Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2)


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1 Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) For all calculations in this book, you can use the MathCad software or any other mathematical software that you are familiar with. 1. Determine which of the following differential equations represents linear and which one represents nonlinear systems. a. b. c d d d d y + 5 y + 10 y + 25 y = 6x dt dt dt dt d d d d y y y y dt ( ) + 10 dt ( ) + 50 dt ( ) ( ) = 60 dt X + 200x d dt 2 2 d 2 ( y) ( y) ( y) = 1 x ( x) dt d. d dt d (y) + (y) y + 100y: = x dt Find a linearized model for the following function around the point x = 5 y: = 120x 2 3. Find a linearized equation for the following equation which may represent a flow equation in a valve. Find the linearized coefficients for a valve for four extreme positions of around fully closed and fully open positions, and at valve opening of 6 mm and extremely low and extremely high pressure. Assume that the fluid is at pressure of 150 bars and the valve displacement is maximum of 6 mm. (Ps P) Q: = Cd A X 2 ρ Where Q is the flow rate and ρ = 0.95 kg/l and C d are the fluid density and valve shape factor and P is the back pressure. A is the crosssection area of the valve. Assume that C d is R. Firoozian, Servo Motors and Industrial Control Theory, Mechanical Engineering Series, DOI / , Springer International Publishing Switzerland
2 162 Appendix A Also determine the absolute flow rate at back pressure of 50 bars and valve opening of X = 3 mm. 4. The following diagram represents a magnetic bearing in which the attractive force is nonlinear in the form of, a F: = x Where a is constant and its value depends on the number of turns of wire on each side of magnetic poles and the amount of current flowing through the wires. It can be seen that for small sizes of gap, the attractive force is inversely proportional to the gap. Find the linearized model for this two variables problem. Assume that the current is also variable. For this purpose, assume that a = ki, where k for a given magnetic bearing is constant. V1 x Rotor Magnetic Pole V2 Schematic diagram of a magnetic bearing Find the linearized coefficients for I = 10 amp, k = 100, and x = 0.5 mm. 5. The diagram below shows fluid discharging at flow rate Q which is a function of the fluid level h of the reservoir and the crosssection area of the outlet port. Determine first the nonlinear equation for the flow rate and find a linearized model which relates the output flow rate to the level h. Determine the constant of linearity for h = 1 m and the fluid density of 0.8 kg/l and crosssection area of m 2. And assume that the outlet valve shape factor is Qi h Schematic diagram of a fluid level control system. The control part is not shown Q
3 Appendix A Prove the nonlinear equation of problem 3. If you have difficulties, refer to a fluid mechanics book. 7. Derive the Laplace Transform for the following step, velocity, acceleration, and exponential inputs that often are used as test functions in control systems. Use the definition of Laplace Transform and for the second and third functions, use integral by parts. Compare your results with those stated in the table of Laplace Transform. x: = A x: = At 2 x: = At x: = Ae τt where A and τ are constants. 8. A simple but complicated control system is the driver and his car. Identify several input and output variables that have to be controlled by the driver. Show every control loop in descriptive block diagram form. Show the driver as the controller and discuss what functions he performs. Identify what kind of time delay is present in every block. 9. There are several control systems in an airplane. Identify several input and output variables that have to be controlled by the pilot. Show each control system by descriptive block diagram. And discuss what methods can be used to control this system in the autopilot position. There are various possibilities of measuring the output variables needed for control in the feedback loop. Discuss these possibilities. 10. The diagram below shows schematic crosssection view of an electric oven. The purpose is to keep the temperature of the oven at constant value. The diagram shows the open loop control. The closed loop properties of the oven will be discussed later in the book. Qi To Ta Qo Where Q i is the input heat to the heater (watt/sec) T o is the oven temperature (centigrade) T a is the outside temperature (centigrade) Q o is the heat loss through the heaters walls Show that the transfer function of the oven relating the oven temperature to the input heat and the outside temperature is,
4 164 Appendix A T : = o Qi + T a K τs+ 1 where mc τ : = K where m is the mass of fluid or air inside the oven and C is the corresponding heat capacity. And K is the conductance constant of the wall. For τ = 2 s and the numerator of 50 when T a = 0, find the time response of oven temperature for step input of the input heat by two methods of partial fraction and the solution given in the theory of control. Calculate the temperature for t = 0.2 and t = 1 and t = 2 and t = 4 and t = 6 and t = 10 s. Discuss the effect of outside temperature to the required temperature. Notice that the outside temperature appears as positive disturbance. 11. Show that a junction in the feedback loop as, G(s) H(s) can be moved to the front of a transfer function with careful consideration as, G(s) G(s) H(s) where G(s) and H(s) are the corresponding transfer functions. 12. Show that a junction in the feedback loop as, G(s) H(s) can be moved in front of a transfer function with careful considerations as, G(s) H(s) G(s)
