Questions tagged [adaptive-control]
This tag is for questions relating to Adaptive control. The adaptive controller is to be designed so that the plant output follows the model output as closely as possible. It is the capability of the system to modify its own operation to achieve the best possible mode of operation. The area of adaptive systems has been one of the most active in identification and control theory of the past decade.
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Control a non-affine system
I have a question about the control of a non-affine system.
Here is my system
$\dot{x} = a(u) + b(u) . u$
\begin{equation}\label{a_beta}
a(u) = 0.22 \left(
\frac{116(u^3 + 1)-4.06 \lambda }{(\lambda +...
1
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0
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62
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Model Reference Adaptive Control for Linear Algebraic Plants
This is a homework problem from my adaptive control course:
Given the plant $y_p = a_pu(t)$ ($a_p\neq 0$) and the reference model $y_m = a_mr(t)$, where $r(t)$ is bounded and continuous. Design a ...
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How to design/add a new controller to a system without breaking the existing controller in the system? [closed]
Please help me to find related topics/books for this problem:
For example, assume we have a water heater, and a tank of water. We can design a controller to heat the water in the tank and keep it in ...
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0
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61
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Derivation of solution for simple control problem
While trying to understand the fundamental concepts in control theory reading the following article Dual Control
for Approximate Bayesian Reinforcement Learning
(chapter 3.1, "A toy problem")...
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0
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Issue with definition of adaptive control and its classification
I have a question about the definition of adaptive control, since I´m researching about making a model-free adaptive control system. I will appreciate your help.
The definition I found says that an ...
0
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1
answer
250
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Proof of the Lyapunov Matrix Equation
Assuming that $A^TP+PA = -Q$ holds, I want to prove that $P = e^{A^Tt} P e^{A^Tt} + \int_{0}^{t} e^{A^T\tau} Q e^{A^T\tau}$ is a solution. After doing the substitutions, I end up with:
$A^TP+PA = A^T (...
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0
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Adaptive control
I have a question regarding the derivation of the adaptive law. Why do we derive the adaptive law-based parameter estimation algorithm in continuous time? Can we derive it in discrete time?
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Adaptive step size for nonlinear static problem
Let's assume $F$ is an external load for a nonlinear static finite element problem. Normally, the problem will not converge if you apply $F$ fully. Instead, we multiply the load $F$ with a scaling ...
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1
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A question about Comparison Principle in Nonlinear Systems?
A question about Comparison Principle
For a general system, we have
$$
V=x^{2}+y^{2}
$$
where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ are two independent states, and $V$ is a Lyapunov function. ...
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The product of the two positive definite matrices
I met a problem in Lyapunov stability proof:
$\dot{V}\leq -c\delta^T\bigg[\mathbb{T}\bigg((\mathbb{L}+G)\otimes(BR^{-1}B^T)\bigg)\bigg]\delta $
where $\mathbb{T}$ is a symmetric and positive definite ...
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1
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PID Controller, Integral term doesn't revert to 0 (or close to 0) once Error is 0 (or close to 0)
I'm having a hard time figuring out the Integral portion of the PID Controller.
The below pic is my simulation. The Setpoint is pTarget and the Input is ...
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2
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1
answer
189
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Control that stabilizes an uknown unstable equilibrium point?
Give a non-linear ( if it helps, multi-linear ) system for the variable with $\mathbf{Z} = [\mathbf X_1, \ldots, \mathbf{X_n} ]^T$:
$$ \dot{\mathbf{Z}} = F ( \mathbf{Z}, \mathbf{u} ) $$
and an unknown ...
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1
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How can I do Model Reference Adaptive Control for MIMO systems?
This is MRAC - Model Reference Adaptive Control for SISO systems.
$G_m(s)$ is our reference model. It's is a first order system because they don't have overshoot. $G_m(s)$ is a desired wish how then ...
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0
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Nonlinear system with time-optimal control
Given nonlinear system:
\begin{cases} \dot{x_1}=x_3+u \\ \dot{x_2}=-x_2+\dot{f} \\ \dot{x_3}=-x_3+x_2 \cdot \alpha \sin(\omega t) \\ \dot{x_4}=-x_4+x_2 \cdot (\frac{16}{\alpha^2}(\sin(\omega t)-\frac{...
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Finite-time criterion for ODE
In article Finite-Time Stability of Continuous Autonomous Systems i found this [page 4].
That's what I don't understand:
Can (2.7) $\dot{y}(t)=-k \cdot {\rm sign}(y(t)) \cdot \lvert y(t) \rvert^{\...
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0
answers
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Which way of solving from nonlinear control to choose?
I have a nonlinear system:
\begin{cases} x'=f(x)+u \\ y=f(x) \end{cases}
where $f(x)$ - gradient of some one-extremal function (for example $f=e^{-(x)^2}$), i.e. $\frac{df}{dx}$.
Task:
I want ...
