Combined Observer and Lyapunov-Based Control for Switching Power Converters With LC Input Filter

This paper presents a control technique for a boost converter that is connected to a power source via an LC input filter in a continuous conduction mode. The proposed method employs the Lyapunov stability criterion to determine the switching command that guarantees system stability. An estimator is introduced to track the load resistance and losses of the system in real-time. Our control strategy regulates all state variables of the system and additional state variables in the estimator concurrently. Additionally, the start-up transient response is considered in the control algorithm. Finally, experimental results are provided to validate the effectiveness of the proposed control method.


I. INTRODUCTION
Power electronic converters are commonly used in various applications, including renewable energy [1], [2], [3], [4], [5], microgrids [6] and transportation [7], [8], [9], [10].To minimize current and voltage ripples in the DC grid, an LC filter is typically placed at the input of the switching converter, as shown in Fig. 1.However, in the early 1970s, it was discovered that an incremental resistance at the input of the converter could be negative and interact with the LC filter, leading to instability [11].A method was proposed to prevent this oscillation, and a guideline for designing the LC input filter was published to avoid unstable operation in the DC grid [12].Negative resistance in tightly controlled power switching converters can be considered a constant The associate editor coordinating the review of this manuscript and approving it for publication was Snehal Gawande .
power load (CPL) if the input current decreases to maintain constant input power as the input voltage increases [13].As a result, the input resistance is negative.CPLs are common in transportation systems where DC microgrids power tightly controlled converter-driven motors [14], [15].Researchers have developed various techniques to control CPLs and prevent unstable operations.
In the literature, researchers have primarily focused on four categories of research related to stability and instability zones in switching converters.The first category is focused on finding these zones to restrict the operation point of the switching converters [15].The second category aims to increase the damping factor of the LC input filter using passive elements [16] or a passive stabilizer, which can lead to a low-efficiency system.The third category involves introducing special terms in the control algorithm, known as an active stabilizer, to provide a larger stability operation zone [17], [18], [19], [20], [21], [22].This method typically mimics passive stabilizers, such as virtual impedance [26], [27] and virtual capacitors [17].The final category focuses on controlling all state variables of the system, including those of the LC filter and switching converters.
Classically, to control a switching converter, the twoloop cascade feedback control method with pulse-width modulation (PWM) is commonly used [18].The voltage outer-loop bandwidth must be lower than that of the current inner loop.The active stabilizer may be added to the control loop in several manners, e.g., at the outer loop [22], current inner loop [20], [21], [22], [23], [24], [25], and duty cycle [18], [19], [21].In [26], the authors proposed a control strategy using the state feedback and pole placement to stabilize PWM DC-DC buck converters with input filters.To use this method, a linearized model was first developed.Therefore, the limitation of this control depends on the validity of that linearized model.In [27], the authors proposed a method to control all state variables that were inspired by [28], [29], and [30] using a switched model without linearization.However, because the control requires an exact value of components including the load, steady-state errors occur, particularly on the output voltage, which is one of the most important objectives.Moreover, the sampling frequency affects the steady-state error, as detailed in [31].A higher sampling frequency makes the system operate with less steady-state error.However, a high-performance controller is required.Parts of the uncertainties of the system are present in the losses in the system, which depend on the operating point.If these losses are known, the controller can effectively compensate the losses, and the output voltage static error is consequently eliminated.Unknown parameters such as the capacitance, inductance, or losses in the switching power converters can be estimated using an estimator [32], [33].
Another drawback of the proposed control in [27] is the matrix P, which is the solution of the Lyapunov function depending on the operating point.When the operating point of this controlled system changes, the matrix P must be changed to stabilize the system.Therefore, P must be either calculated online or pre-calculated offline and put in a lookup table.
In contrast to [34], there are several significant differences in the underlying concept of the proposed control scheme.First, the control scheme provides guidelines for the startup, calculation of matrix P in detail, and modification of the control block diagram.Additionally, simulation results for the startup response, steady-state response, and transient response are presented.Furthermore, the experimental setup is described in detail, and more experimental results are included.
This paper presents an estimator that can estimate losses and load resistance, thereby reducing errors associated with these factors.Furthermore, the need for a current sensor to measure the output current is eliminated by embedding an observer based on Lyapunov stability criteria into the controller algorithm.Additionally, a start-up process method is proposed, and linear matrix inequalities (LMI) are utilized to reduce the matrix P to a single value.
This paper is organized as follows.Section II of the article describes the model of the boost converter with the LC input filter and the observer used in the study.In Section III, simulation results are presented to verify the effectiveness of the observer and controller.These results are then experimentally validated in Section IV.

