Abstract: This study aims to effectively restrain the effect of parametric perturbations on operating performance for Contactless Power Transfer (CPT) system. A generalized state-space averaging method based on frequency domain decomposition is applied. By this method, CPT system is approximately a linear time-invariant system and then a linear fractional transformation of GSSA model and its parametric uncertainties is discussed to generate a linear dynamic system with the perturbed feedback. Aimed at this generalized system with the structured parametric uncertainties, the D-K iteration method is proposed for designing a μ-synthesis controller based on the Structured Singular Value (SSV).The simulation and experimental results show that μ-synthesis control system has a certain dynamic performance and robustness and can not only rapidly restrain the effect of frequency and load perturbations on the system itself but also accurately track the steady value of output voltage. The results also verify that μ-synthesis method is effective for the CPT technology.
INTRODUCTION
Contactless Power Transfer (CPT) technology utilizes high frequency magnetic coupling to realize wireless energy transfer in short distance. Since there doesn’t exit electric contact in the power transfer process, the CPT technology has been widely used in special power utilization environments such as inflammable, explosive, humid and underwater environments. Furthermore, it can provide flexible power supply for biomedical and medical devices and electric transportation vehicles. In the last decade, CPT technology has attracted more and more attention from power electronics researchers (Chaoui et al., 2005; Ghariani et al., 2006; Hmida et al., 2007; Casanova et al., 2009; Kissin et al., 2009; Keeling et al., 2010; Low et al., 2010; Moradewicz and Kazmierkowski, 2010).
In CPT system, due to parameter variations, especially for the disturbance on frequency and load parameter, system performances which include power transfer efficiency and distance will drop accordingly. Specially, when the frequency variation exceeds some extent, it will bring multi period operation problem (Tang et al., 2009) which will influence the system stability. A typical CPT system includes high frequency inverter, resonant tank, magnetic coupling and rectifier. It makes the CPT system is a typical high order nonlinear system which is easily affected by parameters disturbance.
Due to the system complexity, classic control methods mainly focus on single objective control (such as frequency and excitation current) which are independent of system model. For example, aimed at frequency stability control, dynamical and static resonant compensation methods were proposed by Ping et al. (2006, 2008), respectively. And a piecewise control method, which can select different control strategies according to control error and its derivative, was proposed to maintain excitation current constant (Guodong et al., 2008). In article of Yang et al. (2010), considering frequency uncertainty, a numerical iteration computing method to acquire the range of frequency variation and an EMI filter design method were proposed to maintain frequency stable. It is obvious that above methods are limited by high order, nonlinear and multi disturbance characteristics of CPT system.
At present, robust control methods (Bai et al., 2005; Asseu et al., 2008; Soltanpour et al., 2008; Li and Fang, 2009) for multi objectives mainly include H∞ and μ approach. The H∞ optimal method (Gaviria et al., 2005) is designed to maintain system robust stability under worst disturbance condition, which ignoring the robust performance of whole system and resulting in system conservation. The H∞ control method is mainly designed based on non-structured uncertainty and small gain theory. While the μ-synthesis control method (Li et al., 2010) mainly focuses on structured uncertainty. It utilizes system structured singular value to depict the effect of structured uncertainty on system stability and robust performance. It can solve the conservation problem brought by H∞ control method.
Based on SP type CPT system, considering the structured uncertainty of system operation frequency and load parameter, the paper put forward a μ-synthesis controller with structured singular value. The method can improve system dynamical and robust performance under the parameter disturbance of frequency and load. The method models the whole system with Generalized State Space Averaging (GSSA) method. And the uncertainty part of the model is detached by upper fractional transformation. Furthermore, with D-K numerical iteration algorithm, a μ-synthesis output feedback controller is proposed and verified by the simulation and experimental results.
OPERATING PRINCIPLE OF SP-TYPE CPT SYSTEM
SP-type CPT system is a typical contactless power transfer system with the circuit topology of the primary side series resonance and the secondary parallel resonance, its circuit diagram can be shown in Fig. 1.
