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Switching Controller Design for a Class of Markovian Jump Nonlinear Systems Using Stochastic Small-Gain Theorem
Advances in Difference Equations volume 2009, Article number: 896218 (2009)
Abstract
Switching controller design for a class of Markovian jump nonlinear systems with unmodeled dynamics is considered in this paper. Based on the differential equation and infinitesimal generator of jump systems, the concept of Jump Input-to-State practical Stability (JISpS) in probability and stochastic Lyapunov stability criterion are put forward. By using backsetpping technology and stochastic small-gain theorem, a switching controller is proposed which ensures JISpS in probability for the jump nonlinear system. A simulation example illustrates the validity of this design.
1. Introduction
Stability of dynamic systems has been the primary study topic for system analysis. After Lyapunov's second method was created in 1892, it has been developed and applied by many researchers in the past century with fruitful classical stability results achieved. Among these important developments is the Input-to-State Stability (ISS) property, which was firstly formulated by Sontag [1] and has found wide use in engineering by incorporating the idea of nonlinear small-gain theorem [2, 3]. The ISS-based small-gain theorem has some advantages over the earlier passivity-based small-gain theorem and is currently becoming a desirable tool for nonlinear stability analysis, especially in the case of nonlinear robust stabilization for systems with nonlinear uncertainties and unmodeled dynamics [4, 5]. Among all the practical nonlinear systems with uncertainties, the systems of lower-triangular form are of great importance; such systems have several special properties so that they are recently attracting great attention. Firstly, the "lower-triangular" form has close connection with feedback linearzation method; therefore this provides convenience to designers. Second, many real-world dynamic systems are of lower-triangular form [6, 7], and some general systems can be transformed to lower-triangular form via mathematical method [8]. For this reason, lower-triangle nonlinear system can find its wide applications in lots of practical dynamic systems: turbine generator and water turbing generator [9], intelligent robot [10] and missile [11], and so forth. However, dynamic processes in this field are very difficult to describe exactly and depend on many factors. For example, imagining an attacking missile tracking a moving target, this dynamic process is a classical problem of model following and tracking; till now, different control algorithms have been put forward by using ideal assumptions [12, 13]. However, the missile itself may have variable structures subject to random changes and/or failures of its components or environments during its flight; such problems also occur in the case of the moving of robots or the operation of generators. Therefore, an urgent requirement appears to remodel such dynamic processes to meet the need of accuracy and precision.
On the other hand, Markovian jump systems, which were firstly put forward by Kac and Krasovskii [14], have now become convenient tools for representing many real-world systems [15, 16] and therefore arouse much research attention in recent years. In the case of fault detection, fault-tolerant control, and multimodal control, discrete jumps in the continuous dynamics are used to model component failures and sudden switch of system dynamics. With further study of Markovian jump systems, many achievements have been made in the last decade, among which Shaikhet and Mao performed foundational work on stochastic stability for jump systems [17–19] and jump systems with time delays [20–23]. Based on the stochastic stability, more efforts are devoted to applications of jump system model: system state estimation [24, 25], controller design [26–28], and hierarchical reinforcement learning for model-free jump systems [29, 30]. However, in the referred works concerned with controller design problems, assumptions are firstly made that system models considered only consist of static uncertainty. This is an ideal approximation of real situation and not the case nevertheless. As we know, Markovian jump systems are used to represent a class of systems which are usually accompanied by sudden change of working environment or system dynamics. For this reason, practical jump systems are usually accompanied by uncertainties, and it is hard to describe these uncertainties, with precise mathematical model. Thus, how to stabilize Markovian jump systems with unmodeled dynamic uncertainties is a concernful work in our view.
In this paper, we focus on the switching controller design for a class of Markovian jump nonlinear systems with dynamic uncertainties. The control strategy ensures robustness property of systems in the presentence of dynamic uncertainties. And our main contributions are composed of three aspects.
