2.2 Adaptive Robust Fault Tolerant Controller (ARFTC)
2.2.4 Stability Analysis
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.
Figure 2.1: Schematic representation the proposed fault tolerant control scheme
ofV1 is given by,
V˙1=z1z˙1 =z1(ξ2−y˙r) =z1(z2+α1+ ˙yr−y˙r) Using (2.10) yields,
V˙1 =z1z2+z1(−c1z1) =−c1z12+z1z2 =−κ1V1+z1z2 (2.24) where, κ1 = 2c1, c1 >0. Clearly, if z2 = 0, thenV˙1 =−κ1V1 and hence z1 is guaranteed to converge to zero asymptotically.
Now, fori= 2,· · ·, ℘−1, concerning the stability of each error variablez2, ..., zi, ...z℘−1, the Lyapunov function is defined as, Vi = 12z2i. Taking the time derivative of Vi and using equality (2.10) results in,
V˙i=ziz˙i=zi zi+1+αi+y(i)d −
i−1
X
k=1
∂αi−1
∂ξk ξk+1−
i−1
X
k=0
∂αi−1
∂yr(k) y(k+1)r −yr(i)
!
=zi(zi+1−cizi−zi−1) =zizi+1−ciz2i −zi−1zi (2.25) V˙i=−κiVi+zizi+1−zi−1zi, for κi= 2ci, ci>0 (2.26)
Now,
˙ z1
˙ z2
˙ z3
...
˙ z℘−1
=
−c1 0 0 · · · 0 0 −c2 1 · · · 0 0 −1 −c3 · · · 0 ... ... ... . .. ... 0 0 · · · −1 −c℘−1
| {z }
Az
z1 z2 z3 ... z℘−1
+
0 0 0 ... 1
| {z }
Bz
z℘ (2.27)
Here (2.27) describes the(℘−1)thorder tracking error dynamics controlled by the℘therror variable. It is obvious that, z℘= 0 guarantees an asymptotically stable behavior for the solutions of the dynamics in (2.27) since the matrix Az is Hurwitz. Alternatively, stability can also be proved through the choice of a total Lyapunov function given by V =P℘−1
i=1 Vi and subsequently proving its first time derivative to be negative as,
V˙ =
℘−1
X
i=1
V˙i =
℘X−1 i=1
cizi2+z℘−1z℘ =−
℘−1
X
i=1
κiVi+z℘−1z℘=−min{κi}℘i=1−1
℘X−1 i=1
Vi+z℘−1z℘ (2.28) Considering z℘ = 0, the negative definiteness of (2.28) is proved. Hence, it can be stated that finite time stability of z℘ ensures asymptotic stability of the (℘−1)th order error dynamics. Now, the finite time stability ofz℘and other extended state variables explored in the accomplishment of the proposed design are proved. This objective was achieved through the design of an integral sliding manifold (2.17), which when proved to be attractive and converge to zero in finite time, ensured the convergence of the state z℘ or conversely the pair {s1,s˙1}, to the origin in finite time. To study the stability properties of the auxiliary sliding surface σ = 0 and parameter convergence, a Lyapunov function candidate is chosen as,
Vσ = 1 2σ2+ 1
γ Xm j∈B
parH
|bj|Kj
2 Γ˜2+ Xm j∈B
