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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22 views

Describing function of a non linearity with memory

Can anyone help me on finding the correct methodology to compute the describing function of the following NL function? ...
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2answers
79 views

Riccati & Lyapunov equations

Hope to ask: Lyapunov Eq: $A'P + PA + Q = 0$ Algebraic Riccati Eq: $A'P + PA + Q + PB*inv(R)*B'P= 0$ It seems that the difference between the two lies in $B = 0$ (zero input) in Lyapunov Eq and both ...
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34 views

Constructing error state kalman filter

I am trying to construct an error state kalman filter for GPS/INS integration using simulated data and I am having problem on a few steps. My error state vector is $\delta x = [\delta\alpha \, ...
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1answer
56 views

How does the z transform work in practice?

What I've found I've implemented a PID controller using the equations 7 and 9 of this article, which states that: $$\frac{U(s)}{E(s)}=K_p+\frac{K_i}{s}+K_ds$$ Translates to ...
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1answer
32 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
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31 views

Prospect of research in some stochastic optimization/approximation field

This question is a not a technical one. Sorry for that. As I am new to the area of stochastic optimization/control, I want to know the active prospect of research in the following areas 1) ...
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25 views

Extension of Schur-Cohn for quadratic matrix equation

Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$: $z^2 + bz+c=0$ and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq ...
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40 views

How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
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41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
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17 views

What about the HUM for a finite-dimensional system?

We consider the finite-dimensional system \begin{equation}\begin{cases}y'(t)=Ay(t)+Bv,\ t\in (0,T)\\ y(0)=y^0\end{cases}\end{equation} Where $v$ is the control, $A\in Mat(N\times N); B\in Mat(N\times ...
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35 views

Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb ...
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55 views

how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$
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1answer
430 views

deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
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16 views

Controlling a random variable

I've got a system the output of which is a random variable with a certain distribution, which for the purpose of this discussion can be assumed to be normal. The input variable is voltage. The ...
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43 views

designing a controller for an unstable plant?

How do would anyone make a controller for a system which is $G(s) = \frac{(s-2)}{(s-1)(s-6)}$ I do not see how this system can ever become stable, without using pole/zero cancelation. So how ...
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51 views

Bode plot of an unstable system?

I am a bit confused on how to sketch a bode plot for an unstable system? (being a/all pole(s) lies on RHP). I tried plotting it in matlab, but it doesn't resemble the output i was expecting using ...
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24 views

Design feedback control law to make the whole matrix Hurwitz

Suppose $(A_1, B_1)$ and $(A_2, B_2)$ are both stabilizable. Then we know that we can find some $K_1$ and $K_2$ to make $A_1+B_1K_1$ and $A_2+B_2K_2$ Hurwitz, respectively. Now, for non-zero constant ...
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25 views

Control stability problem

Design controllers $u_1$ depending on $x$ and $u_2$ depending on $y$ such that the following system is exponentially stable: $$\dot x = A_1 x + B_1 u_1 + C_1 y \\ \dot y = A_2 y + B_2 u_2 + C_2 x ...
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1answer
24 views

write all functions as rational functions

my question is short and simple : could we write all functions that have the formula : $1 + G(s)H(s)$ as $$\frac{(s+b_0)(s+b_1)\dots(s+b_n)}{(s+p_0)\dots(s+p_m)}$$ if the answer is yes , could you ...
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1answer
45 views

it it possible to solve these equation for their root.

I am trying to solve an equations such as the roots of $$k*x(11*x + 1) + d*x(11x + 1)$$ has to match the roots of this function $$x^2 + 0.1x + 6 + k*x(11*x + 1) + d*x(11x + 1)$$, where I have to ...
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1answer
31 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
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9 views

A confusion on adaptive algorithm

Consider flight trajectory control problem i.e. find out the control parameter for which the average error of the actual output and desired output is minimized. Can we call any algorithm for solving ...
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107 views

Using overshoot and settling time formula to determine pole location?

Is it possible to use the formula for overshoot and settling to determine where where ones pole should. by using the overshoot and settling time formula i mean, using it to define what $\zeta$ and ...
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44 views

From position to velocity?

I have an transfer function which tells what the angular displacement of an DC motor. This transfer function is in the S-domain, and normally when you differentiate (*s) the angular position you would ...
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22 views

Adding poles and zeroes due to PID

I've been wondering for a long time if a system is adding a pole or a zero to the close loop system, when a PID controller is added to the control system???.. I tried google it everywhere and i ...
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87 views

System stability for adding a pole and Zero?

