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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2
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3answers
131 views

Weird condition for a transfer function

I am reading a paper that presents the following system, represented by a second-order transfer function: $G(s) = \frac{K\times(1+0.036s)}{(1+0.0018s)(1+as)}$, where the gain $K$ is a known ...
4
votes
1answer
245 views

Solving for specific entries in a Lyapunov Equation

Let $A$ be a $2n\times 2n$ real matrix with the following structure \begin{equation} A = \left(\begin{matrix} 0 & -I \\ K & S \end{matrix}\right) \end{equation} with all sub-matrices of size ...
0
votes
2answers
314 views

Controllability and observability of a transfer function

I have the following transfer function: sys = 1/(s+1)^10 (s+0.8)^10 There is no pole zero cancellation, but still when i convert this into state-space form using ...
1
vote
2answers
417 views

Approximated Laplace transform of a non-linear system

Assume a system with dynamics: $\dot{\omega}(t) = \alpha \omega^2(t) + \beta i(t)$, where $\dot{\omega}(t), \omega(t)$ are system's states and $i(t)$ is the system's input. I'd like to approximate ...
7
votes
4answers
3k views

Why use a Kalman filter instead of keeping a running average

I've been trying to understand Kalman filters. Here are some examples that have helped me so far: http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx ...
1
vote
1answer
69 views

Parametric uncertainty in conditional term of piecewise nonlinear dynamical system

Consider a Hammerstein nonlinear dynamical system of the form $\mathbf{\dot{x}} = \mathbf{Ax} + \mathbf{Bu}$, where the non-linearity is in the control term $\mathbf{u}$, and has a piecewise ...
2
votes
1answer
149 views

When are attracting sets invariant?

Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the ...
3
votes
1answer
69 views

Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
2
votes
1answer
325 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
0answers
155 views

Matrices made negative semidefinite, but not simultaneously

Consider matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, such that $A$ has at least $1$ strictly positive eigenvalue. Let $X_1, X_2 \in \mathbb{R}^{n \times n}$ such that ...
1
vote
1answer
305 views

Perturbation Analysis for Linear Linearly-Perturbed ODEs

I have been struggling with an ostensibly simple problem, that is how to apply perturbation analysis principles on a system of linear differential equations with linear perturbation of the following ...
-1
votes
2answers
173 views

Feedback characteristics of nonlinear dynamical systems

I am trying to understand the following article: A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco, Europhys. Lett. 81, 60001 (2008). In the beginning, it was quite easy to follow: ...
0
votes
1answer
294 views

Numerically calculating inverse Laplace via the inverse Laplace transformation formula

I'm trying to simulate a control system whose transfer function is $H(s)$. I'm comparing different numerical methods for this. I have already used these two methods: - Converting the transfer function ...
3
votes
2answers
295 views

Practical physics problem about distance between lines.

I am a software engineer and I’m developing a soccer game. I have a solution for this problem based on Newton law and I’m using Newton method to solve the equation I got. I’m here because I think ...
2
votes
1answer
721 views

Convert a linear difference equation into a controllable state-space model

I have the following linear difference equation (which is an discrete-time SISO ARX model): $y(k)+\sum_{i=1}^{n}a_iy(k-i)=\sum_{i=1}^{n}b_iu(k-i)$ and I need to transform it in an equivalent ...
2
votes
1answer
369 views

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ $x ...
1
vote
1answer
132 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 ...
1
vote
2answers
206 views

Is the integral in a PID controller definite or indefinite?

Would the integral calculated by the PID controller be considered definite or indefinite?
1
vote
1answer
218 views

Property of dynamical system and transformation

EDIT2: After some discussion here's the original problem: Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^{n\times m}, u_1 \in \mathbb{R}^{m}$ be a control system evolving on M (F is the ...
-1
votes
1answer
67 views

What happens at step 3 of this Feedback Control Loop proof [closed]

So input $$= R(s)$$ and output $$= C(s)$$ and forward transfer $$= G(s)$$ and feedback transfer $$= G_2(s)$$ deriving feedback now $$= G_2(s)C(s)$$ forward signal $$= R(s) - \text{feedback}$$$$ =R(s) ...
7
votes
2answers
556 views

Difference between Bellman and Pontryagin dynamic optimization?

