��BASIC CONTROL THEORY�LECTURE 6��
TOPICS
and in the z-operator domain with the pulse transfer function
THE STRUCTURE OF A SAMPLED DATA CONTROL SYSTEM
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
2
2019
Nowadays, it is becoming increasingly commonplace for control systems to be implemented by digital devices with adequate real-time services. Depending on the number and complexity of controls within a given application, the range of digital control devices ranges from single-chip microcontrollers to more powerful midi / mini digital computers. Networking of digital equipment also allows for the creation of distributed control systems.
ADVANTAGES OF THE DIGITAL TECHNOLOGY
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2019
THE STRUCTURE OF A SAMPLED DATA CONTROL SYSTEM
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4
2019
THE STRUCTURE OF A SAMPLED DATA CONTROL SYSTEM
In computer process control, the basic tasks of continuous process control are performed by the computer in real time. The output signal must be digitized for computer processing. At certain sampling intervals, a continuous signal is sampled and, after quantization and digitization, is assigned a numerical value that can be received by the computer. These operations are performed by the A / D analog/digital converter. Through this the continuous process is connected to the computer.
In the computer, in each sampling interval so-called „tasks” are running, which, when implementing closed-loop control, form a digital reference signal, receive the signal from the process and possibly filter it, compare the reference signal with the measured output signal value, and then generate an actuator signal based on a discrete control algorithm. The discrete control algorithm is implemented by a program.
Then the computer forwards the control signal toward the process. This is a digital signal, which must be converted into an analog physical that can actuate the actuator (e.g., voltage, valve stem position, etc.). This operation is performed by the D / A digital/analog converter, which connects the computer to the continuous process.
It should also be ensured that the input signal of the actuator signal is available between the sampling points. This is provided with a hold element that most often holds the value of the previous signal until a new signal arrives at the next sampling point (zero order hold, ZOH).
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
5
2019
SAMPLING
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6
2019
Physical sampling
Mathematical sampling
Impulse modulation
Sampling entails information loss.
No information is available between sampling points.
In A / D converter, quantization and coding also mean information loss.
SHANNON THEOREM
The sampling time must be carefully selected: it must be consistent with the dynamics of the process being controlled and with the capabilities of the real-time processing system (operational speed, accuracy of numerical representation).
SHANNON THEOREM:
The sampling frequency should be at least twice the frequency of the component with the highest frequency in the signal. In the time range, this means that at least one sample has to be taken every half period of an oscillating signal.
The smaller the sampling time, the closer the operation of the control system approaches the continuous operation. With larger sampling times, fewer calculations are required. By choosing the sampling time we can influence the dynamics, the speed of the control, the magnitude of the control signal.
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7
2019
HOLDING
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8
2019
Function of the holding element: A signal should be provided between two sampling times.
Zero order holding
First order holding
In practice, the use of a zero order holding is common.
ZERO ORDER HOLDING (ZOH)
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2019
Taylor approximation:
Sampling and holding adds extra time delay to the system,� whose value is approx. half sampling time.
ELEMENTS AND SIGNALS IN A SAMPLED DATA CONTROL SYSTEM
Continuous and discrete signals appear in the sampled control system.
The control algorithm is implemented with a real-time program.�New types of sensors have been developed transforming analog inputs to digital outputs.
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
10
2019
REQUIREMENTS SET FOR A CONTROL SYSTEM
The requirements are the same as for continuous systems.
Stability
Prescribed static accuracy for reference signal tracking and
disturbance rejection
Prescriptions for dynamic behaviour (overshoot, settling time)
Keeping the control signal within the prescribed limits
Features different from the continuous case:
Impact of sampling (loss of Information)
The value of the control signal is constant between two sampling
points
Investigation of behaviour between sampling points
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
11
2019
EXAMPLE FOR A SAMPLED DATA CONTROL SYSTEM
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12
2019
The process is an integrator, the controller is a constant element. The sampling time is Ts, zero order hold is applied at the input of the process.
