0
votes
2answers
40 views

Is an Invariant set Connected?

Let an autonomous dynamical system is characterized by the state equation $$ \dot x(t) = f(x(t)),\quad x(0)=x_0 $$ with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, ...
0
votes
0answers
10 views

Backstepping analysis of multi-input system

Suppose I have a simple system that's like following: $\dot{x}_1 = A x_2 + Bx_3 \\ \dot{x}_2 = u_1 \\ \dot{x_3} = u_2$ I am familiar with a standard method of backstepping if there was only one ...
0
votes
0answers
28 views

Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
7
votes
2answers
111 views

Rigorous mathematical treatments of engineering topics

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I ...
0
votes
1answer
51 views

Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
1
vote
0answers
35 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
0
votes
1answer
34 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?
0
votes
2answers
37 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 & ...
1
vote
1answer
77 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}
0
votes
1answer
104 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.
1
vote
1answer
126 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 = ...
1
vote
1answer
126 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
votes
1answer
74 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 ...
2
votes
1answer
133 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
votes
1answer
103 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 ...
1
vote
1answer
53 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\\ ...
2
votes
0answers
31 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
0
votes
1answer
41 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
0
votes
1answer
83 views

Found transition matrix and state matrix

I tried to found a solution for this problem but I can't! Any suggestion?Thank you. Given that A is a 2x2 matrix and that dx/dt=Ax(t) suppose that x(0)=[1 ; -3] implies x(t)=[e^-3t ; -3e^-3t] and ...
0
votes
2answers
86 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
0
votes
1answer
206 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
2
votes
1answer
126 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
1
vote
0answers
31 views

How can I prove this theorem about differential inclusions?

Consider the following differential equations with initial conditions at time $t_0$ specified: $\dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, ...
0
votes
0answers
144 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
3
votes
2answers
126 views

Prove that $x \equiv 0$ of $\dot{x}(t)=a(t)x$ is Uniformly Asymptotically Stable

I have a problem: Consider the scalar equation: $$\dot{x}(t)=a(t)x \tag{I}$$ where $a(t) \in C(\mathbb{R}^+)$. Prove that $x \equiv 0$ of $(I)$ is Uniformly Asymptotically Stable iff ...
3
votes
2answers
267 views

Show that System $(I)$ is stable iff $X(t)$ is bounded.

I have a theorem: For a linear homogeneous system: $$\dfrac{dx}{dt}=A(t)x \tag{I}$$ Where $A(t)=(a_{ij}(t))_{n \times n} \in C(\mathbb{R}^+,\mathbb{R}^{n \times n})$ Suppose that $X(t)$ be the ...
2
votes
1answer
105 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
3
votes
1answer
140 views

Construct a Liapunov function for this system

Construct a Liapunov function for the system (Determine the stability of $x \equiv 0$): I have an example:$$\begin{cases} & \mathrm { } \dot{x}= -x^3+xy^2\\ & \mathrm { } \dot{y}= ...
2
votes
1answer
347 views

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system.

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system: I have an example: $$\begin{cases} & \mathrm{ } \dot{x}= \tan(y-x)\\ & \mathrm{ } ...
2
votes
0answers
74 views

Stability of linear systems with singular state matrix

Given a linear time invariant system $\dot X(t) = AX(t)$ where $X \in {R^{n \times 1}}$ and $A \in {R^{n \times n}}$ is a singular matrix ($A$ has at least one zero eigenvalue). How can I study the ...
2
votes
2answers
102 views

How does the singularity of a system matrix affect the system's stability?

