CONTROL THEORY 2.1 Fundamentals of Control Theory 2.1.1 Introduction The object of this section is to provide an introduction to control engineering principals by firstly considering the operating characteristics of the individual elements used in typical control engineering systems. It then further considers the performance of these elements when combined to form a complete control engineering system. The text includes the development of control theory relating to servo mechanism control in velocity and positional control systems. This is considered essential in ensuring that the student both understands and is able to explain the results obtained from the practical investigations contained in Section 4 of this manual. This also allows the initial controller setting for the individual systems to be set or established as directed. It then helps in the analysis of how the systems actually respond to various steady state and transient operating criteria. The primary object of the CE110 Servo Trainer, of which this manual forms an important part, is to provide a practical environment in which to study and understand the control of a servo-system. These systems occur widely throughout all branches of industry to such an extent that a grounding in servo mechanism control forms a basic component of a control engineer's training. A simple but widespread industrial application of servo control is the regulation at a constant speed of an industrial manufacturing drive system. For example, in the production of strip plastic, a continuous strip of material is fed through a series of work stations. The speed at which the strip is fed through must be precisely controlled at each stage. Similar examples exist where accurate position control is required. A popular example is the position control of the gun turret on a battle tank, which must be capable of both rapid aiming, target tracking and rejection of external disturbances. The following theory and examples are based upon the need to maintain a selected speed or position of a rotating shaft under varying conditions. 2.1.2 Control Principles Consider a simple system where a motor is used to rotate a load, via a rigid shaft, at a constant speed, as shown in Figure 2.1. Figure 2.1 Simple Motor & Load System The load will conventionally consist of two elements, 1 2 A flywheel or inertial load, which will assist in removing rapid fluctuations in shaft speed and, An electrical generator from which electrical power is removed by a load. Under equilibrium conditions with a constant shaft speed, we have Electrical = Mechanical power absorbed by the power supplied generator and frictional to motor losses When this condition is achieved the system is said to be in equilibrium since the shaft speed will be maintained for as long as both the motor input energy and the generator and frictional losses remain unchanged. If the motor input and/or the load were to be changed, whether deliberately or otherwise, the shaft speed would self-adjust to achieve a new equilibrium. That is, the speed would increase if the input power exceeded the losses or reduce in speed if the losses exceeded the input power. When operated in this way the system is an example of an open-loop control system, because no information concerning shaft speed is fed back to the motor drive circuit to compensate for changes in shaft speed. The same configuration exists in many industrial applications or as part of a much larger and sophisticated plant. As such the load and losses may be varied by external effects and considerations which are not directly controlled by the motor/load arrangement. In such a system an operator may be tasked to observe any changes in the shaft speed and make manual adjustments to the motor drive when the shaft speed is changed. In this example the operator provides; a) The measurement of speed by observing the actual speed against a calibrated scale. b) The computation of what remedial action is required by using their knowledge to increase or decrease the motor input a certain amount. c) The manual effort to accomplish the load adjustment, required to achieve the desired changes in the system performance, or by adjusting the supply to the motor. Again, reliance is made on the operators experience and concentration to achieve the necessary adjustment with minimum delay and disturbance to the system. This manual action will be time consuming and expensive, since an operator is required whenever the system is operating. Throughout a plant, even of small size, many such operators would be required giving rise to poor efficiency and high running costs. This may cause the process to be an uneconomic proposition, if it can be made to work at all! There are additional practical considerations associated with this type of manual control of a system in that an operator cannot maintain concentration for long periods of time and also that they may not be able to respond quickly enough to maintain the required system parameters. A more acceptable system