A LINEAR QUADRATIC CONTROL APPLIED TO BUCK CONVERTERS WITHH-INFINITY CONSTRAINTS

Luiz Antonio Maccari Junior?, Vinícius Foletto Montagner?

and André Augusto Ferreira?Federal University of Santa Maria, GEPOC, Santa Maria, RS, BrazilFederal University of Juiz de Fora – UFJF, Juiz de Fora, MG, Brazil

e-mail: luizmaccari@gmail.com

Abstract—This paper provides a control design al-gorithm to get state feedback controllers for DC-DCbuck converters with input voltage disturbances. Thealgorithm is based on the linear quadratic regulator andon the H∞ norm, ensuring a prescribed settling timefor the transient responses and a prescribed attenuationfor disturbances on the input voltage. Results show thatthe proposed algorithm can provide control gains withsuitable time and frequency responses for the closed-loopsystem.

Keywords—Linear Quadratic Regulator, Linear MatrixInequalities, DC-DC Converters.

I. INTRODUCTION

Power converters are undoubtedly important in renewableenergy source systems [1], [2]. In this context, one problemthat deserves attention is the regulation of the voltage fromthe primary source. DC-DC converters can be used in thisstage, and the control of such systems is an issue addressed byseveral works [3]–[9]. One can also observe the importanceof DC-DC converters for several other applications, asfor instance in their use in electrical machine drives, inuninterruptible power supplies [10], [11], in power factorcorrection devices [12]–[14] and, more recently, also inelectrical vehicle applications, as in [15]–[17], and in windenergy systems [18], in photovoltaic panels in solar energysystems [19], [20] and in fuel cells [17], [21].

From the point of view of the control design, one problemto be solved is to provide output voltage regulation for a DC-DC converter under input voltage disturbances. This problemcan be solved using, for instance, state feedback control,whose theory is well established for linear systems [22]–[24]. Among the state feedback controllers, one can use thelinear quadratic regulator (LQR), known as a popular optimalcontroller, which can be designed with the help of availablesoftwares as MATLAB. The LQR minimizes a quadratic costfunction in terms of the system states and of the control input,with application in power converters [25]–[29]. For instance,in [30], a good regulation with rejection of small disturbancesis achieved. In [26], the results are extended to the problemof state observer . In [31], an LQR is applied to a DC-DCconverter considering a large range of load, and implementedin a digital signal processor in [28].

This work deals with the control of the output voltage ofa DC-DC buck converter whose input voltage is assumed asaffected by disturbances. The control objectives are to providesuitable transient responses, and also suitable rejection of

disturbances. This is achieved by means of an LQR designedto meet specifications of maximum real part of closed-loopeigenvalues and of the H∞ norm of the closed-loop system[32], [33]. The H∞ norm is obtained based on linear matrixinequalities (LMIs) to certify the performance of a set ofgains for the controller that ensure stability and optimalrejection of disturbances [33], [34]. The advantage of theuse of this technique is that one gets the value of the H∞norm of the closed-loop system in a fast way, and alsogets a Lyapunov function certifying the closed-loop stability.Several other constraints can be included in design or analysiswith LMIs, as for instance pole location, bounds on energyof the control signal, decay rate of the transient responses,etc [35]–[42]. The results obtained with the proposed designprocedure are of good quality, as demonstrated by time andfrequency simulations. It is also illustrated that an optimalH∞ controller leads to high control gains, not suitable forpractical applications with this system. In the sequence, inSection II, one has the state space model of the converter,in Section III, one has the control design problem and theprocedure for its solution, in Section IV, one has an exampleof application to a DC-DC buck converter and in Section Vone has the mains conclusions of the paper.

II. MODELING

Consider the DC-DC buck converter affected by distur-bances, in Fig. 1, where x1 (the input capacitor voltage), x2

(inductor current) and x3 (output capacitor voltage) are thestate variables, and the objective is to regulate the outputvoltage x3 under disturbances in u1.

