Second Order Systems in Standard Form

Introduction to Second Order Systems in Standard Form

  • Second-order systems in standard form are ubiquitous in real-world applications.

  • The primary goals are to characterize their properties and design control systems to meet predetermined closed-loop objectives.

  • The standard form of a second-order system is given by the transfer function:

    G(s)=ω<em>n2s2+2ζω</em>ns+ωn2G(s) = \frac{\omega<em>n^2}{s^2 + 2\zeta\omega</em>n s + \omega_n^2}

  • Key terms in this standard form are:

    • ζ\zeta (zeta): The damping factor.

    • ωn\omega_n (omega n): The undamped natural frequency.

  • These two factors directly relate to the system's poles (natural modes).

Visualizing Damping and Pole Locations
Undamped Systems (ζ=0\zeta = 0)

  • Definition: An undamped system is characterized by a damping factor ζ=0\zeta = 0.

  • System Equation: When ζ=0\zeta = 0, the term 2ζω<em>ns2\zeta\omega<em>n s vanishes, leaving: G(s)=ω</em>n2s2+ωn2G(s) = \frac{\omega</em>n^2}{s^2 + \omega_n^2}

  • Pole Locations: The poles are the roots of the characteristic equation s2+ω<em>n2=0s^2 + \omega<em>n^2 = 0, which are: s=±jω</em>ns = \pm j\omega</em>n

    • These poles are located on the imaginary axis of the s-plane, specifically at +jω<em>n+j\omega<em>n and −jω</em>n-j\omega</em>n.

  • System Behavior: Poles on the imaginary axis immediately imply non-decaying sinusoids, characteristic of an oscillator.

  • Step Response Calculation:

    • To find the step response, we multiply the system's transfer function by the Laplace transform of a unit step (1s\frac{1}{s}):

      Y(s)=ω<em>n2s(s2+ω</em>n2)Y(s) = \frac{\omega<em>n^2}{s(s^2 + \omega</em>n^2)}

    • The inverse Laplace transform of this yields the output in the time domain:

      y(t)=1−cos⁡(ωnt)y(t) = 1 - \cos(\omega_n t)

  • Step Response Characteristics:

    • The response is a continuous, non-decaying sinusoid oscillating around the reference value of 1.

    • Associated Frequency: The oscillation frequency is ωn\omega_n, hence its name, the undamped natural frequency.

    • Overshoot: The system exhibits a 100% overshoot. The response peaks at 2, which is 100% greater than the unit step reference of 1.

    • Peaking Time (t<em>pt<em>p)**: The first peak occurs at half the period, which is: t</em>p=πωnt</em>p = \frac{\pi}{\omega_n}

      • At this time, y(t<em>p)=1−cos⁡(ω</em>nπωn)=1−cos⁡(π)=1−(−1)=2y(t<em>p) = 1 - \cos(\omega</em>n \frac{\pi}{\omega_n}) = 1 - \cos(\pi) = 1 - (-1) = 2.

    • Period (TT): The period of oscillation is:

      T=2πωnT = \frac{2\pi}{\omega_n}

  • Practical Implications: Such a design is generally undesirable for control systems aiming to track a reference command (e.g., a robot arm that oscillates indefinitely instead of settling).

Underdamped Systems (0<ζ<10 < \zeta < 1)

  • Definition: An underdamped system has a damping factor ζ\zeta strictly between 0 and 1.

  • System Equation:

    G(s)=ω<em>n2s2+2ζω</em>ns+ωn2G(s) = \frac{\omega<em>n^2}{s^2 + 2\zeta\omega</em>n s + \omega_n^2}

  • Pole Locations: Using the quadratic formula, the poles are: s=−2ζω<em>n±(2ζω</em>n)2−4ω<em>n22=−ζω</em>n±ζ2ω<em>n2−ω</em>n2s = \frac{-2\zeta\omega<em>n \pm \sqrt{(2\zeta\omega</em>n)^2 - 4\omega<em>n^2}}{2} = -\zeta\omega</em>n \pm \sqrt{\zeta^2\omega<em>n^2 - \omega</em>n^2} s=−ζω<em>n±ω</em>nζ2−1s = -\zeta\omega<em>n \pm \omega</em>n\sqrt{\zeta^2 - 1} Since 0<ζ<10 < \zeta < 1, ζ2−1\zeta^2 - 1 is negative. We rewrite ζ2−1=j1−ζ2\sqrt{\zeta^2 - 1} = j\sqrt{1 - \zeta^2}. Thus, the poles are: s=−ζω<em>n±jω</em>n1−ζ2s = -\zeta\omega<em>n \pm j\omega</em>n\sqrt{1-\zeta^2}

    • These are complex conjugate poles located in the left-half of the s-plane.

  • System Behavior: Associated with sinusoids multiplied by a decaying exponential envelope. The step response will exhibit oscillations that gradually decrease in amplitude.

  • Key Features of Poles:

    • Real Part: −ζωn- \zeta\omega_n

      • Determines the rate of decay of the exponential envelope. A larger ζ\zeta (closer to 1) or larger ωn\omega_n will cause the envelope to decay faster, moving the poles further into the left-half plane.

    • Imaginary Part (Damped Natural Frequency): ω<em>d=ω</em>n1−ζ2\omega<em>d = \omega</em>n\sqrt{1-\zeta^2}

      • Determines the frequency of the oscillations in the system's response. This term is so special it's given its own name.

