Questions tagged [control-theory]
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The desired trajectory of the output 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 should manipulate the inputs to the system to obtain the desired effect on the output of the system
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Linearized model of a nonlinear dynamic system
I have following dynamic system
$$
\frac{\mathrm{d}v_C}{\mathrm{d}t} = -\frac{1}{R_b\cdot C}\cdot v_C\cdot\alpha,
$$
where $v_C$ is the system state and output, $\alpha$ is the system input and $R_b, ...
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vote
1
answer
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Rank of the Lyapunov equation
Consider the Lyapunov equation $A^TP+PA=Q$. I know the matrix $P$ has the integral form $$P=\int_{0}^{\infty} \exp(A^Tt) Q \exp(At) dt$$ if the eigenvalues of matrix $A$ are negative. I want to know ...
- 313
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Applicability of Robust Optimization Techniques
I have a control problem in which I want to minimize a quadratic function subject to the uncertain equality constraints
$$
(\mathcal{A} + \Delta\mathcal{A})z_k = 0, \ k = 0,\ldots,N-1,
$$
where $N\in\...
- 267
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1
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45
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Nyquist plot for arbitrary path
Suppose we have the following transfer function:
$G(s) = K \frac{s+3}{s(s+1)}$
Given the above Nyquist path
I want to sketch Nyquist plot. By using the Nyquist plot rules, I made the following sketch:
...
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30
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How to design PI controller for a non-linear dynamic system?
I have following dynamic system
$$
\frac{\mathrm{d}v_C}{\mathrm{d}t} = -\frac{1}{R_b\cdot C}\cdot v_C\cdot\alpha,
$$
where $v_C$ is the system state and output, $\alpha$ is the system input and $R_b, ...
- 303
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votes
1
answer
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Are there any concrete application of the Lyapunov theorem for LTI systems?
Consider a LTI system $\dot x = Ax$.
This system is globally asymptotical stable iff given any $Q \succ 0$, there exists a unique $P \succ 0$ such that $A^{T}P+PA+Q=0$ holds.
https://en.wikipedia.org/...
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Can an equilibrium be unstable and asymptotically stable at the same time?
Consider the autonomous dynamical system,
$$\dot x = f(x)$$
An equilibrium $p$, ($f(p) = 0$), is unstable in the sense of Lyapunov means:
$\exists \epsilon > 0$ such that $\forall \delta > 0$, ...
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$Im(F) \subset Im (G) \iff \exists \gamma >0 : \| F^{\prime}z \|_{V\prime} < \gamma \|G^{\prime}z \|_{W\prime} \; \forall z \in Z^{\prime}$
I found this lemma which it used for the principal theory without proof
I try hard to proof this lemma or searches it in functional analysis books but i dont found it.
The lemma is :
Let Z ,W and Z ...
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Steady state speed and steady state error.
I got this first order block diagram below.
Below is the transfer function for the motor model.
I tried to get an equation for the steady state speed and steady state error, but my answer doesn't ...
- 11
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1
answer
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Sufficient conditions for stabilizability of [A,B]
Let $\mathbf{A}$ and $\mathbf{B}$ be $n\times n$ and $m\times n$ matrices, respectively, the pair $(\mathbf{A},\mathbf{B})$ is stabilizable if there exists an $n\times m$ matrix $\mathbf{K}$ such that ...
- 67
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Determine stable set via simultaneous stability
Consider a system of n nonlinear ODE's $\dot{x} = f(x)$, where $x \in \mathbb{R}^{n}$, with only a few equilibrium points scattered around (The one in my mind right now is the Kuramoto model ) Given ...
- 1,056
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Lyapunov Control vs Sliding Mode Control?
I think I have a good understanding of the idea behind each, i.e., Lyapunov control drives a system to an equilibrium and attempts to keep it there while sliding mode control drives a system to a ...
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What does the laplace transform do really?
From the solution method of second degree differential equations we know that a system can be modelled as $ce^{(\lambda + \omega i)t}$. The complex constant contains information about the decay rate (...
- 1,116
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PI Controller Design
Suppose we have a negative unity feedback system with a plant transfer function $G(s) = \frac{1}{s(s+2)}$. I want to design a PI controller for this system in order to meet some time domain ...
- 391
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2
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Stability of discrete-time dynamical systems using Lyapunov stability
I am studying the use of LMIs as an analysis tool for discrete-time dynamical systems. Consider the autonomous discrete-time system given by
$$
x_{_{k+1}} = A x_{_k} \tag{1} \label{sys}
$$
where $ x \...
- 147
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50
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Turning a constrained optimal control problem into unconstrained (Lagrangian)
If I wish to minimize the cost function
$$
J(x(\cdot),u(\cdot)) = \int_0^TL(x,u)dt
$$
with dynamics constraint $\dot{x}(t) = f(x(t),u(t))$ $\forall t$, many textbooks state that this constrained ...
