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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Controllability of a system

How can I show that all solutions of $x(t)'=\pmatrix{0&-1\\ 1&0}x(t)+\pmatrix{\cos(t)\\ \sin(t)}u(t)$ are within the area $x_1sin(t)-x_2cos(t)=0$ ?
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D - Part in PID doesn't impact steady state error?

Why does the D-part in a PID controller not do any impact on the steady state error. I mean if tries to resist changes, should it not then make it stay at wanted configuration?
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3answers
29 views

transformation of a difference equation

How can I translate the difference equation $$x_{k+3}+4x_{k+2}+3x_{k+1}+x_k=2u_{k+2}$$ into a state-space representation of the following form (A and B are matrices) $$x_{k+1}=Ax_k+Bu_k$$
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is Lyapunov direct method applicable for infinite dimensional nonlinear system?

after linearize the infinite dimensional system, I have an A matrix in which each element is in terms of the dimension index k. And as k goes to infinity, A matrix has some positive eigenvalue. But if ...
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54 views

Equal number of poles and zeros for square transfer matrix

In page 10 of this document (MIT Courseware on control) it is stated that since the transfer function is square, there is an equal number of poles and zeros. Does this hold and if so under which ...
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71 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 ...
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27 views

Discrete time state-space

$G = \left[ \begin{array}{c|c} A & B \\ \hline C & D \\ \end{array} \right] = C(zI-A)^{-1}B+D$ Suppose: 1. $A,B,C,D$ are all real matrix. 2. $z = e^{j\theta}$, i.e. $r=1$ for simplicity. 3. ...
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237 views

How do I determine the transfer function of a plant?

I sitting here with a system which I have to determine the transfer function. The unit receives a velocity and position, and move towards that position with the given velocity. What kind of test ...
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38 views

BIBO stable but not stabilizable

Consider a system: $$\dot x = \begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}x+ \begin{bmatrix}0\\1 \end{bmatrix}u$$ $$y = \begin{bmatrix}1 & 1\end{bmatrix}y$$ Its transfer function is: ...
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25 views

How to check if a matrix transfer function is in Hardy-infinity space?

Just like the question says. For instance if I have a matrix transfer function $$\mathbf{G}(s) = \begin{bmatrix}s & -s \\ T & s \\ \end{bmatrix}$$ where $T$ is a positive constant, how can I ...
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70 views

Is a feedback system with an unstable component and the other component being zero internally stable?

So let's consider a system like I described, say looking like such: Where $K, P_{1}$ and $P_{2}$ are all multivariable transfer function matrices. In this case technically it could be presented as ...
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Roll PID Control for quadcopter

I'm trying to implement with simulink a PD controller for my quadcopter. I use a simplified model, and for the roll case I have $ I_x * \phi = L $, where L is the roll torque. So, the transfer ...
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35 views

A $w$ system to stabilize.

I have the following system to be stabilized: \begin{equation} \begin{aligned}\dot{w}=Aw+Bv \\& A=\left( \begin{array}{ccc} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 1 & 2 & 0 \\ ...
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25 views

Oscillations of a feedback interconnection

I have a feedback interconnection described via the following transfer function $G(s)=\frac{1}{s^3+5s^2+6s+1}$ and the nonlinearity $\psi(e)=\text{sgn}(e)$. I have used the describing function method ...
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46 views

How is an ODE a consistency condition?

I was reading a text on Optimal Control Theory by E. Todorov, when I came accross this passage (on page 10): An ODE is a consistency condition which singles out specific trajectories without ...
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2answers
124 views

What is the largest invariant set?

I think the largest invariant set is on other than $\{x:\dot{x}=0\}$, is this correct, is there other way to establish the largest invariant set? Please give an example.
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51 views

Is this system stable?

I got this control system with such dynamics: \begin{equation} \dot{x}(t)=-\frac{\partial{H(x)}}{\partial{x}},~H\geq 0,~H(x)=0\Rightarrow x=0 \end{equation} $x(t)$ is a $n$-dimension vector, ...
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1answer
44 views

How to show that the delay margin is zero if the open loop gain $|L(i\infty)| \geq 1$?

