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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Product of Observability and Controllability matrices

I noticed in Linear Control Theory, we may multiply matrices for controllability $\mathcal{C} \in \mathbb{R}^{n \times np}$ and observability $\mathcal{O} \in \mathbb{R}^{nq \times n}$ as follows $\...
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Is the delay tracking a ramp for a unit gain stable causal LTI system equal to the group delay calculated at frequency (or pulsation) zero?

I am trying to get my head around the following problem: I have a unit-gain stable LTI system (we can assume it is a discrete-time one) of which I need to calculate the delay in tracking a ramp signal ...
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In control systems, What inputs will make system state unbounded while system output bounded?

Suppose I have a discrete LTI state space system: $$x(t+1)=Ax(t)+Bu(t),$$ $$y(t)=Cx(t)+Du(t),$$ What i am courious is that under what conditions the input $u(t)$ satisfies would make $lim_{t\...
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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 ...
Matt Morrison's user avatar
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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#...
Matt Frank's user avatar
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Linearized model of a nonlinear dynamic system

I have following dynamic system $$ \frac{\mathrm{d}v_C}{\mathrm{d}t} = -\frac{1}{R_b\cdot C}\cdot v_C\cdot\alpha, $$ where $v_C$ is the system state and output, $\alpha$ is the system input and $R_b, ...
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How to place a zero of the PI controller?

Let's say I have a plant with following transfer function $$ G(s) = \frac{\frac{L_M\cdot R_R}{L_R}}{s + \frac{R_R}{L_R}} = \frac{0.0129}{s + 1.935} $$ and I would like to design a PI controller for it ...
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The Lyapunov inequality for a given matrix $P$

The famous Lyapunov theory says if a system matrix $A$ is stable, then the Lyapunov inequality $$A^TP+PA<0, \qquad P>0$$ is unique which depends on the negative definite matrix $-Q$, which I ...
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Globally exponentially stable point

consider this linear, non-autonomic system: $x_1 ̇=-x_1-f(t)(x_2-x_3 )$, $\ x_2 ̇= -x_2+x_1$, $x_3 ̇=-x_3-x_1$ where $f(t)$ is continuously differentiable and satisfies $0≤f'(t)≤f(t)≤k$ for all $0≤t ...
Alon Vain's user avatar
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On sufficient conditions to determine observability when the number of outputs $p$ is the same as the number of states $n$

A system of linear differential equations $$ \dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t)\\ \mathbf{y}(t) = \mathbf{C}\mathbf{x}(t), $$ where $\mathbf{x} \in \mathbb{R}^{n\times 1}$, $\mathbf{y}\in\...
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How to manually find discrete-time LQR gains using Algebraic Riccati Equation / Hamiltonian?

For a continuous-time optimal feedback controller, I'm manually computing LQR gains using the Algebraic Riccati Equation, using the Hamiltonian method. This seems to works fine, as I compare to gains ...
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Are there formal definitions of "state" and "state variable" in the context of state space models in control theory?

I'm taking a class on control theory and I thought I understood the state space representation of linear systems -- it seemed like essentially just extra syntax (or, "syntactic sugar" as ...
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Block Diagram Reduction Error?

I am reducing a block diagram manually and am not sure whether the following step is a valid step? I am using the rule of moving a summing ahead of a block in reverse. Reduction Step in Question I've ...
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How to pronounce $<A| Im B>$?

I'm studying linear control theory (by "Control theory for linear systems" by Trentelman et al.) and there appears subspaces $$ \langle A \vert im \, B\rangle = im \,B + A \,im\, B + \ \...
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State-space realization of transfer function — controllable canonical form

Given the following transfer function $$ G(s) = \frac{100(s-2)}{(s-3)(s+4)}$$ find matrices $A, B, C, D$ that realize the system. I know that: should be the solution, but I am trying to derive it ...
Edward Josef's user avatar
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When is a solution of a differential matrix Riccati equation positive definite?

The question is in the context of linear control systems. Let $A(t)$, $R(t)$, and $Q(t)$ be time-varing square $n\times n$ matrices of reals. For all $t$, $R(t)\ge 0$ and $Q(t)\ge 0$ are semi ...
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Same input for both Reachability and Controllability

For LTI systems, both Reachability and Controllability are equivalent. Now, this may be a stupid question, but does the same input works for both the cases? I explain myself better : for example, ...
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Equivalent Condition of Asympotic Stability of Positive Linear System

I am reading about Positive Linear System in the book Positive Linear Systems Theory and Applications by LORENZO FARINA & SERGIO RINALDI. On chapter 5, page 40, I encounter Theorem 13, which the ...
Hoan Nguyễn's user avatar
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Is this dynamical system coupled?

A LTI system is described by $$\dot{x}=Ax+Bu$$ If $A$ is of the form: $$A=\begin{pmatrix} A_1(x_1,…,x_j)\\ A_2(x_{j+1},…,x_n) \end{pmatrix}$$ The meaning of this structure of $A$ is that only the ...
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Why does MPC controller performance decrease when control horizon increases?

I am currently doing an academic study to study the effect of various tuning parameters on the MPC controller performance. I have a model of a chemical reaction systems (van de Vusse reaction) in a ...
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Expecting but not getting steady state error proportional motor controller

I am designing a proportional controller for a DC motor. I am expecting to get an error of $$\dfrac{K_p}{1+K_p}$$ but the step response of my closed loop system has practically no steady state error (<...
Ton's user avatar
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Stability and controllability of a system (check my answer)

Let´s study some control properties of the following system with $a,b,d,c > 0$ : $$ \pmatrix{x\\y\\z\\v}'=\pmatrix{-a&a&0&0\\0&0&1&0\\d&-d&0&0\\c&0&0&...
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Condition for 3*3 block matrix to be stable

Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times ...
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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 ...
dodo's user avatar
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2 answers
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How can I approach the target temperature as fast as possible given the transfer function?

I have an electric heater and close to it a temperature sensor. And I could use some help with controlling the heater. The amount of power supplied to the heater can be changed at any moment. Let $0 \...
Paul's user avatar
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Safe set term in control theroy

The general process in the discrete-time linear time-invariant (DLTI) direct feedthrough state-space model form is considered: \begin{equation} \begin{cases} \boldsymbol{x}(k+1) = \...
ConT's user avatar
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Is it possible to convert a linear, time-variant (LTV) dynamical system into a time invariant system (LTI)?

Lets consider a dynamical system, whose state transition matrix is defined by $A(t)$, where size of A is $2$ x $2$ matrix. Can't we change $A(t)$ into $\hat{A}$, where $\hat{A}$ is a $3$x$3$ constant ...
Afaq Ahmad's user avatar
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How can I determine the closed loop transfer function matrix algebraically for this block diagram?

I am checking some algebra from this paper James Anderson, John C. Doyle, Steven Low, Nikolai Matni, System Level Synthesis, 2019. Specifically, I want to understand how transfer functions from $(\...
js9's user avatar
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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 ...
Steve's user avatar
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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 ...
Steve's user avatar
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1 answer
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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 ...
learn design's user avatar
-1 votes
2 answers
90 views

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 $P$ ...
Jeremy's user avatar
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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 $$\dot{x}(t) = Ax(t) + Bu(t) \hspace{1cm} x(0) =0 $$ ...
Jason's user avatar
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1 answer
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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 ...
Hwang's user avatar
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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 $A$ ...
William's user avatar
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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 ...
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