# Questions tagged [linear-control]

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

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### Limiting behaviour of state vector under delayed (open-loop) inputs

Given a discrete-time LTI system as $$x_{i+1} = A x_{i} + B u_{i}$$ and suppose the feedback law $u_{i} = \mathcal{K}(x_{i})$ assymtotically stabilizes the closed-loop system. Now, consider the ...
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### How to determine the resonant peak magnitude of the closed loop via the Nichols chart?

I have been studying the control theory and I have recently found the Nichols chart theme. Among others this chart is useful for determining the resonant-peak magnitude of the closed loop based on the ...
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### How can I get the inverse of the transfer function $\Phi$? a question from Page 56 of Lemma 4.5 in the book "ESSENTIALS OF ROBUST CONTROL" [closed]

Let $\Phi(s) = = \gamma^2 I + B^*\left( (sI-A)(C^*C)^{-1}(sI+A^*) \right)^{-1}B + B^*(sI+A^*)^{-1}C^*D - D^*C(sI-A)^{-1}B - D^*D.$ How to compute the inverse of $\Phi$, in Page 56 of Zhou's book "...
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### How to design/add a new controller to a system without breaking the existing controller in the system? [closed]

Please help me to find related topics/books for this problem: For example, assume we have a water heater, and a tank of water. We can design a controller to heat the water in the tank and keep it in ...
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### Two CSTR in series (glucose to acetate)

Problem Hello, I am studying modeling and control course and I'm struggling with drawing neat figure describing the reactor system and to write the differential equations in detail describing the ...
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### A transfer function relating jet engine thrust to pitch angle during a phugoid motion

Crossposted on Aviation SE I am doing a school project that requires me to find a transfer function that relates the thrust of a jet engine (which could change with time in one way or another) and ...
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### control theory: Does proportional control add energy to a system?

I am looking at a good tutorial on PID control, and I am a little confused about how control works. My question is really whether a proportional controller adds energy to the system to obtain some ...
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### Spectrum of a discrete-time observability Gramian from system matrices

Suppose that $A$ is an asymmetric matrix that has all eigenvalues inside the unit circle. Let $Q$ be a symmetric, positive semidefinite matrix. Let $W=\sum_{t=0}^{\infty} (A’)^t Q A^t$ a discrete time ...
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### Proof of Existence of a Solution of a Periodic Boundary Value Problem System with Switching

I have the following vector PDE: $${\dot {\mathbf {x} }}(t)=\mathbf {A_{i(\gamma,x(t))}} \mathbf {x} (t)+\mathbf {B_{i(\gamma,x(t))}}\mathbf {u},$$ where $\mathbf{x(t)} \in \mathbb{R}^{n}$. Some of ...
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### Derivation of solution for simple control problem

While trying to understand the fundamental concepts in control theory reading the following article Dual Control for Approximate Bayesian Reinforcement Learning (chapter 3.1, "A toy problem")...
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### The controllability of a system whose adjacency matrix has a combination of negative and positive eigenvalues

I'm currently working on a multidisciplinary research project about the structural controllability of brain networks. Specifically, I have constructed the adjacency matrix of brain networks and ...
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### Show that two definitions of the controllability Gramians are equivalent

Given a dynamical system $\dot x = Ax + Bu$, I've noticed that several authors define the controllability Gramian as: $$W = \int_0^te^{A(t-\tau)} BB^T e^{A^T(t-\tau)}d\tau$$ while others define it as, ...
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### Equivalent Statements for a Discrete-Time System

I have a discrete-time system x_(k+1) = A*x_k, x(0) = x_0 where A is in n x n dimensional space and is a real constant matrix. How do I show that the following statements are equivalent? All ...
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### would Kalman Filter capture consistent drift in observation model?

I am studying Kalman filter and I wonder how it handles the case of consistent bias in observation model? Let's take this example from wikipedia: https://en.wikipedia.org/wiki/Kalman_filter#...
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### Minimum time control

Given the dynamical system : $\dot{x}_1=x_2\\ \dot{x}_2=u\\ \dot{x}_3=x_4\\ \dot{x}_4=\alpha x_3+\beta x_4 + u$ where $\alpha,\beta \in R-\{0\}$ and $|u| \leq 1$. My goal is to find the minimum ...
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### Stability of a linear time-varying system?

I am interested in finding out the stability of the system $\dot{x} = -a \begin{bmatrix} \cos^2(t) &\cos(t)\sin(t) \\\cos(t)\sin(t) &\sin^2(t) \end{bmatrix}x$ with $a > 0$, via Lyapunov ...
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### What is the capability of the control loop to compensate offset in measurement?

Let's say I have a control loop where $D$ is the transfer function of the controller and $G$ is the transfer function of the plant. The $R$, $Y$, $E$, and $V$ are the Laplace transforms of the ...
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### Root locus asymptotes

I have been studying the root locus method from the Feedback Control of Dynamic Systems. Here in Chapter 5 I have found a derivation of the rule for finding the origin point of the asymptotes. The ...
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### Implementing digital controller in the time domain

I have simulated a digital control system in the Z domain using MATLAB and I have got satisfactory results. However, when I converted the plant and the digital controller to difference equations and ...
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### Lyapunov criteria for discrete linear system with noise

Consider constant model matrix $A$ and $B$, the Lyapunov criteria for system $x_{k+1}=Ax_k+Bu_k$ with state feedback input $u_k=Kx_k$ (K is designed matrix) is $P-(A+BK)P(A+BK)^\top>0$, where $P$ ...
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### Bilinear Transform vs Standard Numerical Methods

I am not very familiar with control theory but have a decent bit of experience with classical numerical integration. I am looking at a the equation $$\dot{x}(t) = Ax(t) + Bu(t) \hspace{1cm} x(0) =0$$ ...
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### Derivation of the discrete-Time Algebraic Riccati Inequality (DARE)

everyone. I'm interested in control theory and I'm studying the topic of "discrete-time algebraic Riccati Inequality (DARE)". However, I have one question regarding the matrix inequality on ...
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### How to verify if the kalman gain matrix K is working properly?

If I have a state space model. $$x(k + 1) = Ax(k) + Bu(k)$$ $$y(k) = Cx(k) + Du(k)$$ And a kalman gain matrix $K$. Then, how do I know if the kalman gain matrix $K$ is properly designed for my state ...
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### Asymptotic stability of linear time varying system

Consider a linear time varying homogeneous system: $$\dot{x}=A(t)x$$ where $x\in\mathbb{R}^n$ and $A(t)$ is a $n\times n$ real symmetric matrix satisfying $A(t)\to -I_n$ as $t\to\infty$. Suppose $A$ ...
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### Bode Stability Criterion satisfied but not stable?

I encountered a question when using Bode stability criterion to analyze the closed-loop stability of a system. In a word, the Bode stability criterion says the system is stable but it turns out to be ...