- 21
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vote
1
answer
82
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Linearize and then substitute vs. Substitute and then linearize
If I have some nonlinear function $f(x,u)$, I can linearize it as
$$
f(x,u) \approx f(x_{ss},u_{ss})+\left.\frac{\partial f(x,u)}{\partial x}\right|_{x_{ss},u_{ss}}(x - x_{ss}) + \left.\frac{\partial ...
- 21
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0
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How large the error ball of $\|x\| $is, when using the Lyapunov function $x^\top Px$?
I would like to know how large the error ball of $\|x\|$ is when using the Lyapunov function $x^\top Px$:
Assumption: I have an almost linear closed-loop system $\dot{x}=(A-BK)x+\epsilon(x)$ with ...
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1
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Find a Lyapunov function and prove that an almost linear closed-loop system is stable
I would like to find a Lyapunov function and prove the following closed-loop system is stable:
$\dot{x}=(A-BK)x+(z(u)+\epsilon)$,
where a function of control input $z(u)$ satisfies $\|z\| < \rho \|...
- 141
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votes
1
answer
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How is the damping equation obtained?
I modified some online notes in the internet and prepared this illustration for later use.
Why does the formula fail?
If $B=0$ (it is a sinusoidal system with no exponential components, undamped ...
- 1,116
1
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1
answer
90
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Finding a single equilibrium point of the Kuramoto model
The Kuramoto model is used to describe a large variety of synchronization phenomena. It describes the time evolution of a group of n oscillators with $\omega_{i} \in\mathbb{R}, \frac{K}{n} > 0$:
$$
...
- 1,056
2
votes
2
answers
121
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How to place a zero of the PI controller?
Let's say I have a plant with following transfer function
$$
G(s) = \frac{\frac{L_M\cdot R_R}{L_R}}{s + \frac{R_R}{L_R}} = \frac{0.0129}{s + 1.935}
$$
and I would like to design a PI controller for it ...
- 303
1
vote
1
answer
90
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Feedback and system linearity
The following notes have a nice discussion of how feedback can make the response of a nonlinear static system more linear, i.e. reduce nonlinear distortion (at the expense of gain).
https://web....
- 125
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0
answers
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Transforming determining $\exists x \in \mathbb{R}^m, A(x) \succ 0$ into least squares possible?
Consider a linear operator $A: \mathbb{R}^{m} \to S^{n \times n}$, where $S^{n\times n}$ are the symmetric n by matrix.
Can we turn the problem of determining if there exists $x \in \mathbb{R}^{m}$ s....
- 1,056
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votes
1
answer
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Real-Valued Error Function on SO(3)
In some geometric control papers, the author usually defines the real-valued error function to be:
$\Psi(R,R_d)$ = $\frac{1}{2} Trace[I - R_d^T R ]$. (1)
where $R_d$ is the arbitrary smooth attitude ...
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1
answer
73
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Pole-zero cancelation method for PI controller design
I have a simple LTI system with following transfer function
$$
G(s) = \frac{K}{s + p} = \frac{0.0128647}{s + 1.935}
$$
and I would like to design a PI controller for this system i.e.
I have been ...
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votes
1
answer
80
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State-space transformation of differential equation with a single input that includes delay and non-delay
I have a second-order vector differential equation
$$\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K}\mathbf{x}(t)=\mathbf{P}\mathbf{u}(t)$$
in which the input vector is of the ...
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votes
1
answer
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On sufficient conditions to determine observability when the number of outputs $p$ is the same as the number of states $n$
A system of linear differential equations
$$
\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t)\\
\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t),
$$
where $\mathbf{x} \in \mathbb{R}^{n\times 1}$, $\mathbf{y}\in\...
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How to model the update-delay of observable variable in a model?
INFO: I have asked the same question last week at CS (here), but there is not even a single comment, so I am posting it here in hope someone might help with literature or approach.
I have a problem ...
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votes
1
answer
25
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Show that sector bounded function has unique global minimum
Fix constants $0 < m < L$. We say the function $f \in C^1(\mathbb{R}^n;\mathbb{R})$ is bounded in the sector $[m, L]$ if there exists a reference point $y^* \in \mathbb{R}^n$ such that for all $...
- 4,734
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53
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How to manually find discrete-time LQR gains using Algebraic Riccati Equation / Hamiltonian?
For a continuous-time optimal feedback controller, I'm manually computing LQR gains using the Algebraic Riccati Equation, using the Hamiltonian method.
This seems to works fine, as I compare to gains ...
- 25
2
votes
1
answer
76
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Does the flow identify the vector field?
Setup
Let $d\in\mathbb N$. Let $f\colon\mathbb R^d\to\mathbb R^d$ be $L$-Lipschitz continuous and bounded by $C$ for some fixed constants $L,C\in(0,\infty)$, i.e., $\Vert f(x)\Vert \leq C$ and $\Vert ...
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1
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Is state space representation useful for nonlinear control systems?
