The proposed control integrates the finite time current observer with an adaptive backstepping control (FTCO-ABSC) mechanism for a current sensorless control.
5.2.1 Adaptive Backstepping Control (ABSC) Briefly, the steps involved are recalled below:
Step 1: The term xR1 in (5.1) is assumed to be an unknown function. It is estimated by framing an updation law to yield Θ∗TΦ where Θ∗ is the optimum weight and Φ is the regressor. Replacing xR1 in (5.1) with Θ∗TΦ yields
˙
x1 =−Θ∗TΦ C +x2
C. (5.3)
Now, (5.3) and (5.2) define the dynamics of the DC-DC buck converter which will be utilized in control design.
Step 2: The error variables are defined as
z1=x1−vr z2= xˆC2 −α
)
(5.4) where ˆx2 is the estimated inductor current obtained by using finite time current observer discussed in the next subsection andvr is the reference output voltage.
Step 3: Using the output voltage dynamics in (5.3), thez1-dynamics is stabilized using a virtual-control input α as,
α=−c1z1+ΘˆTΦ
C + ˙vr (5.5)
where c1 > 0 is the controller gain and ˆΘTΦ is the adaptive estimate of unknown function xR1. Moreover, ˜Θ = Θ∗−Θ. Subsequently, the next errorˆ z2 can be rewritten as
˙ z2= ξ˙˜2
C − x1 LC −
X2 k=1
∂α
∂xkx˙k− X1 k=0
∂α
∂vr(k)vr(k+1)−Θ˙ˆT ∂α
∂ΘˆT + E
LCu (5.6)
where ˜ξ2= ˜x2 = ˆx2−x2 is the error in current state estimation. Appropriate selection of control law u to stabilize z2 is to be guided properly. Hence,u is found from (5.6) as
u=LC
E −z1−c2z2+ x1
LC + ∂α
∂x1 x2
C
− ∂α
∂x1
ΘˆTΦ C
! +
X1 k=0
∂α
∂vr(k)vr(k+1) +Θ˙ˆT ∂α
∂ΘˆT + ¨vr+ ∂α
∂x1γϑ2
(5.7) wherec2 >0 defines the controller gain,γ is the adaptation rate and ϑ2 is a tuning function described as ϑ2 =−ΦCz1+∂x∂α
1
Φ Cz2.
Step 4: Further, to find the optimal weight required for estimation of the uncertain term xR1, an online Lyapunov based adaptive learning law is formulated to yield a close approximation. The optimum
5.2 Proposed Controller Design
weight vector estimate ˆΘ(t) as explained in (2.50) is given by Θ(t) = ˆˆ Θ(t0)−γ
Z t t0
Φ(x1(ν)) C
z1(ν)− ∂α
∂x1z2(ν)
dν. (5.8)
where γ >0 is the adaption rate. In order to compute the control lawu in (5.7) precisely, the exact knowledge of system states is essential. Hence, the proposed methodology proposes the control strat- egy with a minimal usage of sensing units. The inductor current state x2 is reconstructed from the finite time current observer.
Remark 11. The choice of adaptation gain parameter γ is crucial in order to attain a fast and satisfactory dynamic response of the output voltage state of the converter. A high value of γ results in a faster adaptation and enhanced transient response. However, a very high value of γ also results in a decreased stability margin of DC-DC buck converter system. Therefore, it is recommended to judiciously prescribe the value of γ in such a way that a desired and satisfactory control response is achieved, besides preserving a safe stability margin.
5.2.2 Finite Time Current Observer (FTCO)
Following the design philosophy in Section A.1, herein a finite time current estimator is presented.