Give the difference equation of the closed loop control system, calculate the values of the output signal in the first 5 sampling points for unit step reference signal, if K=1 and Ts=0.5, 1, 1.5 and 3 sec.
Rearranging:
If Ts tends to 0,
The continuous system is structurally stable.
EXAMPLE: THE VALUES OF THE OUTPUT SIGNAL
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13
2019
For Ts=0.5: 0; 0.5; 0.75; 0.875; 0.9375
For Ts=1: 0; 1; 1; 1; 1
For Ts=1.5: 0; 1.5; 0.75; 1.125; 0.9375
For Ts=3: 0; 3; -3; 9; -15
The sampled system did not preserve the structurally stable nature
of the continuous system.
At Ts=1 the settling is reached in finite time within one sampling step.
Such a phenomenon does not occur for continuous systems.
DESCRIPTION OF A SAMPLED SIGNAL IN THE TIME DOMAIN. � Z-TRANSFORMATION
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14
2019
The Laplace transform of the discrete signal:
Introduce the following notation:
This is the z – transformation.
This is the shift operator.
or
This is the z –transform of the signal.
Z-TRANSFORMATION MEANS COMFORM MAPPING
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15
2019
The z-transformation maps the complex plane s to another complex plane z. The imaginary axis of plane s is formed into a unit radius circle. The left half plane is inside the circle, the right half plane is outside the circle.
Because the entire left half plane "shrinks" to the inside of the unit radius during mapping, greater precision is required in the z-domain when specifying each point than in the s-domain.�(Coefficients, poles, zeros in the s-domain are sufficient to give with 2 decimal accuracy, z-domain requires at least 4 decimals.)
SOME PROPERTIES OF Z-TRANSFORMATION
Initial value:
Final value:
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16
2019
Z – TRANSFORMS OF SOME BASIC SIGNALS
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17
2019
Z – TRANSFORMS OF SOME BASIC SIGNALS
BARS RUTH*, KEVICZKY LÁSZLÓ**, HETTHÉSSY JENŐ*, MAX GYULA*, VÁMOS TIBOR**, *BME AAIT, **SZTAKI
18
2019
But this is not a geometric series, it cannot be converted into a closed form.�The signal can be specified as the sum of shifted jumps.
Z – TRANSFORMS OF SOME BASIC SIGNALS
can be derived from the z-transform of the exponential signal.
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19
2019
A METHOD TO DETERMINE THE Z-TRANSFORMS
The form of a continuous signal:
Its Laplace transform:
Its z-transform:
and then bring it to a common denominator.
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2019
There are z-transformation tables.
(See the Lecture notes.)
INVERSE Z-TRANSFORMATION
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21
2019
Inversion integral
is located inside the circle of radius C sugarú around the origo.
Each pole of
Practical methods:
With polynomial division
With partial fractional expansion
Example
PULSE TRANSFER FUNCTION
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22
2019
Calculation of step equivalent pulse transfer function:
PULSE TRANSFER FUNCTIONS OF BASIC ELEMENTS
Integrator
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23
2019
Or with step equivalent calculation: for unit step the output of the integrator is a ramp.
Remark: In continuous case the pole is in s1=0, in discrete case it is in z1=1.
PULSE TRANSFER FUNCTIONS OF BASIC ELEMENTS
First order lag element
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2019
Remark: In continuous case the pole is at s1=-1/T1, in discrete case the pole is at z1=exp(-Ts/T1)
PULSE TRANSFER FUNCTIONS OF BASIC ELEMENTS
Second order lag element
If there is a sampler and a holder between two first order lag elements connected in series, the pulse transfer functions are simply multiplied.
If there is no sampler in between:
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2019
Remark:
There are two poles and one zero,
Because of sampling and holding an additional zero is introduced!
EXAMPLE
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2019
The sampling time is Ts=5 sec
If there is a sampler and holder between the two first order lags:
If there is no sampler and holder between the two first order lags:
PULSE TRANSFER FUNCTIONS OF BASIC ELEMENTS
Pulse transfer functions of elements containing more lags, pulse transfer functions of elements given by rational can be calculated similarly, by partial fractional expansion.