5 Appendix A Using block diagram algebra, reduce the following closed loop control system to a single block y i + G 1 (s) G 2 (s) y o H(s) 14. Using again the block diagram algebra, reduce the following two negative feedback control systems to a single block. x + + G 1 (s) y H 1 (s) H 2 (s) 15. Reduce the following block diagram to a single block. Hint: you should move a junction to another point or split the summing junction. x G 1 (s) G 2 (s) y H 1 (s) H 2 (s) 16. The transfer function of some transducers and element of control systems may be modeled as first order lag. The following transfer function is a typical of such elements. yo 10 := y 2s+ 1 i Determine the time constant and sketch the step input response by finding two or three important points of the response. 17. For a first order lag system and for a step input, determine the slope of the response at the origin. For this use, the theoretical response for a gain of A and time constant of τ. 18. An experiment conducted on an element of a control system gave the following response. It can be seen that the system can be modeled by a first order lag. Find the time constant and the gain by two methods of (1) finding the slope at
6 166 Appendix A the origin and (2) finding the response when it reaches 63 % of its final values. You should know that this occurs when t: = τ response y(t) t time (sec) 19. A transducer is governed by the following transfer function. Determine the frequency response of the transducer. You should calculate both amplitude ratio and phase lag. Plot both diagrams with amplitude ratio in decibels and use log scale for variation of frequency. Plot both diagrams in the range of interested frequency. yo 20 : = y 01. s+ 1 i 20. An experiment conducted on an element of a control system gave the following graphical frequency response results. From the diagrams, calculate the time constant and gain and hence determine the transfer function and show it in block diagram form. response F(ω) ω rad/sec phase lag degrees ψ(ω) ω rad/sec Discuss why the transfer function can be modeled by a first order lag transfer function.
7 Appendix A A system is governed by the following second order lag transfer function. Determine the natural frequency and damping ratio and the steady state gain. Sketch the step input response by finding several important points. Plot the frequency response for the interested frequency range. You may use the MathCad or other mathematical software. 22. A step input response of a system is given below. Discuss why this system may be represented by a second order lag transfer function. Using the equation for percentage overshoot and frequency of oscillation, find the steady state gain, natural frequency, and damping ratio and hence find the transfer function of the system. 2 y 100 : = o 2 yi 0.01 s s + 1 output amplitude y(t) t time (sec) The exact unit step input response for a second order lag transfer function is, 2 ( 1 ξ ) θ : = atan + ξ π 2 ( ξ ) 1 ξ ω n t 2 y(t): = 1 + e cos ω 2 n t 1 +θ 1 ξ 23. An experiment conducted on an element of a control system gave the following frequency response. You should know that when frequency response is mentioned both the amplitude ratio and phase angle are plotted against the frequency. Discuss why the element can be modeled by a second order lag transfer
8 168 Appendix A function. Determine the natural frequency, damping ratio, and the gain. Show the transfer function in block diagram form. amplitude ratio (db) M(ω) ω frequency (rad/sec) phase lag ψ(ω) ω frequency (rad/sec) To find the damping ratio, you must find the maximum values of amplitude ratio from the theoretical solution of the amplitude ratio. 24. It is desired to control the temperature value of an oven discussed in problem 10, in closed loop control. For this reason, the temperature of the oven is measured by a thermocouple and a negative feedback is used. A proportional controller is used. The controller is constructed in such way to convert low level signal to large output voltage with capability of high currents. Thermocouple Ti K V Qi To T2 Qo Using the parameters defined in problem 10, draw the closed loop block diagram. Then reduce the block diagram to a single block. Determine the time constant and the steady state gain. Remember that this is a two inputs and one output variable and two output variables problem. Use the principle of superposition to find the two transfer functions. Discuss the effect of gain K on the time constant and the two step input steady state errors. For calculating the steady state errors, find first the error function and then use the final value problem. The heat generated by the element inside the oven is proportional to RI 2 which is nonlinear and in order to determine the transfer function you need to linearize this equation.
