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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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39 views

Semidefinite programming relaxation of linear dynamical system to find Lyapunov function

I am considering a linear dynamical system of the form $$x_{k+1} = Ax_k$$ I know that when we have stability (that is, that $x_k$ goes to $0$ as $k$ approaches infinity), there exists an $n$-by-$n$ ...
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1answer
29 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
60 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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2answers
35 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 ...
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1answer
70 views

Controllability of cascade connection of two systems

I have two linear control systems that are represented by their state space models $$\left( \begin{array}{c|c} A_1 & B_1 \\ \hline C_1 & D_1 \\ \end{array} \right), \left( \begin{array}{c|c}...
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1answer
23 views

Why the angle is $-180°$ at $\omega = 0$ for this system

I'm trying to plot the nyquist from the analytical expression of the system but the bode plot generated by matlab yields an angle -180 whereas the analytical expression yields zero when $\omega=0$. ...
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1answer
48 views

Exosystem for reference generation

Recently I have read a paper where an LTI system of the form $$ \begin{align} \dot{x}_p &= A_p x_p + B_p u \\ y &= C_p x_p \end{align}\tag{1} $$ for the control plant was considered. In ...
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33 views

Questions about LQG with full information

I have implemented LQG in MATLAB software. But, now I do not know how to determine the value of optimal cost. Each way of calculating cost, returns a different value. Which one should I trust to ...
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1answer
51 views

Show that if a linear dynamical equation is controllable at $t_0$, then it is controllable at any $t<t_0$.

Consider a $n$-dimentional $p$-input equation: $$\dot{x}=Ax+Bu$$ where $A$ and $B$ are constant $n\times n$ and $n\times p$ real matrices. By definition, the latter state equation is said to be ...
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1answer
46 views

Lyapunov stability of 4x4 matrix.

Consider the following continuous-time state space representation of the form: $\frac{d}{dx}x(t) = Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t\in \mathbb{R}^{+}$ $A=\begin{bmatrix}-1&3&0&0\\-...
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1answer
42 views

Achievable performance for systems with RHP zeros/poles

I have often read that RHP zeros and poles set limits on the maximum achievable performance of LTI systems. However, what does that exactly mean and how can you compute these performance limits? For ...
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25 views

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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0answers
10 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
26 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
37 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
36 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
28 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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1answer
29 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
30 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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29 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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0answers
15 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
48 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
63 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
129 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
88 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
86 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
126 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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37 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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1answer
46 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
63 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
64 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
27 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
80 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
25 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
86 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
56 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
50 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
42 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
63 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
61 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
94 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
35 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
54 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
56 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
43 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
26 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
31 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^...