The idea is to consider the unknown current as a disturbance rather than a state variable. Hence, the proposed current observer design utilizes only the voltage tracking error dynamics instead of considering the full dynamics of the buck converter. This would render computational simplicity and enhance output transient performance by reconstructing the inductor current under both nominal and perturbed situations. Hence, its applicability to the control design problem of DC-DC buck converters is well suited. The current observer is designed followed by the finite time stability analysis. The dynamics of the proposed finite time current observer (FTCO) is given as,
ξ˙1 =−λε1|ξ1−x1|12sgn(ξ1−x1)−ΘˆCTΦ +ξC2 ξ˙2 =−2ελ22sgn(ξ2−υ1)
υ1 =−λε1|ξ1−x1|12sgn(ξ1−x1) +ξC2
(5.9)
where ξ1 and ξ2 denote the estimates of x1 and x2 respectively. The terms λ1, λ2 are the observer gains andε >0 is a small number close to zero. In subsequent analysis, a bound on εwill be found so as to achieve a finite time convergence of the observer error to the origin. Proceeding further, let us define ∆(x1, t) =x2, ˜ξ1 =ξ1−x1 and ˜ξ2 =ξ2−∆(·). Hence, the error dynamics are written as,
ξ˙˜1=−λε1|ξ˜1|12sgn(˜ξ1) + ˜ξ2 ξ˙˜2=−2ελ22sgn(|ξ˜1|12sgn(˜ξ1)) + ˙∆(·)
)
. (5.10)
Next, the finite time stability of the observer error dynamics described in (5.10) is stated in Theorem 5 below.
Theorem 5. Considering the finite time current observer error dynamics given by (5.10) and as- suming that the disturbance ∆(x1, t) is at least once continuously differentiable, the resulting observer
error variables ξ˜1 and ξ˜2 converge to the origin in finite time provided the gain 2ελ22 >sup{∆(.)˙ }=L, L>0 yielding∆(ˆ ·) =ξ2.
Proof: For analyzing the finite time stability of observer error dynamics in (5.10), the degree of homogeneity of the vector fields associated must be obtained first. Applying a homogeneity transfor- mationTr: (t, ξ˜1)7−→(rt, r3−iξ˜1) to the observer error dynamics yields the degree of homogeneity of the associated vector fields to be−1<0. Therefore, as a next step, to ensure the finite time stability of the observer, let us consider a Lyapunov function V0 =ζTPζ, where, ζ = [ζ1 ζ2]T = [⌈ξ˜1⌋1/2 ξ˜2]T and ⌈ξ˜1⌋ν =|ξ˜1|νsgn(˜ξ1). The first time derivative of ζ is given by,
ζ˙=
" 1
2|ξ˜1|−1/2ξ˙˜1 ξ˙˜2
#
=
" 1
2|ξ˜1|−1/2(−λε1⌈ξ˜1⌋12 + ˜ξ2)
−2ελ22sgn(˜ξ2−ξ˙˜2) + ˙∆(·)
#
=
" 1
2|ξ˜1|−1/2(−λε1⌈ξ˜1⌋12 + ˜ξ2)
−2ελ22sgn(λε1⌈ξ˜1⌋12) + ˙∆(·)
#
(5.11)
= 1
2|ξ˜1|−1/2
"
−λε1⌈ξ˜1⌋1/2+ ˜ξ2
−22ελ22⌈ξ˜1⌋1/2+ 2 ˙∆(·)|ξ˜1|1/2
#
= 1
2|ξ˜1|−1/2
"
−λε1⌈ξ˜1⌋1/2+ ˜ξ2
−(λε22⌈ξ˜1⌋1/2−∆(˙ ·)sgn(˜ξ1))⌈ξ˜1⌋1/2
#
(5.12)
=|ξ˜1|−1/2
"
2λ1
ε 1
2
−(2ελ22 −∆(˙ ·)sgn(˜ξ1)) 0
# "
⌈ξ˜1⌋1/2 ξ˜2
# .
As argued in [108], using the fact that sup{∆(.)˙ }=L, gives
=|ξ˜1|−1/2
" 2λ
ε1 1
2
−(2ελ22 − L) 0
#
| {z }
A
"
⌈ξ˜1⌋1/2 ξ˜2
#
=|ξ˜1|1/2Aζ. (5.13)
Thereby the first time derivative of V0 can be written as,
V˙0 =|ξ˜1|−1/2ζT(ATP+PA)ζ =−|ξ˜1|−1/2ζTQζ <0 (5.14) It can be noted that (5.14) above is satisfactory because the matrixAis Hurwitz and the matrixPis a positive definite symmetric matrix satisfying the Lyapunov criterion given byATP+PA=−Qwith Q>0. The matrixAis guaranteed to be Hurwitz if and only if the observer gain λ2
2ε2 >sup( ˙∆(.)) =L and λ1
ε >0. Using Rayleigh principle [41], |ξ˜1|1/2 ≤ ⌈ξ˜1⌋1/2 ≤ kζk2 < βmin−1/2(P)V01/2 and then ˙V0 can
5.2 Proposed Controller Design
be rewritten as,
V˙0 ≤ −|ξ˜1|−1/2βmin(Q)kζk2
≤ −βmin1/2(P)V0−1/2βmin(Q)kζk2 (5.15)
≤ −βmin1/2(P)βmin(Q)
βmax(P) V01/2 ≤ −ΓV01/2 (5.16)
where the term Γ = βmin1/2(P)βmin(Q)
βmax(P) is the observer gain parameter andβmax(·) and βmin(·) denote the maximum and minimum eigen values of a square matrix. Though the transformation ζ is con- tinuous, it follows thatζ reduces to zero in finite time which means that the observer error variables ξ˜1 and ˜ξ2 converge to the origin in finite time. Further, (5.16) can be solved to obtain an explicit expression describing the maximum finite time required for the estimation error to converge to the origin.
