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BASIC CONTROL THEORY�LECTURE 2

TOPICS

  • Analysis in the frequency domain. Relationship with time domain analysis. Nyquist and Bode diagrams. Characteristic functions of basic elements (step and impulse responses, transfer functions, Nyquist and Bode diagrams). Approximating Bode amplitude diagrams.
  • Series and parallel connection, negative feedback and its resulting transfer functions. Properties of negative feedback. Resulting transfer functions in the control system between different output and input signals. Block diagram algebra.
  • Calculation of steady state for systems of different type with different input signals. The role of the integrator.
  • Relationship between the Bode diagrams of the open- and the closed-loop systems. Relationship with the time domain.

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THE AIM IS THE ANALYSIS AND DESIGN OF THE CONTROL SYSTEM

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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CONTROLLER C IS DESIGNED FOR PROCESS P

TO MEET THE QUALITY SPECIFICATIONS

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ANALYSIS OF THE CONTROL SYSTEM

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DOMAINS OF ANALYSIS

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ANALYSIS IN THE TIME DOMAIN�

  • Differential equation and its solution
  • Typical input excitations
  • Impulse and step responses
  • State equation

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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RELATIONSHIP OF THE TIME AND THE FREQUENCY DOMAIN

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Demonstration of the relationship between the analysis in the time and in the frequency domain

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THE FREQUENCY FUNCTION

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(Let us prove it)

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GRAPHICAL INTERPRETATION OF THE FREQUENCY FUNCTION

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Nyquist diagram

Bode diagram

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TRANSFER FUNCTION

With zero initial conditions the Laplace transform of the differential equation:

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The transfer function is the ratio of the Laplace transforms of the output and the input signals, respectively (polynomial/polynomial form):

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BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Its further forms (zero-pole, partional fractional representation, time constant form):

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GENERAL FORM OF THE TRANSFER FUNCTION

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General form of the differential equation:

General form of the transfer function in time constant form:

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RELATIONSHIPS OF THE TRANSFER FUNCTION, THE WEIGHTING FUNCTION, THE STEP RESPONSE, POLES AND TRANSIENTS

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BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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The location of the poles on the complex plane determines the transient behaviour in the time domain.

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CHARACTERISTIC FUNCTIONS OF THE BASIC ELEMENTS

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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FIRST ORDER LAG ELEMENT

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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It can be interpreted as a feedbacked integrator.

Differential equation:

Transfer function:

Frequency function:

Nyquist diagram

Bode diagram

The approximate Bode diagram

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SECOND ORDER LAG ELEMENT

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Differential equation

Transfer function

Poles

a./ Aperiodic case:

b./ Aperiodic border case:

c./ Oscillating case:

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SECOND ORDER LAG ELEMENT

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Step response

Overshoot

Time point of the overshoot

Settling time within ~5%

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SECOND ORDER LAG ELEMENT

The frequency function

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BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Nyquist diagram

Bode diagram

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CHARACTERISTIC FUNCTIONS OF PROPORTIONAL, INTEGRATING AND DIFFERENTIATING LAG ELEMENTS

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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SERIALLY CONNECTED INTEGRATOR AND A SECOND ORDER OSCILLATING ELEMENT

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BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Bode diagram

Nyquist diagram

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EFFECT OF THE ZEROS

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Inserting a zero accelerates the system. This is reached by overexcitation.

Inserting a zero modifies the Nyquist and the Bode diagrams.

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NON-MINIMUMPHASE SYSTEMS

The system has zero on the right side of the complex plane.

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BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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The absolute values are

the same, the phase

angles are different.

Step response of a

non-minimumphase system

Plot the phase-frequency diagrams of both systems.

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FAST DRAWING OF ASYMPTOTIC BODE DIAGRAMS

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APPROXIMATE MODELS. DOMINNANT POLE PAIR. APPROXIMATION OF DEAD TIME.