9 Appendix A Using OPAMP sketch, an amplifier to give a gain of 100. You should remember that to get the sign of the signal correct you should use two OPAMPs in cascade. Select the input resistance in the range of kilo ohms to minimize the current drawn through the amplifier. It is not required to design the system circuits just determine the input and feedback resistors. 26. Using OPAMP, define a proportional controller to have a gain of 100. The system equation must be as, 27. Using OPAMP, design a system to have the following first order lag transfer function. θ o 10 : = θ 3s + 1 Select the required capacitance in the range of microfarad. 28. Using OPAMP, design a proportional plus integral controller. Again select the resistance in the range of kiloohms and capacitor in the range of microfarads. The transfer function must be in the form of, or V : = 100 ( o V 1 V ) 2 i θ o : = Kp + θ i θ o = θ i : K s (Kp s + K i) s i where, K K p i : = 100 : = Again using OPAMPs, design a system to have a leadlag network as, Vo ( 2s+ 1) : = V 5s + 1 i 30. Again using OPAMPs, design a system to have the properties of a PID controller. The system transfer function should be the form of, θ o Ki : = K p + + K d S θ i s 2 o p + d + i θ (K s K s K ) : = θ s i
10 170 Appendix A where, K K K p i d : = 1000 : = 50 : = 20 It should be noted that the gain of derivative cannot be selected too large because it will amplify the noise. Hint: design three separate OPAMPs to give the gain of three terms separately and then design a Summation junction to add the three terms together. 31. In problem 24, with only proportional controller there will be always a steady state error. In order to achieve zero steady state error, an integral must be added to the proportional control. A typical block diagram as discussed in problem 24 is shown below. Ta Ti + K p + K i s s + 1 To Where Ti, Ta, and To are the desired, atmosphere, and output temperature (Degree centigrade) respectively. Kp and Ki are the proportional and integral gains. Using the principle of superposition, determine two transfer functions which relate the inputoutput and outputdisturbance respectively. Note that the closed loop transfer function will be of second order characteristic equation. Determine the proportional and integral gains so that the systems have a natural frequency of 10 (rad/sec) and a damping ratio of 0.7. For these values, determine the steady state error for step and ramp input functions. Use the final theorem. Also find the effect of atmosphere temperature on the steady state errors. 32. A simple control system is a level control of a fluid in a tank. There are various methods of controlling a fluid level in a tank. The simplest system is to use a float where as the fluid level is increased the flow of fluid in the valve is reduced. In this method, only a small variation of fluid level can be controlled. For conducting fluid level to be controlled over a wide range, a simple resistance at the side of the tank is attached. The resistance changes as fluid level is changed. This signal together with the electronics control system can be used to control the flow of fluid from the valve. Usually a proportional control is sufficient to control the fluid level. The diagram below shows a simple control of fluid level control using a float to control the fluid level at a constant level. First determine a relationship between the command level and the float level. You can obtain this relationship by assuming a small variation from the steady state position. Show the equation in block diagram form which should be similar to a proportional control.
11 Appendix A 171 P L Float h Q Assume that the inlet pressure is constant at 50 (bar) and the distance between the pivot and float is 50 cm. The fluid must be controlled at h = 1 m. The crosssection area of the tank is 0.25 m 2. The output flow rate is nonlinear function of the fluid level. The fluid discharged at h = 1 m is 3 lit/min. The input flow rate at fully open position (x = 2 cm) is 9 lit perminute. Draw the block diagram as you derive the differential equation. Assume that the command fluid level is h i and the actual fluid level is h o. Remember that although you drive the linearized equation for small variation of variables, but the mathematical block diagram assumes no small variation and may be used for large variation of variables. If you notice from the block diagram, a negative flow rate is also possible even though in practice it has no meaning. The only problem will be that the result does not indicate a real system. The model is only valid from small deviation from the operating points. After drawing the block diagram, substitute numerical values and find the closed loop transfer function for this two inputs and single output system. Determine the time constant and steady state gain and sketch the step input response characteristic by finding two or three important points. Discuss the steady error for step and ramp inputs and determine it using final value theorem. Discuss what parameters reduce the steady state and increase the speed of response. If your derivation is correct, you have a first order lag transfer function. Also sketch the frequency response of this simple control system. Determine the frequency band width. A frequency bandwidth is defined as the frequency at which the amplitude ratio becomes 0.7 or 3 db. 33. Using the RouthHurwitz stability criteria, determine which one of the characteristic equations given below represents a stable and which one represents an unstable system. If a system is unstable how many roots are in the right hand side of the splane. 3 2 s + 15 s + 66 s + 80 : = s + 19 s + 78 s 280 s 1600 : = s + 17 s + 87 s + 95 s 200 : = s + 13 s + 92 s : = s 6 s s s : = 0.0
