Sensitivity Analysis of Exact Tracking Error Dynamics Passive Output Control for a Flat/Partially Flat Converter Systems

In this paper, identification of sensitive variables is attempted for second-order (flat/partially flat) and fourth-order partially flat converters with dynamic loads. The sensitivity nature of each state variable to the output speed variable of the DC motor for the above-mentioned systems was analyzed via the frequency domain technique. Further, in continuation of this, we aimed to confirm that the variables that are used in the control law exact tracking error dynamics, passive output feedback control (ETEDPOF) are sensitive. To verify the sensitivity property, an experimental case study was done using ETEDPOF and compared with the proportional-integral controller (PIC) for a flat system, and the results are presented.


Introduction
Stable trajectory tracing is essential in power converter applications such as power quality and drives, etc. Frequently, DC-to-DC power converters are used to achieve the required voltage for a DC drive based on demanded trajectory profiles. Based on the restrictions of a ramp's magnitude and constant values which act as a reference profile, conventional feedback controllers cannot perform better. In order to alleviate this problem, trajectory tracing control of rest-to-rest maneuvers becomes mandatory. The passivity based controller (PBC) for power converters is designed to trace the required voltage/current effectively. The passivity theory was initially proposed in circuit analysis.
Due to the promising stability [1,2] and robustness [3] features, the PBC has found applications in fuel cells [3], UPFCs [4] and bilateral teleoperations. Further, the performance of two types of PBCs, i.e., energy-shaping and damping injection (ESDI) and exact tracing error dynamics passive output feedback (ETEDPOF) controllers are compared with various controllers and presented below: Chie-chi Chu and Hung-Chi Tsai proved the dominating characteristics of ESDI over the conventional proportional-integral controllers (PIC) in the control of power flow [4]. Tzann-Shin Lee investigated the behavior of ETEDPOF+ PIC and PIC, and the author indicated that the performance of ETEDPOF with PIC is better than PIC [5].
Tofighi et al. achieved better performance with ESDI in comparison with PIC in PV systems [6]. The authors demonstrated the robustness of ESDI in photovoltaic power management system for the change in reference to DC voltage, solar irradiance, as well as load resistance [6].
Dynamic response and realization complexity properties for a 1Φ inverter have been compared with adaptive control, sliding mode control (SMC) and ESDI methods. The comparative result shows

General Procedure for ETEDPOF Development
The Hamiltonian of any system represents the energy function (H) and can be written in terms of state vector 'x', which is given in Equation (1): where M is a positive definite, and constant matrix. On taking the partial derivative of Equation (1) with respect to 'x', Equation (2) is obtained: Generally, any averaged state model can be represented in Equation (1): Matrix J is skew-symmetric in nature, which will not affect system stability, as: The term 'bu' is the energy acquisition term. External disturbances like load torque are introduced in '∈'. R is symmetric and positive-semi-definite, i.e., The skew symmetry matrix J(u) can be written as: The main objective of the present work is to regulate the speed of a DC motor for a given desired speed profile (ω * ) under no-load and load conditions. The desired speed profile is obtained in an offline fashion using bezier polynomials. For this desired speed profile (ω * ), it is assumed that a state reference trajectory x*(t) satisfies the desired open-loop dynamics, and it is given by: The matrix J(u) is skew-symmetric, which is related to an average control input 'u', and it satisfies the following expansion property: where, ∂J(u) ∂u is a skew-symmetry constant matrix. Under steady-state conditions, state trajectory 'x' will reach x* and control input 'u' becomes u*. Now Equation (7) is modified to: In order to implement a closed-loop operation, errors in the system's state trajectories should be identified to modify the control input. These errors in the state trajectory (e) and control input (e u ) are calculated as given below: On differentiating Equation (10) with respect to time 't': . e = .
x (12) On adding and subtracting the required steady-state values to Equation (12), yields: The value for ∈ obtained from Equation (9) is substituted in Equation (13), then the error dynamics will become: ∂H(e) ∂e On using Equation (7) in Equation (14), error dynamics of the system is modified into; . .
In Equation (13) "J(u) ∂H(e) ∂e T " is a conservative term which will not affect the stability property of the system: Other remaining terms exactly coincides with the tangent linearization part of the dynamics [2]. The passive output tracking error is given by: A linear time-varying average incremental passive output feedback controller is simply given by: On substituting Equation (20) in Equation (16), the error dynamics of the system will become: .
Now with the skew symmetry property of J(u), .
H(e) is given by: .
Equation (23) is negative definite, if the dissipation matching condition (24) is satisfied.
In general, the satisfaction of dissipation matching condition will be in two forms, and both forms are explained below: Case I: R > 0 This indicates that the dissipation matching condition is strictly satisfied, and Equation (18) will become negative definite.
Case II: R ≥ 0 Here, the dissipation matching condition is not strictly satisfied. For these cases, LaSalle's theorem [43] is used to establish the global asymptotic stability of the origin of the tracking error space.
LaSalle's theorem can be stated as follows: Suppose that a positive definite function H exists: R n →R whose derivative satisfies the inequality . H ≤ 0 and moreover, if the largest invariant set contained in the set {x: . H = 0} is equal to {0}, then the system is globally asymptotically stable. Due to the bounded nature of control input [0 and 1] in power electronic converters, the system can be semi-globally asymptotically stable. Now, the dissipation matching condition and the average control input are given by: where 'γ' represents the damping injection coefficient whose value can be considered of low value to avoid noise amplification. Thus, the control input for regulating the speed of the DC motor is obtained. Control function (u) in Equation (26) clearly indicates the absence of the derivative term, which makes the controller simpler.

