761 Mapping from the splane into the zplane
Just as transient analysis of continuous systems may be undertaken in the s-plane, stability and transient analysis on discrete systems may be conducted in the z-plane. It is possible to map from the s to the z-plane using the relationship z e a Jw r earejwr using the positive jw value 7.62 Fig. 7.16 Mapping from the s to the z-plane. If eaT z and T equation 7.62 can be written Equation 7.63 results in a polar diagram in the z-plane as shown in Figure 7.16. Figure 7.17 shows mapping of lines of...
942 The Kalman filter single variable estimation problem
The Kalman filter is a complementary form of the Weiner filter. Let Ax be a measurement of a parameter x and let its variance Pa be given by where Ax is the mean and E is the expected value. Let Px be a measurement from another system of the same parameter and the variance P is Assume that x can be expressed by the parametric relationship where K is any weighting factor between 0 and 1. The problem is to derive a value of K which gives an optimal combination of Ax and Px and hence the best...
372 Step response performance specification
The three parameters shown in Figure 3.21 are used to specify performance in the a Rise time tr The shortest time to achieve the final or steady-state value, for the first time. This can be 100 rise time as shown, or the time taken for example from 10 to 90 of the final value, thus allowing for non-overshoot response. b Overshoot The relationship between the percentage overshoot and damping ratio is given in equation 3.68 . For a control system an overshoot of between 0 and 10 1 lt C gt 0.6 is...
46 Case study examples
Example 4.6.1 CNC Machine-Tool Positional Control See also Appendix 1, examp461.m The physical configuration and block diagram representation of a CNC machinetool is shown in Figures 1.10 and 1.11. The fundamental control problem here is that, by design, the lead-screw by the use of re-circulating ball-bearings is friction-free. This means that the positional control system will have no damping, and will oscillate continuously at the undamped natural frequency of the closed-loop system. Damping...
911 Types of optimal control problems
a The terminal control problem This is used to bring the system as close as possible to a given terminal state within a given period of time. An example is an automatic aircraft landing system, whereby the optimum control policy will focus on minimizing errors in the state vector at the point of landing. b The minimum-time control problem This is used to reach the terminal state in the shortest possible time period. This usually results in a 'bang-bang' control policy whereby the control is...
354 Experimental determination of system time constant using step response
Method one The system time constant is the time the system takes to reach 63.2 of its final value see Table 3.2 . Method two The system time constant is the intersection of the slope at t 0 with the final value line see Figure 3.13 since This also applies to any other tangent, see Figure 3.13. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Time Constants Fig. 3.13 Unit step response of a first-order system. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Time Constants Fig. 3.13 Unit step response of a...
355 Ramp response of firstorder systems
Find an expression for the response of a first-order system to a ramp function of slope Q. Zo s _ s2 1 Ts s2 s 1 T _7 2 i T T 3 2 See partial fraction expansion equation 3.13 . Multiplying both sides by s2 s 1 T , we get i.e. Q- As2 As Bs B Cs2 3.33 Equating coefficients on both sides of equation 3.33 Substituting into 3.35 Hence from 3.34 Fig. 3.14 Ramp response of a first-order system see also Figure A1.1 . Fig. 3.14 Ramp response of a first-order system see also Figure A1.1 . Fig. 3.15 Unit...
422 Block diagram manipulation
There are occasions when there is interaction between the control loops and, for the purpose of analysis, it becomes necessary to re-arrange the block diagram configuration. This can be undertaken using Block Diagram Transformation Theorems. Table 4.1 Block Diagram Transformation Theorems Table 4.1 Block Diagram Transformation Theorems Moving a summing point ahead of a block. Z GX Y Z X G Y G 4.9 A complete set of Block Diagram Transformation Theorems is given in Table 4.1. Example 4.3 Find the...
441 DC servomotors
One of the most common devices for actuating a control system is the DC servomotor shown in Figure 4.13, and can operate under either armature or field control. a Armature control This arrangement is shown in schematic form in Figure 4.14. Now air gap flux is proportional to z'f, or where Kfd is the field coil constant. Also, torque developed Tm is proportional to the product of the air gap flux and the armature current Fig. 4.14 DC servo-motor under armature control. ea t Armature excitation...
Predictive SelfOrganizing Fuzzy Logic Control PSOFLC
This is an extension of the SOFLC strategy discussed in section 10.2.5 and illustrated in Figure 10.17. Predictive Self-Organizing Fuzzy Logic Control is particularly useful when the plant dynamics are time-varying, and the general architecture is shown in Figure 10.32. In Figure 10.30 the predictive neural network model tracks the changing dynamics of the plant. Following a suitable time delay, em kT is passed to the performance index table. If this indicates poor performance as a result of...
Case study Xqy
The laser guided missile shown in Figure 5.26 has an open-loop transfer function combining the fin dynamics and missile dynamics of Fig. 6.33 Nichols chart for uncompensated laser guided missile. Fig. 6.33 Nichols chart for uncompensated laser guided missile. Design a cascade lead compensator that will ensure stability and provide a phase margin of at least 30 , a bandwidth greater than 5rad s and a peak closed-loop modulus Mp of less than 6dB. The open-loop transfer function is third-order...
431 Principle of superposition
A dynamic system is linear if the Principle of Superposition can be applied. This states that 'The response y t of a linear system due to several inputs x t , x2 t , , xn t , acting simultaneously is equal to the sum of the responses of each input acting alone'. Find the complete output for the system shown in Figure 4.9 when both inputs act simultaneously. The block diagram shown in Figure 4.9 can be reduced and simplified to the form given in Figure 4.10. Putting R2 s 0 and replacing the...
The Adaptive Network based Fuzzy Inference System ANFIS
The ANFIS neurofuzzy controller was implemented by Jang 1993 and employs a Takagi-Sugeno-Kang TSK fuzzy inference system. The basic ANFIS architecture is shown in Figure 10.31. Square nodes in the ANFIS structure denote parameter sets of the membership functions of the TSK fuzzy system. Circular nodes are static non-modifiable and perform operations such as product or max min calculations. A hybrid learning rule is used to accelerate parameter adaption. This uses sequential least squares in the...
The magnitude criterion
If a point 1 lies on a locus, then the value of the open-loop gain constant K at that point may be evaluated by using the magnitude criterion. Equation 5.56 can be expressed as Product of pole vector magnitudes Product of zero vector magnitudes Fig. 5.13 Application of the magnitude criterion. Fig. 5.13 Application of the magnitude criterion. For Example 5.7, if s1 lies on a locus, then the pole and zero magnitudes are shown in Figure 5.13. From Figure 5.13 and equation 5.61 , the value of the...
134 Ship autopilot control system
A ship autopilot is designed to maintain a vessel on a set heading while being subjected to a series of disturbances such as wind, waves and current as shown in Figure 1.3. This method of control is referred to as course-keeping. The autopilot can also be used to change course to a new heading, called course-changing. The main elements of the autopilot system are shown in Figure 1.12. The actual heading is measured by a gyro-compass or magnetic compass in a smaller vessel , and compared with...