Control Theory & Applications PID (, )

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1 : (2009) Control Theory & Applications Vol. 26 No. 12 Dec PID,, (, ) : PID (PIDNN), PID,, (BP).,, PIDNN PIDNN (MPIDNN), (CPSO) BP, MPIDNN CPSO MPIDNN CRPSO MPIDNN PSO MPIDNN BP4,, CPSOMPIDNN. BP, CPSO. : PID ; ; ; ; ; : TP273 : A Nonlinear control system of PID neural network based on cooperated particle swarm optimization(pso) PIAO Hai-guo, WANG Zhi-xin, ZHANG Hua-qiang (Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai , China) Abstract: The PID neural network(pidnn) model is a novel neural network model with the advantages of PID and artificial neuron network. This model has been used for complex control systems to achieve desirable control performances. However, the conventional backward-propagation(bp)algorithm restrains the model s wide applications to control field. To control the nonlinear MIMO system efficiently and to extend the application range of PIDNN, we develop the MIMO PID neural network(mpidnn) controller based on PIDNN, and propose the cooperated PSO(CPSO) algorithm to take the place of BP algorithm. Simulation results of the MPIDNN controllers based on BP, PSO and CRPSO algorithms indicate that CPSO-based MPIDNN controller is more effective than the other three in controlling the MIMO systems. The CPSO algorithm makes MPIDNN controller better in performances than BP algorithm in accuracy, stability and robustness. Key words: PID neural network; PSO algorithm; nonlinear dissymmetrical control; stability; robust; CPSO 1 (Introduction) PID,, PID [1]. MIMO,,,,.,,.,,, [2] :, ;,, ;,.,,. PID, [2] PID (PIDNN)., PID.,,,,. BP. BP, ;,,,,, : ; : : (08R214134); ( ); ( a).

2 , BPPIDNN,. 2.1 (Forward algorithm) PIDNN,,. (AIA) (ACO) (GA)(PSO).. PSO MPIDNN., Eberhart and Kenne 1995 [3],,., [4,5]., Eberhart Shi [6], PSO. PSO, [7] (CRPSO), PSO,, BP PSO GA( ),. MIMO MPIDNN, (CPSO), CRPSO, MPIDNN, CPSO PIDNN. BP PSO CRPSO,, CPSO MPIDNN, PIDNN. 2 PIDNN (The structure and algorithm of PIDNN neural network) PIDNN 3 :.,,.,. PIDNN, 3 : (P)(I)(D). 1 : 1, PIDNN3., : x i (k) = u i (k), q, u i (k) q, q <u i (k) <q, u i (k) q. (1) : i =1, 2, ; k ; q ; q q ; u i (k) i k ; x i (k) i k., : u j = 2 w ij x j (k). (2) i=1 : j =1, 2, 3, ; w ij. : x 1(k)= u 1(k), q, x 2(k)= x 2(k 1) + u 2(k), q x 3(k)= u 3(k)+u 3(k 1), q, u 1(k) q, q <u 1(k) <q, u 1(k) q; u 2(k) q, q<u 2(k)<q, u 2(k) q; u 3(k) q, q<u 3(k)<q, u 3(k) q. (3) (4) (5) (3) (5).,, : u h = 3 w jh x j(k). (6) j=1 : h =1, ; w jh. : x h(k)= u h(k), q, u h(k) q, q <u h(k) <q, u h(k) q. (7) Fig. 1 1 PIDNN The basic structure of PIDNN control system 2.2 (BP) (Cost function and BP algorithm),,. [2] MSE.

