�BASIC CONTROL THEORY�LECTURE 7�
TOPICS
2
DEMONSTRATING EXAMPLE: INTEGRATOR
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correction
The approximation is valid, if ω<1/Ts
Thus, in the low frequency range, the amplitude-frequency diagram of the continuous and sampled systems are
roughly the same, the discrete phase-frequency diagram differs from that of the continuous one with the angle
of the additional dead time of value Ts / 2.
SAMPLING AND HOLDING INTRODUCES ADDITIONAL DEAD TIME
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Taylor approximation:
Sampling and holding introduces additional dead time whose value
is approximately half of the sampling time.
DEMONSTRATING EXAMPLE: FIRST ORDER LAG ELEMENT
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If Ts/T1<1 and ωTs<1
Example: T1=0.1, Ts=0.1
Bode diagram
THE SAMPLED CONTROL SYSTEM
Continuous and discrete signals appear in the sampled control system.
The control algorithm is implemented with a real-time program.�When designing the controller, the additional dead time resulting from sampling should be taken into account.
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REQUIREMENTS SET FOR THE CONTROL SYSTEM
The requirements are the same as for continuous systems.
Stability
Prescribed static accuracy for reference signal tracking and
disturbance rejection
Dynamic prescriptions (overshoot, settling time)
Keeping the the control signal within the prescribed limits
Features different from continuous case:
The effect of sampling and signal conversion (loss of information)
The value of the control signal is constant between two sampling
points
Investigation of behavior between sampling points
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THE CONTROLLER IS DESIGNED� FOR THE MODEL OF THE PROCESS
The controller should be designed so that the controller meets the quality specifications.
When designing the controller the model of the process is considered.
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 domain.
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DISCRETE PID CONTROLLERS
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CONTINUOUS CONTROLLER | DISCRETE CONTROLLER | DIFFERENCE EQUATION | STEP RESPONSE |
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Ideal PD: ----- | | | |
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THE IDEAL DISCRETE PD CONTROLLER IS REALIZABLE
as its overexcitation (the ratio of its initial and final value for step input) is not infinity. (In continuous case the overexcitation is infinity, therefore the continuous ideal PD controller is not realizable.)
Calculate the value of the overexcitation for unit step input.
Initial value:
Final value:
The overexcitation:
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DESIGN OF DISCRETE PID CONTROLLER BASED ON CONSIDERATIONS IN THE CONTINUOUS FREQUENCY DOMAIN
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Pole cancellation technique can be used.
For example
With PI controller the biggest time constant is „cancelled” and instead an integration effect is introduced.
With PD controller the second biggest time constant is cancelled and instead a smaller one is introduced.
The gain factor of the regulator kc is chosen from the Bode diagram of the open loop ensuring
phase margin of ~60°. For this the cut-off frequency should be located at the straight line
of slope -20dB/decade and . Td is the continuous dead time, Ts is the sampling time.
The discrete Bode amplitude diagram approximately coincides with the continuous one until ω ~1/Ts, it is drawn until this point. From control point of view, this range is of interest.
FIRST EXAMPLE FOR CONTROLLER DESIGN
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The transfer function of the continuous process:
The sampling time: Ts=0.5 sec
The pulse transfer function:
Requirements: follow the unit step reference signal without static error;
The control should be as quick as possible; the phase margin should be approx. 60 °.
The task can be solved with PI controller using pole cancellation.
The dead time together with the additional dead time: Th=2+0.25, the cut off frequency should be located
approximately at 1/2Th=0.218.
(Draw the Bode diagramot until ω=2.)
FIRST EXAMPLE FOR CONTROLLER DESIGN (continued)
The pulse transfer function of the open loop:
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Its frequency function:
Its absolute value at the cut off frequency is 1.
whence kc=0.1724
The phase margin: 61.89°. u(0)=kc; u(∞)=1.
The difference equation of the controller:
The behavior of the control system over time can be simulated with matlab and with simulink.
SECOND EXAMPLE FOR CONTROLLER DESIGN
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Design a discrete serial PID controller, fulfilling the
following quality specifications:
- Phase margin ≅ 60°
- The control should be as quick as possible;
- The static error for step reference signal should be 0.
The transfer function of the continuous process:
The sampling time: Ts=1sec
The pulse transfer function:
The pulse transfer function of the PID controller
with pole cancellation:
The pulse transfer function of the open loop:
SECOND EXAMPLE FOR CONTROLLER DESIGN (continued)
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The frequency range is considered till 1/Ts=1. Draw the Bode diagram.
The cut off frequency should be located approximately at 1/(2(1+1)=0.25 to ensure the required phase margin.
This can be calculated by Matlab. Kc is obtained appr. as 15.
Behavior between sampling points
can be checked with the simulink program.
CHECKING THE INITIAL AND FINAL VALUES� OF THE OUTPUT AND THE CONTROL SIGNALS
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The control signal should be kept within limits for practical reasons. (Avoid saturation of the actuator.)
The values can be calculated from the limits of the resulting transfer functions, or based on simple considerations.
For step reference signal:
With simple considerations: if the process contains lags, the output value will initially be 0. If the controller
has an integrator, the output signal is set exactly to 1 in steady state. The final value of the control signal
is the reciprocal of the gain of the process. In t = 0, the feedback signal value is 0, u(0) can be calculated
as if the reference signal is directly exciting the controller.
OTHER CONTROLLER DESIGN METHODS
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whose parameters can be derived from the parallel connected discrete P, I and D effects.
a./ Intermediate continuous controller design and its discretization.
The model of the continuous process is supplemented with the holding element, and for this we design
a continuous controller. Then the discrete equivalent of the controller is determined.
b./ Design of discrete controller based on the discretised process model.
The pulse transfer function of the continuous process is determined. A discrete controller is designed
considering this pulse transfer function which ensures that the pulse transfer function
of the closed loop system fulfills given design specifications. (See Lecture 8.)
c./ Design of discrete controller based on the continuous process model.
(based on frequency domain considerations, this method was dealt with previously.)
STATE FEEDBACK IN SAMPLED DATA SYSTEMS
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The discrete state equation:
The poles of the system are obtained by the roots of the characteristic equation:
The aim of control is to accelerate the dynamic behavior, one of the ways of which is to feed back the state variables. By specifying the poles of the closed loop system, we determine the degree of acceleration.
The control signal is obtained by feeding back of discrete state variables:
The characteristic equation of the closed loop system:
F and g are known, the prescribed poles are given. The state feedback vector can be calculated (Ackermann formula).
As there is no integrating effect in the control system, static compensation is required.
CALCULATION OF THE STATIC COMPENSATION
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whence
and the correction factor:
STATE FEEDBACK IN SAMPLED DATA SYSTEMS
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The block scheme of the state equation
of the discrete process
Block scheme of state feedback in discrete system
It can be enhanced with an additional integrating state variable.
STATE FEEDBACK WITH STATE ESTIMATION
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The state feedback loop should be faster
than the process.
The state estimation loop should be faster
than the state feedback loop.
Example in matlab exercises material.
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