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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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Jordan form corresponding to Discrete time impulse response.

Which of the following discrete-time state-space models $(A,B,C,D)$ of the form $x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t)+Du(t), \quad t\in \mathbb{N}$ with $A$ in jordan form has its impulse response ...
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6 views

Causality of linear singular systems

According to Dai(1989, p. 234), the following system: $Ex(k+1)=Ax(k)+Bu(k)$ $y(k)=Cx(k), $ $k=0, 1, ..., L$ where $ x(k) \in \mathbb{R}^n$, $ u(k) \in \mathbb{R}^m$, $y(k) \in \mathbb{R}^r$ and $...
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1answer
16 views

How to divide an uncontrollable LTI system into controllable and uncontrollable parts?

Consider this linear system $\frac{dx}{dt}=Ax+Bu$ Assume that $B\neq 0$ and the system is uncontrollable. It’s easy to show the existence of an invertible state transform $x=Ty$ satisfying $$\frac{dy}...
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1answer
25 views

Dynamic order of transfer function matrix.

Consider the transfer function matrix $G(s)$ of a continuous-time system given by: $G(s) = \begin{bmatrix}\frac{1}{s^2+2s}&\frac{s+1}{s} \\ -\frac{1}{s+1} & \frac{1}{s^2+4s+3} \end{bmatrix}$ ...
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1answer
26 views

'Modes' in Control Theory

What is the meaning of 'Mode' in control theory , in many places while studying linear system theory and control specially controllablity,observability,stabilizability and detectability i saw people ...
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1answer
27 views

Discretization before or after closing the feedback loop?

Say I have a continous plant which is controlled by a digital controller. In order to apply methods from discrete control, I can change from the continous $s$-domain to the discrete $z$-domain. Now ...
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27 views

Discrete time-invariant MIMO systems with a multidimensional state

Consider discrete time-invariant MIMO systems with a multidimensional hidden state (or simply state) as the recursive system $$ h_{t+1}=Ah_{t}+Bx_t+\eta_t $$ $$ y_t=Ch_t+Dx_t+\xi_t $$ Where $h_t$ ...
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1answer
22 views

Summation of polynomial matrix multiplication in terms of vector outer product

Consider the following summation $$ \sum_{i=1}^{T-1}C(A^i-A^{i-1})Bx_{t-i} $$ where $A$ is a $d \times d$ diagonal matrix, i.e. $A=\text{diag}(\alpha_1,\cdots,\alpha_d)$, $C$ is an $m \times d$, $B$ ...
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26 views

Finding relationship between input and output

I am just trying to figure out what key words I should look up to help me with the following problem. I have a control system to control a PWM motor and a sensor to detect the motors frequency for ...
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14 views

Discrete transfer functions with different sample rates

is there a way to deal with a connection of two discrete transfer functions with different sample rates? E.g.: $G = G_1G_2$ with sampling rate of $G_1$ being $T_1=1$ and for $G_2$ being $T_2 = 2$. ...
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1answer
38 views

prove/show algebraic equivalence of 2 3x3 systems.

Consider the following two continuous-time state-space representations of the form $\frac{d}{dt}x(t) = Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t \in \mathbb{R}^+$ With their matrices given by $1) \...
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1answer
54 views

0-controllability of three simple systems.

Consider the following three discrete-time state-space realizations $(A_1,B_1,C_1), (A_2,B_2,C_2) \ \text{and} \ (A_3,B_3,C_3)$ with $A_1=\begin{bmatrix}0&1\\0&1 \end{bmatrix}, \ \ \ \quad ...
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1answer
49 views

Similarity transformation to controllable and observable canonical form.

Consider the system $\dot{x}=Ax+Bu, \quad y=Cx$ with: $A = \begin{bmatrix}2&4&-5\\3&1&-3\\4&4&-7\\ \end{bmatrix}, \quad B=\begin{bmatrix}4\\1\\3 \end{bmatrix}, \quad C = \...
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1answer
58 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
37 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
72 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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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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1answer
40 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 $\...
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1answer
37 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
61 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
48 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
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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
65 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
24 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
60 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
33 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
37 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
41 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
58 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
33 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
52 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
67 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
29 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
50 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 ...
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1answer
55 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
42 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
23 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
30 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
27 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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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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19 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
95 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
16 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
28 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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35 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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22 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
30 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
27 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)...