CHAPTER 6. CALCULATION OF TUNING PARAMETERS FOR VIBRATION CONTROL USING LabVIEW

Transcription

1 130 CHAPTER 6 CALCULATION OF TUNING PARAMETERS FOR VIBRATION CONTROL USING LabVIEW 6.1 INTRODUCTION Vibration control of rotating machinery is tougher and a challenging challengerical technical problem. There are two major categories of vibration control techniques for rotating machinery such as direct control using vibration isolators and balancing control using vibration dampers. This chapter presents the simulation of a PID control scheme and calculation of tuning parameters. These parameters have been derived by using certain fuzzy rules. In this research, obtaining the tuning parameters were formulated by using a change in error for different continuous control mode gains. 6.2 FUZZY BASED PID CONTROLLER Fuzzy logic system and its implementation in control algorithms are illustrated in Figure 6.1. Fuzzification converts the crisp values into fuzzy variables and defuzzification converts the fuzzy variables into crisp values. This concept is being configured with membership functions, rule base, error and change in error mapping, inference engine for executing computational algorithm. Alessandro Ferrero et al (2010) described fuzzy inference system to indicate about uncertainties of few methods. Jia Ma et al (2009) discussed

2 131 about controller for vibration isolation. Lianqing et al (2009) have also discussed about control method for vibration isolation. The following sections in this thesis describe a fuzzy logic system to calculate tuning parameters or gains for PID controller. Figure 6.1. A common fuzzy logic based PID controller is depicted in Error FUZZIFICATION RULE Change in Inference Engine BASE Error DEFUZZIFICATION K p K i K d Error PID CONTROLLER CONTROLLER O/P Figure 6.1 Fuzzy logic based PID controller The basic model of the process to be controlled by a PID is a system of n-th order (linear or nonlinear) with the state Equation (6.1) and (6.2) x (n) = f(x, x`,, x (n-1), t, u) (6.1) y = g(x) (6.2)

3 132 with x - process state vector x = ( x 1,x 2,..,x n ) = (x, x`,..,x (n-1) ) T y - process output u - control variable t - time parameter f, g - linear or nonlinear functions The basic idea of a PID-controller is to choose the control law by considering Equations (6.3), (6.4), (6.5) and (6.6) error e = x x d (6.3) change of error = x` - x`d (6.4) integral of error =. (6.5) u PID = K P.e + K D.e` + K I.., (6.6) where x d is the desired value (set-point). For a linear process the control parameters K P, K D and K I are designed in such a way that the closed-loop control is stable. Gustavo et al (2003) discussed about gain and phase margins of PI controller design. The corresponding analysis can be done by means of the knowledge of process parameters (e.g. mass, damper, spring of a mechanical system) taking into account special performance criteria. In case of non linear processes which can be linearized around the operating point, conventional PID-controllers also work successfully.

4 133 PID-controllers with constant parameters in the whole working area are robust but not optional. In this case, tuning of the PID-parameters has to be performed. A conventional PID-controller is better where, The process is either linear or can be piecewise linearized. The process can be stabilized taking into account selected performance criteria. 6.3 DESIGN REQUIREMENT Rao and Sreenivas (2003) studied the dynamic behaviour of misaligned rotor system to perform harmonic analysis to detect the dominant harmonic between two critical speeds. With the derivation of Gibbon, results were analysed to find axial forces due to harmonics and hence it became possible to predict the presence of misalignment. The resultant method is based on harmonics study only and hence the proposed methods resolve for simple controller. David York et al (2011) presented a vibration isolator with single DOF and vibration control mount investigation. The analysis and experimental results were compared and for parametric identification and system identification. Though the study was made for single DOF, vibration is tougher to control instantaneously. Fixed gain values were used in the control scheme. In the proposed control auto tuning, parameters are computed with respect to the instantaneous requirement of the controller. Since Fuzzy logic is a method of rule-based decision making used for expert systems and process control, it has been selected in the proposed

