3.3 Adaptive Multiple Model Fault Tolerant Control (AMMFTC) for Uncertain Multi-
3.3.1 Problem Formulation
Let us consider a MIMO nonlinear system as follows, X
p:
˙
x=f0(x) + Pp i=1
κ∗ifi(x) + Pm
j=1
gj(x)uj y=h(x)
(3.133)
The state vector of the system (3.133) is denoted by x := [x1, x2, . . . , xn]T ∈ Rn. The vector y :=
[y1, y2, . . . , yq]T ∈ Rq represents the outputs while h(x) are defined as known smooth functions.
Further,f0,fi,gj :Rn→Rnrepresent the vectors of known smooth nonlinear functions fori= 1, . . . , n and j = 1, . . . , m. The terms uj defines the control inputs, collectively denoted by the vector u :=
[u1, u2, . . . , um]T and κi, i= 1, . . . , pare unknown constants.
Considering the system dynamics in (3.133), the following assumptions are necessary.
Assumption 3.7. The system (3.133) has well defined relative degrees ℘ = {℘1, ℘2, . . . , ℘q}, 1 ≤
℘i ≤ n, for i = 1, 2, . . . , n. Therefore the following equalities hold for j = 1, m, i = 1, q and k= 0, ℘i−2.
LgjLkf0hi(x) = 0, LgjL℘fi−1
0 hi(x)6= 0
The operator Lf(·) defines the Lie derivative of the argument in the direction of f(x). Further, the nonlinear functions fi satisfy the conditions given below [102] for k= 1, p, ri= 1, ℘i−1 andi= 1, q, Dx(LfkLrfi−1
0 hi(x))∈span{Dx(h1(x)), . . . , Dx(L℘f1−℘i+ri−1
0 h1(x)), Dx(h2(x)), . . . , Dx(L℘f2−℘i+ri−1
0 h2(x)), Dx(hj(x)), . . . , Dx(L℘fj−℘i+ri−1
0 hj(x)), Dx(hq(x)), . . . , Dx(L℘fq−℘i+ri−1
0 hq(x))},
where, the operator Dx(·) denotes the partial differentiation of a function with respect to x ∈ Rn resulting in a gradient vector Dx(·) :=
"
∂
∂x1, ∂
∂x2, . . . , ∂
∂xn
#T (·).
In accordance with FTC literature, actuator fault in a system manifests itself in either of the forms as under, (i) partial loss of effectiveness (PLOE);(ii) lock-in-place (LIP);(iii) float failure. However in this work, LIP and float failures have been considered. Such types of actuator faults can be charac- terized by a unified mathematical model with a piecewise defined input-output relationship as given
below [100].
uj =
( KjuHj + ¯uF j, uHj,
∀ t≥tF j
∀ t < tF j j = 1, m (3.134) The term uHj andu¯F j denote the controller output fed to thejth actuator and the value at which the jthactuator is stuck under LIP failure, respectively. Float failure is the condition when thejthactuator fails with u¯F j=0. Apart from the additive consequences of actuator failures, multiplicative actuator faults are modeled using the factor Kj. The actuator failure value u¯F j, the actuation effectiveness index Kj, the failure time instanttF j and the actuator indexj are all unknown. This means that the actuator failures are unknown in time, pattern as well as magnitude. The actuator index set is given by {j1, j2, . . . , jm}. Unlike the actuator fault modeling in multi-input single-output (MISO) systems wherein the actuators were assumed to have same/similar physical characteristics, MIMO systems have multiple inputs with actuators not necessarily reflecting the same physical behavior/features and functioning. Therefore, to design fault tolerant controllers for a MIMO nonlinear system, an actuator grouping scheme G := {G1, G2, . . . ,Gq} is required [32]. Herein, each set Gk ⊂ {j1, j2, . . . , jm} represents the set of actuators with similar operational characteristics and such a classification is done on the basis of ℘ := {℘1, ℘2, . . . , ℘q}. Further to effectively exploit the actuator redundancy, a proportional actuation scheme is utilized similar to [32]. Therefore, considering such a grouping and actuation scheme G, the control input to be fed to each actuator included in the set Gk is given by,
uHj =bjkvk ∀uj ∈Gk, j = 1, m&k= 1, q (3.135) where,bjkdefines a certain design function associated with each actuator groupGk. Further, the choice of such design functions satisfy the relationbjk=bj if thejthactuator belong toGkunder the grouping schemeG. Thereafter, to facilitate control design, under assumption 3.7 and (3.133)-(3.135), there ex- ists a diffeomorphismT(x) := [ξ,η]T = [ξ1,1, ξ1,2, . . . , ξ1,℘1, . . . , ξq,1, ξq,2, . . . , ξq,℘q,η] = [Tc(x),Tz(x)]T such that
Tc(x) := [h1(x), Lf0h1(x), . . . , L℘f01−1h1(x), . . . , hq(x), Lf0hq(x), . . . , L℘f0q−1hq(x)]∈R℘1+℘2+...+℘q and Tz ∈ Rℓ transforms the system (3.133) to the following form with a controllable part P
c and internal dynamics P
z, X
c1
:
ξ˙1,r1 =ξ1,r1+1+ϕT1,r1κ∗
ξ˙1,℘1 =a1(ξ, η) +ϕT1,℘1κ∗+ P
j∈{j1,...,jd}
β1,j(ξ, η)¯uF j+ Pq k=1
β¯1,k(ξ, η)vk (3.136)
...
