Takagi-Sugeno Fuzzy Model-Based Approach to Robust Control of Boost DC-DC Converters

Abstract

This paper considers the robust controller design problem for a boost DC-DC converter. Based on the Takagi-Sugeno fuzzy model-based approach, a fuzzy controller as well as a fuzzy load conductance observer are designed. Sufficient conditions for the existence of the controller and the observer are derived using Linear Matrix Inequalities (LMIs). LMI parameterizations of the gain matrices are obtained. Additionally, LMI conditions for the existence of the fuzzy controller and the fuzzy load observer guaranteeing α-stability, quadratic performance are derived. The exponential stability of the augmented fuzzy observer-controller system is shown. It is also shown that the fuzzy load observer and the fuzzy controller can be designed independently. Finally, the effectiveness of the proposed method is verified via experimental and simulation results under various conditions.

1. Introduction

Precise control of DC-DC converters is not easy due to the nonlinearities of converters, unavoidable uncertainties, parameter or load variations. To get around this problem various control design methods have been proposed in the literature such as [1-7]. Most of the previous control design methods require the good knowledge of the parameter values and the control performances can be severely degraded in the presence of load or parameter variations. On the other hand, the Takagi-Sugeno (T-S) fuzzy model-based control theory has been successfully applied to control of complex nonlinear or ill-defined uncertain systems [8-12].

Considering these facts, a nonlinear control design method for a boost DC-DC converter is proposed based on the T-S fuzzy approach. This paper first gives a T-S fuzzy model-based control design method. This paper introduces an error vector associated with the desired output voltage, and using this error vector a T-S fuzzy error dynamics model of a boost DC-DC converter is obtained to construct a fuzzy controller. In the fuzzy error dynamics model for controller design a boost DC-DC converter is represented as an average weighted sum of simple local linear subsystems. Local linear controllers are designed for each local subsystem and a global nonlinear controller from the local linear controllers is derived via a standard fuzzy inference method. An LMI condition for the existence of the fuzzy controller is derived and an explicit parameterization of the fuzzy controller gain is obtained in terms of the solution matrices to the LMI condition. LMI existence conditions of the fuzzy speed controller guaranteeing α-stability or quadratic performance are additionally derived. Secondly, the fuzzy controller method is applied to design a T-S fuzzy load conductance observer. It should be noted that the previous fuzzy control systems of [13-18] can suffer from lack of systematic and consistent design guidelines to determine design parameters such as fuzzy partition of the input and output spaces, membership function shapes, the number of fuzzy rules [11, 18]. However, in the proposed T-S model-based approach, one can systematically design a fuzzy controller as well as a load observer guaranteeing the asymptotic stability of the closed-loop control system. And one can handle various useful convex performance criteria such as α-stability, quadratic performance. Through simulations and experiments, it is shown that the proposed method can be successfully used to control a boost DC-DC converter under load variations and it is very robust to the model parameter variations.

 

2. System Description

A boost converter shown in Fig. 1 can be represented by the following nonlinear equation [3, 4]:

where iL, vC, urepresent the input inductor current, the output capacitor voltage, the discrete-valued control input taking values in the set{0, 1}, and E, L, C, R are the external source voltage value, the inductance of the input circuit, the capacitance of the output filter, the output load resistance, respectively.

Fig. 1.Topology of boost converter

Following assumptions are used in this research:

A1 : iL, vC are available. A2 : can be neglected and it can be set as = 0 where Y = R-1. A3 : The inductor current is never allowed to be zero, i.e. the converter is in continuous conduction mode.

By introducing the duty ratio input function uc(∙) ranging on the interval [0, 1] and the following error terms [1, 3, 5].

the following approximate averaged model can be derived from (1)

where z = [z1, z2,, z3]T and Vr is the desired reference output voltage such that Vr > E > 0. After all, our design problem can be formulated as designing a fuzzy controller for the system (2). Fig. 2 shows a schematic diagram of the control system.

