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Questions tagged [linear-control]

Linear control theory is the sub-branch of control theory dealing with linearized systems.

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1answer
18 views

BIBO stability of 4 matrices.

The following question is from a System Theory test without answers or solutions. It concerns the BIBO stability of the following 4 systems: $1) \left[ \begin{array}{l|l} A&B\\ \hline C & \...
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1answer
11 views

Kalman decomposition of given system.

The following question is from a System Theory test without answers or solutions: Consider the continuous-time state-space representation $\frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t\in \...
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1answer
36 views

Jordan form of simple 2x2 matrix

Considere the following transfer function: $\frac{1}{s^2+1}$ Calculate the Jordan form, real Jordan form and determine if this system is Lyapunov stable? My approach: The system's $A$ matrix is: $\...
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0answers
29 views

A Fact on the Transfer Function of a Linear System

While doing some work today, I found that the transfer function of the linear system $(A,B,C,D)$ is equal to the upper Schur complement of the $(n+p)\times (n+m)$ block matrix $$ M(s) = \begin{...
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0answers
29 views

stability of a matrix in control theory

let $\dot x=Ax$ and $S$ be an orthogonal basis of matrix $A$ i.e. $AS=0$ and let $S_\perp$ be the orthonormal complement of $S$. Is $S_{\perp}^T A S_\perp$ an stable matrix?
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1answer
33 views

How can I add find the gain from root locus and poles?

I try to find the P-gain from a root locus plot where I know the poles. Assume that we got a reference model: $$G(s) = \frac{\omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2 }$$ Where $\zeta$ and $\...
1
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1answer
32 views

State response given $A$ matrix and eigenvector matrix $M$ without using $M^{-1}$

This question is from a System Theory test without answers or solutions: Consider the system $\dot{x}=Ax$, where $A=\begin{bmatrix}-3&1&2&-1&1&0\\2&-2&0&2&-2&...
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2answers
55 views

Matrix to the power t.

Compute the matrix $A^t$ for the following cases: $A_1=\begin{bmatrix}0&0\\0&1\end{bmatrix}, \quad A_2=\begin{bmatrix}-1&0\\0&-2 \end{bmatrix}, \quad A_3=\begin{bmatrix}0&1\\0&...
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0answers
32 views

MIMO transformation to controller canonical form

I am unable to prove a result concerning MIMO linear dynamical systems. Let $$ \dot{X} = A\cdot X + B\cdot U$$ be a linear time invariant dynamical system, with $A \in \mathbb{R}^{n\times n}$, $B \in \...
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0answers
24 views

from multi input to single input in linear dynamical systems

I am working with some linear multi input dynamical systems. There is a result here which reduces the problem to single input linear systems. Given the following linear system: $$ \dot{X} = A\cdot X + ...
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1answer
64 views

Lyapunov equation.

This question is from a system theory test without answers or solutions: Let the following two cases be given $A) \quad A=\begin{bmatrix}-2&1\\-1&0\end{bmatrix} \quad $and$ \quad C=\begin{...
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1answer
23 views

Discrete time impulse response

The following question is from a System Theory exam whitout answers or solutions: Which of the following discrete-time state-space model (A,B,C) of the form $x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \...
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1answer
39 views

Lyapunov, asymptotic and BIBO stability of $4$ given systems

Which of the systems are Lyapunov, asymptotically or BIBO stable: $1) \quad \left[ \begin{array}{c|c} A & B\\ \hline C & \end{array} \right]=\left[ \begin{array}{ccc|c} -2&1&0&2\\ ...
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1answer
21 views

Stability of Discrete Time state space system with eigenvalues 0, 1/2 and 1.

This question is from a system theory exam without answers. So I was wondering if my resoning is correct. Consider the discrete-time state-space realization $x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \...
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1answer
28 views

Transfer function given A,B,C, diagonal and eigenvectors.

Consider the continuous-time state-space representation $\frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t \in R^+,$ with the matrices given by $A =\begin{bmatrix}-4&-5&0&5\\-4&...
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2answers
40 views

Determine Jordan block size.

The following question is from as System Theory test. Let the system matrix $A$ be given as $A = \begin{bmatrix} 0&0&0&1\\0&-1&1&3\\0&1&-1&-1\\0&-1&1&2 ...
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2answers
42 views

Impulse response to Jordan form.

Which matrix $A$ in real Jordan from is such that, for suitable choices of the matrices $B$ and $C$, continuous-time state-space model $(A,B,C)$ of the form $\frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=...
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1answer
32 views

minimal time to reach given state. number of time steps unclear.

