Simulation Study

3.3 Adaptive Multiple Model Fault Tolerant Control (AMMFTC) for Uncertain Multi-

3.3.4 Simulation Study

With the auxiliary error dynamics in (3.206), what remains is to prove the L2-integrability of the auxiliary variable ϑ. This would essentially lead us to proving z(t) ∈ L2[0,∞). In view of this objective, we compute the following.

d dt

1 2kϑk²

≤ϑ^TA¯_zϑ+ϑMcξ˜_s+ϑ^TB_˙ˆ

θ2

θ˙ˆ_2s

≤ −Ckϑk²− Xq i=1

℘i

X

ri=2

g_i,ri

∂α_i,ri−1

∂θˆ_2s

2

ϑ²_i,ri+ Xq

i=1

℘i

X

ri=2

∂α_i,ri−1

∂θˆ_2s

θ˙ˆ_2sϑ_i,ri+ϑ^TMcξ˜s

≤ −Ckϑk²+kθ˙ˆ_2sk²

4¯g +ϑ^TMcξ˜_s (3.209)

Applying Peter-Paul inequality to the third term in (3.209) with the factorC results in ϑ^TMcξ˜_s≤Ckϑk²

2 + 1

2CkMck^2Fkξ˜_sk² (3.210) Substituting RHS of the inequality (3.210) in (3.209) yields

d dt

1 2kϑk²

≤ −C

2kϑk²+kθ˙ˆ_2sk²

4¯g +kMck²Fkξ˜sk²

2C (3.211)

Since, all the closed loop signals are bounded, it is justified to infer that M, M˙ , A¯_z and (A0 − γΦ^TΦP) are bounded. Thus, the matrix Mc is bounded. The following integral if proved to be finite would imply the energy integrability ofϑ.

Z∞

0

kϑk²dt≤ kϑ(0)k² 2C + 1

4¯gC Z∞

0

kθ˙ˆ_2sk²dt+ kMck²F

2C2

Z∞

0

kξ˜_sk²dt (3.212)

Owing to the following properties, that is, θ˙ˆ_2s, ξ˜s ∈ L2[0,∞), the above integral (3.212) is finite.

Thus, ϑ∈ L2[0,∞) follows. With this result, the tracking error z(t)∈ L2[0,∞) is concluded from its definition asz=ϑ+Mξ˜_s.

Since. z(t)∈ L2[0,∞)∩ L_∞[0,∞) andz(t)˙ ∈ L_∞[0,∞), invoking Barbalat’s Lemma [122], the asymp- totic stability ofz(t) = 0 is proved. The proof is complete.

with AVS eliminates body roll and pitch variation during turning at medium/high speeds, cornering, accelerating, and braking. Assuming that the AVS has actuation redundancy , the actuators may be partially effective under unanticipated situations and also some of the actuators may totally fail. If left unattended with no corrective action, such failures unknown in time and magnitude would culminate to severe accidents. To deal with these situations, the AVS must be equipped with an FTC module.

Hence, the suitability of a mass-spring-damper system as a benchmark example to illustrate the prolific features of the proposed adaptive FTC methodology is justified.

Prior to control design, dynamical modeling of the system is necessary. The equations of motion for the concerned mass-spring-damper system are expressed as vector differential equations as

M(y,y) ¨˙ y+F_B(y,y) ˙˙ y+F_K(y) =u(t) +d(t). (3.213)

where FB :=

"

4.2 + 0.05 ˙y₁ −2.2 + 0.3 ˙y₁−0.15 ˙y₂ 2 + 0.2 ˙y₁ 0

#

and FK :=

"

