# Tagged Questions

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input 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 ...

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### Why are Lyapunov functions always quadratic?

Consider stable linear system $\dot x= Ax + Bu$. We’ll show that the Lyapunov bound is tight with $V (z) = z^T W^{−1}z$. Multiply $AW_c + W_c A^T + B B^T = 0$ on left & right by ${W_c}^{−1}$ to ...
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### Differences between Nyquist plot and polar curve

Which are the differences between Polar curve and Nyquist plot in System Control Theory? I wasn't able to figure it out myself. Are they the same thing? As googling for these two concept returned ...
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### Bounds on a system of coupled ODEs

Suppose we have a $1$-dimensional differential inequality $$\frac{dx}{dt} \leq x - x^3$$ We can apply the Comparison principle to claim that if $y(t)$ is the solution to $\frac{dy}{dt} = y - y^3$, ...
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### Bode Sensitivity Integral

The bode sensitivity integral is well known for linear control systems. To state it in simplistic terms, Any system $L$ with relative degree $2$ or more satisfies the following equation for the ...
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### Sequential Loop Closing - SIMO

I have studied Sequential Loop Closing or SLC in context of MIMO systems. Closing the loops sequentially while taking care of interactions make sense for a square MIMO systems. But does any theory ...
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### Is there a strategy for discrete control of a system with dynamics near sample rate?

I'm trying to control a system where the controller sample rate is physically fixed and the plant has significant dynamics on the same order as the sample rate. I understand that one would prefer to ...
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### Find parameter $k_f$ for which steady-state error equals $0$

$u(t)=t, G_0(s)=\frac{k_p}{s}, G_R(s)=k_f$ Find parameter $k_f$ for which steady-state error equals 0 Finding $E(s)$ $$E(s) = U(s)*\frac{1}{1+\frac{k_p}{s}*k_f} = \frac{1}{s^2+k_pk_f*s}$$ Steady-...
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### Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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### How to interpret multi-conditional piecewise functions.

I'm trying to simulate hysteresis and the its inverse for a control problem. This is a model found in [Tao & Kokotovic, Adaptive Control Systems with Actuator and Sensor Nonlinearities] to ...
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### Matrix transformation for linear state-space systems

In http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/lecture-notes/MIT6_241JS11_lec12.pdf on pages 11-12 it is said: For a stable ...
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### asymptotic stability with exact feedback and feedback with measurement errors

I'm trying to show global asymptotic stability (GAS) for a system with a feedback controller. I managed to show GAS for the perfect feedback signal, that is $$\gamma(x(t)) = K_p x_1(t) + K_d x_2(t).$$...
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### What is the difference between moving-horizon DP and MPC?

What is the difference between moving-horizon DP (dynamic programming) and MPC (modelbased predictive control)? In both cases, the system input at time $t$ is determined by solving a finite-time ...