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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16 views

Is this end-point map surjective

Consider the differential equation: $\frac{d U_s}{dt} = (a + w(s)b)U_s$ where $w$ is some unknown, smooth, real and bounded function on the interval $[0,T]$ and $a,b \in \mathfrak{su}(n)$. Let ...
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1answer
22 views

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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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 ...
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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' ...
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1answer
70 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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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
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: ...
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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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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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1answer
44 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 ...
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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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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) = ...
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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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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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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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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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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 ...
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46 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 ...
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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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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 ...
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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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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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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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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 ...
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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
50 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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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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29 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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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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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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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 ...
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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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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, ...
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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
58 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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59 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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32 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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19 views

Reducing uncertainty of a mathematical model with data (Process control)

I know this is a very broad question, but need suggestions, link to good reference papers etc. So here is the question: I have an uncertain model whose parameters are static (not changing with time) ...
2
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1answer
57 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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20 views

Transfer function from state variable expression

I have a 3x3 state variable system. I need to choose where to place my poles according to some criteria. For example: (a) Percent overshoot < 20% (b) SettlingTime < 1.5s, and (c) steady-state ...
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2answers
55 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= ...
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44 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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82 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 ...
2
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1answer
67 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 ...