Adaptive Terminal Sliding Mode Controller
5.2 Design of chattering free adaptive terminal sliding mode con- trollertroller
Let us consider a class of nonlinear system
˙
x=f(x) + ∆f(x) +d(t) +Bu (5.1)
wherex= [x1 x2 x3....xn]T ∈Rnis the state vector. Furthermore, ∆f(x)∈Rn is an uncertain term representing the unmodelled dynamics or structural variation of the system (5.1) and d(t)∈Rn is an external disturbance. Moreover, u∈Rm is the input andB is a known matrix of order n×m. The uncertainties of the system (5.1) are assumed to be bounded and matched such that ∆f(x) and d(t)
∈ spanB. The control objective is to track a given reference signal xd in finite time from any initial state.
Let the desired state vector be xd= [x1d x2d x3d....xnd]T. The tracking error is defined as, e=x−xd = [(x1−x1d) (x2−x2d) ...(xn−xnd)]T
= [e1, e2, ...en]T. (5.2)
The goal is to design a chattering free adaptive terminal sliding mode controller for a given target xd
such that the resulting tracking error satisfies
tlim→∞||e||= lim
t→∞||x−xd|| →0 (5.3)
where || · ||denotes the Euclidean norm of a vector.
The controller is designed in two steps. At first, a linear sliding surface is defined and then using the sliding surface, a terminal sliding manifold is obtained so that the derivative of the control input occurs at the first derivative of the terminal sliding manifold. The actual control input is obtained by integrating the derivative of the control signal which contains the discontinuous function and thus eliminates the chattering [27, 65, 66]. The uncertainty is estimated by using an adaptive tuning law.
A set of sliding surfaces is defined in the error space passing through the origin to represent a sliding
5.2 Design of chattering free adaptive terminal sliding mode controller
manifold as fo1lows:
s = [s1, s2, ..., sm]T =cTe= [c1 c2 ...ci...cm]Te
=
c1nen+c1(n−1)en−1+...+c11e1
c2nen+c2(n−1)en−1+...+c21e1
.
cinen+ci(n−1)en−1+...+ci1e1 .
cmnen+cm(n−1)en−1+...+cm1e1
(5.4)
and ci = [cin ci(n−1)....ci1] be such that all roots of the polynomial
ϕ(λ(ei)) =cinλn−1+ci(n−1)λn−2+...+ci1λ (5.5) are in the open left half-plane [100], i= 1,2, .., m. The choice ofcdetermines the convergence rate to the sliding surface. Let us consider (5.4), where e=x−xd. The first time derivative of (5.4) yields
˙
s = cTe˙
= cT( ˙x−x˙d) (5.6)
Using (5.1) and (5.6) yields,
˙
s=cT(f(x) + ∆f(x) +d(t) +Bu−x˙d) (5.7) Taking the derivative of (5.7) gives rise to
¨
s = cT(d
dtf(x) + d
dt∆f(x) + ˙d(t) +Bu˙−x¨d)
= cT( ˙f(x) + ∆ ˙f(x) + ˙d(t) +Bu˙−x¨d) (5.8) A nonsingular terminal sliding mode manifold is first designed as
σ=s+βs˙
p
q (5.9)
Hereβ = diag(β1, β2, ...., βn) is a positive constant andp/q(pandq are positive odd integers) is chosen in such a way that the condition 1< pq <2 holds [94]. The linear sliding surface s is combined with the nonsingular terminal sliding manifoldσ to realize the terminal sliding mode control. Asσ reaches
zero in finite time, both sand ˙sare bound to reach zero. Then, the tracking error e asymptotically converges to zero.
Taking the time derivative of (5.9) yields,
˙
σ = s˙+β(p
q) ˙spq−1¨s
= β(p q) ˙s
p q−1
(¨s+β−1(p
q)−1s˙2−(
p q)
) (5.10)
Assumption 5.1.The first time derivative of the uncertain term,∆ ˙f(x)and the first time derivative of the disturbance, d(t)˙ are assumed to be bounded and satisfy the following condition:
||cT(∆ ˙f(x) + ˙d(t))|| ≤
∑r i=0
B¯i||x||i r= 0,1, ..n (5.11)
whereB¯iare unknown positive constants, which are not easily obtained due to the complicated structure of the uncertainties in practical control systems. The number of adaptive rules r is determined by the designer in accordance with the knowledge of the relative order of perturbation that the system might encounter. For designing the traditional sliding mode controller, one usually assumes that the upper bound of lumped perturbations satisfies certain conditions. For example, if r = 0, then the nature of the disturbance is periodic and it is well represented by a known constant value. If we choose r = 1, it covers more area in the range space rather than when r = 0 is considered. Thus rest of the control law is designed by considering r = 1.
It will be proven in Theorem 5.1 that since the derivative of the control input contains the discon- tinuous term, the actual control signal which will be obtained after the integration operation will not contain any high frequency switching component. Thus the proposed terminal sliding mode controller will be free from the chattering phenomenon. Moreover, the controller does not need prior knowledge about the upper bound of the disturbance. Instead, the upper bound is obtained by designing an adaptive tuning law.
T heorem 5.1.Considering the uncertain system (5.1), the tracking error dynamics (5.6) can asymp- totically converge to zero if the nonsingular terminal sliding manifold is chosen as (5.9) and the control law is obtained as follows:
˙
u=−(cTB)−1[cTf˙(x) +β−1(p
q)−1s˙2−(pq)+ ( ¯B0+ ¯B1||x||)sign(σ) +K†σ−cTx¨d] (5.12)
5.2 Design of chattering free adaptive terminal sliding mode controller
where (cTB)−1 is nonsingular, B¯0 , B¯1 and K†=diag(K1†....Km†)>0 are the designed parameters.
