2.2 Adaptive Robust Fault Tolerant Controller (ARFTC)
2.2.3 Proposed Actuator Failure Compensator Design
The control objective is attained by designing a controller based on the systematic backstepping procedure [69] in conjunction with an adaptive second order sliding mode control (SOSMC) strategy [97], embedded in an integral sliding mode concept (ISMC) [103]. This yields a transformed system, described by(℘−1)differential equations augmented with an extended uncertain second order nonlinear dynamics. This extended system is firstly stabilized in finite time with the adaptive SOSMC in action and subsequently being acted upon by the backstepping controller, the remaining (℘−1)th order tracking error dynamics pursue an asymptotically stable behavior in the global sense. The benefits provided by the proposed methodology will be evident in the sequel.
The control signaluHj is designed following the procedure proposed in [104] with modifications, since the authors have considered only the case of partial loss of actuators. Moreover, no insight has been provided, pertaining to the tuning of controller parameters for transient performance enhancement and their subsequent effects on the overall stability of the system. The crucial issue of transient improvements have been rigourously dealt with, in this section, in addition to the problem of controller design. The generation of uHj initiates, firstly by defining ℘error variables as,
z1=ξ1−yr (2.8)
zi=ξi−αi−1−y(ir−1) i= 2,· · · , ℘ (2.9) Now, these tracking error variables z1, z2,· · · , z℘ are stabilized by αi serving as the smooth virtual control inputs determined at the ith step as,
αi =−zi−1−cizi+
i−1
X
k=1
∂αi−1
∂ξk ξk+1+
i−1
X
k=0
∂αi−1
∂y(k)r y(k+1)r (2.10)
where, the subscripti= 2,· · · , ℘−1. Finally, retaining the (℘−1) steps of backstepping, the℘thstep of the proposed scheme is different from the conventional backstepping procedure and is elaborated
below. For i=℘, the ℘th error variable dynamics is given by,
˙
z℘= ˙ξ℘−α˙℘−1−yr(℘)=a0(ξ, η) + ∆a(ξ, η) + Xm jk∈BtotF
bjβj(ξ, η)¯uF j
+ Xm j∈BparH
bjβj(ξ, η)KjuHj−
℘−1
X
k=1
∂α℘−1
∂ξk ξk+1+
℘−1
X
k=0
∂α℘−1
∂y(k)r yr(k+1)
!
−y(℘)r (2.11)
The uncertainties introduced into the system at the onset of faults and failures ofjactuators in (2.11), for the sake of simplicity, can be expressed as,
∆1= Xm jk∈B
totF
bjβj(ξ, η)¯uF j, ∆2= Xm j∈B
parH
bjKj
Furthermore,
¨
z℘= ˙a0(ξ, η)−
℘−1
X
k=1
∂
∂ξk
℘X−1 k=1
∂α℘−1
∂ξk ξk+1
!
ξk+1+
℘−1
X
k=0
∂
∂yr(k)
℘−1
X
k=0
∂α℘−1
∂yr(k)
y(k+1)r
! y(k+1)r
!
(2.12)
−y(℘)r + d
dt(∆a(ξ, η)) + d
dt(∆1) + ∆2 d
dt(βj(ξ, η)uHj)
| {z }
v
¨
z℘= ˙a0(ξ, η)−
℘−1
X
k=1
∂
∂ξk
℘X−1 k=1
∂α℘−1
∂ξk ξk+1
! ξk+1+
℘−1
X
k=0
∂
∂yr(k)
℘−1
X
k=0
∂α℘−1
∂y(k)r
y(k+1)r
! y(k+1)r
!
−yr(℘)+ d
dt(∆a(ξ, η)) + d
dt(∆1) + ∆2
|{z}
ζ(·)
v (2.13)
where, all the terms in (2.13) except the last term, are collectively represented by the function ϕ(·).
