Questions about linear or nonlinear methods for manipulating the outputs of a mathematical model that typically represents a dynamical system.
0
votes
0answers
17 views
improving fluid analysis on queuing problems
Discrete-queuing models are hard to solve computationally and can become easily intractable with an increased number of state-action pairs. Markov decision processes can be employed to come up with ...
1
vote
0answers
41 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 ...
0
votes
0answers
11 views
Phase Plane of Digital Systems
I have a nonlinear digital system which can not become differential equation with subtracting the states and deviding them by the time difference on account of being nonlinear. Therefore, I want to ...
1
vote
1answer
48 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 ...
2
votes
2answers
32 views
Finding the steady state error in the Laplace domain
I have the following block diagram:
Now I like to find the steady state error for theta_ref being a step input and for several values of n, Td, K1 and K2.
For the moment we can assume all gains ...
0
votes
0answers
10 views
Pontryagin Principle for Random Process
Can Pontryagin's principle be written for minimizing the expected cost in the case of a stochastic process which is controlled? (cost at each time is function of state and action)?? I have not come ...
4
votes
1answer
61 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
$$\begin{align*}
y(t)&=\text{missile altitude}\\
u(t)&= \text{force}\\
m&= \text{mass}\\
C_d&= ...
1
vote
2answers
62 views
Does negative third derivative imply negative first derivative?
Does negative third derivative imply negative first derivative? For a system, the negative of the derivative of the Lyapunov function means the system is stable. How about the negative third ...
2
votes
1answer
22 views
What is the difference between regulator and stabilization
What is the difference between regulator and stabilization in control theory
don't they both minimize the disturbance to the system?
could answer be elaborated from the view of state and output?
11
votes
3answers
151 views
+50
What is the mathematical foundation of Control Theory?
There is a question which I'm wondering again and again in recent months. I have taken courses like Elementary Differential Equations, Signals and Systems, Linear Control Systems, General Theory of ...
1
vote
0answers
36 views
How do I solve an optimal control problem when the state and control are multiplied?
Suppose I have the following objective function
$$
R = \sum_t^T x_tu_t + ku_t^2
$$
subject to
$$
\Delta x_t = m u_t + n x_t
$$
where $x_t$ is the state and $u_t$ is the control. $x_0$ is known.
How ...
1
vote
1answer
33 views
Nyquist criterion
When using the Nyquist stability criterion, amplitude-frequency characteristic etc. we go from the Laplace image $G(s)$ to $G(j\omega )$. By definition of the Laplace transform, $s=\sigma + j\omega$. ...
0
votes
1answer
21 views
Can linearization of a function around $x=0$ show whether first derivative is positive or negative?
As title says, can linearization of a function $f(x)$ (by the method of taylor series around $x=0$) show whether first derivative of the function ($df/dx$) is positive or negative at $x=0$?
And.. ...
0
votes
1answer
31 views
Understanding the Hamiltonian function
Based on this function:
$$\text{max} \int_0^2(-2tx-u^2) \, dt$$
We know that $$(1) \;-1 \leq u \leq 1, \; \; \; (2) \; \dot{x}=2u, \; \; \; (3) \; x(0)=1, \; \; \; \text{x(2) is free}$$
I can ...
3
votes
1answer
58 views
minimization problem on differential equations - optimal control
I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows:
Given $\lambda< \mu_1, \mu_2$ fixed ...
2
votes
0answers
26 views
How verification argument really works?
Let $C(u,s)$ be cost functional for an admissible control $u$ with initial state of the system being $s$.
Our aim is the solution of the following problem:
$$\inf_u E(C(u,s))$$
We defined the value ...
1
vote
0answers
29 views
Question about proof of bounded real lemma
My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure?
The $H_\infty$ performance is defined as:
\begin{align}
\parallel ...
3
votes
2answers
131 views
Solution of a Sylvester equation?
I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
0
votes
0answers
32 views
References that discuss systems of ODEs on the non-negative orthant of $\mathbb{R}^n$?
Does anyone know of any references discussing initial value problems on the non-negative orthant? More specifically, consider the initial value problem
$\frac{dx}{dt}=f(x),\quad\quad ...
0
votes
1answer
61 views
Solve $AXB=X^\top$
Suppose that $X$ is an $m\times n$ matrix, $A$ and $B$ are $n\times m$ matrices. How can you solve $$AXB=X^\top.$$
Is there an explicit formulation of $X$ in terms of $A$ and $B$ that makes the ...
3
votes
0answers
78 views
Root Locus Diagrams - “Breakaway Point”
Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
0
votes
1answer
49 views
find the general control function
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference
where:
$A = \begin{pmatrix}
and:
So far I have caculated the controlability matrix to be
$
C
the system is ...
1
vote
2answers
83 views
Find a general control and then show that this could have been achieved at x2
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n,$$
where:
$$A = \begin{pmatrix}
3 & 2 & 2 \\
-1 ...
0
votes
1answer
38 views
Book on theoretical computational optimal control
I'm looking for a comprehensive introduction to the theoretical side of optimal control, existence of solutions and so on, including theory behind numerical solution methods. Regarding the latter I'm ...
2
votes
1answer
42 views
Verification for maximum principle
Given optimal control problem
$$
\dot x = f(t,x(t),u(t)), \quad x(0) = x_0,\\
J(u) = \int_0^T f^0(t,x(t),u(t))dt \to \min,
$$
we can apply Pontryagin's maximum principle to get a necessary condition ...
0
votes
0answers
28 views
MATLAB plotting issue
I posted this question on StackOverflow, but did not get any answers, so hopefully this will work better.
Is anyone familiar with plotting in the Matlab SISOtool? For some reason, I cannot access the ...
