The state feedback control method for an Interleaved Boost Power converter is designed which in turn can achieve interleaved current sharing among the parallel connected converters. The state feedback control in continuous time domain is derived using pole placement technique and Linear Quadratic Optimal method. The load estimator is designed by deriving full order state observer to ensure the robustness and optimality of the state feedback control and an observer based controller (control law plus full order state observer) is designed using Separation Principle. The control strategies thus adopted achieve effective output voltage regulation, good dynamic stability and rejects the disturbances. Extensive Simulation has been carried out and the results are illustrated.
Key words: Control law, Full Order State Observer, Interleaved Boost Converter, Pole Placement, Riccati Matrix, Separation Principle
1. Introduction
The dc-dc converters are widely used in many applications such as distributed power supply systems, power factor improvement, harmonic elimination, fuel cell applications and photo voltaic arrays. Among the basic dc-dc converters, Boost converters are more significant and it has several advantages such as simple in construction, step up conversion ratio, higher efficiency and performance. In high power applications, it is often required to associate the converters in series or in parallel due to the unavailability of a single device to withstand voltage stress or current stress. In particular for higher current ratings, interleaved boost converter is the best choice since a fraction of input current flows through the switches [1].
Interleaved boost converters consist of N-paralleled boost converters. The main advantages are: (i) input current ripple is reduced (ii) inductor size is reduced (iii) current rating of the semiconductors are reduced (iv) good current sharing among the converter modules (v) I2R losses and inductor ac losses are reduced (vi) easier system maintenance and expansion (vii) increased system reliability. The main challenge in the field of Power Electronics is emphasized more on the control aspects of the dc-dc converters. The control approach requires effective modeling and a thorough analysis of the converters. In conventional design approaches control problems are more complex and topology dependent [2].
The control of the dc-dc converters are mainly aimed at obtaining desired output voltage regulation. The typical control strategy which is used in many applications is the current mode control. In current mode control, two control loops are provided. The outer loop is slower in which the output voltage is compared with a reference signal, which in turn forms a reference to the inner loop. The inner loop is much faster which forces the inductor current to reach the reference value, provided by the outer voltage loop. This control approach is mainly used in the Boost converters which suffer from an undesirable non-minimum phase response [3].
The control aspect in implementing the Interleaved Boost converter mainly focuses two major drawbacks: (i) when operating in continuous conduction mode current unbalances due to intrinsic device parameter variation occur which is quite critical (ii) complexity in the circuit [2]. This paper is aimed to present a feasible solution for the above mentioned control problems existing in Interleaved Boost converters. The control approach is based on deriving a control law defined as, where k is the state feedback matrix and x (t) is the state vector. The proposed Observer Controller is designed in twofold:
The appropriate Observer poles are chosen and the controller is designed by combining state feedback matrix and Observer poles by using Separation Principle. The main advantage of this principle is that the design of state feedback matrix and the observer can be carried out independently and when both are used concurrently the roots remain unchanged.
The state feedback matrix is optimized by deriving Riccati matrix and using the same observer poles chosen as in step 1, a Linear quadratic optimal regulator (LQR) is designed.
The Observer Controller thus designed for the Interleaved Boost converter results in excellent output voltage regulation, improved dynamic response, robust, rejects the disturbances, highly efficient with much lesser settling time in the range of milli second and good current sharing among the converter modules.
2. Modeling of the Interleaved Boost Converter
The schematic diagram of Interleaved Boost converter is shown in fig.1.Here Vg is input voltage, L1 and L2 are magnetizing inductances, S1 and S2 are semiconductor switches, D1 and D2 are diodes, C is an output capacitor and R is a load resistance respectively.
Fig 1. Schematic diagram of Interleaved Boost Converter
The converter is modeled using state space averaging method. State space averaging method is highly significant for this kind of converters since PWM converters are special type of non linear systems which is switched in between two or more non linear circuits depending upon the duty ratios. Further, control signals include not only the independent voltages and currents but also the duty ratios [4].
As a general case state space averaging method for two switched basic PWM converters is discussed now. The inductance currents and capacitance voltages are state variables and matrix form of equation is mentioned below,
(1)
(2)
where x is a state variable vector, u is a source vector, A1, B1, A2, B2 are the system matrices respectively. The significance of state space averaging technique lies in replacing the above two sets of state equations by a single equivalent set described as follows,
(3)
The A and B matrices are the weighted averages of actual matrices describing the switched system given by the following equations,
(4) (5)
where d is the duty cycle ratio. Based on the above discussion, state model of Interleaved Boost converter is derived and is discussed now.
Mode of operation is assumed as continuous conduction mode in which, only a part of the energy is delivered to the load. In recent researches, continuous current mode is mainly considered since higher power densities are possible only with this mode of operation. But the main disadvantage in going for continuous current mode of operation is the inherent stability problems caused due to the right-half plane zero in converter transfer function [5]. This can easily be solved by the proposed pole placement technique.