Thereafter, in order to accommodate the second order sliding mode strategy within the backstepping procedure, the second time derivative of z℘ in (2.13) is used to yield an uncertain extended second order system ins=z℘ ands˙= ˙z℘ as,
¨
s=ϕ(·) +ζ(·)v (2.14)
where, the sliding variable and its first time derivative are designated by the terms s(ξ, t) and s(ξ, t)˙ respectively. Here the control objective is to ensures(ξ, t) = 0ands(ξ, t) = 0˙ in finite time to guarantee the existence of a second order sliding mode. Hence, before proceeding any further, the definition of SOSMC followed by some remarks are given below.
Definition 2.1. Let us consider the second order nonlinear dynamics in (2.14) closed by a discontinu- ous feedback v. Assuming thatsands˙are continuous functions and the setS2={ξ |s(ξ, t) = ˙s(ξ, t) = 0}, called a second order sliding set, is nonempty and is locally an integral set in the Filippov sense, the motion onS2 is called second order sliding mode with respect to the sliding variables(ξ, t) [97].
Remark 2.1. The solutions of (2.14) are understood in the Filippov sense [105] and the control law v is Lebesgue measurable.
Remark 2.2. The uncertain functions ϕ(·) and ζ(·) in (2.14) are known to be bounded with unknown upper bounds.
Therefore, the second order sliding mode control of (2.14) with respect to sliding variablesreduces to the finite time stabilization of the perturbed double integrator dynamics given by,
˙ s1=s2
˙
s2=ϕ(·) +ζ(·)v=ϕ(·) + (ζ(·)−1)v
| {z }
ψ(·)
+v (2.15)
with [s1 s2]T = [s s]˙T. The following lemma gives an affirmation as to how the dependent variables involved in a double integrator dynamics driven by an external input v, can be steered to the origin in finite time.
Lemma 2.1. Let us consider an n-integrator system described as:
ξ˙i=ξi+1
ξ˙n=vnom i= 1,· · · , n−1 (2.16) Let β1, β2, ..., βn >0 be such that the polynomial φ(λ) =λn+ Pn
k=1
βkλk−1 is Hurwitz. For the system (2.16), ∃ ̺ ∈ (0,1) such that for every ϑk ∈ (1−̺,1), k = 1, ..., n, the origin is a globally stable equilibrium in finite time under the feedback, vnom(x) =− Pn
k=1
βksign(ξk)|ξk|ϑk, where ϑ1, ...ϑn satisfy ϑk−1 = 2ϑϑkϑk+1
k+1−ϑk, k= 2, ..., n andϑn+1= 1.
Proof.
Please refer to the work in [106].
Attaining finite time stability of (2.15) requires the design of an auxiliary integral sliding manifold σ with an embedded nominal control lawvnomand thereafter a discontinuous control law to ensure finite time stability and invariance of (2.15) to perturbations [107] collectively represented by ψ(·). Let the integral sliding manifold be defined as,
σ :=s2(t)−s2(t0)− Z t
t0
vnomdt′ (2.17)
Now under the action of a discontinuous control law, sliding mode is established on the sliding surface defined by the set SR ={ [s1, s2]T ∈R2 |σ = 0} for t≥0 and hence the reaching phase is totally eliminated. Therefore, an invariance towards matched uncertainties is achieved from the very beginning and robustness is guaranteed over the entire state space. Finally, the synthesis of the overall control law v is based on Gao’s reaching law approach [108] and subsequent analysis of the motion equations
on the sliding surface σ = 0 using the equivalent control method proposed in [103]. Hereinafter, the constant plus proportional reaching law is considered as,σ˙ =−εσ−τ sign(σ). Where ε, τ are design constants which characterize the rate at which σ converges to zero.
Now, since the only knowledge available about the uncertainties is their property of boundedness and that too is unknown, the reaching law will be incompetent in establishing a sliding mode along σ= 0 in the event of unknown actuator failures. Therefore, it is quite relevant to append an adaptive counterpart to the control law v, in order to ensure that the constraint σ = 0 ∀ t ≥ 0 is fulfilled.
