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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Find piecewise constant function u for $X'(t)=AX(t) + Bu(t)$ and $X(t)=\begin{pmatrix}10 \\0 \end{pmatrix}$ for some T

Consider the system $$x''(t)=u(t)$$ such that $x(0)=100, \; x'(0)=50$. Find a function $u$ piecewise constant such that $x(T)=0, \; x'(T)=10$ for a time $T$ Using the control theory language, it is ...
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12 views

Are steady-state values in LTI systems differentially influenced by the rate an input is introduced? [migrated]

I have a general question about LTI systems I was hoping someone could clarify. I'm currently learning about steady state behavior in LTI systems, and how you can evaluate them with different inputs. ...
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0answers
11 views

Robust Control of a Linear System with Input Saturation

The following system is considered: $$\dot{x}=Ax+Bu+Qw$$ Where, $x\in R^2$ is the state vector,$u\in R^1$ is the control input and $w\in R^1$ is an unmatched disturbance signal. The goal is to ...
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1answer
36 views

Control Theory: Why is $A+BK$ called a closed loop system?

Given a control system $\dot x = Ax + Bu$ and $y = Cx$. Suppose we use state feedback to create $u = +Kx$ where $K$ is the gain matrix. Subbing into above equation, we have $\dot x = Ax + Bu = Ax + ...
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2answers
37 views

Can someone explain the meaning of asymptotically stability in the following definition?

I am taking this from an online course note. Given a system $\dot x = Ax$, we say that: The origin is asymptotically stable if $x(t) \to 0$ as $t \to \infty \thinspace \forall x(0)$ I am ...
2
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1answer
418 views

Roll PID Control for quadcopter

I'm trying to implement with simulink a PD controller for my quadcopter. I use a simplified model, and for the roll case I have $ I_x * \phi = L $, where L is the roll torque. So, the transfer ...
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2answers
60 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
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2answers
219 views

Feedback characteristics of nonlinear dynamical systems [closed]

$ \newcommand{\e}{\boldsymbol \eta} \newcommand{\h}{\boldsymbol h} \newcommand{\T}{T^{\mathsf{ref}}} \newcommand{\g}{\boldsymbol g} \newcommand{\e}{\boldsymbol \eta} \newcommand{\dt}{\partial_{t}} ...
2
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0answers
30 views

“Dictionary” of linearizations for nonlinear dynamical system

I have recently jumped on a control project that involves predicting output of a nonlinear system given some input. The team has used $N$ training input/output relationships to build a 'dictionary' ...
0
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1answer
43 views

Pontryagin's Maximum Principle as a sufficient condition?

It is know that Pontryagin's maximum principle provides in general only a necessary condition in the following sense: The ODE system which is known to be solved by the optimal control may have ...
2
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1answer
69 views

What is $H^\infty$ norm and why is it used in control theory?

Can anyone knowledgable elaborate on what exactly is a $H^\infty$ norm and why it is used in control theory instead of some other norms? Thanks!
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12 views

Surface fitting: Where to start from?

Often, we deal with identification problems such as identifying the parameters $\alpha_i$ where $z(x) = f_{\alpha_i}(x,y)$, which means simply $z$ is a function of $(x,y)$ and the parameters ...
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2answers
26 views

State space representation for non-proper transfer function

Is there a way to find a state space representation of a non-proper transfer function? In the case of a PID controller the transfer function is: $\frac{K_d s^2 + K_p s + K_i}{s}$ What would be the ...
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0answers
38 views

Please Help Me - Is this conclusion true?