During continuous conduction mode of operation , diodes D1 and D2 are always in complementary state with the switches S1 and S2 respectively, that is when S1 is on, D1 is off and vice versa. Similarly when S2 is on, D2 is off and vice versa. Accordingly four modes of switching states are possible and corresponding state equations are explained as follows:
Mode1: S1 and S2 are on
(6)
(7)
where i1 and i2 are currents flowing through the inductances L1 and L2 respectively and VC is the capacitance voltage.
Mode 2: S1 is on and S2 is off
(8)
Mode 3: S1 is off and S2 is on
(9)
Mode 4: S1 and S2 are off
(10)
where,
(11)
(12)
(13)
(14)
(15) The average state model takes the form described as follows:
(16)
Where , , and the duty cycle ratio is given by .The output equation is defined as follows,
(17)
3. Robust design of State Feedback Matrix
A. Pole Placement Technique
The robust design comprises the derivation of state feedback gain matrix based on control law defined as for the converter under consideration. The required steady state value of the controlled output variable y is a constant reference input r, which is taken as unit step input. The design is carried out using the values shown in Table I. The root locus of an Interleaved Boost converter is drawn from which the desired closed loop poles are chosen. The poles are arbitrarily placed in s-plane in such a way that the output variable y tracks any of the reference r which is considered as a step function in this case.
Table I Circuit Parameters of an Interleaved Boost Converter
Sl. No
Circuit Parameters
Values
1
Switching frequency
20kHz
2
Input Voltage
24V
3
Inductance L1
72µH
4
Inductance L2
72µH
5
Capacitance C
217µF
6
Load Resistance
23Ω
The necessary condition for arbitrary pole placement is that the system should be completely state controllable. When all the state variables are assumed to be accurately measured at all times, then implementation of a linear control law is possible which is defined as With this state feedback control law, state equations of the system under consideration take the form as follows:
(18)
Now the system under consideration is of third order and the desired poles can be easily placed by assuming the following converter specifications,
(19)
From the desired pole locations, characteristic equation of the converter is given as,
(20)
The third root is considered as at least six times the value of [6]. The control scheme for the Interleaved Boost converter is shown in fig.2.
Fig 2. Control scheme for Continuous time system
Here N represents the scalar feed forward gain. It is evident from fig.3, the state feedback gain matrix thus designed, very well tracks the reference step input which proves that the system is completely stable.
Fig 3. Step response of Interleaved Boost Converter
with state feedback control
B. Linear Quadratic Optimal Regulator method
It is necessary to determine the state feedback gain matrix optimally in order to obtain tight output regulation and highly insensitive to system parameter variations or external disturbances. The optimal solution is obtained by choosing appropriate performance index. The desired closed loop poles can be chosen such that the poles are closer to the desired locations by linear optimal control theory. The optimal controller thus designed reduces the sensitivity to plant parameter variations [2]. The optimal regulator problems determine the state feedback matrix k for obtaining optimal control law given by
.
The main objective is to minimize the performance index which is defined as follows,
(21)
Here Q and R are positive definite Hermitian symmetric matrix. The design of control scheme is carried out in following two steps:
The positive definite Riccati matrix, P is determined which should satisfy the following reduced Riccati matrix equation given by,
(22)
For the appropriate P value (A-Bk) should be asymptotically stable.
Substitution of Riccati matrix in the equation described below results in the optimal k value.
(23)
4. Design and Implementation of the full order
Observer controller
The observer is designed using same pole placement technique mainly to estimate the unmeasurable variables. It is desirable that the response of the observer should be faster than the response of the system since the observer tends to act upon the error of the system. The necessary condition for the observer design is that the system should be completely state controllable. By thumb rule the desired observer locations is made by having the following assumption:
Natural frequency of oscillation (Observer Controller) ≈ 2 to 5 times that of the Natural frequency of oscillation of the system. Now, dynamic equation of the system with full order state observer takes the following form,
(24)
where k1 is the element of state feedback gain matrix and r is the step input. The dynamic equation describing state observer (continuous time system) takes the following form,
(25)
where ke is observer gain matrix.
Now, transfer function of the Observer Controller (control law plus full order observer) obtained by pole placement method is as follows,
(26)
The transfer function obtained for the Interleaved Boost Converter with Linear Quadratic Optimal Regulator (optimal control law plus full order observer) is as follows,
(27)
The output voltage thus obtained for the Interleaved Boost Converter with Observer Controller and Linear quadratic optimal regulator is shown in fig.4. It is evident that the settling time is much lesser in range of milli seconds, no overshoots or undershoots are seen and the steady state error is zero. The dynamic performance of the converter is very much improved.