is to use a transducer to produce an electrical signal which is proportional to the shaft speed. Electronic circuits would then generate an Error Signal which is equal to the difference between the Measured Signal and the Reference Signal. The Reference Signal is chosen to achieve the shaft speed required. It is also termed the Set Point (or Set Speed in the case of a servo speed control system). The Error Signal is then used , with suitable power amplification, to drive the motor and so automatically adjust the actual performance of the system. The use of a signal measured at the output of a system to control the input condition is termed Feedback. In this way the information contained in the electrical signal concerning the shaft speed, whether it be constant or varying, is used to control the motor input to maintain the speed as constant as possible under varying load conditions. This is then termed a Closed-Loop Control System because the output state is used to control the input condition. Figure 2.2 shows a typical arrangement for a closed-loop control system which includes a feedback loop. Figure 2.2 Closed-Loop Control System including Feedback Loop The schematic diagram shown in Figure 2.3 represents the closed-loop control system described previously. Figure 2.3 Schematic Representation of Closed-Loop Control System Next, consider the situation when the system is initially in equilibrium and then the load is caused to increase by the removal of more energy from the generator. With no immediate change in the motor input, the shaft speed will fall and the Error signal increase. This will in turn increase the supply to the motor and the shaft speed will increase automatically. As the speed is being returned to the original Set Speed value, the Error signal reduces causing the energy supplied to the motor to also reduce. Eventually the supply to the motor would become so small that it cannot drive the load and so stalls. In practice the actual motor torque would reduce until a new equilibrium was produced where the motor torque equalled the load torque and the Error achieves a new constant value. The difference between the Actual speed and the Set speed is termed the Steady State Error of the system. If the Gain of the amplifier was increased, the Steady State Error would be reduced but not totally removed, for exactly the same reasons as given previously. If the Gain were to be increased too much the possibility of Instability may be introduced. This will become evident by the shaft speed oscillating and the input of the motor changing rapidly. Figure 2.4 Proportional Control Amplifier Gain Characteristic The system described previously is said to have Proportional Feedback since the Gain of the amplifier is constant. This means that the ratio of the output to the input is constant once selected. Figure 2.4 shows the characteristic of a typical Proportional Control Amplifier with the Gain set at different levels, increasing from K1 to K5. In order to maintain a non-zero input to the motor drive, there must always be a non-zero error signal at the input to the proportional amplifier. Hence, on its own Proportional Control cannot maintain the shaft speed at the desired level with zero error, other than by manual adjustment of the Reference. Moreover, proportional gain alone would not be able to compensate fully for any changes made to the operating conditions. Operating with zero Error may, however, be achieved by using a controller which is capable of Proportional and Integral Control - (PI). Figure 2.5 shows a typical schematic diagram of a PI Controller. Figure 2.5 Schematic of PI Controller. The Proportional Amplifier in this circuit has the same response as that shown previously in Figure 2.4 (K1 to K5). An Integrating Amplifier is designed such that its output is proportional to the integral of the input. Figure 2.6 shows the typical response of an Integrating Amplifier supplied with a varying input signal. Figure 2.6 Typical Response of an Integrating Amplifier supplied with Varying Input Signal From Figure 2.6 it can be seen that, a) b) c) d) e) f) When the input is zero the output remains constant. When the input is positive the output ramps upwards at a rate controlled by the actual magnitude of the input and also the gain of the integrator. When the input is negative the output ramps downwards at a rate controlled by the actual magnitude of the input and also the gain of the integrator. If the input itself is damping or changing in any way then the outpu will follow an integral characteristic, again following the criteria givci in (a) and (b) above. When a change in input polarity occurs the output responds in the manner described above, starting at the instantaneous output value a which the change occurred. The magnitudes achieved at the output are dependent on the magnitude of the input signal and also the time allowed for the damping to occur. In a practical integrator, the output signal is also limited by the voltage of the power supply