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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Stability of non-homogeneous and non-autonomous first-order difference equation

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by \begin{eqnarray*} x_i(t+1) = ...
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23 views

Pontryagin's Maximum Principle as a sufficient condition?

It is know that Pontryagin's maximum principle provides in general only a necessary condition in the following sense: The ODE system which is known to be solved by the optimal control may have ...
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1answer
36 views

Controller design for an exponential plant of the form $y=a\exp(bx)$

I have a stationary model for a plant(a valve) given by $y=a \exp(bx)$. I linearised this by taking log on both sides: $$\ln(y) = b\cdot x + \ln(a).$$ Then, I estimated the plant transfer ...
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25 views

How to Compute Arrival and Departure Angles in Root Locus for break-in and break-away points?

I have difficulty to find the angles in the point from which poles go away from real-axis. The transfer function L(s) of open-loop is: $$ L(s) = \frac{\mu_r(s+10)}{s^2(s+100)} $$ where $C(s) = ...
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1answer
33 views

Discrete PID controller Laplace formula

I saw the following formula: the transfer function is: $$Gr(s) = K_p \bigg(1 + \frac{1}{T_n s}+ \frac{T_v s}{1 + T_d s}\bigg) $$ From my understanding: $K_p$ is the proportional gain $T_n$ is the ...
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14 views

How can I measure the dispersion of a discrete time system?

I have a discrete time system: x[n+1]=A*x[n]+B[n], with B[n] as an excitation signal. How can I measure the dispersion of this system? I guess this is related to some property of A, which depicts the ...
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1answer
26 views

Simplification of $H(s)=(4+2/s)(3/(1+2s))$

I had the following problem: We combine a parallel PI-control system ($H_{1}(s)=P+\frac{I}{s}$ with $P=4, I=2$) with a 1st order process ($H_{2}(s)=\frac{3}{1+\tau s}$ with $K=3$, $\tau=2$) This ...
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1answer
31 views

A question about Deadbeat Observers based on Deadbeat State feedback

In digital control systems we can place both observer and system in zero to have a deadbeat performance but there is a Ambiguity in the ratio of performance: We need observer to be faster than ...
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24 views

About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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1answer
36 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
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0answers
38 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
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1answer
18 views

Derivation for state equation linearization

In the following notes, how to linearize a state equation is described. The part I don't understand is why you can just remove the $\delta$ like that. I think the state equation should be: ...
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2answers
34 views

Non-minimum phase systems

I wanted to clear this doubt I have since a long time and for which I am not able to find a clear answer since different sources say differently or ambiguously. $\textbf{Does a system have to be ...
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32 views

Solution to a state-space equation

In my notes on non-linear linearization there is the following example. It asks to verify the solution to the state-space equation. My understanding is that the solution is where the equilibrium point ...
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3answers
57 views

In control theory, why do we linearize around the equilibrium for a nonlinear system?

For example, in these notes: In the first example with the pendulum, they define the equilibrium as where the pendulum is at the vertical position (x=0), with a angular velocity of 0 (x'=0) and the ...
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1answer
48 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: ...
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2answers
76 views

matrix inequality proof [completion of squares]

Can someone help me to prove this? $\begin{bmatrix} 0 & B^\top W^\top \\ WB & 0 \end{bmatrix} \leq \begin{bmatrix} B^\top Q B & 0 \\0 & W^\top Q^{-1}W \end{bmatrix}$ with $Q$ ...
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1answer
44 views

From multivariable system transfer function matrix to state space representation

I have the transfer function matrix $H(s) = \begin{bmatrix} {1\over s+1} & {2\over s+2} \\ {-2\over s^2+3s+2} & {2s\over s+1} \\ \end{bmatrix}$ And I want to ...
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15 views

Hints with predictive control problem

I'm studying a course in MPC and found a problem which is giving me a hard time. I kind of understand how the programming stuff works, but problems that involve just theoretical concepts are a little ...
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32 views

Given dynamic system $\dot x = Ax + Bu$, how can we prove that $M$ = {$(A,B)$: system is controllable} is an open set on an Euclidean space?

I wish to show that $M$ = {$(A,B)$: system is controllable} is an open set in some Euclidean space. Equivalently, how can we show that the complement to this set is closed? Here Controllability ...
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2answers
36 views

controllability of a system (control theory)

For a system in state-space representation: $\dot{x}(t) = Ax(t) + Bu(t)$; $y(t) = Cx(t) + Du(t)$, we say that a system is controllable if for $\gamma = [B \quad AB \quad \cdots \quad A^{n-1}B]$, ...
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2answers
45 views

Nyquist Stability Criterion

The rational function $b(s) = (s+3)(s-4)^{-1}$ is the frequency response function (FRF) of a system $B$. Is $B$ stable? I understand that a system is unstable if there are poles in the closed right ...
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15 views

Linear quadratic regulator with output tracking

I have the following linear system: $$ x_n = Ax_{n-1} + Bu_n $$ $$ y_n = C^Tx_n $$ And I want to find the control $u_1, ..., u_N$ such a: $$ \min_u J$$ $$ J = u_1^2 + ... + u_N^2 + \sum_{n = 1}^{N} ...
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1answer
36 views

Feedforward control - Developing an understanding

In general, for a feedforward controller design of a motion system, it is essential to recognize the various components (acceleration feedforward, viscous friction and dry friction) present in the ...
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23 views

how can I determine if a MIMO system is ''similar'' to another one?

I'm talking about closed loop regime. I have a complex model and a simplified one, I want to show that they behave similarly in closed loop. Both of them are MIMO. for SISO I could use the ...
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13 views

What are the deduction of $\sin\left(\omega_0t+\theta\right)$ for Laplace form?

I have the following function in time domain $\sin\left(\omega_0t+\theta\right)$, which is $\left(\frac{s\sin\left(\psi_1\right)+\omega_0\sin\left(\psi_1\right)}{s^2+\omega_0^2}\right)$ in Laplace ...
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25 views

Frequency response of unstable systems

Frequency response theorem's corollary ensures that the theorem can be applied to unstable systems as well, as long as you find a proper initial condition x(0). Is that right? Now, if I've an ...
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1answer
229 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$ \dot x = (-A+M)x, $$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
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22 views

Discrete sinusoidal to state space

I'm looking to apply an optimal LQR filter to a discrete signal of the form $x[n]=Asin(ω_0n+ϕ)+v[n]$ The amplitude $A$ and the phase $ϕ$ are unknown variables I want to estimate using the filter, ...
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1answer
23 views

Synchronization of Rossler system - the Rossler Attractor

I am studying synchronization of Rossler system given by the following set of two linear ODEs and one nonlinear ODE: $\dot{x_1} = -x_2 - x_3$ $\dot{x_2} = x_1 + ax_2$ $\dot{x_3} = c + x_3(x_1 - b)$ ...
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1answer
51 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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26 views

Example of BIBO stable system that is not internally stable

In the theory of system, we know that a system can be BIBO stable but not internally stable (if there is a pole-zero cancellation in the transfer function for example). I find this concept quite ...
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31 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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19 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) ...
2
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1answer
54 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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0answers
19 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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2answers
49 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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43 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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73 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
66 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
171 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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1answer
22 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) + ...
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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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1answer
31 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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2answers
55 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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39 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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1answer
45 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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1answer
33 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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23 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 ...