Let us consider,V0(t0) =V0(0), and the final convergence timeT satisfyingV0(T) = 0 due to negative definiteness of ˙V0(t) and proceed as follows.
V˙0+ ΓV01/2≤0
⇒
Z V0(T) V0(0)
dV0
V01/2 ≤ −Γ(T −0)
⇒2(V0(T)1/2−V0(0)1/2)≤ −ΓT.
AsV0(T) = 0, it is substituted in the above inequality to get closed form expression for the convergence timeT in terms of initial conditions and observer gains as defined below.
T ≤ 2V0(0)1/2
Γ (5.17)
Therefore, (5.17) reflects that the relevant parameter influencing the finite time convergence is the bound on the convergence time given by T.
Using Theorem 2.1 given in [106], the observer error dynamics are inferred to be finite time stable from (5.16) . This completes the proof. .
Next, to justify the exactness in the estimation of inductor current using the proposed current observer, the results have been summarized as Theorem 6.
Theorem 6. The current observer given in (5.9) yields an exact estimation of the current provided that it is at least once continuously differentiable, i.esup{∆˙} exists. This means that at steady state, the exact ultimate bound on the observer error variables is given byξ˜1 = 0 and ξ˜2 = 0.
Proof: Following the procedure of finding the ultimate bounds presented in [109], the observer error dynamics described in (5.10) are now considered and using a specific diffeomorphism, error dynamics have to be transformed to a more suitable form to make the analysis convenient. Utilizing the diffeomorphism [ψ1, ψ2]T = [|ξ˜1|12sgn(˜ξ1), εξ˜2]T and ¯D = D
ψ1′
(˜ξ1), whereDi = ˙∆(·), the transformed
dynamics are obtained as
"
ψ˙1 ψ˙2
#
= ψ′(˜ξ1) ε
( "
−λ1 1
−λ2 0
#
| {z }
F
"
ψ1 ψ2
# +ε2
"
0 1
#
| {z }
G
D¯ )
. (5.18)
Now let us consider the auxiliary dynamics as,
"
ψ˙1 ψ˙2
#
=
"
−λ1 1
−λ2 0
# "
ψ1 ψ2
# +ε2
"
0 1
#
D.¯ (5.19)
As given in [110], it is understood that the dynamics described in (5.18) and (5.19) follow the same trajectories, in case ψ
′ 1
ε is a positive definite function. Hence, analysis of current estimation error using the auxiliary dynamics in (5.19) is now carried out. Finally, the original estimation error can be obtained using (5.19). Next, using the concept of linear control theory and Lemma 1 mentioned in [109], the ultimate bound on the estimation error is derived as |ψk| ≤ε2{F}kkD¯k∞, where,
F=
"
F1
F2
#
= Z ∞
0 |eFτG|dτ (5.20)
Since kD¯k∞< K with the change of co-ordinates, the original estimation error can be obtained as 2ε2|ξ˜1|12F1K=|ξ˜1|12sgn(˜ξ1)
|ξ˜1|12(2ε2F1K−1) = 0 )
. (5.21)
By solving (5.21), it is found that if ε2 < 12F1K, the estimation error ˜ξ1 exactly converges to zero.
Similarly, the ultimate bound on the current estimation error ˜ξ2 can be found using the relation, ψ˜2 ≤2ε2F2K|ξ˜1|12, implying ˜ξ2 = 0. Therefore, it is proved that the finite time current observer (5.9) achieves an exact estimation of the current state.