Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 ...

0
votes
0answers
11 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 ...
0
votes
0answers
10 views

Binary controller transfer function

I have a simple controller that turns off when my variable is over the setpoint and on when it's below. Is there any transfer function for modelling this?
0
votes
1answer
21 views

How to find a transformation matrix which will make the system a chain of integrators?

Consider a system of the form $$\dot{x}(t)=Ax(t)+Bu(t)+\phi(t)+D(t)$$ I have $$\dot{x}(t)=\begin{bmatrix} -p_1 &G_b & 0 & 0 &0 \\ 0& -p_2 & p_3 & 0 & 0\\ 0& ...
0
votes
1answer
39 views

An inequality with constraints.

I came across a result in a control theory book (without proof), which states that: Given two variables $x,z \in \mathbb{R}$ and four parameters $c_{1}, c_{2}, k_{1}, k_{2}$ with $c_{1}, c_{2} > 0$...
0
votes
0answers
24 views

Proof of preservation of non-collinearity of a triangular formation under a distance-based control law

I am studying a 3-agent distance-based formation control problem. The position of each agent is denoted $\pmb{p}_1$, $\pmb{p}_2$, $\pmb{p}_3 \in \mathbb{R}^2$. The desired distance between $i$ and $j$ ...
0
votes
1answer
28 views

How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis? \begin{equation} \left.\begin{aligned} \dot{x_1} &= x_2 \\ \dot{x_2} &= -\frac{x_1}{1 + x_2^2} ...
3
votes
2answers
59 views

Stability of a Degenerate Equilibrium Point in a Planar ODE

Consider the planar ODE $\dot x_1 = x_2$ $\dot x_2 = - x_1^2 - 2 x_1 - 1$ Obliviously, $(x_1,x_2)=(-1,0)$ is an equilibrium point. The Jacobian matrix at this point is $$J = \begin{bmatrix} 0 &...
0
votes
0answers
16 views

Controllability of multiple Input system (proof)

Prove the following statement for multiple Input system: If $(A,B)$ is controllable then for any $x_0 \neq 0$ there exists a sequence $(u_0, u_1, \cdots, u_{n-1})$ such that span $\{x_0, \cdots, ...
2
votes
1answer
41 views

Existence of Periodic Orbit

Consider the planar system $\dot x_1 = x_2 - x_1^3$ $\dot x_2 = -x_1$ Prove that there exists no periodic orbit in this system. I tried to use the Bendixson criteria. The divergence is equal to $-...
0
votes
0answers
18 views

Different equations of motion for same system?

So I’ve derived the equation of motion for an inverted pendulum on a cart (X is the position of the cart) using the Euler-Lagrange equations and measuring the angle theta anti-clockwise from the ...
2
votes
1answer
16 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}...
0
votes
1answer
25 views

Dynamic order of transfer function matrix.

Consider the transfer function matrix $G(s)$ of a continuous-time system given by: $G(s) = \begin{bmatrix}\frac{1}{s^2+2s}&\frac{s+1}{s} \\ -\frac{1}{s+1} & \frac{1}{s^2+4s+3} \end{bmatrix}$ ...
-2
votes
0answers
30 views

Non-linear controller for 2nd order closed-loop system

Consider the following non-linear system: $$2y'' + 3yy' - 2(y')^2 + 1 =\tau $$ Design a non-linear controller such that the closed-loop system is always critically damped with $k_{CL} = 10$. How can ...
1
vote
1answer
26 views

State space model with a constant disturbance term

I have a question related with the problem shown in the title. I got a state space model of my system through calculations and it turns out to have a form as follows $\dot x(t)=Ax(t)+Br(t)+c$ ...
1
vote
1answer
52 views

How would I compute this matrix in Matlab?