to the integrator itself. Effectively, when a constant DC signal is supplied to the input of an Integrating Amplifier its output will ‘ramp’ at a constant rate. Whether it ramps up or down is determined by whether the input polarity is either positive or negative. By arranging the polarity of the Error signal in a control system correctly, the output from the integrator can be configured to always drive the system in the correct direction so as to minimise (zero) the Error. In practice, an integrator would be used, as shown in Figure 2.5, with proportional amplification to give an overall system response of the required characteristic. The overall response of the PI Controller to a step change in Set Speed (or the shaft speed conditions due to the load increasing) is the combined effects of its two circuits, as shown in Figure 2.7. Figure 2.7 Overall Response of the PI Controller to a step change in Set Speed Consider the system described previously by Figure 2.2, where the load rate is increased by the load generator, but now with a PI Controller in the Feedback Loop. As before, the Proportional Amplifier on its own will leave an Error at the instance of the change in speed. However, with the Integrator output signal increasing, ramping upwards in response to this error, the supply to the drive motor and the motor torque will correspondingly increase. The shaft speed will rise until the Set speed is achieved and the Error is zero. At this condition the motor and loads are equal and the system is in equilibrium. This new operating condition will be maintained until another disturbance causes the speed to change once again, whether upwards or downwards, and the controller automatically adjusts it’s output to compensate. In practice the PI Controller constantly monitors the system performance and makes the necessary adjustments to keep it within specified operating limits. The amount of Integral Action will affect the response capability of the system to compensate for a change. Figure 2.8 shows the typical response of a system with constant Proportional and varying levels of Integral Action. Figure 2.8 Typical System Response with Constant Proportional and Varying Integral Action. With an intermediate level of Integral Action the system moves quickly, with minimum overshoot, to the Set Level value. In the example shown, the value of Integral Action chosen is said to achieve Critical Damping. With a low level of Integral Control there is a very slow response giving rise to a distinct time delay between when a change occurs and when the control circuit re-establishes the Set Level again. This type of system is said to be Over Damped. With a high level of Integral Control the response of the system may be so fast that it overshoots the required value and then oscillates about that point under Integral Action until it finally settles down to the steady state condition, if at all. Note that, in the example given, the time for the system to settle down is greater than when the Integral Control value was small. This type of system is said to be Under Damped. For large levels of Integral Control, the system oscillations of the under damped system might grow and become unstable. In general, a) b) Any increase in the amount of integral action would cause the system to accelerate more quickly in the direction required to reduce the Error and have a tendency to increase instability. Decreasing the integral action would cause the system to respond more slowly to disturbances and so take longer to achieve equilibrium. Where fast response is required with minimum overshoot a Three-Term Controller is used. This consists of the previous PI Controller with a Differential Amplifier included to give a PID ( or Three-Term ) Controller. The performance of a Differential Amplifier is that the output is the differential of the input. Figure 2.9 shows the characteristic of a Differentiator supplied with a square wave input. Each time the input level is reversed the output responds by generating a large peak which then decays to zero until the next change occurs. In a practical Differentiator the maximum peak value would be achieved at the power supply rail voltage levels to the Differentiator itself. In a PID Controller the polarity of the output would be configured to actually oppose any change and thereby dampen the response of the system. The gain of the Differentiator would control the amount of damping provided, both in amplitude and duration. Figure 2.9 Differentiator supplied with Squarewave Input The damping required for the situation described in Figure 2.8 could also, therefore, be achieved by including a Differentiator in the control loop to suppress the high acceleration caused by the Integrator without affecting it's ability to remove the Error. It is the balance between the Integral and Differential Action which now controls the overall system response to a step change in Set Level. The speed and manner with which a system can overcome disturbances is termed the Transient