12 172 Appendix A 34. Using the RouthHurwitz stability criteria, determine the condition of stability as a parameter of the system is changed. Usually the gain of the system is changed. Other parameters of the system may also be variable. The power of this technique is to determine the condition of stability as one or several parameters are variable. 3 2 s + 19 s s K:= d 3 2 s s s K:=0.0 s s s + ( K ) s K : = 0.0 p 35. Using the mathematical software such as MathCad or any other mathematical software that you are familiar with, determine the roots of the characteristic equations given in problem 33 and hence confirm the accuracy of the Routh Hurwitz method. 36. Calculate the roots of the characteristic equation given below as the gain K changes. Calculate a few points and sketch the rootlocus. Determine which root dominates the response for step input and hence determine the value of K for which the damping ratio of the dominant root is 0.7. You may use MathCad for this purpose. 3 2 s + 80 s s K: = The following characteristic equation is a function of two parameters. Again using the MathCad program calculate the roots of the characteristic equation as two parameters change. Keep one parameter constant and vary the other parameter and find the resulting root locus. Change the first parameter and repeat the process until you find a clear indication how the roots change as both parameters are changed. Determine the dominant root(s) in the step input response and make sure that at least the system has a damping ratio s s s + ( K d) s K p : = The following block diagram shows a control system with three first order lags in cascade with unity feedback. The overall gain of the system is shown by K. Using MathCad draw the Nyquist plot. First set the gain to unity and remember that the Nyquist Plot is the frequency response of the open loop transfer function. Find the gain and phase margins. Determine the maximum value of gain which gives a gain margin of 6 db and phase margin of 60. θ i + K θ o 20s s s + 1
13 Appendix A The following block diagram shows a feedback control system where the feedback transducer is governed by first order lag transfer function. Determine the open loop transfer function. You do not need to do the manipulation by hand. The MathCad software is capable of performing complex number algebra. Plot the Nyquist diagram and find the gain and phase margins. Determine the maximum value of the overall gain as shown by K in the block diagram so that the control system has a minimum gain margin of 6 db and phase margin of 60. y i + K 0.01s + 0.1s + 1 y o 1 2s + 1 For the gain obtained previously plot the frequency response of the closed loop system and hence determine the frequency bandwidth of the system. 40. The following block diagram shows a simple feedback control system. Note that this represents a slow system with time constant in the range of 5 20 s. Determine the open loop transfer function and with the help of MathCad draw the Bode Diagram for K = 1. Remember that when referred to Bode Diagram, it means that two diagrams, one of amplitude ratio in db against the frequency in logarithmic scale and other one is the diagram phase angle in degree against the frequency in logarithmic scale. Determine the gain and phase margins for the overall gain as one. Determine the maximum value of the gain which gives a gain margin of 6 db and phase margin of 60. Note that both conditions of gain margin and phase margin must be met. θ i + θ K 10s s + 1 θ o 1 5s A system is governed by a first order lag in the form shown below. Prove that when the frequency response is plotted in amplitude ratio in db against frequency in logarithmic scale, it can be approximated with two straight lines, 1 τ one at low frequency ω << with 0 db line and with another straight line at large frequency much larger than 1 with slope of 20 db per decade. And τ
14 174 Appendix A the maximum error occurs at frequency of 1 with 3 db (amplitude ratio of τ 0.7). And also show graphically that the phase lag can be approximated by a straight line with slope of 45 /decade. The maximum error is at both ends of the straight line with 5/6. θ i 1 τs + 1 θ o 42. The following block diagram shows a system with transfer function in the form of a general second order lag. Similar to the previous example shown, the frequency response when plotted in db against the frequency in logarithmic scaled can be approximated by two straight lines. One at low frequency much smaller than the natural frequency ω n, with 0 db line. Another straight line at high frequency much larger than ω n with slope of 40 db per decade. The only correction that must be added is at frequency near the natural frequency which depends on the damping ratio. The correction can be made by knowing the damping ratio (show that the maximum amplitude ratio is 1 2ξ and this occurs at frequency of (( ω ) 2 n) 1 ξ for damping ratio smaller than one. Also show that the phase lag may also be approximated by a straight line with slope of 90 per decade. The error depends on the damping ratio and it is minimum with damping ratio of approximately between 0.5 and 1. θ i 1 1 s ξ 2 ω s + 1 ω n n θ o 43. The block diagram shown below represents a simple negative feedback control system. Set K to one and find the open loop transfer function. Remember that when the amplitude ratio calculated in db the multiplications become summation and hence do not reduce the open loop transfer function. Using the straight line approximation draw the Bode Diagram and investigate the stability of the closed loop system. You only need to calculate few points to sketch the Bode Diagram. Determine the maximum gain that gives a gain margin of 8 db and a phase margin of 60. θ i +  K s s + 1 θ o s + 1 Suppose that for position accuracy the gain must be set to 100. Investigate by using the Bode Diagram drawn above, the stability of the closed loop system. If unstable design a leadlag network so that the system has a gain margin of 6 db and phase margin of 60. Hint: add a leadlag network such as shown below. K( τ1 s + 1) τ s+ 1 2
15 Appendix A 175 Find the lag time constant τ 2 so that the system gain margin increases to 6 db. Then find the lead time constant so that the phase margin increases to 60. Again use the straight line approximation to correct the system Bode Diagram. And correct the maximum errors which occurs at the corner frequency (frequency break point). 44. The system shown below represents a spring and damper system where x is the input and y is the output variables. Show the transfer function is a first order lag with time constant of τ. Determine the time constant for k = 1000 and C = 10. Sketch the frequency response by straight line approximations. Find the frequency bandwidth of this simple system. x y 45. The diagram below shows gear systems for transmitting the input power to the output power. N 1, N 2, N 3, N 4 is the number of tooth on each gear. Assume that T 1, T 2 are the input and disturbance torques respectively. Also assume that the inputs to this control system are the torque applied to the first and second inertial loads and the output variable to be controlled is ω i. Assume the friction is applied to the input inertial load as C. Determine the transfer function that relates the output variable ωi to the input torque T 1 and T 2. Assume that the three shafts are flexible with stiffness of K 1, K 2, K 3 respectively. The load inertia on the input and output positions are J 1, J 2 respectively. To obtain the transfer function, draw the free body diagram for each load inertia and gears. You should try to obtain the equivalent inertia and torque referred to the input side. T1 wi J1 K1 N1 N2 K2 N3 N4 K3 J2 T2 Assume that the inertia of shafts and gears are negligible.