In practice the bounds of the uncertain term ( ¯B0+ ¯B1||x||) in (5.12) is often difficult to know. Hence an adaptive tuning law is designed to determine ¯B0 and ¯B1. So the control law is represented as,
u = −(cTB)−1
∫ t
0
[cTf(x)dτ˙ +β−1(p
q)−1s˙2−(pq)+ ( ˆB¯0+ ˆB¯1||x||)sign(σ) +K†σ−cTx¨d]dτ(5.13) where ˆB¯0 and ˆB¯1 estimate the bounds of uncertainty, i.e. cT(||∆ ˙f(x) + ˙d(t)||)≤B¯0+ ¯B1||x||. Defining the adaptation error as ˜B¯0 = ˆB¯0−B¯0 and ˜B¯1 = ˆB¯1−B¯1, the parameter ˆB¯0and ˆB¯1are to be estimated by using the adaptation law
B˙ˆ¯0= 1 ν0(p
q)||β|| ||s˙(
p q)−1
σ|| (5.14)
and
B˙ˆ¯1 = 1 ν1
(p
q)||β|| ||s˙(pq)−1σ|| ||x|| (5.15) whereν0 and ν1 are the positive tuning parameters.
P roof : Let us consider the following Lyapunov function V(t) = 1
2σTσ+ 1
2ν0B˜¯02+1
2ν1B˜¯12 (5.16)
Using (5.8 - 5.15), the time derivative of the Lyapunov functionV(t) is obtained as, V˙(t) = σTσ˙ +ν0B˜¯0B˙ˆ¯0+ν1B˜¯1B˙ˆ¯1
= β(p q) ˙s(
p q)−1
σT(¨s+β−1(p
q)−1s˙2−(
p q)
) +ν0( ˆB¯0−B¯0)B˙ˆ¯0+ν1( ˆB¯1−B¯1)B˙ˆ¯1
= β(p q) ˙s(
p q)−1
σT[cT(d
dtf(x) + d
dt∆f(x) + ˙d(t) +Bu˙−x¨d) +β−1(p
q)−1s˙2−(
p q)
] +( ˆB¯0−B¯0)(p
q)||β|| ||s˙(pq)−1σ||+ ( ˆB¯1−B¯1)(p
q)||β|| ||s˙(pq)−1σ|| ||x||
≤ ||β||(p q)||s˙(
p q)−1
σT||[ ¯B0+ ¯B1||x|| −K†||σ|| −( ˆB¯0+ ˆB¯1||x||)sign(σ)]
+( ˆB¯0−B¯0)(p
q)||β|| ||s˙(
p q)−1
σ||+ ( ˆB¯1−B¯1)(p
q)||β|| ||s˙(
p q)−1
σ|| ||x||
≤ −K†||σ|| (5.17)
The above inequality holds ifB˙ˆ¯0= ν1
0(pq)||β|| ||s˙(
p
q)−1σ||andB˙ˆ¯1 = ν1
1(pq)||β|| ||s˙(
p
q)−1σ|| ||x||. Moreover,
||s˙(
p
q)−1||>0 for any ˙s̸= 0 and ˙s(
p q)−1
= 0 only when ˙s= 0. Therefore, the convergence to a domain σ= 0 is guaranteed from any initial condition [30].
Suppose that tr is the time when σ reaches zero from σ(0) ̸= 0 , i.e. σ = 0 for all t ≥ tr. Once σ reaches zero, it will stay at zero using the control law (5.12). Thus the sliding surface swill converge to zero in finite time tf. The total time fromσ(0)̸= 0 to stf can be calculated by using the equation s+βs˙(
p q)
= 0 (5.9) from which the time taken from str tostf [94] is obtained as, tf =tr+ (pq)
(pq)−1β−(pq)||str||(pq)−1 (5.18) Hence the error (5.2) asymptotically converges to zero and the system reaches the equilibrium. This completes the proof.
Remark 5.1. Practically, ||σ|| cannot become exactly zero in finite time and thus the adaptive pa- rameter B˙ˆ¯i may increase boundlessly. A simple way of overcoming this disadvantage is to use the dead zone technique [30] and modify the adaptive tuning law (5.14 - 5.15) as,
B˙ˆ¯0=
1
ν0(pq)||β|| ||s˙(
p
q)−1σ||, ||σ|| ≥ε
0, ||σ||< ε
(5.19)
and
B˙ˆ¯1=
1
ν1(pq)||β||s˙(
p
q)−1σ|| ||x||, ||σ|| ≥ε
0, ||σ||< ε
(5.20)
where εis a small positive constant.
Remark 5.2.The parameter ϵ in controller (5.13) is very important and it is one of the parameters determining the convergence rate of the sliding surface. It is clear that a large ϵ will force the system states to converge to the origin with a high speed. However, a very large value of ϵ will require a very high control input but in reality it is always bounded. Thus the parameter ϵ cannot be selected to be too large. In practice, a compromise has to be made between the response speed and the control input.
Remark 5.3.The parameters ν0 and ν1 in (5.14 - 5.15) determine the convergence rate of the esti- mated bounds Bˆ¯0 and Bˆ¯1. Large values of ν0 and ν1 can be chosen to force the estimated bounds Bˆ¯0 and Bˆ¯1 to rapidly converge to the actual bounds.
Remark 5.4. An exact robust differentiator is available for accurately measuring or estimating the derivative of variables.