3 12 : PID PIDNN : : m ; r(k) y(k). J = E h = 1 m [r(k) y(k)] 2. (8) n 0,BP : m k=1 2 m [r(k) y(k)] m k=1 dx w jh(k) 1 x i, P neuron, Δw ij = η dj = 2 m [r(k) y(k)] dw ij m k=1 dx w jh(k) u j x i, I neuron, 2 m [r(k) y(k)] m k=1 dx w jh(k) dx j du x i, D neuron, j Δw jh =η dj = 2 m [r(k) y(k)] dw jh m k=1 dx x j (k), w ij (n 0 +1)=w ij (n 0 ) Δw ij, w jh (n 0 +1)=w jh (n 0 ) Δw jh. (9) (9) : η ; dx dx j du j y(k +1) y(k) sgn( x (k) x (k 1) ) sgn(x j (k) x j (k 1) u j (k) u j (k 1)). dx dx j du (9), j,,,,. 3 CPSO MPIDNN (CPSO algorithm and MPIDNN control system) PSO, CRPSO, g best,. PSO. g best,, g best,. CRPSO,, CPSO. 3.1 CPSO (CPSO algorithm) PSO (Conventional PSO algorithm) PSO, [8,9]. PSO,,. p best g best. : v(k +1)=ω v(k)+c 1 r 1 [p best x(k)]+ c 2 r 2 [g best x(k)], x(k +1)=x(k)+v(k +1). (10) : v x ; k ; ωpso ; c 1 c 2 ; r 1 r ; p best ; g best CRPSO(CRPSO algorithm), [7]PSO CRPSO,,. CRPSO, g best. CRPSO (11) : v i (k +1)=ω v i (k)+c 1 r 1 [p best x i (k)]+ c 2 r 2 [g best (r) x i (k)], x(k +1)=x(k)+v(k +1). (11) : i =1,, n, ; r 1 n,

4 g best,, CPSO (CPSO algorithm), CRPSO PSO, [7],,,, PSO., CPSO g best, g best,. CPSO : v i (k+1)=ω v i (k)+0.5 c 1 r 1 [p best x i (k)]+ 0.5 c 2 r 2 [g best x i (k)]+ 0.5 c 2 r 2 [g best (r) x i (k)], x(k+1)=x(k)+v(k +1). (12) 0.5 p best, g best g best (r). 3.2 PIDNN(MPIDNN) (MPID- NN control system) MPIDNN (The structure of MPIDNN control system) MIMO, m m Δw ij = η J w sij = 2 m 2 m Δw jh = η J = 2 w sjh m 2 s=1 h=1 k=1 2 3 s=1 h=1 k=1 2 3 s=1 h=1 k=1 m s=1 k=1 (14) y s x x sj h u sj sgn( y s(k +1) y s (k) sj (k) x sj (k 1) x h (k) x h (k 1)) sgn(x u sj (k) u sj (k 1)) CPSO (The training of CPSO) MPIDNN : Step 1 MPIDNN CPSO ;. MPIDNN. 2, 2, 3 2, w sij w sjh, spidnn ; i, j, h,. Fig. 2 2 MPIDNN The structure of MPIDNN control system PIDNN, MPIDNN PIDNN., J = 2 E s = 1 2 m [r s (k) y s (k)] 2, (13) s=1 m s=1 [r s (k) y s (k)] y s x h x h k=1 w sjh (k) 1 x si, P neuron, m [r s (k) y s (k)] y s w sjh (k) u sj x si, I neuron, m [r s (k) y s (k)] y s x h w sjh (k) x sj u sj x si, D neuron, [r s (k) y s (k)] y s x x sj (k). (14) h Step 2. Step 3, ; p best g best, g best g best. Step 4 1 nr. Step 5 (11)i. Step 6 f(x i ) <f(p best ), i p best.

5 12 : PID 1321 Step 7 f(p best ) <f(g best ), f(x i ),f(p best ),f(g best ) i g best. Step 8 g best g best. p best g best. Step 9 Step4 8, 4 (Simulation and analysis). Step 10 g best 4.1 (Object system and the parameters of control system) MPIDNN. : [ ] z z z 1 V Y1 (z) = z z z (z) Y 2 (z) z z z 1 V 2 (z) z z z 1 V3 2 (z) (15) (14). MPIDNN CPSO MPIDNN CRPSO : s =2, h =3, q =1, q =1, = 200, = 50, = 3, = 30; : w sij = [ ], w sjh = [ 2 2], = 30; Xmax sij = Xmin sij = 1.5, Xmax sjh = Xmin sjh =2, : V max sij = V min sij = 0.1 (Xmax sij Xmin sij ) = 0.3, V max sjh = V min sjh =0.1 (Xmax sjh Xmin sjh )=0.4; : { pos = X min +(X max X min) rand, (16) vel = V min +2 V max rand. rand 0 1. MPIDNN PSO, = 90, MPIDNN CPSO. MPIDNN BP : s = 2,h =3,q =1,q =1, = 200, = 4500, η=0.01; : w sij (1 : s, 1, 1) = 2,w sij (1 : s, 2, 1) = 2, w sij (1 : s, 1, 2) = 1,w sij (1 : s, 2, 2) = 1, w sij (1 : s, 1, 3) = 2,w sij (1 : s, 2, 3) = 3, w sjh (1 : s, 1:3, 1:h)=0,w sjh (1, 1:3, 1:3)=0.1.,. 4.2 (The experiment of simulation and the analysis of results) 3, 4 MPIDNN MATLAB (System input with constant values) : { r 1 =0.55, (17) r 2 =