16 176 Appendix A 46. Repeat problem 45 for applications where the position of input shaft must be controlled instead of angular velocity. In this case you should write all the equations in terms of angular position θ i. 47. In the transfer function obtained in problem 45, select some realistic engineering values for the parameters involved in the transfer function. Discuss on the order of transfer function and sketch the frequency response and obtain the frequency bandwidth of the system. Find the roots of characteristic equation and obtain the corresponding damping ratio. Investigate the effect of viscous damping coefficient C on the damping ratio of each of the complex roots. 48. The diagram below shows a mechanical massspringdamper system. Assume that input is the force F and the output is the displacement of the mass, x. Determine the transfer function of the system for the parameters shown on the diagram. Select realistic engineering values for the parameters involved in the transfer function and find the natural frequency and damping ratio of the system. Discuss the effect of damping in the damper on the natural frequency and damping ratio. Select the value of C so that the damping ratio 0.7 and find the frequency bandwidth of the system. Using MathCad software plot step input response of the system. Find a relation between natural frequency and settling time for step input response. Remember that the settling time is defined as the time when the output reaches 95 % of its final value. K1 F X C M K2 49. It was shown both in the book and previous problems that OPAMP is often used to introduce compensation in the control system. In the controller network often capacitors and resistors can be used in the different form to produce the required transfer function. One commonly used circuit to reduce noise in the electronics is the following form.
17 Appendix A 177 R Vi C Vo It should be noted that different transfer functions may be obtained by defining different output variables. In the diagram shown above, the input variable is the voltage input to the resistor and the output variable is the voltage across the capacitor. R and C are the corresponding resistance and capacitance. In Laplace domain, C can be replaced by 1/Cs. Then the circuit can be used to obtain the amount of current flowing through the circuit. With this in mind, show that the transfer function for the above circuit diagram may be obtained as, Vo 1 : = V RCs + 1 i The corner frequency is given by RC. This means that signal with frequency below the corner frequency passes through the network and signal with frequency above the corner frequency will be attenuated and the effect on the output signal will be negligible. For an exercise assume that the input is Vi and the output is the current I flowing through the circuit. For this condition find the transfer function. You will find the transfer function is different from the above mentioned transfer function. 50. The following circuit diagram may be used to generate derivative action. It also produces a filter in the form of first order lag transfer function. Derive the transfer function assuming the input is Vi and the output is Vo. C1 Vi C2 R Vo Remember for capacitors, they can be replaced by their equivalent effect which in Laplace domain can be replaced by 1/C1s and 1/C2s. Then the whole system can be used as pure resistive circuit. Find first the equivalent resistance of the system. The current then can be calculated by dividing the input voltage to the equivalent resistance. Once the transfer function is found for current the output voltage may be obtained by calculating the current flowing in each arm of the circuit. Once you have calculated the current through the resistive load the output voltage can be obtained by multiplying the current by the resistance R. 51. A DC motor is used to control the position of a rotary load. The motor is connected directly to the load without a gear box in order to control the position fast and efficiently. The following block diagram shows the action of various parts of the system. Acceleration feedback provides compensation for large
18 178 Appendix A gain, together with a first order lag to reduce the effect of noise that may be present in the system is used. θ + i K V 1 + R I K t T 1 s 2 + C c s θ o C m s R K s s τ f s Where V is the output voltage from the proportional controller, I is the current flowing through the motor and T is the output torque from the motor. θ i, θ o are the input and output angular positions. Note that the block diagram is produced as the equations for different parts are written. For practice, write the equations and confirm the accuracy of the model. In derivation of the block diagram it is assumed that the inductance and the stiffness of the transmission mechanism is negligible. It should be noted that the application of external torque is not included in the model. Assume the following numerical values for the parameters, R: = 0.6 ohms (N M) K t: = 0.8 amp 2 J: = 0.1 Kg m C m: = 0.8 (N M) C s: = 0.1 rad sec τ : = 0.01 f It should be noted that the cut off frequency of the lag network of the acceleration feedback is chosen as 100 (rad/sec) to eliminate the noise above this frequency. This must be experimentally obtained by measuring the noise frequency in the acceleration feedback. It also depends on the quality of the sensor. It is better to measure the rotational velocity and differentiate this signal. Derive the closed loop transfer function and determine the value of proportional gain and the acceleration feedback gain so that the system has a natural frequency of 80 (rad/sec) and a damping ratio of 0.7 in the fundamental mode of the response. Use MathCad to derive the roots of the characteristic equation for various values of gains. You also should make sure that the higher modes must be stable and you can assume they decay rapidly from the response of the system.
