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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7 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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34 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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2answers
39 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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1answer
42 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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0answers
74 views

Kalman Filter application to non-linear system.

I want to use the Kalman filter to have a better estimate of the state of a system which I know its equations of motion: ...
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1answer
22 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 $$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+3y(t)+2\frac{dz}{dt}+z(t)=U_1(t)$$ ...
3
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1answer
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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1answer
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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2answers
106 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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10 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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1answer
30 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
3
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2answers
88 views

How to use mathematically the I and D of a PID controller

I am trying to mathmatically understand how the $P$, $I$, and $D$ parameters work on a system, quite having a hard time doing so. I've only been able to show that the Steady State Error (SSE) never ...
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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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2answers
36 views

Having trouble forming the initial matrices for a positioning problem

The question asks me to solve the positioning problem where: $$ \dot{x_1} = x_2 $$ $$ \dot{x_2} = u_1 \in U_{bb} $$ $$ x_1(0) = - \text{X} (<0) $$ $$ x_2 (0) = 0 $$ $$ x_1(t_1) = 0$$ $$ x_2(t_1) = ...
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35 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
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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2answers
44 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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0answers
44 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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0answers
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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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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1answer
27 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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31 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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2answers
111 views

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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1answer
38 views

Is this a discrete time Lyapunov function?

I have an algorithm to optimize a process. It is a discrete time algorithm. Every iteration of this algorithm changes the state of the process. I found a function, say $f(s)$, where $s$ is the state ...
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1answer
21 views

characteristic equation of transfer function

$$\frac{K}{s(s+1)(s+5)}$$ Find the characteristic equation of this transfer function. The book gives this answer: $$\frac{K}{s(s+1)(s+5)} +1=0$$ or $$s^3 +6s^2 +5s +K =0.$$ I don't get how the ...
2
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2answers
118 views

What is the difference between regulator and stabilization

What is the difference between regulator and stabilization in control theory don't they both minimize the disturbance to the system? could answer be elaborated from the view of state and output?
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1answer
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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57 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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1answer
24 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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2answers
105 views

Stability of unit feedback LTI system (s-1)/(s(s+1)) vs. Nyquist Criterion

Consider a unit feedback system $$ X(s) = \frac{G(s)}{1+G(s)} $$ where the open loop transfer function of the system is $$ G(s) = \frac{s-1}{s(s+1)} $$ Open loop Bode & Nyquist plots: ...
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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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0answers
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 ...
1
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1answer
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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2answers
55 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 ...
0
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1answer
30 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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0answers
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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0answers
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 \, ...
0
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1answer
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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0answers
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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0answers
29 views

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) ...
19
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5answers
964 views

What is the mathematical foundation of Control Theory?

There is a question which I'm wondering again and again in recent months. I have taken courses like Elementary Differential Equations, Signals and Systems, Linear Control Systems, General Theory of ...
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0answers
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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0answers
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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0answers
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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0answers
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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1answer
189 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 ...
2
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0answers
32 views

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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0answers
41 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?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$