Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input 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 ...

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If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$

If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show ...
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2answers
148 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
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2answers
201 views

eigenvalues of the sum of a diagonal matrix and a skew-symmetric matrix

Suppose $A$ is a skew-symmetric matrix (i.e., $A+A^{\top}=0$) and $D$ is a diagonal matrix. Under what conditions, $A+D$ is a Hurwitz stable matrix?
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1answer
281 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
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0answers
120 views

(Still open) Convergence of a certain matrix product representing a class of piecewise linear dynamical systems.

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline ...
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6answers
3k views

Control / Feedback Theory

I am more interested in the engineering perspective of this topic, but I realize that fundamentally this is a very interesting mathematical topic as well. Also, at an introductory level they would be ...
9
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2answers
168 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
3
votes
1answer
81 views

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ...
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2answers
121 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
3
votes
1answer
131 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
3
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1answer
635 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
3
votes
1answer
736 views

Existence of global solution of Riccati equation

Consider a Riccati differential equation $$ \dot P + A(t)^{T}P + PA(t) -PB(t)R(t)B(t)^{T}P + Q(t) = 0,\;\;\; P(t_0) = P_0 = P_0^{T} \geqslant 0 $$ where $Q(t) = Q(t)^{T} \geqslant 0$, $R(t) = ...
2
votes
2answers
65 views

How to drive a vehicle (limited by acceleration) on a flat ground to a given point as fast as possible?

So I have a function $\mathbf{x}(t): \mathbb{R} \rightarrow \mathbb{R}^2$, which is supposed to mean the path of the vehicle (time mapped to position). The initial conditions $\mathbf{x}(0)$ and ...
2
votes
1answer
498 views

Inverse of State-space representation

Ask two questions from a paper (2012 ACC): Consider the plant: Let X be the stabilizing solution of the Riccati equation: where . Define the LQR gain by . The transfer matrix has a ...
1
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1answer
57 views

Hamiltonian question, find optimal controller, simple question

An object is moving with accordance to Newton's laws: $\begin{pmatrix} \dot{y} \\ \dot {v} \end{pmatrix} = \begin{pmatrix} v \\ u \end{pmatrix}$ where $y$ is the objects location and $v$ is its speed. ...
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1answer
162 views

What is the purpose of the transfer function in PID?

In a SciLab project wherein they build a PID controller they include a CLR/continuous transfer function between the output of the PID and the multiplexer, which was used to combine the step and the ...
0
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1answer
91 views

The inverse of a state space matrix

First of all, I would like to link this question to another one about the inverse of a state space representation: Inverse of State-space representation I understand the prove as given on the ...
3
votes
2answers
342 views

significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
3
votes
3answers
97 views

showing any controllable system can be put in 'controller' form.

I am looking at the proof of the following theorem: Let $\dot{x} =Ax + Bu$ be a controllable single input system, where $\Delta_A:= \det(\lambda I -A) = \lambda^n + a_1\lambda^{n-1} + \ldots + ...
3
votes
1answer
208 views

Optimal control

Consider the growth equation: $ \dot{x} = tu $, with $x(0)=0$ and $x(1)=1$, and with the cost function: $ J= \int_0^1 u^2 dt $. Show that $u^*=3t$ is a successful control, with $x^*=t^3$ and $J^*=3$ ...
3
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1answer
960 views

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
2
votes
2answers
161 views

How does the singularity of a system matrix affect the system's stability?

What can be said about system stability, given a singular system matrix below? \begin{align} A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
2
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1answer
102 views

Derivation of a formula for discrete time case

Given is the following discrete system $$\begin{align*} &x(k + 1) = Ax(k) + Bu(k)\\ &x(0) = x_0\;. \end{align*}$$ How do we prove that the explicit solution formula for $x(k)$ (analogously ...
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1answer
48 views

$X= \int_0^S e^{(S-s)A} B u(s) ds \Rightarrow X= \int_0^T e^{(T-s)A} B \bar{u}(s) ds$

Consider the ODE system $$X'(t) = AX(t)+Bu(t)$$ where $X(t) \in R^n, \; A \in R^{n \times n} \text{ and } B \in R^{n \times m}$. In control theory, we define the set of states reachable as $$A(0,T) = ...
1
vote
1answer
106 views

What is the Laplace transform transfer function of affine expression $\dot x = bu + c$?

For the one dimensional case, with $a, b, c$ being real constants, $u$ being the system input, $x$ the state, what is the Laplace transfer function of: $$\dot x = bu + c$$ Ideally I'm looking for ...
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vote
2answers
66 views

Problem with understanding theorem on Riccati Equation.

`The matrices $A,B,C,D,X$ are real, square, $n \times n$. I have trouble understanding theorem 7.1.2 from Lancaster & Rodman "Algebraic Riccati Equations". The part that I understand is as ...
1
vote
2answers
379 views

Model Predictive Control

I have a few confusions about Model Predictive Control (MPC). Since they are all minor questions related to the same category, I ask them under one topic. In an article, the cost function is defined ...
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1answer
247 views

Quadratic approximation of a cost function with a Taylor expansion

See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92. Given an instantaneous cost ...
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1answer
125 views

Multiplication of state space transfer function (state-space form)

If I know the following transfer function (ss-form) How to obtain the following efficiently: Thanks
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2answers
204 views

Find the control function

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...