Adaptive Integral Sliding Mode Controller
3.3 Design of adaptive integral sliding mode controller
where 1≤i≤r−1 and [z1, z2, ..., zr]T = [σ(x),σ(x), ..., σ˙ (r−1)(x)]T
Assumption 3.3.Matrices a(x) andb(x) consist of the nominal parts (¯a(x),¯b(x)) which are known apriori and uncertain parts (∆a(x), ∆b(x)) which are bounded and unknown [62].
Thus the following can be written,
a(x) = ¯a(x) + ∆a(x)
b(x) = ¯b(x) + ∆b(x) (3.4)
σ(r)(x) = (¯a+ ∆a)(x) + (¯b+ ∆b)(x)u
= a(x) + ¯¯ b(x)u+ ∆F(x, t) (3.5) where ∆F(x, t) = ∆a(x) + ∆b(x)u includes all the uncertain parameters and external disturbance.
Using (3.4) and (3.5) withz as the state variable, the r-th order sliding mode control for the system (3.1) can be written as,
˙
zi = zi+1
˙
zr = ¯a(z) + ¯b(z)u+ ∆F(z, t) (3.6) In the regular form, the above can be written as,
˙
z= ¯A(z) + ¯B(z)u+ ∆F(z, t) (3.7)
wherez= [z1 z2...zi..zr]T, and ¯A(z), ¯B(z) are matrices with proper dimension. The uncertainties in the system due to modeling error and parameter variation are denoted by ∆F(z, t) which is assumed to be differentiable with respect to time. In this problem, the uncertainties in the system (3.6) are assumed to meet the matching conditions. Then ∆F(z, t)∈span{B(z)¯ } [30] meaning that ∆F(z, t) is a matched uncertainty.
3.3 Design of adaptive integral sliding mode controller
The design procedure for the overall control signal is carried out in two parts, design of the nominal control wnom and then design of the overall control law u. At first, the nominal control lawwnom is designed that guarantees finite time stabilization of the chain of integrators in absence of uncertainties.
Then the reaching law based overall control law is designed to reject the uncertainties and maintain the sliding mode.
3.3.1 Finite time stabilization of an integrator chain system
Let us consider the nominal system which is represented by the single input single output (SISO) integrator chain as described below,
˙
z1 = z2
˙
z2 = z3
.
˙
zr = wnom (3.8)
The control objective is to drive the states of (3.8) toz= 0 at the fixed finite time [63].
T heorem 3.1. Let k1, k2, ...kn>0 be such that the polynomial ϕ(λ) =λn+knλn−1+...+k2λ+k1
is Hurwitz. For the system (3.8), there exists a value ε∈ (0,1) such that for every αi ∈ (1−ε,1), i= 1,2, ...n, the origin is a globally stable equilibrium in finite time under the feedback
wnom(z) =−k1sign(z1)|z1|α1 −k2sign(z2)|z2|α2−...−knsign(zn)|zn|αn (3.9) where α1, ...αn satisfy
αi−1 = 2ααiαi+1
i+1−αi, i= 2, ..., n withαn+1= 1 [63].
3.3.2 Design of integral sliding mode controller
However, when the system is perturbed or uncertain, the finite time stabilization is not ensured [63]. In this section a reaching law based discontinuous control law is developed which rejects the uncertainties of the system and ensures that the control objectives are fulfilled [62].
Let us consider an integral sliding surface,
s(z) =zn−zn(0)−
∫
wnom(z)dt (3.10)
The initial condition of the system is defined by zn(0). The nominal controlwnom ensures the conver- gence of the chain of integrators in finite time as given in Theorem 3.1.
3.3 Design of adaptive integral sliding mode controller
By taking the time derivative of (3.10), the following is obtained,
˙
s(z) = z˙n−wnom
= a(z) + ¯¯ b(z)u+ ∆F(z, t)−wnom (3.11) Using (3.11) and the constant rate reaching law ˙s(z) = −Gsign(s(z)) [30] such that it satisfies the reachability condition s(z) ˙s(z)≤ −η|s(z)|where η being a positive constant yields,
−Gsign(s(z)) = ¯a(z) + ¯b(z)u−wnom (3.12) Here Gis the switching gain. The control law described above ensures finite time stabilization of the system states and also rejects the uncertainties if G >|∆F(z, t)|. Hence the overall control law can be obtained as [62],
u= ¯b(z)−1{−a(z) +¯ wnom−Gsign(s(z))} (3.13) However, the high frequency chattering is always present in the control signal.
In order to remove the undesired chattering in the control input, an adaptive integral chattering free sliding mode controller is developed. In the proposed controller, the time derivative of the control input, ˙uwould be designed to act on the higher order derivatives of the sliding variable [64, 65]. Hence instead of the actual control u, the time derivative of the control, ˙u would be used as the control input. The new controlv = ˙uwould be designed as a discontinuous signal, but its integral (the actual controlu) would be continuous thereby eliminating the high frequency chattering.