5 134 research to obtain tuning parameters. To implement real-time decision making or control a physical syste, the controller output of a PID controller is the summation of the proportional, integral, and derivative actions. These are controlled by respective gains, by considering the actual controller output limited to the range specified for control output. 6.4 DESIGN ALGORITHM In this research, it has been simulated with a fuzzy based PID controller in LabVIEW for controlling the vibration. The fuzzy-pid has two inputs, three output for fuzzy and four inputs and one output for PID. For vibration control applications, since the set point or desired output is drastically varying with larger and random amplitudes, a continuous disturbance is found in the system. PID controllers are found suitable in this aspect to bear with the disturbance which is abrupt in nature. But tuning the controller for obtaining quick settling, it is necessary to adopt an algorithm to correlate with the change in error. Based on the following algorithm presented in this proposed methodology, tuning parameters are calculated for PID controller. The algorithm has been implemented in LabVIEW. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Input the controller with sensory units Perceive the present values of K p, K i, K d Compute the error Compute the change in error Execute the error mapping by membership function Determine the new values of K p, K i, K d

6 CONTROL FLOWCHART The control flow chart is illustrated in 6.2 for calculating the required tuning parameters for the PID Controller. START INPUT TO THE SYSTEM FROM SENSORS REFER THE PRESENT GAIN VALUES K p, K i, K d COMPUTE e COMPUTE e EXECUTE THE ERROR MAPPING BASED TRIANGULAR MEMBERSHIP FUNCTION TO DETERMINE THE NEW VALUES FOR K p, K i, K d IS e = 0? NO YES NO IS e = 0? YES STOP Figure 6.2 Flowchart for estimating tuning parameters

7 TUNING PARAMETER CALCULATION FOR PID CONTROLLER In the proposed method, calculation of tuning parameter or different gain values required for achieving control action is being calculated. This procedure is simulated using LabVIEW as illustrated in the succeeding sections. The error and the change in error are given to the fuzzy system which calculates the appropriate K p, K i, K d parameters that will be given to the PID controllers Determining Triangular Membership Function of Error The triangular membership function is used for all input and output mapping. The error mapping is done using triangular membership function as indicated in Figure 6.3. The error range is set from -10 to 10. The error is always expected to lie around 0 (i.e. as minimum as possible) and the range is mapped from -10 to -4, -5 to -1, -1.5 to 1.5, 1 to 5, 4 to 10. The error is said to be critical if it lies between -10 to -4 or 10 to 4. So a corresponding action has to be taken to reduce it. The error in middle membership function must be maintained. The area between -5 to -4, -1.5 to - 1, 1.5 to 1, 4 to 5 are dual mapped by two membership functions. These values l choose any value of the two membership functions. When a linguistic variable is created to represent an input or output variable, it decides the linguistic terms or categories of values. Linguistic variables usually have an odd number of linguistic terms, with a middle linguistic term and symmetric linguistic terms at each extreme. In most of the

8 137 applications, three to seven linguistic terms are sufficient for categorizing the values of a system. Figure 6.3 Triangular membership function of error Depicting Triangular Membership Function Change in Error The change in error mapping is done using triangular membership function. The error range is set from -10 to 10. The change in error is always expected to lie around 0 (i.e., as minimum as possible) and the range is mapped from -10 to -2, -5 to -1, -1.5 to 1.5, 1 to 5, 2 to 10. The error is said to be critical if it lies between -10 to -2 or 10 to 2. So a corresponding action has to be taken to reduce it. The change in error, as shown in Figure 6.4, must be maintained in the middle membership function. The area between -5 to -2, -1.5 to -1, 1.5 to 1, 2 to 5 are dual mapped by two membership functions. These values will choose any value of the two membership function.

9 138 Figure 6.4 Triangular membership function change in error Proportional Gain (K p ) Calculation The membership function for K p is mapped in three membership function from 0 to 8 as indicated in Figure 6.5. The value of 0 to 4 is chosen if the value has to be low, the value 3 to 7 is chosen if the value has to be medium and the value 8 to 8 is chosen if the value has to be high. Figure 6.5 Triangular membership function for K P

10 Integral gain (K i ) CALCULATION The membership function for Ki is mapped in three membership function from 0 to 0.7. The value of 0 to 0.4 is chosen, as referred in Figure 6.6, if the value has to be low, the value.3 to Figure 6.6 Triangular membership function for K i Derivative Gain (K d ) Calculation The value 0.8 is chosen if the value has to be medium and the value 0.8 to 0.7 is chosen if the value has to be high. The membership function for K d is mapped, as indicated in Figure 6.7, in three membership function from 0 to The value of 0 to.02 is chosen if the value has to be low, the value.01 to.03 is chosen if the value has to be medium and the value 0.03 to 0.05 is chosen if the value is to high.

