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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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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745 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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17 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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45 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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58 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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26 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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26 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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26 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 ...
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46 views

it it possible to solve these equation for their root.

I am trying to solve an equations such as the roots of $$k*x(11*x + 1) + d*x(11x + 1)$$ has to match the roots of this function $$x^2 + 0.1x + 6 + k*x(11*x + 1) + d*x(11x + 1)$$, where I have to ...
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34 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
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10 views

A confusion on adaptive algorithm

Consider flight trajectory control problem i.e. find out the control parameter for which the average error of the actual output and desired output is minimized. Can we call any algorithm for solving ...
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1answer
159 views

Using overshoot and settling time formula to determine pole location?

Is it possible to use the formula for overshoot and settling to determine where where ones pole should. by using the overshoot and settling time formula i mean, using it to define what $\zeta$ and ...
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54 views

From position to velocity?

I have an transfer function which tells what the angular displacement of an DC motor. This transfer function is in the S-domain, and normally when you differentiate (*s) the angular position you would ...
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26 views

Adding poles and zeroes due to PID

I've been wondering for a long time if a system is adding a pole or a zero to the close loop system, when a PID controller is added to the control system???.. I tried google it everywhere and i ...
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176 views

System stability for adding a pole and Zero?

How come does a system become more stable when a zero is added to a system.. I mean i doesn't not change the location of the pole, it is still the same? An example: Looking a closed loop system ...
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20 views

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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472 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
98 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
37 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
93 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
79 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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107 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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43 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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66 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
82 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 ...
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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
53 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
136 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
202 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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38 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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45 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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1answer
75 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
50 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
54 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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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
38 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
104 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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58 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
56 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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36 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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47 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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46 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
54 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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36 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
35 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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41 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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44 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 ...