Response. By careful selection of the parameters of the proportional, integral and differential amplifiers it is possible to produce a system Transient Response to suit the specific application. This section so far has only dealt with control engineering principles in a very basic way so that the CEIIO Servo Trainer can be used by students and engineers new to control engineering without them having to be familiar with the mathematics. It is possible to verify these principles by setting up suitable test circuits with the CEIIO and the CE120 and then confirming the various system responses. The Section 2.2 builds upon these fundamental principles and introduces the advanced topics of mathematical modelling, system tuning and predicting system performance. This includes the more complex control of shaft position in an output shaft of a reduction gearbox by varying the motor drive in the input shaft of the gearbox. 2.2 Advanced Principles of Control 2.2.1 Introduction In this Section we build on the introductory material of Section 2.1 and describe more advanced methods for the analysis and control of the Servo Trainer. The ability to analyse a system, real or otherwise, is especially important in establishing the relevant design parameters for new plant or in predicting the performance of existing equipment which is to operate under new conditions. Being able to predict the performance of any complex engineering system in advance of its construction and operation will both reduce costs and also minimise project development time. The ability to represent a control situation using mathematical equations also allows computers to be used as an invaluable development tool for the engineer. The computer, once programmed to respond in exactly the same way as the chosen system, can thoroughly ‘test’ or simulate that system under all possible operating conditions, both quickly and cheaply. For some equipment it may only be possible to simulate certain operating conditions since in real life the actual condition cannot be safely or economically reproduced, e.g. the landing on the moon could only be achieved after the equipment had been designed and built, and yet the engineers had the confidence to commit vast resources to the development and construction project as well as gaining experience in advance through the use of simulators. Most importantly, they were able to commit the safety of humans to man the vehicles. Figure 2.10 Servo Control System: Clutch Disengaged 2.2.2 Servo System Modelling: Speed Control System Initially, consider the servo control system with the clutch disengaged. In this configuration the system is a speed control process which can be represented as shown in Figure 2.10 The system model is determined by relating the torque supplied by the motor ( T ) to that required to drive the load generator, the flywheel and frictional losses. This can be expressed as, τm = Load Torque + Frictional Torque + Inertial Torque The load torque can be considered as a torque which is proportional to the load control voltage (v/) while the frictional torque can be considered as a torque which is proportional to the shaft speed (o. The inertial torque is determined by the flywheel dω inertia and the shaft acceleration . Thus dt τm = bω + k ι v ι + I dω dt 2.1 Where b = Friction coefficient of rotating components k ι = Gain constant of load/generator I = Inertia of flywheel The motor electrical circuit is governed by the equation v(t) = Ri + L di + vbemf dt 2.2 Where v(t) is the motor input voltage R is the motor armature resistance L is the armature inductance i is the armature current and v bemf is the motor back emf The back emf and the motor torque can be written in terms of the motor constant km, thus v bemf = k m ω τ m = k mI 2.3 Combining Equations 2.1, 2.2 and 2.3 by taking Laplace transforms and eliminating variables yields the transfer function relating the output speed ω (s) to the input voltage v(s) and the load voltage vι (s) ω (s) = k ' m v( s) ( sI + b)( sL + R) + k 2 m - kι ( R + sL) ( sI + b)( sL + R) + k 2 m vι (s) 2.4 The transfer function simplifies if the inductance L of the armature circuit is assumed to be small compared with the inertia of the flywheel. This gives the first order transfer function ω (s) = k ' m v( s ) k 'ι vι ( s ) Ts + 1 Ts + 1 2.5 Where time constant T is given by T= IR bR + k m2 and k’m = km bR + k m2 k’ι = kι R bR = k m2 2.5b Frequently, we will consider the situation when the servo-control system onl has an inertial load. In this case v/(s) = 0 and Equation 2.5 simplifies to ω (s) = k 'm v( s) (Ts + 1) 2.6 2.2.3 Servo-System Modelling: Position Control System With the electric clutch engaged, the gearbox and output position shaft are connected to the main shaft as shown in Figure 2.11 Figure 2.11 Servo Control System: Clutch Engaged The output shaft position (θ), is related to the main shaft velocity (ω ) by: θ (s) = ω ( s) 30 s 2.7 Where the constant ‘30’ is associated with the 30:1 reduction in speed through the gearbox. Note that the addition of the gearbox load will also change the gain and time constant characteristics of Equations 2.5 and 2.6. Equations 2.5 and 2.6 are used together to provide the system model of the servo-control system dynamics. 