C. Adaptive sliding mode controller design
III. Simulation Results
The performance index to be minimized for the optimal control is defined as J =
Z ∞ 0
(e(t)TQe(t) +u1(t)TRu1(t))dt (2.20) where Q∈ Rn×n and R∈ R are positive definite weighing matrices.
The optimal control lawu1(t) is given by
u1(t) =−R−1BTP e(t) =−Ke(t) (2.21) where K = R−1BTP and P is a symmetric positive definite matrix which is the solution of the algebraic Riccati equation
ATP+P A+Q−P BR−1BTP = 0 (2.22)
Now for the uncertain system (2.18), an integral sliding surface is designed as s(t) =Ce(t)−Ce(0)−
Z t 0
C[Ae(τ) +Bu1(τ)]dτ = 0 (2.23) where C∈ R1×n andCB is nonsingular. The sliding mode control u2(t) is designed as
u2(t) =−(CB)−1Ψ(t)sign(s(t)) (2.24)
where Ψ(t) is adaptively tuned controller gain [95]. The tuning law used for stabilization and discussed in the previous section is also applied for the tracking problem.
2.2 First order optimal adaptive sliding mode controller for the linear uncertain system
The proposed first order optimal adaptive sliding mode controller (OASMC) is now used for sta- bilization of the above linear uncertain system (2.25).
For the stabilization problem, the weighing matrices Q and R in (2.4) are chosen as follows,
Q=
1 0 0 0 1 0 0 0 1
,R= 1.
The state feedback gain matrix K in (2.5) for the given system is found as K=
0.0990 0.7033
.
In the first order OASMC (2.8), the value of C is chosen as [2 1]. Parameters in the gain adaptation law (2.9) are chosen as Ψ(0) = 0.9, Ξ = 0.5, ǫ= 0.8, ν =.08.
The performance of the proposed OASMC is compared with that of a conventional SMC. The conventional sliding surface θ(t) = 0 is designed as
θ(t) =Cx(t) = [2 1]x(t) = 0 (2.26)
and the sliding mode control law u(t) is chosen as
u(t) =
−(CB)−1CAx(t), θ(t) = 0 ;
−(CB)−1sign(θ(t)), θ(t)6= 0.
(2.27) The performance of the proposed OASMC is also compared with that of the integral SMC proposed by Laghrouche et al. [1] which is described in Appendix A.1. The statesx1 and x2 obtained by using the proposed OASMC, the conventional SMC and the integral SMC proposed by Laghrouche et al. [1]
are shown in Figure 2.1. The control inputs obtained by using these three controllers are shown in Figure 2.2. It is observed in these plots that although the convergence time taken by the states to reach the origin is almost the same for all the controllers, the proposed OASMC spends lesser control input compared to the other two controllers.
Now the output tracking problem for the same system (2.25) is considered. The desired output for tracking is chosen as xd1(t) = 0.5 sin(√
5t). The system can be defined in the error domain as
0 2 4 6 8 10 12 14 16 18 20
−2
−1 0 1 2
Time (sec)
States
data1x1(t) data2x2(t)
(a) Proposed OASMC
0 2 4 6 8 10 12 14 16 18 20
−1 0 1 2
Time (sec)
States
x1(t) x2(t)
(b) Conventional SMC
0 2 4 6 8 10 12 14 16 18 20
−3
−2
−1 0 1 2
Time (sec)
States
x1(t) x2(t)
(c) Integral SMC proposed by Laghrouche et al. [1]
Figure 2.1: States obtained by applying the proposed OASMC, conventional SMC and integral SMC proposed by Laghrouche et al. [1] to stabilize the linear uncertain system
0 2 4 6 8 10 12 14 16 18 20
−14
−10
−6
−2 0 2 6 10
Time (sec)
Controlinput
(a) Proposed OASMC
0 2 4 6 8 10 12 14 16 18 20
−14
−10
−6
−2 0 2 6 10
Time (sec)
Controlinput
(b) Conventional SMC
0 2 4 6 8 10 12 14 16 18 20
−14
−10
−6
−2 0 2 6 10
Time (sec)
Controlinput
(c) Integral SMC proposed by Laghrouche et al. [1]
Figure 2.2: Control input obtained by applying the proposed OASMC, conventional SMC and integral SMC proposed by Laghrouche et al. [1] to stabilize the linear uncertain system
˙ e(t) =
0 1
−5 −0.5
e(t) +
0 1
u(t) +
0 0.5 sint
+
0
−5xd1(t)−0.5x(1)d1(t)−x(2)d1(t)
(2.28)
2.2 First order optimal adaptive sliding mode controller for the linear uncertain system
whereA(t) =e
0
−5xd1(t)−0.5x(1)d1(t)−x(2)d1(t)
For the proposed OASMC, weighing matrices Q and R in (2.20) are chosen as follows:
Q=
1 0 0 0 1 0 0 0 1
,R= 1.
