Now we seek the switching controller
for jump system (4.1) so that the closed-loop system could be JISpS in probability. Perform a new transformation as
For simplicity, we just denote
,
by
,
,
,
, where
, and the new coordinate is
.
According to stochastic differential equation (2.11), one has:
Here we define
From assumption (A2), one gets that there exist nonnegative smooth functions
,
satisfying
The inequality (5.4) could easily be deduced by using Lemma 4.1.
Now we turn to the martingale process
; according to Lemma 4.2, there exist a function
and an
-dimensional standard Wiener noise
satisfying
, where
, and
is a positive bounded constant. Therefore we have
Remark 5.1.
Differential equation of new coordinate
is deduced as above. The martingale process resulting from Markov process is transformed into Wiener noise by using Martingale representation theorem, and it will affect the Lyapunov function construction and affect the remainder of the control design process; for nonjump systems with uncertainty, a quadratic Lyapunov is chosen to meet control performance in most cases [32, 35, 36]. However, for jump systems, this choice will fail because of the existence of martingale process (or Wiener noise). To solve this problem, we suggest using quartic Lyapunov function instead of quadratic one, and this will increase largely the difficulty of design.
Choose the quartic Lyapunov function as
In the view of (5.5) and (5.6), the infinitesimal generator of
satisfies
The following inequalities could be deduced by using Young's inequality and norm inequalities with the help of changing the order of summations or exchanging the indices of the summations:
where
, and
are design parameters.
Based on assumption (A2) and (5.4), we obtain the following inequality by applying Lemma 4.1:
Here
,
are design parameters.
Submit (5.9) into (5.7), there is
Choose the virtual control signal as
Thus the real control signal
is
such that
where
,
, and
function
is chosen to satisfy
.
Theorem 5.2.
If Assumptions (A1) and (A2) hold and a switching control law (5.11) is adopted, the interconnected Markovian jump system (4.1) is JISpS in probability, and all solutions of closed-loop system are ultimately bounded. Furthermore, the system output could be regulated to a small neighborhood of the equilibrium point with preset precision in probability within finite time.
Proof.
From Assumption (A1), the
subsystem is JISpS in probability. There exist
such that
Considering (5.12), for any given
, there is
Notice the fact that
stands up as long as
and vice versa. Thus we have
In (5.15), appropriate parameter
can be chosen to satisfy
.
According to Theorem 3.6 and (5.12), with switching controller
adopted, the
subsystem of jump system (4.1) is JISpS in probability with
as system state and
as input, which means for any given
, there exists
function
,
function
, and
such that
On the other hand, according to Assumption (A1), there is
Similarly, by choosing parameter
, for any given
, there exist
function
,
function
, and
such that
Here parameter
can be chosen to satisfy
.
By combining (5.16) and (5.18) we choose parameters
guaranteeing that
According to stochastic small-gain theorem, for any given
, there exists
function
such that
where
,
is given as in [32]. From (5.20) it can be seen that all solutions of closed-loop system are ultimately bounded in probability.
According to (5.20) and the property of
function, for any given
, there exists
. If
, there is
. At the same time by choosing approximate parameter, it can be guaranteed that
.
Let
, thus we have that for any given
, there exists
and
such that if
, the output of jump system
satisfies
meanwhile
can be made as small as possible by choosing approximate parameters
. The proof is completed.
Remark 5.3.
Theorem 5.2 shows that if both
subsystem and
subsystem are JISpS in probability, the jump system (4.1) is JISpS in probability with appropriate control parameters chosen. Meanwhile the system output can be regulated to a small region in probability with preset precision within finite time. In the following simulation, we will show how different parameters affect the system states and output.