I'm trying to compute the H$_\infty$ norm of a matrix using the paper: Bruinsma, N. A.; Steinbuch, M., A fast algorithm to compute the $H{\infty}$-norm of a transfer function matrix, Syst. Control ...
0
votes
1answer
27 views

Function bounded at large values of t?

using laplace I've calculated the output of my system as: $$-\frac1{104} \exp(-5t) + \frac18\exp(-t) +\frac1{26}(2\sin(t)-3\cos(t))$$ I've been told that this is bounded from above and below at ...
2
votes
1answer
115 views

strongly connected $L$, then what are the eigenvalues of $L+L^T$?

I asked a similar question before. Now things become a bit different here. Suppose $L$ is a non-symmetric Laplacian matrix, of which the corresponding graph is strongly connected. Is it true that $L+...
0
votes
1answer
26 views

PID controller for 2 DOF system [closed]

I have system : 2 dof system Speed of the second mass wanted to be controlled by a PID. The transfer function of the system is : s/( s^4 - 1.89e-17 s^3 + 2 s^2 - 1.408e-16 s - 3.263e-32) I could ...
3
votes
2answers
56 views

How to pick a Lyapunov function and prove stability?

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for autonomous systems. Say we are given the nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_1(t)x_2(t)...
1
vote
1answer
38 views

How to find stability of a third order non-linear system

Suppose we have a third order system, reduced to three first orders in the form $\dot x_1 = x_2 \\ \dot x_2 = x_1 + x_3F(x_1) \\ \dot x_3 = x_3F(x_1)$ Suppose we know $F(0) = 0$ How do we find the ...
2
votes
1answer
16 views

Relationship between transfer functions in a transfer matrix.

I'm taking a MIMO control theory course and this is my first exposure to control theory as an academic topic. Well for this course, I think I understand the concept of finding state space realizations ...
0
votes
1answer
44 views

Linear Control Theory and the set where the control lives

In linear control theory, we consider a system of the form $$ \dot x = A\,x(t) + B\, u(t) \qquad \star $$ where $x(t)$ is the state function, $u(t)$ is the control function and $A,B$ are matrices. ...
1
vote
1answer
37 views

Show that the internal dynamics is unstable

Let's consider the following nonlinear system, $$\dot{x}_1=x_2^3+u$$ $$\dot{x}_2=-u$$ $$y=x_1$$ Taking the derivative of $y$, $$\dot{y}=\dot{x}_1=x_2^3+u \qquad \text{Equation 1}$$ Let $f_1(x) = ...
0
votes
1answer
26 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 ...
1
vote
1answer
63 views

How to deal with non-equilibrium operating point

Given the nonlinear system $$ \begin{align} \dot{x}_1 &= -4x_1 + 10x_2 + u \\ \dot{x}_2 &= -x_1 - 2x_2 - \log(1 + x_1^2) \\ y &= x_1 + x_2 \end{align} $$ Assume the system should be ...
0
votes
0answers
16 views

Feedback loop, control throttle to land on target

I'm working on a coding challenge in which I have to control a car to go to different spots on the map as quickly as possible. The commands I can give to the car at each turn is a target position <...
0
votes
1answer
21 views

State-space representation

I have 2 differential equations. I need to find state-space representation of the equations. Input should be $\theta(t)$ and outputs should be $x(t)$ and $y(t)$. $M,m,D_{x},D_{y},K_{x}$ and $K_{y}$ ...
0
votes
1answer
27 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 ...
1
vote
1answer
47 views

radially unbounded functions

Is the following function radially unbounded or not? $$V(x) = \frac{x_{1}^2}{1 + x_{1}^2} + x_{2}^2$$ I know that if $x_{2} \to \infty$ in which case $||x|| \to \infty$ and $V(x) \to \infty$ but if $...
0
votes
1answer
18 views

Using PI control to eliminate steady state errors

In a negative feedback loop i understand the mistake of canceling unstable poles. But take for example a plant $G(s)=\frac{1}{s+1}$ and an I-control $F(s)= \frac{1}{s}$ Then the system has the ...
0
votes
1answer
27 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 $h_t$ ...
1
vote
1answer
22 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 $m \times d$, $B$ ...
0
votes
0answers
20 views

