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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Calculate the rank of a matrix expression
Consider the matrices $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$ with dimensions $n$ by $n, n$ by $m$, and $p$ by $n$, respectively. Also, assume that $\mathrm{N}=\mathrm{n}$. These matrices are such ...
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Asymptotic stability by comparison with another system
Consider a nonlinear system of the form
$$ \dot{x} = f(x)x, $$
with $x \in \mathbb{R}^n$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$. I know that there exists an asymptotically stable ...
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Control a non-affine system
I have a question about the control of a non-affine system.
Here is my system
$\dot{x} = a(u) + b(u) . u$
\begin{equation}\label{a_beta}
a(u) = 0.22 \left(
\frac{116(u^3 + 1)-4.06 \lambda }{(\lambda +...
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How to compute equilibrium of a discrete time system?
I know that a continuous time system $\dot{x}= f(x) $ has an equilibrium at $x_0$ s.t $f(x_0)=0.$
But I cannot translate the same idea in discrete-time systems since there we are given the system as a ...
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Uniqueness of optimal control in infinite horizon LQR
Consider the following discrete finite horizon LQR problem with dynamics
$$
x_{t+1} = Ax_t + B u_t$$
and cost matrices $Q$ and $R$. The goal is to find a linear control $K$ which optimizes
\begin{...
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Show that if $(A, B)$ is controllable then the D.T described by $x(k+1) = Ax(k-1) + Bu(k)$ with initial states $x(0), x(1)$ is also controllable
I show this statement on some lecture notes :
It is obvious that if $(A, B)$ is controllable then the D.T with initial conditions $x(0), x(1)$ described by
$$ x(k+1)= Ax(k-1) + Bu(k)$$ is also ...
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About a property of solutions to the delay equation $\frac{dx}{dt} = Ax(t) + F(t,\,x_t)$ in a Banach space $E$
I'm dealing with the equation \begin{equation*} \tag{1} \frac{dx}{dt} = Ax(t) + F(t, \, x_t),\end{equation*} in which $A$ is the generator of a $C_0$-semigroup $(T(t))_{t \, \geqslant \, 0}$ on a ...
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Discrete time linear quadratic regulator - uniqueness of feedback gain given Riccati solution
Suppose we have a controllable discrete time linear system
\begin{align*}
x_{t+1} = Ax_t + Bu_{t}
\end{align*}
In order to design a stabilizing LQR controller with respect to supply rate $l(x,u)=x^{\...
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What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?
I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.''
Here's an attempt at a ...
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Relative degree of a transfer function
Consider the (scalar) transfer function
$$T(s)=\frac{p_ms^m+p_{m-1}s^{m-1}+\cdots+p_1s+p_0}{s^n+q_{n-1}s^{n-1}+\cdots+q_1s+q_0},$$
with $p_m\neq0$ and $m<n.$ The coefficients $p_i$ and $q_i$ are ...
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Mathematical interpretation and Fuzzy logic interpretation of d err/dt or change of error
Thank you for the possibility to ask a question.
I am new at this forum.
Currently I am scratching the surface of fuzzy logic with the idea to go deeper an deeper.
From calculus, I understand that a ...
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Approximation of z-transform transfer function
It is common in control theory to approximate a transfer function neglecting the high order terms, in example,
a transfer function with two poles:
$$
P=\frac{1}{(as+1)(bs+1)}=\frac{1}{abs^2+(a+b)s+1}...
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How to find an exact solution for $X=X^T \in \mathbb{R}^{n \times n}$ satisfying $AX=XA^T$ and $B=XC^T$
Assuming that I know that the following pair of equations has an exact solution:
$$\exists X=X^T \in \mathbb{R}^{n \times n}: AX=XA^T,\ B=XC^T$$
For some matrices $A \in \mathbb{R}^{n \times n}$, $B \...
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Space state representation from given differetial equation
I have the ODE:
$\ddot{\alpha}+4\dot{\alpha}+3\alpha=2u-\dot{u}.$
Define $y=[\alpha\ \ \ \dot{\alpha}]^T$. Find $A,B,C,D$ such that:
$$\dot{x}=Ax+Bu$$
$$y=Cx+Du$$
When $y=\alpha$, I get:
$$A=
\begin{...
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Observable and unobservable subspace
In control theory, the set of unobservable states is defined as $\operatorname{Ker}(W)$, where $W$ is the observability Gramian. Thus, the set of unobservable states is also a subspace.
Is it correct ...
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