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- Theory and Application of Digital Control - 1st Edition
- Computer-Controlled Systems: Theory and Design, 3rd Edition
To use Algorithm 5. Step 1. Solution with higher order observer: The solution above is not the only one solving the original problem. To solve the equation for the closed loop characteristic polynomial we must increase the order of R by one. In this case there is a delay of one sample from the measurements y to the control signal u.
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In both cases we get the same closed loop pulse-transfer function from uc to y since the observer polynomial is cancelled by T z :. The desired closed loop pulse-transfer function is. Cancellation of process poles and zeros is handled by Algorithm 5.
Thus, we must increase the order of the controller by one and, consequently, add an observer pole which is placed at the origin, i. Output 1 1.
Control 0. The ringing in the control signal in Case a is due to the cancellation of the process zero on the negative real axis. Case a should probably be avoided because of the ringing in the control signal. Using the controller. Further the pole of the process is cancelled. Bm must contain the zeros of B that are outside the unit circle.
Am , Bm must contain the poles of the process that are outside the unit circle. Equation 5. By forcing the factor z , 1 into R z we thus have obtained integral action in the controller.
The design problem is solved by using the general Algorithm 5. The reference value is zero and there is an initial value of the state in the process.
Publications 2000 – present
The design in Problem 5. The right hand side of the inequality is also shown solid. The right hand side is also shown. Further the desired model is stable, i. The rule of thumb on p. Using Example 2. The controller polynomials are now given by 5. This will increase the degree of the closed loop by one compared to Problem 5. Using Theorem 5. Problem 6. A disturbance in the process will propagate in the direction of the ow. Figure 6. The system is then represented with the block diagram in Fig.
Notice the order of the states. The system will remain stable for all positive values of Ki.
A disturbance will now propagate in the direction opposite the ow. Cascade control loops are found for the cooling media ow and for the output product ow. Feedforward is used for the level control loop where the input ow is used as a measurable disturbance. The input ow is also used as feedforward for the cooling of the jacket. Nonlinear elements are used in the ow control loops of the product out- put and the coolant ow.
The square root device is thus used to remove the nonlinearity of the measurement device. An intentional nonlinearity is introduced in the selector. Either the tem- perature or the pressure is used to control the coolant ow depending on the status of the process.
Theory and Application of Digital Control - 1st Edition
Problem 7. Since sampling is a linear operation we consider each component of f t sep- arately. The sampled signal has the Fourier transform, see 7. The Fourier transform of sin! Thus, if the signal sin! All frequencies in the interval ,f1 ; f1 should be possible to reproduce from the samples of the continuous time signal.
Problems 7. Figure 7. The frequency of the camera shutter! The picture will not move if! A correct picture will be seen, if! The wheel will appear to rotate with a frequency lower than r if! For instance let! Aliasing will give a frequency! The wheel then appears to rotate three times slower and in the wrong direction.
Compare the stroboscope. Sampling the signal u t gives the alias of sin 6! Problem 8.
Forward differences Backward differences Tustin Figure 8. Eulers method, b. Tustin's approximation. Euler's method gives. Zero order hold sampling gives. The shorter sampling period gives a better approximation. This approximately corresponds to a decrease of the damping by , This gives! The derivative part of 8. Euler's approximation gives a delay of one sampling interval before an error will in uence the integral part. The sampling interval is, however, usually short for digital PID-algorithms. At the desired cross over frequency we have jG i! The sampling interval is 6 dash- dotted and 12 seconds dashed.
The zero of the closed loop system is , Using the rule of thumb from Section 8. Using 8. A discrete time 8 state9 space representation of the motor is given in A. Modifying L using 8. The response for the continuous-time controller is also shown solid. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit.
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Computer-Controlled Systems: Theory and Design, 3rd Edition
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