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Adaptive Control of PWM Dc-to-Dc Converters Operating in Continuous Conduction Mode Said EL BEID*, Said DOUBABI*, Mohammed CHAOUI**
*: Laboratoire d'Automatique et d'Informatique Industrielle, FSTG, Université Cadi Ayyad, Marrakech, Morocco.
[email protected] ;
[email protected] **: Laboratoire d'Automatique et de Productique, ENSEM, Université Hassan II, Casablanca, Morocco.
Abstract— This paper presents the application of adaptive control to PWM Dc-to-Dc Converter Operating in Continuous Conduction Mode. The system under study is a buck-boost with parasitic as non linear and variable structure plant. First the model of converter is discussed; the non linearity is handled with state space averaged modeling into the linear time-varying representation; one adopted consequently an on-line identification of the converter parameters which rests on the exploitation of the Recursive Least Squares (RLS) algorithm to take into account great variations of the operating point, and the controller adaptation is based on a pole placement strategy. The synthesis of the regulator lends particularly well to be put under the shape of an intelligent algorithm which the self-tuning can be elaborated on-line. Simulation results show the improvement of the dynamic responses and the robustness against load variations or parameters variations.
In this communication, an application of the techniques of adaptive control for a Buck-Boost PWM Dc-to-Dc Converter is described. II. BUCK-BOOST CONVERTER MODELING We consider the buck-boost converter shown in Fig 1 where Rl, and Rc denote the internal resistors of inductor, and capacitor, respectively. Vg, and iL, denote supply voltage and current in inductance respectively. R, Y, designate, respectively load and output voltage. Te and Ts denote respectively the sampling period and the switching period. D is the duty cycle where D=ton/Ts and D’=1-D, D ∈[0, 1]; (ton = switch on duration). Let us consider xc = [iL, vc ]T as instantaneous state vector of converter. In continuous conduction mode, two configurations following the state of switch sw will be obtained as shown in fig.2:
Index Terms— Adaptive control, PWM Dc-to-Dc Converter, Recursive Least Squares (RLS) algorithm.
I.
INTRODUCTION
I
n many situations, the increasing complexity of processes has for consequence to see the dynamic properties of the system evolving, would be it that slowly, during time. The recursive identification algorithms are particularly adapted to this type of problem. Several adaptive methods were proposed since the 70s [1] to provide models which parameters vary with regard to the process changes, their main objective consists in estimating, in a substantial and on-line way, a model of the process studied to make decisions in real time according to the last information acquired on the process while minimizing, memory resources and time of calculation, and by supplying a reliable and robust model of the process. PWM Dc-to-Dc Converters have recently aroused an increasing deal of interest in modern electronic equipments due to their compactness and high efficiency [2]; they are used for almost all analogue and digital electronic systems especially as regulated DC voltage. In the realm of control engineering, they represent an interesting study case as they are variable structure non linear systems, and can be subject to disturbances of different causes (electromagnetic, offsets, etc.).
Figure 1. Buck-boost converter scheme Figure 2.
Figure 2. Buck-Boost converter configurations
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represent respectively the mean state vector and the mean output voltage:
A. Steady-state: In steady state, all the variables take static values, and the state space system for every configuration can be written as: ɺ = Ai X + bi X i=1 for fig.2.a Y = ci X i=2 for fig.2.b With: − RL L A1 = 0
−1 C ( R + Rc ) 0
− R L + R c // R L A2= R ( R + Rc ) C
c2=[R//RC
R R + Rc
1
, b1= L
, c1= [0
0
R R + Rc
]
,b2= 0 , −1 0 ( R + R c ) C −R
L ( R + Rc )
1 Ts
∫
t+Ts
t
∫
t
x(τ)dτ = X + xˆ m (t)
t+Ts
y(τ)dτ=Y+yˆ m (t)
System in transient state is governed by the state space system according to:
xɺ m = [ dA1 +d'A 2 ] x(t)+ [ d.b1 +d'.b 2 ] v g y m = [ d.c1 +d'.c 2 ] x d' = 1-d
(1)
].
− R + (1 − D ) R // R c L L A= R (1 − D ) R + Rc ) C ( C = (1 − D ) R // Rc
− R (1 − D )
L ( R + Rc )
D
non linear model is obtained according to: ˆ ˆ xɺˆ m (t)=A(D)xˆ m (t)+B1 (D)xˆ m (t)d(t)+B 2 (D)d(t) ˆ ˆ yˆ m (t)=C(D)xˆ m (t)+D1 (D)xˆ m (t)d(t)+D 2 (D)d(t)
R + Rc R
(1 − D )
D
2
R
1 − D RL + D (1 − D ) R // Rc + R (1 − D )2
α
(1)
β
α : Ideal transfer function. β : Correction factor due to the capacitor and inductor uncertainties. B. Transient state: In this section, the developed small signals model, allows obtaining a good approximation of the converter behaviour around the operating point towards a load variation. We assume that each variable can be written as the sum of a constant component (noted in upper-case letter) and a small varying one (noted in hat lower-case letter). Hence:
ˆ et d'(t)= ˆ ˆ , vg(t) = Vg+ vˆ (t) iL(t)=IL+ d(t) = D+ d(t) - d(t) g vc(t)=Vc+ vˆ c (t) , R(t)=R+ ˆr .
