6

Anti-windup is a concept for feedback controllers with integral terms, e.g. PID, to keep the integral term from „overcharging“ when regulating a large set point error. It basically saturates the integral term to keep the system from overshooting the set point. The classic form of anti-windup, as described above, does not actually ensure satisfaction of ...


5

At the very high level you will need 24V Power supply or a method to generate 24V 24V Motor controller Microcontroller - Arduino is a good place to start There are also prebuild motor controllers that can be programed via computer. These tend to be expensive. I would suggest following web sites similar to the ones listed below. They tend to have blogs, ...


5

In nonlinear control theory, you will recognize most concepts such as controllability and observability where the linear case is often a special case of the nonlinear case. I would highly recommend digging into linear control theory first if you have not done so. Depending on the course you take, concepts such as Lyapunov stability are discussed here, which ...


4

So, just to formally repeat your question, we consider an infinite horizon continuous-time optimal control problem $J^*(x_0) = min \int_0^{\infty} x(t)^TQx(t)+u(t)^TRu(t)dt$ which is subject to some constraints, basically just the system dynamics and the initial condition $\dot x (t) = A x(t) + B u(t) \\ x(0) = x_0 $ We assume the cost matrices $Q = C^...


3

The transient response (or homogeneous solution) of a linear ODE is $$y_h(t) = \sum_{i=1}^N C_i e^{p_i\cdot t} $$ where $p_i$ is the i-th pole of your system. Assume you have a pole at location $p_i = \lambda + i\omega$, with $i = \sqrt{-1}$. The real part ($\lambda$) of this pole will determine the convergence rate towards zero, while the imaginary part $\...


3

A LTI system is nonminimum phase if it has one or more zeros in the right half plane. When using a LTI controller for feedback also then the bandwidth of this controller will be limited to roughly to smallest magnitude of zeros in the right half plane. This is because a right half plane zero gives a phase drop and slope increase. But near the bandwidth a ...


3

Why would $z$ need to be external to $g$? $$g'(t,x(t),y(t))=g(t,x(t),y(t),z(x(t)))$$ now use $g'$ as $g$ $g$ can be any arbitrary function, so any function $z$ could just be incorporated into $g$. Regarding your restriction $h$ mentioned in the comment section. Any restrictions on the control input could be enforced via the cost function: $$f_{new}(t,x(...


3

You can prove it using the PBH test for controllability. It states that $rank \left( \begin{array}{cc} \lambda I -A & B \\ \end{array} \right) = n$ for all values of $\lambda$. For the uncertain system this becomes $\left( \begin{array}{cc} \lambda I -A -B \phi & B \\ \end{array} \right) = \left( \begin{array}{cc} \lambda I -A & B \\ \end{...


3

I think the other answer is not complete. Model predictive control (or 'receding-horizon control' is a technique in which a predictive system model is used to evaluate a sequence of future control inputs; of all such control input sequences, an optimization algorithm chooses the best one. Typically, the first input of the sequence is implemented. After a ...


2

There are a few issues to consider. Dedicated off road vehicles tend to have ladder or in some cases space-frame chassis. There are robust and make it easier to add specialist bodies but tend to be heavier than equivalent monocoque designs. Equally they may also need roll protection which adds further weight. They tend to have high displacement engines ...


2

One way might be to try to model and understand the physical problem. You are however clear in your question that you would prefer an empirical approach where constants are tuned to experience data. I would suggest that the easiest and most general way is to use a piece-wise approximation of the velocity curve. Generically this may be made by using basis ...


2

In short: it is both. Model Predictive Control is a repeated open-loop control in a feedback fashion. The explanation comes not from the general concept of open-loop and closed-loop, but from how MPC works. We solve an optimization problem with a certain prediction horizon N, i.e. we predict the next N states until a certain point in the future (that's why ...


2

The idea that you describe is basically Model Predictive Control with successive linearization (the optimization problem that is solved in MPC is finite-horizon while LQR is infinite-horizon, but the concept is still the same). This requires a lot of computational power, because you need to be able to linearize the system and solve the optimization problem ...


2

The first equation you wrote is called discrete-time system. They are often also written as $$x_{k+1} = Ax_k + Bu_k$$ Recall that MPC solves an optimization problem every time-step, so you need your system to have discrete time-steps in the first place. In fact, you can use your Euler integration scheme to discretize a continuous time-system of the form $$\...


2

Using a different state space model, which is equivalent to the same transfer function, means that you are applying a similarity transformation. The new state vector will therefore have a different meaning. For example when given a state space model $(A,B,C,D)$ and applying a similarity transformation $\hat{x}=T\,x$ then the new state space model $(\hat{A},\...