ETEDPFO for Buck Converter with Dynamic Load
To initiate the implementation of ETEDPOF, the average model of the buck converter-fed DC motor is developed (Figure 1), and it is given in Equations (27)- (30).

ETEDPFO for Buck Converter with Dynamic Load
To initiate the implementation of ETEDPOF, the average model of the buck converter-fed DC motor is developed (Figure 1), and it is given in Equations (27)- (30).
The above equations can be written in matrix form with the state vector: and it is given in Equation (32):  The above equations can be written in matrix form with the state vector: and it is given in Equation (32): Buck converter-fed DC motor model shown in Equation (32) clearly exhibits that it is an energy management system with conservative and dissipative forces. As ETEDPOF involves energy-based operations, it is mandatory to modify Equation (32) further in terms of energy function and Equation (32) can be written as: where Matrix M is given by: The matrices 'b' and ' 'are given by: ∈ T = 0, 0, 0, −T L J (37) and the matrices J and R are given by: The matrix 'J' is independent of 'u', and it is of skew-symmetry in nature. Matrix R is symmetric and positive semi-definite: i.e., R T = R ≥ 0 Thus, the modified average model of buck converter-fed DC motor is obtained using the basic principles of the energy management structure of the system. This derived model is used in the development of ETEDPOF control for buck converter-fed DC motor with or without loading. To regulate the speed, ETEDPOF control is essential, and it is derived based on the error stabilization dynamics, which is presented in the next section. Under loading conditions, load torque estimation is required to regulate the speed.

ETEDPOF Design
The main objective of the present work is to regulate the speed of a DC motor for a given or desired speed profile (ω * ) under no-load and load conditions. The desired speed profile is obtained in an offline fashion using bezier polynomials. By following the procedure explained in Section 2.4, the control function (u) is derived as: where the constant 'γ' must be >0, and it is termed as damping injection coefficient. In this case, the dissipation matching condition is verified using Equation (24). Final dissipation matching matrixŘ for buck converter system is given by: It is realized that,Ř matrix is not strictly satisfying the dissipation matching condition. Hence, the control law (41) makes the origin of error space as an asymptotically stable equilibrium point by virtue of LaSalle's theorem [43] e 4 = 0. On substituting these values in Equation (16), then e 2 = 0. Hence, the error dynamics of the system becomes converging, and the system becomes globally asymptotically stable by virtue of LaSalle's theorem. Due to the bounded nature of control input [i.e., 0, 1], system stability is not a global one.
In Equation (41), the inductor current plays an important role in control operation, and hence, it can be termed as the more sensitive variable. This sensitive nature of the inductor current is analyzed in the next section. Equation (41) indicates the control input derived by using ETEDPOF. This control law makes the system to become semi-globally asymptotically stable for the given desired speed profile (ω*).

