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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A doubt on markov decision process

Given that a policy is a function from a state action pair to probabilities, the set of policies for a MDP forms a POSET (the partial order is due to value function for a policy). Why there should be ...
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416 views

Effect of adding a Pole and Zero due to PID

I am kinda confused on how Adding a D(which adds a zero to the complete system) decreases the speed of the system. But when we normally add a zero to the system, it normally causes the system ...
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1answer
97 views

Integration in PID controller

I am trying to understand how come there is a phase difference is from the error signal and the output of my PID controller which consisting of I = 1. As far i've understood should the integration ...
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1answer
36 views

BIBO stable system

I would like to ask if this system is Bounded Input Bounded Output stable : $$y[n] = r^nx[n],\quad r\in \mathbb{R}$$ And why? I think this system is stable because $$| x[n] | ≤ B,\quad B < ...
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2answers
92 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
76 views

No repeated eigenvalues or the real part of any eigenvalue is not zero

I have an $n$ x $n$ matrix $M=\begin{bmatrix}-1 & -1\\ \frac{1}{2} & 0 & -\frac{1}{2}\\ & \ddots & \ddots & \ddots\\ & & \frac{1}{2} & 0 & -\frac{1}{2}\\ ...
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1answer
95 views

Difference between Variation of Calculus problems and Control Theory problems?

Variation of Calculus seems to have problems without the control with variables such as state and time. Then again Control Theory problems seems to have problems with one extra variable that is ...
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42 views

Reachable set using constant control input

Consider a linear time-invariant control system given by the differential equation \begin{align*} \dot{x}(t) &= Ax(t) + Bu(t), \;\; x(0) = x_0, \end{align*} where $x\in \mathbb{R}^n$ and $A,B$ ...
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64 views

Design control law

Consider the function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ which is continuously differentiable and strongly convex. Let $x^*$ to be the unique global minimizer of $f$. Assume that $L_1\|x\| ...
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3answers
81 views

Rank of the matrix product $C e^{At} B$

Let $A \in \mathbb R^{n \times n}$. Fix $m<n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. What I am interested in is the matrix ...
2
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1answer
75 views

How to prove these relations

I have following three basic recurrence relations $$\mathcal U_{k+1}=A(I+\mathcal U_{k}Q)^{-1}\mathcal U_{k}A^{\mathrm T}+G\\ \mathcal V_{k+1}=\mathcal V_{k}(I+Q\mathcal U_{k})^{-1}A^{\mathrm T}\\ ...
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1answer
44 views

Full rank of the matrix $Ce^{At}B$

Let $A \in \mathbb R^{n \times n}$. Fix $m <n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. I want to find algebraic conditions (which ...
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1answer
52 views

laplace transform of a sine function

I'm a little confused about how to find Laplace transforms of a sine function when it is a function of time. As in, suppose the function is $x(t)=\sin(at)$ , then I can proceed to get ...
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2answers
115 views

Rewriting differential equation into state space

I have some problems rewriting the following differential equation into state space form. I know the general principle of how it is done, but I'm getting confused of how the states are being defined, ...
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1answer
162 views

Why does the Final Value Theorem not hold for a transfer function with more than one pole at the origin?

The Wikipedia article on the Final Value Theorem states the following for cases where it does not hold: There are two checks performed in Control theory which confirm valid results for the Final ...
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37 views

Nyquist diagram of transfer function

Transfer function of a system is given as $$G(s) = \frac{100(s+5)}{s^2(s+3)(s^2+4)}$$ Sketch the Nyquist diagram and find if the system is stable. Also find the gain margin and phase margin. Please ...
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1answer
40 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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0answers
43 views

Simple Stochastic Control Problem

Consider $dX_t = \pi_t X_t dt + \pi_t X_t dW_t, X_0 = x$, where $W_t$ is a standard brownian motion, and $\pi$ is some real valued process. Let T>0. How can we calculate $P[X_T\geq 2x]$, where ...
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71 views

How to plot $\dot{x}= Ax + Bu$ (x versus t, by matlab)

I am junior in control. If $\dot{x} = Ax$ where $A$ is a $n\times n$ matrix and $x$ & $\dot{x}$ are $n\times 1$ vectors, by $x = \exp(At)$, we can draw $x$ versus $t$. If $\dot{x} = Ax + Bu$, ...
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1answer
48 views

Determine the transient response

How do i Determine the transient response of an transfer function if it's in the s domain?? the obvious answer would be using inverse laplace transform, but how come?? consider i have system like ...
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1answer
49 views

Damped systems - deriving equation

I am having some troubles deriving the formula for the roots for different types systems.. I am not quite sure if they are correct (pretty sure they aren't). $y(s) = \frac{s+2\zeta\omega_n}{s^2 + ...
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1answer
53 views

How are the tracking error and plant input related in a PID controller?