1

Taking the time derivative of $y$ yields: $$ \dot{y}=\frac{\partial f}{\partial x}(f(x)+u) $$ We need $y(t)=y(0) e^{-\beta t}$ but this is possible if and only if $\dot{y}=-\beta y $ , if the hessian is invertible then this is possible if and only if $u=-f(x)-\left(\frac{\partial f}{\partial x}\right)^{-1}\beta y$. To get an intuition on why the hessian must ...


1

Adding to the other answers. I just so happen to have done exactly this. I used a windshield wiper motor and a potentiometer but the principal is the same. Here's my arduino source code: https://pastebin.com/0ezsmi4y And a short video I took of it in action. This is an alternate version that takes RC PWM input instead of serial. I think all the talk about ...


1

You are at the right track. As the DC motor is rather fast for a potential slow microcontroller, using a discrete controller will improve the reliability and stability of the closed-loop system. Even though a DC motor is rather easy to model (speaking of the basic dynamics upto the 3rd order), using system identification can improve parameter estimation ...


1

In control systems, the main focus is the design of a controller for machines or robots, here we mainly deal with linear systems application of linear control theory. While non-linear systems is an advanced topic, where we deal with advance and mathematically more complex systems. If its ur first course in control systems or automatic control. Go for the ...


1

Found out about Response Surfaces and using all the points that I already created, created a Response Surface Optimization which let me easily find the answer without a significant amount more computation.


1

I would think that this refers to using mathematical and physical principles and equations to predict the behaviour of a control system. The opposite would be to empirically design a control system, by implementing it and measuring it.


1

What you need to do is use Laplace transform to $U(t)$ so you would get $\mathcal{L}(U(t))=U(s)$ for example, if you are rusty on Laplace transforms (or their inverse), you can use wolfram alpha multiply $H(s)*U(s)$ that will get you a function of s $Y(s)$ use the inverse laplace transform to Y(s) to get the $y(t) = \mathcal{L}^{-1}(Y(s))$


1

MPC finds the optimal input $u^*$ which is the input that minimizes the cost function $J$ or $c$. This means that regardless of what this actual value is, its proven to be its minimum. As such, multiplying the cost function with any constant value does not change this minimum, it just scales the value. Therefore, $\frac{t_{hor}}{N}$ does not affect the ...


1

I think I have found an answer and please correct me if there are any other reasons apart from the following justification. Since $x^{*}(t)=\{x(t)\in\Omega|J(x)<\min J(y), \forall { y \in \Omega}\}$ is optimal between all admissible trajectories $x(t)\in\Omega$ for $t\in[t_0,t_1]$ with $x^{*}(t_0)=x(t_0)=x_0$ and $x^{*}(t_1)\neq{x(t_1)}$, it is still ...


1

Normally state space models who are equivalent to the same transfer function are also equivalent to each other, such that there exists a similarity transformation between them. However if the considered transfer function has pole zero cancellations then an equivalent state space model would be a rank deficient controllability- or observability matrix, but it ...


1

I think the book is using "infinite" to mean "a number that is too big for the computer software to represent", not in the strict mathematical sense. See near the top of page 1-6: Open-loop unstable plants: if p*Ts is too large, such that the plant step responses become infinite during this amount of time, key parameters needed for MPC calculations ...


1

Control problems are usually easier if you specify The measured variables, I assume $T_{in}$ and $T_{out}$ The manipulated variables, I assume the pump speed according to your description The objectives of the controller. Do you need to be exactly at 90 degC or would and outlet with a certain offset be acceptable? Or is the temperature very important and ...


1

Without knowing the transfer function of the plant you try to control it might be that a PID-controller would suffice, but for now I will assume this is not the case. Assuming that your graph is a step response; I am however a little confused by the drop back to zero, since there all systems seem to exert the same behavior. For the initial step the system ...


1

I assume that you also want to minimize some function of $t$, $T$, $\vec{x}(t)$ and $w(t)$. In this case it depends on how you define a solution to this problem. If you define a solution as minimization of that function, then you will have multiple solutions if there are multiple local minima, but the global minima would be the best solution. If there are ...


1

Using this reference on linear discrete Kalman Filters, it looks like you can apply a standard observability model. Namely, for a linear Kalman Filter system defined as $$ \begin{align*} x_{k+1} &= A x_k + B u_k \\ y_k &= C x_k + D u_k, \end{align*} $$ the system is observable if $M_{obs}$ is full rank, where $M_{obs}$ is defined as: $$ M_{obs} = ...


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