How to show that the delay margin is zero if the open loop gain $|L(i\infty)| \geq 1$ ? Where $L(s)$ is the open loop transfer function and the delay margin is the amount of time delay for the system ...
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1answer
25 views

Decibel adjustment on Bode diagram

Say we have the system $G(s) = 1/(s+1)^3$ with break frequency $\omega_b = 1$. Can someone explain to me why we should expect $|G(\omega_b)|$ to be $3$ dB below the low frequency asymptote, rather ...
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2answers
112 views

Why doesn't superposition imply linearity? Why is homogeneity needed?

If I have a function which satisfies superposition I know $f(x_1+x_2)=f(x_1)+f(x_2)$. If I had now an element making f inhomogeneous ($f(0)\neq0$) this element would occur once on the left hand side ...
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69 views

state-space reduction

I am confused about state-space reduction. I learned it in the class but am not skilled in it. If $A,B,C,D$ matrices are given with values, we can 1. find its controllability matrix to see if ...
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451 views

State space representation involving derivatives of input

We have the system $y''=-7y'-12y-u'-2u$ If we choose $x_1=y,x_2=y'$ we can write the system as $x'=Ax + Bu \\ y= Cx$ Finding A is easy, but how do I find expressions for $B$ and $C$ when we have ...
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0answers
36 views

Does this violate the notion of positive definiteness?

From a previous course in linear algebra, I was taught that a function is positive definite if it satisfies $$\left< \vec x \mid v(\vec x) \mid \vec x\right> > 0 $$ In simpler notation, ...
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141 views

Geometric interpretation of PBH test

I need to find geometric interpretation of PBH test i.e. for any space X isomorphic to R^n and U isomorphic to R^m. A is a linear operator from X to X and B is a linear operator from U to X. Prove ...
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86 views

Final value theorem for closed system

We have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, ...
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1answer
129 views

Determine bounds for BIBO stable system

Let $\dot{x} = A x + B u$, $y = x$ be a BIBO (bounded input, bounded output) stable system. Given an output bound $y_l \leq y(t) \leq y_h$, how can we determine the maximum input bound $u_l \leq u(t) ...
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1answer
59 views

Final value of 1/(( s+2 )² * (s² - s + 1)) in the time domain

The original question is given as $$\frac {d^3y}{dt^3}+y=u=(1-t)e^{-2t}$$ The initial value y(0) = 0 and the same for all derivatives of y. Determine Y(s) What happens to u(t) and y(t) when ...
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1answer
43 views

How to find the $H_2$ norm of MIMO system in matlab

$K$ is given. $A,B_1, B_2, C_1, C_2, D$ are also given. I want to find the following: $P_{11}, P_{12}, P_{21}, P_{22}$ And use the resulte in 1., to find the $H_2$ norm of the system: ...
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173 views

Question about nested conditional expectation.

Question 1: I am interested in finding out, why the following is or is not the case: $\mathbb{E}\left[ \mathbb{E} \left[( X_k|(Y_0,Y_1,...,Y_{k-1}) )|( Y_k|(Y_0,Y_1,...,Y_{k-1}) ...
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38 views

How to find a partial derivative with respect to a matrix?

Let we have a $2\times2$ matrix $A=\begin{bmatrix}a_1&a_2\\a_3&a_4\end{bmatrix}$, a $1\times2$ matrix $C$, and a $2\times1$ matrix $X$. How can we calculate derivative of $CAX$ with respect to ...
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1answer
131 views

What exactly is a random disturbance in control theory?

I'm embarrassed to even ask but... I frequently see this word used in articles about dynamical systems, but not until now have I questioned what it really is. I understand that it is opposite of a ...
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81 views

stabilizable ,detectable and regulator

Assume that $(A_2,B_2)$ is stabilizable and $(C,A)$ is detectable then there exist aregulator if the equation $TA_1-A_2T-B_2V=A_3$ $D_1+D_2T+EV=0$ have solution (T,V).if $A_1$ is antisatable the ...
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1answer
31 views

Kronecker Product and state space

I am reading a paper, and one step of it seems like the following: If $S_1 = $ Then, $I \otimes S_1$ = How to show it? (Suppose dimension of all of them are correct. $I$ is the ...
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1answer
73 views

