Small-signal simulation applied to calculation of the parameters of nonlinear dynamics processes control systems for DC/DC converters

The article discusses the issues of choosing the control system parameters for pulse-width DC/DC converters with the target-oriented control. This takes into account the nonlinear dynamic processes control system, which is important when constructing power conversion systems with increased requirements for speed. A nonlinear dynamic model of the system is developed taking fully into account the dynamic properties of the control system. A technique for selecting control system parameters based on small-signal modelling is proposed, which greatly simplifies the calculation process. The suggested approach can be applied in developing DC/DC converters of a wide class.


Introduction
Designing control systems for pulse-width DC/DC converters is a specific task from the control theory perspective.Such converters are built on the basis of pulse-width modulation (PWM), when the output voltage is determined by the width of the power switch control pulse [1][2].
From the control theory perspective, such converters are a nonlinear dynamic system, which determines the features of calculating their control system parameters.At present, there are quite a lot of works devoted to designing such systems [1][2][3][4][5][6][7][8][9] and considerable experience has been accumulated in the relevant field, which makes it possible to generalize the results obtained.
On the one hand, small-signal modelling is the most frequently used in their calculation, when a system of differential equations is linearized in a small neighbourhood of a equilibrium point, then small-signal transfer functions are constructed [1,[4][5][6][7].In this case, equilibrium point stability of the system, which it enters after finishing the transient process, is ensured.The point coordinates are determined by fixed values of the system state variables.This approach is widely used in practice, since it is characterised by simplicity specific to the linear system theory, but at the same time, it does not always give acceptable results due to accuracy limitations.In particular, when using small-signal average dynamic models, the system oscillatory mode is not taken into account, and in the steady state, phase variables do not change with time.
On the other hand, non-linear dynamic models are used, which are the most accurate and are able to predict the system specific behaviour, which is not available for small-signal models [3,8,9].As it is known, the main mode of the converter operation is an oscillating mode [3], when the output voltage, in addition to the constant component, has an alternating component, with a frequency equal to that of the pulse-width modulator.This mode is called a 1-cykle and is desirable because it provides the minimum amplitude of oscillations.Undesirable m-cykle modes appear in certain parameter areas with an increased amplitude of oscillations (m is the cycle multiplicity), which is unacceptable.In these modes, the ripple frequency of the output voltage is m times less than the frequency of the pulsewidth modulator.
To analyse systems using nonlinear dynamic models, the point mappings method is applied [3], where a secant surface is used to study a periodic phase trajectory.The specified trajectory crosses this surface transversally a certain number of times at specific coordinates.These points are called fixed points of the dynamic mode, and in this case the stability of these points is analysed.
The main task when choosing the regulator parameters of pulse-width converters is to ensure the system stability.In this case, as practice shows, the stability of the equilibrium point of the smallsignal dynamic model of the system does not always coincide with the stability domain of the fixed point of the desired 1-cykle, calculated using the nonlinear dynamic model [3].So, in [10] a buck converter based on a proportional controller is considered.In this case, the small-signal open-loop model has the properties of a second-order low-pass filter, i.e. the phase shift of such a link asymptotically approaches -180 0 , i.e. at any value of the proportional controller coefficient the system will be stable.However, the calculation using a nonlinear dynamic model reveals complex dynamic modes at large values of the coefficient, which is not explained by the theory of linear dynamic systems.
Although nonlinear dynamic models are more accurate, they do not allow using frequency methods of the system analysis and are usually reduced to analysing a large number of time series obtained with different sets of system parameters, leading to significant computer time.This results in the necessity for having an optimal combination of these approaches.For example, to select the controller parameters, small-signal transfer functions and frequency design methods can be used at the beginning.Then the results can be verified applying a nonlinear dynamic model by calculating and analysing transient processes in the selected ranges of the system parameters, which can change significantly during the system operation.This approach reduces the amount of calculations using a nonlinear dynamic model.
The specificity of pulse-width converters as non-linear dynamic systems ultimately led to the necessity to consteuct specific control systems that allow these non-linearities to be taken into account [11,12].As a rule, methods developed by specialists in the field of cybernetic physics are used, but their application to pulse-width converters has its own peculiarities.To date, the following methods have been tested: the linearization method of the Poincare map [11]; the time-delay feedback control [12] and the target-oriented control [13], which allow having the desired dynamic operation of the system.Such control systems are the development of classical closed-loop automatic control systems and differ from them in the use of an additional control subsystem for nonlinear dynamic processes.The subsystem provides stability of the fixed point of a single mode.
Choosing the optimal parameters of such control systems is usually carried out by a combined method.First, the optimal parameters of the main control system controller are calculated based on small-signal simulation, and then, using the point mappings method combined with the first Lyapunov method, the fixed point stability of the desired dynamic mode is ensured in a given range of system parameters by choosing the optimal parameters of the additional control system.This paper will consider a half-bridge converter with the target-oriented control system.This system was previously considered in [14].To the disadvantages of this work the author refers the assumption of large time constants of filters as part of an additional control system, which made it possible to reduce the system order of differential equations.However, when developing high-speed voltage converters, such an assumption is incorrect, which requires using the differential equation systems of higher orders.To select the parameters of an additional control system, this article proposes a technique based on a small-signal system model, which in some cases will significantly speed up the design process.However, additional verification with a non-linear dynamic model is recommended.