Consider the discrete-time state-space realization $$x(t+1)=Ax(t)+Bu(t), \qquad y(t)=Cx(t)$$ with $$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & 0 \end{bmatrix}, \...
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1answer
45 views

Stability of system of differential equations.

The following question is from a System Theory test without answers or solutions. Let a continuous-time LTI system be given by the following differential equations: $\frac{d^2}{dt^2}y_1(t)+4\frac{d}{...
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1answer
57 views

Discretization of continuous-time state-space system.

This question is from a Systems Theory test without answers or solutions. Consider the folowing continuous-time state-space system $\dot{x}=Ax+Bu, \quad y=Cx.$ The continuous-time system given above ...
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1answer
28 views

Can I build an adaptive controller by using an ODE solver and a 3D graphics engine? [closed]

Let's assume that you're using a 3D graphics engine with built in physics. You create a inverted pendelum in a 3D designing software, e.g Blender, and then import the model into your 3D grapics engine ...
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1answer
48 views

Transfer function is $0$?

Given the continuous time state space model: $\dot{x}(t)=Ax(t)+Bu(t)$, $\quad y(t)=Cx(t), \quad t\in R^{+}$ with: $\left[ \begin{array}{c|c} A & B \\ \hline C & \\ \end{array} \right]$...
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0answers
37 views

What's wrong with this robust control scheme?

I'm learning how to control a double integrator with $H_\infty$. my model is simply $ \dot{r} = v $ $ \dot{v} = F/m $ $ r(t_0) = 0$ m, $v(t_0) = 0 $ m/s, $m = 1000 $ kg so I want to be able to ...
1
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1answer
54 views

Determining eigenvalues with limited information

The following question is from a System Theory test with only answers (no solutions). Maybe someone here knows how to tackle it. Consider the discrete time system $$x(k+1) = Ax(k)$$ with a matrix $$...
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2answers
38 views

stability of linear system

I have a discrete system of the form $x(k+k_o) = Ax(k+k_o-1) + Bx(k)$ where $A$ and $B$ are $n\times n$ matrices ($k_o>0$). I want to know about the stability of the system when both A and B ...
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0answers
20 views

Is realisability a necessary and sufficient condition for physical implementation in hardware?

A state-space model can be physically implemented in hardware, Rugh. If a transfer function is realisable then there exists a corresponding state-space formulation. Hence, realisability is a ...
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1answer
29 views

Saturation limit compared to constrained limit

I have a simple question. What's the difference in behaviour between saturation limit and constrained limit in control theory? We say that we got this objective function: $$J_{min} = \frac{1}{2}x^...
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1answer
24 views

How can I handle the delays in Generalized/Model Predictive Control?

I trying to handle delays in a model who is poorly damped but I haveing som issues to estimate its parameters due to the delay. Assume that we got a state space model, which is poorly damped: $$x(k+...
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0answers
24 views

What's the idea behind having internal integration in predictive control?

According to lots of books about predictive control, they recommend to having internal integration inside the model. For example if we have a state space model: $$x(k+1) = Ax(k) + Bu(k) \\ y(k) = Cx(...
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0answers
18 views

Create a Kalman filter from ARMAX model?

Assume that I have a ARMAX model: $$A(z)y(t) = B(z)u(t) + C(z)e(t)$$ I going to use the Algebraic Riccati Equation(ARE) to find the LQR control law $L$ by selecting the weighting matrices $Q$ and $R$...
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2answers
80 views

What's the difference between Generalized Predictive Control and Model Predictive Control?

As I know, the Generalized Predictive Control(GPC) is older than Model Predictive Control(MPC). But what is the real difference between them? I know that GPC contains some kind of system ...
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0answers
15 views

How do I choose the polynomials for a stochastic filter? - Transfer functions + Extended Least Square

I'm buildning a Mimimum Variance Controller(MVC) but I having som trouble to select the stochastic filter. First of all! To build a MVC, you need a ARMAX model, in other words polynomial who look ...
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0answers
21 views

What's the applications of Minimum Variance Controller?

I going to show how to create a Minimum Variance Controller(MVC) and then ask what's the applications of MVC. First! Let's say that we have a stochastic transfer function model, ARMAX in other words. ...
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0answers
27 views

How can I choose the disturbance model if I know the plant and controller - Transfer functions

I going to select the disturbance models $C_f$ and $H$. I know my plant $P$ and the controller $C_b$. I also know that the disturbance $d$ is step formed. The noise $v$ is $v = 0$. Question: How ...
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0answers
20 views

Integral action or correction factor on disturbance?