The state feedback gain matrix K in (2.21) for the above controller is found as K =
0.0990 0.7033
.
In the first order OASMC (2.23), the value of C is chosen as [2 1] and parameters in the gain adaptation law (2.9) are chosen as Ψ(0) = 1.25, Ξ = 0.5, ǫ= 0.8, ν =.08.
The proposed OASMC is compared with the conventional SMC whose sliding surface θ(t) = 0 is designed as
θ(t) =Ce(t) = [2 1]e(t) = 0 (2.29)
and the sliding mode control law is chosen as
u(t) =
−(CB)−1(CAe(t) +CA(t)), θ(t) = 0 ;e
−(CB)−1sign(θ(t)), θ6= 0.
(2.30) The performance of the proposed OASMC is next compared with that of the integral SMC proposed by Laghrouche et al. [1] discussed in Appendix A.1. Figure 2.3 shows the statex1obtained by using the proposed OASMC, the conventional SMC and the integral SMC proposed by Laghrouche et al. [1] for the tracking problem. The control inputs obtained by using these controllers are shown in Figure 2.4.
From these figures it is evident that the proposed OASMC uses lesser control input compared to the other two controllers while offering similar convergence speed of the state. Tables 2.1 and 2.2 compare the control energies by computing the second norm of the control input till 20 sec for both the stabilization and tracking problem applications. It is clear from Tables 2.1 - 2.2 that the proposed OASMC is able to reduce the control effort for comparable performance standard.
0 2 4 6 8 10 12 14 16 18 20
−1 0 1 2
Time (Sec)
Output
Desired output Actual output
(a) Proposed OASMC
0 2 4 6 8 10 12 14 16 18 20
−1 0 1 2
Time (sec)
Output
Desired output Actual output
(b) Conventional SMC
0 2 4 6 8 10 12 14 16 18 20
−1 0 1 2
Time (sec)
Output
Actual output Desired output
(c) Integral SMC proposed by Laghrouche et al. [1]
Figure 2.3: State x1(t) obtained by applying the proposed OASMC, conventional SMC and integral SMC proposed by Laghrouche et al. [1] for tracking the linear uncertain system
0 2 4 6 8 10 12 14 16 18 20
−20
−15
−10
−5 0 5 10 15
Time (Sec)
Controlinput
(a) Proposed OASMC
0 2 4 6 8 10 12 14 16 18 20
−20
−15
−10
−5 0 5 10 15
Time (sec)
Controlinput
(b) Conventional SMC
0 2 4 6 8 10 12 14 16 18 20
−20
−15
−10
−5 0 5 10 15
Time (sec)
Controlinput
(c) Integral SMC proposed by Laghrouche et al. [1]
Figure 2.4: Control input obtained by applying the proposed OASMC, conventional SMC and integral SMC proposed by Laghrouche et al. [1] for tracking the linear uncertain system
2.3 Disturbance observer based first order optimal sliding mode controller
Table 2.1: Comparison of control energy of conventional SMC, integral SMC proposed by Laghrouche et al. [1]
and the proposed OASMC for the stabilization problem
Method Control Energy
Conventional SMC 41.81
Integral SMC proposed by Laghrouche et al. [1] 22.59
Proposed OASMC 10.41
Table 2.2: Comparison of control energy of conventional SMC, integral SMC proposed by Laghrouche et al. [1]
and the proposed OASMC for the tracking problem
Method Control Energy
Conventional SMC 33.60
Integral SMC proposed by Laghrouche et al. [1] 25.56
Proposed OASMC 14.02
2.3 Disturbance observer based first order optimal sliding mode controller
A first order optimal sliding mode controller is proposed for the linear system affected by mis- matched uncertainty. The optimal controller is designed for the nominal nonlinear system using the LQR technique as discussed in the previous section. As the sliding mode controller (SMC) cannot tackle the mismatched uncertainty, it is estimated by using a disturbance observer. A first order sliding mode methodology is proposed based on an integral sliding surface.