In a SP-type CPT system, the DC voltage source Edc is the input. And two switching pairs (S1, S4) (S2, S3) compose high frequency inverter network. With the complementary conduction of the two switching pairs, high frequency energy is injected into the series resonant network (composed by CP LP and RLp). To obtain high quality sinusoidal oscillation, the zero-crossing detection of primary resonant current iLp is utilized to ensure zero-current switching condition of the inverter network. The contactless transmission of high frequency energy is realized by magnetic coupling between the primary exciting coil Lp and the secondary pickup coil LS. In the secondary side, the parallel resonant network (composed by LS, CS and RLs) is utilized to produce resonance to acquire maximum power transfer. Furthermore, the high frequency energy is transformed to DC energy output with rectifier and filter (Lf, Cf).
| Fig. 1: | Circuit topology of SP-type CPT system |
To ensure maximum efficiency in the power transmission process, the inherent resonant frequency of primary series resonant network and secondary parallel resonant network should be consistent (Nayak and Reddy, 2011) and the inherent resonant frequency can be expressed as:
| (1) |
According to the magnetic coupling principle, the RMS value of the open-circuit voltage of secondary pickup coil can be expressed as:
| (2) |
where, ω0 is inherent resonant frequency, ILp is the RMS value of primary resonant current, M is the mutual inductance between the primary side and the secondary side coils.
On the condition that the secondary side adopts parallel resonant network, the relation between output voltage UO and secondary open-circuit voltage UOC is approximately UO = QSUOC, which QS is the quality factor of load resonant circuit and can be defined as:
| (3) |
To satisfy the power demand of load, the maximum output power should fulfill:
| (4) |
where, ISC is short-circuit current, PO is output power of load. Therefore, short-circuit current of the secondary side should satisfy:
| (5) |
MODELING OF SP-TYPE CPT SYSTEM
The linearization approximation model of SP-type CPT system: As SP-type CPT system exhibits high order nonlinear characteristics, therefore, with GSSA method, the nonlinear model of the high frequency inverter and rectifier can be transformed to linear model.
According to Kirchhoff's voltage and current laws, the differential equation model of equivalent circuit of SP-type CPT system can be described as:
| (6) |
where, g1(t) and g2(t) are binary logic switching functions, defined as:
| (7) |
| (8) |
For AC signals iLp, uCp, iLs, uCs, the even harmonics are approximately zero and the odd harmonics are conjugate symmetric, when the system works close to the inherent frequency point of the LC resonant network, fundamental components of the AC signals can be used to approximate the sinusoidal oscillation components. And for DC signals iLf, uCf, zero-order harmonic component can be used to describe the steady-state characteristics and transient characteristics, the state variables can be defined as:
| (9) |
Based on the differential characteristics of Fourier coefficients (Shittu and Shangodoyin, 2008) and the time-domain differential equations of the SP-type CPT system, the linear differential equations described by Fourier coefficients can be obtained as:
| (10) |
The generalized state variables of GSSA model can be detached to the real and imaginary part of the Fourier coefficients:
| (11) |
On the basis of convolution property of Fourier coefficients, the generalized state space averaging model of system can be established by linearizing the nonlinear switching term of Eq. 10:
| (12) |
where,
The uncertain model of SP-type CPT system: To restrain the influence of frequency and load perturbation for the system effectively, the uncertain model with respect to frequency and load of SP-type CPT system is established.
First, operating frequency and load parameter of system are defined as:
| (13) |
where, ω and RL are uncertain parameters but their variation range are bounded. ω* and RL* are the nominal values of frequency and load parameter, respectively, pωδω and pLδL represent perturbation range of frequency and load parameter, respectively, where δω and δL are normalized uncertain parameters, namely satisfy ||δω||≤1, ||δL||≤1, pω and pL which directly determine system performance, are coefficients of uncertain parameters δω and δL, respectively. The parameter uncertainty model of CPT system is to separate two uncertain parameters δω and δL from the linear approximation model of system.
