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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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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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32 views

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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21 views

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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32 views

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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7 views

the system controllability and observability

If a linear system is controllable, does it mean we can find the control standard form in state space, but it doesn't mean all the forms of state space representations is controllable?
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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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2answers
42 views

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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41 views

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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Optimal basis generation using simplex

Given the objective function $\sum_{i=0}^{i=n} t_i$ (which I want to minimize), constraints $At = u, t \geq 0$ where $A \in m \times n$, and $ n>m$, I'm trying to determine all of the possible ...
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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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53 views

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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52 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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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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2answers
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 ...
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1answer
76 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
25 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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49 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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36 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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53 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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56 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
51 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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27 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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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
34 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
41 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
49 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
31 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
31 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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44 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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35 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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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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60 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
68 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
160 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
109 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
106 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
163 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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38 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
51 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
38 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
66 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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82 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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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
106 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
119 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
94 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
58 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&...