How come does a system become more stable when a zero is added to a system.. I mean i doesn't not change the location of the pole, it is still the same? An example: Looking a closed loop system ...
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17 views

A doubt on markov decision process

Given that a policy is a function from a state action pair to probabilities, the set of policies for a MDP forms a POSET (the partial order is due to value function for a policy). Why there should be ...
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323 views

Effect of adding a Pole and Zero due to PID

I am kinda confused on how Adding a D(which adds a zero to the complete system) decreases the speed of the system. But when we normally add a zero to the system, it normally causes the system ...
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1answer
90 views

Integration in PID controller

I am trying to understand how come there is a phase difference is from the error signal and the output of my PID controller which consisting of I = 1. As far i've understood should the integration ...
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1answer
35 views

BIBO stable system

I would like to ask if this system is Bounded Input Bounded Output stable : $$y[n] = r^nx[n],\quad r\in \mathbb{R}$$ And why? I think this system is stable because $$| x[n] | ≤ B,\quad B < ...
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2answers
88 views

How to use mathematically the I and D of a PID controller

I am trying to mathmatically understand how the $P$, $I$, and $D$ parameters work on a system, quite having a hard time doing so. I've only been able to show that the Steady State Error (SSE) never ...
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1answer
75 views

No repeated eigenvalues or the real part of any eigenvalue is not zero

I have an $n$ x $n$ matrix $M=\begin{bmatrix}-1 & -1\\ \frac{1}{2} & 0 & -\frac{1}{2}\\ & \ddots & \ddots & \ddots\\ & & \frac{1}{2} & 0 & -\frac{1}{2}\\ ...
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1answer
87 views

Difference between Variation of Calculus problems and Control Theory problems?

Variation of Calculus seems to have problems without the control with variables such as state and time. Then again Control Theory problems seems to have problems with one extra variable that is ...
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40 views

Reachable set using constant control input

Consider a linear time-invariant control system given by the differential equation \begin{align*} \dot{x}(t) &= Ax(t) + Bu(t), \;\; x(0) = x_0, \end{align*} where $x\in \mathbb{R}^n$ and $A,B$ ...
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63 views

Design control law

Consider the function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ which is continuously differentiable and strongly convex. Let $x^*$ to be the unique global minimizer of $f$. Assume that $L_1\|x\| ...
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3answers
81 views

Rank of the matrix product $C e^{At} B$

Let $A \in \mathbb R^{n \times n}$. Fix $m<n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. What I am interested in is the matrix ...
2
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1answer
74 views

How to prove these relations

I have following three basic recurrence relations $$\mathcal U_{k+1}=A(I+\mathcal U_{k}Q)^{-1}\mathcal U_{k}A^{\mathrm T}+G\\ \mathcal V_{k+1}=\mathcal V_{k}(I+Q\mathcal U_{k})^{-1}A^{\mathrm T}\\ ...
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1answer
41 views

Full rank of the matrix $Ce^{At}B$

Let $A \in \mathbb R^{n \times n}$. Fix $m <n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. I want to find algebraic conditions (which ...
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1answer
51 views

laplace transform of a sine function

I'm a little confused about how to find Laplace transforms of a sine function when it is a function of time. As in, suppose the function is $x(t)=\sin(at)$ , then I can proceed to get ...
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2answers
88 views

Rewriting differential equation into state space

I have some problems rewriting the following differential equation into state space form. I know the general principle of how it is done, but I'm getting confused of how the states are being defined, ...
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1answer
105 views

Why does the Final Value Theorem not hold for a transfer function with more than one pole at the origin?

The Wikipedia article on the Final Value Theorem states the following for cases where it does not hold: There are two checks performed in Control theory which confirm valid results for the Final ...
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35 views

Nyquist diagram of transfer function

Transfer function of a system is given as $$G(s) = \frac{100(s+5)}{s^2(s+3)(s^2+4)}$$ Sketch the Nyquist diagram and find if the system is stable. Also find the gain margin and phase margin. Please ...
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1answer
39 views

Is this a discrete time Lyapunov function?

I have an algorithm to optimize a process. It is a discrete time algorithm. Every iteration of this algorithm changes the state of the process. I found a function, say $f(s)$, where $s$ is the state ...
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42 views

Simple Stochastic Control Problem

Consider $dX_t = \pi_t X_t dt + \pi_t X_t dW_t, X_0 = x$, where $W_t$ is a standard brownian motion, and $\pi$ is some real valued process. Let T>0. How can we calculate $P[X_T\geq 2x]$, where ...
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1answer
67 views

How to plot $\dot{x}= Ax + Bu$ (x versus t, by matlab)

I am junior in control. If $\dot{x} = Ax$ where $A$ is a $n\times n$ matrix and $x$ & $\dot{x}$ are $n\times 1$ vectors, by $x = \exp(At)$, we can draw $x$ versus $t$. If $\dot{x} = Ax + Bu$, ...
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1answer
46 views

Determine the transient response

How do i Determine the transient response of an transfer function if it's in the s domain?? the obvious answer would be using inverse laplace transform, but how come?? consider i have system like ...
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1answer
45 views

Damped systems - deriving equation

I am having some troubles deriving the formula for the roots for different types systems.. I am not quite sure if they are correct (pretty sure they aren't). $y(s) = \frac{s+2\zeta\omega_n}{s^2 + ...
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1answer
44 views

How are the tracking error and plant input related in a PID controller?

I am having trouble understanding a basic relationship in control theory - how the output of the controller is interpreted by the plant. Most control theory tutorials and introductions include a block ...
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1answer
26 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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1answer
34 views

The range of the controllability matrix

Consider a matrix $M$, the range of $M$, denoted by $R(M)$: $R(M) = \{b | b = Mx\}.$ Now, consider the controllability matrix $$C = \begin{bmatrix}B&AB & \dots& A^{n-1}B\end{bmatrix}=\\= ...