ETEDPOF for Boost Converter
Armature voltage (v) and boost inductor current (i) are used as feedback signals for ETEDPOF ( Figure 2) implementation with ω * as the desired speed profile. Using Kirchhoff's laws and Newton's laws, the average model for boost converter-fed DC motor can be derived. The derived average model is given by: The Equations (45)- (48) can be written as below: Electronics 2020, 9, x 8 of 27

ETEDPOF for Boost Converter
Armature voltage (v) and boost inductor current (i) are used as feedback signals for ETEDPOF ( Figure 2) implementation with ω * as the desired speed profile. Using Kirchhoff's laws and Newton's laws, the average model for boost converter-fed DC motor can be derived. The derived average model is given by: The Equations (45)- (48) can be written as below:  ( with the state vector where L-Boost converter inductance (Henry) C-Boost converter capacitance (Farad) R m -Motor armature resistance (Ohm) L m -Motor armature inductance (Henry) u-Average control input i-Input current (Ampere) v-Armature voltage or converter output voltage (Volt) By using the Hamiltonian principle, which was explained in Equation (3) can be written as .
Matrix M is given by: The matrix is given by: and the matrices J and R are given by: Thus, the average model of boost converter-fed DC motor is modified based on the energy management structure. By following the procedure explained in Section 2.1, a natural feedback law may be written as: where the constant 'γ' must be >0. Here dissipation matching condition is strictly satisfied, and the final R matrix is given by: R is a positive definite matrix. Since R is positive definite whenever t ≥ 0, the origin of the error space is asymptotically stable, and due to the bounded nature of u between 0 and 1, the result is not a global one. When 'u' semi globally stabilizes to u*, output voltage, current of the boost converter and speed of the motor will be stabilized.

ETEDPOF for a Luo Converter System
Earlier differentially flat and non-flat system of order four were considered for sensorless load torque estimation and ETEDPOF implementation. In this paper, a sixth-order non-flat system is proposed for analysis. One example of a non-flat sixth-order system is the luo converter-fed DC motor. Out of many pump circuits, the fundamental luo converter is taken for analysis. The relative degree of luo converter-fed DC motor is three, which is less than the order of the system. This confirms the unstable internal dynamics of luo converter inductor current and capacitor voltage with respect to the speed. Hence, reference trajectories are developed in an indirect manner. Figure 3 represents the luo converter with a dynamic load for ETEDPOF implementation. ETEDPOF can be implemented with inductor currents (i 1 & i 2 ) and capacitor voltage (v 1 ).
R is a positive definite matrix.
Since R is positive definite whenever t ≥ 0, the origin of the error space is asymptotically stable, and due to the bounded nature of u between 0 and 1, the result is not a global one. When 'u' semi globally stabilizes to u*, output voltage, current of the boost converter and speed of the motor will be stabilized.

