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Questions tagged [control-theory]

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The desired trajectory of the output 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 should manipulate the inputs to the system to obtain the desired effect on the output of the system

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existence of solution of wave equation with a feedback term

If we have a system of coupled wave equations with a feedback acting on one equation,that is $u_{tt}-\Delta u+py-\alpha(t)d^{2}(x)u_{t} =0$ $y_{tt}-\Delta y+pu=0$ $u=y=0 $ on boundary $\Gamma$ with d ...
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How can I design a (PID) Controller if I don't have a reference signal?

I have been trying to control lateral and longitudinal movement of a robot for an autonomous lane keeper project. I have no problem with the lateral movement, however I couldn2t figure out exactly how ...
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Model reduction of estimated state space models - System identification

Assume that we have a dynamical model in form of this simple transfer function $$G(s) = \frac{1}{2s^2 + 5s + 4}$$ G = tf(1, [2 5 4]) We do a step response with ...
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How can I reduce noise from measurement without a Kalman Filter?

I'm going to create an adaptive Model Predictive Controller (MPC). The model is a state space model. Due to noise, it's very difficult to determine the model order. I'm using subspace identification ...
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Transversality condition for infinite time horizon; Pontryagin Maximum principle

I want to solve a optimal control problem with the Pontryagin-Maximum-Principle, given following differential equation: $\dot x(t)=(a+b-c(t))x-c(t)\\ x(0)=x_0$ Regarding to the conditions: $x\geq0$ ...
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46 views

Controllability of LTI Networks

Let us assume a 4-node network, described by $\dot x = A x + B u $, where $$ A=\begin{pmatrix} 0 & 0 & 0 & 0 \\\ b & 0 & 0 & 0 \\\ c & 0 & 0 & e \\\ d & 0 ...
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Why is the 'controllable subspace' actually controllable?

I am looking at the Kalman decomposition of a linear system into 'controllable' and 'uncontrollble' subspaces. The references I am using are these lecture notes and section 3.3 of 'Robust and Optimal ...
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Analytical calculation of control input for given system [closed]

In the equations given below is it possible to calculate $u$ analytically to drive system from the initial condition of $x_{1} = 5$ ,$x_{2} = 0$ to the equilibrium point in the origin $x_{1} = 0$ ,$x_{...
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How do I solve this integral? Comes from Lyapunov equation and the derivative being negative definite

I am sure this is a simple solution, but I just cant seem to get this solution In sliding mode, following along with this video, https://www.youtube.com/watch?v=x9WxwM6Ebvo&list=PLv8cjLiRoYbivwv0-...
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Conditions for solving generalized Sylvester matrix equation XA + BX + CXD = E

In relation with an observation problem I have the matrix equation (1) $XA + BX + CXD = E$ where all the matrices $A$, $B$, $C$, $D$, $E$ can be assumed real, square and known, whereas $X$ is the ...
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Beginner's question about fuel control of a rocket

I am very new to control and mostly just reading Bellmann's stuff. He has some nice examples and writes really clearly, although there are times when his notation gets a little crazy. Does anyone ...
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Linear algebra based proof that if there exists $P\succ 0$ and $P\succ A^TPA$ then $|\lambda_i(A)|<1$

Is there a proof based on linear algebra that shows the following? If there exist $P \succ 0$ and $P \succ A^TPA$, then $| \lambda_i (A) | < 1$ for all $i$. Here, $|\lambda_i(A)|$ denotes the ...
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29 views

$H_\infty$ norm for transfer function

For a given scenario in the context of control system, I'm trying to invesigate how the $H_\infty$ norm can be calculated for a transfer function as follows: $$G(s)= \frac{w_n^2}{s^2 +2\zeta w_ns +w_n^...
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Example of a toy ODE control problem for chemistry/chemical engineering application?

