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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characteristic equation of transfer function

$$\frac{K}{s(s+1)(s+5)}$$ Find the characteristic equation of this transfer function. The book gives this answer: $$\frac{K}{s(s+1)(s+5)} +1=0$$ or $$s^3 +6s^2 +5s +K =0.$$ I don't get how the ...
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38 views

Relationship between output and poles/zeros in the complex plane

Context There are lots of videos online which explain the time domain equivalent of poles depending on their place in the complex plane, but it's only useful for the simplest examples for which we ...
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39 views

A particular DE as first order system

When does a differential equation $\frac{d^2y}{dt^2}+a\frac{dy}{dt}+by=cu(t)+d\frac{du}{dt}$ admit a solution? If $d=0$, the existence is answered by Picard-lindelöf, and we can write it as a system ...
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36 views

Can I always put a system in modal form?

Given the transfer function of a system, can I always put the system in modal form? Are there exceptions?
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74 views

Derivation of the Mexican wave as Mean field equilibrium of an ergodic control problem

I'm dealing with a Mean Field system introduced by Gueant, Lasry, Lions in their "Mean Field Games and Applications" which admits as solution the so-called Mexican wave. Without going into the details ...
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121 views

significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
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57 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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20 views

Stein's theorem and the solution of Q=H-B*HB

Stein's (1952) Theorem 1 says: "A necessary and sufficient condidion that B is convergent is that there exists a positive definite Hermitian matrix H, for which H-B*HB is positive definite." ...
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53 views

finding the input sequence of a discrete-time dynamical system

I am studying Dynamical Systems, actually linear systems and I came across the following question: Consider the following discrete-time dynamical system: $x_{i+1}= \left( \begin{array}{ccc} 2 & ...
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2answers
101 views

Reference for LPV controls

I am looking for a good mathematical introduction to LPV (Linear Parameter Varying) methods in control theory. I would like it to be more on the mathematical side of things, instead of something aimed ...
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1answer
31 views

Literature study for Optimal Estimation Theory

It seems Optimal Estimation/Control Theory requires a lot more than undergraduate maths. Any good book that would help me get started? I have so far referred the following books but found them quite ...
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59 views

analytic solution to structured algebraic Riccati equation

In solving a model I have written down for a research paper, I am left with two Algebraic Riccati Equations, that is I need to solve for $X$ in the equation \begin{align*} X = A^\top (X + XB(R + ...
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1answer
528 views

Pole-zero cancellation Paradox

Suppose we have an open-loop transfer function $$G(s) = \frac{1}{s(s+a)(s+b)}$$ If we plot the root locus for the closed-loop system we will get roughly something like this : Now the question is ...
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1answer
88 views

Is there a formalized control-theory-like approach for uncontrollable systems?

I'm basically trying to control a system to achieve a given set of outputs, but I don't actually have enough inputs to control all the outputs. Is there any formalized theory on how to achieve the set ...
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2answers
54 views

Why does this phase calculation go to 180 instead of 90?

This is all coming from the following video I am studying from http://www.youtube.com/watch?v=XSS6L42ce88 So I am working from this system $$ G(s)\,=\,\frac{4}{s^{2}+s+2}$$ and the video states the ...
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0answers
24 views

Is the ordered list of controllability indices invariant, or the unordered list?

The list of controllability indices of a linear time-invariant system is invariant under state feedback and change of variables. What is invariant exactly though: the ordered list, or the unordered ...
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1answer
86 views

Calculate step response from impulse response LTI-system.

Can someone please give me a few pointers on how to calculate the step response for an LTI system with this impulse response?? \begin{equation} h[n] = 2^nu[n]. \end{equation}
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3answers
209 views

An introduction to control systems

I am looking for an introduction on control systems in the context of engineering, but treated from a more mathematical point of view. Does anybody have a good reference?
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0answers
18 views

Controlled system question optimisation

Consider the controlled system $x_{t+1} = x_t + u_t + 3\epsilon_{t+1}$, where the $\epsilon_t$ are independent $N(0,1)$ variables. The instantaneous cost at time t is $x_t^2 + 2u_t^2$. Assuming that ...
3
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1answer
98 views

