Questions tagged [control-theory]

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The desired trajectory of the output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller should manipulate the inputs to the system to obtain the desired effect on the output of the system

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"Spanning" of solutions of ordinary differential equations

Suppose we have a switched ODE $$\dot{x} = A_{\sigma(t)}x,$$ where $A_{\sigma(t)}$ is a constant matrix given $\sigma(t)\in\mathcal{M}=\{1,2,\cdots,m\}$. If we fix the initial condition and can ...
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Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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Why an LTI system with some zero eigenvalues still stable?

The textbook says an LTI system $\dot x=Ax$ is stable if and only if the eigenvalues of $A$ have the strictly negative real part. However, I found a counterexample. If $$A= \begin{bmatrix}-3 & -1 ...
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Are these equations "properly" defined differential equations?

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it Intro Recently, in this answer I figure out that the following autonomous ...
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Counterexample for a convex problem

The convex optimization problem is as follows: \begin{align} \underset{\mathbb{X},\mathbb{Y}\in\mathbb{S}_n^+}{\min}\quad &\operatorname{Tr}(X)+ \operatorname{Tr}\left(D Y \right)\nonumber\\ \...
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LQR definitions

I have to define the choice of parameters I have chosen to create an LQR controller for a drone, and I have written the following: High penalties in the Q matrix mean that the state will try to ...
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6 votes
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A direct proof of the Vandermonde decomposition of a nonsingular Hankel matrix?

I have been doggedly searching for a direct proof of the following theorem: Theorem 1: Let $H$ be a complex nonsingular $n\times n$ Hankel matrix. Then $H$ can be factorized $H = V^\top DV$ where $V$ ...
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Mathematical notation on a matrix

Matrix notation question (in reference to "simultaneous interconnection and damping assignment passivity based control"): We select an $n\times n$ matrix called $F_d(x)$. Let $G(x) := F_d(x)+...
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Stabilizing a transfer function With a PD controller

I have the following Plant function: $$ P(s) = \frac{50(s-1)}{(s-5)(s+5)} $$ And a controller $ C(s) = K_p + sK_d$ I want to Check if I can stabilize the closed loop system with this controller. My ...
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3 votes
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Condition for finite $\infty$-norm of a transfer function

I am reading "Feedback control theory" by Doyle, Francis and Tannenbaum DFT. Lemma 1 on page 16 states: the $\infty$-norm of a transfer function $G$ is finite iff $G$ is proper and has no ...
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Model is observable, but why doesn't the EKF implementation converge to correct values?

I have a vector field (unable to provide the details, unfortunately) $$\frac{dx}{dt} = f(x,u)$$ and measurements $$y = g(x,u)$$ Linearizing around any $\bar{x}$ and $\bar{u}$, I verified that the ...
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Algorithms/Solvers for Hard Constrained Non-Linear Optimization Problems - Model Predictive Control Example

I have an autonomous robotic swarm path planning/control problem where a set of "leader" robots have predefined (nontrivial) dynamics in the control set, and "follower" robots are ...
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The latest best approach to determine the order of the system model

The classical maximum likelihood estimation using Akaike's criteria is defined by $$\text{AIC}=-2\log^-\text{(maximum likelihood)} + 2 \text{(no. of independently adjusted parameters within the model)}...
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On the positive definiteness of the observability Gramian

Given the system $\dot{x} = Ax$, $y = Cx$, it is known that the Gramian, given by $$ W({t_0},{t_1}) = \int_{t_0}^{t_1} e^{A^T(\tau -t_0)}C^TC e^{A(\tau -t_0)} \,{\rm d}\tau $$ is positive definite for ...
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Feedback linearization with integral action - How?

Assume that you know sort of the dynamics of the system. It's not 100% perfect, but it's at least 90% perfect. $$\dot x = f(x, u)$$ I want to find a control law that suits this system. I have been ...
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Any geometric interpretation for the adjoint system of a linear dynamical system?

On page 26, Section 1.3, of his book on linear dynamical systems1, Professor Roger Brockett asks: If $$\dot{\mathbf{x}}(t) = A(t) x(t) , \qquad \mathbf{x}(0) = \mathbf{x}_0$$ and $$\dot{\mathbf{p}}(t)...
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A matrix $A$ is Hurwitz, $e^{Ah}$ is Schur. Discretization of continuous-time linear stochastic system and its stability.

Suppose we have a real-valued square matrix $A$ that is Hurwitz, i.e., all eigenvalues of $A$ have strictly negative real parts. I want to show that given a scalar $h>0$, if A is Hurtiwz, then $e^{...
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1 answer
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A question about Comparison Principle in Nonlinear Systems?

