Chapter-6 Performance Analyi of Cuk Converter uing Optimal Controller 6.1 Introduction In thi chapter two control trategie Proportional Integral controller and Linear Quadratic Regulator for a non-iolated Cuk converter have been tudied a in [17]. The output of the Cuk converter remain contant throughout it operation by varying the input parameter. SSA technique i ued to deign the dynamic model of the ytem. Thee two control technique are applied to thi converter and the bet controller can be elected by comparing the performance pecification uch a ettling time, peak overhoot of the ytem. Cuk regulator i a pecial type of DC-DC converter or it i a cacaded combination of boot converter followed by buck converter with capacitive energy tranfer. The output voltage i either tep-up or tep-down, depending on the duty ratio d. The eential circuit configuration of Cuk regulator i hown in Figure: 6.1. The converter ha two tage of operation namely witch ON tage and witch OFF tage. L 1 act a a filter inductor which prevent the large harmonic content at the input ide. The energy tranfer in Cuk converter depend on capacitance C 1. The output capacitance C 2 i ued to maintain output voltage contant. il1 ic1 il2 io L1 C1 L2 Vi D C R Figure: 6.1 Circuit configuration of Cuk converter The advantage of Cuk converter i continuou input and output current. The operation i baed on capacitive energy tranfer who i more efficient than inductive energy tranfer. Though the current in the input and output ide are continuou and ripple free, there i very little chance of the ripple current falling to zero. It can be applied where very low input and output noie are eential. In thi chapter two controller are deigned uch a PI controller 78
and LQ regulator which give the bet poible performance w.ith repect to ome pecified meaure of performance. The PI controller conit of proportional and integral element, which i often ued in feedback control and for indutrial application due to their implicity in tructure and eay operating principle. By uing Proportional controller the ytem become table with a le teady tate error but peak overhoot i very high. So PI controller i ued, which overcome the diadvantage of Proportional controller and provide a good et point tracking. In thi chapter PI controller i modelled to improve the behaviour of a DC-DC converter. It can be oberved that ome problem faced by uing a P+I controller can be nullified by uing another controller named a Linear-Quadratic Regulator (LQR), which i an optimal controller. The function of thi controller i to minimize the cot function and improve the overall performance of the ytem. The performance index i a quadratic function compoed of tate vector and control vector. The baic concept of PI and LQR control cheme are explained in thi chapter and the imulation reult of the ytem with each controller are preented below. 6.2 Dynamic Model of Cuk Converter A Cuk converter i a buck boot converter whoe output i either more or le than the input voltage. It conit of four energy toring element two inductor (L) and capacitor (C), a witch and a diode. The capacitor i ued to tranfer energy and two inductor L 1 and L 2 are connected to convert input voltage ource (V i ) and output voltage ource (V c ) into current ource. Here a non-iolated Cuk converter i conidered operating on tep up mode. The witch on and witch off tate are already decribed in chapter-4.the dynamic model of the Cuk converter i obtained uing variou tep: Though the converter contain inductor and capacitor a energy toring element it i a non-linear ytem. The non-linear ytem i converted into a linear ytem by uing SSA technique. In SSA technique plant i repreented by tate pace model then converted into tranfer function model. Subequently by uing reduction technique the order of the model i reduced to a lower order becaue a lower order model i eaier to analyze. All thee tep are decribed in chapter-4. Here output voltage i regulated by controlling the duty ratio of the witch. The Cuk converter decribed in thi chapter operate in continuou conduction mode with an operating frequency of 5 khz and duty ratio of.8.the deign data for thi converter i pecified in Table-4.1. 79