The frequency and load parameter can be expressed in the form of upper linear fractional transformation ω = FU (Mω, δω):
system structure with upper fractional transformation can be shown in Fig. 2.
where, both Mω and ML are invariant transfer function matrixes, respectively are:
| (14) |
| (15) |
The input and output of uncertain parameters δω and δL can be expressed as yiω, uiω and yjL, ujL, are xi and xj are system state variables, hence there exist viω = xiω, vjL = xj/RL. Mω and ML both have 2 inputs and 2 outputs, the relationship between input and output can be expressed as:
| (16) |
| (17) |
By separating uncertain parameters δω and δL relating to each state variables, uncertain matrix Δ of SP-type CPT system can be obtained as:
| (18) |
where, ΔεR9x9,
And input and output of uncertain matrix Δ are used as output and input of nominal model and can be expressed as:
| (19) |
The uncertain model of SP-type CPT system can be expressed as a linear dynamic system with perturbation feedback G = FU (Gmds, Δ), as shown in Fig. 3.
In Fig. 3, Gmds is generalized nominal model, perturbation block Δ is uncertainty part, input vector u includes all external signals, such as input and output perturbation, noise, reference input, etc.
| Fig. 2: | Linear fractional transformation for uncertain parameters ω and RL |
| Fig. 3: | Uncertain model of SP-type CPT system |
The state space for generalized nominal model Gmds can be described as:
| (20) |
Which, xεRn is state variable, uεRm is control input, yεRr is measured output (load output voltage), upεCp and ypεCp are external signals of model uncertainty.
μ-SYNTHESIS CONTROLLER DESIGN
In SP-type CPT system, to overcome the influence to system performance from frequency, load perturbation and output disturbance, the output feedback controller based on μ-synthesis method u(s) = K (s)e(s) can be shown in Fig. 4.
In the figure, ref is reference value of load output voltage in; Gmds is generalized nominal object model; Δ is norm-bounded structured uncertainty block; d is load output disturbance with limited energy, that is dεL2; Wp and Wu are performance weighting functions They reflect robust performance requirements at different frequency.
μ-synthesis problem: First closed-loop control system can be transformed into the standard linear fractional description for μ-synthesis control, as shown in Fig. 5.
Z = [ep, eu]T is controlled output, P represents
open-loop connection, including generalized nominal model and performance weighting
function;
| (21) |
μ-synthesis for system robust stability and performance is to find a steady controller K, such that:
| Fig. 4: | Block diagram of μ-synthesis control system |
| (22) |
Considering the optimal case, above problem can be attributed to find the minimum
amplitude and frequency peak value of
| (23) |
When solving μ-synthesis problem in practice, upper bound of μ can
be used to replace
| (24) |
where,
μ-synthesis controller: The effective approximate method for solving the μ-synthesis problem is D-K iterative algorithm, its principle is to alternately the minimize Eq. 24 with fixing metric matrix D and controller K, respectively.
The first step of D-K iteration is to select D (s) = 1, an output feedback controller is solved. However, within the frequency range of [10-2, 106], the peak value of μ for closed-loop system reaches 118.004, which will dissatisfy the design requirements for robust stability and robust performance.
By going on with D-K iteration, the robustness of system is continuously improved until there exit μ<1 in the entire frequency band of [10-2, 106].
| Fig. 5: | Standard block diagram of μ-synthesis control system |
The result at tenth iteration is shown in Fig. 6.
Obviously, in the observed frequency range, the amplitude-frequency curves of the μ value in closed-loop system are always below 1, which indicate the robust performance requirements are met under the μ-synthesis control. In the frequency range of [10-2, 106], the peak value of μ is 0.571, indicating that to maintain the stability of closed-loop system, the permissible perturbation are ||Δ||∞<1/0.571 with structured parameter uncertainty.
In the process of D-K iteration, characteristic parameters of controller and D-matrix are shown in Table 1.