parH
|bj|Kj
2 θ˜TP−1θ˜ (2.29) where, Γ := ˆ˜ Γ−Γ∗ and θ˜:= ˆθ−θ∗. Additionally let us defineΓ∗ :=θ∗1ω∗ ≥θ∗1|dtd(∆a(ξ, η))|. NowV˙σ is obtained as,
V˙σ =σσ˙ + Xm j∈B
parH
|bj|Kj
γ Γ˜˙ˆΓ + Xm
j∈B
parH
|bj|KjθP˜ −1θ˙ˆ=σσ˙ + 1
θ∗1γΓ˜˙ˆΓ + 1
θ1∗θP˜ −1θ˙ˆ (2.30)
=σ
θ∗TΦ θ∗1 +
Xm j∈BparH
bjKjv+θ∗1 θ∗1
d
dt(∆a(ξ, η))
+ 1
θ∗1γΓ˜˙ˆΓ + 1
θ1∗θP˜ −1θ˙ˆ (2.31)
Substituting the control lawv from (2.19) yields, V˙σ =σ
θ∗TΦ θ1∗ +
Xm j∈BparH
bjKj
(−θˆTΦ(ξ)−εσ−τ sgn(σ)−Γsgn(σ))sgn(bˆ j) + θ1∗
θ1∗ d
dt(∆a(ξ, η))
+ 1
θ1∗γΓ˜˙ˆΓ + 1
θ∗1θP˜ −1θ˙ˆ (2.32)
=σ θ∗TΦ θ∗1 + 1
θ1∗
−θˆTΦ(ξ)−εσ−τ sgn(σ)−Γsgn(σ)ˆ +θ∗1
θ∗1 d
dt(∆a(ξ, η)
! + 1
θ∗1γΓ˜˙ˆΓ + 1
θ1∗θP˜ −1θ˙ˆ Utilizing the adaptive laws from (2.21) and (2.22),
V˙σ =−σ 1
θ1∗(ˆθT −θ∗T)Φ−σ 1
θ∗1Γsgn(σ)ˆ − 1
θ1∗εσ2−σ 1
θ∗1τ sgn(σ) + 1
θ∗1Γ∗σ+ 1
θ∗1Γ˜|σ| − 1 θ1∗ǫ1Γˆ˜Γ + 1
θ∗1θ˜TP−1PΦσ− 1
θ1∗θ˜TP−1P ǫ2θˆ
≤ −1
θ1∗εσ2− 1
θ∗1τ σsgn(σ)− 1
θ∗1ǫ1Γˆ2+ 1
θ∗1ǫ1Γ∗Γˆ− 1
θ∗1ǫ2θˆTθˆ+ 1
θ∗1ǫ2θ∗Tθˆ
≤ −1
θ1∗εσ2− 1
θ∗1τ σ− 1 θ1∗ǫ1
Γˆ−1
2Γ∗ 2
− 1 θ1∗ǫ2(
θˆ1−1
2θ∗1 2
+
θˆ2,1−1 2θ∗2,1
2
+
θˆ2,2−1 2θ∗2,2
2
+· · ·+
θˆ2,(m−p)− 1
2θ2,(m∗ −p) 2
) + 1 4
1
θ1∗ǫ1Γ∗2+1 4
1 θ∗1ǫ2
θ1∗2+θ2,1∗ 2+θ2,2∗ 2+· · ·+θ2,(m∗ −p)2
≤ −1
θ1∗εσ2− 1
θ∗1τ|σ| − 1 θ1∗ǫ1
Γˆ−1
2Γ∗ 2
− 1
θ1∗ǫ2kθˆ−1
2θ∗k2+ 1 4
1
θ1∗ǫ1Γ∗2+1 4
1
θ∗1ǫ2kθ∗k2
≤ −1
θ1∗εσ2− 1
θ∗1τ|σ|+1 4
1
θ1∗(ǫ1Γ∗2+ǫ2kθ∗k2)
≤ −1
θ1∗εσ2− 1
θ∗1τ|σ|+1
4 ǫ1Γ∗2
θ1∗ +ǫ2 θ1∗+θ2,1∗ 2
θ1∗ +θ∗2,22
θ∗1 +θ∗2,32
θ1∗ +· · ·+ θm∗−p2 θ1∗
!!
≤ −1
θ1∗εσ2− 1
θ∗1τ|σ|+ 1 4θ1∗
ǫ1Γ∗2+ǫ2(θ1∗2+θ∗2,12+θ2,2∗ 2+θ2,3∗ 2+· · ·+θm∗−p2)
| {z }
θ¯∗
(2.33)
Now, negative definiteness ofV˙σ in (2.33) can only be attained on satisfying the conditions, kσk>
rθ¯∗
4ε and |σ|> θ¯∗
4τ (2.34)
Hence, it can be inferred that the trajectories of the closed loop system are upper bounded in the region defined as a compact subset and a positively invariant set inR given as,
V :=n
σ:R2×R+−→R, z1, z2, . . . , z℘+1∈R
int( ˙Vσ = 0),
kσk2≤ θ¯∗ 4ε
∩
|σ| ≤ θ¯∗ 4τ
o
(2.35) Application of the control therefore first drives the pair {s1, s˙1} into this small set around the origin defined in (2.35) and then it attains a finite time stable behavior to the origin. The analysis of the
closed loop signals is commenced after the sliding mode is procured. As per Lemma 2.1, when real sliding mode is attained, i.e. |σ| < |σmax|, finite time convergence of s1 and s˙1 is guaranteed. The term σmaxdesignates a small boundary region around the sliding surfaceσ= 0 defined asV in (2.35).