II. SYSTEM DESCRIPTION AND LOSS MODEL
The system shown in Fig. 2 is comprised of an LC input filter (L f , C f ), a boost converter with a switching command u∈{0,1}, an inductor L, an output capacitor C, and a resistive load R. The considered model accounts only for the parasitic resistances of the filter and converter inductors, denoted as r f and r, respectively.In Fig. 3, an ideal converter is considered, and losses are represented by a series independent voltage source V T and a parallel independent current source I P , inspired by [33].V T represents losses that depend on the level of input current, while I P represents losses due to the output voltage.The differential equations governing the behavior of the studied boost converter, connected to the power supply V i , are given by: The behavior of the LC input filter is described by the following differential equations: The control method proposed in [27] involves defining the switching command for the boost converter, u, to ensure the stability of the entire system at each sampling time, T s = 1 f s , where f s is the sampling frequency.The first step is to select a Lyapunov function candidate, followed by evaluating the time derivative of the Lyapunov function candidate to determine when to turn on and off the switch.The switching command that stabilizes the system is then selected.This method guarantees the global stability of the system during operation.

A. CONTROLLER
Considering Fig. 3, one can express the following equation depending on the switching command u ∈ {0, 1} as: The considered system is described by state variables Here, i f represents the input current, v f represents the filter capacitor voltage, i L represents the boost inductor current, and v o represents the output voltage.The system is powered by input voltage V i .When the switching command u = 1, the system is described by matrices A 1 ∈ R n×n , B 1 ∈ R n as follows (equation ( 4)), with When u = 0, the system is described by matrices , where m = 2, and p ∈ R m represents the vector of voltage V T and current I P .Further details regarding these matrices are provided below: with When the converter operates at the equilibrium point where x = x ref , assuming that the estimated value of p, denoted as p ∈ R m , where m = 2, p = V T I P T , is known, it can be verified that: with Now, for simplicity, we introduce a new state variable z.Then, we can define the corresponding variable of z : ẑ represents an estimated value of z, and ż is the time derivative of z: Using ( 3) and ( 9), one can find the following expression: Noted, 10 implies that the time derivative of x ref , denoted by ẋref , is zero when the system operates under a constant x ref in steady-state conditions.However, during the startup 24896 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
process, the state variable references will vary, and the timedependent x ref will be utilized.From this equation, when u = u ref , it leads to: Moreover, when the switching command u = {0, 1}, the system can be expressed as: where A nonlinear estimator comprising the following two equations is proposed to estimate the value of the vector p, which consists of voltage source V T and current source I P : where the errors between the estimated value and the measurement value of the state variables z and loss variables p are defined as: where K 1 ∈ R n×n is a positive-definite matrix.K 2 ∈R n×m , K P ∈R n×m , η∈R n×m , and χ∈R n×m .
To realize the estimator as described in ( 13) and ( 14), the time derivative of state variable z is typically required, but measurement noise can cause issues as mentioned in references [32], [35].Due to limited estimator gain, the quick estimation of unknown parameters can be difficult, leading to overshoot inductor current and a large output voltage drop.In this paper, a change of variable method is proposed to circumvent the need for measuring the signal derivative.
Using ( 12) and ( 13), one can find the differential equation of ξ z : Suppose that the variation of p is very slow compared to the state variable, then: We can find the differential equation of the error ξ p using ( 14) and ( 18) as follows: The integration of (19), which is ξ p with the initial condition ξ p = 0, is used to implement (17).Then, we replace ξz obtained from ( 17) in ( 19) and ( 14) to calculate p.The block diagram of this estimator will be provided in Section II-E.