Description of the Voltage Converter
This section considers the principle of operating the system under study.The Structural diagram is presented in Fig. 1.

Figure 1. Structural diagram of the pulse-width converter
The power section of the converter is constructed on the basis of a half-bridge circuit developed on transistors VT1, VT2, diodes VD1-VD3, transformer TV with the transformation ratio Ktr, choke CHK with inductance L and resistance RL, capacitor CAP with capacitance C and resistance Rc.This scheme is usually used in power supplies with an output power of up 1 kW.
Main control system (MCS) in Fig. 1 is constructed on the basis of pulse-width modulation of the Ikind (PWM-I), which is used in the microprocessor implementation of the control system.The reference-input signal uref is supplied to the subtractor SB the second input of which receives the signal uvs from the voltage sensor VS.This subtractor calculates the error uerr, which is fed to the PI-controller (PIC).Sample-and-hold circuit SH samples the control signal ucon on the master clock umc, which comes from the master clock (MC).The output signal SH is supplied to the non-inverting input of the PWM-comparator «==», and the inverting input of the comparator is supplied with a ramp signal urg from the RG generator output.The comparator signal ucmp is fed to the phase splitter, consisting of two elements «AND1» and «AND2», performing the logical operation «AND».The upper input of these blocks receives a signal from the PWM-comparator.The lower input of the AND1 block receives a signal from the non-inverting output of the counting trigger T, and the lower input of the AND2 block receives a signal from the inverting output of the trigger T. Further, the output signals AND1 and AND2, are fed to the power switch drivers DR1 and DR2, which are control signal amplifiers.The main purpose of the phase splitter is to generate switch control signals, i.e. on the first clock interval, the control pulse is supplied to VT1, and on the second it is supplied to VT2, so the duration of the complete conversion cycle is two cycles.
Also, the non-linear dynamic process control system (NDPCS) is used as part of the control system when applying target-oriented control.In this case, two phase variables are used to form a correction action, namely: the choke current iL, read by the current sensor CS with the transmission gain βcs and the capacitor voltage uc, read by the voltage sensor VS with the transmission gain βvs.The correction signal of the NPDCS, entering the circuit of the main control system is determined by the expression ( ) where ufik -is the output signal of filter Fi at discrete times, uik is the output signal of the sample-andhold circuit SHi, Ki -are constant coefficients (Fig. 1).
The main task of the filters is to give the average value of the system state variables (choke current iLavg and capacitor voltage ucavg).These values are used to calculate, with a certain accuracy, two coordinates of the fixed point of the desired single mode, which are counted as where ΔiL -is the coordinate deviation of the fixed point for the choke current from the average value [14].
The computed fixed point coordinates are the reference to the fixed point position for the NDPCS.With the help of subtractors SB1 and SB2 the deviations of the current coordinates uik from the given values uspik are calculated and scaled with coefficients Ki.Then they are summed up using AD and the resulting correction action uck is fed additively into the control loop (Fig. 1).Obviously, if the converter is operating in the desired single mode, then uck=0, i.e. blocks Ki actually act as controllers for the corresponding coordinates of the fixed point of the desired dynamic mode.