ETEDPOF for a Luo Converter System
Earlier differentially flat and non-flat system of order four were considered for sensorless load torque estimation and ETEDPOF implementation. In this paper, a sixth-order non-flat system is proposed for analysis. One example of a non-flat sixth-order system is the luo converter-fed DC motor. Out of many pump circuits, the fundamental luo converter is taken for analysis. The relative degree of luo converter-fed DC motor is three, which is less than the order of the system. This confirms the unstable internal dynamics of luo converter inductor current and capacitor voltage with respect to the speed. Hence, reference trajectories are developed in an indirect manner. Figure 3 represents the luo converter with a dynamic load for ETEDPOF implementation. ETEDPOF can be implemented with inductor currents (i1& i2) and capacitor voltage (v1). With the available wide variety of pump circuits, a fundamental positive output luo converter is taken for the present work. Using Kirchhoff's laws and Newton's laws, a linear averaged model for a luo converter feeding DC motor can be derived. Because of the selection of the armature control method for speed control, field circuit equations are omitted. The derived linear averaged model is given by: With the available wide variety of pump circuits, a fundamental positive output luo converter is taken for the present work. Using Kirchhoff's laws and Newton's laws, a linear averaged model for a luo converter feeding DC motor can be derived. Because of the selection of the armature control method for speed control, field circuit equations are omitted. The derived linear averaged model is given by: Using matrix notation and with the help of the Hamiltonian system and as it is given in Equation (3) with the state vector shown in Equation (65).
and the matrices b, ∈, J(u), R are given from Equation (66) to Equation (69): Matrix J is skew-symmetric in nature. R is symmetric and positive-semi-definite, i.e., The skew symmetry matrix J(u) can be written as: where J 0 and J 1 are skew symmetry constant matrices which are given in Equations (72) and (73) The total stored energy of the system is given as: where the matrix 'M' is given by: Which is positive definite, and constant. Thus, the average model of luo converter-fed DC motor is modified. ETEDPOF control law can be derived by substituting the necessary matrices in Equation (26), and it is given by: In the present case, the dissipation matching condition is not strictly satisfied, and the final dissipation matching matrix is given by: Since R is positive semi-definite whenever t ≥ 0, the control law (76) makes the origin of error space as an asymptotically stable equilibrium point by virtue of LaSalle's theorem.

Stability Proof:
Error dynamics in the luo converter and first derivative of energy can be written as: .
and it is calculated as When .
H(e) = 0 and e 1 = e 2 = e 3 = e 5 = e 6 = 0 → e 4 = 0 . This indicates that LaSalle's theorem is established, and the origin of error space is globally asymptotically stable. Therefore, due to the bounded nature of control input between 0 and 1, the origin of error space is semi-globally asymptotically stable.

Sensitivity Analysis
Sensitivity analysis plays a vital role in power systems, power electronics and control engineering. In power systems, sensitivity is used for identifying influential parameters for the generation of slow oscillations [44]. Further sensitivity provides better insights into system behavior, which cannot be obtained from traditional simulation [45].
In power electronics, sensitivity is used for the determination of state variable variation and steady-state determination [46]. Furthermore, sensitivity analysis efficiently computes the tolerance region between the actual characteristics and characteristics obtained through the design procedure in a power electronic circuit.
Due to the above-mentioned advantages, sensitivity analysis is preferred as a tool in the Posicast control of buck converters and in optimal control [47]. In continuation of this control engineering application, sensitivity is used for the comparison of different control techniques, which are based on parameter variations. On top of these applications, sensitivity is used in discrete systems for studying the local and global effects of discrete systems due to perturbations in sampling frequency on system performance [48] and in sampled systems.
From the above, it can be concluded that sensitivity analysis is an important tool used in various areas like power systems, power electronics, control engineering and so on. To date, sensitivity analysis is not utilized for identifying more sensitive variables used in the ETEDPOF control law, and it is discussed in this paper for the control law ETEDPOF developed for fourth and sixth order systems. Sensitivity analysis can be performed for power converters by following the algorithm shown below: Step 1: Identify the output variable. In the present case, it is the speed of the motor; Step 2: Identify the state variables of the given system; Step 3: Number of equations required for analysis = (Order of the system) − 1; Step 4: Obtain the relation between the output variable and each state variable in the frequency domain. For a flat system, derived expressions are obtained in terms of converter and motor parameters only. In contrast, for a partially flat system, expressions will be in terms of control input in addition to converter and motor parameters; Step 5: Find the gain margin and phase margin values for the various values of load torque in the case of a flat system. For a partially flat system, margin values are obtained for the various values of load torque as well as various values of control input; Step 6: Sensitive variables can be identified based on the margin values obtained through the bove-mentioned procedure, and it is presented in the following subsections.