Moreover, the partitioning of modeling uncertainties from the impact of faults and failures will be clearly visible in the course of design. Consequently, from (2.13) and (2.17), the first time derivative ofσ can be derived as,
˙
σ = ˙a0(ξ, η)−
℘X−1 k=1
∂
∂ξk
℘−1
X
k=1
∂α℘−1
∂ξk ξk+1
! ξk+1+
℘−1
X
k=0
∂
∂yr(k)
℘−1
X
k=0
∂α℘−1
∂y(k)r yr(k+1)
! yr(k+1)
!
−yr(℘)+ d
dt(∆a(ξ, η)) + Xm j∈B
totF
bju¯F jβj(ξ, η) + Xm j∈B
parH
bjKjv−vnom
=
Xm j∈BparH
|bj|Kj
θ∗TΦ(ξ) + Xm j∈BparH
bjKjv+ d
dt(∆a(ξ, η)) (2.18)
where, θ∗ := [θ1∗ θ∗2]T and θ2∗ := [θ2,1∗ θ2,2∗ · · · θ2,m∗ −1]T. These unknown parameters are now defined below as,
θ∗1 := 1 Pm j∈BparH
|bj|Kj
, θ2,j∗ :={bju¯F j}j∈BtotF
Pm j∈BparH
|bj|Kj
Furthermore, the regressor matrixΦ(ξ) := [Φ1(ξ) Φ2(ξ)]T is known and whose components are defined as follows.
Φ1(ξ) = ˙a0(ξ, η)−
℘X−1 k=1
∂
∂ξk
℘X−1 k=1
∂α℘−1
∂ξk ξk+1
! ξk+1+
℘−1
X
k=0
∂
∂yr(k)
℘−1
X
k=0
∂α℘−1
∂y(k)r yr(k+1)
! y(k+1)r
!
−y(℘)r
−vnom
Φ2(ξ) = [β1(ξ, η) β2(ξ, η) . . . βm−1(ξ, η)]T
Further, (2.18) clearly reveals the separation between modeling uncertainties and the fault induced perturbations. Accordingly, this demands two adaptive laws to counteract the effect of uncertainties due to faults and that of modeling in a decoupled fashion. Thereafter, analyzing the dynamics along the sliding surfaceσ = 0, the total control law v satisfying the µ-reachability condition is derived as,
v= (−θˆTΦ(ξ)−εσ−τ sgn(σ)−Γsgn(σ))sgn(bˆ j) (2.19) The first and the third part on the right hand side of (2.19) characterize the adaptive part of v. In
consequence, the actual control law for the system (2.1) is straightaway found to be, uj =uHj =sgn(bj) 1
βj(ξ, η) Z t
t0
(−θˆTΦ(x)−εσ−τ sgn(σ)−Γsgn(σ))dtˆ ′ (2.20) forj = 1,2, ..., m. Moreover,θˆandΓˆ are the estimates ofθ∗ andΓ∗ respectively. Γ∗ signifies the upper bound of system uncertainties, that is
d∆adt
≤Γ∗ andθ∗ denotes the failure induced adversities on the plant. Moreover, the adaptive parameter update laws are given as,
θ˙ˆ=−P ǫ2θˆ+PΦσ (2.21)
˙ˆΓ =−γǫ1Γ +ˆ γ|σ| (2.22)
The matrix P is user defined and is a solution of the Lyapunov equation, ATP +P A = −Q. Here, A is a Hurwitz stable matrix chosen by the designer and the matrix Qis positive definite. The finite time stabilizing nominal control law vnom is formulated from Lemma 2.1 as,
vnom =−β1sign(s1)|s1|ϑ1 −β2sign( ˙s1)|s˙1 |ϑ2 (2.23) The terms γ, ǫ1, ǫ2, β1, β2 > 0 and ci > 0 for i = 1, 2,· · · , ℘−1 are positive constants and depend on the designer’s choice. The block diagrammatic representation of the proposed control scheme is shown in Figure 2.1.