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1566 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 • Mismatch model In order to create performance degradation condition, A disturbance, Δ A was added to the nominal model (2), to construct a man-made mismatch model as, ⎛ 0 1 0 0 ⎞ ⎜ 0 0 1 0 ⎟ A' = A+Δ A= ⎜ ⎟. (36) ⎜ 0 0 0 1 ⎟ ⎜ ⎟ ⎝−600 −300 −100 −30⎠ • MPC tuning parameters Based on the extended MVC 3 algorithm combined with LQR inverse optimal, the MPC controller matrixes of above mismatch model can be gotten. R = 0.0388 , Q = diag(0.0706 0.0002 0.0357 0.000) Through MVC 3 performance evaluation, controller performance declining was detected. If it dropped below threshold, ψ , (here, ψ is set to 0.8), then, weight parameter of manipulated variable in MVC 3 , R ,is updated by R' = R+ ξ I ,(here,ψ is set to 0.5). Repeat the MPC weight matrix calculation process to update Q, R, the new MPC controller parameters as R = 0.0952 , Q = diag(0.1703 0.0006 0.0552 0.0000) can be gotten. Apply the new controller parameters to operation to restore the desired operational performance. V. SIMULATION AND ANALYSIS OF MPC CONTROLLER A. Simulation and Comparison of MPC Controller and the LQR Controller • Simulation of LQR controller LQR controlled system in the Simulink of Matlab was shown in Fig. 3. Parameter of LQR block was set to K = ( −31.623 − 20.151 72.718 13.155) , which was provided by Googol Technology LTD. Figure 3. Simulation block of LQR control loop • Simulation of MPC controller MPC controlled system in the Simulink was shown in Fig. 4, where, parameter of Acker block was set to stabilizer feedback gain. Weighted matrixes of MPC block were set the parameters calculated by the nominal model. Figure 4. Simulation block of MPC control loop • Analysis of simulations results Simulation curve charts of MPC controller and LQR controller are shown in Fig. 5, where, u denotes manipulative variable, which is cart angular velocity. angle, pos denote outputs, they are pendulum angle and cart position. The maximum deviation of MPC controller and LQR controller are shown in Table I and comparison bar chart is shown in Fig. 6. TABLE I. MAXIMUM DEVIATION COMPARISON u angle pos MPC 0.8240 0.0096 0.0093 LQR 34.4589 0.3101 0.3533 Obviously, due to the introduction of steady state manipulative variable and outputs covariance constraint, MPC controller can make the maximum deviation significantly reduced than LQR controller, which greatly improved the system dynamic performance. B. Simulation of MPC Controller Tuning • Simulation of MPC controller tuning MPC controller tuning dynamic curves was obtained by replace original weight matrix Q , R of MPC controller with mismatched and Controller-tuned parameters, shown in Fig. 7, where, the legend (good, bad and tuned) means controller running under, nominal model, mismatch model and MPC controller tuned. Through curves, if using original controller parameters to control mismatch model, it would lead an increase on manipulative variable and outputs deviation. After adjusting controller parameters by controller tuning system, the deviation reduced to some extent. This proves the feasible of MPC controller tuning algorithm. • Extended MVC 3 performance evaluation method The extended MVC 3 performance evaluation curve is shown in Fig. 8. Set λ = 1 in the MVC 3 performance evolution object function (18) and get p m i= 1 i i j= 1 j j −7 benchmark J 3 = ∑qY + ∑ rU by LMI method, MVC here, J 3 = 1.3217 × 10 .Then actual run-time MVC variance was compared with this benchmark, get η © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1567 40 30 MPC LQR 0.35 0.3 MPC LQR 0.3 0.2 MPC LQR 0.25 20 0.2 0.1 u 10 0 angle 0.15 0.1 pos 0 -0.1 -10 0.05 0 -0.2 -20 -0.05 -0.3 -30 0 20 40 60 T -0.1 0 20 40 60 T -0.4 0 20 40 60 T Figure 5. Simulations curves charts of MPC and LQR control loop 35 0.35 0.4 30 0.3 0.35 25 0.25 0.3 20 0.2 0.25 u 15 angle 0.15 pos 0.2 0.15 10 0.1 0.1 5 0.05 0.05 0 MPC 1 LQR 0 MPC 2 LQR 0 MPC 3 LQR Figure 6. Bar charts of MPC and LQR simulations curves maximum deviation 1.5 1 good bad tuned 12 10 good bad tuned 0.01 0.005 good bad tuned 8 u 0.5 0 angle 14 x 10-3 T 6 4 pos 0 -0.005 2 -0.5 0 -0.01 -2 -1 0 20 40 60 T -4 0 20 40 60 -0.015 0 20 40 60 T Figure 7. Dynamic curves of nominal model, mismatch model and MPC controller-tuned as: J 3 MVC η 1 = = 0.9803 ; J arh 1 J 3 MVC η 2 = = 0.7551 ; J arh 2 J 3 MVC η 3 = = 0.9143 , where, J arh 3 J , J , J denote arh1 arh2 arh3 © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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