k₁₀y₁+k₁₁y₁³

k₂₀(y₂−y₁) +k₂₁(y₂−y₁)³

#

represent the matrices related to friction forces and the spring forces, respectively. The control inputs are defined by u= [u₁, u₂]^T and the unknown exogenous disturbance is given by d= [d₁, d₂]^T. The matrix M(y,y) :=˙ diag{m1, m2} ∈R^2×2 defines the inertia of the system. Further, the parameters of the system are totally unknown to the control designer. The output vector of the system is denoted by y= [y₁, y₂]^T. For simulations, the values of unknown parameters are assumed to bem₁ = 0.25kg, m2= 0.2kg,k10= 1N/m, k11= 0.1N/m,k20= 2N/m andk21= 0.12N/m. As had been mentioned earlier, the system considered exhibits strong output coupling and also captures the nonlinearities of the spring as well as damping through the nonlinear terms. The system has two outputs driven by two inputs, hence the case of total actuator failure has not been considered in this example, to satisfy the design assumptions. We show the effectiveness of the proposed FTC approach in improving the start up and post fault transient/steady state performances in the event of partial loss of effectiveness of each of the actuators. Further, such improvements are obtained without any substantial increment in the input usage when compared to single model adaptive control and modular backstepping control [6].

K

₁

F

_K1

B

₁

M

₁

F

_B1

K

₂

F

_K2

B

₂

F

_B2

u

₁

u

₂

y

₁

y

₂

M

₂

Figure 3.13: Schematic diagram of a coupled mass-spring-damper system

In simulations, the actuator failure occurs as per the following piecewise definition.

u₁(t) =

( 0.5u_H1, t≥30s

u_H1, otherwise (3.214)

u₂(t) =

( 0.5uH2, t≥60s

u_H2, otherwise (3.215)

The reference trajectories to be tracked by the system outputsy1andy2are given byyr,1= 0.2 sin(0.5t) andy_r,2= 0.5 sin(0.5t). The relative degree of the system with respect to each of the outputs is found to be ℘₁ = ℘₂ = 2. In order to translate the proposed control design procedure, let us first choose the state variables as ξ_1,1 = y₁, ξ_2,1 = y₂, ξ_1,2 = ˙y₁ and ξ_2,2 = ˙y₂ such that ξ₁ := [ξ_1,1, ξ_2,1]^T and ξ₂ := [ξ_1,2, ξ_2,2]^T. With this choice of states, the state vector ξ := [ξ₁, ξ₂] and the partial actuator failure model, the entire dynamical system is rewritten as

ξ˙₁ =ξ₂ (3.216)

ξ˙₂ =ϕ^Tθ₂^∗+ X2 j=1

θ_1,j^∗ N_ju_H. (3.217)

Where ϕ^T := [diag{ϕ^T_1,2, ϕ^T_2,2}]^T and u_H = [u_H1, u_H2]^T. The regressor matrices are defined as ϕ_1,2 := [−ξ_1,1, −ξ³_1,1, −ξ_1,2, −ξ_1,2² , (ξ_2,2−ξ_1,2), (ξ_2,2−ξ_1,2)²]^T and ϕ_2,2 := [−(ξ_2,1−ξ_1,1), −(ξ_2,1− ξ_1,1)³, −ξ_1,2, −ξ_1,2² ]^T. Further, the estimation model for the considered nonlinear dynamical system (3.213) is framed as

ξ˙=







 ξ_1,2 ξ_2,2 0 0







+









0 0_1×6 0 0_1×4 0 0_1×6 0 0_1×4 uH1 ϕ^T_1,2 0 01×4

0_1×9 0_1×6 u_H2 ϕ^T_2,2









| {z }

Φ^T

θ^∗. (3.218)

Hereinθ^∗ := [θ^∗_1,1, θ^∗_2,1, . . . , θ^∗_2,6, θ_1,2, θ_2,7, . . . , θ_2,10]^T. The adaptive controller is thereafter developed in accordance with (3.160). Whereas the unknown parameter vectorθ^∗is estimated using (3.162)-(3.163) followed by the second layer adaptation. The total number of unknown parameters is 12. Hence, as per the proposed design, the number of identification models to be considered isN = 13to satisfy the conditions of a convex hull. In simulations, the initial conditions of the system areξ(0) = [0,0,0,0]^T. The controller gains are selected as, c_1,1 = 15, c_1,2 = c_2,1 = c_2,2 = 5 and the damping factors as,

¯

c_1,2 = ¯c_2,2 = 5. The matrices A_µ0 = [diag{−14,−14,−16,−16}] and γ = 2are chosen to be the same for allµidentification models; however with different initial parameter estimates. Lastly, the adaptive rate matrix for the parameter update law is considered to beΓ_µ= 20I₁₂.

