Digital Motion Control System: Lecture1- Review

in Robotics / Robotics & Control on Digital_motion_control

Table of Contents

  1. Introduction
  2. Frequency Responce
  3. Convolution
  4. Stability
    1. Stability of the Zero-Input Response: Asymptotic stability
    2. Stability of the Zero-State Response: BIBO stability

Introduction

An n th order linear time-invariant(LTI) can be described by a linear constant coefficient differential equation of the form \[\begin{aligned} y^{(n)} +a_{n-1} y^{(n-1)} + ... + a_1 \dot{y} + a_0 y = b_m u^{(m)} + b_{m-1} u^{(m-1) } +...+ b_1 \dot{u} + b_0 u \end{aligned}\]

where apprpriate initial conditions are specified for the output, y(t) of the system. Laplace transforming the above equation, we get \[\begin{aligned} Y(s) = \frac{N(s)}{D(s)} U(s) + \frac{IC(s)}{D(s)} = G(s)U(s) + \frac{IC(s)}{D(s)} \end{aligned}\]

where \[\begin{aligned} G(s) = \frac{b_m s^m +b_{m-1} u^{m-1 + ... + b_1 s b_0}}{s^n + a_{n-1} s^{n-1} +...+ a_1 s a_0} = \frac{N(s)}{D(s)} \end{aligned}\]

IC(s) : a polynimial in s, a complex variable, containing the terms arising from the initial conditions.

The total response(Y(s)): sum of the contributions from the initial conditions and the system input.

If the system is unforced(i.e., U(s) = 0), we call the resulting response the zero-input response(ZIR) of the system. Likewise, if all initial conditions are zero(i.e., IC(s) = 0), the forcing function produces the zero-state response(ZSR).

Frequency Responce

  • s(\(=\sigma + j \omega\)): complex variable, transfer function(G(s)) are also complex.
  • complex: can be expressed with a magnitude and angle.
  • To draw such graph, particularly with the help of a computer
    • This give us sinusoidal steady state responce, which fully characterizes the system.
    • Frequency response plots: Plots of magnitude and phase of a transfer function vesus frequency.
    • Magnitude: 입력/출력의 신호세기를 보여준다.
      • -3db가 되는 지점을 bandwidth라 하며 energy level이 절반이 되는 시점이다(\(-3 = 20 log (\frac{V_2}{V_1}) = 20 log (\frac{1}{\sqrt{2}})\))
      • 작은 입력으로 큰 출력이 발생할 때, 이것은 공진(resonance)이라 하며 sensor에서 많이 사용된다.
    • Phase:
      • 시간지연을 나타낸다, tracking control을 하기 위해서는 phase를 줄여야만 한다.
      • stability를 파악할 수 있다, 180도의 위상차(phase margin) 이상은 상당히 위험하다.
      • 저주파일때는 위상차 지연이 크지 않으나, 고주파일때는 위상차 지연이 크게 발생할 수 있다.

Convolution

  • Convolution: fundamental concept in systems analysis.
    • Time domain technique that provides the basis for transfer function analysis.
    • Time domain에서의 convolution 은 frequence domain에서 곱셈 과 동일.
  • The zero-state response of a linear time invariant system can be represented as \(Y(s) = H(s)X(s)\)
    • where H(s) is the system transfer function. The output, \(Y(s)\), is the Laplace transform of
\[\begin{aligned} y(t) = \int^\infty_{-\infty} h(\tau) x(t-\tau) d\tau = \int^\infty_{-\infty} x(\tau) h(t-\tau) d\tau \end{aligned}\]
  • The equation can be abbreviated as
\[\begin{aligned} y(t) = h(t) * x(t) = x(t) * h(t) \end{aligned}\]
  • If the input to the system is an impulse, \(\delta(t)\), then X(s) = 1 and
    • This property is used in system identification.
\[\begin{aligned} Y(s) = H(s) \cdot 1 \\ \therefore y(t) = h(t) \end{aligned}\]
  • Higher order systems can also be analyzed using this technique. The transfer function of an n th order system can be expanded as
\[\begin{aligned} H(s) = \frac{R_1}{s-p_1} + ... + \frac{R_n}{s-p_n} \end{aligned}\]
  • An alternative approach is to represent the n th order system with a first order matrix equation.

Stability

  • There are many definition of stability.
  • But, we define a stable system as one in which the output of the system does not grow without bound for any initial condition (natural response) or for any bounded input.
  • The second type os commonly known as bounded input-bounded output (BIBO) stability.

Stability of the Zero-Input Response: Asymptotic stability

Consider the Zero-Input Response of a system. This is determined only by the system’s characteristic equation and its initial conditions. That is, \[\begin{aligned} Y(s) = \frac{IC(s)}{D(s)} \end{aligned}\]

Recall that the roots of the characteristic equation are called system modes. There are 3 possibilities.

\(1\). Re(\(p_i\)) < 0 for all i

where \(p_i\) are roots of the characteristic quation. In this case, \(y(t) \rightarrow 0\) as \(t \rightarrow \infty\), and the system is asymptotically stable.

\(2\). Re(\(p_i\)) >0 for any i

Now, \(y(t) \rightarrow \infty\) as \(t \rightarrow \infty\), and the system is unstable.

\(3\). Re(\(p_i\)) = 0 for any i

This means that a root of the characteristic equation(system modes) is 0 or purely imaginary. In this case, the output either remains constant or is sinusoidal. Therefore, the output neither returns to zero nor goes to infinity. This is known as marginal stability.

Stability of the Zero-State Response: BIBO stability

The output of a linear time invariant system with general input x(t) is given by the convolution integral \[\begin{aligned} y(t) = \int^\infty_{-\infty} h(\tau) x(t-\tau) d\tau \end{aligned}\]

If the input is bounded(i.e., $$x(t)\(\)\leq M < \infty$$), then
\[\begin{aligned} |y(t)| = | \int^\infty_{-\infty} h(\tau) x(t-\tau) d\tau | \leq M | \int^\infty_{-\infty} h(\tau) d\tau | \leq M \int^\infty_{-\infty} |h(\tau)| d\tau \end{aligned}\]

Therefore, the system if BIBO stable if and only if \[\begin{aligned} \int^\infty_{-\infty} | h(\tau) | d\tau < \infty \end{aligned}\]

Is a marginally stable system BIBO stable? No

Reference:

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