Calculate state transition matrix

I have a question to the following problem: "There is a linear, time-invariant System with the form $\frac{d\mathbf{x}}{dt}=\mathbf{A}x$. The Eigenvalues of the matrix $A$ are $s_1=-1$ and $s_2=-2$, ...
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1answer
66 views

Optimal control question: find value function and optimal input

If we have a continuous-time system with a scalar state variable, plant equation $$\dot{x}= u,$$ and cost function $$Q\int_o^h u^2 dt + x(h)^2,$$ then by writing the dynamic programming equation in ...
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1answer
43 views

Reference request: “initial” PDE control

I'm somewhat familiar with the ideas of boundary and internal (source) control for PDE (in particular, for wave equations) but am wondering if the following type of problem is classified/studied; if ...
2
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1answer
102 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
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1answer
28 views

Stability of system with gain K

It is asked to sketch the root loci of the system . I have done that . Then it is asked to determine the stability of the system as function of $K$ . After plotting I see that for $K>0$ it is ...
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1answer
23 views

Determination of the modulus of continuity

I'm trying to prove the uniqueness of the viscosity solution of an Hamilton-Jacobi-Bellman equation. Thanks to a classical result, I'm left to check if it exist a modulus of continuity $\omega_1$ --- ...
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1answer
111 views

Relationship between controllability and null space of A, B

For the dynamic system $\dot x = Ax + Bu$ There's a saying that this system is controllable when $Ker(B) \in Ker(A)$, which means that $u$ have the control in every dimension of $x$. I have no ...
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1answer
51 views

Second order control system

We have the open loop transfer function $$ G(s)H(s) = \frac{k(s+2)}{s^2+2s+3}$$ There were three parts to this question : a. The value of $k$ for which repetitive roots occur b. The range of $k$ ...
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207 views

Proof of Impulse response of a BIBO stable system

I was wondering if anyone here could expand upon or provide the proof of the boxed theorem which I have shown in the image below? Any help would be greatly appreciated.
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1answer
49 views

Frequency response of Continous-time system

Not sure where to start on this one: $$H(s)={(s-j\omega_0)(s+j\omega_0)\over(s+\omega_0\cos\theta+j\omega_0\sin\theta)\left(s+\omega_0\cos\theta-j\omega_0\sin\theta\right)}$$ Sketch the frequency ...
2
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1answer
100 views

stability of a closed-loop system

We have $$G(s)H(s) = \frac{Ke^{-s}}{s(s^2+5s+9)}$$ We have to determine the maximum value of K for the closed-loop system to be stable . We have to do it using the Routh hurwitz criterion . I ...
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1answer
193 views

Block Diagram Reduction

i am trying to simplify this systems block diagram. I calculated something but I am not sure about it, is my reduction true? Thank you.
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2answers
43 views

Horizon selection

$$x(t+1) =Ax(t)+Bu(t)=\left(\begin{matrix}0&0&0.8\\1&0&0\\0&1&0 \end{matrix}\right)x(t)+\left(\begin{matrix}1\\0\\0 \end{matrix}\right)u(t)$$ Basically the excercise is to find ...
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1answer
148 views

Basic question about my understanding of the Lyapunov equation

Consider the system $\dot{x}(t) =Ax(t)$ where $A \in \Bbb R^{n\times n}$. Now let $P$ be a symmetric matrix and define $V(x) = x^T Px$. Then $V(x)$ satisfies $$\frac{d}{dt}V(x) = -x^TQ x,$$ where $Q = ...
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1answer
85 views

Is Lyapunov equation always solvable with A as a negative definite matrix?

Given a negative definite matrix $A$ and $Q=I$, is the Lyapunov equation for $P$, that is, $PA+A^TP=-I$ always solvable? what kind of form does the solution have? I will appreciate if examples can be ...
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1answer
208 views

Local stability + global attractivity = global asymptotic stability?