C. SWITCHING COMMAND AND STABILITY ANALYSIS
A Lyapunov stability analysis is employed to demonstrate the exponential stability of the proposed estimator.To do this, a Lyapunov function candidate is chosen and expressed in (20).
Note that the matrix P in the Lyapunov function candidate expressed in ( 20) is a symmetric positive definite matrix that satisfies the Lyapunov equation, where A(u ref ) should be the Hurwitz matrix and Q 1 is a symmetric positive definite matrix.
A(u ref The switching command is given by a control law, which is defined to control the system as: which is a global stability feedback for system (12).Indeed, the derivative of V along the trajectories of ( 12) yields: where and when u = u ref , (23) becomes: To ensure that the system is exponentially stable, we propose that: where Q 2 ∈R m×m is a positive-definite matrix.Note that because g is not a square matrix,K p is calculated using the pseudo-inverse of g [36].With the conditions ( 27)-( 31), when u = u ref , equation ( 26) becomes: Equation ( 32) is strictly negative when z ̸ = 0, ξ z ̸ = 0, and ξ p ̸ = 0 because Q 1 > 0, K 1 > 0, and K p g > 0.
Observe that for u = u ref and with (32), equation ( 23) becomes: which implies by linearity that min Thus, there always exists a subscript k ∈ {1, 2} such that

D. CALCULATION OF P
In [30], the Lyapunov equation shown in ( 21) is solved to obtain the matrix P using the given symmetric positive definite matrix Q 1 and Hurwitz matrix A(u ref ), which depends on the operation point u ref .
As a result, the matrix P is not constant and changes when the operating point changes.In practice, the matrix P can be calculated online by solving (21) with the Lyap solver in MATLAB, or it can be pre-calculated and stored in a lookup table for each operating point.However, embedding this approach in the practical controller can be challenging.On the other hand, by considering ( 7) where u ref ∈ 0, 1, and assuming that A 1 and A 2 are Hurwitz matrices andP > 0, then we know that there exist: where Q 1 is a symmetric positive definite matrix.Therefore, the first term of (20) becomes: As can be observed, the expression (A k T P + PA k ) with k ∈ {1, 2} does not depend on u ref , and generally, it is independent of the operating point of the system.The solution for the set of equations (37) -( 38) is the matrix P, which remains constant for all operating points, making it easy to implement.The matrix P can be pre-calculated offline using the linear matrix inequalities (LMI) solver in MATLAB with only the constant matrices A 1 and A 2 of the system.However, the value of P obtained using LMI may differ significantly from that obtained using (21) for the same value of matrix Q 1 at the nominal load value.Therefore, matrix Q 1 for LMI is adjusted to match the value of P obtained using (21) for the nominal operating point.

E. START-UP PROCESS
Before initiating the system, the power source must supply power to the converter.If the output voltage of the boost converter is equal to the input voltage, the converter's losses can be neglected.At the equilibrium point where u = u ref , the state variables are equivalent to the state variable references obtained from the constant output voltage reference V oref ., which is generally unchanging.Conversely, during system startup, all state variables transition from the initial equilibrium point to the final equilibrium point.The trajectories of these state variables can be easily computed based on the trajectory of the output voltage.Therefore, the output voltage trajectory should be chosen first, and in this case, we select the response of a second-order system to reduce the inrush current of the output capacitor.The time derivatives of state variable references are required, as shown in equation (10).The block diagram of the proposed controller and observer is depicted in Fig 4 .The state variables X mes are sampled at intervals of T s , and the state variable references X ref are calculated using the desired output voltage V oref and system parameters.Notably, X ref is derived from the output voltage reference, so the time derivative of X ref does not contribute to control noise.An estimator is employed to provide the values of ẑ and p, and the LMI solver in Section D determines a constant value of matrix P used to determine the switching command.

FIGURE Proposed control and diagram.
24898 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

III. SIMULATION RESULTS
To verify the proposed system, a simulation of the switched model was developed using MATLAB/Simulink.Table 1 shows the parameters of the converter used in the simulation, while the voltage and current sources were included in the model to demonstrate the real effects of the proposed estimator and control algorithm.The currents i f , i L and voltages v f , v o were sampled in every period T s = 1 f s , and the control and estimator algorithms were executed every T s .In order to select the appropriate switching command u, which is the output signal of the proposed controller in Fig. 4, several signals had to be sampled.
Matrix Q 1 was chosen, and matrix P was calculated using the system matrices that do not depend on the references, and the LMI solver in MATLAB.For the estimator, an element of matrix Q 2 was selected.The estimated values were then fed into the proposed controller to select the switching command that minimized the candidate Lyapunov function.The control algorithm, which includes the proposed estimator, is illustrated in Fig. 4.
The system and controller parameters are listed in TABLES 1 and 2, respectively.