11 140 Figure 6.7 Triangular membership function for K d K d Mapping with Error and Change in Error The graph in the right indicates the K d value that the fuzzy logic controller will choose for the value of error and change in error, as illustrated in Figure 6.8. The error and change in error can be varied to note the K d value that will be chosen, its weight and invoked rule. Normally rules will describe, in words, the relationships between input and output linguistic variables based on their linguistic terms. A rule base is the set of rules for a fuzzy system. Based on this theory controller gains are mapped with error and change in error and this the rule based mapping is done for gains.

12 141 Figure 6.8 K d parameter with error and change in error K p Mapping with Error and Change in Error The graph in the right indicates the K p value that the fuzzy logic controller will choose for the value of error and change in error, as presented in Figure 6.9. The error and change in error can be varied to note the K p value that will be chosen, its weight and invoked rule. Figure 6.9 K p parameter with error and change in error

13 K i Mapping with Error and Change in Error The graph in the right indicates the K i value that the fuzzy logic controller will choose for the value of error and change in error, as displayed in Figure The error and change in error can be varied to note the K i value that will be chosen, its weight and invoked rule. Figure 6.10 K i parameter with error and change in error 6.7 RULES FOR FUZZY CONTROLLER The following rules are framed for setting the rule base for the Fuzzy PID controller: 1. IF Error is very high AND change in error is very high THEN K p is high Also K i is low ALSO K d is medium

14 IF Error is very high AND change in error is medium THEN K p is high Also K i is medium ALSO K d is medium 3. IF Error is very high AND change in error is low THEN K p is medium Also K i is medium ALSO K d is medium 4. IF Error is very high AND change in error is high THEN K p is high Also K i is low ALSO K d is medium 5. IF Error is very high AND change in error is very low THEN K p is medium Also K i is medium ALSO K d is medium 6. IF Error is high AND change in error is very high THEN K p is medium Also K i is low ALSO K d is medium 7. IF Error is high AND change in error is high THEN K p is medium Also K i is low ALSO K d is medium 8. IF Error is high AND change in error is medium THEN K p is medium Also K i is low ALSO K d is medium 9. IF Error is high AND change in error is low THEN K p is medium Also K i is medium ALSO K d is medium 10. IF Error is high AND change in error is very low THEN K p is medium Also K i is medium ALSO K d is medium 11. IF Error is medium AND change in error is very high THEN K p is low Also K i is medium ALSO K d is low

15 IF Error is medium AND change in error is high THEN K p is low Also K i is medium ALSO K d is low 13. IF Error is medium AND change in error is medium THEN K p is low Also K i is low ALSO K d is low 14. IF Error is medium AND change in error is low THEN K p is low Also K i is low ALSO K d is medium 15. IF Error is medium AND change in error is very low THEN K p is low Also K i is low ALSO K d is medium 16. IF Error is low AND change in error is very high THEN K p is low Also K i is low ALSO K d is low 17. IF Error is low AND change in error is high THEN K p is low Also K i is low ALSO Kd is low 18. IF Error is low AND change in error is medium THEN K p is low Also K i is low ALSO K d is low 19. IF Error is low AND change in error is low THEN K p is low Also K i is medium ALSO K d is low 20. IF Error is low AND change in error is very low THEN K p is low Also K i is medium ALSO K d is low 21. IF Error is very low AND change in error is very high THEN K p is low Also K i, is low ALSO K d is low 22. IF Error is very low AND change in error is high THEN K p is low Also K i is low ALSO K d is low 23. IF Error is very low AND change in error is medium THEN K p is low Also K i is low ALSO K d is low

