# Sliding modes after the first decade of the 21st century : by Leonid Fridman, Jaime Moreno, Rafael Iriarte

By Leonid Fridman, Jaime Moreno, Rafael Iriarte

Half I VSS and SM Algorithms and their Analysis.- half II Sliding Mode regulate Design.- half III functions

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In addition, and without loss of generality, let us assume that rankC = p, rankB = m, rankD = q and that the triplet (A, D,C) is strongly observable, such that the state x(t) may be recovered in finite-time using only the output and its derivatives (through the use of the HOSM differentiator). e. |w(t)| ˙ ≤ L, an extra derivative of the estimated state can be computed, thus obtaining an estimate for x. ˙ Under these considerations, an estimate for the unknown input may be obtained as wˆ = D+ [x˙ − Ax − Bu], With this estimate of the unknown input, it is natural to try to compensate the effect of the unknown input in the system as much as possible.

5k3|x1 | 2 sign(x1 ) + k32 x1 Under the assumption that the disturbances satisfy the bounds |g1 (t, x)| ≤ ρ1 (t, x)|φ1 (x1 )|, |g2 (t, x)| ≤ ρ2 (t, x)|φ2 (x1 )| for some known continuous functions ρ1 , ρ2 and that the gains are selected to satisfy 2 1 4ε [2ερ1 + ρ2 ] + 2ερ2 + ε + [2ε > 4ε 2 + 2ε k1 (t, x) , k1 (t, x) > k2 (t, x) + ρ1 (t, x)] β + 4ε 2 , for some ε > 0, then the finite-time stability of x1 = 0 is ensured. This approach may be widely exploited in the near future because it allows: • designing absolutely continuous SM controllers capable of compensating Lipshitz continuous perturbations/uncertainties which may grow together with the states; • adaptation of the control law.

0 I Thus, the state vector x can be expressed by the identity ⎡ ⎤ y ⎢ y[1] ⎥ d n−1 ⎢ ⎥ x = n−1 Mn−1 Jn ⎢ . ⎥ dt ⎣ .. 30) y[n−1] By defining H (t) as ⎡ ⎢ ⎢ H (t) = Mn−1 Jn ⎢ ⎣ y y[1] .. 31) y[n−1] d n−1 we obtain that x (t) = n−1 H (t). Then, vector x (t) can be obtain by means of a dt high order differentiator [92]. Thus, the j-th term of x (t) can be estimated in the following way z˙ j,0 = λ0 z j,0 − H j z˙ j,1 = .. 32) 1/2 z˙ j,n−2 = λn−2 z j,n−2 − z˙ j,n−3 sign z j,n−2 − z˙ j,n−3 + z j,n−1 z˙ j,n−1 = λn−1 sign z j,n−1 − z˙ j,n−2 With the proper selection of constants λi (i = 0, · · · , n¯ H − 1), there exists a finite n−1 time t j such that the identity z j,n−1 (t) = dtd n−1 H j (t) is achieved for all t ≥ t j .