Control Systems Transfer Function Quiz

Study Notes

This presentation covers modelling and simulation of control systems, focusing on system identification techniques.
Dr. Hatice Doğan is the presenter.

System identification is about creating models of dynamic systems from observed data.

Nonparametric models describe system behavior using curves or functions, not parameters.
They require a large number of measurements for an acceptable representation.
Examples of nonparametric models include impulse or step response records in the time domain and frequency records like Bode diagrams.

Parametric models concentrate all the information into a model structure with limited parameters.
This approach makes the model more economical and powerful, contrasting with nonparametric models that require numerous measurements.

The unit impulse function, also known as the Dirac delta function, is defined heuristically.
The function has a value of zero for all 't' not equal to zero, and its integral over all time is equal to one.
The impulse response of a linear time-invariant (LTI) system is a time-shifted impulse input to the system, this is g(t-t0).
The practically applicable convolution model is derived based on assumptions of linearity and time-invariance of the system.
If the impulse response g(t) is known, the output (y(t)) of the system can be calculated easily given a specified input (u(t)).

The Laplace transformation of the convolution model describes an algebraic relationship between the transformed output signal (Y(s)) and the transformed input signal (U(s)).
The function G(s) represents the Laplace-transformed impulse response function and is referred to as the transfer function.
The relationship Y(s) = G(s) U(s) is the transfer function model representation.

Direct impulse response identification: If the input is a unit pulse, the coefficients of g(t) can be directly determined from the measured output.
Algorithm I:
- Generate a pulse input (with a specific magnitude).
- Apply the pulse to the system.
- Determine the impulse response using the formula: y(t)/a = g(t)
Impulse response identification using step responses:
- Algorithm II:
  - Generate a step input.
  - Apply the step to the system.
  - Determine the impulse response using the formula: (y(t)-y(t-1))/a = g(t).

Several examples are demonstrated, illustrating different scenarios of system identification.
- These examples include both discrete-time and continuous-time systems.
- They show the use of difference equations and convolution sums for calculating outputs.

A thermal system example using experimental data collected from Daisy database is presented.
The example shows how to estimate system parameters using nonparametric models followed by validation.

The general form of a first order system is represented as H(s) = K/(Ts+1) where:
- K: the gain
- T: the time constant

The general form of a second-order system is represented as H(s) = Kw^2/(s^2+2ξw_ns+w_n"2) where:
- K: the gain
- ξ: the damping factor
- w_n: the natural frequency

The damping factor ξ shapes the step and impulse responses.
Different ξ values create distinct response curves.
- Different damping cases exist as 0 (undamped), 0<ξ<1 (underdamped), ξ=1 (critically damped), ξ>1 (overdamped)

Graphical methods showing how to determine if a system is first or second order.
Considerations relating to derivatives when determining the type of order.

The methods for practical impulse responses are discussed with relation to rectangular signals.
The relation to practical limitations in impulse responses.

Relation between impulse and step response in the Laplace domain.
- Impulse response is the derivative of the step response.

The state space model is shown for both first-order and second-order systems.
Definitions and characteristics of the matrices (A, B, C, D) are reviewed.
Example calculation of the approximate state-space matrices is shown for both first and second order systems using the identified parameters (gain, time constants for first order and gain, damping factor, natural frequency for second order).