As it can be seen, with every iteration, μ-value and γ-value of closed-loop system decrease accordingly, indicating that the robustness of system is improved constantly; the order of μ-controller is directly related to metric matrix D, after the tenth iteration, the order of μ-controller will be 46, it is necessary to reduce the order for the controller.
The contrast frequency response curves of μ-controller before and after reduced-order are shown in Fig. 7. It can be seen that the two frequency response curves are consistent.
| Table 1: | Iteration parameters |
| Fig. 6: | μ values at 10th iteration |
| Fig. 7: | Frequency response curves of full-and reduced-order controllers (46 order-real line, 8 order-dash line) |
It should be noted here that the solving result of D-K iterative algorithm only is the optimal solution in the frequency band of [10-2, 106], hence the designed μ-synthesis controller can’t guarantee a global optimal solution but its effectiveness has been verified by coming simulation and experimental results.
EXPERIMENTAL AND SIMULATION RESULTS
According to the operating principle of SP-type CPT circuit, the main circuit parameters for the simulation are shown in Table 2.
Simulation result: With the MATLAB/SIMULINK simulation platform, simulation model has been established with parameters in Table 2, shown in Fig. 8.
The voltage source Edc is realized by a buck chopper. In the simulation, the output voltage is sampled and sent to discretized μ-synthesis controller. And the control output is transformed to PWM signal by PWM block to drive the buck chopper.
First, this paper simulates and analyses the startup process of SP-type CPT system. The simulation time is set at 0.05 sec, the system starts up at 0.02 sec and required output voltage of load is set at 40 V, the waveforms are shown in Fig. 9.
It can be seen from the figure that μ-synthesis controller has fast response at startup, the load output uo can be stabilized on 40 V after a small overshoot by adjusting input voltage Edc. The adjusting time is about 10 m sec. However, since buck converter exists inherent output ripple, the waveform of uo at steady state is affected relevantly. The small overshoot of startup process and short adjusting time show the closed-loop system has a good dynamic response characteristic.
The closed-loop system’s ability to restrain the perturbation of circuit parameters directly reflects the robust performance of system. Assuming that the reference value of output steady voltage is 40 V, under the condition of load perturbations: the load value is 100 Ω in the range of [0.01, 0.03) sec; the load value varied to 300 Ω in the range of [0.03, 0.05) sec; the load value reverts to 100 Ω in the range of [0.05, 0.07] sec. The system waveforms with whole load perturbations are shown in Fig. 10.
From the figure, ripple amplitudes of output voltage Edc of buck converter with load perturbations are slightly different, ripple with 300 Ω load is small; the envelope of the primary resonant current iLp tends to be smooth with the reduce of ripple of Edc; the current through load changes obviously, while the amplitude of output voltage uo has little change, which shows that the system output has a certain robustness for load perturbations with μ-synthesis control.
| Fig. 8: | Block diagram of closed-loop control system |
| Fig. 9: | System waveforms in startup process (200 Ω) |
| Fig. 10: | System waveforms with load perturbations |
| Table 2: | Parameters of SP-type CPT system |
| Fig. 11: | Startup process of CPT system (1: uo, 50.0 V/Div, 2: Edc, 20.0 V/Div, 3: iLp, 5.00A/Div, Time: 50.00 msec/Div) |
| Fig. 12: | Steady state of CPT system (1: uo, 50.0 V/Div, 2: Edc, 10.0 V/Div, 3: iLp, 5.00 A/Div, Time: 50.00 μsec/Div) |
Experimental results: To verify the effectiveness of μ-synthesis controller, the experimental device of system is established on the basis of principle block diagram shown in Fig. 8. To obtain high-precision control effect and good sine wave oscillation, we use DSP (Digital Signal Processor) TMS320F2812 as the main control chip to achieve μ-synthesis discrete control algorithm; the buck chopper link uses 20 V dc-source. The system realizes steady voltage robust control of uo by high-speed sampling, processing output voltage uo and exporting PWM switching signals of buck link.
| Fig. 13: | Process with the load perturbations (1: uo, 50.0 V/Div, 2: Edc, 20.0 V/Div, 3: iLp, 5.00A/Div, Time: 100.00 msec/Div) |
Figure 11 is circuit waveform in startup process, including output voltage uo of load, output voltage Edc of buck converter and primary resonant current iLp.