The state space equations for the mode of operationdescribed by the switch on and for the mode of operationdescribed by the switch off are given, respectively, by

A1 =

− 1RiCi

− 1Ci

01L 0 − 1

L0 1

Co− 1RoCo

, B1 =

1RiCi

00 00 − 1

Co

(1)

and

A2 =

− 1RiCi

0 0

0 0 − 1L

0 1Co

− 1RoCo

, B2 =

1RiCi

00 00 − 1

Co

(2)

In these equations, x = [x1 x2 x3]′ and u = [u1 u2]

′.An averaged model can be written as

978-1-4799-0272-9/13/$31.00 ©2013 IEEE 339

L

DS

+

-

u2

+

-

x2

x3

Ri

Ro

x1

+- Ci

Co

u1

Figure 1. Buck converter with disturbances

x = [A1d+A2(1− d)]x+ [B1d+B2(1− d)]u (3)

where d is the duty cycle. Considering that the converteroperates around an equilibrium point, the variables can berewritten as

x = X + x, x = ˙x, u = U + u (4)

where, for instance, X is the equilibrium point for x and x isthe small perturbation on x, and U is the nominal value forthe inputs and u are the small perturbations on the vector u.

This leads to

˙x = [A1(d+ δ) +A2(1− d− δ)](x+X)

+[B1(d+ δ) +B2(1− d− δ)](u+ U)

˙x = [A1δ +A2(1− δ) + (A1 −A2)d](x+X)

+[B1δ +B2(1− δ) + (B1 −B2)d](u+ U)

(5)

where δ is the nominal duty cycle and d is a smallperturbation on the duty cycle, used as control signal. Onecan express

A = A1δ +A2(1− δ)B = B1δ +B2(1− δ) (6)

This results in

˙x = [A+ (A1 −A2)d](x+X)

+ [B + (B1 −B2)d](u+ U)

˙x = AX +BU

+Ax+Bu+ [(A1 −A2)X + (B1 −B2)U ]d

+ (A1 −A2)dx+ (B1 −B2)du

(7)

Assume that dx and du are sufficiently small to beneglected. Moreover, observe, from the steady state solution,that

AX +BU = 0 (8)

Then one has the state equilibrium vector X = −A−1BU .Thus (7) can be rewritten as the linear model

˙x = Ax+Bcd+Bu (9)

where

Bc = [(A1 −A2)X + (B1 −B2)U ] (10)

In order to ensure zero steady state error for the outputvoltage x3, one can include an integrator over the error onthis variable. For that, consider

e = r − x3 (11)

and taking into account

x3 = X3 + x3, X3 = r (12)

one hase = −x3 (13)

This allows to write the augmented system

˙ξ = Gξ +Hd+ E1u1 + E2u2

y = Cξ(14)

where

ξ =

[xe

], G =

[A 0

[ −1 0 0 ] 0

],

H =

[Bc0

], E =

[B0

]= [E1 E2] ,

C = [0 0 1 0]

(15)

This model is suitable for state feedback control, as given inthe next section.

III. CONTROL DESIGN

Consider the state feedback control law

d = −Kξ (16)

It is known that the LQR controller allows to compute thegain vector K which minimizes the cost function [24]

J =

∫ ∞0

(ξ′Qξ + d′Rd

)dt (17)

using for that the solution of Ricatti equation

ATS + SA− SBR−1BTS +Q = 0 (18)

This solution is efficiently provided by softwares as MAT-LAB, with the lqr function. The control designer can choosea positive definite matrix Q, to inform the relative importanceof the state variables, and R to, inform the weight of

340

the control signal in this cost function. This allows to gettradeoffs between performance and energy of the controlsignal [24], [25].

The design problem addressed in this paper is given by:get the gain vector K such that the closed-loop system has• i) maximum real part of all eigenvalues less than α,

that is,max(Re(λ(G−HK))) < α (19)

• ii) an upper bound γ for the frequency magnituderesponse, for the entire frequency range, that is,

x3(jω)

u1(jω)< γ, ω ∈ [0,∞] (20)

The aim of i) is to provide a limit α for the settling timeof the transient responses for the closed-loop system; andthe aim of ii) is to provide a limit γ for the gain from thedisturbance input voltage to the output voltage, indicating thecapacity of rejection of disturbances.

The proposed control design algorithm is given by:• Step 1) inform α (real negative scalar) and γ (real

positive scalar);• Step 2) choose the LQR weighting matrices Q and R;• Step 3) solve the LQR problem using, for instance, the

lqr function in MATLAB;• Step 4) solve, for instance, in the LMI Control Toolbox

from MATLAB, the convex optimization problem

µ∗ = minµs. t.