What can be said about system stability, given a singular system matrix below? \begin{align} A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
0
votes
2answers
121 views

Closed loop stability

Regarding the Lyapunov stability, we check if a nonlinear system stays near the equilibrium point or approaches to e.p. as time goes to infinity, when it is disturbed. Let's assume that we have a ...
0
votes
1answer
84 views

Finding Transfer functions for linearised systems

I'm using Nise for my control systems class. Finding a linearised system is all gravy baby, but when it comes to finding the transfer function Nise does some stuff which confounds me: See page 6/7, ...
0
votes
1answer
76 views

which book teaches analysis of nyquist, bode and rlocus diagram

would like to use knots to get a formula for nyquist diagram however, no crossing, and have no experience in analysis of graph related to control, as i have no books mentioning this and i observe ...
3
votes
1answer
312 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
4
votes
2answers
1k views

Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
1
vote
2answers
97 views

Stability analysis $\dot{x}=-\gamma x + \alpha$

Suppose that $\alpha(t)$ is an infinitesimal as t goes to infinity, i.e., $\lim_{t\rightarrow\infty}\alpha(t)$=0. Consider the ODE $$ \dot{x}(t)=-\gamma x(t) + \alpha(t), \quad \gamma>0 $$ Can we ...
1
vote
1answer
121 views

Choose $\mathbf{B}$ such that eigenvalues are un/controllable

I have the state space system $\dot{x} = Ax + Bu$ with $A = \begin{bmatrix} 1 & -5 \\ -5 & 1\end{bmatrix}$. I have to find a $B$ vector such that the system has $\lambda = 6$ as controllable ...
0
votes
1answer
1k views

Solving lyapunov equation, Matlab has different solution, why?

I need to solve the lyapunov equation i.e. $A^TP + PA = -Q$. With $A = \begin{bmatrix} -2 & 1 \\ -1 & 0 \end{bmatrix}$ and $Q = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. Hence... ...
2
votes
2answers
2k views

Why do we want to know the poles and zeros of a linear system?

I know that I already asked this kind of question on the website, but meanwhile I have much more knowledge about the subject and ready to describe my real problem with enough background information I ...
3
votes
3answers
1k views

Poles and Zeros of Linear Systems

This period I follow a course in System and Control Theory. This is all about linear systems $$\frac{dx}{dt}= Ax + Bu $$ $$y = Cx + Du $$ where A,B,C,D are matrices, and x, u and y are vectors. ...
2
votes
0answers
46 views

Question regarding continuous time systems

If $S_u$ is the set of all solutions to the continuous time problem x(dot) = Ax + Bu and $\phi_h :(R^n)^R to (R^n)^N$ be an operator which maps every state x into the sequence (x(0), ...
1
vote
1answer
82 views

Derivation of a formula for discrete time case

Given is the following discrete system $$\begin{align*} &x(k + 1) = Ax(k) + Bu(k)\\ &x(0) = x_0\;. \end{align*}$$ How do we prove that the explicit solution formula for $x(k)$ (analogously ...
1
vote
0answers
96 views

generalizing superfunctions of entire functions

Let $z$ be complex and $x$ real. Define $f(z,0) = f(z)$ where $f(z)$ is an entire function. Define $f(z,x)$ as the $x$ th superfunction of $f(z)$. We know that $f(z,x-1) = f(f^{-1}(z,x)+1)$ where ...
2
votes
3answers
145 views

Weird condition for a transfer function

I am reading a paper that presents the following system, represented by a second-order transfer function: $G(s) = \frac{K\times(1+0.036s)}{(1+0.0018s)(1+as)}$, where the gain $K$ is a known ...
1
vote
1answer
70 views

Parametric uncertainty in conditional term of piecewise nonlinear dynamical system

Consider a Hammerstein nonlinear dynamical system of the form $\mathbf{\dot{x}} = \mathbf{Ax} + \mathbf{Bu}$, where the non-linearity is in the control term $\mathbf{u}$, and has a piecewise ...
2
votes
1answer
178 views

When are attracting sets invariant?

Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the ...
3
votes
1answer
75 views

Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
-1
votes
2answers
182 views

Feedback characteristics of nonlinear dynamical systems

I am trying to understand the following article: A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco, Europhys. Lett. 81, 60001 (2008). In the beginning, it was quite easy to follow: ...
1
vote
1answer
221 views

Property of dynamical system and transformation

EDIT2: After some discussion here's the original problem: Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^{n\times m}, u_1 \in \mathbb{R}^{m}$ be a control system evolving on M (F is the ...