In a sliding mode control, we have : $ s = \dot e + \Lambda e $ we know that e and $ \dot e $ are independent variables. Now in order to find control effort from the system dynamics, we rewrite the ...
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1answer
315 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 ...
1
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1answer
38 views

Inverse Laplace

I want to calculate the inverse laplace of $$F(s)=e^{-3s}\frac{1+s}{s^3+2s^2+2s}$$ And i'm wondering if applying the derivative theorem is correct. To keep it simple it split them up: ...
4
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1answer
2k views

Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
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1answer
18 views

Stability of non-homogeneous and non-autonomous first-order difference equation

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by \begin{eqnarray*} x_i(t+1) = ...
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58 views

Example of BIBO stable system that is not internally stable

In the theory of system, we know that a system can be BIBO stable but not internally stable (if there is a pole-zero cancellation in the transfer function for example). I find this concept quite ...
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1answer
47 views

Controller design for an exponential plant of the form $y=a\exp(bx)$

I have a stationary model for a plant(a valve) given by $y=a \exp(bx)$. I linearised this by taking log on both sides: $$\ln(y) = b\cdot x + \ln(a).$$ Then, I estimated the plant transfer ...
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59 views

Eigenvalues of a matrix written in controllable canonical form

Let the following equation represent a stable (marginally) dynamical system in discrete time domain \begin{equation} \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k \end{equation} ...
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0answers
28 views

How to Compute Arrival and Departure Angles in Root Locus for break-in and break-away points?

I have difficulty to find the angles in the point from which poles go away from real-axis. The transfer function L(s) of open-loop is: $$ L(s) = \frac{\mu_r(s+10)}{s^2(s+100)} $$ where $C(s) = ...
1
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1answer
50 views

Discrete PID controller Laplace formula

I saw the following formula: the transfer function is: $$Gr(s) = K_p \bigg(1 + \frac{1}{T_n s}+ \frac{T_v s}{1 + T_d s}\bigg) $$ From my understanding: $K_p$ is the proportional gain $T_n$ is the ...
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0answers
17 views

How can I measure the dispersion of a discrete time system?

I have a discrete time system: x[n+1]=A*x[n]+B[n], with B[n] as an excitation signal. How can I measure the dispersion of this system? I guess this is related to some property of A, which depicts the ...
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1answer
28 views

Simplification of $H(s)=(4+2/s)(3/(1+2s))$

I had the following problem: We combine a parallel PI-control system ($H_{1}(s)=P+\frac{I}{s}$ with $P=4, I=2$) with a 1st order process ($H_{2}(s)=\frac{3}{1+\tau s}$ with $K=3$, $\tau=2$) This ...
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2answers
104 views

How do I determine the transfer function of a plant?

I sitting here with a system which I have to determine the transfer function. The unit receives a velocity and position, and move towards that position with the given velocity. What kind of test ...
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1answer
35 views

A question about Deadbeat Observers based on Deadbeat State feedback

In digital control systems we can place both observer and system in zero to have a deadbeat performance but there is a Ambiguity in the ratio of performance: We need observer to be faster than ...
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0answers
25 views

About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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1answer
38 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
2
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0answers
45 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
0
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1answer
19 views

Derivation for state equation linearization

In the following notes, how to linearize a state equation is described. The part I don't understand is why you can just remove the $\delta$ like that. I think the state equation should be: ...
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2answers
36 views

Non-minimum phase systems

I wanted to clear this doubt I have since a long time and for which I am not able to find a clear answer since different sources say differently or ambiguously. $\textbf{Does a system have to be ...
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2answers
52 views

What is the largest invariant set?

I think the largest invariant set is on other than $\{x:\dot{x}=0\}$, is this correct, is there other way to establish the largest invariant set? Please give an example.
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39 views

Solution to a state-space equation

In my notes on non-linear linearization there is the following example. It asks to verify the solution to the state-space equation. My understanding is that the solution is where the equilibrium point ...
2
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2answers
78 views

matrix inequality proof [completion of squares]

Can someone help me to prove this? $\begin{bmatrix} 0 & B^\top W^\top \\ WB & 0 \end{bmatrix} \leq \begin{bmatrix} B^\top Q B & 0 \\0 & W^\top Q^{-1}W \end{bmatrix}$ with $Q$ ...
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3answers
63 views

In control theory, why do we linearize around the equilibrium for a nonlinear system?