19 Appendix A 179 You may find difficulties to improve the damping of the system. In this case, add a velocity feedback with gain of K v. The velocity must be obtained from the output position and fed back to the proportional controller. In practice, a small DC motor known as tachometer is attached to the motor and the signal is fed back to the controller. Now with three parameters of the gain of proportional, derivative, and acceleration you should be able to locate the roots in proper position in the s plane. 52. In process industry it is often required to control the level of fluid in tanks. The figure below shows a two tank system. In reality it is required to control the level of the fluid in both tanks which means that this system is two inputs and two output variables problem. This problem will be investigated in the next section where multivariable control systems are studied. At this stage assume that only the level of fluid in the second tank must be controlled. With this in mind derive the transfer function of the system. Note that you should linearize the relevant nonlinear flow equation around an operating point. Remember that the problem becomes two inputs and single output variable problem. First draw the block diagram when you write the differential equations. Using principle superposition find the two transfer functions which relates the output variable h2 to the command signal (hi) and the transfer function which relates the fluid level (h2) to the output flow rate which is controlled by a variable valve. Assume a proportional controller with gain K is used Q1 h1 Q2 Electric valve V K h2 Qo hi Level Sensor Select some engineering values for the parameters involved in the transfer function. To have proper control the flow rate from the main valve should be greater than the output flow from the first tank and also the input flow rate to the second tank should be greater than the output flow rate. The output flow rate from the second valve is controlled by the operator. Remember that the output from the first tank is only proportional to fluid level as the output flow rate from the second tank is determined by the user. Having substituted numerical values for the parameters, discuss the effect of gain K on the transient behavior of the system and steady state values for single step input of the output flow rate and step input of demand signal (hi).
20 180 Appendix A 53. In problem 52, the two tanks were not interconnected and the derivation of the governing differential equations were simpler. The diagram below shows two interconnected tanks. Assume the aim is to control the fluid level in the second tank (h2), the input flow rate to the first tank is controlled by a proportional control valve. The fluid level in the second tank is measured by an electronic transducer and is fed back to the controller for comparison with the demand fluid level (hi). To have proper control the input flow rate must be greater than the output flow rate from the second tank (Qo). Derive the transfer function which relates the fluid level (h2) to the demand signal (hi). Also derive the transfer function which relates the fluid level (h2) to output flow rate (Qo). Note that you should linearize the relevant nonlinear flow equations around the operating points. Determine the effect of the gain (K) on the transient response of the output fluid level when the input level (hi) is changed in step fashion. Choose some engineering numerical values for the parameters involved in the transfer functions and discuss the effect of the gain on the transient behavior of the system. Plot the step input (hi) response for step input. Choose an appropriate gain K for the system to have maximum speed of response and determine the steady state error when step input of (hi) is applied and for the case of step input of the output flow rate. Qi h1 h2 Qo V K h2 + electronic level transducer hi Assume that the inlet side of the system has a valve that produces flow rate proportional to the input voltage. 54. The following figure shows a massspringdamper system of a different kind. Assuming that the measurements are taken from static equilibrium position, this means that the weight of the mass can be ignored, determine the transfer function which relates the mass displacement (x) to the force (F). Also assume small displacements. For the following values of parameters determine the value of (c) so that there are about a damping ratio of 0.7. m = 10kg, K = 1000N/m
21 Appendix A 181 L3 L2 L1 F m K X c 55. An electrohydraulic device as shown below is used to control the position of a mass of 100 kg against the spring with stiffness of 100 N/m. The flow rate and its direction are controlled by an electrohydraulic servo valve. The servo valve is of spool with internal feedback (not shown in the diagram) form. The spool displacement is a maximum of 10 mm and with supply pressure of 150 bars it can control a maximum flow rate of 9 lit/min. Note that at high pressure large volume of fluid can be displaced by a small opening. xi +  Ps v K x m K P1 P2 There is an internal feedback in the servo valve so that the spool displacement (x) is proportional to the applied voltage at steady state and the transient behavior can be approximated by the following transfer function. K s x : = V τs+ 1 where τ = 0.01 s. Note that the servo valve is quite fast. The actual time constant for a particular application must be obtained from manufacturers. Determine the gain of servo valve when 10 V of applied voltage produces 10 mm of spool displacement. Note that the unit of all parameters must be consistent. Write the nonlinear equation of the flow rate through the servo valve. Linearize the two variables flow equation and find the parameters assuming that with valve opening of 0.5 mm and with zero back pressure, the flow rate is 9 lit/min.