How to design a Robust observer for a 2D system

Consider the second order system given by $\dot{x}=Ax+Bw(t)$, where $x\in\mathbb{R}^2$, $$A = \begin{bmatrix} {0},{6}\\ {-1} {-6} \end{bmatrix}, \quad B = \begin{bmatrix} {0}\\{1}\end{bmatrix}...
1
vote
0answers
27 views

Adaptive Pole Placement and Canonical forms

I have a linear system in the observable canonical form as \begin{align} \dot{x} &= \underbrace{\begin{pmatrix} -a_2 & 1&0\\ -a_1 & 0&1\\ -a_0 & 0&0 \end{pmatrix}...
0
votes
0answers
33 views

Dynamic programming's principle of optimality as an abstract construct

In dynamic programming, the principle of optimality (refer to Bertsekas's Optimal Control, volume 1, page 18) is a statement that says: For any optimal policy $\pi$ , we always have a suboptimal ...
2
votes
0answers
64 views

On the solution of a matrix optimization problem

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues $\{\lambda_i\}_{i=1}^n$ such that $\mathrm{Re}(\lambda_i)<0$. Moreover, let $\succeq$ denote the standard partial order in the cone of ...
1
vote
1answer
29 views

If A(t) is a continuously-differentiable n×n matrix function that is invertible at each t, show that

If A(t) is a continuously-differentiable $n \times n$ matrix function that is invertible at each $t$, show that: $$ \frac{d}{dt}A^{-1}(t) = -A^{-1}(t)\,\dot{A}(t)\,A^{-1}(t) $$
1
vote
1answer
18 views

Finding alternate transformation matrix for similarity transformation

A pair of square matrices $X$ and $Y$ are called similar if there exists a nonsingular matrix $T$ such that $T^{-1}XT=Y$ holds. It is known that the transformation matrix $T$ is not unique for given $...
0
votes
0answers
26 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 ...
0
votes
0answers
14 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 $T_2 = 2$. ...
2
votes
1answer
102 views

State-space representation of a nonlinear MIMO system

Question: Obtain a state-space representation of nonlinear multiple-input multiple-output (MIMO) system: $$\dddot{y}_1 + 2\dot{y_1} + 3y_2 + 2 = u_1 y_2 \tag{1}$$ $$\ddot{y}_2 - 2 \dot{y}_2 + \dot{y}...
0
votes
0answers
19 views

What is the difference between Direct Shooting and Direct Collocation?

I'm started learning control theory as I'm reviewing papers in human motion dynamics modeling that uses control theory. There are two methods that are covered namely Direct Shooting and Direct ...
0
votes
1answer
45 views

Solving a matrix differential equation using Runge-Kutta methods

I could not find a control systems forum on stack exchange and so I am doing this here. Is it possible to solve the state space variable form of a system $\dot{x}=A\,x + B\,u$ using any order of Runge-...
0
votes
0answers
12 views

X-Bar and R charts after removing data

I have a question regarding the creation of X-Bar and R charts after removing data that plots out of control (above or below the control limits). Given a dataset of n grades and k subgroups, we can ...
0
votes
1answer
38 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) \...
1
vote
1answer
54 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 ...
0
votes
1answer
49 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 = \...
4
votes
1answer
95 views

What's an example where Lyapunov fails to find the bounds of stability

In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum ...
3
votes
1answer
59 views

Existence of Hamiltonian for Planar Ordinary Differential Equations

Consider the following planar ODE $$ \begin{cases} \dot x = f(x,y) \\ \dot y = g(x,y) \end{cases} $$ and suppose $ \frac{\partial f }{\partial x} + \frac{\partial g }{\partial y} = 0$. Is this a ...
1
vote
0answers
20 views

Price of a stochastic game between an agent and the market

In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as \begin{align} V(x,C) = \dfrac{1}{\...