TU/e EE Notes

Control Systems - 5ESD0

14 min read

5ESD0 Control Systems

Summary

5esd0-summary-samenvatting-control-systems.pdf

Notes

control-systems-summery.pdf

Module 1: Course Introduction (Modifying open-loop transfer function properties using feedback)

Topic: Laplace Transform
Systems - 5ESB0

Module 2: Root Locus (Modifying closed-loop transfer functions using Feedback)

Topic: Feedback and Stability
The Root Locus Rules
(VERY USEFUL) How To Sketch a Root Locus
Departure Angle Example








Departure Angles




same just without

Module 3: Frequency Response Functions / Bode Diagrams (Understanding the relevance of frequency-domain control-design methods and frequency-responses on Bode plots)

Topic: Transfer Functions and Frequency Response
Final Exam Tutorial - Bode Plot Example - YouTube

Bode Plots by hand
Bode Plots Explained - YouTube
ChatGPT



















$\text{break point: }\omega=\omega_{n}$
$\text{magnitude asymptote deviations: } \frac{1}{2\zeta}\text{ so }=\pm 20\log(2 \zeta)$
$\text{magnitude slope: }\pm40db$
$\text{phase slope: }\pm180 \degree$


Bode Plots Explained - YouTube

Module 4: Nyquist Stability & Stability Margins (Assessing closed-loop stability and its robustness via Nyquist plots)

Topic: Feedback and Stability
Cauchy’s argument principle

  • A clockwise contour map of a complex function will encircle the origin times, with the number of zeros and is the number of poles of the function inside the contour.
  • Control Design: 1) Zeros of are closed-loop poles, 2) contour of interest is one the covers the RHP.
  • Encirclements of around the origin is equal to encirclements of around -1.

  • is the number of RHP zeros of (i.e, the closed-loop poles).
  • is the number of (clockwise) encirclements of the point −1.
  • is the number of RHP (unstable) poles of .






    thee amount by which the phase can be decreased before reaching instability.



Phase and Gain Margins Example
Final Exam Tutorial - Nyquist Plot Example - YouTube



Module 5: Frequency Response Controller Design (Designing single-input-single-output feedback controllers by shaping the open-loop frequency-response function)

Topic: Controller Design

performance specifications


how to tune D for L.
Design feedback controller using open loop response functions.

gain margin is a margin of gain that can be applied to a system since systems IRL can vary a margin is good to have since some have stronger or weaker gain due to mechanical wear-outs.

adding more than needed amount of integrator poles effects the phase margin and puts more requirements on the lead/lag compensator.

stability margins

Nyquist plot, having a certain PM and GM

steady state error tracking

controlled by system type.
steady state error → 0 if slope of the bode diagram of L(s) is steep.

overshoot

enough phase margin

closed loop bandwidth

crossover frequency of



compensators




lead lag intuitive understanding

lead: adds phase lead to the system increasing gain (between zero → pole)
(typical goal to provide gain in high frequency range).

lag: adds phase lag to the system decreasing gain (between pole → zero)
(typical goal to provide gain in low frequency range).

(max phase margin)

lead lag compensators
1 pole and 1 zero

zeros add of phase and amplify high frequencies
poles subtract and attenuate high frequencies

example of lead (opposite for lag), cancels if pole zero at same:


lead/lag: to amplify at lower and attenuate at higher…
Designing a Lead Compensator with Bode Plot - YouTube
Designing a Lag Compensator with Bode Plot - YouTube



which means natural frequency in lead is higher which implies faster response and therefor shorter rise and settling times. which we can see above.
usually lead filter have higher cross over frequency…

other compensator pics




Module 6: Fundamental Limitations in Control Design (Identifying the limitations of linear control design, and the trade-off between disturbance rejection and sensor noise amplification)

Topic: Controller Design

General knowledge



By analyzing the sensitivity function in the logarithmic bode scale we have derive robustness and performance limitations.

If I want to reduce sensitivity with regards to tracking references, then I can make system susceptible to measurement noise.

Peak of sensitivity and how it relates to nyquist


The peak of the sensitivity function represents how much the system amplifies disturbances. It is the reciprocal of the smallest distance from the Nyquist plot to the critical point , which indicates the system’s proximity to instability. If the plot gets close to , the system becomes sensitive to disturbances, meaning a higher peak value.

