Input Voltage and Current Sensorless Control of a Single Phase ACDC Boost Converter Mehdi Tavan and Kamel Sabahi Abstract— It is well known that measuring the input voltage and current, as the feedforward and feedback terms, are vital for the controller design in the problem of power factor compensation (PFC) of an AC-DC boost converter. Traditional adaptive scenarios corresponds to the simultaneous estimation of these variables are failed because the system dynamics is not in the classical adaptive form. In this paper, the system dynamics is immersed to a proper form by a new filtered transformation to overcome the obstacle. The phase and amplitude of the input voltage along with the input current is exponentially estimated from the output voltage with global convergent. An application of the proposed estimator is presented in conjunction with a well-known dynamic controller. and power factor quality . It is worth noting that, the ACDC boost converter belongs to second-order bilinear systems group and both of its states are not observable when the control signal is zero. These features pose an interesting state and parameter estimating problem. I. INTRODUCTION Consider the single-phase full-bridge boost converter shown in Fig. 1. The circuit of the converter combines two pair of transistor-diode switches in two legs to form a bidirectional operation. The switches in each leg operate in complementary way and are controlled by a PWM circuit. The dynamic equations describing the average behavior of the converter can be obtained using the Kirchhoff’s laws as (1) ( ) (2) where describes the current flows in the inductance , and is the voltage across both the capacitance and the load conductance . The continuous signal operates as a control input and is fed to the PWM circuit to generate the sequence of switching positions . The switch position function takes the values in finite set . ( ) represents the voltage of the Finally, ( ) AC-input. The problem of power factor compensation (PFC) for an AC-DC boost converter has widely been studied by many researchers due to its applications for interfacing renewable energy sources to hybrid microgrids , flexible AC transmission systems , motor drive systems , LED drive systems  etc. Knowledge about the phase and amplitude of AC-source is vital to provide the feedforward control signal in the problem of PFC of an AC-DC boost converter; see control signals in  and . Besides an accurate measuring of these parameters, feedback from input current improves output voltage regulation, total harmonic distortion (THD) M. Tavan is with the Department of Electrical Engineering, Mahmudabad Branch, Islamic Azad University, Mahmudabad, Iran (e-mail: m[email protected]). K. Sabahi is with the Department of Electrical Engineering, Mamaghan Branch, Islamic Azad University, Mamaghan, Iran. Figure 1. AC-DC full-bridge boost converter circuit . In this paper the effect of the phase difference between the input voltage and current on the power quality is investigated. Specially, the DC error and the amplitude of harmonic in the output voltage increase for big values of the phase shift. Hence, the problem of phase difference estimation is interested in this paper. A new nonlinear, globally convergent, and robust estimator is designed via immersion and invariance (I&I) based filtered transformation (see [9-11] for some applications of the method). The input current and voltage is exponentially estimated from the output voltage via a fifth- dimensional estimator. An application of the estimator is presented in conjunction with a well-known dynamic controller. II. PROBLEM STATEMENT For the system, the control objective is to regulate in average the output (capacitor) voltage in some constant desired value with a nearly unity power factor in input side. The objective can be achieved indirectly by means of stabilizing input (inductor) current in phase with the source voltage. This is done due to unstable zero dynamic with respect to the output to be regulated which imposed a secondorder harmonic on the output in steady-state . Let us to assume that the input current has the following steady-state form ( ) ( ) (3) for some constant as the phase difference between input voltage and current, and some yet to be specified. Replacing (3) into (2) yields ( ) ( ) (4) where and are the control input and output voltage in steady-state. Now, replacing (3) and (4) into (1) yields the following dynamics in steady-state ( ̇ ) ( ( ) ( ( ) (6) ) ( ) ( ( ) ( )) (7) ( ) ( ( ) ( )) (8) Finally, doing a basic trigonometric simplification (6) takes the form ) ( ) (9) with ( ) ( ) √ (10) ( ) (11) (16) which