Sensitivity Analysis of Buck Converter
Here, the order of the flat system is four, and thus three expressions are derived, which are shown in the Equations (81) Table 1. Gain and Phase margin values are shown in Table 2 and in Figure 4, from which it is found that negative margin values obtained for the inductor current made that variable more sensitive.  (1), it is found that inductor current is more sensitive than other variables and this sensitive variable is chosen as a control variable inherently in the ETEDPOF. To validate, a comparative study is performed between the PIC and ETEDPOF.  Comparison of ETEDPOF and PIC Figure 5 shows the experimental setup for the fourth-order system, i.e., dynamic load fed via a buck converter to study the performance comparison between ETEDPOF and PIC. Comparison of ETEDPOF and PIC Figure 5 shows the experimental setup for the fourth-order system, i.e., dynamic load fed via a buck converter to study the performance comparison between ETEDPOF and PIC. Comparison of ETEDPOF and PIC Figure 5 shows the experimental setup for the fourth-order system, i.e., dynamic load fed via a buck converter to study the performance comparison between ETEDPOF and PIC. Specifications of interest are shown in Table 1. Experiments were conducted for various load torque and various speed references ( Figure 6). For PIC, Kp and Ki values are obtained using the Ziegler-Nichols method. Simulation and experimentation study were completed, and the following observations are made from the results, which are shown in Figures 5 and 6: Specifications of interest are shown in Table 1. Experiments were conducted for various load torque and various speed references ( Figure 6). For PIC, K p and K i values are obtained using the Ziegler-Nichols method. Simulation and experimentation study were completed, and the following observations are made from the results, which are shown in Figures 5 and 6: Speed references and applied load torques for the present analysis are shown in Figure 6. Comparing the speed regulation performances of ETEDPOF and PIC (Figure 7), it can be concluded that the former produces a good dynamic response than the latter for all load variations and speed variations. Further, Integral Square Error (normalized) for ETEDPOF is lesser than PIC (Figure 8). This is due to the fact that ETEDPOF utilizes the most sensitive variable, i.e., inductor current, for control operation. Speed references and applied load torques for the present analysis are shown in Figure 6. Comparing the speed regulation performances of ETEDPOF and PIC (Figure 7), it can be concluded that the former produces a good dynamic response than the latter for all load variations and speed variations. Further, Integral Square Error (normalized) for ETEDPOF is lesser than PIC (Figure 8). This is due to the fact that ETEDPOF utilizes the most sensitive variable, i.e., inductor current, for control operation. Speed references and applied load torques for the present analysis are shown in Figure 6. Comparing the speed regulation performances of ETEDPOF and PIC (Figure 7), it can be concluded that the former produces a good dynamic response than the latter for all load variations and speed variations. Further, Integral Square Error (normalized) for ETEDPOF is lesser than PIC (Figure 8). This is due to the fact that ETEDPOF utilizes the most sensitive variable, i.e., inductor current, for control operation.

Sensitivity Analysis of Boost Converter
Earlier sensitivity analysis is completed for a flat system, whereas for fourth-order partially flat boost converter with dynamic load is challenging. For a partially flat system [49], both gain and phase margin values are calculated from the expressions (84)-(86), which are related to control input and load torque, whereas for a flat system discussed earlier is related, with only load torque. Margin values are computed for the specifications shown in Table 3

Sensitivity Analysis of Boost Converter
Earlier sensitivity analysis is completed for a flat system, whereas for fourth-order partially flat boost converter with dynamic load is challenging. For a partially flat system, both gain and phase margin values are calculated from the expressions (84)-(86), which are related to control input and load torque, whereas for a flat system discussed earlier is related, with only load torque. Margin values are computed for the specifications shown in Table 3