Bode sensitivity integral



improving characteristics in one range make them worse in the other range…
Cauchy’s integral:

numeric effects when placing poles or zeros in specific spots based on waterbed effect…

Control system types





sensitivity is looked at when asked about maximal disturbance amplification.
complemetary sensitivity is looked at when asked about maximal disturbance amplification.

overall:
fight disturbances D controller should be large…
fight measurement noise D controller should be small…

Performance Limitations of Right Half Plane (RHP) Zeros


  • Explanation: When there are RHP zeros, closed-loop poles eventually approach these zeros as the loop gain increases. This causes limitations in control performance, especially at higher frequencies.
  • Math:

Performance Limitations of Right Half Plane (RHP) Poles


  • Explanation: RHP poles require a minimal gain for stability. As frequency approaches the RHP pole, the system needs higher bandwidth to maintain stability.
  • Math:

exercise

20241019_185055.pdf

Module 7: State-Space Models (Understanding the relevance of state-space control design, and the analysis of the state-space equations)

Topic: State-Space Models
State-space representation - Wikipedia
Math I - 2DE20
transpose, product, inverse, rank, eigenvalues, eigenvectors

deriving state space models

Simulation Diagram, Modal Decomposition, Control Canonical Form.









block diagram to state space - YouTube

state space analysis

TF from SS:

  • poles of the transfer function are the eigenvalues of the system matrix




    controllability: reach a specific state (not maintain)
    observability: core states in A matrix


if you look above

we mainly work with control canonical form but there is also observer canonical form

linearisation of SS models

  • Write
  • Use taylor series to obtain:

extra


Module 8: State-Space Control Design (Designing state-space controllers via the pole-placement)

Topic: State-Space Models

pole placement is a fancy root locus with placement across matrix not just locus lines…

8.1 Recap Eigenvalues / Eigenvectors / Determinants

Topic: Linear Algebra

  • Eigenvector condition: For , vector is an eigenvector if it satisfies .
  • Eigenvalue computation: (non-trivial ) → solve .
  • Key points:
    • Eigenvalue : scalar associated with eigenvector .
    • Eigenvector is in the nullspace of .

8.2 State-Space Control Design

  • Plant model: ,
  • State Feedback:
  • State Observer:
  • Integral Action (when ):

8.3 Controllability / Observability

  • Controllability: Ability to control each state (actuator/sensor location).
  • Controllability Matrix :
  • Observability Matrix :
  • Conditions:
    • Controllable if
    • Observable if

8.4 State Feedback

Topic: Controller Design

  • Controller Design: to achieve .
  • Design of Matrix :
    • Transform plant to control canonical form.
    • Use desired characteristic polynomial .
    • Compute , or use MATLAB place command.
  • Matrix for steady-state reference tracking:

    where

8.5 Observer Design

  • Controller Equations:

  • Observation Error:

  • Matrix : Controls transient response of observation error.

    • Duality of control and estimation: closed-loop poles of
    • Separation Property: Stable state feedback + stable estimation error = stable closed-loop.

  • Matrix for tracking:

    • If , estimation error is independent of reference.
    • Set and for “classical control scheme”.

8.6 Reference Tracking


  • State Feedback:
    • No feedback path with ; no integral action.
  • Inexact Plant Model Tracking:
    • Augment plant with integrator:
  • Controller Structures:
    • Case 1: and :
    • Case 2: and :
    • Case 3: , , add integral action:

Module 9: Digital Control (Addressing the limitations of controller design due to sampling, and the limitation of experimental platforms)

Discrete control #5: The bilinear transform - YouTube


The s-domain’s left half-plane maps to the area inside the z-domain’s unit circle, while the s-domain’s right half-plane maps to the area outside of the z-domain’s unit circle.

9.1 Artifacts Introduced in Digital Control

Topic: Sampling

  • Sensor quantization: perceived as noise.
  • Zero-Order Hold (ZOH) creates a delay of .
    • Transfer function of this delay:
    • Linear phase shift:
    • Approximation:
    • Rule of thumb: Choose sample frequency

9.2 The Z-domain

Topic: Z-Transform

  • Stability: Unit circle instead of Left-Hand Plane.
  • Mapping:
    • represents a forward time shift, is a derivative in Laplace.

9.3 Three Different Mappings

Topic: Z-Transform

  • Matched pole-zero: (rarely used).
  • ZOH: Used for discretizing plant to create discrete-time controller.
    • Solve .
    • Compute , then do partial fraction expansion.
  • Tustin: Used for discretizing controller.
    • Replace
    • Lead/lag filter:

9.4 Two Different Design Approaches

  1. Design CT controller, then discretize:
    • Take sampling delay into account (increases phase margin, PM).
    • Use .
    • Discretize controller using Tustin/Bilinear transform.
  2. Discretize CT plant to DT, then design DT controller:
    • Example:
      • , let and .
    • Sampling delay is considered; DT is not stable for all because of sampling (in CT: RHP zero).
    • Design controllers using standard techniques (Root locus, Nyquist, loop shaping).

9.5 Compensation in Discrete-Time

  • Proportional compensation:
    • CT:
    • DT:
    • Apply ZOH rules.
  • Derivative compensation:
    • CT:
    • DT:
    • Apply ZOH rules.
  • Integral compensation:
    • CT:
    • DT:
    • Apply ZOH rules.
  • Lead/lag compensation:
    • CT:
    • DT:
    • The relation between is not straightforward but not critical for design.

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