indicates a steady state error in output voltage. Remark 2: The minimum amplitude of the second-order harmonic ( ) does not occur for . From (10) it can be computed that for which satisfies the following inequality ) ( )( ( ( (15) √ ( with ( ) ( )) (5) ) The steady-state solution of (4) can be calculated using Fourier series and is given by ( ) ( √ with for some constant . Replacing the amplitude and phase of the current in (12) yields to ) ( ( ) ) ( ) ( ( )) ( )) (17) the magnitude of ( ) is less than ( ). From (17) it can be concluded that when the converter operates in leading form, i.e. , the magnitude of ( ) increases, but, it can be decreased when the converter operates in lagging form with a small near to zero. For instance consider the experimental system used in  with the set of parameters; , , , , and . The ratio of ( ) to ( ) is shown in Fig. 2 for ( ) ( ) and . Fig. 2 justifies the above statements and shows that a small lagging phase shift can be beneficial to reduce the harmonic amplitude. This may be induced by time delays or unmodeled dynamics in the actual system. On the other hand, a big phase difference, in leading or lagging form, causes high harmonic amplitude as shown in Fig. 2. The new representation, given by (9), demonstrates the relationship between the phase difference and the steady-state amplitude of the input current and the second-order harmonic on the output. Remark 1: The DC-component of concluded from (9) and is given by ( ) √ ( in steady state can be ) (12) As a result, in order to provide the desired output voltage the input current amplitude must be forced to achieve ( ) ( ) , (13) which justifies the fact that for any phase shift (positive or negative) the input current and then the converter losses increases. For , the minimum value of can be achieved and is defined as (14) As an example, consider the static control law in  which causes the following input current in steady-state Figure 2. The influence of and on the amplitude of the second-order harmonic. With respect to the remarks above which cite the importance of phase difference in the power quality, in this paper we are interested to estimate the parameter vector ( ) ( ) (18) from the output voltage . The phase and the amplitude can be obtained from the above vector as with ( ) ( ) (20) ) ( ) ( ̇ (21) ( ( ) ( (22) ) In this section an estimator is designed for the system (1), (2) which requires the knowledge about , , , and . It should be noted that the only measurable state is the output voltage . This cancels the use for input current sensor and makes the practical implementation of the proposed procedure attractive. In the design procedure the following assumptions are considered. where Assumption 1: The control signal indicated as . in (23) where is an auxiliary dynamic vector, which its dynamics are to be defined. The equation above admits the global inverse (24) ( ) is the new unavailable vector. The where dynamics of the new variable can be obtain as ( ) [ ] (25) ) ̅ ̅ which ̅ ( ̅ ̅) ̅, (29) ( ̅ (30) ̅ ( and , and ) with ) . To obtain the dynamics of ̅ differentiate (29), yielding ̇̅ A. Design procedure In order to form a proper adaptive structure the following input-output filtered transformation is considered ̇ ( ̇̅ is bounded and Assumption 1 is needed so that the input current become observable from the output voltage . The assumption is satisfied in operation mode when the system is forced to track a positive constant as the desired output. Although Assumption 2 seems to be somewhat restrictive, a dynamic control law, like the one introduced in Proposition 8.9 of , satisfies the assumption. The final assumption is standard in the estimator design and is extremely milder compared to the boundedness of trajectories. ( ) (28) is the estimator state, and the mapping yet to be specified. From (24) and (29), the current estimation error can be obtained as ( Assumption 3: The system (1), (2) is forward complete, i.e. trajectories exist for all . ( ) ) ̅ is PE, which will be (27) Now, under the inspiration of the I&I technique, let us to define the estimation error as III. ESTIMATOR DESIGN Assumption 2: The time derivative of available. ) where to get ( ) (26) where and are the zero matrix and zero vector, respectively. The dynamics of the output voltage (1) can be rewritten in terms of the new variable as and the correspond regressor can be defined as ( ] (19) √ ( ) ̇ [ ) [ ̇ ] ̇ ̇ (31) Let ̇ )( ( [ ) ] ̇ ̇ (32) which cancels the known terms in the error dynamics and yields to ̇̅ ( ) ̅ (33) The following proposition put forward a selection of and ̇ , which renders the origin of the