Óbudai Egyetem Dr. Neszveda József Open and Closed loop Control II. Block diagram model

Modelling dynamical systems Engineers use models which are based upon mathematical relationships between two variables. We can define the mathematical equations: Measuring the responses of the built process (black model) Using the basic physical principles (grey model). In order to simplification of mathematical model the small effects are neglected and idealised relationships are assumed. Developing a new technology or a new construction nowadays it’s very helpful applying computer aided simulation technique. This technique is very cost effective, because one can create a model from the physical principles without building of process.

LTI (Linear Time Invariant) model The all physical system are non-linear and their parameters change during a long time. The engineers in practice use the superposition’s method. x(t)y(t) x(j  ) y(j  ) First One defines the input and output signal range. In this range if an arbitrary input signal energize the block and the superposition is satisfied and the error smaller than a specified error, than the block is linear.

Block model (Classical method) w(t)y(t) W Y w(t) y(t) t t The steady-state characteristic. When the transient’s signals have died a new working point WP 2 is defined in the steady-state characteristic. The dynamic behaviour is describe by differential equation. Using the Laplace transform method the transfer function replace the differential equation. WP 2 WP 1 We assume the variables are within a range and the output remain in this range in steady-state.

Performance of a block Frequency transfer function: All block has frequency transfer function, but not all time signal - x(t), y(t) - can be converted to frequency form. Using Laplace transform it is possible. If the system investigation is started from WP 1 in steady-state than the x(t) and y(t) signals form are such as they are multiplied by 1(t) unit step. y(t) t W Y x(t) t WP 2 WP 1

Correlation between frequency and time domain of linear systems Fourier and inverse Fourier transform Laplace and inverse Laplace transform The above is true if:

Laplace transform Rules of Laplace transformLaplace transform of standard signals If pole of sY(s) are in the left half of the s-plane the final value theorem: is available.

Using Laplace transform Using the rules of Laplace transform to convert a differential equitation to operator frequency form. x(t)y(t) x(j  ) y(j  )

The steady-state characteristics and the dynamic behavior x(t) y(t) t t The steady-state characteristic. When the transient’s signals have died a new working point WP 2 is defined in the steady-state characteristic. The dynamic behavior is describe by differential equation or transfer function in frequency domain. WP 2 WP

Transfer function in frequency domain Amplitude gain: Phase shift:

The graphical representation of transfer function The M-α curves: The amplitude gain M(ω) in the frequency domain. In the previous page M(ω) was signed, like A(ω) The phase shift α(ω) in the frequency domain. In the previous page α(ω) was signed, like φ(ω)! A Nyquist diagram: The transfer function G(jω) is shown on the complex plane. A Bode diagram: Based on the M-α curves. The frequency is in logaritmic scale and instead of A(ω) amplitude gain is: A Nichols diagram: The horizontal axis is φ(ω) phase shift and the vertical axis is The a(ω) dB.

The basic transfer function In the time domain is the differential equitation In the frequency domain is the transfer function

Block representation Actuating path of signals and variables One input and one output block represents the context between the the output and input signals or variables in time or frequency domain Summing junction Take-off point (The same signal actuate both path) G1G1 G2G2 G1G2G1G2 G1G1 G2G2 G 1 +G 2 G1G1 G2G2

P proportional Step response Bode diagram t

I Integral Step response Bode diagram

D differential Step response Bode diagram The step response is an Dirac delta, which isn’t shown

PT1 first order system Step response Bode diagram

PT2 second order system Step response Bode diagram

PH delay Step response Bode diagram

IT1 integral and first order in cascade Step response Bode diagram

DT1 differential and first order in cascade Step response Bode diagram

PI proportional and integral in parallel Step response Bode diagram

PDT1 Step response Bode diagram

Block diagram manipulation G1G1 G1G1 G2G2 G1G2G1G2 G2G2 G 1 +G 2 G1G1 G2G2 G1G1 G1G1 G1G1 G1G1 G1G1 G1G1 G1G1 G1G1 G1G1 G1G1

Block diagram reduction example G5G5 G2G2 G7G7 G6G6 G1G1 G3G3 G4G4