.5.Simulation Results and Discussion
The design and performance of Interleaved Boost Converter is accomplished in continuous conduction mode and simulated using MATLAB/ Simulink. The ultimate aim is to achieve a robust controller in spite of uncertainty and large load disturbances. The converter specifications under consideration are rise time, settling time, maximum peak overshoot and steady state error which are shown in Table II.The results thus obtained are in concurrence with the mathematical calculations. The simulation is also carried out by varying the load not limiting to R load and it is illustrated in Table III. In order that the dynamic performance has to be ensured both methods show tight output regulation with much lesser settling time, no steady state error without any undershoots or overshoots which is illustrated in fig.4. It is evident that the optimal solution for control law thus obtained shows improved results when compared with pole placement method in terms of the performance specifications as listed in Table II.
Fig 4. Comparison of Observer Controller with LQR
Simulation has been carried out in two modes. In mode1 the inductances are chosen as L1 = L2 and in mode2 inductances are chosen as L2 = 2L1.The added advantage is that the efficiency is higher even with high input to output ratios. From Table IV, it is very well understood that the control scheme offers a robust control and good current sharing among the converters. Fig.5 shows the efficiency as a function of output load current and it is seen that the state feedback control method achieves higher efficiency for a wide range of load variations and the maximum efficiency achieved is 95.63% at a 176W load condition. Fig.6 shows the load current in which ripple is very minimum. The inductor currents and corresponding duty cycle ratios are shown in the fig.7 and fig.8 for mode1 and mode2 respectively. It is evident from the current waveforms that the controller provides an effective current sharing among the converter modules irrespective of the values of the inductances.
Fig 5. Efficiency of the Interleaved Boost Converter
Fig 6. Load Current of the Interleaved Boost converter
Table II Comparison of the Performance Parameters of Interleaved Boost Converter
Sl.No
Controller
Settling Time
(s)
Peak Overshoot (%)
Steady State Error
(V)
Rise Time
(s)
Output Ripple Voltage
(V)
1
Observer Controller
0.15
0
0
0.075
0
2
Linear Quadratic
Optimal Regulator
0.005
0
0
0.001
0
Table III Output Response of Interleaved Boost Converter for Load Variations
R (Ω)
L (mH)
E (V)
Reference
Voltage (V)
Output
Voltage (V)
10
-
-
60
60
15
-
-
60
60
23
-
-
60
60
10
50
-
60
60
15
100
-
60
60
23
100
-
60
60
30
50
5
60
60
23
100
10
60
60
15
100
15
60
60
10
100
20
60
60
Table IV Performance calculations of the Converter with Proposed control scheme
Sl.No
Mode
Vref
(V)
Vout
(V)
IL
(A)
Iin
(A)
Vin
(V)
Pin
(W)
Pout
(W)
Efficiency
(%)
1
I
60
60
2.5
6.5360
24
156.8640
150
95.62
II
2.499
6.5360
24
156.864
149.94
95.59
2
I
65
65
2.708
7.6700
24
184.08
176.024
95.62
II
2.708
7.6735
24
184.164
176.112
95.63
3
I
70
70
2.916
8.9034
24
213.6816
204.115
95.52
II
2.916
8.9050
24
213.7200
204.129
95.51
4
I
75
75
3.124
10.23
24
245.52
234.33
95.44
II
3.124
10.229
24
245.5008
234.345
95.46
5
I
80
80
3.28
11.652
24
279.648
258.202
92.33
II
3.332
11.651
24
279.622
266.373
95.26
Fig 7. Inductor current and duty ratio for mode 1
Fig 8. Inductor current and duty ratio for mode2
6. Conclusion
A state feedback control approach has been designed for the Interleaved Boost converter in continuous time domain using pole placement technique and Linear Quadratic Optimal Regulator method. The load estimator has been designed by deriving full order state observer to ensure robust and optimal control for the converter. The Separation Principle allows designing a dynamic compensator which very much looks like a classical compensator since the design is carried out using simple root locus technique. The mathematical analysis and the simulation study shows that the controller thus designed achieves good current sharing among the converters, tight output voltage regulation and good dynamic performances and higher efficiency. This method is topology independent and also can be extended for any of the applications such as power factor preregulation, photovoltaic cell and speed control applications.
7. Nomenclature
x(t) State vector
A, C State Coefficient matrices
B, D Source Coefficient matrices
VS Input Voltage
VO Output Voltage
d duty cycle ratio
x1 average inductor current
x2 average output capacitor voltage
R Load Resistance
C Capacitance
L1,L2 Inductances
y(t) Controlled Variable
u control law
k state feedback gain matrix
r reference input
N scalar feed forward gain
k1 element of the state feedback gain
matrix
ke observer gain matrix
ζ damping ratio
ωn natural frequency of oscillation
J performance index
Q, P Positive definite Hermitian
Symmetric matrix
R Riccati matrix