  • Geometric Representation in the s-plane:

    • Consider a right-angled triangle formed by the origin, a pole (e.g., −ζω<em>n+jω</em>d-\zeta\omega<em>n + j\omega</em>d), and its projection onto the real axis (−ζωn-\zeta\omega_n).

    • The hypotenuse of this triangle has length ωn\omega_n.

    • The horizontal leg (distance from origin to real part of pole) has length ζωn\zeta\omega_n.

    • The vertical leg (imaginary part of pole) has length ω<em>d=ω</em>n1−ζ2\omega<em>d = \omega</em>n\sqrt{1-\zeta^2}.

    • If θ\theta is the angle between the positive imaginary axis and the line connecting the origin to the pole:

      • ζ=sin⁡(θ)\zeta = \sin(\theta) (opposite side ζω<em>n\zeta\omega<em>n over hypotenuse ω</em>n\omega</em>n).

  • Step Response Characteristics:

    • Exhibits decaying sinusoidal oscillations, eventually settling to the reference value.

    • Overshoot: Present, but decreases as ζ\zeta increases.

    • Relationship between ζ\zeta and Response:

      • Increasing ζ\zeta (while keeping ωn\omega_n constant) causes the poles to move closer to the real axis (less oscillatory) and further into the left-half plane (faster decay).

      • This generally leads to decreased overshoot and faster damping of oscillations.

  • Design Trade-offs: There is a common trade-off between speed (getting to the reference faster, often with lower ζ\zeta) and acceptable overshoot.

    • Surgical Robot: Requires zero overshoot, tolerating a slower response.

    • High-Performance Ground Robot/Quadcopter: Might tolerate some overshoot for quicker response times.

Critically Damped Systems (ζ=1\zeta = 1)

  • Definition: A critically damped system has a damping factor ζ=1\zeta = 1.

  • System Equation: Plugging ζ=1\zeta = 1 into the standard form:

    G(s)=ω<em>n2s2+2ω</em>ns+ω<em>n2G(s) = \frac{\omega<em>n^2}{s^2 + 2\omega</em>n s + \omega<em>n^2} This factors perfectly into: G(s)=ω</em>n2(s+ωn)2G(s) = \frac{\omega</em>n^2}{(s + \omega_n)^2}

  • Pole Locations: The system has repeated real poles at: s=−ωns = -\omega_n

    • The poles are located on the negative real axis of the s-plane, merging from the complex conjugate locations.

  • System Behavior:

    • No oscillations.

    • No overshoot. This is the critical value for eliminating overshoot.

    • Represents the fastest possible response without any overshoot.

  • Geometric Representation in the s-plane: The poles lie directly on the negative real axis. The angle θ\theta from the positive imaginary axis would be 90∘90^\circ.

  • Step Response Characteristics:

    • Smooth, exponential rise to the final value without any oscillations.

Overdamped Systems (ζ>1\zeta > 1)

  • Definition: An overdamped system has a damping factor ζ\zeta strictly greater than 1.

  • Pole Locations: Using the quadratic formula, the poles are: s=−ζω<em>n±ω</em>nζ2−1s = -\zeta\omega<em>n \pm \omega</em>n\sqrt{\zeta^2-1} Since ζ>1\zeta > 1, ζ2−1\zeta^2 - 1 is positive, resulting in two distinct real poles located on the negative real axis:

    • Slow Pole (s<em>1s<em>1)**: s</em>1=−ζω<em>n+ω</em>nζ2−1s</em>1 = -\zeta\omega<em>n + \omega</em>n\sqrt{\zeta^2-1}

      • This pole is closer to the origin (less negative) and is typically the dominant pole, dictating the slower part of the response.

    • Fast Pole (s<em>2s<em>2)**: s</em>2=−ζω<em>n−ω</em>nζ2−1s</em>2 = -\zeta\omega<em>n - \omega</em>n\sqrt{\zeta^2-1}

      • This pole is further from the origin (more negative) and contributes to the faster initial decay.

  • System Behavior: Consists of two decaying exponentials.

    • No sinusoids are present, meaning no oscillations and no overshoot.

  • Step Response Characteristics:

    • Smooth, exponential rise to the final value, similar to critically damped but generally slower.

    • As ζ\zeta increases further (e.g., from 1.1 to 1.8), the dominant (slow) pole moves closer to the origin, making the overall system response slower.

    • For damping greater than or equal to 1, there will be no overshoot.

Summary of Damping Effects on Step Response

The lecture concludes by visually comparing the step responses for all four damping cases on a single plot (assuming a fixed ωn\omega_n):

  1. Undamped (ζ=0\zeta = 0): Exhibits continuous, non-decaying oscillations with 100% overshoot. (e.g., robot arm constantly overshooting its target).

  2. Underdamped (0<ζ<10 < \zeta < 1): Shows decaying oscillations. As ζ\zeta increases, the overshoot decreases, and oscillations die out faster.

  3. Critically Damped (ζ=1\zeta = 1): Represents the threshold where overshoot is completely eliminated. The response is the fastest possible without any oscillations.

  4. Overdamped (ζ>1\zeta > 1): No overshoot is present. However, as ζ\zeta increases beyond 1, the response becomes progressively slower due to the dominant pole moving closer to the origin.

These general ideas are crucial for understanding and intelligently designing control systems for second-order systems. The next segment will delve into underdamped step response characteristics in more detail.