2.2.4 Actuator and Sensor Characteristics When the servo-control system is used as a feedback control system the motor speed, ω , is controlled (or actuated) by adjusting the applied voltage to the motor drive amplifier, v. Likewise, the shaft speed and angular position are sensed by transducers which produce output voltages, yω and yθ which are proportional to the shaft velocity,ω, and position, θ, respectively. Figure 2.12 Schematic Representation of Servo Control Feedback System The overall system may be represented schematically as shown in Figure 2.12. The motor voltage, v, and the shaft speed, ω, are related by a steady state actuator characteristic which is assumed to be linear (more will be said of this assumption in section 2.3). The velocity sensor and angular position sensor also have linear characteristics, as shown in Figures 12.13a, b, and c. Figure 2.13a Speed vs Motor Input Voltage Figure 2.13b Sensor Output vs Shaft Speed Figure 2.13c Sensor Output vs Shaft Position If ki, kω, kθ are the motor, velocity sensor and angle sensor gain constants respectively, then ω = ki v yω = kω ω yθ = kθ θ 2.8 Note that k is, as stated previously a steady state gain constant which, from Equation 2.5, is equal to the gain k'm obtained from the modelling exercise. Combining Equations 2.6 and 2.8 gives the standard first order system transfer function. yω (s) = G1 v( s) (Ts + 1) 2.9 Where G1 = ki kω, is steady state gain of the transfer function from the input drive voltage, v, to the sensed shaft position, yω. In addition, the sensed output shaft position yθ is related to the sensec velocity yω by yθ (s) = G2 yω ( s ) s 2.10 where G2 = kθ 30kω 2.10b Then the overall transfer function for the servo-control system can be drawr as in Figure 2.14 and written thus: yθ (s) = G1G2 v( s) s (Ts + 1) 2.11 Figure 2.14 Overall Transfer Function for Servo Control System Again it should be noted that, because of the increase in loading due to the gearbox, the value of G1, T will be changed when the clutch is engaged and the gearbox and output position shaft are connected. 2.2.5 Measurement Of System Characteristics Motor and Sensor Characteristics The motor steady state characteristic, and the speed sensor characteristics are obtained by running the motor at various velocities and recording the corresponding voltages. These are then plotted to obtain the characteristics, as shown in Figure 2.13. The angular position sensor is likewise obtained by rotating the output shaft (using the motor) to various positions, recording the corresponding voltages and plotting to obtain the characteristics. Note that all the servo-control system characteristics are approximately linear. The output and gains will, however, change slightly over a period of time. This phenomenon is known as drift and is not uncommon in industrial sensors and actuators. The motor characteristics will change significantly according to operating conditions. Specifically, the gain G,, and the time constant T will change when the clutch connecting the gearbox and output position shaft is activated. Also, the servo-control system allows for the inertial load to be varied by altering the flywheel thickness (mass) by adding or removing discs. This will alter the inertia I and hence (via Equation 2.5b) the system time constant, T. System Dynamic Characteristics: Step Response Method. For a first order system, like the servo-control transfer function for shaft speed, the gain G, and time constant T can be obtained from a step response test as follows: With reference to Figure 2.15, the gain is determined by applying a step change, with amplitude U, to the input of a system. The final, or steady state, value of the output will be the product U x G1, from which the gain can be relatively determined. The time constant T is defined as the time required for the step response of the system to reach 0.632 of its final value. Figure 2.15 Step Response This method is generally easy to use, and gives reasonably accurate result provided the system characteristic is known to be first order. System Dynamic Characteristics: Direct Calculation An alternative to step response testing is to measure the system characteristics individually and then use Equations 2.5b to calculate the gain and tin constant of the process. This method requires a knowledge of the system. model (from Sections 2.2.2 and 2.2.3) and the ability to make basic measurements of system parameters. In the case of the servo-control system and with reference to Equations 2.5b, it is possible to determine the parameters by either experimentation, direct measurements, or use of manufacture's data sheets (in the case of the motor characteristics). In practice however, the time required and inaccuracy of certain measurements (especially the friction coefficient, b) mean that direct calculation of the system dynamic characteristics would only be undertaken if a detailed simulation of the process was required. We will use step response testing methods in this manual. 