Control Systems 1
Modeling physical systems
Mechanical models
Heat exchanger
Energy balance of steam
Energy balance of water
Mechanical Impedance (Frequency-domain)
Electrical models
Two-Port Impedance (Frequency-domain)
Electro-mechanical models
DC Motor
Example used for introduction of feedback and feedforward control
Demographic models
Population balance
System identification
Input data u(t)
Output data y(t)
Determine blackbox in-between
Mathematical Basics
Laplace Transform
Remarks
The final value theorem only applies if there exists a finite value for y(t) with t going towards infinity.
Eigendomain
Eigenvalue Equation
Where:
Multiplicity
Algebraic multiplicity
multiplicity of an eigenvalue
Geometric multiplicity
number of linearly independent solutions of the eigenvalue equation (v)
Diagonalization
Condition
All eigenvectors must be linearly independent
Caley-Hamilton
Modes
Eigenvalues and Eigenvectors correspond to modes
Theorems
Final Value theorem
Can only be used if the function is bounded in the time-domain.
System representations
Describing models
Higher-order ODE:
State-space model
Transfer function in the Laplace domain
Nonlinear systems & Linearization
General state-space model of a dynamical system
Example: Inverted pendulum
Conservation of angular momentum
Equilibria
Linearization
Example: The water tank
Mass balance
Bernoulli equation
Time-invariance
Causality
LTI Systems
General solution to LTI Systems
General LTI System
Step 1:
Step 2:
Step 3:
Step 4:
Stability of LTI Systems
Open-Loop
Example
Open-Loop Dynamics
Stability
Closed-Loop
Example
Closed-Loop Dynamics
Stability
Root Locus
A root locus is the plot of the eigenvalues of the closed-loop dynamics as a function of k. It can be used to determine the behaviour of the system.
I/O maps
Example: Model of a Spring-Damper System with two sprung masses.
Transfer Function:
Zeros:
System Response to fixed frequency excitation. The result seems counterintuitive, because the output doesn't diverge.
One can even choose the initial conditions to produce an all zero response:
Output zeroing condition
General formula
Here we plug in:
Matrix form
Connection to "zeros" of the transfer function
Control Approaches
Feedforward vs. Feedback control
Transfer function of the motor example:
Steady state simplification (s=0)
Feedforward Control
Feedback Control
Feedback control reduces the effect of disturbances by a factor 1 + AK
Model uncertainty
Applied to the motor example with feedforward contrl:
Model uncertainty of 1 means, that you have zero additional control over the behaviour of the motor by turning up the K value.
Applied with feedback control:
In conclusion, with feedback one can regulate the uncertainty with a higher K.
Impact on dynamic performance
Feedforward control: Gain k does not affect the performance of the sysem
Feedback control: Gain K does affect the dynamics
Designing a Closed Loop Controller
Good dynamic performance is desirable
Desirable behaviour of the open-loop gain
MIMO Systems
Convert to superposition of SISO Systems
Set up:
To analyze the stability of f.e. the system response y(s). First check the stability of the transfer functions
If all sub-tranfer-functions are stable, one can assume the MIMO system to be stable aswell
Lyapunov Stability
Conditions for the Lyapunov function
Confirm positiveness of the lyapunov function
Nyquist Plots
From Plot to Transfer function
The phase determines where the nyquist plot should enter the point you consider.
From Transfer function to Plot
Go via Bode Plot
Choose frequencies in 45° steps and choose more in between if not monotonic in between.
Extract magnitudes at those points
Plot nyquist using the magnitude and phase of the points just like any other polar coordinate plot.
Directly
Decompose transfer function into real and imaginary part
Set Imaginary part to zero, to find the intersections with the real axis and vice-versa.
Infer the rest from the bode plot -> Derps
Useful variations
F(s) = 1 + k*L(s) where L(s) is the open-loop transfer function
Going via F(s) = k*L(s) and counting encirclements of -1 gives a good idea on gain margins
Going via F(s) = L(s) and counting encirclements of -1/k gives a good idea on how to choose k to stabilize the system
Time-Delay Systems
By plotting the nyquist plot of a system with a delay one can tell if adding a delay causes instability
Phase Margin
= Robustness against Delays
Gain Margin
= Robustness against Gain Uncertainties
Bode Criterion
Assume: - open-loop gain L(s) is stable - Gain & Phase should be monotonically decreasing
Subtopic 1
Definitions
Strictly proper
The order of the denominator is strictly higher than the numerator.
Steady-state
After transients, ergo at time infinity.
Delays
Time delays can destabilize a system
Important with networked control systems that involve communication delays
PID-Control
Proportional Gain
Integral Gain
System has de facto no steady-state error.
Derivative Gain
Ziegler-Nichols
Oscillation method
Increase P gain until sustained oscillations are observed
Go to stability boundary
Astrom & Haglünd
Excite the system using a square wave
Note
Map 1
500 % 400 % 300 % 200 % 150 % 120 % 100 % 80 % 50 % 20 %