# 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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### "Spanning" of solutions of ordinary differential equations

Suppose we have a switched ODE $$\dot{x} = A_{\sigma(t)}x,$$ where $A_{\sigma(t)}$ is a constant matrix given $\sigma(t)\in\mathcal{M}=\{1,2,\cdots,m\}$. If we fix the initial condition and can ...
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### Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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### On the positive definiteness of the observability Gramian

Given the system $\dot{x} = Ax$, $y = Cx$, it is known that the Gramian, given by $$W({t_0},{t_1}) = \int_{t_0}^{t_1} e^{A^T(\tau -t_0)}C^TC e^{A(\tau -t_0)} \,{\rm d}\tau$$ is positive definite for ...
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### Feedback linearization with integral action - How?

Assume that you know sort of the dynamics of the system. It's not 100% perfect, but it's at least 90% perfect. $$\dot x = f(x, u)$$ I want to find a control law that suits this system. I have been ...
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### Dynamical systems with control input

Please I have been trying to write the mathematical formulation of my nonlinear dynamical system for quite some time and I will appreciate any input. ** Problem Description** Assuming, I am traveling ...
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### LQI for angular speed control in MATLAB

With the following state space system and setting for the Linear Quadratic Integrator (LQI) Q = diag([1,1,1]) and r = 1; for ...
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### Nyquist plot- is G(s) stable?

I got very confused with the nyquist plot. I have a basic question that would suffice: say I got a general nyquist plot of some transfer function $G(s)$: Is $G(s)$ stable? what information apart ...
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### Control of flow rate to be 'fair'?

We have a pipe transporting discrete particles from a to c with max flow capacity of $C$ particles/second. It takes $d_{min}$ seconds to get from a to c when the pipe is not fully utilized (details ...
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### Z transform using Convolution Integral

Let our transfer function be $G(s) = \frac{10e^{-s}}{5s+1}$. We know that for sampling period of $T = 1$, we have $G(z) = \frac{2}{z-0.8187}$ (You can verify this in MATLAB using c2d function). What I ...
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### Estimate $\Vert \Delta u \Vert_{2}$ for wave equation [closed]

We consider the wave equation \begin{equation}\label{1} \left\{ \begin{array}{ll} u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\ u=0, \quad u \in \partial \Omega, t>0 \\ u(0,x)=u_{0}(x), \quad ...
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### How to pick a Lyapunov function and estimate PID gains? [closed]

I am currently trying to estimate the range of PID gains by developing a Lyapunov function for a nonlinear 6-Dof quadrotor system. The system is of the following form: M(q)\ddot{q}+C(q,\dot q)\dot q+...
The following lemma is stated without proof in the lectures notes on Model Reduction by M. Voigt (available here) Lemma 4.4 Let $[A,B,C,D]\in\Sigma_{n,m,p}$ be asymptotically stable. Let \$T\in\mathbb{...