Sliding Mode Applications to Optimal Filtering and Control

In practical applications, a control system operates under uncertainty conditions that may be generated by parameter variations or external disturbances. Consider a real trajectory of the disturbed control system

$$ \dot{x}(t)=f(x(t))+B(t)u+g_1(x(t),t)+g_2(x(t-\tau ),t). ~~ (5.1) $$

Here,

x

(

t

) ∈ 

R

n

is the system state,

u

(

t

) ∈ 

R

m

is the control input,

f

(

x

(

t

)) is a known function determining the proper state dynamics, the rank of matrix

B

(

t

) is complete and equal to

m

for any

t

 > 

t

0

, and the pseudoinverse matrix of

B

is uniformly bounded:

$$ \|B^{+}(t)\| \leq b^{+},\,b^{+}=const>0,\ B^{+}(t) := [B^{T}(t)B(t)]^{-1}B^{T}(t), $$

and

B

 + 

(

t

)

B

(

t

) = 

I

, where

I

is the

m

-dimensional identity matrix. Uncertain inputs

g

1

and

g

2

represent smooth disturbances corresponding to perturbations and nonlinearities in the system. For

g

1

,

g

2

, the standard matching conditions are assumed to be held:

$g_1,g_2\in \mbox{span} B$

, or, in other words, there exist smooth functions

γ

1

,

γ

2

such that

$$g_1(x(t),t)=B(t)\gamma_1(x(t),t),~~ (5.2)$$

$$g_2(x(t-\tau ),t)=B(t)\gamma_2(x(t- \tau),t),$$

$$||\gamma_1(x(t),t)||\le q _1||x(t)|| +p_1,\ q_1,p_1>0,$$

$$||\gamma_2(x(t-\tau ),t)||\le q _2||x(t-\tau )||+p_2,\ q_2,p_2>0.$$

The last two conditions provide reasonable restrictions on the growth of the uncertainties.

Let us also consider the nominal control system

$$ \dot{x}_0(t)=f(x_0(t))+B(t)u_0(x_0(t-\tau ),t), ~~ (5.3) $$

where a certain delay-dependent control law

u

0

(

x

(

t

 − 

τ

),

t

) is realized. The problem is to reproduce the nominal state motion determined by (5.3) in the disturbed control system (5.1).

The following initial conditions are assumed for the system (5.3)

$$ x(s)=\phi(s), ~~ (5.4) $$

where

φ

(

s

) is a piecewise continuous function given in the interval [

t

0

 − 

τ

,

t

0

].

Thus, the control problem now consists in robustification of control design in the nominal system (5.3) with respect to uncertainties

g

1

,

g

2

: to find such a control law

u

 = 

u

0

(

x

(

t

 − 

τ

),

t

) + 

u

1

(

t

) that the disturbed trajectories (5.1) with initial conditions (5.4) coincide with the nominal trajectories (5.3) with the same initial conditions (5.4).