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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20 views

Symmetric Matrix Sign equivalence and rank of extenden matrix

I have the following matrix strict inequality: where $X,Y,A,B \in \mathbb{R}^{n\times n}$, $X,Y$ are symmetric matrices $(X=X^T, Y=Y^T)$, there are no conditions imposed on $A$, nor on $B$ $\begin{...
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50 views

Solving the matrix differential equation $\dot \Delta P(t) = (A + P(t)C^{T}R^{-1}C)\Delta P(t) + \Delta P(t)(A^{T} + C^{T}R^{-1}CP(t))$

Here $P, \Delta P \in \mathbb{R}^{N X N}$ The initial condition $\Delta P(0)$ is given and the dynamics of $P(t)$ is known. $ A,C,Q,R$ are constant matrices of compatible dimensions. Since it is a ...
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65 views

Singular transfer function matrix and System singularity

I have a linearized dynamic system that can be summarized as: [ΔY] = [A][ΔX] The transfer function matrix, [A], is singular for steady state. My question is ...
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53 views

Barrier certificate for simple system

I'm trying to prove that for the system $x' \leq -x$, if the system starts with $x \leq 0$, then it is always the case that $x \leq 0$. I'm sure that there are easy ways to prove this, but I'm ...
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53 views

How to get the peak value of a second order system that has non-zero initial condition with step input?

It is known that the peak value of a second order system which is excited with a step input can be expressed by $y(t_p)=1+e^{({-\pi\zeta}/{\sqrt{1-\zeta^2}})}$ In my case, the initial conditions are ...
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51 views

How to apply Picard-Lindelof existence and uniqueness theorem for autonomous LTI dynamical system $\dot x = Ax$?

In nonlinear dynamical system, we have the picard-lindelof existence and uniqueness principle which guarantees existence of unique solution to problem of the type $\dot x = f(x,t), x(0) = x_o$ ...
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55 views

Why do high-frequency dynamics quickly go away in a step response?

As we know, a step input hits all the frequencies of a dynamical system. However, my professor told me today that the high-frequency response is only present for a short time at the very start, and ...
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80 views

When is a system called linear?

In real time systems / control engineering we have to solve exercises like this: Check if the following systems are linear: 1) $0.2\ddot{x}(t) - (t^2 + 2t -1) x(t) = 3 w(t)$ 2) $\ddot{x}(...
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94 views

Asymptotic stability of cascade control

For many systems, it seems to be common practice to stack controllers on top of each other. For example, in a quadcopter, one first builds an attitude controller, then builds a velocity controller ...
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148 views

Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE ...
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19 views

If state is reachable in time T_1, then it is reachable in time $T > T_1$

Consider a Linear Time System with the admissble control set $$U = \left\{ u: R \rightarrow R^m \;|\;\text{u is integrable in any finite interval} \right\} $$. Show that, if starting on $x_0=0$ we ...
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50 views

Expression for polynomial of companion matrix

I am rather stuck on an exercise concerning the companion/controllability matrix (the exercise stems from a course in control theory). Given the companion matrix \begin{equation} A=\left(\begin{...
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30 views

Finding an admissible control such that $(0,0) \to (1,e^2)$

Verify the controllability of the system $(A,B)$, for $$A=\begin{pmatrix}1&0\\1&-1\end{pmatrix}, \; B=\begin{pmatrix}1\\1\end{pmatrix}$$ Find a control $u \in L([0,1];R)$ such that $(0,0)\...
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34 views

How to understand “maximal ”, bounded“, and ”complete" for solutions of a (hybrid) dynamical system?

I am trying to read some text books about hybrid dynamical system, in which maximal solution, bounded solution, and complete solution are mentioned frequently. The following passage is a description ...
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31 views

Controllability of $x' = Ax + Bu(t)$ implies controllability of $\left \{ \begin{matrix} x' = Ax + By \\ y'=u(t) \end{matrix} \right.$

Suppose that the system $$x'(t)=Ax(t)+Bu(t)$$ is controllable in $R^n$, where $A$ is $n \times n$, $B$ is $ m \times n$ and $u(t)$ is $m \times 1$ Show that the system $$\left \{ \begin{matrix} x'(t) ...
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102 views

Observability of a System in State Space form

as we all know, each system has infinite state space representations. My question is, whether the observability/controllability (and other common properties) are a properties of the SYSTYEM or ...
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48 views

$X= \int_0^S e^{(S-s)A} B u(s) ds \Rightarrow X= \int_0^T e^{(T-s)A} B \bar{u}(s) ds$

Consider the ODE system $$X'(t) = AX(t)+Bu(t)$$ where $X(t) \in R^n, \; A \in R^{n \times n} \text{ and } B \in R^{n \times m}$. In control theory, we define the set of states reachable as $$A(0,T) = \...
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159 views

Functional optimization: maximize a double integral where the functional appears twice

Please help me solve the following optimization problem. Suppose that you have to choose a function $U: [0,1]\mapsto [0,1],$ which must be nondecreasing ($U'\geq 0$) to maximize the following integral:...
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32 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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24 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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68 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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180 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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50 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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114 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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14 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 $\alpha_i$...
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52 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: $$F(s)=F_1(s)....
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72 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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59 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) = \frac{1}{t+...
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114 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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69 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 function(...
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58 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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93 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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22 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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29 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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110 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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27 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
40 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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59 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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34 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: \begin{...
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130 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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77 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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365 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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122 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
89 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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104 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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154 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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2answers
82 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
62 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 ...