ˆ xɺˆ m (t)=[A+(DAˆ 1 +D'Aˆ 2 )]xˆ m (t)+[A1 +Aˆ 1 -A 2 -Aˆ 2 ]xˆ m (t)d(t) ˆ ˆ ˆ +(b1 +bˆ 1 -b 2 -bˆ 2 )vˆ g (t)d(t)+[A 1 +A1 -A 2 -A 2 ]X ˆ ˆ +D'bˆ )vˆ (t) +(b1 +bˆ 1 -b 2 -bˆ 2 )vˆ g ]d(t)+(Db 1 2 g (2) ˆ ˆ ˆ ˆ +(DA1 +D'A 2 )X+(Db1 +D'b 2 )Vg ˆ yˆ m (t)=[c+(Dcˆ1 +D'cˆ 2 )]xˆ m (t)+[c1 +cˆ1 -c 2 -cˆ 2 ]xˆ m (t)d(t) ˆ +(c1 +cˆ1 -cˆ 2 -cˆ 2 )Xd(t)+ Dcˆ1 +D'cˆ 2 )X Between nTs and (n+1)Ts, we suppose that vˆ g (t) =0 and
ˆ = 0,bˆ =0, cˆ = 0 ; after some manipulations, ˆr =0 then A i i i
, B =L, −1 0 ( R + Rc ) C
That gives the transfer function relating the output voltage to the input voltage:
ˆi (t) , L
y m (t)=
1 Ts
For a load variation, one replaces respectively Ai,bi,ci, ˆ ,b +bˆ ,c +cˆ in the model (1) (i=1,2), thus the by Ai +A i i i i i transient state space becomes so:
During each sampling period Ts, the state space system becomes: ɺ = A X + B Vg X Y=CX with : A = D.A1 + D’.A2 B = Db1+ D’.b2= Db1 C = D.c1 + D’.c2 After some manipulations one obtains:
Y = Vg
x m (t)=
xˆ m (t) and yˆ m (t)
(3)
With: A(D) = D.A1 + (1-D).A2 B1(D) = A1- A2 B2(D) = (A1- A2).X + (b1-b2).Vg=(A1- A2).X + b1Vg C(D) = D.c1 + (1-D).c2 D1(D) = c1 – c2 D2(D) = (c1 – c2).X The next task is to synthesise a model giving the relation between the output voltage and the duty cycle around D. By neglecting the second order terms in equations set (3), the dynamic state space system becomes: ˆ xɺˆ m (t)=A(D)xˆ m (t)+B2 (D)d(t) ˆ yˆ m (t)=C(D)xˆ m (t)+D 2 (D)d(t)
(4)
Where from the control to output voltage transfer function around the operating point is given by: H(s) =
yˆ m (s) = C(D)(sI-A(D))-1B2(D) + D2(D) ˆ d(s)
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The buck-boost converter given in figure 1 has the following parameters: L = 18.45mH Rl = 1.23Ω, Rc = 0.12Ω, R = 50Ω, C = 470µF, Thus: Hd(s) =
(s-z 0 )(s-z1 ) yˆ m (s) =k ˆd(s) (s-s 0 )(s-s1 )
y*(t) = [q-1BM(q-1)/ AM(q-1)] u(t)=Hm(q-1)u(t)
Where: k=0.0306; z0=-6230.9; z1=17730; s1=56.992+i253.67; s2= 56.992 - i253.67. s is the Laplace operator. It is about a system with nominimum phase feature [3],[7], so one must take into account it during the elaboration of the control law. In order to implement digital control, the transfer function (Hd(s)) is discretized, with the following sampling period: Te=10Ts=500µs. Hd(z-1) =
yˆ m (z -1 ) = ˆ -1 ) d(z
-1
b 0 +b1 z +b 2 z -1
-2
-2
=
B(z-1 ) -1
A(z ) 1+a1 z +a 2 z Where: b0 = 0.4, b1 = -0.8, b2 = 0.2, a1 = 0.7497, a2=0.2431. III.