I wonder what's the difference between having a integral action or correction factor when it comes to disturbances? Ofc I know how to apply then, the reason for this question is: What's suits best ...
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1answer
28 views

Controllability of a convex polytope of matrices of LTI system

Given Linear time invariant (LTI) dynamic system: \begin{align*} \dot{x}(t)=Ax(t)+Bu(t) \end{align*} where $A \in R^{n \times n}$ and $B \in R^{n}$ are system matrices, $x(t)$ is the system's state ...
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2answers
29 views

How do I find a controller for non-minimum phase systems?

Assume that we have a transer function $G(s) = \frac{B}{A}$ which has stable poles, but unstable zeros. We use the controller $Q(s) = \frac{A}{B} = G^{-1}(s)$ and we want that the loop transfer ...
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0answers
24 views

How to find a matrix with eigenvalues of different signs using LMI tools?

Suppose I am given two $n \times n$ real matrices $A_1$ and $A_2$ and I am wondering if there exists a positive definite matrix $P=P^\top$ such that (this is related to the stability of linear systems)...
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1answer
60 views

How do I place the poles and zeros form a disired system? Adaptive control

If I have a transfer function of a system $G(s)$ $$G(s) = \frac{4 - 2s}{4 + 0.8s + s^2}$$ $G(s)$ has the poles and zeros and is a stable system. And the step answer look like. It has a delay as you ...
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1answer
75 views

What's the difference between non-minimum phase systems and minimum phase systems?

I wonder if you can explain what's the difference between non-minimum phase systems and minimum phase systems? How can I recognize them in bode/time plots? Is this a minimum phase system? $$G(s) = \...
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1answer
42 views

How should I write the difference equation of a polynomial equation?

I going to estimate the polyomial $R^*$ and $S^*$ from $$ y(t)= \frac{R^*}{A_o^*(z^{-1}) A_m^*(z^{-1})}u(t) + \frac{S^*}{A_o^*(z^{-1}) A_m^*(z^{-1})}y(t)$$ $A_o^*(z^{-1})$, $A_m^*(z^{-1})$ polynomals ...
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1answer
46 views

How can I estimate a discrete transfer function? Recursive Least Square

This is going to be a large fun question about practical estimation for real world problems. Assume that we have a poor damped system described with this transfer function. $$G(s) = \frac{4.5}{1 + 0....
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3answers
51 views

What's the point of creating discrete control laws for analog processes?

Assume that we have a state space model of a real system e.g mass-spring system. $$\dot x = Ax + Bu$$ Then we want to implement this in a micro controller. The controller has a sampling rate of $h = ...
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1answer
47 views

Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: \begin{array}{c} \varphi _{t}+\...
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0answers
23 views

Redesigning MPC augmented state for integrating action

In model predictive control for a system $$ \begin{cases} x_m(k+1)=A_m x_m(k)+B_m u(k)+\xi_k\\ y_m (k)=C_m x_m(k)+\eta(k) \end{cases} $$ where $u$ is the manipulated variable (dimension $y$ is the ...
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2answers
38 views

How do I find the state space representation of a Linear Fraction Transformation (LFT)?

I am having a problem with solving this question: As you can see, I have the filtering problem and I need to find a state space representation. I know how to get the state space of a SISO transfer ...
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0answers
41 views

What's the point of having a reference model in Model Reference Adaptive Control?

The MRAC(or called MRAS where S = 'System') controller is called a adaptive controller. I don't know why, because it have no system identification process loop. But what's the point to having a ...
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0answers
22 views

Stability of delayed linear time invariant differential equation

When considering a delayed differential equation of the form $$ \dot{x}(t) = \sum_{n=1}^N A_n\,x(t-\tau_n) \tag{1} $$ A solution to $(1)$ is assumed to be able to be written as $$ x(t) = e^{M\,t}\,...
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0answers
34 views

Augmented nonlinear state space model for nonlinear model predictive control?

Assume that we have a discrete state space model: $$x(k+1) = Ax(k) + Bu(k)\\ y(k) = Cx(k) + Du(k)$$ And we want to use optimization to minimize this cost function. $$J =\sum_{k=0}^{n}(x_k^TQx_k + ...
2
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1answer
61 views

Linearization of System of ODEs around Operating Point / Transfer Functions and State Space

I have this system of ODEs and I'm trying to get a linearized version of it around the "operating point" $\overline{x}_1 = 1$ $$ \left\{\begin{matrix} \ddot{x_1}(t)+2\dot{x_1}(t)+2x_1^2(t)-2\dot{x_2}(...