The strongly coupled mass-spring-damper system with unknown parametric uncertainties was simu- lated in MATLAB. To illustrate the robustness of the proposed AMMFTC scheme to actuator failures, the system is deliberately subjected to abrupt and unknown actuator faults consistent with equations

Time [s]

0 20 40 60 80 100

Trackingerror(y1−y1r)

×10^-3

-5 -3 -1 1 3 5

Single Model Multiple Model

(a)

Time [s]

0 20 40 60 80 100

Trackingerror(y2−y2r)

-0.04 -0.02 0 0.02 0.04

Single Model Multiple Model

(b)

Figure 3.14: (a) Tracking error comparison in displacement output y1;(b) Tracking error comparison in displacement outputy2; using the proposed adaptive FTC using multiple models and under a single model adaptive FTC strategy.

(3.214)-(4.174). The simulation results obtained are depicted in Figures 3.16-3.15. As claimed in our theoretical developments, stability of the closed loop system is evident from the time evolution of the outputs y₁,y₂ and their respective velocity profiles y˙₁ and y˙₂ in Figure 3.15 for all time even inspite of occurrence of unknown faults at t= 30s andt= 60s. As can be observed from Figure 3.15(c), the control inputsu₁ andu₂ are also bounded. A comparison of output tracking errors and control inputs obtained using the proposed AMM(Adaptive Multiple Model) FTC strategy and single identification model based adaptive control are provided in Figure 3.16 and Figure 3.15(c)-(d) to further substantiate our theoretical claims.

Strong output and input interactions betweeny₁−subsystem andy₂−subsystem allow the propaga-

Table 3.3: Tabular comparison of output and input performances under proposed AMMFTC and adaptive FTC using a single identification model

Control Schemes

Output Performance metrics Input Performance metrics ITAE RMSE Control Energy Total Variation

y₁ y₂ y₁ y₂ u₁ u₂ u₁ u₂

Multiple Models 1.1727 3.5234 0.00036 0.0015 285.26 637.09 14.85 29.04 Single Model 13.01 116.66 0.0028 0.0231 285.26 573.73 14.20 27.04 tion of adverse effects of actuator faults occurring in any of the subsystems to the other. These are widely known as cross coupling effects. In addition to the accommodation of uncertain and unknown actuator faults, these cross coupling effects should also be minimized which otherwise may jeopardize the stability of the system. In this simulation study, the system has been so considered that fault induced perturbations and subsystem interactions are categorically placed under the same class of

Time [s]

0 20 40 60 80 100

Displacementy1=x1,1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

y₁ y_1r

(a)

Time [s]

0 20 40 60 80 100

Displacementy2=x2,1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y2

y_2r

(b)

Time [s]

0 20 40 60 80 100

Control inputs

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

u1

u2

(c)

Time [s]

0 20 40 60 80 100

Control inputs (Single Model)

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

u1

u2

(d)

Figure 3.15: (a) Time evolution ofy1(t)under the proposed adaptive FTC using multiple models;(b) Time evolution of y2(t) under the proposed adaptive FTC using multiple models;(c) Control inputsu1 andu2 under the proposed FTC scheme using multiple models; (d) Control inputsu1 andu2 under adaptive FTC scheme using a single model.

uncertainties affecting system performance. Therefore, improvement of transient performance in terms of L∞ bounds at start up and post fault scenarios in both the outputs would definitely reflect better mitigation of coupling effects. The tracking error comparison in Figure 3.16 demonstrate superior transient and steady state performance as claimed in theory over single model adaptive FTC; besides achieving substantial alleviation of coupling effects. The output tracking error performance in both the outputs, that is, displacements of massesm₁ andm₂ denoted byy₁ andy₂is indeed promising. On the contrary, as evident from Figures 3.16-3.15, the single model adaptive FTC suffers from a fairly unsatisfactory output tracking while its multiple model counterpart shows remarkable performance without any considerable increase in the control usage. Furthermore, the output tracking performance and the extent of control usage or rather the input energy spent under the proposed control and its single model equivalent, are quantified in terms of relevant and suitable performance metrics in Table 3.3 . Thereafter, as an added testament to our claims, a tabular comparison of such calculated input and output performances are drawn. In fact, as anticipated, the comparison reverberates the prolific features of the proposed adaptive FTC methodology.