22 182 Appendix A With back pressure of 50 bar and with valve opening 0.5 the flow rate is 5 lit/ min. Determine the coefficients of the linearized equation. Also assume that the leakage coefficient of the piston is one lit/min when the pressure is 150 bars. The diameter of the jack in the cylinder is 0.2 m. By writing the governing differential equation for each part derive the block diagram of the system. Assume that the compressibility of hydraulic oil for this particular application is negligible and that the friction on the load is negligible. Any parameters that have not been specified use some engineering value for them. Determine the transfer function of the system which relates the output displacement (x) to the command signal (xi). Find the appropriate gain (K) so that the damping ratio of the dominant root is 0.7. Study the steady state errors of the system for step and ramp inputs. Repeat the analysis when an external force of F is applied to the mass. Determine the steady state error of the system for step and ramp input of force on the mass. Repeat the calculation for when the hydraulic oil compressibility is not negligible and it has a bulk modulus of compressibility is 100,000 (N/(m 2 ). Determine the maximum steady state speed of the piston when it is operated in open loop with maximum pressure and maximum valve opening. 56. The figure below shows a massspringdamper system in closed loop control form. The force applied to the mass determines the position of the mass. In this case it is required to operate the system in closed loop form so that the position of the mass is kept at a certain value determined by the command signal and in the face of disturbance that may be applied to the mass. The force applied to the mass and it is relationship from output voltage of controller can be approximated by first order lag as, F 10 : = V 05. s+ 1 The mechanism which generates the force is not shown. It may be a hydraulic jack or a servo motor. But it should be noted that because for this particular application the loading mechanism can be approximated by a gain and a first order lag with time constant of 0.5 s. The diagram also shows a PID controller for this particular application. Determine the gains of the controller so the system with dominant roots of the characteristic equation has at least a damping ratio of 0.7 with natural frequency of 20 rad/sec. Use the following numerical values to calculate the transfer function. Use both root locus and Bode diagram to study the stability of the system. M = 100 kg K = N/m C= 10N/(m/s)
23 Appendix A 183 PID Controller xi +  K p + K i s + K d s V Loading Mechanism F M Xo K C Any parameters that have not been defined use an engineering value and continue the analysis.
24 Appendix B: Exercise Problems on State Variable Feedback Control Theory (Chap. 3) You should remember that the general form of state variables feedback control theory is as follows, In the above equations, X is the vector of n state variables, A is a matrix with dimension n n. B is the input matrix with dimension of n m, U is the input with dimensions of m l. For a single input system U is just a single variable. Y is the vector of L output variables. The matrix C is known as the output matrix and its dimension is L n. For some systems there may be direct contribution from the input variables to the output variable. In this case a second term must be added to the output equation. In the following problems, you should use mathematical software such as MathCAD or any other mathematical software you are familiar with. When a problem must be solved by calculation by hand, they will specify explicitly. 1. A system transfer function is given by second order lag transfer function as, For ω n = 1000 and ξ =.5 d d X : = AX + BU t y : = CX θ ω θ + ω ω 2 o n : = 2 2 i s 2 ξ n s + n a. Find the roots of the characteristic equation. b. Write the transfer function in state space form. c. Determine the eigenvalues and eigenvectors of the system matrix using MathCAD or other mathematical software, and show that the eigenvalues are the same as the roots of characteristic equation. Discuss the properties of the eigenvectors. d. Determine the dynamic matrix and show by hand that the determinant of this matrix is the same as the characteristic equation. R. Firoozian, Servo Motors and Industrial Control Theory, Mechanical Engineering Series, DOI / , Springer International Publishing Switzerland
25 186 Appendix B 2. Repeat problem 1 for ω n = 10 and ξ = 2 and discuss the difference between the two problems. 3. A system has the following third order lag transfer function, θ o : = i 10 θ s 25s 600s 2500 a. Using MathCAD or other mathematical software determine the roots of the characteristic equation. b. Write the system transfer function in state space form. c. Determine the eigenvalues of the system matrix and show that they are the same as roots of the characteristic equation. d. Determine the dynamic matrix of the system and using the symbolic expansion facility of the MathCAD software determine the determinant of the dynamic matrix and show that it is the same as the characteristic equation of the transfer function. 4. A system is governed by the following transfer function, θ o : = i ( s s + 32) θ s 110s 13500s a. Write the transfer function in state space form. b. Determine the eigenvalues of the system and show that they are the same as the roots of the characteristic equation. c. The effect of the numerator on the transient and steady state behavior is very complex but there are indications that the numerator does not affect the frequency of oscillation but it only changes the overshoot and undershoot behavior of the system. Discuss this point on the system state equation. 5. Consider a simple mass spring damper system as shown below, F x M K C This problem is extensively studied in vibration and classical feedback control theory.
26 Appendix B 187 a. Derive the second order governing differential equation for the above system. Assume that the x is measured from the static equilibrium position and hence you can ignore the weight of the mass (M). For K= (N/m), M = 50(kg),and C= 10(N/(m/sec)) Solve both problems 5 and 6 by hand because the order of system is two and it is convenient to understand the principle of state variable control theory. b. Write the governing differential equation in state space form. c. Determine the eigenvalues of the system from the determinant of the dynamic matrix. Calculate the natural frequency and damping ratio of the system. d. Determine the eigenvectors of the system and discuss the meaning of the eigenvectors. e. Find the steady state value of x for a given step input of force from the state equation. 6. In problem 5, assume that it is required to control the position of the mass very fast with over damped transient response characteristic. Assume that the desired roots of the characteristic equation should be, Also assume that the system is controllable. Design state variable feedback so that the closed loop system has the above mentioned roots. Note that the closed loop system is quite fast with the dominant time constant 0.02 s. Determine the two gains of state variable feedback. Assume that the demand position is x i. Determine the error for step input of the position. 7. The open loop a system has the following transfer function θ o 100 : = 3 2 θ s + 40s s i s : = 50 s : = Write the transfer function in state space form and using MathCAD or any other mathematical software determine the eigenvalues of the system matrix. Suppose that the system must have an over damped characteristic behavior with the following roots, s1 : = 50 s2 : = 100 s : = Check the controllability of the system and if controllable design a state variable feedback controls so that all roots move to the required position. Find the steady state value of the output for a unit step input.