16 IF Error is very low AND change in error is low THEN K p is low Also K i is medium ALSO K d is low 25. IF Error is very low AND change in error is very low THEN K p is low Also K i is medium ALSO K d is low The above said rules have been used to derive auto tuning parameter of PID controller. These parameters K p, K i, K d will be determined instantaneously with respect to the present status of the variable to be controlled. LabVIEW front panel for the above controller has been shown in Figure CALCULATED TUNING PARAMETERS FROM THE PROPOSED ALGORITHM Derived tuning parameters from the proposed algorithm are tabulated in Table 6.1. These values indicate a small instantaneous change in the values to adopt the continuous changes in the system. This fetches the advantages as an adaptive auto tuning of gains over the fixed value controllers. Table 6.1 Adaptive Tuning parameters for PID controller S.No K p K i K d

17 146 Figure 6.11 and 6.12 depicted as graphs, present the K p, K i and K d values over a stipulated number of iterations in the system. It is found to be continuously adopting within the range Kp Kp Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Figure 6.11 K p Estimation Ki Kd Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Figure 6.12 K i, K d Estimation

18 SIMULATION ON CALCULATED CONTROLLER GAINS WITH PIDCONTROLLER Tuning parameters calculated in the above said algorithm can be implemented in a real time system. In this proposed methodology, it is simulated with a randomly simulated value as shown in the front panel, in Figure This LabVIEW front panel has two modes for manually entering tuning parameters and tuning the PID controller with autotuned parameters. Set Point (SP), PV (Process Variable) and MV(Manipulated Variable) are graphically shown for their responses. Besides, in a separate numeric indicator, three tuning parameters have also been displayed. The total loop is configured in an iterative loop (WHILE structure in LabVIEW) to control the execution without affecting the hardware interface. Figure 6.13 LabVIEW front panel for PID controller with tuning parameters

19 ENHANCEMENT COMPARISON OF PROPOSED CONTROL SCHEME Rao and Sreenivas (2003) studied the misaligned rotor system and control estimation was done by considering harmonics present in the signal. The proposed method is well organised with 6 sensors to perceive the complete signal with spectral density and it proposed a continuous mode control scheme. David York et al (2011) dealt with a vibration isolator with fixed gain controller. The proposed control scheme describes a commonly specified controller with auto tuning parameters for taking care of instantaneous Change in Error and Error in the system. It adjusts the gains in an adaptive fashion MAJOR VIBRATION CONTROL COMPONENT FOR ROTATING MACHINERY The next section enlists the components for vibration control and they serve suggestions Vibration Absorbers Under certain conditions, the amplitude of vibration of the mass that is being excited can be reduced to zero, while the second mass continues to vibrate. If a particular system has a having large vibration under its excitation, this vibration can be eliminated by coupling a properly designed auxiliary spring-mass system to the main system. This forms the principle of undamped dynamic vibration absorber where the excitation is finally transmitted to the auxiliary system, bringing the main system to rest.

20 149 Absorber is extremely effective at one speed only and thus is suitable only for constant speed machines. A damped dynamic vibration absorber can take care of the entire frequency range of excitation but at the cost of reduced effectiveness. This will not be treated in this work as it is not within its scope. Various other types of absorbers have been used under various other conditions. These are subsequently introduced in the following sub-sections Vibration Isolation The principal of vibration isolation is an ideal case since the motion of the block was considered in one direction only. In practice, the exciting forces, in a machine produce motions in all directions and it becomes necessary to consider the coupling between different modes for an effective study. Chida et al (2004) discussed about vibration isolation controller by frequency shapping and Giuseppe et al (2008) explained about dampers. Jia Pengxiao et al (2010) explained about fuzzy control for vibration isolation. Lin Yan et al (2012) narrated about vibration isolation study. Song et al (2009) presented about magnetic suspension isolator. Yue Wenhui et al (2011) correlated the role of computer based instrument and Zhan Xueiping et al (2011) described about damper characteristics. This isolation system has been designed to produce a symmetry in two planes. Such a system is not only simpler to analyse but it also gives isolation characteristics. The system is symmetrical about both the vertical planes x-y and y-z. The transmissibility characteristics for this mode are identical for the single degree of freedom system.

21 SUMMARY Though the vibration in rotating machinery is tougher to control mechanically, a control scheme has been suggested in this research. This chapter depicted the methodology and rule base used in fuzzy inference system to calculate tuning parameters in a dynamic fashion. LabVIEW based program has been highlighted to invoke the way of control scheme with tuning parameters with respect to the instantaneous vibration signals.