In the pulse transfer function of element containing more lags, the number of zeros is 1 less than the number of poles. There will be zeros also outside the unit circle.
There are tables providing the pulse transfer functions of the most important elements.
Dead time element:
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2019
EXAMPLE
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28
2019
The pulse transfer function of the closed loop system:
Notation:
For unit step reference signal:
K=1 and for Ts=0.5 :
The result is the same as obtained with the difference equation.
ABOUT STABILITY
The roots of the characteristic equation (the poles of the pulse transfer function of the closed loop) in the z-domain should be inside the unit circle.
For our example:
The range of stability:
(The same result was obtained analysing the difference equation.)
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2019
TRANSITION FROM THE PULSE TRANSFER FUNCTION TO THE DIFFERENCE EQUATION, DISCRETIZATION OF THE INTEGRATOR
Example: Integrator
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2019
Transition to the difference equation:
This is a recursive relationship,
which is easily programmable.
Zero order hold corresponds to the left hand rectangle rule.
According to the right hand rectangle rule:
The corresponding pulse transfer function:
According to the trapezoid rule:
The corresponding pulse transfer function:
DISCRETE STATE SPACE MODEL
The discrete state equation can be determined by sampling the continuous state variables.
The continuous state equation and its solution:
Zero order hold is applied. Let us execute the integration between two consecutive sampling points.. Let be the beginning of the sampling interval and the end of the interval.
In this range is constant.
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2019
DISCRETE STATE SPACE MODEL
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2019
where
If matrix A is invertible,
The pulse transfer function on the basis of the discrete state equation:
RECURSIVE SOLUTION OF THE STATE EQUATION
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2019
Recursive solution:
PREDICTIVE STATE SPACE MODEL
With a so called predictive model, we intend to determine and predict the values of the future output signal or state variables of the system based on the available information and data on the input signal and the state variables (or the output signal) up to the given k sampling moment. Assume d = 0.
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2019
PREDICTIVE STATE SPACE MODEL
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2019
The vertical lines mean that the predicted value of the signal is determined based on the information available up to the current time k. In vector / matrix form up to 4 predicted points:
The first part of the right side of the equation is the so-called free response. The second part gives the forced response. The free response is the effect of past interventions on the future time horizon, and we cannot change this, we must bear that effect. The generated forced response gives the effect of the current and next input interventions on the output signal. These interventions can be chosen so that the output signal follows a given reference signal as closely as possible over a specified future time horizon. We can only influence the future behavior of the system with input effects at current and future times.
EXAMPLE: DISCRETE STATE SPACE MODEL OF A DOUBLE INTEGRATOR
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2019
The continuous state equation:
The state matrices of the discrete state equation:
The pulse transfer function:
For Ts=1:
DETERMINING THE STATE EQUATION FROM THE PULSE TRANSFER FUNCTION
For a given pulse transfer function, an infinite number of state space descriptions can be generated which give the same output to a given input.
Consider a third-order pulse transfer function.
Observable canonical form
(Remark: d is not zero only if the degree of the numerator and denominator of G are equal.)
Restructuring:
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2019
or
OBSERVABLE CANONICAL FORM
Further conversion supposing d=0:
Rearranging:
Based on this the block scheme
and the state equation:
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2019
The state variables can be recorded at the output of the shift elements.
CONTROLLABLE CANONICAL FORM
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2019
Conversion of the pulse transfer function by introducing the auxiliary variable V :
Based on this expression:
The block scheme:
The controllable form of the state equation:
CANONICAL FORM
With partial fractional expansion:
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2019
Introduce the state variables as follows:
The block scheme:
The canonical state equation:
THE CONTROLLER IS DESIGNED� FOR THE MODEL OF THE PROCESS
The controller must be designed so that the controller meets the quality specifications.
When designing the controller, we consider the process model.
The process model can be the pulse transfer function or the state equation.
The control loop can be examined and designed
in the time, the z-operator, and the frequency domains.
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2019
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