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1
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Changing the quality of the transient process in a nonlinear system (Part II)
My question is a continuation of the topic.
Changing the quality of the transient process in a nonlinear system (in Mathematica)
Unfortunately, last time I didn’t get any help, so I decided to ...
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1
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Changing the quality of the transient process in a nonlinear system (in Mathematica)
I urgently need advice and help.
I have a system of differential equations like this:
$\begin{cases} \frac{dx}{dt} == y[t] \cdot \alpha \cdot sin(\omega t) + \frac{d}{dt}(\alpha \cdot sin(\omega t))
...
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0
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Synthesis gradient observer
I ask the advice of specialists on control systems. We have the following system.
Where $u(t)$ and $y(t)$ time-varying input and output. The characteristic $y(t)=f(u(t))$ is assumed to be non-...
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1
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Gramien operator of controllability!
please I have confusion if someone can help me:
We define the Grammian operator as follows:
\begin{equation}Q_{T}:=L_{T} L_{T}^{*}=\int_{0}^{T} S(T-s) B B^{*} S^{*}(T-s) d s, \quad T>0\end{equation}...
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1
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How am I suppose to estimate the next state vector if the model have internal integration? - Kalman filter
Assume that we have a state space model with no integration (no poles at 1)
$$x(k+1) = Ax(k) + Bu(k)\\y(k) = Cx(k)$$
And we know our kalman gain matrix $K$. To compute the next state $\hat x(k+1)$, ...
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0
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104
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Lyapunov Candidate function to derive parameter estimation law
I have a system and a reference model represented in state space in the following form:
\begin{gather}
\dot{x} = Ax+Bu \\\
u = -Kx+k_rr \\\
K,k_r : constants - controller \ gains \\
A_m = A-BK \\\
\...
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0
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Adaptive Control Norm inequality proof
This is a follow-up question to this.
In my adaptive control book, we are using the following inequality.
$$\left\|\Phi^T\left(\mathbf{x}\right)\Phi\left(\mathbf{x}\right)\right\| \geq \left\|\Phi\...
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1
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How can $\int_{t}^{t+T}xx^Td\tau \geq \alpha_1 I$, where $x \in \mathbb{R}^n$, especially when the determinant of outer product or $xx^T$ is 0?
I am quite perplexed, I wish to prove one of the fundamental lemmas in adaptive control, i.e.,
\begin{equation}
\alpha_1 I \leq \int_t^{t+T}xx^Td\tau \leq \alpha_2 I
\end{equation}
where, $\alpha_1,\...
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0
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Backstepping control of second order nonlinear system
$\dot{x_{1}}=x_{2}^2-3\sin(x_{1})x_{2}$
$\dot{x_{2}}=x_{1}^3-3x_{2}\cos(x_{1})+u^{1/2}$
Question: Using the backstepping method and Lyapunov function, design the controller $u$ that will make the ...
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2
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Why is high-gain controller undesirable?
In control theory and for example the scalar plant:
$\dot{x}=ax+u+d$
where $x$ is the state and $u$ is input and $d$ is disturbance
If the following control law is chosen:
$u=-kx$
where
$k\geq |...
- 103
1
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1
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Matrix Norms, and Integrals; why is the norm and integral inequality true?
In my adaptive control textbook that I am using, we are using the following:
$$\left\|\Phi^T\left(\mathbf{x}\right)\Phi\left(\mathbf{x}\right)\right\| \geq \left\|\Phi\left(\mathbf{x}\right)\Phi^T\...
- 255
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0
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Derivative of Matrix with respect to Matrix notation
I was reading my adaptive control textbook, and I noticed that my professor used a notation that is somehow confusing me.
$$J\left(\Theta\right) = \frac{1}{2}\epsilon^T\epsilon$$
where $\Theta \in \...
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1
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633
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Barbalat's Lemma Proof Typo and Clarification
I am trying to study the proof of Barbalat's Lemma by Hao Liu as shown in this link
However, I realized that there is a typo in there:
$$\lim_{t\rightarrow\infty}\left|f\left(t_n + \delta \right)- f\...
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0
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Backpropagation with linear algebra - How?
I have an intresset in control theory - optimal control. But unfortunately all control theories are for linear models. That's become a very difficult issue when to implement a linear controller for a ...
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0
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Extremum Seeking Convergence Rate: Perturbation Based vs Sliding Mode
Not so long ago i experimented with control systems that are looking for extremum of functions and, as i believe, users are familiar with them to one degree or another.
As sources, i refer to two ...
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Deterministic Control Problem with first exit limitation
Let $f:\mathbb{R}^d\times \mathbb{R}^n\rightarrow \mathbb{R}^d$ be a $C^2$-function and $u_t$ be a "sufficiently well-behaved smooth map from $[0,T]$ to $\mathbb{R}^n$ such that the (controlled) ODE:
$...
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0
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Is there a way to check true/false stability in a discrete transfer/state space model?