I work in control theory. I have a student who is interested in chemistry and chemical engineering problems. I thought it would be somewhat easy to find such a problem, but apparently coming up with ...
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Explain the controllability distribution for a pendulum

As i was reading a book & i came across this statement- As the a pendulum passes through horizontally the controllability distribution doesn't have any constant rank- what does this mean ?? ...
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Solution to Average of Several Trails of Dicrete Time LQR with Noise

The solution to discrete time finite horizon LQR problem is well studied. We have the linear system $$x_{k+1}=A x_{k}+B u_{k}+w_k$$ where $w_k$ is a random variable with mean $0$ and finite second ...
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Linear quadratic regulator via least squares

In this set of slides, the finite horizon LQR problem is stated as a least-squares problem (slide 11), and using a naive method (e.g., QR factorization), the cost to solve this problem is $O(N^3nm^2)$ ...
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Model Predictive Control: Why the horizon size, $N$, must be equal or larger than 2?

If you read "Nonlinear Model Predictive Control" by L. Grune and J. Pannek (and anywhere else), everyone says that the prediction horizon size $N$ must be larger or equal to $2$,$ N\geq2$. Why?
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1answer
50 views

How can I solve the discrete algebraic Riccati equations?

I have heard that Schur decomposition $$A = USU^{-1}$$ can be used to solve discrete algebraic Riccati equations $$X = A^T X A -(A^T X B)(R + B^T X B)^{-1}(B^T X A) + Q$$ and also continuous ...
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Is LQR obsolete compared to non constrained MPC?

I have heard that LQR and MCP have common similarities. The difference is that MPC is using QP-programming and LQR using Riccati Equations. With QP-programming, constraints can be applied. If we ...
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60 views

Minimum in a non-linear system

I have the linear system: $$\begin{cases}\dot{x}=y\\ \dot{y}=-ay+x-x^3\end{cases}$$ where $a\geq 0$. I want to prove that this dynamical system has two minimum. I found the 3 equilibrium points $(...
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Discrete-time linear control with linear state/input constraints

Given a controllable discrete-time linear system $x(k+1) = A x(k) + B u(k)$ the input sequence leading from state $x_0$ to $x_f$ is given by $C^{-1} (x_f - A^n x_0)$ where $C$ is the controllability ...
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So what are state space representations actually doing? How are the state vectors related to the output?

I'm currently learning state space representations but I am really struggling with the concept. More specifically, I'm not seeing how the selection of state variables is made based off of an output ...
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The Center Manifold of an ODE

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a vector field given by $f_i(x)=\sum_{k=1}^n \sin(x_i-x_k), \forall x\in \mathbb{R}^n $. Now consider the ODE $\dot x=f(x)$. We observe that the Jacobian of $f(x)...
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Gramian Observability of Nonlinear System

I understand that for a linear system, the observability Gramian is given by $W_{o}=\sum_{\tau=0}^{\text{inf}}(A^{T})^{\tau}C^{T}CA^{T}$. However, I am wondering, how would be the calculation of $W_{...
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48 views

control with poles at unit circle

Could anyone explain me the paragraph? How they have defined the $PI$ controller in discrete time and how they got such transfer function? and why the matrix will not be Schur? Thanks.
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tracking error state space, non-linear control example

I am trying to understand an example from [1]. In detail I do not understand how the equation for the dynamic of the tracking error is chosen. I am not a mathematician so please forgive me if I may ...
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50 views

If (A,B) is controllable, is $(A^2,B)$ controllable as well?

I'm assuming a 3x3 matrix with controllability matrix as $[B\ AB\ A^2B]$. I feel that if A is a nilpotent matrix with n=4, then controllability of $(A^2,B)$ would be $[B\ A^2B\ A^4B]=[B\ A^2B \ 0]$ ...
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Proving electrostatic analogy for root locus

My teacher told us that there is a provable mathematical analogy between root locus and the lines of force generated by electric charges, where every pole can be associated to a positive charge and ...
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43 views

Are the path grid problem and Mason's Gain Formula fundamentally interconnected?

While working on a more computationally efficient solution for a path grid problem (according to this formulation: https://www.codeabbey.com/index/task_view/paths-in-the-grid ), I found an interesting ...
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1answer
33 views

Integral for expected value for Witsenhausens CE

For Witsenhausens counterexample, I want to compute the first term of: $$ J = k^2E[u_1(x_0)^2] + \text{something positive} \tag{1} $$ where $x_0 \sim N(0,\sigma^2)$ is a random variable drawn from a ...
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scaling eigenvalues in matrix A (state space form)