X
cq
:
ξ˙q,rq =ξq,rq+1+ϕTq,rqκ∗
ξ˙q,℘q =aq(ξ, η) +ϕTq,℘qκ∗+ P
j∈{j1,...,jd}
βq,j(ξ, η)¯uF j + Pq k=1
β¯q,k(ξ, η)vk (3.137)
X
z : ˙η=Λ0(ξ,η) +ΛT1(ξ,η)κ∗+ΛT2(ξ,η)¯u (3.138) y= [ξ1,1, ξ2,1, ξ3,1, . . . , ξi,1, . . . , ξq,1]T (3.139) where ϕi,ri(ξ,η) := [Lrfi
1hi(x), . . . , Lrfpihi(x)]T,ai(ξ,η) :=L℘fi
0hi(x), βi,j(ξ,η) :=LgjL℘fi−1
0 hi(x), βi(ξ,η) := [{LgjL℘f0i−1hi(x)}mj=1], β¯i,k:= P
j6∈{j1,...,jd}∩Gk
bjkKjβi,j(ξ,η) ∀k= 1, . . . , q
Λ0(ξ,η) := ∂∂xTzf0(x),ΛT1(ξ,η) := ∂Tz
∂x[f1(x), f2(x), . . . , fp(x)] and ΛT2(ξ,η) := ∂Tz
∂x[g1(x), g2(x), . . . , gm(x)]for i= 1, q, ri = 1, ℘i−1 and j= 1, m. At this point, we are well equipped to state the following additional assumptions.
Assumption 3.8. The zero dynamics (3.138) of the system (3.133), given by P
z
:= Λ0(ξ,η) + ΛT1(ξ,η)κ∗ +ΛT2(ξ,η)¯u is input-to-state-stable (ISS) for each failure pattern with ξ as the input.
With a supposition that there areM possible actuator failure patterns, the equivalent actuation matrix given by the argument of det operator in (3.140), for the entire system corresponding to an actuator failure pattern Ξ ∈ Ξ, = 1, M under the grouping scheme G should be non-singular ∀x∈ Rn, that is,
det
P
j6∈{j1,...,jd}∩G1
bj1KjLgjL℘f1−1
0 h1(x) . . . P
j6∈{j1,...,jd}∩Gq
bjqKjLgjL℘f1−1
0 h1(x) P
j6∈{j1,...,jd}∩G1
bj1KjLgjL℘f2−1
0 h2(x) . . . P
j6∈{j1,...,jd}∩Gq
bjqKjLgjL℘f2−1
0 h2(x)
... . .. ...
P
j6∈{j1,...,jd}∩G1
bj1KjLgjL℘fq−1
0 hq(x) . . . P
j6∈{j1,...,jd}∩Gq
bjqKjLgjL℘fq−1
0 hq(x)
G(Ξ)
6
= 0.
(3.140) Herein, Ξ defines a set with cardinality M = mq
, whose elements are m ×m diagonal matrices representing all possible actuator failure topologies. Further, the diagonal elements of each of such matrices inΞare either zero or unity depending on the state of health of the corresponding actuator. It is to be mentioned at this point in time that all of the M actuator failure patterns are not compensable and hence it is required to classify failures as compensable and non-compensable ones. The basis of such a classification are the actuation matrices corresponding to all possible failure topologies. The non- singularity of the actuation matrix for an actuator failure pattern is the deciding factor which defines an actuator failure pattern to be compensable or not. In other words, this classification criterion, if satisfied, guarantees the existence of an adaptive fault tolerant controller which can ensure stability of the closed-loop system in the event of actuator failures.
Assumption 3.9. The nonlinear dynamical system in (3.133) is so designed that the desired output performance can only be achieved in the event of atmost d ≤ (m−q) actuator failures unknown in time, pattern and magnitude. Further, under the grouping scheme G, since the actuators are grouped on the basis of their similarity in structural characteristics, atleast one actuator in each of the groups Gk should be operational corresponding to an actuator failure pattern.
Assumption 3.10. The respective reference trajectories yr(t) := [yr,1, yr,2, . . . , yr,q]T ∈ Rq to be tracked by the system outputs are known, piecewise continuous and bounded. Further, ˙yr, ¨yr, . . . ,y(℘)r ∈ Rq are also piecewise continuous, bounded and belong to a known compact set.
Having stated all the necessary definitions and assumptions, we are now well equipped to state the FTC design problem. Considering the nonlinear uncertain MIMO system (3.133) and the assumptions 3.7-3.10, the control problem and its objectives can be framed as follows:
• To design the control inputs vk corresponding to each actuator group Gk for k= 1, q such that the closed loop system is stable and the output vector y(t) tracks the desired reference vector yr(t) in the event of atmost d ≤(m−q) unknown actuator failures ∀ t ∈ [0, ∞). The design assumes no prior knowledge of the magnitude, pattern and timetjF of the occurrence of actuator failures/faults.
• To ensure an improved start-up and post failure output transient performance without any sig- nificant expense of control energy
• Asymptotic stability of output tracking error vector, i.e. lim
t→∞y(t)−yr(t) = 0 under no failure and past fault scenarios.
• Guaranteeing robustness towards parametric uncertainties, external perturbations and conse- quences of strong unknown cross coupling arising from subsystem interactions through estimation and subsequent compensation instead of merely alleviating their adverse effects.
3.3.2 Adaptive Multiple Model Fault Tolerant Control (AMMFTC) Design