Uing State pace averaging technique the mall ignal model with repect to duty cycle i obtain a in (6.1) V o 2.7781 8 1 7.6541 34 38 ( ) 7.911 4.391 12 11 7 d( ) 66.67 2.77 1 7.327 1 27 1 9 5.5641 1 3 2.1451 31 2 1.3841 15 8 9 1.6561 9 18 7 3.37 1 2.4691 32 35 3.5821 7.231 14 22 8 1.3731 6 6.2351 16 24 7 7.8281 5 5.431 2 28 6 3.8781 4 1.8361 22 29 3 33 9.121 5 26 2.211 4 2 (6.1) Thi i the actual tranfer function of the converter with repect to duty cycle. The order i very large for analyi; hence the order i reduced by uing Model Order Reduction (MOR) technique. After applying the reduction technique the tranfer function with repect to duty cycle i obtained a in (6.2) Vo( ) d( ) 3 8 2 1 14.154 2.7781 8.8891 5.5561 4 3 6 2 8 22.22 9.221 1.5111 8.8891 11 (6.2) 6.3 Control Method In thi chapter two control mechanim are introduced and dicued in detail, that i PI controller and LQ Regulator (LQR). The conecutive deign parameter have been calculated to figure out the performance of both thee control mechanim. Thee deign objective are (a) Reducing the peak overhoot (b) Decreaing the rie time (c) Decreaing the ettling time (d) Reducing the teady tate error 6.3.1 Claical PI Controller P+I controller i a type of feedback controller whoe deign depend on the error between deired value and actual value. The error i reduced when paing through the controller in order to get the deired et point value. Two parameter, proportional gain (k p ) and integral gain (k i ), mut be evaluated for the PI controller and each gain parameter ha a conequence on the error value. The tranfer function P+I controller i repreented in equation (6.3) G ( ) K c p K i (6.3) 8
The tuning of PI controller i baed on Zeigler-Nichol method firt approach, where the dynamic are known to u. 6.3.2 LQR Controller LQ regulator come under optimal control problem which provide the bet poible performance with repect to ome pecified performance level. State feedback approach i employed for deigning a LQ controller. From tate feedback approach a feedback gain matrix i calculated which minimize the performance index J to accomplih ome ettlement between the ue of control application, magnitude and peed of repone, which will inure an abolutely table ytem. For a linear continuou-time varying ytem tate equation can be decribed a. x Ax Bu (6.4) With a cot function defined a J T ( x Qx u T Ru) dt (6.5) Where Q and R repreent weight matrice, Q i a poitive definite or poitive emi-definite ymmetric matrix; R i a poitive definite ymmetric matrix. The feedback gain matrix K in 1 T LQR i olved uing the equation K R B P. The P matrix can be calculated by Algebraic Riccati Equation (ARE) which i ued in modern control theory. Three tep can be formulated to find feedback gain matrix K for LQR. (a) The matrice Q and R mut be elected properly. (b) P mut be olved by uing ARE. 1 T (c) The feedback gain matrix K mut be calculated by uing the relation K R B P. 6.4 Deign of PI Controller The aim for deigning a controller for the Cuk converter i to enure both tability of the ytem and guarantee a le teady tate error with minimum overhoot in pite of the perturbation in the input voltage. The control law preented in thi chapter how tabilized regulation of the output voltage by reducing the tranient and tracking of the et point value. A ucceful cloed-loop deign i poible by Zeigler-Nichol method firt approach. The output voltage of the power converter i controlled by a feedback loop between output voltage and duty ratio a input. Figure 6.2 how the cloed loop Simulink model where output voltage of the power converter i matched with a et-point value and the teady tate 81