A have matrix A. The values and eigenvalues are: ...
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29 views

lie derivative and vector fields

I want to prove that [X,Y]=LxY where X=Ax and Y=Bx. What i have done is this: $\phi_t(x)=e^{At}x$, $\phi_{-t}(x)=e^{-At}x$, $Y (\phi_t(x))=Be^{At}x$ , $\phi_{-t}(Y (\phi_t(x)))=e^{-At}Be^{At}x$, ...
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57 views

Inequality manipulation for a vector norm

How can I manipulate the following inequality to reach from $$\dot{V}\leq -4x_1^2 +4x_1x_2 -2x_2^2 $$ to $$\dot{V}\leq -(3-\sqrt{5}) \|x\|^2 $$ where $x=[x_1 \;x_2]^T$ is a 2D vector and $\|x\|$ is ...
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results from limit of a complex sequence

the limit of a complex sequence An+iBn is zero as n tends to infinity, where An and Bn are two real sequences. Does this implies that the limit of An and Bn is zero
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33 views

Steady state error using final value theorem

From the continuous time control system above, i need to find the steady state error using the final value theorem in response to a unit ramp input signal. How do I begin solving this kind of question....
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Is there always at least one attracting set in an invariant set?

(Edited) Suppose we have an invariant set for a dynamical system. Can we claim there exists an attracting set (including the strange attractors) in the invariant set? If so, is there any specific ...
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Solution for first order ODE with discontinuous right hand side

I'm now approaching for the first time to first order differential equations with discontinuous right hand side. Let $A\subset \mathbb{R}\times\mathbb{R}^n$ and $f=f(t, x):A\to \mathbb{R}^n$. Fix $(...
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Eigenvalues of normalized vs unnormalized Laplacian of weighted digraph

Let $G$ be a weighted digraph. What is the connection between the eigenvalues of the normalized and unnormalized Laplacians of $G$. I think there is no explicit connection. We can at most find some ...
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1answer
31 views

How to decompose a MIMO system into a multiplication minimum and nonminimum phase part

suppose the transfer function of m-input-m-output system is $$G\left( s \right) = \left[ {\begin{array}{*{20}{c}} {{g_{11}}\left( s \right)}& \cdots &{{g_{1m}}\left( s \right)}\\ \vdots & ...
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Smooth endpoint map

Consider a control system $$ \dot x(t) =f(x(t),u(t)) \qquad (\star) $$ where $f$ is a smooth vector field and $x\in \mathbb{R}^n$. The endpoint mapping is defined by $$ E:\mathbb{R}^n\times \mathbb{...
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How to identify a system with no overshoot in terms of poles?

I'm trying to find a standard in order to identify a system with no overshoot in term of poles. I know that there is something related to dominant poles. I was looking into a particular 4th-order ...
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2answers
51 views

Stablility of a linearized time-delay system

I have a linearized time-delay system as follows: $$\frac{\mathrm d X}{\mathrm d t} = a[X(t)-X^*] + b [X(t-R) - X^*], $$ where $a$, $b$ are constants, $R$ is the constant delay, and $X^*$ is the ...
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Q learning for more than one goal state

I have recently implemented a reinforcement learning (TD method) problem consisting of 19 states and 2 actions (Increase/decrease relative to previous time step action) with one goal state. Now I want ...
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50 views

is this function regular?

I am learning nonsmooth analysis for discontinuous dynamical systems. An important concept in this field is the regularity of functions. A function $f$ is regular at $x$ if its right directional ...
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25 views

How to find the transfer function for advection equation?

The Problem: I have to find the transfer functions of the two equations in my system and of the whole system but I am not sure about something and as I haven't had mathematics lessons for my three ...
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1answer
35 views

A simple quadratic optimizer for only constraints on input

I'm going to implement an quadratic optimizer with C for embedded systems. I will do that because I need speed. But I have some trouble to find a quadratic optimizer for C that works with embedded ...
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1answer
66 views

Property of interconnected feedback systems

In the figure you can see the statespace form of a feedback interconnection system. Very quick question: is there a reason they have taken $D_1=0$ and $D_2=0$? It makes workings a lot easier but I ...
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60 views

Is there any rule of thumb when it comes to selecting control/predict horizon for MPC?

I have a simple question: Is there any rule of thumb when it comes to selecting control/predict horizon for MPC? Normaly I set control and predict horizon equals, but I have heard that's not good ...