Dominant pole pair

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Approximation of a higher order proportional with first order lag and dead time

Approximation of dead time by rational transfer functions

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PADE APPROXIMATION OF A DEAD TIME ELEMENT

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Dead time element

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First order Pade approximation

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Second order Pade approximation

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Third order Pade approximation

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Approximation by a non-minimumphase rational fraction.

The first elements of the Taylor series of the dead time element and those of the rational fractions are the same.

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EXAMPLES FOR DESCRIPTION OF CONTINUOUS TIME SYSTEMS

  • DC motor
  • Fluid tanks
  • Heat process
  • Inverted pendulum
  • etc.

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Sysbook: Case studies

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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BASIC CONNECTIONS OF ELEMENTARY BLOCKS

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Serial connection

Parallel connection

Feedback

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BLOCK DIAGRAM ALGEBRA

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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EXAMPLE

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NICE PROPERTIES OF NEGATIVE FEEDBACK

  • Stability should be ensured
  • Tracking of the reference signal
  • Stabilising of an unstable process
  • Disturbance rejection
  • Improving the transient response
  • Feedback decreases the sensitivity of the system to parameter variations
  • In the range of high gains the feedback creates the approximate inverse of the feedback element
  • Feedback has a linearising effect
  • Feeding back an integrator by a static nonlinear element results in the inverse of the nonlinear characteristics

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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DEMONSTRATION OF SOME PROPERTIES OF NEGATIVE FEEDBACK

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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Calculate the value of y, if A=9.8 and if A1=980 and u=10

Sensitivity

Linearization effect

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RESULTING TRANSFER FUNCTIONS OF THE CONTROL SYSTEM

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TYPE NUMBER. STATIC SIGNAL TRANSFER PROPERTIES.

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The general form of the transfer function of the open loop:

The type number is the number of the integrators, it is denoted by i, its value can be 0, 1 or 2.

It does not influence the steady state. It influences the course of the transients.

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STATIC SIGNAL TRANSFER PROPERTIES

Steady state error in case of 0-type control system for unit step, ramp and parabolic reference signals:

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Type number

0

1

2

Unit step reference signal

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0

0

Ramp reference signal

∞

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0

Parabolic reference signal

∞

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∞

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INITIAL AND FINAL VALUES OF THE SIGNALS IN THE CONTROL SYSTEM

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The analytical expressions of the signals can be calculated from the resulting transfer functions.

The initial and final values of the signals can be calculated applying the final value theorem

of the Laplace transformation.

The initial and final values can be calculated also from physical considerations.

Example:

F=1, the reference signal is a unit step, the value of the disturbances is zero. Calculate the initial and final values of the output and the control signals, respectively.

Give the resulting transfer function of the closed loop. Determine the value of gain K to ensure that the value of the damping factor of the closed loop transfer function be 0.7.

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SENSITIVITY OF NEGATIVE FEEDBACK TO PARAMETER VARIATIONS. SENSITIVITY FUNCTION.

The parameters of the process may change.

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BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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A small change of the open loop, L=CP is

The relative change:

The resulting transfer function and its small change is:

The relative change:

Define the sensitivity function S as:

The sensitivity function shows how much the relative change of the process influences the relative change of the resulting transfer function.

T is the complementary sensitivity function.

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REQUIREMENTS SET FOR THE CONTROL SYSTEMS

  • Stability
  • Prescribed static accuracy for tracking of the reference signal and for disturbance rejection
  • Attenuation of the effect of measurement noise
  • Insensitivity to parameter changes
  • Prescribed transient behaviour
  • Consideration of the restrictions due to practical realization

BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI

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FORMULATION OF THE QUALITY SPECIFICATIONS IN THE TIME AND IN THE FREQUENCY DOMAIN

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Overshoot, settling time, integral criteria

In the time domain

In the frequency domain

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QUALITY SPECIFICATIONS IN THE FREQUENCY DOMAIN

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The controller should be designed for the given process

to fulfill the the quality specifications (loop shaping).

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