-
(i)
Stochastic differential equation for Markovian jump system is given according to generalized Itô formula, and the similar result is achieved by Yuan and Mao [19] with a different method. Based on differential equation, the martingale process caused by Markovian process is incarnated in the procedure of controller design by applying mathematical transform.
-
(ii)
We introduce the concept of Jump Input-to-State practical Stability (JISpS) and give stochastic Lyapunov stability criterion.
-
(iii)
By composing backstepping technology and stochastic small-gain theorem, a switching controller is proposed. It is shown that all signals of the closed-loop system are globally uniformly ultimately bounded. And the closed-loop system is JISpS in probability.
The rest of this paper is organized as follows. Section 2 begins with some mathematical notions and Markovian jump system model along with its differential equation. In Section 3, we introduce the notion of JISpS and stochastic Lyapunov stability criterion. Section 4 presents the problem description. In Section 5, a switching controller is given based on backstepping technology and stochastic small-gain theorem. In Section 6, an example is shown to illustrate the validity of the design. Finally, conclusions are drawn in Section 7.
2. Stochastic Differential Equation of Markovian Jump System
Throughout the paper, unless otherwise specified, we denote by 
a complete probability space with a filtration 
satisfying the usual conditions (i.e., it is right continuous, and 
contains all 
-null sets). Let 
stand for the usual Euclidean norm for a vector 
and 
stand for the supremum of vector 
over time period 
, that is, 
. The superscript 
will denote transpose, and we refer to 
as the trace for matrix. In addition, we use 
to denote the space of Lebesgue square integrable vector.
Take into account the following Markovian jump nonlinear system:
where 
, and 
is the state vector and input vector of the system, respectively. 
is a right-continuous Markov chain on the probability space taking values in finite state space 
, and 
is 
-dimensional independent standard Wiener process defined on the probability space, which is independent of the Markov chain 
. The functions 
and 
are locally Lipschitz in 
, for all 
, namely, for any 
, there is a constant 
such that
Consider the right-continuous Markov chain 
, and we introduce 
, the indicator process for the regime 
, as
And 
satisfies the following differential equation [15]:
with 
, an 
-martingale satisfying 
, and 
the chain generator, an 
matrix. The entries 
are interpreted as transition rates such that
where 
and 
satisfies 
. Here 
is the transition rate from regime 
to regime 
. Notice that the total probability axiom imposes 
negative and
Let 
denote the family of all functions 
on 
which are continuously twice differentiable in 
and once in 
. Furthermore, we will give the stochastic differentiable equation of 
.
Fix any 
; by the generalized Itô formula, we have
According to (2.4), the differential equation of the indicator 
is as following:
Submit (2.8) into (2.7) and notice that
There is
Therefore, the stochastic differentiable equation of 
is given by the following:
We take the expectation in (2.11), so that the the infinitesimal generator produces [18, 19, 23]
Remark 2.1.
The differential equation of Markovian jump system (2.1) is given as above and, the similar result is also achieved by Yuan and Mao [19]. Compared with the differential equation of general nonjump systems, two parts come forth as differences: transition rates 
and the martingale process 
, which are both caused by the property of Markov chain 
(see (2.4)). Up till now, switching controller design for jump systems contains only the transition rate 
in most cases. In this paper, the controller design will take into account the martingale process as well since the jump systems considered here are of the form of lower triangular. The detailed description will be given in Section 4.
3. JISpS and Stochastic Small-Gain Theorem
Definition 3.1.
Markovian jump system (2.1) is 
-moment Jump Input-to-State practically Stable (JISpS) if there exist 
function 
, 
function 
, and a constant 
such that
Definition 3.2.
Markovian jump system (2.1) is JISpS in probability if for any given 
there exist 
function 
, 
function 
, and a constant 
such that
Remark 3.3.