I was wondering how could I prove such a property stated in [Angeli, 2004]. For instance, consider the system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is Lipschitz continuous. Claim. ...
3
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1answer
83 views

Stability of $\dot{x}=-A(t)x$

I already have an ODE of $A(t)$, that is $\dot{A}=-G(A(t)-A^*)$, where $G$ and $A^*$ are constant positive definite matrices. Thus I can deduce that $A(t)$ exponentially converge to $A^*$. Now I take ...
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1answer
110 views

Continuity of solution of Riccati equation with negative source term

Let $t_0 > 0$ and consider the scalar Riccati differential equation $ p'(t) + 2 a(t) p(t) - r(t) \, p(t)^2 + q = 0 \; , $ with initial condition $ p(t_0) = 0 \; , $ in which $a$ is a function, ...
2
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1answer
167 views

Lyapunov stabilty, elementary question

Let’s say I have a system 1/(T1s+1) or any other n-th order polynomial and a PI controller (KP and TI). I already know that the system is stable but for, let’s say, educational purposes (not ...
2
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0answers
79 views

Cellular Automata Method to Reach Equilibrium on a Network of Numbers

Here's a puzzle that I'm curious if anyone can attack with a simple method without resorting to simulation. I can tell you the answer (well, an answer) from running some programs. But I'm writing a ...
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1answer
78 views

Given a system $\dot{x}(t)=Ax(t) + Bu(t)$ find $S$ such that $s^{-1}AS=\bar{A}$, $S^{-1}B=B$ is in controller form.

I am given the system $\dot{x}(t)=Ax(t) + Bu(t)$ where $$A = \left( \begin{matrix} -1 & 0& 2\\ 0 & -3 & 0 \\ 1&0&0 \end{matrix} \right), \quad B = \left( ...
2
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1answer
63 views

Exponential and uniform stability

I really don't understand how resolve this exercise...with Lyapunoff? can someone help me? Thanks Consider the state equation: $$ \frac{\partial}{\partial t}x(t)= A(t)x(t), \: x(\tau)=x_0 $$ ...
2
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2answers
69 views

showing any controllable system can be put in 'controller' form.

I am looking at the proof of the following theorem: Let $\dot{x} =Ax + Bu$ be a controllable single input system, where $\Delta_A:= \det(\lambda I -A) = \lambda^n + a_1\lambda^{n-1} + \ldots + ...
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1answer
29 views

whether a Gramin matrics is symetric or not?

A Gramin matrix is defined as $$ G^c(t_0,t_f)=\int_{t_0}^{t_f} \exp\bigl((t_0-t)A\bigr) BB^T \Bigl(\exp\bigl((t_0-t)A)\Bigr)^T \,dt$$ where ${}^T$ is for transpose of matrices. How can i prove it?
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45 views

Bode phase calculus with frequency to $0$ and $\infty$ in a quadratic term

The tangent to calculate is: $$\phi=-\tan^{-1}\left[\frac{2\zeta\frac{\omega}{\omega_n}}{1-(\frac{\omega}{\omega_n})^2}\right]$$ Where $\omega_n$ is constant. If $\omega=0 \to\phi=0$ ...
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5answers
178 views

Does Interpolation/Extrapolation the crucial thing happening in our brain while driving a motor vehichle?

Does Interpolation/Extrapolation is the crucial thing happening in our brain while driving a motor vehichle? What I'd like to know is the mathematics happeing while we are behind the wheel. PS : I am ...
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1answer
30 views

Extensions to linear control with output constraints

Does anybody know which extensions to the linear controller exist that can cope with constraints in the output value and its derivative? Usually, the plant being controlled have some limits and I ...
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1answer
87 views

Affine Non Autonomous State Space system

Normally We all know the state space model of the the form der(x) = F*x(t)+G*u(t) y = H*x(t)+J*u(t). However I came across a state space model which has the following form der(x) = F*x(t)+G*u(t) + ...
3
votes
1answer
108 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
2
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2answers
81 views

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$?

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$? Now to check controllability I usually check the rank of the matrix $$R = \left( B\ \ \ \ ...
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1answer
71 views

Show that the system $\Sigma(SAS^{-1},SB,CS^{-1},D)$ is observable/controllable iff $\Sigma(A,B,C,D)$ is observable/controllable

I am given the two linear systems: \begin{eqnarray} \Sigma_1: \dot{x}&=&Ax+Bu\\ y&=&Cx+Du \end{eqnarray} and \begin{eqnarray} \Sigma_2: \dot{x}&=&\bar{A}x+\bar{B}u\\ ...