27 188 Appendix B 8. The figure below shows two degrees of freedom vibrating system with no damping. F1 X1 M1 F2 K1 X2 M2 K2 Assume the following values for the parameters involved, M 1 : = 10 Kg M 2 : = 20 Kg N K 1 : = 5000 m N K 2 : = 7000 m Write the governing differential equations and convert them to state space form. Note that there are two input and two output variables. From state equations find the eigenvalues and eigenvectors of the system. Discuss the meaning of both eigenvalues and eigenvectors for this system. Check the controllability of the system and assume that the force F1 must be manipulated to control the position of both masses in the system. In this case, it is assumed that the device that generates the force is very fast and there is no time delay. Design a state variable feedback control strategy so that the closed loop system has the following eigenvalues, s 1 : = 50 s 2 : = 60 s 3 : = 90 s : = Show the state variable feedback control strategy in block diagram. Note that for state variable control strategy you should measure four state variables with different gains and compare them with the desired position value xli.
28 Appendix B Repeat problem 8 when the required position of the roots of characteristic equation must be set to have the same value as, s1, s2, s3, s4 := 50 It means that all roots must have a time constant of 0.02 s. The gains of the feedback signals for this problem will be different from the previous problem. Note that for both problems you need to expand the determinant of the dynamic matrix symbolically. 10. Solve both problems of 8 and 9 by numerical integration technique and compare the transient response for step input of xi. You should observe that both responses have a dominant time constant of 0.02 s and the remaining roots only affect the transient response behavior. Discuss the differences and make some conclusions. 11. In problem 8, obtain the steady state error for step input of xil. You should notice that there is a large steady state error. Add an integrator after the controller and before the position where the force is generated. Obviously, the order of state equation increases to five and you should now obtain five gains for state variables so the all rootsmove to the following over damped positions, s1 : = 50 s2 : = 75 s3 : = 100 s4 : = 160 s : = The two noninteracting tanks system which was discussed in Sect. 1 is shown below. Linearize the nonlinear flow equations and write the governing differential equations in state space form. Use the numerical values that was considered in problem 52 in Section 1 and find the eigenvalues of this two outputs and two inputs variables system in open loop conditions. Q1 h1 Electric Valve Y h1 h2 Q2 Q0 Level Sensor h2
29 190 Appendix B Check the controllability of the system with respect to Q i. Suppose that it is required to control the system with state variable feedback control strategy so that the closed loop system has the following roots of characteristic equation. In this case note that you have to measure two fluids levels in the tanks and feed them back with deferent gains to the controller and assume it is required to control the level of second tank with the desired value of h 2i. s1 : = 20 s : = 80 2 Where numerical values required for some parameters assume some engineering values for them and insert them in the state equation so that you can solve the problem. Note that you should use correct units in the equations. Determine the steady state values of h 1 and h 2 for step input of h 2i and Q o. The large steady state error that may be present in the system poses no problems as the desired input may be calibrated in terms of output values. Investigate the possibility of adding an integrator in the controller so that the system has zero steady state errors. You should note that the order of the system increases by one and suppose that the system with integrator must have the eigenvalues as, s1 : = 20 s2 : = 50 s : = Consider the following two interacting tanks where the aim is to control the fluid level in both tanks. Write the governing differential equations and convert them to state space form. Linearize the nonlinear flow equations and assume that the two variables to be controlled are the fluid levels in the tanks and the input variables are the input flow rate in the first tank and the output flow rate from the second tank. Use the numerical values given in problem 53 in the first section. First, assume that the fluid level at the second tank must be controlled by the input voltage to the valve on the inlet side. Find the eigenvalues and eigenvectors of the system matrix and check the controllability of the two output variables with respect to the input flow rate. Design state variable control strategy that the two eigenvalues move to the following position. s 1 : = 20 s : = 70 2
30 Appendix B 191 Q h1 h2 Q0 Y h1 h2 Electronic level transducer Investigate whether the two output variables can be controlled by the single input variable Q i. If not study the possibility of adjusting the two variables Q i and Q o. Investigate the possibility of designing two parallel state variable feedback control strategies. This may be set as final year project for students. 14. Consider an upward pendulum which is connected to a motor driven cart. This system is inherently unstable. As shown below, by an application of a force on the cart the position of pendulum must be controlled. In this example, it is assumed that the motor driven cart moves by the force generated by the motor. It is also assumed that the force is generated quite fast and its dynamics compared the response time of the pendulum is negligible. This upward pendulum is the model for space craft and missiles that have to move in a defined angle. θ m L F M Write the governing differential equations in terms of (sin (θ) and cos (θ)) and then assume that the variation of θ is very small and convert the governing differential equations to linear form. You should drive two second order differential equations. Assume that the inertia of connecting rod, which connects the
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