A. STARTUP RESPONSE AND STEADY-STATE RESPONSE
The trajectory of the output voltage is selected to reduce an inrush current from the power source and used to derive other references in the controlled system.The simulation results in this section are shown in the following figures.The trajectory of the output voltage is created by feeding the output step voltage reference from 63 V to 150 V through a secondorder filter with unity damping factor ζ = 1, and an angular frequency of ω voref = 50 rad/s.In this test, the load resistance is 45 .The responses of the state variables are shown in Fig. 5.In this section, the examination is focused on the dynamics of the state variables with ω voref set to 50 rad/s.The investigation comprises two tests, namely: 1) modifying the output voltage reference and 2) adjusting the resistive load.

1) CHANGING OF THE OUTPUT VOLTAGE REFERENCE
Using the same control parameters as in the previous section, the output voltage tracking was evaluated by conducting a test.The test involved altering the output voltage reference from 100 V to 150 V, and the results are illustrated in Fig. 6, depicting the behavior of the state variables.Fig. 6 (a) displays the input current i f and its reference i fref , while Fig. 6 (b) presents the input filter capacitor voltage v f and its reference v f .In both cases, the state variables i f and v f are able to track their respective references.Fig. 6 (c) shows the inductor current of the boost converter i L and its reference i Lref , while Fig. 6 (d) displays the output voltage v o and its reference v oref .

2) SIMULATION BEHAVIOR OF THE ESTIMATION FOR A LOAD STEP
This test simulates the evolution of the estimated voltage, VT , and estimated current, ÎP , during a load step.The output voltage remains constant at 150 V throughout the test, while the load resistance changes from 160 to 45 at t = 0.05s.As shown in Fig. 7, the estimated values of VT and ÎP , respond to the load step and eventually converge to constant values.Prior to t = 0.05 s, the load resistance R = 160 , which is lower than the nominal load resistance R N = 102 , causing ÎP to be negative.However, after the load step, with R = 45 , ÎP becomes positive since the load resistance is now below R N (R < R N ).

3) CHANGING THE RESISTIVE LOAD
In this experiment, the output voltage is regulated at 150 V, while the load resistance is varied from 160 to 45 .The response of the system's state variables is presented in Figs. 8.As shown, the inductor current of the boost converter increases rapidly from its initial value to the final value, limited by the input voltage and inductance.The output voltage decreases as the current increases, and then returns to its reference level.
To compare our proposed method with a classic control method, we employed a two-loop cascade PI control strategy with a virtual capacitor stabilizer, as described in [20].The switching frequency F sw = 30kHz used in this paper is equal to the sampling frequency f s .The control strategy achieved a gain margin (GM) greater than 6 dB and a phase margin (PM) of 45 degrees at 2πF sw 10 rad/s and 2πF sw 300 rad/s for the inner current loop and outer voltage loop, respectively.
We observed that the settling time with 2% error of the output voltage using the classic control method was approximately 55 ms, which is higher than the design expectation of 8.8 ms due to the effect of the stabilizer.In contrast, our proposed method achieved a settling time of approximately 6 ms, demonstrating its superior performance.

IV. EXPERIMENTAL RESULTS
To validate the proposed control system, a test bench was set up in the laboratory.The control and estimator algorithms were developed using MATLAB/Simulink and then implemented on dSPACE DS1103, as depicted in Fig. 9.The test bench consisted of various equipment such as a digital oscilloscope, a programmable power supply (63 V/85 A), two variable resistance loads, digital voltmeters, current probes, and a dSPACE unit.
The system utilized an IGBT SKM195GB126D module as a switch and a diode.To measure all necessary signals for the proposed controller, two differential voltage probes and three current probes were employed.This setup ensured accurate measurement of the system's performance and allowed for effective validation of the proposed control system.
The presentation will begin by showcasing the startup and steady-state responses, followed by the transient responses.