Multiplication of transfer function

If I have the following: How do I show the following: $P_{11} = G_{11} + G_{12}\hat Y\tilde MG_{21}$ is: I am stuck in this complicated system. Or, the other simpler one: ...
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34 views

transfer matrices times a real-valued matrix in Matlab

I am a beginner in matlab. My question is: I want to do the following thing: where $A,B,C,D$ are all $3 \times 3$ matrices. So the transfer matrix should be a $3 \times 3$ matrix. $K$ is a $3 ...
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1answer
67 views

Is multiplication commutative in the laplace domain?

I'm studying control theory and saw this picture explaining some of the basic rules. My question is if we could also say that Y(s) = (G2(s) * G1(s)) * U(s) Or Y(s) = U(s) * G2(s) * G1(s) I'm ...
2
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1answer
45 views

Shortest path in the plane under derivative constraint

A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0. This ...
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1answer
23 views

Control theory: when does $G(s) = \frac{1}{P_\lambda(A)}$

In other words, under what condition is the system transfer function G(s) = Y(s)/U(s) equivalent to the reciprocal of the characteristic equation of the $A$ matrix in state space realization?
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1answer
34 views

Z-transform and$ H_2$ space

The following is from the preliminaries of a paper. Let $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ be the unit disc of complex numbers. A function $G:(\mathbb{C} \cup \{\infty\})\backslash ...
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1answer
63 views

Transfer function of differential eqaution

I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform ...
2
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1answer
215 views

Transfer function for double cart system

System: Define X2 = Y2; I've described the system with the following diff equation: $$f_{tot} = m_1\ddot{x_1} + k(x_2-x_1)+m_2\ddot{x_2}+B(\dot{x_2}-\dot{x_1})$$ where m1, m2, k and B are Cart ...
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Does the iteration $e_i^\top x_{t+1} = \max_j e_i^{\top} (\alpha A^j x_{t} + b^j)$ converge?

Given a constant $0 < \alpha < 1$, a matrix $A \in R^{n \times n}$ and a vector $b \in \mathbb{R}^n$, it is well-known that a sufficient condition for the iteration $x_{t+1} = \alpha A x_t + b$ ...
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1answer
277 views

Direct vs Indirect Learning Control

What is the difference between direct and indirect learning control? I found the following comments on direct and indirect control in this paper by Wang, Gao, and Doyle: "Survey on iterative learning ...
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42 views

Upper bound for affine differential equation

Let $\frac{dx}{dt} = a x + b$ be a stable affine differential equation where $a \in \mathbb{R}^-,b \in \mathbb{R}$ and let $c \in \mathbb{R}^-, d \in \mathbb{R}^+$. How can we determine a maximum ...
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1answer
216 views

Understanding controllability indices

I'm teaching myself linear control systems through various online materials and the book Linear Systems Theory and Design by Chen. I'm trying to understand controllability indices. Chen says that ...
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3answers
198 views

If $\|u\|^2 = c^2$, then $\|\dot x\|^2 \to a $ for $\ddot x = -\dot x + u(t)$?

I am dealing with the next simple equation $$ \ddot x = -\dot x + u(t), $$ where $u, x\in\mathbb{R}^m$, with $m \geq 1$, and I am wondering if for $\|u\|^2 = c^2 > 0$ then $\|\dot x\|^2\to a$, ...
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1answer
203 views

What does linearization do in MATLAB's PID tuner?

I noticed that the PID tuner from MATLAB has a linearizatin step before tuning. What does this linearization step do? And why we have to linearize a model in PID tuning?
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2answers
223 views

Convolution of unit impulse with impulse response

I have a question that's been bothering me. If we can convolve any arbitrary input with a system's impulse response to get the system's total response for that input, then if we convolve the impulse ...
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1answer
70 views

Controllability of a pair of matrices

If the pair $(A,B)$ is controllable, then is the pair $(A^{2},(A+I)B)$ controllable? The question becomes more interesting if there exists $(A,B)$ is uncontrollable, but the pair $(A^{2},(A+I)B)$ ...
3
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Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...