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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109 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
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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 ...
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1answer
92 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
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1answer
129 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 ...
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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
95 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 ...
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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
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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} ...
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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
124 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 = ...
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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
103 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
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0answers
263 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 ...
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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. ...
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0answers
53 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
177 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, $\mathbf{x}$ ...
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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... ...
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votes
1answer
63 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 \\ ...
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vote
1answer
501 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
260 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
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0answers
178 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
483 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
80 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
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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
80 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
82 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
560 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: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ ...
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0answers
75 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
79 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 ...
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1answer
8k views

How can I find the time constant of a first order system transfer function?

How can I obtain the time constant of the transfer function of a first order system, such as the example below? $$ \frac{C(s)}{R(s)} = \frac{2}{s + 3}$$ Where $C(s)$ is the output of the system and ...
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2answers
2k views

What is the physical meaning of Bode plot in case of unstable system?

I know that from the mathematical point of view it doesn't matter if we plot Bode diagram of stable or unstable system. It's just a function of complex value. However from the physical point of view, ...
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1answer
163 views

Solution of matrix equation or matrix inequality

When there exist a positive definite solution $S$ of the following matrix equation or matrix inequality: $ SA^{T}+AS+\alpha S-\beta BB^{T}=0 $, that is, what condition on $A,B, \alpha, \beta$ can ...
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0answers
74 views

Is this a common (nonlinear?) optimal control problem?

I have the following optimal control problem, which can be considered in a two period discrete time setting. It has this generic form. Let $w: \mathbb{R} \rightarrow \mathbb{R}$ be a given function, ...
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1answer
810 views

MDP quadcopter stabilization

I've designed a quadcopter and have it printed out on a 3D printer. Now I need to control it. I have formulated an MDP (Markov decision process) and want the helicopter to learn when it is in a ...
2
votes
1answer
2k views

From set of differential equations to set of transfer functions (MIMO system)

I want to know how I can get from a set of differential equations to a set of transfer functions for a multi-input multi-output system. I can do this easily with Matlab or by computing $G(s) = C[sI - ...
1
vote
1answer
135 views

Quadratic approximation of a cost function with a Taylor expansion

See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92. Given an instantaneous cost ...
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vote
1answer
60 views

Quadraticize a generic function

I have a generic function: $g(s,u)$. Now I want to have a local approximation near the point $(s^{\star}, u^{\star})$ in the quadratic form $$s^{T} Q s + u^{T} R u$$ to apply an optimal control ...
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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 ...
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vote
1answer
66 views

Modification of the continuous time Lyapunov Equation

For my research I have been working with different continuous-time Lyapunov equations of the form \begin{equation} M R + R M^\text{T} = G \end{equation} where all matrices are real and $n\times n$ ...
2
votes
3answers
141 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 ...