Nonlinear dynamic model of a converter
This section will present a nonlinear dynamic model and a linear average dynamic model, which is necessary to solve the problems.Signal timing diagrams are presented in Fig. 2.This converter belongs to I-type converters, so its power section model is almost identical to that of a buck converter with the PIC [14], but in this case, due to the task, it is also necessary to take into account the dynamics of the NDPCS, which was not previously taken into account when studying similar systems.
In this converter, as it has already been mentioned earlier, the conversion cycle has duration of two clock intervals.These two clock intervals differ from each other only by the turned on switch at the stage of transferring energy to the load (Fig. 2).In Fig. 2 T is the clock interval duration, k is the clock interval number.

Figure 2. Timing diagrams of main signals
In the intermittent mode, the clock interval is divided into three sections [1], each of which is characterised by a certain combination of switched on power section valves: 1) the section of energy transfer from input to output (transistor VT1 is open (k-1)T...tk-1,1 in Fig. 2) or VT2 is open (kT...tk,1 in Fig. 2)); 2) the section of the locked state of both switches, while the choke current flows through the diode VD3 (sections tk-1,1...tk-1,2 and tk,1...tk,2 in Fig. 2); 3) the section when the choke current is zero and diode VD3 is locked ((sections tk-1,2…kT and tk,2…(k+1)T in Fig. 2)).
The transfer function of a real PIC has the form where Kpi -is the transmission gain of the PIC proportional part, Ki=[s -1 ] -is the coefficient inversely proportional to the integrator time constant, K -is the coefficient taking into account the controller imperfection.
The transfer functions of filters Fi as part of the NDPCS have the form where Tfi -is the time constant of the first order low pass filter.
Taking into account the above, the system under consideration is described by the following combination of differential equations where Aiis the system matrix in the i-th section (Fig. 2), Biis the control matrix in the i-th section (Fig. 2), ( ) ( ) X=(iL,uc,uipi,uif1,uif2) T -is the vector of state variables, uipi -is the variable characterising the integrator state as part of the PIC, uifi -variable characterising the state of the i-th filter as part of the NDPCS, ui=(uin, 0, uref) T -is the vector of control signal on the i-th section.
From the description in form (3), a transition is usually made to a nonlinear discrete mapping, which for I-type converters has the form where Xk=(iLk,uck,uipik,uif1k,uif2k) T -is the vector of state variables at the k-th discrete moment of time, zki=(tki-(k-1)T)/T -are commutation moments in relative time.
At the same time The commutation moment zk2 is calculated based on the transcendental equation ( ) ( )

