Final Report for CS5611 (Fuzzy Sets: Theory and Applications):

Mixed H₂/H_inf Optimal PID Control of Induction Motor Using Derivative-Free Approach

曾仲熙 (857909, 電博一)
黃怡碩 (853903, 電碩一)
詹益光 (853908, 電碩一)
李哲穎 (853909, 電碩一)

Table of Contents

Abstract	Problem Definition	Our Approach	Simulation Results
Conclusions	Computer Programs	Division of labor	References

• The object function:
  
  
  We can plot the minimal cost with three different values of rho as follows :
  
  1. Rho = 0.01
     
     
     
     
  2. Rho = 0.05
     
     
     
     
  3. Rho = 0.1
     
     
     
     
• The constrained functions:
  
  1. robust stability condtion :
     
     
  2. disturbance constrain :
     
     
  3. measurement noise constrain :
     
     
  According to three condtions above, we can plot the boundary of KP,KI.
  
  
  

• Genetic Algorithm
• Random Search
• Downhill Simplex Search
• Simulated Annealing

• Genetic Algorithm:
  
  1. The bound of KP is between 0 and 22,
     and KI is between 0 and 500
  2. Randomly generating a population of 3000 strings( 13 bits each)
  3. Through 3 generations
  4. Crossover rate = 1 ; Mutation rate = 0.01
  5. Simulation results

     The simulation results of Genetic Algorithm

     Rho	KP	KI	J
     0.01	7.3016	15.2381	0.0078
     0.05	4.4440	15.8730	0.0169
     0.10	3.4921	17.1429	0.0247

     
     
  6. The simulation time takes about one and half hours for pentium 90
  
  
• Random Search
  1. The bound of KP is between 0 and 8
     The bound of KI is between 0 and 400
  2. Maximum number of function evaluation(eval_n) = 500
  3. Maximun number of actual parameter changes(tran_n) = 100
  4. Using reverse step and bias term
     • If J(x+b-dx) < J(x), then x=x+b+dx and b=0.2b+0.4dx
     • If J(x+b-dx) < J(x), then x=x+b-dx and b=b-0.4dx
     • otherwise set b=0.5b
  5. Simulation results (using intel pentium 100)
     • For rho = 0.01

       The results of Random search at rho=0.01

       start point	KP	KI	J	Els_time(s)
       (1,1)	7.3747	13.4693	0.00775816	123.47
       (3,200)	7.369039	13.153779	0.00775852	96.89
       (7,13)	7.37453	12.86189	0.00775582	114.14

       
       
     • For rho = 0.05

       The results of Random search at rho=0.05

       start point	KP	KI	J	Els_time(s)
       (1,1)	4.6617	15.6455	0.0169278	261.12
       (7,50)	4.6235	15.6775	0.0169267	59.6
       (4,15)	4.5918	15.7234	0.0169272	64.92

       
       
     • For rho = 0.1

       The results of Random search at rho=0.01

       start point	KP	KI	J	Els_time(s)
       (1,1)	3.4002	16.7922	0.0246569	213.55
       (5,200)	3.3629	16.8727	0.0246592	62.06
       (3,16)	3.3329	17.0116	0.0246693	67.56

       
       
  6. In the process of simulation, we can find that a starting ponit must satisfy above checking conditions otherwise RA will take long time to run(maybe very long).
  
  
• Downhill simplex search
  1. Select a=1, b=0.5, r=2
  2. Initial values KP and KI must satisfy the constrains
  3. Iterate until the diameter of the simplex is less than the tolerance(=0.0001)
  4. The calculated value must be checked

     The simulation results of Downhill simplex search

     Rho	KP	KI	J
     0.01	7.3781	14.7374	0.0078
     0.05	4.5198	15.7565	0.0169
     0.10	3.6264	17.3436	0.0248

  
  
• Simulated annealing
  1. Initial temperture = 1000; Annealing constant=0.999
  2. The simulation result :

     The simulation results of simulated annealing

     Rho	KP	KI	J
     0.01	6.8233	15.1052	0.0080
     0.05	4.8117	15.8860	0.0170
     0.10	4.3056	16.9140	0.0255

     
     
  3. Simulation result depends on annealing schedule.
  
  
• Comparisons between these methods:
  
  • Simulation Time : RA < DS < SA < GA
    
  • Complexity of program : RA < SA < DS < GA
    
  • We also find the result of RA depends on the starting point
  • Result
    

    The Value of KI, Kp, J

    	KP	KI	J
    Roh = 0.01
    Genetic algorithms	7.3016	15.2381	0.0078
    Random search	7.37453	12.86189	0.00775582
    Down simplex search	7.3781	14.7374	0.0078
    Simulated annealing	6.8233	15.1052	0.0080
    Roh = 0.05
    Genetic algorithms	4.444	15.873	0.0169
    Random search	4.6235	15.6775	0.0169267
    Down simplex search	4.5198	15.7565	0.0169
    Simulated annealing	4.8117	15.8860	0.0170
    Roh = 0.1
    Genetic algorithms	3.4921	17.1429	0.0247
    Random search	3.4002	16.7922	0.0246569
    Down simplex search	3.6264	17.3436	0.0248
    Simulated annealing	4.3056	16.9140	0.0255

    
    
  • We find that different approaches have slightly different results. This is because different methods have different stop conditions, i.e. GA stop after three generations, RA stop if eval time longer than eval_n.
  
  
• The simulation response:
  
  • Step response without uncertainty, disturbance and noise
    
    
    
    
  • Control input without uncertainty, disturbance and noise
    
    
    
    
  • Step response with uncertainty, disturbance and noise
    
    
    
    
  • Control input with uncertainty, disturbance and noise
    
    
    
    

• Concluding Remarks:
  Randsearch seems better because it is fast and easy programing.
• Future Work:
  We have build up the system, so we plan to realize the Mixed H₂/H_inf control for the induction motor in practice.
  
  
  
  
  
  

• All the computer programs for this project are available by clicking here
• How to run the program :
  
  If you want to run Random search,
  you just type the "RA(startpoint)" in matlab command windows
  Genetic algorithms just type "GA(number)",
  Downhill simplex search type "DS(startpoint)",
  Simulated annealing just type "SA(eta)"
  
• others : all these programs are written by matlab

References

• J.S.R.Jang,'Neuro-Fuzzy And Soft Computing',Prentice-Hall,1997.
• B.S. Chen,Y.M.Cheng,C.H.Lee,'A Genetic Approach to Mixed Optimal PID Control' IEEE Control System.,1995.
• B.K.Bose, 'Power Electronics And AC Drives', Prentice-Hall, 1986.