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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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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32 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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37 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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52 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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40 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
40 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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31 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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23 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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29 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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89 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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49 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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37 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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33 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
41 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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42 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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33 views

control engineering transfer function vibration

What does vibrational mode even mean? How do you tell it from the poles?
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1answer
49 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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31 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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38 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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33 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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59 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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29 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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36 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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34 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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49 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
74 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 ...
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46 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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15 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
33 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 & ...
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2answers
94 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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22 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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43 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 + ...
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1answer
137 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
48 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 ...
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2answers
49 views

Why does this phase calculation go to 180 instead of 90?

This is all coming from the following video I am studying from http://www.youtube.com/watch?v=XSS6L42ce88 So I am working from this system $$ G(s)\,=\,\frac{4}{s^{2}+s+2}$$ and the video states the ...
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20 views

Is the ordered list of controllability indices invariant, or the unordered list?

The list of controllability indices of a linear time-invariant system is invariant under state feedback and change of variables. What is invariant exactly though: the ordered list, or the unordered ...
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66 views

Calculate step response from impulse response LTI-system.

Can someone please give me a few pointers on how to calculate the step response for an LTI system with this impulse response?? \begin{equation} h[n] = 2^nu[n]. \end{equation}
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Root counter of a closed loop system

How can we determine the root locus counter of the following closed loop system $$ s^3 + K_2s^2+K_1s+K_1 = 0 $$ where $K_1$ and $K_2$ are parameters which vary from $0$ to $\infty$ . I have tried ...
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3answers
165 views

An introduction to control systems

I am looking for an introduction on control systems in the context of engineering, but treated from a more mathematical point of view. Does anybody have a good reference?
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13 views

Controlled system question optimisation

Consider the controlled system $x_{t+1} = x_t + u_t + 3\epsilon_{t+1}$, where the $\epsilon_t$ are independent $N(0,1)$ variables. The instantaneous cost at time t is $x_t^2 + 2u_t^2$. Assuming that ...
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1answer
68 views

Calculate state transition matrix

I have a question to the following problem: "There is a linear, time-invariant System with the form $\frac{d\mathbf{x}}{dt}=\mathbf{A}x$. The Eigenvalues of the matrix $A$ are $s_1=-1$ and $s_2=-2$, ...
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50 views

Continuous time optimisation question- find value function and optimal control

If we have a continuous-time system with a scalar state variable, plant equation $\dot{x}= u$, and cost function $Q\int_o^h u^2 dt + x(h)^2$, then by writing the dynamic programming equation ...
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1answer
36 views

Reference request: “initial” PDE control

I'm somewhat familiar with the ideas of boundary and internal (source) control for PDE (in particular, for wave equations) but am wondering if the following type of problem is classified/studied; if ...
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1answer
57 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
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1answer
23 views

Stability of system with gain K

It is asked to sketch the root loci of the system . I have done that . Then it is asked to determine the stability of the system as function of $K$ . After plotting I see that for $K>0$ it is ...
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1answer
17 views

Determination of the modulus of continuity

I'm trying to prove the uniqueness of the viscosity solution of an Hamilton-Jacobi-Bellman equation. Thanks to a classical result, I'm left to check if it exist a modulus of continuity $\omega_1$ --- ...
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1answer
63 views

Relationship between controllability and null space of A, B

For the dynamic system $\dot x = Ax + Bu$ There's a saying that this system is controllable when $Ker(B) \in Ker(A)$, which means that $u$ have the control in every dimension of $x$. I have no ...
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1answer
46 views

Second order control system

We have the open loop transfer function $$ G(s)H(s) = \frac{k(s+2)}{s^2+2s+3}$$ There were three parts to this question : a. The value of $k$ for which repetitive roots occur b. The range of $k$ ...
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1answer
123 views

Proof of Impulse response of a BIBO stable system

I was wondering if anyone here could expand upon or provide the proof of the boxed theorem which I have shown in the image below? Any help would be greatly appreciated.