system (33) attractive. Proposition 1: Consider the system (33) verifying Assumption 1-3, with [ ̇ ( ) ] ( (34) ) (35) where is the identity matrix, and are constant and positive definite matrixes, and is a constant value. Then the system (33) has a uniformly globally stable equilibrium at the origin, ̅ and ̇ ̅ asymptotically converges to zero. Moreover, the regressor is PE and then ̅ converges to zero exponentially. Proof 1: The proof is constructed in three steps. In Step 1 the convergence of ̇ ̅ is proved. Step 2 shows that . Finally, the convergence of ̅ is proved in Step 3. Step 1: Substituting (34), (35) into (33) yields the error dynamics ̇̅ ( ) [ ] ̅ (36) Consider now the following Lyapunov function candidate ( ̅ ̅) ̅ (̅ ̅) ̅ ( ) (̅ ( ) (̅ ̅) ̅) ) ( (38) Step 2: Notice that, for the asymptotic stable filter (35), if . The PE condition implies the existence of a constant such that ∫ (39) ‖ ‖ for all and all . With respect to (21), a straightforward check for the inequality above ( ( ) ( )) with obtains by parameterizing ) , which yields to ( ∫ for all . This implies that ) (40) . Step 3: As shown in Step 1, ̅ converges to zero. Let us to assume that after the convergence time of ,̅ there exists a time instance such that ‖ ̅‖ ‖ ̅‖ (41) for any constant . By virtue of the assumption above, using (41) and (37) in (38) and applying some basic bounding, results in ̅) ( ‖̅‖ ̅ ‖ ̅‖ (42) ̅) ( ‖̅‖ for some constant ,̅ (42) can be written as . After the convergence time of ̇ ̅ ( Integrating (43) when ( ) ‖̅ ‖ ) (43) yields the solution ∫ ( ( ) ( ) ∫ ( ( ) ( ) ( ) ) (44) ) ̅ ‖ ̅ ‖ and it is well known that ‖ ‖ with . Considering (39), (40) and invoking Proposition 2 in  for some constant and all implies that ( ( ) ( ) ) ∫ for some constant . This implies that the system (36) has a uniformly globally stable equilibrium at the origin, ̅ and ̅ then ̅ , and finally ̅ ̅ and then ̇ ̅ . It is clear that, the dynamic (35) is input-to-state stable with respect to which is given by (21). Hence, due to then and ̇ . Now with respect to Assumption 2 and (36) it can be concluded that ̇ ̅ and then ̈ . Therefore, the uniformly ̅ and then ̇ ̅ to zero follows directly by convergence of ,̅ Lemma 1 in . ̅ ( ) (37) whose time-derivative along the trajectories of (36) satisfies ̇ ̇ (45) due to . Therefore, there exists some time instants , such that (44) satisfies ( ) (46) for some constant , which implies that ‖ ̅‖ ‖ ̅‖ (47) This contradicts with (41) and induces attractiveness of the equilibrium ̅ by contradiction. This property is uniform with respect to . Therefore the equilibrium ̅ is uniformly asymptotically stable. Indeed, due to linear form of the time-varying system (36), the convergence property is exponential by Theorem B.1.4 in . Remark 3: With respect to (29) and (30), the equilibrium point ̅ admits to estimating and by ̂ (48) ̂̇ ( ) Indeed, from (19) and (20) an estimate of obtained by and ̂ ( ) ̂ ̂ ̂ (49) √̂ ̂ can be (50) (51) IV. OUTPUT FEEDBACK CONTROL The PFC main function is to achieve unity power factor. This happens if the input current to be in the phase with the AC-source. This can be achieved by the following dynamic control law  ̇ where ( ) (52) ( ) is derived from the filter ( ) ( ) with some positive constants , (53) phase shift. A new nonlinear, globally convergent, robust estimator is designed via I&I based filtered transformation. The input current and voltage is exponentially estimated from the output voltage via a fifth-dimensional estimator. An application of the estimator is presented in conjunction with a well-known dynamic controller. , , and ( REFERENCES ) (54) ( ) with positive constant value , where and is given by (14). For sufficiently large , the closed loop system is asymptotically stable, and then converges to zero and converges to . Note that, from analysis in Remark 1, for in (12) with zero phase difference, i.e. , the output voltage converges in average to . On the other hand, and in (54) are sinusoid signals, hence we can conclude that is sinusoid in the steady state. As a result and Assumption 1 is satisfied. From (54), it is clear that an accurate measuring of the input voltage and current is vital to achieve control objective. Hence, the state and parameter estimation given by (49)-(51) can be used in conjunction with the aforementioned dynamic control law. Notice that, with respect to (14), singularity problem can be occurred when ̂ is used to estimate in . In order to circumvent the problem, we assume that is lumped in the arbitrary gain in (53) and the signal        (55) is fed to the filter (53) instead of . 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