P(q-1) = 1+p1q-1+…..+pnpq-np is an asymptotically stable polynomial which specifies the desired dynamic of regulation. y*(t) : Reference sequence defining the system dynamics during tracking, it is generated by the following transfer function of reference model:
POLE PLACEMENT ADAPTIVE CONTROL
Generally Hm(q-1) is determined from wished dynamic performances by choosing w0 and ζ. β is a scalar introduced to obtain unitary dc-gain in closed loop between y(t) and y*(t) such as: 1/b 0 If b 0 ≠ 0 1 else
β =
So, the control objective in (7) can be accomplished by a pole placement strategy [5] such as: S(q-1)D(q-1)u(t)+R(q-1)y(t) = P(q-1)β y* (t+d) =T(q-1)y*(t+d) Where S(q-1) and R(q-1) are the solution of the following Diophantine equation: P(q-1)=A(q-1)D(q-1)S(q-1) + q-1B(q-1) R(q-1)
A. Law control: The Buck-Boost converter is considered as a CARIMA model [4], having a representation of the follow form:
Thus, we finally get the control law [5]: u(t) =
A(θ(t),q-1) y(t) = B(θ(t),q-1) q-d u(t) + w(t)
(5)
1 [T(q -1)y*(t+d)-R(q -1 )y(t)] (8) S(q -1)D(q -1)
This can be represented by “fig.3”:
A(θ(t),q-1) = 1+a1(t)q-1 + …+ana(t)q-na ;. B(θ(t),q-1) = b0+b1(t)q-1 + …+bnb(t)q-nb ;. θ(t)=[a1(t),..,ana(t),b0(t),..,bnb(t)]T
B. Pole placement Adaptive control :
Where θ(t) is the parameters vector. y(t), u(t) and w(t) denote the output, input and disturbance sequences, respectively. A(θ(t),q-1), B(θ(t),q-1), are polynomials in the backward-shift operator q-1 with respective orders na, nb. The sequence w(t) is assumed to be modelled as: w(t) = C(q-1)[D(q-1)]-1ξ(t) where : D(q-1)=1-q-1 introduces an integral action in control law. ξ(t) is a white noise process. The noise polynomial C(q-1)=1 here for simplicity of exposition.. In order to develop the control law, we assume besides that ξ(t) is identically zero, (Afterward one will take into account disturbances during the on-line identification). Thus, the system model (5) becomes: A(θ(t), q-1) D(q-1) y(t) = B(θ(t), q-1)u(t-d).
The control law that we have just elaborated supposes that the converter is without uncertainties, in that follows one is going to use an adaptive control to on-line estimate process parameters of which will be used afterward to establish an adaptive control. The controller scheme is given in “fig.4” The (RLS) algorithm is used as parameters estimator using output/input signals. Let us:
(6)
Let us consider the following control objective that contains both the regulation case and the tracking one: Figure 3. PP regulator
P(q ) [y(t)- βB(q ) y*(t)] = 0 -1
Where :
-1
(7)
ϕ(t-1) = [-y(t-1),….., -y (t-na), u(t-d-nb)]T, θ(t)=[a1(t),..,ana(t),b0(t),..,bnb(t)]T
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ϕ(t-1): Being the observation vector of the converter. The RLS estimation consists in determining the ˆ parameters vector θ(t) which minimizes criterion according to [6]: T 2 [y(k)- θˆ (t )ϕ (k)] k=1
• •
Implanting the adaptive control law in (9). Wait for the end of sampling period to begin again.
t
J(t) = ∑
That leads to the following algorithm:
θˆ(t ) = θˆ(t − 1) +
T F ( t − 1)ϕ ( t − 1) y ( t ) − θˆ ( t − 1) ϕ ( t − 1)
T 1 + ϕ ( t − 1) F ( t − 1)ϕ ( t − 1)
1 F(t)= F ( t − 1) − λ1 (t )
T F ( t − 1)ϕ ( t − 1)ϕ ( t − 1) F ( t − 1)] λ1 ( t ) T + ϕ ( t − 1) F ( t − 1)ϕ ( t − 1) λ2 ( t )
Where: λ1,λ2 are forgetting factors, F is matrix of adaptation gain.
IV. SIMULATION AND RESULTS To validate our approach, we present in that follows the results of simulation for a dc-dc Buck-boost converter of figure1 which characteristics are previously mentioned, for the following operating point: Y=30v, D=0.5. The tracking performances are chosen by a polynomial which has: w0 = 0.5 and ζ = 0.9, which gives after discretisation: BM(q-1)= 0.0928 + 0.0687 q-1 AM(q-1)= 1-1.2451 q-1 + 0.4066 q-2 The dc performances are chosen by a polynomial which has w0=0.4 and ζ = 0.9, which gives after discretisation:
1
F(0)= I npr 0