A question about Comparison Principle For a general system, we have $$ V=x^{2}+y^{2} $$ where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ are two independent states, and $V$ is a Lyapunov function. ...
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Can we apply fundamental lemma of calculus of variations

Let us assume that $t_0$ and $t_1$ are some constants, and $f(t)$ and $u(t)$ is a continuous mapping from $[t_0,t_1]$ to $\mathbb{R}^p$ and $\mathbb{R}^m$, respectively. Suppose that $$\int_{t_0}^{t_1}...
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Dynamical systems with control input

Please I have been trying to write the mathematical formulation of my nonlinear dynamical system for quite some time and I will appreciate any input. ** Problem Description** Assuming, I am traveling ...
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1 answer
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LQI for angular speed control in MATLAB

With the following state space system and setting for the Linear Quadratic Integrator (LQI) Q = diag([1,1,1]) and r = 1; for ...
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1 vote
1 answer
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Nyquist plot- is G(s) stable?

I got very confused with the nyquist plot. I have a basic question that would suffice: say I got a general nyquist plot of some transfer function $ G(s) $: Is $ G(s) $ stable? what information apart ...
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Confusion About Laplace transform to Z transform

Suppose we want to find the z transform of $\frac{G(s)}{s}$ where $G(s)$ is a general transfer function. Since we know that $\frac{1}{s}$ is the integrator operator in Laplace domain, then we have $\...
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proof of the existence of a bang bang control

I was reading the proof of the existence of the bang bang control throughout the theory of the L infinty space; however i got stuck at a piece of the proof. I am sure im overcomplicating the problem ...
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1 answer
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Gradient of a (Lyapunov) function

For $x\in\mathbb{R}^n$, define $\hat{r}(x) = \left\{ \begin{array}{ll} \vec{0} ,& x = 0\\ \frac{x}{||x||} ,& x \ne0\\ \end{array} \right. $ and for some $r>0$, $x_0\in\...
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1 answer
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Show uniform stability of the system

the LTV system $\dot x(t)=A(t)x(t)$ is called uniformly stable if $\exists \gamma>0$ such that $\left\| {\Phi \left( {t,{t_0}} \right)} \right\| \leqslant \gamma $ for all $t\ge t_0$ where ${\Phi \...
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0 votes
1 answer
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How do I add a steady-state offset to my transfer function

I have been trying to do some simple system identification. I have some input and output data from a system and I was trying to manually tune a transfer function that would behave in a similar manner. ...
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3 votes
1 answer
60 views

Controllability for second order coupled system

I have the following system: $\ddot{y_1}=-y_1+\alpha y_2+u_1$ $\ddot{y_2}=-y_2+\alpha y_1-2u_2$ I am trying to answer 4 questions: For what values of $\alpha$ is the system controllable For what ...
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Model Predictive Control with Linear Programming VS Quadratic Programming

Model Predrictive Control is often used with Quadratic Programming. But I have tried Model Predictive Control with Linear Programming and it works very well. Let's begin with the discrete SISO state ...
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How to carry out the expected value of the cost function in a LQG problem to tackle path tracking?

I have a system, whose state is defined by $x_t$. The transition state mapping (STP) for the systems is defined as: $$x_{t+1} = A_t x_t + B_t u_t + w_t$$ where, $x \in \mathbb{R}^{n \times 1}$, $A_t \...
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State-space initial conditions

I have a state space system represented by the following system of equations. The state vector is defined by: $$ \dot{X}\left(t\right)=\left[\begin{matrix}0&1\\a&b\\\end{matrix}\right]\left[\...
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2 votes
2 answers
116 views

Why are repeated poles at the origin regarded as unstable?

I thought you needed the poles and zeroes to be at the right hand side to make the system unstable. Therefore here is the question in two parts: Why are repeated poles unstable at the origin? What ...
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Root locus plot of coupled parameter

I try to plot the root locus plot of $G(s) = \frac{(s+1)(s+\beta)}{(s-1)(s+3)^2}$ with respect to $\beta$, but could not factor out the parameter $\beta$ which is required for the sketch. Any idea how ...
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Range of $a$ for observability.

I am trying to determine the range of $a$ for observability for the following system: $$\dot{x}(t)=\begin{pmatrix}1.8&0&3a\\ 1-a&-3a^2&19\\ -4&0&1.4\end{pmatrix}x(t)+\begin{...
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0 answers
102 views

Control of flow rate to be 'fair'?

