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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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: \begin{...
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137 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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80 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 ...
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397 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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125 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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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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107 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 ...
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36 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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168 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 $\gamma = [B \quad AB \quad \cdots \quad A^{n-1}B]$, rank$(...
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83 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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63 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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30 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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289 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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111 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, ...
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45 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
172 views

Relative Gain Array of a singular matrix

I am a masters student in controls and would like to get insight into the concept of relative gain array for multivariable feedback control. In general what I have come across from the book on the ...
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461 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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37 views

How to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions?

I want to know how to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions. Please help me. Thanks very much.
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78 views

How to prove for a system with rational stable transfer function, the output is square integrable?

I want to know for a system with rational stable transfer function, i.e. H(jw)=1/[(a1+jw)(a2+jw)...(an+jw)] (a1,a2,..,an>0), why a square integrable (L2 integrable) input must generate a square ...
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181 views

Why algebraic Lyapunov equation has an unique solution?

In the following text book (p.47): Optimal Control (Lewis 2nd edition) There is a theorem: (Zero input case) If $A$ is stable, and $(A,\sqrt Q )$ is observable, then $S_\infty= \sum_{i=0}^\...
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78 views

Control theory- basic question on stabilizabilty

Studying some control theory but having difficulty learning because my lecturer doesn't provide solutions to any of his exercises AT ALL. Below I've attached a problem I've just done and my answers ...
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129 views

Stability of zero-order hold controllers for linear systems

From what I've read, Lyapunov functions provide a very nice mechanism for verifying stability of continuous-time linear systems (non-linear as well, but that's not my concern at the moment). For ...
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94 views

Control theory with state and input constraints

What are some control theory tools for solving problems of the following form?: Given a system model, control input constraints $I$, and control output constraints $O$, what is the largest set ...
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473 views

Physical interpretation of transfer function in control theory

I'm learning about transfer functions in control theory. I'm struggling to find a physical interpretation for the input and output of a transfer function, both of which may be complex numbers. In the ...
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28 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) + ...
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1answer
39 views

Controllability of a system

How can I show that all solutions of $x(t)'=\pmatrix{0&-1\\ 1&0}x(t)+\pmatrix{\cos(t)\\ \sin(t)}u(t)$ are within the area $x_1sin(t)-x_2cos(t)=0$ ?
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81 views

D - Part in PID doesn't impact steady state error?

Why does the D-part in a PID controller not do any impact on the steady state error. I mean if tries to resist changes, should it not then make it stay at wanted configuration?
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3answers
31 views

transformation of a difference equation

How can I translate the difference equation $$x_{k+3}+4x_{k+2}+3x_{k+1}+x_k=2u_{k+2}$$ into a state-space representation of the following form (A and B are matrices) $$x_{k+1}=Ax_k+Bu_k$$
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1answer
62 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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98 views

The inverse of a state space matrix

First of all, I would like to link this question to another one about the inverse of a state space representation: Inverse of State-space representation I understand the prove as given on the ...
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31 views

Discrete time state-space

$G = \left[ \begin{array}{c|c} A & B \\ \hline C & D \\ \end{array} \right] = C(zI-A)^{-1}B+D$ Suppose: 1. $A,B,C,D$ are all real matrix. 2. $z = e^{j\theta}$, i.e. $r=1$ for simplicity. 3. ...
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336 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
46 views

BIBO stable but not stabilizable

Consider a system: $$\dot x = \begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}x+ \begin{bmatrix}0\\1 \end{bmatrix}u$$ $$y = \begin{bmatrix}1 & 1\end{bmatrix}y$$ Its transfer function is: ...
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33 views

How to check if a matrix transfer function is in Hardy-infinity space?

Just like the question says. For instance if I have a matrix transfer function $$\mathbf{G}(s) = \begin{bmatrix}s & -s \\ T & s \\ \end{bmatrix}$$ where $T$ is a positive constant, how can I ...
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1answer
96 views

Is a feedback system with an unstable component and the other component being zero internally stable?

So let's consider a system like I described, say looking like such: Where $K, P_{1}$ and $P_{2}$ are all multivariable transfer function matrices. In this case technically it could be presented as ...
2
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1answer
2k 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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1answer
37 views

A $w$ system to stabilize.

I have the following system to be stabilized: \begin{equation} \begin{aligned}\dot{w}=Aw+Bv \\& A=\left( \begin{array}{ccc} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 1 & 2 & 0 \\ \end{...
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1answer
50 views

How is an ODE a consistency condition?

I was reading a text on Optimal Control Theory by E. Todorov, when I came accross this passage (on page 10): An ODE is a consistency condition which singles out specific trajectories without ...
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2answers
227 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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53 views

Is this system stable?

I got this control system with such dynamics: \begin{equation} \dot{x}(t)=-\frac{\partial{H(x)}}{\partial{x}},~H\geq 0,~H(x)=0\Rightarrow x=0 \end{equation} $x(t)$ is a $n$-dimension vector, ...
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1answer
47 views

How to show that the delay margin is zero if the open loop gain $|L(i\infty)| \geq 1$?

How to show that the delay margin is zero if the open loop gain $|L(i\infty)| \geq 1$ ? Where $L(s)$ is the open loop transfer function and the delay margin is the amount of time delay for the system ...
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1answer
28 views

Decibel adjustment on Bode diagram

Say we have the system $G(s) = 1/(s+1)^3$ with break frequency $\omega_b = 1$. Can someone explain to me why we should expect $|G(\omega_b)|$ to be $3$ dB below the low frequency asymptote, rather ...
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2answers
143 views

Why doesn't superposition imply linearity? Why is homogeneity needed?

If I have a function which satisfies superposition I know $f(x_1+x_2)=f(x_1)+f(x_2)$. If I had now an element making f inhomogeneous ($f(0)\neq0$) this element would occur once on the left hand side ...
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1answer
79 views

state-space reduction

I am confused about state-space reduction. I learned it in the class but am not skilled in it. If $A,B,C,D$ matrices are given with values, we can 1. find its controllability matrix to see if ...
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1answer
767 views

State space representation involving derivatives of input

We have the system $y''=-7y'-12y-u'-2u$ If we choose $x_1=y,x_2=y'$ we can write the system as $x'=Ax + Bu \\ y= Cx$ Finding A is easy, but how do I find expressions for $B$ and $C$ when we have ...
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36 views

Does this violate the notion of positive definiteness?

From a previous course in linear algebra, I was taught that a function is positive definite if it satisfies $$\left< \vec x \mid v(\vec x) \mid \vec x\right> > 0 $$ In simpler notation, $$\...
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2answers
145 views

Geometric interpretation of PBH test

I need to find geometric interpretation of PBH test i.e. for any space X isomorphic to R^n and U isomorphic to R^m. A is a linear operator from X to X and B is a linear operator from U to X. Prove ...
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1answer
101 views

Final value theorem for closed system

We have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, ...
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166 views

Determine bounds for BIBO stable system

Let $\dot{x} = A x + B u$, $y = x$ be a BIBO (bounded input, bounded output) stable system. Given an output bound $y_l \leq y(t) \leq y_h$, how can we determine the maximum input bound $u_l \leq u(t) \...
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1answer
79 views

Final value of 1/(( s+2 )² * (s² - s + 1)) in the time domain

The original question is given as $$\frac {d^3y}{dt^3}+y=u=(1-t)e^{-2t}$$ The initial value y(0) = 0 and the same for all derivatives of y. Determine Y(s) What happens to u(t) and y(t) when $t\...