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 calculate the transfer function from a group of equations below?

The group of equations below describe the relationship of variables from a circuit(C stands for capicator, L is for inductor etc.). And the equation at the bottom shows how the transfer function is ...
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51 views

Can the root locus of a minimum phase plant become unstable?

I have a discrete system for which the root locus equation is given as: $$A(z) + K\cdot B(z) = 0$$ They are such that $A(0) = 1, B(0) > 0$, and $K>0$. $\frac B A$ is minimum phase and a ...
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46 views

Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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2answers
49 views

Stability criteria for linear systems with auxiliary variables

Classical texts for control theory show the linear system $\dot x=A \,x$, is stable if the real parts of the eigenvalues are negative. Does the same criteria apply for a system of the following form:...
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27 views

Relationship between delay and fractional order differential equation

There are control systems with delay [1] There are differential equations with fractional order: $$D^\alpha x(t)=f(t,x(t))$$ I am wondering why we see control systems with delay and fractional ...
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30 views

How to obtain a stabilization problem in linear system with controller?

The scheme of system: The equasion after Laplace transform: $$Y(p) = \frac{PID(p)\cdot H(p)}{1 + PID(p)\cdot H(p)} Y^d(p)$$ Now I want to make inverse Laplace transform and then plot $y(t)$, but $y^...
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1answer
31 views

Centroid Root Locus

I can't figure out how to find the root locus centroid for the poles of this simple equation in a positive feedback system. $$ H(s)=\frac{s}{s^2+3s+1} $$ I have read in many places that the centroid ...
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26 views

How to Optimize PID Gains Non-Heuristically

In robotics, variables that control processes (usually voltage output to actuators) are continually adjusted by a PID (Proportional Integral Derivative) Control algorithm to improve the result of a ...
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20 views

Routh stability criterion ranges of k

Hi I need to find the ranges of k and $\beta$ such that the steady state error will be less than 10% for unit step input%. I'm currently at the point where I know that the steady state error is given ...
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1answer
25 views

steady state output transfer function

Hi If I'm given an input: $$r(t) = 2 \sin(3t).$$ And If I'm supposed to get the steady state response $$y(t)$$ for the given transfer function $$G(s) = \frac{1}{(s+1)(s+1)}$$ Is the following ...
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1answer
17 views

Root loci transfer function

Hi I asked a question earlier about this problem: $$T(s) = \frac{G(s)}{1+G(s)H(s)}$$ Where $$H(s) = \frac{s}{s+1}$$ $$G(s) = \frac{k(s+4)}{(s+2)(s^2+s+6)}$$ . Poles resulting: $$-1, -2, -0.5 + 2....
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Transfer function Root loci

Hi I was given a closed system as: $$T(s) = \frac{G(s)}{1+G(s)H(s)}$$ Where $$H(s) = \frac{s}{s+1}$$ $$G(s) = \frac{k(s+4)}{(s+2)(s^2+s+6)}$$ This is where I'm unsure of. When I calculated the ...
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40 views

How to discretize state space with uniform grid

Let us consider a general continuous time stochastic differential equation represented by *dx* = A(x)dt + B(x)udt + $\sigma$ dw where A(x) represent the ...
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38 views

Optimisation/Control in lieu of knowing the future …

I’m interested to hear about the methods that would be best applied to an optimization/control problem. Context: I’m looking at the concept of ‘peak shaving’ in power networks, which are achieved by ...
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1answer
37 views

Trajectories for a Second Order System with negative real eigenvalues

Consider the Second-order System $$ \dot{\mathbf{x}} = A \mathbf{x} $$ $a_1$ and $a_2$ are positive real numbers, with $a_1 > 2*a_2$. The matrix A is given as $$A = \begin{pmatrix} ...
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35 views

Polar form of the Fourier transform of $\sin(t)$

I'm studying signal processing, and I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain ...
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3answers
54 views

modal truncation of state space system while preserving certain eigenvalues

Given a state space system $(A,B,C)$: $$\dot{x}=Ax+Bu\\y=Cx$$ Is there any method to obtain a reduced system $(A_r,B_r,C_r)$, where $$\dot{x}=A_rx+B_ru\\y=C_rx,$$ such that the eigenvalues of $A_r$ ...
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26 views