2.2.6 Controller Design: Angular Velocity Control Figure 2.16 represents a velocity control system in block diagram form. Figure 2.16 Velocity Control System The aim of the feedback controller is twofold. First it is to bring the output speed, yω, into correspondence with the reference speed yr,. This necessitates finding ways of making the error, e, under steady operating conditions. The second aim of the controller is to alter the dynamic behaviour of the servo system to improve the speed of response to changes in the reference speed This requires us to find ways of altering the system dynamic response vie feedback control. We will consider the steady operating performance separately below. Steady State Errors A main reason for applying feedback control to a system is to bring the system output into correspondence with some desired reference value. The theory developed in Section 2.1.2 'Control Principles' has already explained that there is often some difference between the reference and the actual output. In this Section we see how these errors are quantified when the steady state has been reached. The steady error ,ess, is a measure of how well a controller performs in this respect. The steady state is defined as, ess = lim [e(t)] t→ oo 2.12 Where the error, e(t), is the difference between the reference Set Speed value and the actual output, as shown in Figure 2.16. Equation 2.12 can be re- written in the frequency domain as, ess = lim[s.e( s )] s→o 2.13 For a constant set speed or reference input y,, the steady state error (froir Figure 2.16) is, ess = lim s→o yr 1 + K ( s ).(G )1 ( s ) 2.14 Where K(s) is the controller transfer function and G(s) is the servo systeir transfer function. If proportional control only is used then, K(s) = Kp and, ess = lim s→o yr 1 + K p .G1 ( s ) 2.15 Thus proportional control for the servo control system will involve a steady error which is inversely proportional to the gain, Kp. If proportional plus integral (PI) control is used, K(s) = Kp Ki s ess = lim s. y r =0 s + ( K p .s + K i ).G1 ( s ) and, s→o 2.16 Thus, with proportional plus integral (PI) control, for the servo system speed transfer function the Steady State Error is zero. Dynamic Response The effect of feedback upon the dynamic response of the servo control system velocity controller can be seen from a consideration of the closed-loop transfer function. From Figure 2.16 it is possible to write yω (s) = K ( s )G1 ( s ) y r ( s ) 1 + K ( s )G1 ( s ) 2.17 Recall that the speed transfer function is, from Equation 2.9 G1 (s) = G1 Ts + 1 2.18 With proportional control, the closed-loop transfer function is obtained by combining Equations 2.17 and 2.18 to give yω (s) = k p G1 Ts + 1 + k p G1 yr (s) 2.19 yω (s) = Gc11 Tc11 s + 1 y r (s) 2.20a Where the closed-loop gain is G c11 G c11 given by: K p G1 = 1 + k p G1 2.20b and the closed-loop constant is T c11 = T c11 given by: T 1 + k p G1 2.20c From Equation 2.20c it can be seen that the closed-loop speed of response can be increased by reducing the time constant T . This in turn is achieved by increasing the proportional gain k . If the system controlled by a proportional plus integral controller, the closed loop system is given by yω (s) = (k i + k p s )G1 Ts + (1 = k p G1 ) s + k i G1 2 yr (s) 2.21 By comparing the denominator of Equation 2.21 with the standard expression for the denominator of a second order transfer function:y(s) = ω n2 y r ( s) s 2 + 2ξω n s + ω n2 2.22 it is possible to show that ω n2 = k i G1 T 2.23 and 2 ξω n = 1 + k p G1 T Thus by use of Equation 2.23, it is possible to achieve a desired increase in system transient response performance in terms of a second order closed-loop response. This is done by selecting ki,and kp to give desired values of ω n and ξ 2.2.7 Controller Design: Annular Position Control Figure 2.17 represents the possible block diagram configuration for feedback control of angular position. Figure 2.17 Feedback Control of Angular Position Notice that the control system has two feedback loops. An inner loop feeds back a proportion, kv, of the system velocity, while an outer (position) loop feeds back the sensed output position yθ(s). The role of the inner velocity loop is to improve transient performance of the overall system. This can be seen by considering the overall closedloop transfer function with proportiona] control, such that K(s) = kp ; yθ (s) = k p G1G2 y r ( s ) s 2T + s (1 + k v G1 ) + k p G1G2 2.24 Again this can be compared with the standard second order Equation (Equation 2.22) and the following results obtained: ω n2 k p G1G2 = T 2 ξω n = 1 + k v G1 T 2.25 By selecting kp and kv appropriately it is possible to obtain the desired dynamic performance, as specified by ω n and ξ · Note that when kv=0 (i.e. there is no velocity feedback) it is not possible to specify the damping factor; this can lead to very oscillatory behaviour when the system proportional gain is increased. 