0
votes
0answers
24 views

How to plot $\dot{x}= Ax + Bu$ (x versus t, by matlab)

I am junior in control. If $\dot{x} = Ax$ where $A$ is a $n\times n$ matrix and $x$ & $\dot{x}$ are $n\times 1$ vectors, by $x = \exp(At)$, we can draw $x$ versus $t$. If $\dot{x} = Ax + Bu$, ...
0
votes
0answers
27 views

Designing a state feedback law for a nonholonomic system

Consider the set \begin{equation*} A_r=\left\{(e_x,e_y,L)\in\mathbb{R}^3:e_x=e_y=0,L(t)=\sqrt{\dfrac{\mu}{p_0^3}}t,t\in\mathbb{R}_{\geq0}\right\} \end{equation*} I have been trying to design a state ...
0
votes
1answer
28 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
0
votes
1answer
33 views

Determine value a in system matrix

I'm trying to solve the following problem: "Look at the image of trajectories of a linear, time-invariant system with the form: $\frac{d\textbf x}{dt}=\textbf {Ax}:$ Determine possible eigenvectors ...
0
votes
1answer
35 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 ...
3
votes
1answer
60 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 $s_2=-2$, ...
2
votes
1answer
44 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: ...
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 ...
1
vote
1answer
79 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, ...
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 ...
0
votes
0answers
21 views

DE: $pU^2 = (rx-1)U' + Ur + \frac{1}{2}\sigma^2U''$?

I have been trying to solve the following ordinary differential equation that results from a problem in stochastic control theory. $U$ is a function of $x$. $pU^2 = (rx-1)U' + Ur + ...
0
votes
1answer
37 views

Help finding the transfer matrix for this system

Find the transfer function of the following system: \begin{eqnarray} \dot{x}_1&=&ax_1+bx_2 + u\\ \dot{x}_2 &=&-bx_1 +ax_2\\ \dot{x}_3&=&cx_3\\ y &=& x_1+x_3 ...
0
votes
1answer
17 views

Check if a system admits solutions of period 2

I have the following problem. Let $r \geq 0$ be a parameter in the discrete time system $x(k + 1) = r − rx(k)$. Verify whether there exist $r \geq 0$ such that this system admits solutions of period ...
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{ } ...
1
vote
2answers
90 views

Determine the stability of $(x,y)=(0,0)$

Determine the stability of $(x,y)=(0,0)$: 1/$$\bf{\begin{cases} & \mathrm{ } \dot{x}= -2x-y+2xy^2-3x^3\\ & \mathrm{ } \dot{y}= \dfrac{x}{3}-y-x^2y-7y^3 \end{cases} \tag {1}}$$ 2/ ...
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{ } ...
1
vote
1answer
48 views

If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of $\frac{dz}{dt}=[A(t)+B(t)]z$ are bounded

In the book: http://www.mediafire.com/download/gqlo8iqa5b4pd95/Richard_Bellman_Stability_theory_of_differential_equations_2008.djvu Richard_Bellman_Stability_theory_of_differential_equations__2008 ...
3
votes
2answers
86 views

Show that all the roots of $\frac{dx}{dt}=A(t)x$ are bounded in $[t_0, \infty)$.

For real system of equations$$\frac{dx}{dt}=A(t)x,(1)$$ where $A(t) \in C[t_0, +\infty)$. Prove that if $\int_{t_0}^{\infty} \|A(t_1)+A^T(t_1)\|< +\infty$ then all the roots of (1) are bounded in ...
2
votes
2answers
126 views

Show that root $x\equiv 0$ of $\dfrac{dx}{dt}=F(t,x)$ is uniformly stable (uniformly asymptotically stable)

I have a problem: For the system of equations: $$\bf \dfrac{dx}{dt}=F(t,x) \tag 1$$ where $F$ is continuous in $I \times D \subset\mathbb{R}\times \mathbb{R}^n$ and $F(t,0)\equiv0$, ...
2
votes
0answers
87 views

Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
1
vote
2answers
97 views

Prove that, every solution of the scalar system: $\dfrac{dx}{dt}=y,\dfrac{dy}{dt}=-\dfrac{2y}{t},(t \ge 1) $ is bounded in domain $[1, +\infty)$

EDITED I have a problem: Prove that, every solution of the scalar system: $$\dfrac{dx}{dt}=y,\dfrac{dy}{dt}=-\dfrac{2y}{t},(t \ge 1) $$ is bounded in domain $[1, +\infty)$, but this system's not ...
2
votes
4answers
166 views

Stability theory: Every solution of the scalar equation: $\ddot{x}+\left [a+b(t) \right ]x=0$ is bounded in $\left [t_0, +\infty \right )$.