It can be seen from the Fig. 11 that uo can be stable at 40 V after a certain overshoot with μ-synthesis controller after power on, the entire process last about 30 msec, a little longer than simulation time. Consistent with the simulation result, Edc also has ripples of a certain amplitude.
To observe the waveform quality of circuit, the waveforms of steady state in Fig. 11 are expanded, as shown in Fig. 12.
Figure 12 shows output voltage uo can be stable at about 40 V and steady state error is small although Edc has a certain ripple; the primary resonant current iLp is close to standard sine wave oscillating characteristic, indicating that SP-type CPT system with μ-synthesis control has good waveform quality.
Similar to the simulation process, we study and evaluate the robust performance of system by observing the effect of load perturbation for closed-loop system. Figure 13 shows the relevant waveforms with different loads, experiment processes are divided into three steps with different load values 100, 300 and 100 Ω.
Obviously, when load value is 300 Ω (intermediate regime), the amplitude of Edc and iLp decrease correspondingly. But in the entire process, output voltage uo is barely affected by load perturbation. The experimental results are in accordance with simulation results.
In addition, the steady waveforms with different load values above are shown in Fig. 14, including output voltage uo, PWM gate signals and primary resonant current iLp.
| Fig. 14(a-b): | Steady states of CPT system with different load values (1: uo, 50.0 V/Div, 2: PWM signal, 10.0 V/Div, 3: iLp, 5.00A/Div, Time: 50.00 μsec/Div), (a) Steady state 1 and (b) Steady state 2 |
| Fig. 15: | Process with the reference values of steady-voltage changed (30, 20, 40 V) (1: uo, 20.0 V/Div, 2: Edc, 10.0 V/Div, 3: iLp, 5.00A/Div, Time: 200.00 msec/Div) |
Figure 14 show that uo and iLp have good waveform quality at different steady states, the duty cycle of PWM signal is basically stable: the stable duty cycle is D = 0.67 with RL = 300 Ω; the stable duty cycle is D = 0.82 with RL = 100 Ω.
In the application of CPT technology, to improve operating efficiency, output voltage should be varied dynamically with actual demand of loads. The reference values of output voltage 30, 20, 40 V is given successively to observe the following features of μ-synthesis control system in the experiment.
Figure 15 shows that the system can follow the variation of reference voltage value quickly with μ-synthesis control, due to this, the real-time duty cycle calculated by μ-controller is comparatively small when the reference voltage is lower which may cause zero flow of iLp. Thus, to ensure the normal operation, the minimum of duty cycle should be limited in the algorithm.
CONCLUSION
GSSA is a modeling method using low order Fourier series for approximating time-domain variables of system, which is not only suitable for dc variables with slow time-varying characteristics but also suitable for ac variables with rapid oscillation characteristics, besides, transient and steady process of electronic devices can be analyzed faster and better.
Therefore, SP-type CPT system separates uncertainty of GSSA model with GSSA modeling method by linear fractional transformation for the system uncertainty. The μ-synthesis control system designed for uncertainty of system parameters can quickly startup and accurately track the output steady voltage value of load and can restrain the effect of load perturbation for system effectively, so that the robust stability of system can satisfy certain robust performance. The simulation and experimental results have verified the feasibility of μ-synthesis method on the field of CPT.
ACKNOWLEDGMENTS
This study was supported in part by the Fundamental Research Funds for the Central Universities under Grant CDJXS11172238, by the research funds for the National Natural Science Foundation of China under Grant 50807057 and by the Specialized Research Funds for the Doctoral Program of Higher Education under Grant 20090191110021. The authors would like to thank all reviewers for their valuable comments.