2.2.8 Controller Design: Disturbance Rejection Figure 2.18 Velocity Control System Consider the velocity control system discussed in Section 2.2.6, but with the servosystem model extended as indicated by Equation 2.5a to include the effect of the generator load. Figure 2.18 shows this situation. The load disturbance transfer function is (from the second term on the right hand side of Equation 2.5a): G1 (s) = Gi (Ts + 1) where Gι , = kι . The closed-loop equation for the system, including the influence of the generator load is, from Figure 2.18, given by yω (s) = G1 ( s ) K ( s ) G1 ( s ) y r ( s) v1 ( s ) 1 + G1 ( s ) K ( s ) 1 + G1 ( s ) K ( s ) 2.27 Proportional Compensation: If proportional control is applied, then K(s) = kp and if kp is large then the effect of the load change upon the output will be small. In fact the larger kp is the smaller the effect of the load change upon yω. Integral Compensation: If integral plus proportional control is applied, then if a load is applied the integral term will integrate any non-zero error until the effect of the load is removed. This can be seen by writing the closed-loop equation for Figure 2.18 with proportional and integral control:yω (s) = (k i + sk p )G1 Ts + (1 + k p G1 ) s + k i G1 2 y r ( s) - sG1v1 ( s ) Ts + (1 + G1 k p ) s + k i G1 2 2.28 The numerator of the load disturbance term contains a term s (i.e. a zero at the origin) which indicates that for constant load voltages vι (s), the effect upon yω (s) will be zero in the steady state. Feed Forward Compensation: From the previous paragraphs it is seen that proportional control reduces the effect of load changes and integral control action removes the steady state effect of loads. There is, however, a way of reducing the effect of load changes even more. This involves feeding a signal proportional to the load demand into control action. This is called Feed Forward control and is shown in block diagram form in Figure 2.19 Figure 2.19 Feed Forward Control The idea of Feed Forward control is to take a proportion of the load voltage v and after passing it through a suitable controller Kf(s), add it to the motor input voltage, v, such that it compensates for the effect of the load upon the speed, yω. By correctly selecting Kf(s) it is possible to completely compensate for the influence of the load voltage vi. This is done by selecting Kf(s) such that, Kf (s) = G1 s G1 s 2.29 From the equations defining Gι (s) and G1(s), (Equations 2.26 and 2.29 respectively) the feed forward controller required to exactly cancel the load disturbance is a constant Kf, given by Kf = G1 G1 2.30 Thus by calculating K, according to Equation 2.30 it is possible to exactly cancel the effect of changing load upon the speed signal. In practice, K, is often selected experimentally, to approximately remove disturbances and combined with a proportional plus integral controller which removes the remainder of the load effects. 2.3 Advanced Principles Of Control: Non-Linear System Elements The treatment of the servo-control problem thus far has considered the system to be linear. In a practical servo-system, however, a number of non- linearities occur. The most frequently occurring forms of non-linearity are incorporated into the servo-system in a block of simulated non-linearities. The non-linear elements can be connected in series with the servo-motor in order to systematically investigate the influence which nonlinearities have upon practical system performance. 2.3.1 Amplifier Saturation Figure 2.20 Saturation In a practical electronic amplifier for a servo-motor drive there are maximum and minimum output voltages which cannot be exceeded. These maximum and minimum values are due to the limitation imposed by the values of the amplifiers. For example, if the power supply to an amplifier provides ±15 V, then the amplifier output cannot exceed these limits, no matter what the gain of the amplifier. This feature is termed 'Saturation' and is illustrated in Figure 2.20. The saturation amplifier works normally with a specified linear gain relationship between the input voltage, vi, and the output voltage, vo, for inputs in the range -vmin and vmax. Beyond these limits, the output voltage, vo is constant at either vmax or vmin. The servo motor drive amplifier saturates at ± 10V, but in order to show separately the effects of saturation the non-linear element block incorporates a saturation element (Figure 2.21). Figure 2.21 Saturation Element The saturation block is switched into the circuit using the enabling switch With the saturation disabled the input signal passes through the saturation block unmodified. The gain of the saturation amplifier is unity and the voltage at which the amplifier saturates is controlled by a calibrated ‘level control’. 