I am having trouble understanding a basic relationship in control theory - how the output of the controller is interpreted by the plant. Most control theory tutorials and introductions include a block ...
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1answer
27 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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1answer
36 views

The range of the controllability matrix

Consider a matrix $M$, the range of $M$, denoted by $R(M)$: $R(M) = \{b | b = Mx\}.$ Now, consider the controllability matrix $$C = \begin{bmatrix}B&AB & \dots& A^{n-1}B\end{bmatrix}=\\= ...
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1answer
101 views

Linearity and nonlinearity of systems

My teacher of Control Systems did some exercises at the seminar and I don't get it why he said that this system is not linear: $x_1'= x_1 + 2x_2 + 3x_2u_1$ $x_2'= x_2 + 3u_2$; $y_1 = x_1$ ...
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57 views

Determine Gain using root locus

I have this closed loop transfer function $T(s) = \frac{KG(s)}{1+KG(s)}$ Where G(s) is given, and K is Gain. I've to calculate the gain for which the damping ratio is 0.707. I've done that by ...
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1answer
54 views

Breakaway Point in Root-Locus

Can anyone explain me why the breakaway points in Root-Locus are only on the real axis?
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35 views

Designing a state feedback law for a nonholonomic system

Consider the set \begin{equation*} A_r=\left\{(e_x,e_y,L)\in\mathbb{R}^3:e_x=e_y=0,L(t)=\sqrt{\dfrac{\mu}{p_0^3}}t,t\in\mathbb{R}_{\geq0}\right\} \end{equation*} I have been trying to design a state ...
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1answer
46 views

How to control a System that tends to Zero

For a design project in a Control course, my classmates and I must create a Controller that steers an unknown system to a given trajectory within certain constraints. The system is given to us in ...
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1answer
43 views

On the first Lyapunov method, when the linearization fails

I have been trying to apply the first Lyapunov method to decide about the stability of the origin for the following system \begin{equation*} \dot{x}=\sqrt[3]{-x}. \end{equation*} However, the ...
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1answer
53 views

Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
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35 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
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1answer
34 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
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1answer
40 views

Determine value a in system matrix

I'm trying to solve the following problem: "Look at the image of trajectories of a linear, time-invariant system with the form: $\frac{d\textbf x}{dt}=\textbf {Ax}:$ Determine possible eigenvectors ...
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43 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
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77 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
25 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 ...
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0answers
36 views

Relationship between output and poles/zeros in the complex plane

Context There are lots of videos online which explain the time domain equivalent of poles depending on their place in the complex plane, but it's only useful for the simplest examples for which we ...
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1answer
37 views

A particular DE as first order system

When does a differential equation $\frac{d^2y}{dt^2}+a\frac{dy}{dt}+by=cu(t)+d\frac{du}{dt}$ admit a solution? If $d=0$, the existence is answered by Picard-lindelöf, and we can write it as a system ...
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1answer
36 views

Can I always put a system in modal form?

Given the transfer function of a system, can I always put the system in modal form? Are there exceptions?
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1answer
69 views

Derivation of the Mexican wave as Mean field equilibrium of an ergodic control problem

I'm dealing with a Mean Field system introduced by Gueant, Lasry, Lions in their "Mean Field Games and Applications" which admits as solution the so-called Mexican wave. Without going into the details ...
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2answers
111 views

significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
3
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0answers
53 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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17 views

Stein's theorem and the solution of Q=H-B*HB

Stein's (1952) Theorem 1 says: "A necessary and sufficient condidion that B is convergent is that there exists a positive definite Hermitian matrix H, for which H-B*HB is positive definite." ...
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2answers
49 views

finding the input sequence of a discrete-time dynamical system

I am studying Dynamical Systems, actually linear systems and I came across the following question: Consider the following discrete-time dynamical system: $x_{i+1}= \left( \begin{array}{ccc} 2 & ...
2
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2answers
100 views

Reference for LPV controls

I am looking for a good mathematical introduction to LPV (Linear Parameter Varying) methods in control theory. I would like it to be more on the mathematical side of things, instead of something aimed ...
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1answer
27 views

Literature study for Optimal Estimation Theory

It seems Optimal Estimation/Control Theory requires a lot more than undergraduate maths. Any good book that would help me get started? I have so far referred the following books but found them quite ...
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55 views

analytic solution to structured algebraic Riccati equation

In solving a model I have written down for a research paper, I am left with two Algebraic Riccati Equations, that is I need to solve for $X$ in the equation \begin{align*} X = A^\top (X + XB(R + ...
2
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1answer
456 views

Pole-zero cancellation Paradox

Suppose we have an open-loop transfer function $$G(s) = \frac{1}{s(s+a)(s+b)}$$ If we plot the root locus for the closed-loop system we will get roughly something like this : Now the question is ...
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1answer
77 views

Is there a formalized control-theory-like approach for uncontrollable systems?

I'm basically trying to control a system to achieve a given set of outputs, but I don't actually have enough inputs to control all the outputs. Is there any formalized theory on how to achieve the set ...