A. STARTUP RESPONSE AND STEADY-STATE RESPONSE
The simulation results demonstrate that the output voltage reference is generated by passing the output step voltage reference from 63 V to 150 V through a second-order filter 24900 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.with unity gain and an angular frequency of ω voref = 50 rad/s.As depicted in Fig. 10, the response of the state variables when the load resistance is 45 indicates that the proposed system can be initiated with no input inrush current.Furthermore, by selecting the appropriate angular frequency ω voref , the peak current of the input current can be regulated.

B. OUTPUT VOLTAGE TRACKING AND LOAD STEP RESPONSE
In this section, we examine the dynamics of the state variables.Two tests were conducted to achieve this: 1) a change in output voltage and 2) a change in resistive load.The response of the currents i f , i L and their references are depicted in Fig. 11 (a) for the first test, while the voltage responses are shown in Fig. 11 (b).All state variables were able to track their reference during both the transient and steady states.In the second test (Fig. 12), the load resistance R was connected to the converter and abruptly reduced from

V. CONCLUSION
This paper presents a novel control strategy for a boost converter with an LC input filter by utilizing Lyapunov function.To accurately reflect the real-world scenario, the losses resulting from inductor current and output voltage are estimated online, depending on the operating point.The loss attributed to the output voltage reduces the steady-state error of the control state variables.Furthermore, the proposed loss estimator eliminates the need for a load current sensor.The approach enables the simultaneous control of all converter variables, and the switching command is selected to ensure system stability at all times.The simulation and experimental results demonstrate the effectiveness of the proposed control technique.

FIGURE 1 .
FIGURE 1.Example of a DC microgrid system with several loads including CPLs.

FIGURE 2 .
FIGURE 2. Example system: the power source is connected to a boost converter through an LC filter.

FIGURE 3 .
FIGURE 3. Studied system that represents losses by voltage V T and current I P .

FIGURE 5 .
FIGURE 5. Response of (a) input current (i f ) and its reference (i fref ) (b) filter capacitor voltage (v f ) and its reference (v fref ) (c) inductor current (i L ) and its reference (i Lref ) (d) output voltage (v o ) and its reference (v oref ).ω voref = 50 rad/s.

FIGURE 6 .
FIGURE 6. Response to output voltage change (a) input current (i f ) and its reference (i fref ) (b) filter capacitor voltage (v f ) and its reference (v fref ) (c) inductor current (i L ) and its reference (i Lref ) (d) output voltage (v o ) and its reference (v oref ).ω voref = 50 rad/s.

FIGURE 7 .
FIGURE 7. Simulation waveforms of VT and ÎP change of a load resistance from R = 160 ohm to R = 45 ohm.

FIGURE 8 .
FIGURE 8. Response to load change (a) input current (i f ) and its reference (i fref ) (b) filter capacitor voltage (v f ) and its reference (v fref ) (c) inductor current (i L ) and its reference (i Lref ) (d) output voltage (v o ) and its reference (v oref ).ω voref = 50 rad/s.

FIGURE 10 .
FIGURE 10.(a) Response of the input current and its reference; (b) response of the filter capacitor voltage and its reference.

FIGURE 11 .
FIGURE 11.(a) Response of the input current and its reference; (b) response of the filter capacitor voltage and its reference.

FIGURE 12 .
FIGURE 12. Experimental waveforms with the step change of a load resistance from 160 ohms to 45 ohms.(a) Input current and its reference.(b) Response of the filter capacitor voltage and its reference.
160 to 45 .The Math function of the oscilloscope was used to determine the output voltage error.In this test, we examine the evolution of the voltage VT and current ÎP during a step load.The output voltage remained constant at 150 V, while the load resistance changed from 160 to 45 .The estimated values of VT and ÎP changed initially and then converged to constant values, as demonstrated in Fig.13.During the step load of the output voltage reference, the estimated values of VT and ÎP converged to their final values.ÎP was utilized to compensate for the variation in load resistance from the nominal value R N .

FIGURE 13 .
FIGURE 13.Experimental waveforms with the step change of a load resistance from 160 ohms to 45 ohms.