Controllability problem for linear systems

I need to prove: The pair (A,B) is controllable iff (A-BK,B) is controllable. A,B and K are the real matrices: nxn, nx1, 1xn, respectively. Thank you.
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51 views

Matched pole zero discretization

There are several techniques to discretize continuous-time transfer functions to discrete-time transfer functions. Some of them, such as, zero-order-hold, forward euler or Tustin, are well known. ...
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42 views

Mathematical control and differential equations

The differential equation $$\frac{dx}{dt}= Ax(t)+ Bu(t)+ f(t,x(t))+ g(t,u(t))$$ subjected to $x(0)=a$ defined on $[0, t], t>0$, where $u$ is a control and $x$ is a state. Does this differential ...
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24 views

How ca I discretize nonlinear ode ODEs?

I am trying to discretize a nonlinear set of ordinary differential equations. Usually I got good results with Euler, nevertheless, in this case Euler is not fitting the system response for a constant ...
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1answer
30 views

How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
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21 views

About the limitations of linear analysis in comparison with Lyapunov

Consider the connection of two electrical circuits. Both circuits, Z1 and Z2, are stable and only one of them is non-passive. I.e., the eigenvalues are located in the LHP but $\Re e\{Z_{2}(j\omega)\}$ ...
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1answer
57 views

Minimal time trajectory optimization for body in 2D with waypoints

I am trying to control a spacecraft moving in 2D space through a number of waypoints as fast as possible. The spacecraft has directional acceleration (with an upper limit on its magnitude, and a limit ...
3
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1answer
60 views

What is the difference between moving-horizon DP and MPC?

What is the difference between moving-horizon DP (dynamic programming) and MPC (modelbased predictive control)? In both cases, the system input at time $t$ is determined by solving a finite-time ...
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1answer
59 views

Are block diagrams valuable for non-LTI systems?

Do block diagrams provide any value in analyzing non-LTI systems? For LTI systems, block diagrams permit a powerful algebra for manipulation/reduction of a system. Are there classes of non-LTI systems ...
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2answers
69 views

Why do we ignore the real part of the transfer function while calculating Frequency response? [closed]

The transfer function obtained from the differential equation is a function of $s $ which $x + iy$, then why do we ignore the real part while finding out the frequency response magnitude and phase. ...
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31 views

Force Control on a Tether

On the ground, I have a winch driven by a synchronous motor to which I can command a torque input $u$. Parameters of this winch, such as friction $d$ and moment of inertia $J$ are well understood and ...
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1answer
42 views

Feedback in Sampled Data System

I have a feedback loop as shown in figure. Broken link represents sampling and signal with asterisk as superscript represents discrete signal. ZOH is the abbreviation for zero order hold operation. ...
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1answer
40 views

Interpretation of a reaction diffusion equation

I have a reaction-diffusion equation in 1-dimensions of the typical form: $$\frac{\partial }{\partial t} u(x,t)= \frac{\partial^2 }{\partial x^2} u(x,t)+ \alpha(x) u(x,t), \,\qquad (x,t)\in (0,1)\...
3
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1answer
28 views

Transfer function of controller

I am solving this question given in book (Automatic control system). As asked in (a) part $G_c(s)$ of the controller. I solved it and getting answer$$G_c(s) = \frac{F(s)}{E_c(s)}=\frac{100}{s}-\frac{...
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2answers
102 views

Initial conditions that converge to an unstable equilibrium

Consider the discrete-time dynamical system \begin{align} x_{k+1}=T(x_k), \end{align} where $k\in\mathbb{N}$, $x_k\in\mathbb{R}^n$, $x_0$ is the initial condition, and $T:\mathbb{R}^n\to\mathbb{R}^n$ ...
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1answer
25 views

Show exponential stability quadratic form

Please help me with the following proof: Suppose $\dot x=f(x(t))$ and suppose that we have: $$ \frac{d}{dt}\left( x(t)^TPx(t) \right)\le -x(t)^TQx(t) $$ where $P$ and $Q$ are symmetric ...
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27 views

Does every $\mathcal{L}_{2}$ signal have bounded $\mathcal{L}_{2}$ derivative?