2.3.2 Amplifier Dead-Zone A further feature of a practical amplifier is the dead-zone (or dead-band as its is sometimes called), whereby the amplifier output is zero until the input exceeds a certain level at which the internal losses are overcome, i.e. mechanical losses such as ‘stiction’ Hereafter, the amplifier behaves normally. Figure 2.22 shows a typical dead-zone amplifier characteristic. Figure 2.22 Typical Dead-Zone Characteristic Amplifier dead-zone characteristics are inherent in motors in which a certain (minimum) amount of input is required in order to turn the motor against friction and other mechanical losses. Once the motor begins to turn, the amplifier and motor respond in the normal linear way. The servo motor amplifier has a small dead-zone, but in order to show separately the effects of dead-zone the non-linear element block incorporates a dead-zone element (Figure 2.23). Figure 2.23 Dead-Zone Controls The dead-zone block is switched into circuit using the enable switch. With the dead-zone disabled the input signal passes directly through the dead-zone block unmodified. The gain of the dead-zone element is the linear region is unity, and the dead-zone width and location can be controlled by ‘width’ control and ‘location’ control (Figure 2.23). 2.3.3 Anti-Dead-Zone (Inverse Dead-Zone) Figure 2.24 Inverse Non-Linearity Anti-Dead-Zone Characteristic One way in which the non-linear characteristics can be compensated for is by using an inverse of the non-linearity characteristics. In the specific case of a dead-zone nonlinearity, the corresponding inverse non-linearity is the anti- dead-zone characteristic shown in Figure 2.24. By selecting the anti-dead-zone levels vap, and -van to correspond to the dead- zone levels vdp and vdn the two non-linearity cancel exactly. In order that the effects of anti-dead-zone can be demonstrated the non-linear element block incorporates an anti-dead-zone element (Figure 2.25). Figure 2.25 Anti-dead-Zone Block The anti-dead-zone block is switched into circuit using the enable switch. With the antidead-zone disabled the input signal passes directly through the anti-dead-zone block unmodified. The gain of the anti-dead-zone element in the linear region is unity and the anti-dead-zone ‘width’ and ‘location’ can be adjusted by the ‘width’ control and the ‘location’ control . These are shown in Figure 2.25. 2.3.4 Hysteresis (Backlash) A common and yet unwelcome form of non-linearity in mechanical drives is hysteresis or backlash. This form of non-linearity is caused by worn or poor tolerance mechanical couplings (usually gearboxes) in which the two elements of the coupling separate and temporarily lose contact as the direction of movement changes. This can be illustrated with reference to Figure 2.26, in which the worn or incorrectly meshed gears temporarily lose contact during a change in direction of the driving gear. As a result the driven (or output) gear remains stationary until the driving (or input) gear has traversed and made contact again with the driven gear. The region where no contact exists is termed the backlash gap'. a) Driving gear turning clockwise b) Driving gear changes direction and temporarily losses contact with the driven gear c) Driving gear turning anti-clockwise with contact to driven gear re-established Figure 2.26 Hysteresis or ‘Backlash’ Figure 2.27 Input/Output Characteristic of a Hysteresis/Backlash Device The input/output characteristic of a hysteresis/backlash device is shown in Figure 2.27. Notice that the hysteresis is a 'directional' non-linearity in that the output signal depends upon the direction of change of the input signal and (during the backlash gap) the post direction of change. The servo-system gearbox has been selected to have a small hysteresis characteristic, such that backlash in the servo-system should not be a problem. However, in order to show the effects of hysteresis, the non-linear element block incorporates a hysteresis element to add realism to the system. The hysteresis block is switched into the circuit using the enable switch. With the hysteresis disabled the input signal passes directly through the hysteresis block unmodified. The magnitude of the hysteresis is adjusted by the ‘backlash’ control, as shown in Figure 2.28. Figure 2.28 Backlash Gap Control 2.3.5 Composite Non-Linearities The phenomena of dead-zone, saturation and hysteresis often, unhappily, occur together in a system. The combined effects of these non-linearities can be introduced with the non-linear blocks by switching in the desired combination of non-linearities. For example, a saturating non-linearity with dead-zone can be produced by enabling these blocks and adjusting the controls appropriately. Care should be taken to ensure that the composite non-linearity is practically reasonable. For example, the dead-zone width should always be less than the level at which saturation occurs. Used together with the servo-system motor the non-linear blocks enable the demonstration of important limitations to control system design caused by non-linearity.
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