6 , CPSO CRPSO PSO, BP., 3, 4,. 4, 3, MPIDNN CPSO, MPIDNN CPSO. MPIDNN BP,,, MPIDNN BP. 3 Fig. 3 MPIDNN The simulation curves of MPIDNN control system with constant system inputs 3,, 4 MPID- NN,. 50, [7],., PSO CRPSO CPSO,, ;, CRPSO PSO, CPSO ; 4 BP, CPSO, 3, BP ; BP 1,, ; CPSO, MPIDNN, (System input with step values) 2 (17) : { r 1 = (k 150), r 2 = (k 50). (18), 2, r 1 r , 50,,, CPSO CRPSO PSO ; BP, 4 Fig. 4 MPIDNN The simulation curves of MPIDNN control system with step system inputs

7 12 : PID (System input with sine and cosine values) 3: { r 1 = sin(2 π t/100), (19) r 2 = cos(2 π t/100). 5. MPIDNN BP,,, MPIDNN BP,,,,.,,, MPIDNN,, MPIDNN CPSO. 5 (Conclusion).. PIDNN (P)(I)(D),,. BPPIDNN,., PIDNN MPIDNN, CPSOBP, MPIDNN CPSO MPIDNN CRPSO MPIDNN PSO MPIDNN BP4,, CPSO MPIDNN. BP, CPSO. MPIDNN CPSO. (References): [1] FARDADI M, SELK GHAFARI A, HANNANI S K. PID neural network control of SUT building energy management system[c] //Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics. New York: IEEE Press, 2005: [2] SHU H L, SHU H. Simulation stu of PID neural network temperature control system in plastic injecting-moulding machine[c] //Proceedings of the 6th International Conference on Machine Learning and Cybernetics. New York: IEEE Press, 2007: Fig. 5 MPIDNN The simulation curves of MPIDNN control system with sine-cosine system inputs 5,, MPIDNN BP.,,.,, , 3, [3] KENNEDY J, EBERHART R C. Particle swarm optimization[c] //Proceedings of IEEE International Conference on Neural Networks. New York: IEEE Press, 1995: [4] LI M, YANG C W. A modified PSO learning algorithm for PID neural network[c] //Proceedings of the 25th Chinese Control Conference. Beijing: Beijing University of Areonautics and Astronautics Press, 2006: [5] CHEN J Y, ZHENG Q. Particle swarm optimization with Local Search[C] //Proceedings of International Conference on Neural Networks and Brain. New York: IEEE Press, 2005:

8 [6] SHI Y, KENNEDY R C. A modified particle swarm optimizer[c] //Proceedings of the IEEE Congress on Evolutionary Computation. New York: IEEE Press, 1998: [7] ZHAO L, YANG Y P. PSO-based single multiplicative neuron model for time series prediction[j]. Expert Systems with Applications, 2009, 36(2): [8] ZOU J, FU X. A particle swarm optimization approach of PID parameters in hydraulic servo control system[c] //Proceeding of the 6th World Congress on Intelligent Control and Automation. New York: IEEE Press, 2006: [9],,. [J]., 2009, 26(1): (FENG Yuanjing, YU Li, FENG Zuren. Sample particle swarm optimization and its namic behavior[j]. Control Theory & Applications, 2009, 26(1): ) : (1977 ),,,, phg0805@sjtu.edu.cn; (1964 ),,,,, ; (1967 ),,,,. -,,,, Petri,, PCA--ELM,,,,,,, LS--SVR4DOF,,,,, SVC,, H,,,,,,,,,,,, PD λ,

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