I understand that the state space representation is mathematically equivalent to the transfer function representation for linear systems, and that it allows us to solve the corresponding DE by finding ...
0
votes
1
answer
57
views
convex optimization of an inequality
The motivation for this question is a relaxation of the well-known Riccati equation that will be introduced as a constraint in a convex optimization.
The variable is $P\succeq0$, and the constraint is
...
- 574
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vote
0
answers
40
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Differential inclusion for piecewise finite-time lyapunov function
For each segment of a piecewise Lyapunov function that exhibits asymptotic stability, we can utilize the LaSalle-Yoshizawa theorem and solve it using a differential inclusion. This allows us to merge ...
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votes
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Locating variables in a transfer function based on block diagram
I am given the following diagram:
Now, I am being asked to find the $p, a, b, k$ variables respectively with given steady-error function $e(t) = \dfrac{1}{2}(e^{-2t}+e^{-8t})$ with $U(s) = \dfrac{1}{...
- 287
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Existence and uniqueness of the solution of a control system
Let $T>0$, $(U,d)$ be a metric space and $\mathcal{V}:=\{u:[0,T]\to U\,|\, u\text{ is measurable}\}$. Consider the control system $$\begin{cases}\dot{x}(t)=f(t,x(t),u(t)),\quad \text{a.e. }t\in[0,T]...
- 843
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1
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96
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Finding the maximum output with step function based on a transfer function
I have an open and closed loop transfer functions as below:
$$
\begin{align*}
L(s) & = \dfrac{k}{s^2 + as + b} \\
G(s) & = \dfrac{k}{s^2 + as + (b + k)}
\end{align*}
$$
Now, I want ...
- 287
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votes
0
answers
28
views
Is the control given by Pontryagin's Minimum Principle suboptimal?
I'm studying nonlinear control systems and I have some questions about the nature of this optimal control given by PMP:
Traditionally, control methods based on Pontryagin's Minimum Principle (PMP) ...
- 148
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vote
1
answer
80
views
Are there formal definitions of "state" and "state variable" in the context of state space models in control theory?
I'm taking a class on control theory and I thought I understood the state space representation of linear systems -- it seemed like essentially just extra syntax (or, "syntactic sugar" as ...
1
vote
1
answer
52
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Locating variables based on block diagram and its transfer function
I have a diagram that looks like below:
I would like to find the variables $a$ and $b$ when the steady-state error becomes $0$ with unit acceleration as input $u(t) = \dfrac{t^2}{2}$.
I have found ...
- 287
1
vote
1
answer
72
views
Finding the inverse laplace transformation
There is a transfer function as below:
$$
\begin{equation}
G(s) = \dfrac{4s}{s^2 + 1}
\end{equation}
$$
Now I would like to find its output inverse laplace transform with input as below:
$$
...
- 287
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votes
0
answers
40
views
Relation between the convergence of a differentiable function and the convergence of its derivative
Is there any relation between the convergence of a differentiable function and the convergence of its derivative?
I understand that whether $f'$ converges to zero does not tell us anything about the ...
- 33
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vote
0
answers
37
views
Draw vector locus and find its stability boundary
I have a block diagram that looks like this:
Now, I am being asked about the following questions:
Find the open-loop transfer function $L(s)$
Find the closed-loop transfer function $G(s)$ from $U(s)$...
- 287
5
votes
1
answer
86
views
Finding the initial value based on a transfer function
There is a system with angular of rotation as output, and its Laplace transform is:
$$
\Theta (s) = \dfrac{4s + 12}{2s^2 + 5s + 1}
$$
Now, I would like to find the following:
Find the initial value ...
- 287
0
votes
1
answer
103
views
Finding the equation of motion based on the diagram and its transfer function
I have a diagram that looks like below:
Description: By applying force p(t), point A is moving with displacement $y = asin𝜔t$. $k_1$ and $k_2$ stand for the coefficients of the corresponding spring, ...
- 287
1
vote
1
answer
71
views
Finding transfer function based on Nyquist Diagram
I have a Nyquist Diagram that looks like this:
The angular frequency at point A is $\sqrt 2$ rad/s. Now I would like to derive the transfer function based on the diagram. Now, I am given the choices ...
- 287
0
votes
1
answer
59
views
Finding Laplace Transform from diagram
I have a diagram that looks like below:
First, I have attempted to express the diagram function regarding f(t) as below:
\begin{equation}
f(t) =
\begin{cases}
t & \text{0 < t ≦ 1}\\...
- 287
1
vote
0
answers
48
views
Entire Complex Function for dynamical systems
I am working on my master Thesis and I was going through this Thesis : https://scholarship.rice.edu/bitstream/handle/1911/89289/RICE0327.pdf?sequence=1&isAllowed=y
Here, the author computes the ...
- 75
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vote
0
answers
45
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Is the Z transform defined on continuous or discrete functions?
I'm a bit confused because I was previously under the impression, from a course on signals and systems, that the Z transform was the discretized version of the Laplace transform. However, I just read ...