This fact is also evident from (2.15) and (2.17) leading to the dynamics,
˙ s1 =s2
˙
s2 =−β1sgn(s1)|s1|2−2ϑ −β2sgn(s2)|s2|ϑ+|σ˙max| (2.36) Assuming,kΦkto be upper bounded, the term,|σ˙max|, is the maximum value belonging to the residing set of σ, given as,˙
˙ σ∈
|σ˙| ≤ 1 θ1∗
s
θ∗−ǫ1Γ∗2
ǫ2 kΦk+ Γ∗−ε|σmax| −τ
Guaranteeing finite time stability of the dynamics in (2.36) requires that the associated vector field has a negative degree of homogeneity in addition to the system dynamics being asymptotically stable.
This notion of finite time stability for nintegrator dynamics has been given by Bhat and Bernstein in their work [106]. The vector field on R2 in (2.36) is given by,
F : =F1(s1, s2) ∂
∂s1 +F2(s1, s2) ∂
∂s2 (2.37)
where, F1(s1, s2) = s2 and F2(s1, s2) = −β1sgn(s1)|s1|2−2ϑ −β2sgn(s2)|s2|ϑ. Now let us consider a variable r >0 and evaluate the degree of homogeneity of the vector fieldF. It is observed that,
F1(r2−ϑs1, rs2) =r(2−ϑ)+(ϑ−1)F1(s1, s2) (2.38) F2(r2−ϑs1, rs2) =r1+(ϑ−1)F2(s1, s2) (2.39) Therefore, the degree of homogeneity of F is equal to(ϑ−1) being negative, sinceϑ <0follows from Lemma 2.1. The negative degree of homogeneity has now been proved. Now, it is to be shown that the dynamics (2.36) is asymptotically stable.
There exists a Lyapunov functionVs1s2 :R2 −→Rwhich isC1 onR2/{0}such thatV˙s1s2 is continuous and negative definite. Let the Lyapunov function Vs1s2 be defined as,
Vs1s2 : = β1 β2
2−ϑ
2 |s1|2−2ϑ + 1
2β2s22 (2.40)
Subsequently, V˙s1s2 = β1
β2
2−ϑ 2
2 2−ϑ
|s1|2−2ϑsgn(s1) ˙s1+ 1
β2s2s˙2 (2.41)
= β1
β2s2|s1|2−ϑϑsgn(s1) + 1 β2s2
−β1sgn(s1)|s1|2−2ϑ −β2sgn(s2)|s2|ϑ+|σ˙max|
=−|s2|ϑsgn(s2)s2+ 1
β2|σ˙max|s2 V˙s1s2 ≤ −|s2|ϑ+1+ 1
β2|σ˙max||s2| (2.42)
For V˙s1s2 < 0, the following condition, |s2|ϑ > |σ˙max|/β2, must be satisfied. Now, let the minimum value thatVs1s2 can achieve be given by Cmin. This implies that,
|s1|2−2ϑ = 1
β1(2−ϑ) 2β2Cmin−
|σ˙max| β2
2ϑ!
(2.43)
For|s1|2−1ϑ to exist in the real spaceR, calls for, Cmin > 2β1
2
|σ˙max| β2
2ϑ
. Now ifCmin = 2β1
2
|σ˙max| β2
2ϑ is achieved, |s1|= 0 in finite time which in turn guarantees the asymptotic stability of the (℘−1)th order tracking error dynamics. The invariant set is hence defined as,
S1 : =n
(s1, s2)∈R2
int( ˙Vs1s2 = 0), |s1|= 0, |s2| ≤
|σ˙max| β2
1ϑo
(2.44) From (2.28) and (2.44), utilizing Barbalat’s Lemma [110] and Lasalle’s theorem [69] yields,
V˙ =−
℘X−1 i=1
κiVi =−min{κi}℘i=1−1
℘X−1 i=1
Vi ≤0 (2.45)
Therefore, from (2.44), the signal vector [z1 z2. . . z℘−1]T ∈ L2∩ L∞ and thereafter[ ˙z1 z˙2. . .z˙℘−1]T ∈ L∞. Utilizing the signal convergence lemma, the closed loop stability is established and asymptotic output tracking is guaranteed, which means, lim
t→∞z1 = lim
t→∞(ξ1−yr) = 0.
