# 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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### 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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### 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 ...
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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 ...
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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....
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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....
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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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 $U(s)$...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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}\\...
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