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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38 views

Showing that three systems define the same behaviour

I have three input/state/output representations of the form $$\begin{cases}\frac{d}{dt}x=Ax+Bu \\ y=Cx\end{cases}$$ with the three systems given by: $$A_{1}=\begin{bmatrix} -1 & 0 \\ 0 & -2 ...
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57 views

Hamiltonian question, find optimal controller, simple question

An object is moving with accordance to Newton's laws: $\begin{pmatrix} \dot{y} \\ \dot {v} \end{pmatrix} = \begin{pmatrix} v \\ u \end{pmatrix}$ where $y$ is the objects location and $v$ is its speed. ...
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21 views

Is it necessary to divide by the pivot entry when using Routh's Array?

I have read that multiplying a row by a positive constant doesn't change the end result of computing Routh's Array, doesn't this mean that dividing each entry by the respective pivot is not necessary? ...
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61 views

Math major transferred to Electrical Engineering, trying to bridge the gap [closed]

I did a minor in mathematics a couple years ago and the non-engineering (i.e. rigorous) math I have been exposed to were two proper courses in prob and statistics, 2 courses in real analysis and 2 ...
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20 views

Stochastic Control, Variable Coefficients

Here is my problem (I'll present it in terms of two maximizations, for simplicity): Given two cost functions, solve simultaneously $V_1(y_1) = {min_{u_1 \in \mathcal{A}(y)}} J(y_1, u_1)$ $V_2(y_1) =...
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47 views

Explain what the teacher did - system of ode, control theory.

There are a few things I'm not clear about in her solution and would appreciate a short explanation. We are given the system $\dot{x}=-ax+bu$. with an initial value $x(0)=x_0$. We want to find a ...
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1answer
73 views

Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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40 views

Minimum norm matrix to stablize a linear system

Suppose all of the eigenvalues of $A$ locate strictly on the right half plane. $(A,B)$ is controllable, $H$ is symmetric and strictly positive definite. I wonder is there a optimal solution $H^*$ ...
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91 views

Finding the input/output representation

I have a system of differential equations which can be written as $R(\frac{d}{dt})w=0$, where $$R(\xi)=\begin{bmatrix}6-5\xi+\xi^{2} & -3+\xi \\ 2-3\xi+\xi^{2} & -1+\xi\end{bmatrix}$$ I want ...
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59 views

Controllability of internal subsystem and input-outpout controllability

I am trying to prove that the state equation $\dot{x} = \begin{bmatrix} A_{11} A_{12}\\A_{21} A_{22}\end{bmatrix}x + \begin{bmatrix} B_{1}\\ 0\end{bmatrix}u$ is controllable if and only if the pair $(...
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103 views

Bounding reachability of damped harmonic oscillator using barrier certificates

I'm trying to prove that, under certain conditions, a damped harmonic oscillator that starts on one side of the equilibrium remains on that side of the equilibrium. More precisely, consider the ...
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1answer
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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67 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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1answer
55 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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54 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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1answer
54 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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57 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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81 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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96 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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149 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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51 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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2answers
103 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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1answer
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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164 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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25 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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70 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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202 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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51 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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115 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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1answer
54 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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2answers
73 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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121 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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115 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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60 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
37 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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2answers
135 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 ...