Control Systems & Engineering

• Signals are represented as arrows. Systems (and subsystems) are blocks
• A block diagram can be used to represent any physical system that has an input signal and produces an output signal.
• Closed Loop:

  There is a feedback loop.

  Can be positive or negative.

  Can be any domain (mech, elec, fluid).

  Often used in control.
• Open Loop:

  Just a way of saying there is no feedback loop.

  Can still be controlled by adding items (e.g. amplifier).
• Summing Junction - Combines signals from multiple inputs into one output.
• Gain Block - Multiplies the input signal by a constant value.
• Integrator - Integrates the input over time.
• Differentiator - Differentiates the input with respect to time.
• C(s) = R(s)G(s)

  Where:

  C(s) = Output

  R(s) = Input

  G(s) = Transfer Function
• Transforming an ODE into the Laplace Domain:

  dx^2/dt -> s^2 X(s)
• Dynamic variables:

  Force (an “effort” which causes motion)

  Velocity (a “flow” which characterises motion) Displacement (velocity integrated)

  Etc
• Elements:

  Mass (an “inertia” element)

  Spring (a “compliance” element)

  Damper (a “resistance” element)
• There are some other common features:

  Ideal Sources of applied effort or flow

  Nodes about which efforts or flows are equal or can be summed: rigid bodies, dynamic equilibrium, junctions in electrical and hydraulic circuits.

  Transformers and Gyrators which transfer energy from one domain to another: actuators, electromagnets.
• Poles:

  1. The values of the Laplace transform variable, s, that cause the transfer function to become infinite, or
  2. Any roots of the denominator of the transfer function that are common to roots of the numerator
• Zeros:

  1. The values of the Laplace transform variable, s, that cause the transfer function to become zero.
  2. Any roots of the numerator of the transfer function that are common to roots of the denominator.
• Stability:

  Poles have negative real parts, (i.e. they appear on the left half of the s-plane).
• Pole Plot

  1. Poles on the Left Hand Side of the plot indicate a time response which decays to steady state i.e. Poles with negative real parts are stable
  2. Poles on y-axis indicate a response which neither increases nor decreases: marginally stable.
  3. Poles on the Right Hand Side of the plot indicate a time response which increases exponentially to infinity i.e. Poles with positive real parts are unstable
• First-Order Systems:

  1. The expression on the denominator is first order: s + a
• Rise Time (Tr): is the time for the waveform to go from 0.1 to 0.9 final value.
• Settling Time (Ts): is the time for the response to reach and stay within 2% final value.
• Second-Order Systems:

  1. 2 poles, second-order polynomial on denominator of transfer function.
• Natural Frequency 𝜔𝑛 (“omega” n) is the oscillation of the system without damping
• Damping Ratio 𝜁 (“zeta”) is the ratio between the exponential decay frequency and natural frequency, with 1 being critically damped.
• $𝐺 𝑠 =$$𝜔𝑛^2/( 𝑠^2 + 2𝜁𝜔𝑛𝑠 + 𝜔𝑛^2)$
• If zero(s) >> pole(s) it basically acts as a gain i.e. increases amplitude.

  If the zero(s) are close to the pole(s) they affect the response too.

  If the zero(s) and pole(s) are very close or equal, they can be cancelled out to yield a lower-order system.
• Routh-Hurtwitz Table:

  Look only at denominator (poles)

  Begin by labelling the rows with powers of s, from the highest to s 0 .

  In the first row write the coefficient of the highest power, then populate the rest of the row with every other coefficient.

  In the second row, write the coefficients that were not in the first.

  For the next rows, each entry is a negative determinant of entries in the previous two rows, divided by the entry in the first column directly above the calculated row.

  No. unstable poles = no. sign changes in first column.
• Cascade Form:

  When elements / systems follow each other in series, they multiply.

  Ge(s) = G1(s)G2(s) ....... Gn(s)
• Parallel Form:

  Subsystems in Parallel are summed

  Ge(s) = +-G1(s)+-G2(s) ...... +-Gn(S)
• Feedback Form:

  Ge(s) = G(s) / (1+-G(s)H(s))

  Where the +- is the opposite of the summation junction & the H(s) is from the feedback loop.
• Steady State Error:

  Difference (error) between input value and final, steady-state output value.

  For steady state error to be zero [when a step input is applied]. The denominator of e() must be zero, which means there must be an s term in transfer function’s denominator i.e. pole at zero
• Root Locus Plot:

  1. We can ‘move’ the poles on the s-plane to give us desired dynamic behaviour
  2. Arrows indicate the direction in which poles ‘move’ as K increases. These arrows are the ‘loci.’
  3. We can identify where the root loci enter or leave regions corresponding to given damping ratios, natural frequencies or time for exponential decay.
• Root Locus Rules:

  1. There are n lines (loci) where n is the order of the numerator or denominator of the transfer function (whichever is greater)
  2. As K increases from 0 to infinity, the roots move from the [open loop] poles of G(s) to the zeros of G(s). i. Where there are more poles that zeros, the extra loci will move to infinity. ii. Where there are more zeros than poles, the extra loci come from infintiy.
• Root Locus Rules:

  3.When roots are complex, they occur in conjugate pairs

  4.At no time will the same root cross over its path.

  5.The portion of the real axis to the left of an odd number of open loop poles and zeros are part of the loci.

  6.Lines leave (break out) and enter (break in) the real axis at 90°

  7.Lines go to infinity along asymptotes
• Application of Root Locus:

  We can add a Gain and Unity feedback loop to the system

  • The poles of the closed loop system vary with gain: we can plot loci on the s-plane
  • We can find a gain that gives us desired behaviour, provided the poles are on the loci
• Iterative Calculation to Find Gain & Unity Feedback:

  Brute force method where you follow through the control system with different values until the final value becomes the one that you want
• Routh-Hurwitz Gain and Unity Feedback Method:

  • Often, we just want to find the range of K that gives a stable response.
  • We can construct a Routh array containing K as an unknown
  • We can then deduce the values of K that cause sign changes in the first column i.e. instability