Now taking the first order time derivative of (3.11) yields,
¨
s(z) = ¨zn−w˙nom (3.14)
Using (3.2), (3.14) can be written as,
¨
s(z) = d
dt(¯a(z) + ¯b(z)u)−w˙nom
= a(z) + ˙¯˙¯ b(z)u+ ¯b(z) ˙u−w˙nom
= a(z) + ˙¯˙¯ b(z)u+ ¯b(z) ˙u−w˙nom+ ∆ ˙F(z, t) (3.15)
Assuming y1=s(z) and y2= ˙s(z), the system dynamics can be written as [66, 67],
˙
y1 = y2
˙
y2 = Φ[z, u] + Ψ[z]v (3.16)
where v= ˙uand Φ[z, u] collects all the uncertain terms not involving ˙u, i.e. Φ[z, u] = ˙¯a(z) + ˙¯b(z)u−
˙
wnom+ ∆ ˙F(z, t) and Ψ[z] = ¯b(z). Thus the system (3.16) becomes a chain of integrators controlled by the inputv. So a sliding mode controller for the above system can be designed to keep the system trajectories on the sliding manifold by using the control input v. To design an SMC for the system (3.16), the sliding function is considered as,
σ =y2+κy1 (3.17)
where κ is a positive constant. The derivative of (3.17) is obtained as,
˙
σ = ˙y2+κy˙1 (3.18)
Using (3.18) and (3.15) yields,
˙
σ= ˙¯a(z) + ˙¯b(z)u+ ¯b(z) ˙u−w˙nom+ ∆ ˙F(z, t) +κ( ˙zn−wnom) (3.19) Using the µreaching law [68] yields,
˙
σ =−ρsign(σ) (3.20)
where ρ >|∆ ˙F(z, t)|to satisfy the reaching law condition σσ˙ ≤ −η|σ| whereη is a positive constant [56]. Using (3.19) and (3.20), the control law is obtained as,
˙
u=−¯b(z)−1{a(z) + ˙¯˙¯ b(z)u−w˙nom+κ( ˙zn−wnom) +ρsign(σ)} (3.21) 3.3.3 Design of adaptive integral chattering free sliding mode controller
In practice, the upper bound of the system uncertainty is often unknown in advance and hence the error term |∆ ˙F(z, t)|is difficult to find. So an adaptive tuning law is proposed to estimate ρ. Then the control law (3.21) can be written as
˙
u=−¯b(z)−1{a(z) + ˙¯˙¯ b(z)u−w˙nom+κ( ˙zn−wnom) + ˆT sign(σ)} (3.22)
3.3 Design of adaptive integral sliding mode controller
where ˆT estimates the value of ρ. Defining the adaptation error as ˜T = ˆT −T, the parameter ˆT will be estimated by using the adaptation law [40] [69, 70] as given below,
T˙ˆ=ν|σ| (3.23)
where ν is a positive constant. A Lyapunov function V is selected as V = 12σ2+ 12γT˜2 whose time derivative is as follows,
V˙ = σσ˙ +γT˜T˙˜
Using (3.19) yields,
V˙ = σ[ ˙¯a(z) + ˙¯b(z)u+ ¯b(z) ˙u−w˙nom+ ∆ ˙F(z, t) +κ( ˙zn−wnom)] +γ( ˆT −T)T˙ˆ Using (3.22) and (3.23) yields,
V˙ = σ[∆ ˙F(z, t)−T sign(σ)] +ˆ γ( ˆT −T)ν|σ| The above equation can be written as
V˙ ≤ |∆ ˙F(z, t)||σ| −Tˆ|σ|+T|σ| −T|σ|+γ( ˆT −T)ν|σ|
≤(|∆ ˙F(z, t)| −T)|σ| −( ˆT−T)|σ|+γ( ˆT−T)ν|σ|
≤ −(−|∆ ˙F(z, t)|+T)|σ| −( ˆT −T)(−γν|σ|+|σ|)
≤ −βσ√ 2|σ|/√
2−βν√
2γ( ˆT−T)/√ 2γ
whereβσ = (T− |∆ ˙F(z, t)|) and βν = (|σ| −γν|σ|) So, V˙ ≤ −min{βσ√
2, βν√
2/γ}(|σ|/√
2 + ˜T√ γ/2)
≤ −βV1/2 (3.24)
whereβ =min{βσ√ 2, βν√
2/γ}withβ >0. The above inequality holds ifT˙ˆ=ν|σ|,βσ >0, βν >
0,T >|∆ ˙F(z, t)|andγ < 1ν. Therefore, finite time convergence to a domainσ= 0 is guaranteed from any initial condition [40, 62].
Remark3.2.Practically,|σ|cannot become exactly zero in finite time and thus the adaptive param- eter T˙ˆ may increase boundlessly [40]. A simple way of overcoming this disadvantage is to modify the adaptive tuning law (3.23) by using the dead zone technique [30, 40] as
T˙ˆ =
{ ν|σ|, |σ| ≥ϵ
0, |σ|< ϵ (3.25)
where ϵis a small positive constant.
As is evident from (3.22), ˙u is discontinuous but integration of ˙u yields a continuous control law u.
Hence the undesired high frequency chattering of the control signal is alleviated. Thus the above adaptive integral sliding mode control method offers two main advantages. Firstly, the knowledge
about the upper bound of the system uncertainties is not required. Secondly, the chattering in the control input is eliminated.