Imaginary Axi error produced i paed through the P+I controller. The output of P+I controller i compared with aw tooth waveform uing PWM technique to generate the pule which i fed to the gate of the converter witch. The block diagram of Figure: 6.2 i given in chapter-4. Figure: 6.2 Simulink model of CUK converter with PI controller The critical gain and critical time period are.152 and.145 ec repectively which are calculated from the Root Locu plot a illutrated in Figure: 6.3. The point at which root locu plot cut the imaginary axi will give the critical gain and correponding frequency, i known a critical frequency. 1 x 14 Root Locu.8.6.4.2 -.2 -.4 -.6 -.8-1 -1-8 -6-4 -2 2 4 6 8 1 Figure: 6.3 root locu plot of converter without controller Uing Zeigler Nichol method of tuning, the parameter of PI controller are obtained. K p and K i value are found to be.76 and.1482 repectively. So the controller tranfer function i obtained a in (6.6). Real Axi 82
Vo(in volt).76. 1482 G C ( ) (6.6) The voltage output of the power converter uing PI controller i illutrated in Figure: 6.4. 14 output voltage of the converter with PI controller Vo 12 1 8 6 4 2.1.2.3.4.5.6.7.8.9 1 time(in ec) 6.5 Introduction to Optimal Controller Figure: 6.4 output voltage of the converter with PI controller Figure: 6.5 linear quadratic regulator (LQR) feedback configuration Figure: 6.5 how a chematic diagram of a plant with a linear quadratic regulator. Here the plant (Cuk converter) i a continuou linear time invariant ytem in the form of. x Ax Bu y Cx z Gx Hu (6.7) Here y repreent the meaured output which i ued for control and z i the controlled output which i to be mall within hortet viable time. Sometime z = y that mean the 83
control objective i to make the meaured output very mall. In thi block diagram reference ignal i abent and it ue negative feedback. A and G repreent the tate matrice. B and H repreent input matrice. The LQR problem can be defined a by equation (6.8), J LQR 2 z( t) dt u( t) 2 dt (6.8) The firt term correpond to the energy of controlled output and the econd term correpond to the energy of control ignal. The aim i to control the plant repone y around zero. The input diturbance i of low frequency with power pectral denity of 1rad/ec. For deigning a LQG regulator, white Gauian noie i modelled which drive a low pa filter with cut off frequency of 1rad/ec. The chematic diagram of LQG regulator i illutrated in Figure: 6.6. Figure: 6.6 Schematic diagram of LQG regulator The LQG regulator i formed by adding linear quadratic regulator and a Kalman etimator [23]. The optimal control law i tated a u=-kx where K i the optimal controller gain. The optimal controller gain for thi ytem i found to be K = 1 1 3 [.411.219.1698 1.597] By uing the command kalman the Kalman etimator i calculated and uing the command F = lqgreg(ket,k) the tate pace model of LQG regulator i formed. The tranfer function of LQG regulator i calculated and i given in equation (6.9) 84
Amplitude.1 1.785 F ( ) (6.9) 16.41 The optimal control law which minimize the cot function i given below u = [ -.7717-1.3815 -.383-2.535] The explanation for minimization of cot function uing optimal control flow i a follow. A cot function (or performance index) i known a the integral of the time-weighted abolute error (ITAE) to an input ignal. To minimize integral time abolute error, the deigner achieve a repone that i optimized with repect to deviation from a et point (provided by the abolute error) and ettling time. Since the goal of the control ytem deigner i to regulate the output voltage of Cuk converter with repect to input voltage diturbance, integral time abolute error performance index (J ITAE ) provide a calar figure to judge controller performance. For regulator problem, the deired output i rejection of diturbance deviation from the nominal operating point. The error between the deired output and the plant output i defined a e(t) = r(t) y(t). Since r(t) = for all time t in a regulator problem, the error e(t) i imply y(t). In thi chapter the ettling time i reduced from.52 ec to.39 and teady tate error i zero.the cloed loop tranfer function without any filter i calculated and the tep repone i plotted in Figure: 6.7. C( ) R( ) 3 8 2 11 14.1554 4.557 1 1.4581 9.1151 4 5 3 8 2 11 1.641 2.7781 4.217 1 3.9721 9.931 14 (6.1) 14 Step Repone 12 1 8 6 4 2.1.2.3.4.5.6 Time (econd) Figure: 6.7 tep repone of the cloed loop ytem uing LQG regulator 85