EDITED Prove that: If $a>0$ and $$\int_{t_0}^{\infty} |b(t_1)|\mathrm{d}t_1<+\infty$$ then every solution of the scalar equation: $$\ddot{x}+\left [a+b(t) \right ]x=0$$ is bounded in $\left ...
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, ...
-1
votes
1answer
303 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 ...
0
votes
0answers
94 views

Solving this optimal control problem

The problem is: $max \int_{-1}^1 (tx - u^2) dt$ where $\dot{x} = x + u^2, u(t) \in [0,1]$ for every $t \in [-1, 1]$ End points: $x(-1) = 0, x(1) = e^2 - e^{1 + \frac{1}{e}} $ I need to find an ...
1
vote
0answers
73 views

optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
3
votes
1answer
260 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 ...
5
votes
1answer
79 views

Linearization of $ m \dfrac{dy^2}{dt^2} = u(t) - C_d \left( \dfrac{dy}{dt} \right)^2-mg $

$$ m \frac{dy^2}{dt^2} = u(t) - C_d \left( \frac{dy}{dt} \right)^2-mg $$ where $$\begin{align*} y(t)&=\text{missile altitude}\\ u(t)&= \text{force}\\ m&= \text{mass}\\ C_d&= ...
3
votes
1answer
121 views

minimization problem on differential equations - optimal control

I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows: Given $\lambda< \mu_1, \mu_2$ fixed ...
0
votes
1answer
91 views

find the general control function [duplicate]

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $A = \begin{pmatrix} 3 & 2 & 2 \\ -1 & ...
3
votes
2answers
134 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
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 ...
0
votes
0answers
34 views

Determine general form of control function andthus show this coul have been achieved earlier [duplicate]

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
0
votes
1answer
116 views

Find the four equilibrium points

I am not sure if my calculation is correct: $x^T = [x_1\;\; x_2]^T$ $$\frac{dx_1}{dt} = (6-0.5x_1-3x_2)x_1$$ $$\frac{dx_2}{dt} = (-3-3x_2+x_1)x_2$$ 1st equilibrium point: $$6-0.5x_1-3x_2 = ...
1
vote
0answers
49 views

What's the shooting algorithm for the mass-spring problem (ode)?

I have the following problem : $$ \begin{aligned} \frac{d x(t)}{dt} &= y(t)\\ \frac{d y(t)}{dt} &= -x(t)+y(t)(1-x(t)^2)+u(t) \end{aligned} $$ with the initial condition $(x,y) = (0,0)$. Those ...
1
vote
1answer
82 views

2nd Order Optimal Control Problem

I'm working on a homework problem in optimal controls and my plant model is described as: $$\ddot{x}(t) = u(t)$$ The performance index (cost function) is described by: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ ...
2
votes
1answer
327 views

Existence of global solution of Riccati equation

Consider a Riccati differential equation $$ \dot P + A(t)^{T}P + PA(t) -PB(t)R(t)B(t)^{T}P + Q(t) = 0,\;\;\; P(t_0) = P_0 = P_0^{T} \geqslant 0 $$ where $Q(t) = Q(t)^{T} \geqslant 0$, $R(t) = ...
1
vote
1answer
306 views

Perturbation Analysis for Linear Linearly-Perturbed ODEs

I have been struggling with an ostensibly simple problem, that is how to apply perturbation analysis principles on a system of linear differential equations with linear perturbation of the following ...
-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: ...
2
votes
1answer
372 views

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ $x ...
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 ...
2
votes
2answers
341 views

Transfer Function, Newtonian Cooling

I have a system that is modeled by the following differential equation: db/dt = j(t)ha + k(Ta-b(t)) where db/dt is the rate of temperature change, j(t) is an input, ha, k, Ta are all constants, and ...