Sensitivity Analysis of Boost Converter
Earlier sensitivity analysis is completed for a flat system, whereas for fourth-order partially flat boost converter with dynamic load is challenging. For a partially flat system [49], both gain and phase margin values are calculated from the expressions (84)-(86), which are related to control input and load torque, whereas for a flat system discussed earlier is related, with only load torque. Margin values are computed for the specifications shown in Table 3, along with the variations in load and controller input (u). From the margin values achieved, Figures 9 and 10 are drawn: ω(s) = i(s) (a 6b s 3 + a 7b s 2 + a 8b s + a 9b ) − T L k(1 − u)s(a 6b s 3 + a 7b s 2 + a 8b s + a 9b )  For a partially flat boost converter with dynamic load, current flow through the boost inductor involves the variation of load and controller input, and the plots are drawn, as shown in Figure 9. From the figure, it can be confirmed that, due to negative margin values, current flow through the inductor is considered as sensitive.  For a partially flat boost converter with dynamic load, current flow through the boost inductor involves the variation of load and controller input, and the plots are drawn, as shown in Figure 9. From the figure, it can be confirmed that, due to negative margin values, current flow through the inductor is considered as sensitive.    For a partially flat boost converter with dynamic load, current flow through the boost inductor involves the variation of load and controller input, and the plots are drawn, as shown in Figure 9. From the figure, it can be confirmed that, due to negative margin values, current flow through the inductor is considered as sensitive.

Sensitivity Analysis of a Luo Converter
Similar to partial flat boost converter, the procedure can be adopted for a sixth-order partially flat luo converter [50] with dynamic loads, and the expressions are shown in (87) ; a 34 = L 2 C 1 (BRmC2+k 2 C 2 +B L m +R m J+J)+JL m ku * (1−u * ) Margin plots are obtained for the specifications shown in Table 4. Sensitivity analysis plots for a luo converter are shown in Figures 11-14. From the responses, the following observations are made: 1.
The frequency response of inductor current (i 1 ) is shown in Figure 11. Though the gain margin remains positive, the phase margin assumes a low value for smaller values of load torque. Therefore, it can be considered as one of the sensitive variables; 2.
The frequency response of capacitor voltage v 1 is shown in Figure 12. The gain margin for v 1 becomes low for smaller values of load torque, and phase margin value becomes positive to negative when the load torque moves from low to high value. As both gain and phase margin variations are opposite to each other, v 1 is considered as a sensitive variable; 3.
Bode plot response for inductor current (i 2 ) is shown in Figure 13, which indicates that both margin values are negative and thus it makes i 2 as a sensitive variable; 4. Figure 14 confirms that the variables 'v 1 and 'i am ' are not sensitive due to positive margin values.
Electronics 2020, 9, x 21 of 27 1. The frequency response of inductor current (i1) is shown in Figure 11. Though the gain margin remains positive, the phase margin assumes a low value for smaller values of load torque. Therefore, it can be considered as one of the sensitive variables; 2. The frequency response of capacitor voltage v1 is shown in Figure 12. The gain margin for v1 becomes low for smaller values of load torque, and phase margin value becomes positive to negative when the load torque moves from low to high value. As both gain and phase margin variations are opposite to each other, v1 is considered as a sensitive variable; 3. Bode plot response for inductor current (i2) is shown in Figure 13, which indicates that both margin values are negative and thus it makes i2 as a sensitive variable; 4. Figure 14 confirms that the variables 'v1′ and 'iam' are not sensitive due to positive margin values. From the analysis made above, it is concluded that capacitor voltage (v 1 ), inductor current (i 1 ) and inductor current (i 2 ) are considered as more sensitive variables and these variables are inherently used in ETEDPOF implementation.
Therefore, the sensitivity analysis made in the cases of fourth-order and sixth-order systems has indicated the sensitive variables.

Conclusions
In this paper, sensitivity analysis for buck, boost and luo converters was carried out. Frequency response analysis is used for investigation. From the sensitivity analysis, the following points are inferred: (a) In a buck converter with a dynamic load, the inductor current is considered as the sensitive variable; (b) In order to verify the sensitivity nature, ETEDPOF is compared with PIC in the above-mentioned system. The results confirm the superiority of ETEDPOF against PIC for various speed references and different load torque conditions; (c) In a boost converter, the current flow through the boost inductor and the voltage across the load are confirmed as sensitive; (d) In a luo converter, the capacitor voltage (v 1 ) or capacitor voltages (v 2 ), inductor current (i 1 ) and inductor current (i 2 ) are considered as more sensitive variables.
From the above, it can be inferred that a greater number of sensitive variables are inherently used in ETEDPOF control, which makes the controller perform better than the other controllers like PIC.