We have a pipe transporting discrete particles from a to c with max flow capacity of $C$ particles/second. It takes $d_{min}$ seconds to get from a to c when the pipe is not fully utilized (details ...
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1 answer
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stability and fixed points of $y_{i+1} = 0.5(a_{i+1}/b_{i+1})y_{i} + 0.5(y_{i} + (b_{i+1}-a_{i+1})c)$?

I have a data set of a time series, and determined that the data fits this equation, where $y_{n}$ is the dependent variable, $a$ and $b$ are independent variables and $c$ is a constant $y_{i+1} = 0.5(...
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3 votes
1 answer
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State-space with second derivative input variable

I have a second order ODE with derivative terms $$ \alpha_1\frac{d^2i}{dt^2}+\alpha_2\frac{di}{dt}+\alpha_3i=-\beta_1V_r-\beta_2\frac{dV_r}{dt}-\beta_3\frac{d^2V_r}{dt^2} $$ I want to transform this ...
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0 answers
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Steady state error in a digital control system

Suppose we have the following open loop transfer function in a discrete system: \begin{equation} G(z) = \frac{0.1z+0.15}{(z-1)(z-0.1)} \end{equation} Under the unity negative feedback, we know that ...
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1 vote
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Regarding an equation in an iteration method

I am going through a paper on an iterative method related to discrete optimal regulator, G. Hewer, "An iterative technique for the computation of the steady state gains for the discrete optimal ...
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1 vote
1 answer
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Lyapunov function for a second order system involving trigonometric functions

I am studying the stability of the following system: \begin{aligned} \dot{x}_{1} &= -x_{1}^{2} - \sin x_{2}\\ \dot{x}_{2} &= x_{1} - \frac{\cos x_{2}}{x_{1}}\\ \end{aligned} The system ...
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1 vote
1 answer
53 views

scaling down the imarinary parts of eigenvalues of a matrix

Let A be an n-by-n complex matrix. Is there a transformation to preserve the real parts of the eigenvalues of A but scale down the imaginary parts of the eigenvalues of A? Actually , I want to have a ...
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0 votes
1 answer
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Controllability of linear systems with positive controls

I am looking at some papers dealing with the controllability of linear systems under positive controls. M. Heymann, 1975, Controllability of Linear Systems with Positive Controls: Geometric ...
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1 vote
1 answer
50 views

show that the nullspace of $P$ is invariant when applying state matrix

Assume $P$ solves the Riccati equation $${A^T}P + PA - PB{B^T}P + {C^T}C = 0$$ show that nullspace($P$) is invariant up to the application of the state matrix and show that $kernel(P) \subset C$ ...
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3 votes
1 answer
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Why does the complex phase plot look exactly like the root locus diagram?

BRIEF BACKGROUND Basically here's an interesting discovery I made (probably not an original discovery) when playing around with MATLAB the other day. I was basically just trying to make a good ...
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1 vote
1 answer
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Optimal Control-- Terminal vs Running Cost

I'm learning about optimal control, and I've got a goofy question. My professor wrote out that the discrete-time form of LQR is given as $ \text{min}\ J = \frac{1}{2} x_N^TH_Nx_N + {\sum}_{k=0}^{N-1} ...
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0 votes
1 answer
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Z transform using Convolution Integral

Let our transfer function be $G(s) = \frac{10e^{-s}}{5s+1}$. We know that for sampling period of $T = 1$, we have $G(z) = \frac{2}{z-0.8187}$ (You can verify this in MATLAB using c2d function). What I ...
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1 vote
1 answer
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Estimate $\Vert \Delta u \Vert_{2}$ for wave equation [closed]

We consider the wave equation \begin{equation}\label{1} \left\{ \begin{array}{ll} u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\ u=0, \quad u \in \partial \Omega, t>0 \\ u(0,x)=u_{0}(x), \quad ...
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0 votes
2 answers
92 views

How to pick a Lyapunov function and estimate PID gains? [closed]

I am currently trying to estimate the range of PID gains by developing a Lyapunov function for a nonlinear 6-Dof quadrotor system. The system is of the following form: $$M(q)\ddot{q}+C(q,\dot q)\dot q+...
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2 votes
0 answers
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Gramians of similar systems

The following lemma is stated without proof in the lectures notes on Model Reduction by M. Voigt (available here) Lemma 4.4 Let $[A,B,C,D]\in\Sigma_{n,m,p}$ be asymptotically stable. Let $T\in\mathbb{...
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2 votes
1 answer
61 views

Dead-beat Feedback Design

Suppose we have the following discrete time system in state space form: \begin{align*} x[k+1] = \begin{bmatrix} 0 & a_k\\ 1 & 2 \end{bmatrix} x[k] + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u[k] \...
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