0
votes
1answer
38 views
Find the maximal switching period that ensures asymptotic stability of the switching system.
I have a time-dependent switched system $\mathbf{\dot{x}} = \mathbf{A}_i\mathbf{x}$. With
$$\mathbf{A}_1 =
\begin{bmatrix}
-0.5 & 1 \\
100 & -1
\end{bmatrix}
\quad
\mathbf{A}_2 =
...
1
vote
2answers
71 views
Stability analysis $\dot{x}=-\gamma x + \alpha$
Suppose that $\alpha(t)$ is an infinitesimal as t goes to infinity, i.e., $\lim_{t\rightarrow\infty}\alpha(t)$=0.
Consider the ODE
$$
\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
54 views
Nonlinear Systems- L2 stability analysis
I hope you are having a good day. I am working on a homework and I was looking for some help.
Can anyone please help me with the next step to prove whether the L-2 stability of the system and the ...
0
votes
1answer
26 views
Problem with getting the state space representation
I have a little problem here:
I need to find the 2 differential equations and build a state space representation of them.
First I get these 3 equations:
$$ u = u_C + u_R $$
$$ u_R = u_L + u_{r1} ...
0
votes
2answers
118 views
Find the control function
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n,$$
where:
$$A = \begin{pmatrix}
3 & 2 & 2 \\
-1 ...
0
votes
1answer
66 views
How can I show that the trajectory is a circle?
I have this system:
$$\frac{dx}{dt} = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}x+\begin{bmatrix}1 \\0\end{bmatrix}i$$
1st: Is this system stable? I think it is stable, but not ...
0
votes
1answer
43 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:
$$6-0.5x_1-3x_2 = ...
1
vote
0answers
63 views
Identifiability of a state space system
I'm trying to solve assignment 4E.5 from this sheet (ship steering dynamics).
My question are:
Do I need to perform the Laplace Transform in order to check for identifiability?
The state space model ...
2
votes
1answer
56 views
How can I solve this control problem?
Consider this control problem in continuous time, known as Representative Agent Model in macroeconomics:
$$ \max_{c_t,t\ge 0}\int_{0}^{\infty}e^{-\rho t}\ln(c_t)\, \mathrm{d}t,~~~\rho\in (0,1) $$
...
4
votes
1answer
65 views
Upper bound concerning Snell envelope
Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the habitual ...
0
votes
0answers
139 views
Getting Transfer Function from a Block Diagram
Question:
I'm a student who has to extract transfer functions from block diagrams quite often. It would help if there was a graphical tool where I could manipulate block diagrams and see their ...
0
votes
0answers
23 views
control definition meaning measurable function topology
I am reading a paper on control theory i am fairly new to the area.
Could you tell me what does it mean by
"control"
"control set is compact"
What is control mathematically? Are they measurable ...
1
vote
1answer
69 views
How to find discrete integral given different time intervals.
I want to implement a PID controller and I'm unsure of how to find the integration part. Normally the integral is calculated as $\sum_{n=1}^{t} e_{n}$, where $e_n$ is the sample error at time n. ...
1
vote
0answers
35 views
What's the shooting algorithm for the mass-spring problem (ode)?
I have the following problem :
$$
\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}
$$
with the initial condition $(x,y) = (0,0)$. Those ...
1
vote
1answer
55 views
Choose $\mathbf{B}$ such that eigenvalues are un/controllable
I have the state space system $\dot{x} = Ax + Bu$
with $A = \begin{bmatrix} 1 & -5 \\ -5 & 1\end{bmatrix}$.
I have to find a $B$ vector such that the system has $\lambda = 6$ as controllable ...
3
votes
1answer
101 views
Maximum principle for a control with mixed constraints
Consider the dynamical system $\mathbf{x}' = f(\mathbf{x,u},t)$ where $\mathbf{x} \in \mathbb{R}^2, \mathbf{u} \in \mathbb{R}^3$ and $f$ has no explicit time dependence. As conventional, $\mathbf{x}$ ...
1
vote
1answer
174 views
Solving lyapunov equation, Matlab has different solution, why?
I need to solve the lyapunov equation i.e. $A^TP + PA = -Q$. With $A = \begin{bmatrix} -2 & 1 \\ -1 & 0 \end{bmatrix}$ and $Q = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.
Hence...
...
0
votes
0answers
61 views
Nyquist plot: Determine the range for BIBO stability
I have got a problem with this Nyquist plot.
$$ T(s) = K *P(s) =K \frac{2 (1+s)(4+s)}{(-1+s) (2+s)^2}$$
My problem is getting the range for K, in which the system is BIBO stable.
I have solution ...
0
votes
1answer
43 views
Reachable points in state-space system
I have the following $(A,B,C)$ state-space sytem:
$$ A =
\begin{bmatrix}
-2 & 0 & 0 \\
-1 & -1 & 2 \\
-1 & 0 & 0 \\
\end{bmatrix},\ B =
\begin{bmatrix}
0 \\
1 \\
...
1
vote
1answer
101 views
Root Locus, Meaning of the Roots?
I'm studying control theory and I encountered the root locus, I know that It plots the roots of the characteristic equation but i've some questions.
What is the physical meaning of the Roots of the ...
3
votes
1answer
151 views
How can I efficiently sketch a Nyquist diagram?
I have the following transfer function:
$$P(s) = \frac{3}{(s-1)(s+2)(s+3)}, s= j\omega$$
I got the starting and endpoints:
$$\omega_0 = -\frac{1}{2}, \omega_\infty = 0$$
When I split the ...
1
vote
2answers
199 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 ...
0
votes
0answers
17 views
How to find and define the objective for control system?
How to find and define the objective for control system ?
such as how to find Q and R for objective?
which book teaching this?