For example, in these notes: In the first example with the pendulum, they define the equilibrium as where the pendulum is at the vertical position (x=0), with a angular velocity of 0 (x'=0) and the ...
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1answer
49 views

From multivariable system transfer function matrix to state space representation

I have the transfer function matrix $H(s) = \begin{bmatrix} {1\over s+1} & {2\over s+2} \\ {-2\over s^2+3s+2} & {2s\over s+1} \\ \end{bmatrix}$ And I want to ...
7
votes
1answer
232 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$ \dot x = (-A+M)x, $$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
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2answers
49 views

controllability of a system (control theory)

For a system in state-space representation: $\dot{x}(t) = Ax(t) + Bu(t)$; $y(t) = Cx(t) + Du(t)$, we say that a system is controllable if for $\gamma = [B \quad AB \quad \cdots \quad A^{n-1}B]$, ...
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0answers
32 views

Given dynamic system $\dot x = Ax + Bu$, how can we prove that $M$ = {$(A,B)$: system is controllable} is an open set on an Euclidean space?

I wish to show that $M$ = {$(A,B)$: system is controllable} is an open set in some Euclidean space. Equivalently, how can we show that the complement to this set is closed? Here Controllability ...
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2answers
49 views

Nyquist Stability Criterion

The rational function $b(s) = (s+3)(s-4)^{-1}$ is the frequency response function (FRF) of a system $B$. Is $B$ stable? I understand that a system is unstable if there are poles in the closed right ...
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1answer
44 views

Feedforward control - Developing an understanding

In general, for a feedforward controller design of a motion system, it is essential to recognize the various components (acceleration feedforward, viscous friction and dry friction) present in the ...
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6answers
2k views

Control / Feedback Theory

I am more interested in the engineering perspective of this topic, but I realize that fundamentally this is a very interesting mathematical topic as well. Also, at an introductory level they would be ...
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0answers
28 views

Linear quadratic regulator with output tracking

I have the following linear system: $$ x_n = Ax_{n-1} + Bu_n $$ $$ y_n = C^Tx_n $$ And I want to find the control $u_1, ..., u_N$ such a: $$ \min_u J$$ $$ J = u_1^2 + ... + u_N^2 + \sum_{n = 1}^{N} ...
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25 views

how can I determine if a MIMO system is ''similar'' to another one?

I'm talking about closed loop regime. I have a complex model and a simplified one, I want to show that they behave similarly in closed loop. Both of them are MIMO. for SISO I could use the ...
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0answers
14 views

What are the deduction of $\sin\left(\omega_0t+\theta\right)$ for Laplace form?

I have the following function in time domain $\sin\left(\omega_0t+\theta\right)$, which is $\left(\frac{s\sin\left(\psi_1\right)+\omega_0\sin\left(\psi_1\right)}{s^2+\omega_0^2}\right)$ in Laplace ...
3
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1answer
45 views

Equal number of poles and zeros for square transfer matrix

In page 10 of this document (MIT Courseware on control) it is stated that since the transfer function is square, there is an equal number of poles and zeros. Does this hold and if so under which ...
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28 views

Discrete sinusoidal to state space

I'm looking to apply an optimal LQR filter to a discrete signal of the form $x[n]=Asin(ω_0n+ϕ)+v[n]$ The amplitude $A$ and the phase $ϕ$ are unknown variables I want to estimate using the filter, ...
0
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1answer
30 views

Synchronization of Rossler system - the Rossler Attractor

I am studying synchronization of Rossler system given by the following set of two linear ODEs and one nonlinear ODE: $\dot{x_1} = -x_2 - x_3$ $\dot{x_2} = x_1 + ax_2$ $\dot{x_3} = c + x_3(x_1 - b)$ ...
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1answer
23 views

A convoluted transfer function (from prof. Dullerud's robust control testbook)

The following proble is from the book: A Course in Robust Control Theory (a convex approach), middle of p. 200 Consider the following general feedback loop: , ie $\dot x(t) = Ax(t) + ...