The concept of Input-to-State Stability (ISS) is a well-known classical tool for designing nonlinear systems, which means for a bounded control input 
, the trajectories remain in the ball of radius as 
; furthermore, as time 
increases, all trajectories approach the smaller ball of radius 
. However, for general nonlinear systems disturbed by noise and/or unmodeled dynamics, it is impossible to obtain such strong conclusion, therefore some generalized results are put forward: Noise-to-State Stability (NSS) [31] and Input-to-State practically Stable (ISpS) [32]. In the definition of ISpS, the trajectories remain in the ball of radius as 
instead of 
. Similar to ISS, as time 
increases, all trajectories approach the smaller ball of radius 
, and the system is still BIBO. As can be seen in the following analysis of this paper, bound 
can be made as small as possible by choosing appropriate control parameters. For some special cases, if 
, the ISpS is reduced to ISS.
Remark 3.4.
The definition of Input-to-State practically Stable (ISpS) in probability for nonjump stochastic system is put forward by Wu et al. [32], and the difference between JISpS in probability and ISpS in probability lies in the expressions of system state 
and control signal 
. For nonjump system, system state and control signal contain only continuous time 
with 
, while for jump system, they concern with both continuous time 
and discrete regime 
. For different system dynamic 
, control signal 
will differ even with the same time 
taken and that is the reason why it is called a switching controller. Based on the idea of switching control, the corresponding stability is called " Jump ISpS" and it is a more general extension of ISpS. By choosing 
, the definition of JISpS will degenerate to ISpS.
Remark 3.5.
This paper introduces two kinds of JISpS in the sense of stochastic stability: 
-moment JISpS and JISpS in probability. According to the knowledge of stochastic process, if one system is 
-moment stochastically stable, it must be stochastically stable in probability by using martingale inequality. Here only sufficient conditions for 
-moment stochastic stability are considered and now introduce the following stochastic Lyapunov stability criterion.
Theorem 3.6.
For Markovian jump system (2.1), let 
be positive numbers. Assume that there exist a function 
, a 
function 
, and constants 
, satisfying
for all 
, then jump system (2.1) is pth moment JISpS and JISpS in probability as well.
Proof.
Clearly the conclusion holds if 
. So we only need proof for 
. Fix such 
arbitrarily, we write 
as 
.
For each integer 
, define a stopping time as
Obviously, 
almost surely as 
. Noticing that 
if 
, we can apply the generalized Itô formula to derive that for any 
Let 
, apply Fatou's lemma to (3.6), and we have
According to Mean value theorems for integration, there is
Noticing the property of 
function, the following inequality is deduced:
Submitting (3.3) into (3.9) gives
Consequently,
In (3.11), define 
function 
, 
function 
, and positive constant 
as:
There is
This completes the proof.
Consider the jump interconnected dynamic system described in Figure 1:
where 
is the state of system, and 
denotes exterior disturbance and/or interior uncertainty. 
is independent Wiener noise with appropriate dimension, and we introduce the following stochastic nonlinear small-gain theorem as a lemma, which is an extension of the corresponding result in Wu et al. [32].
Interconnected feedback system.
Lemma 3.7 (Stochastic small-gain theorem).
Suppose that both the 
-system and 
-system are JISpS in probability with 
as input and 
as state, and 
as input and 
as state, respectively, that is, for any given 
,
hold with 
being 
function, 
and 
being 
functions, and 
being nonnegative constants, 
. If there exist nonnegative parameters 
such that nonlinear gain functions 
satisfy
the interconnected system is JISpS in probability with 
as input and 
as state, that is, for any given 
, there exist a 
function 
, a 
function 
, and a parameter 
such that
Remark 3.8.
Small-gain theorem for nonlinear systems was firstly provided by Mareels and Hill [33] and was extended to the stochastic case by Wu et al. [32]. The above stochastic small-gain theorem for jump systems is an extension of the nonjump case. This extension can be achieved without any mathematical difficulties, and the proof process is the same as in [32]. The reason is that in Lemma 3.7 we only take into account the interconnection relationships between interconnected system and its subsystems, despite subsystems are of jump or nonjump case. If both subsystems are nonjump and ISpS in probability, respectively, the interconnected system is ISpS in probability. In contrast, if both subsystems are jump and JISpS in probability, respectively, the interconnected system is JISpS in probability and so on.