Fig. 2.Schematic diagram of proposed control system

 

3. Fuzzy Controller Design

Based on the T-S fuzzy modeling approach [8-12], the model (2) can be approximated by a second order rc-rule fuzzy model to design a fuzzy controller. The ith rule of the T-S fuzzy model is of the following form:

Plant Rule i: IF xis Fci, THEN

where Fci(i = 1,∙∙∙, rc) denote the fuzzy sets, rc is the number of fuzzy rules, x = [iL, vc]T, Ac and Bci is given by

Each fuzzy set Fci is characterized by a membership function mci(x) and the ith operating point x = (X1i , X2i ). Via a standard fuzzy inference method, the following global nonlinear model can be obtained.

where mci: R → [0,1], i = 1,∙∙∙, rc is the membership function of the model with respect to plant rule i, hci can be regarded as the normalized weight of each IF-THEN rule and it satisfies hci(x) ≥ 0 and

Let the local controller be given by the following linear controller

Controller Rule i: IF x is Fci , THEN v = Ki z

where Ki ∈ R1×3 are gain matrices. Then the final fuzzy controller is given by

and the closed-loop control system is given by

Assume that the following LMI condition is feasible

where Pc ∈ R3×3 and Yci ∈ R1×2 are decision variables.

And assume that the controller gain matrices Ki are given by

Then there exists a matrix Qc > 0 such that

Let us define the Lyapunov function as Vc(z) = zTXcz where Its time derivative along the closed-loop system dynamics (6) is given by

which implies that the origin z=0 is exponentially stable.

Theorem 1 Assume that the LMI condition (7) is feasible for (Pc, Yci) and the gain matrices Ki are given by (8). Then, x converges exponentially to zero.

Remark 1 The LMI parameterization of the controller gain enables one to handle various useful convex performance criteria such as α-stability, quadratic performance, and generalized H2/H∞ performances [12, 19]. For example, if the controller gain is set as Ki of (8) with Pc and Yci satisfying for some αc > 0

then by referring to (9) the following can be obtained

which implies that x converges to zero with a minimum decay rate αc. On the other hand, if the controller gain is set as Ki of (8) with Pc and Yci satisfying

Then the followings can be obtained

where the Shur complement lemma of [19] is used. After all, it can be seen that the fuzzy controller (5) guarantees the quadratic performance bound constraint

 

4. Fuzzy Load Conductance Observer Design

In this section, a fuzzy observer to estimate the load conductance Y = R-1 will be designed. By applying the T-S fuzzy modeling methods [9, 10] to (2) or the equivalent error dynamics (2), the boost converter and the dynamics of , can be approximated by a second order ro-rule fuzzy model. The ith rule of the T-S fuzzy model is of the following form:

Plant Rule i: IF vcis Foi, THEN

where the assumption A2 is used, Foi(i = 1,∙∙∙, ro) denote the fuzzy sets, ro is the number of fuzzy rules, xo = [vc, Y]T is the state, yo = vc is the output, , uo = iL(1 − u)/C, and

Each fuzzy set Foi is characterized by a membership function moi(vc) and the ith operating point Vi. Via a standard fuzzy inference method, the following global nonlinear model can be obtained :

where moi ∶ R → [0,1], i = 1,∙∙∙, ro is the membership function of the system with respect to plant rule i, hoi is the normalized weight of each IF-THEN rule and it satisfies hoi(vc) ≥ 0 and

Let the local observer given by the following linear observer

Observer Rule i: IF vc is Foi, THEN

where Li ∈ R2×1 are gain matrices, Then the final fuzzy observer induced as the weighted average of the each local observer is given by

which gives the following error dynamics.

where

Theorem 2 Assume that the following LMI condition is feasible for (Po, Yoi)

where Po ∈ R2×2, Yoi ∈ R2×1 are decision variables. And assume that the observer gain Li is given by

Then, the estimation error converges exponentially to zero.

Proof: Assume that (15) is feasible. Then there exists a matrix Qo > 0 such that

Let us define the Lyapunov function as Its derivative with respect to time is given by

which implies that is exponentially stable.

Remark 2 The LMI parameterization of the observer gain (16) also provides some degrees of freedom which can be used to handle various useful convex performance criteria such as α -stability, quadratic performance, and generalized H2/H∞ performances [12, 19]. For example, if the observer gain is set as Li of (16) with Po and Yoi satisfying for some αo > 0

then converges to zero with a minimum decay rate αo. On the other hand, if the observer gain is set as Li of (16) with Po and Yoi satisfying for some Qo ≥ 0

Then the followings can be obtained

After all, it can be easily shown that the fuzzy observer (14) guarantees the quadratic performance bound constraint

 

5. Separation Property and Design Algorithm

This section illustrates the exponential stability of the augmented control system containing the fuzzy controller and the fuzzy load observer. The following theorem implies that the separation property holds.