Assume that we have a discrete transfer function $H(z)$ and a discrete state space model $x(k+1) = Ax(k) + Bu(k)$.
I know how to check the stability, but computing the eigenvalues from the ...
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0
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Does this adaptive time-step algorithm have a name?
I'm using a somewhat unconventional technique to iteratively minimize a high-dimensional function $E(\vec\theta)$, and have proposed a simple routine to dynamically adapt its time-step. I am seeking ...
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1
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What can deep neural network be used for in control?
I wonder if it's possible to use a neural network in control?
Let's say that we have a deep neural network:
$$a_i = \sigma(W_i*x_i + b_i)$$
That is trained from the inputs $u(t), r(t), \Delta u(t), ...
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1
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Is chaning the reference gain a good control strategy? Feed forward control with system identification?
Assume that we have a estimated system:
$$\hat G(s)$$
And we want the system $\hat G(s)$ follow the reference $r(t)$. If we add an input $u(t)$ to $\hat G(s)$ we will get a output response:
$$y(t) =...
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0
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How to estimate a delay?
Assume that we are using recursive least squares to estimate a transfer function
$$A(q)y(t) = B(q)u(t)$$
But the input $u(t)$ is delayed with $d$ time, eg:
$$A(q)y(t) = B(q)u(t-d)$$
How can I find ...
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0
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How can I estimate $A(q)y(t) = B(q)u(t) + C(q)e(t)$?
How can I estimate the model
$$A(q)y(t) = B(q)u(t) + C(q)e(t)$$
Where $e(t)$ is the noise and $u(t)$ is input and $y(t)$ is output of the model? I don't know how to find the $e(t)$ noise.
Notice ...
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0
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Adaptive feedforward cancellation (AFC) and least mean squares (LMS) for periodic disturbance cancellation
I want to implement an adaptive feedforward cancellation (AFC) for cancellation of the impact of periodic input disturbances on the output of a multiple-input single-output system. Filter weights are ...
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Can pattern recognition be used in control engineering? - Neural networks
Assume that we have data who looks like this from a dynamical system $G(s)$ e.g mass-spring-damper system. Where the output is the position of the mass and input is the applied force onto the mass.
...
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1
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Nonlinear geometric control for beginners
Please advise literature or articles on nonlinear geometric control (for beginners). Preferably with computational examples. I want to study this topic, but I do not know where to start.
Remark:
...
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Avoid computing the inverse - Extended Least Square
The extended least square estimates this polynomial equation:
$$A(q)y(t) = B(q)y(t) + C(q)e(t)$$
By using:
$$\epsilon(t) = y(t) - \phi^T(t-1)\hat \theta(t-1)$$
$$\hat \theta(t) = \theta(t-1) + P(t)...
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0
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Discrete Lyapunov function candidate - How to find $dV(k)$?
I have a discrete state space model as a desire reference model:
$$x_m(k+1) = A_m x_m(k) + B_m r(k)$$
$$y_m(k) = C_mx_m(k) + D_m r(k)$$
Then I have a discrete state model as real process model:
$$x(...
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1
answer
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What adaptive controller can be used in embedded system with low RAM?
This is not a question for data science, hardware or programming languages. This is a more practical question about adaptive control for embedded systems, but still a math question.
I have tried to ...
- 2,962
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2
answers
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What's the difference between Generalized Predictive Control and Model Predictive Control?
As I know, the Generalized Predictive Control(GPC) is older than Model Predictive Control(MPC).
But what is the real difference between them? I know that GPC contains some kind of system ...
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1
answer
563
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How do I place the poles and zeros form a disired system? Adaptive control
If I have a transfer function of a system $G(s)$
$$G(s) = \frac{4 - 2s}{4 + 0.8s + s^2}$$
$G(s)$ has the poles and zeros and is a stable system.
And the step answer look like. It has a delay as you ...
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1
vote
2
answers
936
views
How can I estimate a discrete transfer function? Recursive Least Square
This is going to be a large fun question about practical estimation for real world problems.
Assume that we have a poor damped system described with this transfer function.
$$G(s) = \frac{4.5}{1 + 0....
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1
answer
1k
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MIT rule VS Lyapunov design - Adaptive Control
I wonder what's the difference between MIT rule and Lyapunov design when it comes to adaptive control law?
As I get it, MIT rule is more like "transfer function"-based control system and Lyapunov ...
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votes
2
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441
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Adaptive Control + Robust Control - Does it work?
I have a qurius question! Is it possible to design a robust controller for a system by using algoritms and system identification, which are adaptive control + robust control?
I know there is a lot of ...
- 2,962
2
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2
answers
2k
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In-depth example or implementation of adaptive control (direct/indirect MRAC)?
I have seen some examples where adaptive control is used to counter sudden changes in a system with great success. Since I find the subject quite interesting, I would like to learn how to actually ...
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