0
votes
1answer
38 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
31 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
29 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
55 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
72 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
87 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
84 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. ...
2
votes
1answer
66 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
96 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 ...
1
vote
0answers
35 views

Can $x'(t)\leq \mu |x(t)|,\forall t\geq 0$ imply $x(t)<\mu |e^{-\mu t}|$?

Assume $\mu>0$ If it can't, is it possible to give a good exponential bond(better with negative constant in front of t.)?
2
votes
1answer
92 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
50 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
30 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
38 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
68 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
68 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
166 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
97 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
27 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
129 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 ...
2
votes
2answers
120 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
259 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
99 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
127 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
261 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
61 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
87 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
111 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
74 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
0answers
85 views

Unit adjoint eigenvectors

Q.What are unit adjoint eigenvectors? Below I give the context where I found the mathematical term 'adjoint eigenvectors': $$\vec{f_u}.\vec{e_s}=\vec{f_s}.\vec{e_u}=0$$ so that by resolving a ...
0
votes
0answers
24 views

How to make future trajectory to be prescribed trajectory

i do not understand the role of control theory about how it make trajectory to be prescribed trajectory. because if future path's parameters are known and input into system, does it mean that it will ...
0
votes
1answer
65 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 ...
-1
votes
1answer
306 views

Show that a set is positively invariant set [closed]

Consider the system: $$\frac{dx_1}{dt} = -x_1$$ $$\frac{dx_2}{dt} = (x_1x_2 - 1)(x_2)^3 + (x_1x_2 - 1 + (x_1)^2)x_2$$ Show that $T= \{x\in \Bbb{R}^2\mathbin{|}x_1x_2 \geq 2\}$ is a positively ...
3
votes
1answer
261 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
764 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
93 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
118 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
998 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
914 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
73 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
134 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
69 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
150 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
69 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
173 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
218 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 ...