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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5
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1answer
80 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
88 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
2answers
115 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?
19
votes
5answers
925 views

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
1answer
81 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
47 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
85 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
132 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
57 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
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0answers
192 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
878 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
1answer
112 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 ...
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 ...
0
votes
1answer
95 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
135 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
95 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 ...
3
votes
1answer
130 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
36 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 ...
1
vote
1answer
95 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
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 ...
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
75 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 ...
-1
votes
1answer
48 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
186 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
128 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
125 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
117 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
87 views

How can I solve this control problem?

Consider this control problem in continuous time, known as Representative Agent Model in macroeconomics: $$ \max_{c_t,t\ge 0}\int_{0}^{\infty}e^{-\rho t}\ln(c_t)\, \mathrm{d}t,~~~\rho\in (0,1) $$ ...
3
votes
1answer
105 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
270 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 ...
1
vote
2answers
124 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
55 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
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 ...
3
votes
1answer
179 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, $\mathbf{x}$ ...
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... ...
0
votes
1answer
64 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
525 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
263 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 ...
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
180 views

how to obtain state space diagram and state space model for transfer function

How do we obtain the state space diagram and state space model for transfer function for example the question is given How to draw state variable diagram for the given transfer function ...
0
votes
1answer
490 views

Calculating the Kalman decomposition of a matrix?

If we are given the matrices $$ A = \begin{bmatrix} -2 & 3 & 4 & 1\\ 1 & 6 & 6 & 3\\ 5 & 6 & 6 & 4\\ 0 & -17 & ...
2
votes
0answers
81 views

Condition for controllability of matrix pairs

I have a question regarding how determine if a matrix pair is controllable. If $A$ and $b$ are given by A = [$\lambda_1 0 0 ...., 0 \lambda_20 0 0,...,00...\lambda_n$] (matrix) and $b = [b_1 b_1 ...
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 ...
2
votes
1answer
83 views

A control problem for the wave equation solved by the HUM

My question is about an article by J.L. Lions 1, where he introduced the "Hilbert Uniqueness Method" (HUM) for finding a boundary control function (dirichlet action) to bring the system to rest within ...
5
votes
1answer
582 views

Smith normal form of a polynomial matrix.

I have the following matrix: $P(s) :=$ \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix} How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
1
vote
1answer
91 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: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ ...
0
votes
0answers
77 views

Mathematical process model from real world?

I am studying control theory (among other things) at a university. I know about linear, non-linear, continuous, discrete, robust, etc. control theory. However one thing that has never been adequately ...
2
votes
1answer
81 views

Quadratic bounds

Consider the quadratic form $x^T(B^TP + PB)x$ where is $B=DA$, where $D$ is diagonal with positive or nonnegative entries, and $A$ is Hurwitz. Now consider a symmetric positive definite matrix $Q$ and ...