Now, ifCminis not achieved ast→tf, a finite time, its consequence on the stability of the closed loop system is investigated below.
In order to circumvent the problem mentioned above, let us define a variable λ > 0, denoting the difference between the steady state and the instantaneous value of Cmin. This yields the modified invariant set S′
1 defined as, S′
1 : =n
(s1, s2)∈R2
int( ˙Vs1s2 = 0), |s1| ≤
2λβ2 β1(2−ϑ)
2−2ϑ
, |s2| ≤
|σ˙max| β2
ϑ1 o
(2.46) From (2.46), it is obvious that the notion of asymptotic stability of the(℘−1)th order error dynamics to the origin will be violated. However, stability is guaranteed in the vicinity of the origin as shown below.
Referring to (2.28), gives,
V˙ =−
℘−1
X
i=1
κiVi+z℘−1z℘ ≤ −
℘−1
X
i=1
κiVi+|z℘−1||z℘| (2.47) Application of Peter-Paul inequality (Appendix A.2.3) to (2.47), for every 0< ν <2c℘−1, yields,
V˙ ≤ −
℘−1
X
i=1
cizi2+ ν|z℘−1|2
2 +|z℘|2
2ν ≤ −2h
c1 c2. . . .
c℘−1−ν 2
i
| {z }
C
0.5z21 0.5z22
... 0.5z℘2−1
+|z℘|2 2ν
≤ −min{C=Ci}℘i=1−1
℘X−1 i=1
Vi+|z℘|2
2ν (2.48)
From the invariant set S′
1, it is known that |z℘|=|s1| ≤
2λβ2
β1(2−ϑ)
2−2ϑ
, therefore (2.48) reduces to,
V˙ ≤ −min{C =Ci}℘i=1−1
℘−1
X
i=1
Vi+ 1 2ν
2λβ2 β1(2−ϑ)
2−ϑ
(2.49)
=⇒V ≤V(t0)exp(−Cmint) + 1 2Cminν
2λβ2 β1(2−ϑ)
2−ϑ
(2.50) Considering the steady state, the following evolves from inequality (2.50),
(z12+z22+. . .+z℘2−1)1/2≤ 1
√Cminν
2λβ2 β1(2−ϑ)
2−2ϑ
(2.51) In consequence, another positively invariant set S2 using (2.51) is defined as,
S2: =n
(z1, z2, . . . , z℘−1)∈R℘−1 (z21+z22+. . .+z2℘−1)1/2≤ 1
√Cminν
2λβ2 β1(2−ϑ)
2−2ϑo
(2.52) It is quite relevant that the visualization of set S2 is possible if and only if ℘ ≤ 4. Thereupon considering ℘= 4 yields a setS2, which is a ball of radius √ 1
Cminν
2λβ2
β1(2−ϑ)
2−2ϑ
centered around the origin. Eventually, it is now possible to define the final compact and positively invariant set S∗ with the help of (2.46) and (2.52). Defining Z := [z1 z2 z3. . . z℘−1 z℘ z℘+1]T, results in,
S∗ :=
(
Z ∈R℘+1
kZk2 ≤ s
1
Cminν + 1 2λβ2
β1(2−ϑ) 2−ϑ
+
|σ˙max| β2
ϑ2)
⊇S′
1×S2 (2.53) Ultimately the setS∗ is the largest possible positively invariant set, where the closed loop trajectories of the system (2.4) evolve finally and live there for all future time under the action of the proposed controller. It is also clear from (2.35) and (2.53) and can also be concluded thatS∗ ⊂V. Considering,
relative degree ℘=2, the invariant setsS′
1,S2,S′
1×S2 and S∗ are shown in Figure 2.2. The points O, A and A′ denote the origin (0,0,0) and the maximum positive and negative bounds on the output tracking error z1. The sets can be made smaller by proper adjustment of the controller parameters yielding a high precision in tracking accuracy.
The proof is not yet complete. Now, it has to be proved that the results derived above hold
Figure 2.2: Visualization of the calculated invariant sets with the proposed scheme, for systems with relative degree
℘= 2
true even in the presence of unknown actuator failure scenarios occurring at unknown time instances T1, T2,· · ·, Th and T0 < T1 < T2 < · · · < Th subject to a finite h ≤ (m−1) and h ∈ Z+. In other words, closed-loop signal boundedness and asymptotic stability of the (℘−1)th order tracking error dynamics have to be proved at each uncertain time instant when failures are encountered.
