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Questions tagged [linear-control]

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

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What adaptive controller can be used in embedded system with low RAM?

This is not a question for data science, hardware or programming languages. This is a more practical question about adaptive control for embedded systems, but still a math question. I have tried to ...
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18 views

PBH Test matrix

Can the matrix occurring from PBH test have rank greater than the dimension n of the state space? What does it mean for the controllability of the system?
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Wrong stability results when using Padé approximation

The Padé approximation of the exponential function, $F(s) = e^{-\tau s}$, is used often in control theory. I wonder whether its use can lead to erroneous results regarding the stability properties of ...
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31 views

Comparing energy and power signal

I am trying to define and graph the signal to noise ratio of my system in the frequency domain. The system is LTI, excited by a pulse and subjected to filtered ZMWN. The only relevant portion of my ...
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1answer
71 views

Necessary conditions for positive realness

Given a linear time-invariant (LTI) system with $$ \begin{align} \dot{x} &= A x + Bu \\ y &= C x + D u \end{align} $$ We know that the transfer function matrix $G(s) = C(s I - A)^{-1}B + D$ ...
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Fractional Sobolev Spaces and Trace Theory

I've been working with fractional Sobolev Spaces for a while and I still don't get how is it connected to trace theory, is there any literature which goes deeper into such relationship? From the ...
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40 views

Tuning Backstepping Controller Gains

I am doing some experiments and I'll attempt to apply a Backstepping controller to a Laser Beam Stabilization System modelled as the following linear System: $$\frac{X(s)}{V(s)}=\frac{K}{s(\tau s+1)}$...
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22 views

Getting system sensitivity from nyquist plot

Given this frequency response: I need to find the maximum sensitivity for closed system for automatic regulation $ M_\text{s} $. I now that $$ M_\text{s} = \max|S(j\omega)| = \max\left|\frac{1}{1+G(...
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57 views

LQR control for scalar discrete-time control system

Consider the following scalar discrete-time control system $$x(k + 1) = 2x(k) + u(k)$$ where $x \in \mathbb R$, $u \in \mathbb R$ and $x(0) = −2$. Let $N > 1$ be some integer. Consider ...
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40 views

Reason to Design LQR for already Stable System

According to my understanding, if the poles of a system are in RHP, LQR controller can be designed to place the pole in LHP with optimal gain 'K' found by minimizing the cost function (Q, R). I am ...
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35 views

Transition Equation in Linear-Quadratic-Gaussian (LQG) Control Problem

Disclaimer: I am from another field, the language in the following problem therefore may be different from the usual optimal control lingo. Consider the system $$x_{t+1} = Ax_t + Bu_t + C\...
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62 views

Why is $(A+BK, B)$ controllable when $(A,B)$ is, but $(C, A+BK)$ is not observable when $(C,A)$ is?

I attempted to use the Popov-Belevitch-Hautus test to prove these, namely $$[A+BK - \lambda I, B]=[A-\lambda I, B]\begin{bmatrix}I & 0\\K & I \end{bmatrix}$$ which are both full rank and ...
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38 views

How can I find the control for a finite system by definition?

Currently I am working on control theory, precisely in controllability but still on the basics, in the following example by E. Zuazua: It says, consider the following problem \begin{equation} \...
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77 views

Design a PID Controller

I need to design a prefilter and PID controller for a plant whose transfer function is given by $$G(s)=\frac{3}{s(s^2+4s+5)}$$ subject to the following performance requirements: An acceleration ...
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50 views

How to make use of the modified z-transform

The modified/advanced z-transform was introduced to analyze the behavior of sampled data systems between the samples. I understand how to derive the z-transform of a given continous transfer function....
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Robust & Linear Control Systems

The characteristic polynomial of a control system is the following uncertain polynomial: $$s^3 + a_2 s^2 + a_1 s + 3.5 $$ Where $a_1 \in [1.5,4.2],a_2 ∈ [1.2,4.25]$ and $ 4.2 ≤ a_1 + a_2 ≤ 6.3$ . Is ...
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21 views

Is there a way to find propotional gain schedule by iteration?

I have been reading two papers about controling a mechanical nonlinear model where the model structure is the same, but its parameters varying over time. The first controller is gain scheduling. ...
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1answer
82 views

Prove that if and only if $(A,B)$ is controllable, then $(A−BK,B)$ is also controllable [closed]

Given $A \in \mathbb R^{n\times n}$, $B \in \mathbb R^{n\times 1}$ and $K \in \mathbb R^{1\times n}$, prove that $(A,B)$ is controllable $\Leftrightarrow$ $(A−BK,B)$ is controllable. Any help would ...
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43 views

Can I have $Q = R = I$ as covariance matrices for a kalman filter?

Assume that we have no noise in our system. We using a low pass filter to filer away some peaks in the measurements. But our goal is just to estimate the state $X_k$. Can we set the $Q_k$ and $R$ to ...
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29 views

Structural controllability of networks

I want to check the structural controllability of a given network from a given input node. In Matlab, controllability can be verified using Kalman rank condition (...
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1answer
38 views

How can I find the initial state vector from state space model?

Assume that we have input $u(k)\in \Re $ and output $y(k) \in \Re$ and we estimate the black box model by using subspace identification method. $$x (k+1) = Ax (k) + Bu(k) $$ $$y (k) = Cx (k) $$ If ...
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45 views

LQR with derivative-dependent performance?

Given a standard LTI system with $$ \dot{x} = A x + B u $$ The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that $$ J_1 = \int_0^\infty \big( x^T Q x + u^T R u \big) ...
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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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62 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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1answer
44 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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41 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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12 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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54 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
66 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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1answer
46 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
71 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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1answer
94 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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1answer
65 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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36 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
72 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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115 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
102 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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1answer
56 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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39 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
56 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
62 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
73 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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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
47 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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56 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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112 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 ...