After deigning the LQG regulator the peak overhoot i reduced to ome extent but it i not zero percent; when the repone pae through the low pa filter the peak overhoot i reduced to zero with le ettling time. The tep repone of the cloed loop ytem after paing through low pa filter i illutrated in Figure: 6.8. 1 Step Repone 8 6 Amplitude 4 2.2.4.6.8 1 1.2 Time (ec) Figure: 6.8 tep repone of the cloed loop ytem uing LQG regulator and low pa filter The tranfer function of low pa filter for thi paper i choen a 1/(+1) which ha a phae margin of 18 degree and gain margin of infinity. The impule repone of the ytem uing LQG controller and low pa filter i hown in Figure: 6.9. 14 Impule Repone 12 1 8 Amplitude 6 4 2.1.2.3.4.5.6 Time (ec) Figure: 6.9 Impule repone of cloed loop ytem with LQG regulator and low pa filter The Simulink model of the power converter uing LQG regulator i hown in Figure: 6.1, where the white Gauian noie i converted into colour noie after paing through the filter. The block LQG repreent both the linear quadratic gain and Kalman filter. The block plant repreent the converter tranfer function. 86
Figure: 6.1 Simulink model of the plant with LQG regulator and low pa filter The voltage output of the power converter conidering LQG regulator i hown in Figure: 6.11. 12 Output voltage waveform 1 8 o/p(volt) 6 4 2 1 2 3 4 5 6 7 8 9 1 Time (in ec) Figure: 6.11 output voltage waveform of converter uing LQG regulator 87
Phae (deg) Magnitude (db) 6.6 Dicuion Table-6.1 how the performance comparion of Cuk converter with PI controller and optimal controller for a fixed duty ratio. Table-6.1: Performance Analyi Performance pecification Open Loop With PI controller With LQG regulator Output voltage (volt) Nearly 1 98.83 1 Peak Overhoot (%) 9.49 25 Settling time (ec.).52.23.39 Steady tate error Nearly Nearly From the above table it can be concluded that by uing PI controller and LQG regulator both maximum overhoot and ettling time are minimized; however the maximum overhoot i very low by uing LQG regulator which i deired for a power electronic converter. The frequency repone characteritic without controller and with LQG regulator i hown in Figure: 6.12 and 6.13 repectively. 1 Bode Diagram Gm = -55.6 db (at 441 rad/ec), Pm = 1.17 deg (at 1.69e+4 rad/ec) 5-5 36 18-18 1 1 1 2 1 3 1 4 1 5 Frequency (rad/ec) Figure:6.12 Bode Plot without LQG controller 88
Phae (deg) Magnitude (db) 5 Bode Diagram Gm = 13.9 db (at 1.16e+3 rad/ec), Pm = 93.3 deg (at 93.3 rad/ec) -5-1 -15-2 36 18-18 -36 1-1 1 1 1 1 2 1 3 1 4 1 5 1 6 Frequency (rad/ec) Figure: 6.13 Bode plot with LQG regulator From the above plot it i found that GM & PM both are poitive and gain croover frequency i le than phae croover frequency. Hence the ytem i abolutely table with all pole lying on the left half of the -plane. Without uing any controller gain margin i found to be negative with gain croover frequency more than phae croover frequency. From the above tudy (chapter-4 to chapter-6) four different type of controller have been dicued for CUK converter. Some advantage and ome drawback are given for each converter in tabular form in table-6.2. Table-6.2 (Comparion of four different controller for Cuk converter) Performance Without PI Fuzzy Lead LQG Specification Controller Controller Logic Compenator regulator Controller Deired voltage (Volt) output Nearly 1 98.83 1 1 1 Peak overhoot (%) 9.49 25 3 29 Settling time (ec).52.23.4.153.39 Steady tate error Nearly Nearly.5 89
6.7 Concluion The influence of PI controller over ytem performance i that the ytem can ettle very quickly with le overhoot. The overhoot i reduced by reducing the gain but on the other hand ettling time i more. The LQG regulator provide a very table and robut output for the ame ytem. The output voltage of the power converter uing LQG regulator i found to be exactly1 volt. There i a dratic degradation in maximum overhoot and ettling time. Overhoot i reduced from 9.49% to % and ettling time i reduced from.52 to.39 econd. The repone i table and robut. So among all thee four controller LQG i the bet one tracking the et point value with minimum overhoot. 9