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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19 views

A convoluted transfer function (from prof. Dullerud's robust control testbook)

The following proble is from the book: A Course in Robust Control Theory (a convex approach), middle of p. 200 Consider the following general feedback loop: , ie $\dot x(t) = Ax(t) + ...
2
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1answer
31 views

Relative Gain Array of a singular matrix

I am a masters student in controls and would like to get insight into the concept of relative gain array for multivariable feedback control. In general what I have come across from the book on the ...
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2answers
68 views

Physical interpretation of transfer function in control theory

I'm learning about transfer functions in control theory. I'm struggling to find a physical interpretation for the input and output of a transfer function, both of which may be complex numbers. In the ...
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0answers
25 views

How to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions?

I want to know how to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions. Please help me. Thanks very much.
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1answer
47 views

How to prove for a system with rational stable transfer function, the output is square integrable?

I want to know for a system with rational stable transfer function, i.e. H(jw)=1/[(a1+jw)(a2+jw)...(an+jw)] (a1,a2,..,an>0), why a square integrable (L2 integrable) input must generate a square ...
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16 views

Reducing uncertainty of a mathematical model with data (Process control)

I know this is a very broad question, but need suggestions, link to good reference papers etc. So here is the question: I have an uncertain model whose parameters are static (not changing with time) ...
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15 views

Transfer function from state variable expression

I have a 3x3 state variable system. I need to choose where to place my poles according to some criteria. For example: (a) Percent overshoot < 20% (b) SettlingTime < 1.5s, and (c) steady-state ...
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1answer
22 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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2answers
40 views

Why algebraic Lyapunov equation has an unique solution?

In the following text book (p.47): Optimal Control (Lewis 2nd edition) There is a theorem: (Zero input case) If $A$ is stable, and $(A,\sqrt Q )$ is observable, then $S_\infty= ...
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0answers
36 views

Control theory- basic question on stabilizabilty

Studying some control theory but having difficulty learning because my lecturer doesn't provide solutions to any of his exercises AT ALL. Below I've attached a problem I've just done and my answers ...
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53 views

Stability of zero-order hold controllers for linear systems

From what I've read, Lyapunov functions provide a very nice mechanism for verifying stability of continuous-time linear systems (non-linear as well, but that's not my concern at the moment). For ...
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1answer
56 views

Control theory with state and input constraints

What are some control theory tools for solving problems of the following form?: Given a system model, control input constraints $I$, and control output constraints $O$, what is the largest set ...
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2answers
703 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 ...
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1answer
29 views

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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17 views

At what frequency should input and read values

I am at the moment trying to identify a system using frequency sweep. I have using matemathica created a frequency sweep as such. ...
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2answers
42 views

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
28 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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30 views

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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33 views

solutions that don't converge to equilibriums

suppose a nonlinear autonomous system has more than two asymptotically stable equilibrium, how do I find a solution that doesn't converge to any of these equilibrium?
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1answer
41 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 ...
2
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0answers
31 views

Equal number of poles and zeros for square transfer matrix

In page 10. of the docuement below (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 ...
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1answer
27 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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0answers
16 views

Changing direction of Nyquist plot with PID-controller

I am trying to understand how nyquist-plots react to different combination of PID-controllers. In some of the problems I'm working with the nyquist-plot is in the wrong direction, clockwise, I want it ...
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1answer
23 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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1answer
29 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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1answer
98 views

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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19 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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1answer
19 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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1answer
30 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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0answers
21 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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1answer
24 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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58 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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1answer
43 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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1answer
39 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 ...
3
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1answer
54 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
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1answer
36 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, ...
3
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1answer
56 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 ...
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1answer
33 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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1answer
34 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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1answer
16 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
43 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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1answer
34 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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109 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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35 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, ...
2
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1answer
69 views

If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of $\frac{dz}{dt}=[A(t)+B(t)]z$ are bounded

In the book by Richard Bellman, Stability theory of differential equations Theorem 6 (p.43) If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of ...
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1answer
42 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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1answer
29 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
60 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
39 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 ...
2
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1answer
92 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 ...