Let us define the Lyapunov function Vh(t) :Th → R+ for each sub-domain Th := [Th, Th+1) from the total time interval [T0, Tm−1+1]in which all possible failures, which can be compensated, occur.
Vh= 1 2
℘−1
X
i=1
zi2+1 2σ2+ 1
γ Xm j∈BparH
|bj|Kj 2 Γ˜2+
Xm j∈BparH
|bj|Kj
2 θ˜TP−1θ˜ for h= 0,1, . . . ,(m−1) (2.54)
From (2.45) and (2.33), it is evident thatV˙h<0. This in turn proves that the positive definite function Vh is a monotonically decreasing function in Th. Now, the piecewise continuity of Vh at various sub- domains [Th, Th+1) has to be established using the continuity criterion for monotone functions. Since
Vh(Th+1− ) = lim
δt→0−Vh(TH+1+δt) (2.55)
Vh(Th+) = lim
δt→0+Vh(Th+δt) =Vh(Th+) =Vh(Th+) =Vh(Th) (2.56) the function Vh(Th) suffices to be an interval and thereafter by virtue of the Intermediate Value The- orem, the piecewise continuity of Vh(t) is proved. Furthermore, the monotonic decreasing nature of Vh during the time interval Th, h = 0,1, . . . ,(m −1), points out the fact that Th < Th+1 im- plying Vh(Th+) = Vh(Th) ≥ Vh(Th+1− ). The interval in which no fault occurs is T0 during which V0(T0+) = V0(T0) ≥V0(T1−) and since V0(T0) is finite, it is inferred that (z1, z2, . . . , z℘−1) ∈ L2∩ L∞ and ( ˙z1,z˙2, . . . ,z˙℘−1)∈ L∞. Now, considering the first time intervalT1 in which the first failure is en- countered and recovered, V1(T1+) =V0(T1−) +δV1. The termδV1 is finite and arises from the torments in the adaptive parameters at the occurrence of failures, subsequently affecting the related terms in V1 from (2.54). Hence V1(T1+) is bounded in T1 if and only if V0(T1−) defined on T0 is bounded. In similar words, it can be argued that boundedness of Vh(Th+1− )over the setTr ensures the boundedness of Vh+1(Th+1+ ) in Tr+1. Subsequently, the boundedness of all the closed loop signals at each interval between the onset and recovery of the failure, is guaranteed. Finally, this completes the proof.
Remark 2.3. The positive gain parameters ε and τ in the control law (2.20) must be chosen large enough in order to ensure a bounded motion around the sliding surface such that V˙σ < 0 is satisfied outside and on the set V containing the equilibrium point. In addition, the adaptive rates ǫ1 and ǫ2
define the span ofV and should be chosen small enough to guarantee ideal sliding mode dynamics along the sliding surface σ= 0. However, contraction of the setV by small enough choice ofǫ1 andǫ2 results in slow convergence of the adaptive parameters thereby adversely affecting the transient performance.
Hence, a trade-off must be made between the adaptive rate parameters and achievement of ideal sliding motion.
Remark 2.4. The transient performance of the output y =ξ1 in the proposed scheme solely depends on that of the auxiliary sliding manifoldσ. Improvements in theL1 sense can be achieved by increasing the values of parameters τ, ε, γ, P and decreasing the magnitude of ǫ1, ǫ2 judiciously without any significant effect on the convergence of adaptive parameters, guaranteeing close to ideal sliding motion.
Furthermore, the gains β1 and β2 must be chosen in a way, such that, β2/β1 > 1 and β2 being sufficiently large in order to ensure a low value of s2 without any noticeable increase in the control energy. This in turn yields low overshoots in z℘ and hence its impact on the output error z1=ξ1−yr is quite insignificant. Therefore, in contrast to backstepping, the transient performance of the output can be improved without increasing the virtual control gains c1, c2, . . . , c℘ and trajectory initialization even when unanticipated actuator failures are encountered. Furthermore, transient performance of the output in sliding mode depends on the sliding surface coefficients, which are equivalent to the virtual control gains c1, c2, . . . , c℘. Increasing the values of these coefficients can initially result in a good transient response but will result in high overshoots at the time instances of uncertain faults and failures
(L2 bound of the output error increases), which is undesirable. Hence, the superiority of the proposed scheme in offering better transient response is well clarified from the stability analysis leading to the inferences as discussed above.