Let a real signal $f(t) \in \mathcal{L}_{2}$. Does it always imply that $\dot{f}(t) \in \mathcal{L}_{2}$? It is assumed that $\dot{f}(t)$ exists for all $t \in \mathbb{R}^{+}$.
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perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ 2.$|...
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27 views

Is constant system a Causal System?

Is y(t) = 1 a causal system? From the definition of causal systems , a causal system is a system where the output depends on past and current inputs. Here the system doesn't depend on any input. So,...
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1answer
72 views

Reachability from non-zero initial state?

I have the following system: $$ \dot x(t)=\begin{bmatrix} -2 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}x(t)+\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}u(t) $$ The ...
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32 views

LTI system: solving for the time at which a system state reaches a given value

Suppose I have the following Linear Time Invariant (LTI) system: \begin{equation} \dot{x}(t) = Ax(t) + Bu(t) \end{equation} where $x(t)=\begin{bmatrix}x_1(t) & x_2(t) &\ldots &x_N(t)\end{...
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1answer
19 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf \{t\...
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1answer
77 views

Clarke's generalized gradient formula computed on functions defined on open sets

In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let $x\in\mathbb{R}^n$...
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1answer
36 views

Why we cannot apply pole placement for the following system?

$$ \begin{cases} \dot{x}=\begin{pmatrix} -2 & 0 \\ 0 & -2 \\ \end{pmatrix} x+\begin{pmatrix} 2 \\ 2 \\ \end{pmatrix} u \\ y=Cx \end{cases} $$ How can I show that if $p_1 \neq -2$ and $...
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1answer
74 views

Minimum infinity norm control problem

I am having trouble understanding Example 2 of section 5.9 of Luenberger's Optimization by Vector Space Methods. The problem is to select a current $u(t)$ on $[0,1]$ to drive a motor governed by $$ \...
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39 views

Matrix comparison depend on one scalar variable

Let $A$, $B$, $K$ be $n\times n$, $n\times m$ and $m\times n$ matrix respectively. $\alpha_i>0$ is a scalar. Consider the following matrix: $H_i=\int_0^\infty e^{(A+\alpha_i BK)^Tt}(\alpha_iI+\...
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0answers
83 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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1answer
28 views

Discrete time system - phase response

I have a question about a discrete time filter. All I have is the pole-zero plot and I have to calculate the impulse, phase and magnitude response. To make this a proper fraction I used polynomial ...
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1answer
86 views

Stability of sampled-data systems using Lyapunov functions

For continuous systems, Lyapunov functions provide a general technique to establish stability. For example, the simple system $x' = -x$, a Lyapunov function is $V(x) = \frac{1}{2}x^2$. It is easy to ...
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2answers
209 views

What is the difference between an impulse response and a transferfunction?

An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?). By knowing the impulse response you know the system. The ...
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46 views

LQR Problem Minimization Prove

Given J for an LQR Problem is $J = \frac{1}{2} \int {z_1}^T \hat Q z_1 + v^T Q_{22} v\,dt $ where $\hat Q$ above is given as $\hat Q = Q_{11} - Q_{12}{Q^{-1}}_{22}Q_{21}$ is minimized if we use ...
3
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3answers
54 views

Relation between controllability and stabilization of a system

Suppose i have a control system which is described as: $$ \left\{ \begin{array}{c} \dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)+Du(t) \end{array} \right. $$ and i know it is controllable. I use a state ...
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1answer
31 views

Explain: Independent Column variables as Linear combination of free variables.

I am reading a book on control systems and stuck on a text in it. We have $Sx = 0$ where S $\in$ $\Re^{m x n}$ and is full rank i.e Rank of S = m. $x \in \Re^n$ Now it states that "Exactly m ...