Can someone please explain the difference between dynamic optimization via the Bellman equation and dynamic optimization via Pontryagin's maximization principle? Thanks
1
vote
1answer
503 views

Using the state transition matrix to recover the state matrix

I have a state transition matrix $\Phi(t,\tau) = \left(\begin{matrix} e^{-(t-\tau)} & t-\tau \\ 0 & 1+t(t-\tau) \end{matrix}\right)$. I am tasked to find the state matrix $A(t)$ that ...
-1
votes
1answer
114 views

Best method for this example to get from transfer function to state space

I have this system here: In this example the state space representation $ \frac{dx}{dt} = Ax + bu $ and the corresponding transition matrix $\Phi(t)$ is asked for. So to get the state space, I ...
0
votes
2answers
130 views

System matrix of a 2nd order state space representation

I am completely stuck on this: The 2nd order system should be in this form: $\frac{dx}{dt}=Ax$ where A is the system matrix. $$x(t) = \begin{pmatrix} 2-e^{-t} \\ 1+2e^{-t} \end{pmatrix}$$ $$x(t=0) =: ...
1
vote
1answer
448 views

For what values does this system show BIBO stability?

I got this system state representation: $$\begin{align} \frac{dx}{dt} &= ...
2
votes
1answer
119 views

Is this the correct way to get the state space representation of this system?

In this exercise the state space representation of the imaged system is asked for. $$G_1(s) = \frac{s-1}{s+2} = 1 - \frac{3}{s+2} G_2(s)=\frac{1}{s-1}$$ I can see that $G_1(s)$ is "able to leap" ...
1
vote
2answers
241 views

State transform from one state space representation to another

I have a state space representation, system S1, in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ This system is transformed with the state transform $$x=T z$$ into the system S2: $$ ...
3
votes
1answer
549 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: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
1
vote
1answer
272 views

How to obtain the state matrix of this trajectory?

Continuous-time LTI case. I have a problem getting the state matrix of this trajectory. One element of the state matrix is known. $$ A = \begin{pmatrix} a & 4 \\c & d \end{pmatrix} $$ I ...
1
vote
2answers
528 views

Decomposition of a unitary matrix via Householder matrices

If $U$ is unitary, how can I show that there exist $w_{1},w_{2},...,w_{k}\in \mathbb{C}^{n}$, $k\leq n$, and $\theta_{1},\theta_{2},...,\theta_{n}\in \mathbb{R}$ such that $U=U_{w_{1}}U_{w_{2}}\cdots ...
8
votes
3answers
1k views

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 ...
4
votes
2answers
755 views

The nature of eigenvectors of a given eigenvalue

I am working on a problem where I have an ($n \times n $) matrix A and an eigenvalue of A, $\lambda$, where $\lambda$ has geometric multiplicity 1. The right and left eigenvectors of A corresponding ...
2
votes
1answer
210 views

The yacht race problem from L.C.Evans's PDE book 2nd edition chapter 10

There is one yacht starting at $(x_1,0)$ when $t=0$, which is sailing toward positive direction of $x$-axis with a constant velocity $b_1$, another yacht is starting at $(0,x_2)$ when $t=0$, and is ...
0
votes
1answer
111 views

Poles placement problem

It is an equations of inverted pendulum. Example of controlling him with pole placement. $$ \text{eqns}=(M+m)x\text{''}[t]-m l \text{Sin}[\theta [t]] \theta '[t]^2+m l \text{Cos}[\theta ...
1
vote
0answers
605 views

Partial differentiation of vector to find Jacobian (extended Kalman filter)

I am working through some coursework on self-tuning control and part of one of the questions requires the use of the extended Kalman filter for joint parameter and state estimation. For completeness, ...
7
votes
4answers
694 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 ...
2
votes
2answers
878 views

In-depth example or implementation of adaptive control (direct/indirect MRAC)?

I have seen some examples where adaptive control is used to counter sudden changes in a system with great success. Since I find the subject quite interesting, I would like to learn how to actually ...
-2
votes
1answer
479 views

What is the input of Lyapunov function?

I am not sure whether I declare a correct type of matrix in matlab, but it can run. Is it a correct way to derive Lyapunov function? Y is a $3\times3$ matrix. According to the paper ...
1
vote
0answers
115 views

What is the difference between various kalman filters?

What is the difference between additive and multiplicative kalman filters, as well as some other kinds? I'm also looking for reference texts and articles that describe the algorithms, so ...
9
votes
2answers
476 views

Lyapunov stability question (Arnold's trivium)

V.I. Arnold put the following question in his Mathematical trivium: Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization? It puzzled me for a ...
2
votes
2answers
341 views

Transfer Function, Newtonian Cooling

I have a system that is modeled by the following differential equation: db/dt = j(t)ha + k(Ta-b(t)) where db/dt is the rate of temperature change, j(t) is an input, ha, k, Ta are all constants, and ...
2
votes
1answer
202 views

good online resources for 2nd-order system dynamics

I'm looking for good online resources for 2nd order system dynamics. Any recommendations? I'm looking for stuff that ideally includes discussion of Q, damping ratio, overshoot, bode plots, for ...