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 ...

learn more… | top users | synonyms

0
votes
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 ...
0
votes
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....
0
votes
1answer
35 views

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 ...
0
votes
0answers
36 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 ...
0
votes
0answers
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 ...
1
vote
1answer
36 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} ...
0
votes
0answers
32 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 ...
1
vote
3answers
53 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 $A_r$ ...
0
votes
0answers
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.
1
vote
0answers
46 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. ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
0
votes
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: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
0
votes
0answers
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 $\Re e\{Z_{2}(j\omega)\}$ ...
0
votes
1answer
54 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
votes
1answer
58 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 ...
1
vote
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 ...
1
vote
2answers
68 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. ...
1
vote
0answers
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 ...
0
votes
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. ...
1
vote
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
votes
1answer
27 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{...
0
votes
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 $T:\mathbb{R}^n\to\mathbb{R}^n$ ...
1
vote
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 ...
0
votes
0answers
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 $t \in \mathbb{R}^{+}$.
1
vote
0answers
37 views

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.$|...
0
votes
0answers
26 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,...
1
vote
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 ...
0
votes
0answers
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{...
1
vote
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\...
1
vote
1answer
69 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 $x\in\mathbb{R}^n$...
0
votes
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 $...
5
votes
1answer
73 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 $$ \...
1
vote
0answers
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+\...
1
vote
0answers
80 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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
2answers
177 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 ...
0
votes
0answers
45 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
votes
3answers
51 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 ...
1
vote
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 ...
0
votes
1answer
29 views

Find the normal form of this function

A second order control theory function looks like: $$\text{H}_{(s)}=\frac{\text{K}_p}{\frac{1}{\omega_0^2}\cdot s^2+\frac{2\beta}{\omega_0}\cdot s+1}$$ Now I've got the function, with $a,r,k\in\...
0
votes
1answer
39 views

How does state transition matrix works

Suppose I have a simple vehicle moving in 2D. The state vector for the vehicle is X=[x y vx vy ax ay], that is, it contains the position (x,y), the velocity (vx, vy) and the acceleration (ax, ay) of ...
0
votes
1answer
64 views

What is the smallest positive value of K which makes the closed-loop system unstable?

You are given a transfer function $\displaystyle G(s)=\frac{1.81K(s+20)}{(s^3+10s^2+32s+32)}$. This system is connnected with unity negative feedback. I've tried so many things but I can't do it . I'...
1
vote
1answer
51 views

Are integro-differential equations considered dynamical systems?

A definition of the dynamical system is that: $\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$, $E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$, then $\...
0
votes
1answer
30 views

PID tuning for system with parameter

How can I tune a PID to control a system with a parameter? Can I know beforehand how to change the PID parameters in function of the value of the system parameter?
0
votes
0answers
15 views

Should the performance of a PID controller be independent of the input?

Assume you designed a PID controller to let a given system track a unit step. Will the controlled system exhibit the same behaviour with regard to step inputs with different amplitudes?
0
votes
2answers
39 views

How is a state disturbance matrix constructed?

Consider the system: $\dot{x}$ = Ax + Bu y = Cx + Du Where x contains 4 states, we have 2 inputs $u = \begin{bmatrix}u_1\\u_2\end{bmatrix}$ and A, B, C & D are known. Now if 2 separate noise ...
2
votes
2answers
65 views

How to drive a vehicle (limited by acceleration) on a flat ground to a given point as fast as possible?

So I have a function $\mathbf{x}(t): \mathbb{R} \rightarrow \mathbb{R}^2$, which is supposed to mean the path of the vehicle (time mapped to position). The initial conditions $\mathbf{x}(0)$ and $\dot{...
5
votes
0answers
117 views

Two definitions of uniformly observable, are they equivalent?

When I do my research, I found that there are two definitions of uniformly observable, I can't help thinking are they equivalent? These two definitions are listed as follows. For a linear stochastic ...