P = P ′ > 0[(G−HK)′P + P (G−HK) + C′C PE1

E1′P −µI

]< 0

(21)• Step 5) if (19) and

√µ? < γ, then K ensures

the performance specifications; otherwise, return toStep 2.

This control design algorithm will be used in the nextsection to provide a suitable controller for the applicationunder consideration here.

IV. EXAMPLE

Consider the parameters of the converter in Fig. 1, shownin Table I.

Table IParameters of the converter in Fig. 1.

Parameter ValueU1 100 Vδ 0.5L 1.1 mHCi 47 µFCo 470 µFRi 1 ΩRo 25 Ωfs 50 kHz

Using the algorithm in Section III for the controldesign, first consider the specifications α = −100 andγ = −15dB(= 0.17782), second, consider the choice basedon heuristic search

Q =

1 0 0 00 1 0 00 0 1 00 0 0 10060

, R = 51 (22)

Then the algorithm leads to the control vector gain

K ′ =

−0.020360.162620.13104−14.04474

(23)

which ensure the performance specifications.It is worth to mention that one also could try to use the

optimal H∞ controller, given by [33]

µ∗ = minµs. t GW +WG′ −HZ − Z ′H ′ E1 WC ′

E1′ −I 0

CW 0 −µI

< 0

(24)with state feedback gains

K = ZW−1 (25)

in order to solve the design problem. For example consideredin this section, the convex optimization problem in (24) leadsto the gains

K ′ =

−0.000007104474453−0.000053550967997−5.0976663293734280.000000000000001

× 109 (26)

which are not suitable for practical application in this system.On the other hand, the gain (23) from the proposed algorithmis feasible in practice and leads to good results, as shown inthe sequence.

Using the same control structure as in [31], and for adisturbance on u1 given by a pulse, one has the result inFig 2. A detailed view is shown in Fig. 3. Notice that thesettling time of the transient respects the limit 5/α = 0.05 s(using the 1% of error criterium in the transient response).

The frequency closed-loop response from u1 to x3 ispresented in Fig. 4. Clearly, one has the peak of themagnitude diagram less than −15 dB, respecting the limitγ for disturbance rejection.

Finally, in Fig. 5 one has an analysis of the closed-loopstability under uncertainties on the parameters Ro and δ.Notice that the closed-loop eigenvalues are all stable for thisdomain of uncertainties.

V. CONCLUSION AND PERSPECTIVES

This paper provided a control design algorithm suitable forcomputation of LQRs applied to DC-DC buck converters. Thealgorithm uses constraints on the closed-loop eigenvalues andon the closed-loop H∞ norm to get suitable performance intime and frequency responses. An example shows that the use

341

0 0.1 0.2 0.3 0.4 0.5 0.648

49

50

51

52

x3

(V)

0 0.1 0.2 0.3 0.4 0.5 0.6

0

5

10

u1

(V)

Time (s)

Figure 2. Transient response for a disturbance u1.

0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.2648

49

50

51

52

x3

(V)

0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26

0

5

10

u1

(V)

Time (s)

Figure 3. Detailed view from Fig. 2.

100

101

102

103

104

105

-150

-100

-50

0

Mag

nit

ud

e (d

B)

Frequência (Hz)

100

101

102

103

104

105

-270

-180

-90

0

90

Ph

ase

(deg

)

Frequency (Hz)

Figure 4. Closed-loop frequency response from input voltagedisturbance to the output voltage. Peak of magnitude diagram is

equal to −15.2 dB.

-2.5 -2 -1.5 -1 -0.5 0

x 104

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Imag

inar

yA

xis

Real Axis

Figure 5. Closed-loop eigenvalues under parametric uncertaintiesof Ro from 20 Ω to 29 Ω and δ of 0.47 to 0.55.

of an optimal H∞ controller is not viable for this application,while the proposed controller allows very good responses.The technique presented here is also under investigation forapplication for systems used in electric vehicles. Extensionsfor MIMO systems are immediate due to the state spaceformulation of the problem of control design.

ACKNOWLEDGMENT

To the Brazilian agencies CAPES and CNPq.

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