Theorem 3 Assume that the LMIs (7) and (15) are feasible, and the controller (5) is replaced with the following load observer-based control law

where and is the estimated output conductance via the fuzzy observer (14). Then z and converge exponentially to zero.

Proof: It should be noted that because the vector can be rewritten as where

Let us define the Lyapunov function as where η is a sufficiently large scalar, Pc and Po satisfy the LMIs (7) and (15). Its derivative with respect to time is given by

where If η is large enough to guarantee , then for all This proves the exponential stability of

Remark 3 From the standard results [19], it can be shown that if one of the pairs (Ac, Bci) is not stabilizable then the LMI condition (7) is not feasible. And it can be easily shown that (Ac, Bci) are stablilzable as long as X1i ≠ 0 or X2i ≠ 0. It can be also shwon that if one of the pairs (Aoi, Co) is not detectable then the LMI condition (7) is not feasible. And it can easily shown that (Aoi, Co) are detectable as long as Vi ≠ 0. These facts imply that the LMI condition (7) and (15) is always feasible for an appropriately chosen set of the operating points.

Remark 4 Theorems 1-2 imply that our design problem is a simple LMI problem which can be solved very easily via various powerful LMI optimization algorithms. Theorem 3 implies that the controller gains and the load conductance observer gains can be independently designed. And Remark3 implies that our design problem is always feasible for an appropriately chosen set of the operating points. Our results can be summarized as the following LMI-based design algorithm.

[Step1] Choose an appropriate set of {V1,∙∙∙, Vro } and obtain the fuzzy model (13). [Step2] Solve the LMIs (15), obtain the gain matrices Li, and construct the fuzzy observer (14). [Step3] Choose an appropriate set of and obtain the fuzzy model {(X11, X21),∙∙∙, (X1rc , X2rc )} and obtain the fuzzy model (4). [Step4] Solve the LMIs (7), obtain the gain matrices Ki, and construct the load observer-based fuzzy control law (20)

Remark 5 Via extensive numerical simulations and experimental studies, it has been found that fuzzy models with ro = 2 and rc = 2 are enough to obtain load observer-based fuzzy control laws with satisfactory performances. As can be seen in the next section a two-rule fuzzy model (13) with the following normal membership functions is enough to design a fuzzy load observer with satisfactory performances

where εoi > 0 . A fuzzy controller with satisfactory performances can also be obtained by using a fuzzy model (4) with rc = 2 and

where εci > 0, and the nominal values of E, L, C, R are used.

Remark 6 To design a fuzzy load observer under the several performance specifications, one has only to gather LMI conditions corresponding to each design performance specification, and form a system of LMIs as a subset of (15), (18), (19), and solve the system of LMIs under the assumption that the Lyapunov matrices ‘Po’ are common. Similarly, one can design a fuzzy controller under the several performance specifications.

 

6. Simulation and Experiment

Consider a boost converter (1) with L=200[uH], C = 220[uF], vc = 20[V], the PWM switching frequency 60[kHz]. Assume that the nominal load resistance is RO = 100[Ω] and desired reference output voltage Vc is Vc = 20[V]. The parameters values used for the simulations and experiments are summarized in Table 1. Let us first design a fuzzy observer guaranteeing the minimum decay rate αo = 10. Here, the following two-rule fuzzy model to design a fuzzy observer is used.

Table 1.Utilized components and parameters

Plant Rule 1: IF vc is about Vr, THEN

Plant Rule 2: IF vc is about 0.5Vr, THEN

where

And ho1 = m01/(m01 + m02), ho2 = m02/(m01 + m02), mo1 = e-(vc-vr)2, mo2 = e-(vc-0.5vr)2 . By solving (18) with αo = 100 the following fuzzy observer (14) with the following gain

Now, let us design a fuzzy controller guaranteeing the minimum decay rate αo = 20. In order to design a fuzzy controller, the following two-rule fuzzy model is used.