Small-signal dynamic model of an open-loop control system
In this section, a small-signal dynamic open-loop model in the form of transfer functions is considered.Figure 3 shows a dynamic open-loop structural model that corresponds to the system in Fig. 1.Sign «^» marks small-signal variables corresponding to the variables in Fig. 1.
The transfer function of sample-and-hold circuits can be represented by [15] ( ) sT SH e Ws sT − − = In addition, coefficients Kβi in Fig. 3 are calculated as Kβ1=K1βcs, Kβ2=K2βvs.Let us obtain expressions for the transfer functions of the BCi control units, circled in Fig. 4 with dashed lines.The transfer function for BCi taking into account (1) and ( 5) has the form On the dynamic modes map, the white domain D1 corresponds to the stability domain of the desired 1-cykle mode, and the coloured domains correspond to those of existing undesired dynamic modes.As it can be seen from Fig. 5(b), a significant range of oscillations is observed in these domains.In this case, the gray domains D∞ are the domains of chaotic oscillations, and in domains D19 and D20 there are 19-and 20-fold modes, respectively.There are also domains on the map with the modes of other multiplicity that will not be focused on.
To eliminate the domain of undesired modes in the selected ranges for changing the system parameters, the NDPCS is used.In this case, the regulator parameters of the main control system are left as they are previously selected (Fig. 5), but it is necessary to select the NDPCS parameters K1 and K2 to eliminate domains of unwanted modes in Fig. 5.
Below the methodology for calculating the optimal NDPCS parameters is proposed.As it was mentioned earlier in this work, a small-signal open-loop dynamic model will be used for this purpose, after which the result will be monitored applying a nonlinear dynamic model.Due to the complexity of the transfer function (5), boundaries of the stability domains will be constructed using numerical methods.
It is obvious that the transfer function parameters (5) are determined by the parameters K1 and K2, since the latter are included in the transfer functions of the WBCi(s) control units, thus the transfer function ( 5) can be designated as ,, It is known that at the stability boundary, the amplitude-phase-frequency response (APFR) intersects (-1, 0) on the complex plane [15].Thus, a system of equations can be compiled Re ω, , Im ω, , 0, where j -is the complex unit, ω -is the circular frequency.The first equation corresponds to the real part of the APFR, and the second one corresponds to the complex part.By changing the frequency ω from 3000 to 10 6 rad/s and finding K1 and K2 for each value of ω based on (7) using numerical methods for solving systems of nonlinear equations, the boundaries of the stability domains of the system open-loop with the NDPCS are constructed (Fig. 6).In this case, the stability domains are located to the right of the presented boundaries.As it was before, Figure 6 shows the shaded domain obtained by the intersection of four stability domains defined by the boundaries in Fig. 6.Selecting parameters K1 and K2 corresponding to the domain, provides the system stability in the specified change ranges of uin and RLD while ensuring the required performance.
Figure 7 presents a family of Bode diagrams representing the frequency dependences of the amplitude in dB and the phase in rad on frequency.In the case under consideration, they characterise the frequency properties of an open circuit at various integrator coefficients Ki, which are indicated next to each characteristic.Moreover, for all diagrams Kpi=0.25.Each diagram is characterised by the cutoff frequency ω1i and the phase stability margin Δφi.
Brown diagrams (Fig. 7) characterise the system at Ki=500 с -1 , it was previously selected based on Fig. 4. In this case, the cutoff frequency is ω11=700 rad/s, and the phase stability margin is Δφ1=1, 87 rad.As it can be seen from Fig. 8, the transient process duration in this case is tt2=0,028 s, which is not a good result.Increasing Ki till 3000 s -1 without using the NDPCS was to have improved the result, but led to unstable behaviour of the system.As it can be seen from Fig. 7 (red diagrams) the cutoff frequency has increased and amounted to ω13=7780 rad/s and the phase stability margin was negative and amounted to Δφ3=-0,91 rad, i.e. the system has become unstable.The voltage timing diagram corresponding to these regulator parameters is presented in Fig. 8, as it can be seen; oscillations with large amplitude are observed in the system.
Figure 9 shows the diagram of total amplitude for a system with the NDPCS at Kβ1=1; Kβ2=0.5.As it can be seen from the figure, the amplitude of oscillations is quite small in the specified change ranges of uin and RLD, which indicates the correctness of the NDPCS parameters, calculated using the proposed method.

Conclusion
The method proposed in the work for selecting control system parameters for nonlinear dynamic processes, based on small-signal modelling, can significantly reduce the calculation complexity, which is important in the practical design of these systems.The results obtained show its effectiveness and the possibility of using common mathematical packages such as MathCad Prime, etc. in its application without significant labour costs.However, it is worth remembering the limitations of small-signal modelling, which requires additional verification of the calculation results using nonlinear dynamic models.
The proposed technique can be extended to a wide class of DC-DC converters and is of practical importance.

Figure 5 .
2023 Journal of Physics: Conference Series 2697 (2024) 012034 demonstrated in the two-parameter diagrams in Fig. 5, constructed using a nonlinear dynamic model when changing uin and RLD within a given range.a b Two-parameter diagrams at Kpi=0,25, Ki=3000 с -1 : a -dynamic modes map, b diagram of the output voltage peak-to-peak ripple

Figure 6 .
Figure 6.To the selection of the NDPCS optimal parameters

a b Figure 7 .
Bode diagrams for various controller parameters and control system structures