Plant Rule 1: IF (iL, vc) is about (0.4, Vr), THEN

Plant Rule 2: IF (iL, vc) is about (0.4, 0.25Vr), THEN

where

and hc1 = mc1/(mc1 + mc2), hc2 = mc2/(mc1 + mc2), mc1 = e-(vc-vr)2, mc2 = e-(vc-0.25vr)2 . By solving (10) with αc = 100 , the following controller gain can be obtained

After all, the following observer-based fuzzy controller can be obtained

where

Figs. 3 and 4 show the proposed fuzzy load observer and fuzzy controller, respectively.

Fig. 3.Block diagram of the proposed fuzzy load observer.

Fig. 4.Block diagram of the proposed fuzzy controller.

A conventional cascade PI controller shown in Fig. 5 is also considered for performance comparisons. The gains of the above PI controller are designed based on the method [21]. The P and I gains of the voltage PI controller are KpV=0.1 and KiV=4. The P and I gains of the current PI controller are KpI=0.8 and KiI=1. Fig. 6 illustrates the Matlab/Simulink simulation model of the proposed controller system.

Fig. 5.PI control block diagram

Fig. 6.Simulation model of the proposed control system implemented with Simulink.

In order to verify the effectiveness of the proposed method, the following four cases are considered :

C1) The input voltage E changes from 10[V]→15[V]→10[V] while the load resistor R is constant at the nominal value R=100 [Ω]. C2) The input voltage E changes from 10[V]→5[V]→10[V] while the load resistor R is constant at the nominal value R=100 [Ω]. C3) The reference voltage Vr changes from 20[V]→13[V]→20[V] while E and R are kept constant at E=10 [V] and R=100 [Ω]. C4) The load resistor R changes from 100[Ω]→20[Ω]→100[Ω] while Vr and E are kept constant at Vr=20 [V] and E=10 [V].

Fig. 7 shows the time responses under the case C1. Fig. 7(a) shows the time histories of E, vc, and output current io by the conventional PI control method. The PI gain values are computed based on the methods given in [1-2]. Fig. 7(b) depicts the time histories of E, vc, and output current io by the proposed load observer-based fuzzy controller (23). Fig. 8 shows the time responses under the case C2. Fig. 8(a) shows the time histories of E,vc, and output current io by the conventional PI control method. The PI gain values are computed based on the methods given in [1-2]. Fig. 8(b) depicts the time histories of E,vc, and output current io by the proposed load observer-based fuzzy controller (23). Fig. 9 shows the time responses under the case C3. Fig. 9(a) shows the time histories of E,vc, and output current io by the conventional PI control method. The PI gain values are computed based on the methods given in [1-2]. Fig. 9(b) depicts the time histories of E,vc , and output current io by the proposed load observer-based fuzzy controller (23). Fig. 10 shows the time responses under the case C4. Fig. 10(a) shows the time histories of E,vc , and output current io by the conventional PI control method. Fig. 10(b) depicts the time histories of E,vc, and output current io by the following load observer-based fuzzy controller. Figs. 7, 8, 9 and 10 imply that our method gives a faster recovery time as well as a less overshoot.

Fig. 7.Simulation results under C1.

Fig. 8.Simulation results under C2.

Fig. 9.Simulation results under C3.

Fig. 10.Simulation results under C4.

The conventional PI control algorithm as well as our method is implemented on a Texas Instruments TMS 320F28335 floating-point DSP. A Tektronix TDS5140B digital oscilloscope is used to measure and plot the signals vC, iL, E. Fig. 11 shows the circuit scheme of power stage. Fig. 12 illustrates the experimental setup. Figs. 13, 14, 15 and 16 show the experimental results. It can be seen that our method outperforms the conventional PI method.

Fig. 11.Power stage circuit scheme of the boost converter

Fig. 12.Experimental setup.

Fig. 13.Experimental results under C1.

Fig. 14.Experimental results under C2.

Fig. 15.Experimental results under C3.

Fig. 16.Experimental results under C4.

 

7. Conclusion

A simple fuzzy load observer-controller design method was proposed for a boost converter under an unknown load resistance. Explicit parameterizations of the fuzzy controller gain and the fuzzy load conductance observer gain were given in terms of LMIs. LMI existence conditions of the fuzzy controller and the fuzzy observer guaranteeing α-stability or quadratic performance were also derived. Finally, the robust performance of the proposed method was verified via numerical simulations and experiments.