4. Problem Description
Consider the following Markovian jump nonlinear systems with unmodeled dynamics described by
where 
is system state vector, 
is system input signal, 
is unmeasured state vector, and 
is output signal. The Markov chain 
is as defined in Section 2. 
is a smooth function, and 
denotes the unmodeled dynamic uncertainty which could be different with different system regime 
. Both 
and 
are locally Lipschitz as described in Section 2.
Our design purpose is to find a switching controller 
of the form 
, such that the closed-loop jump system could be JISpS in probability, and the system output 
could be within an attractive region around the equilibrium point with radius as small as possible. In this paper, the following assumptions are made for jump system (4.1):
(A1)The 
subsystem with input 
is JISpS in probability, namely, there exists a smooth positive definite Lyapunov function 
such that
where 
is 
function, 
is positive integer, and 
are constants.
(A2)For each 
, 
, there exists an unknown positive constant 
such that
where 
is known constant and 
are known nonnegative smooth functions for any given 
.
For the design of switching controller, we introduce the following lemmas.
Lemma 4.1 (Young's inequality).
For any two vectors 
, the following inequality holds
where 
and the constants 
satisfy 
.
Lemma 4.2 (Martingale representation [34]).
Let 
be N-dimensional standard Wiener noise. Suppose 
is an 
-martingale (w.r.t. P) and that 
, for all 
, then there exists a stochastic process 
, such that
5. Controller Design and Stability Analysis
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.
6. Simulation
Consider a two-order Markovian jump nonlinear system with regime transition space 
, and the transition rate matrix is 
.
The system with unmodeled dynamics is as follows:
Here
From Assumption (A2), we have
where 
, 
, and the 
subsystem satisfies
where 
, 
, 
. Thus the control law is taken as follows (here 
).
Case 1.
The system regime is 
:
Case 2.
The system regime is 
:
In computation, we set the initial value to be 
, 
and the time step to be 0.05 second. For comparison, two groups of different control parameters are given. First, we take the parameter with values 
, 
, 
, and the simulation results are as follows. Figure 2 shows the regime transition of the jump system, and Figure 3 shows the corresponding switching controller 
. Figure 4 shows the system output 
which is defined as the system state 
, and Figure 5 shows system state 
.
Regime transition
.
Switching controller
.
System output
.
System state
.
Now we choose different control parameters as 
, 
, 
and repeat the simulation. The simulation results are as follows. Figure 6 shows the regime transition of the jump system, and Figure 7 shows the corresponding switching controller 
. Figure 8 shows the system output 
which is defined as the system state 
, and Figure 9 shows system state 
.
Regime transition
.
Switching controller
.
System output
.
System state
.
Comparing the results from two simulations, all the signals of closed-loop system are globally uniformly ultimately bounded, and the system output can be regulated to a neighborhood near the equilibrium point despite of different experiment samples. As can be seen from the figures, larger values of 
help to increase the convergence speed of system states while larger value of 
and smaller values of 
help to increase the precision. If one wants the system states to converge to the neighborhood of the equilibrium point with fast speed and an acceptable precision, one should increase the value of 
and decrease 
though this choice will increase the cost of control signals.
Remark 6.1.
Much research work has been performed toward the study of nonlinear system by using small-gain theorem [4, 32, 33]. In contrast to their contributions, this paper focuses on the switching controller design for Markovian jump nonlinear system which is a more general form of nonjump systems. For each different regime 
, the actual controller 
is different, and it consists of two parts with obvious difference (see (5.11)): the coupling of regime 
and 
, which are both caused by the Markovian jumps (see (2.4)). By defining regime 
, the above two terms will reduce to zero. Thus this switching controller design is capable of stabilizing the general nonjump system as well.
7. Conclusion
In this paper, the authors take into account switching controller design for a class of Markovian jump nonlinear system with unmodeled dynamics. Based on the differential equation and infinitesimal generator of jump systems, the concept of Jump Input-to-State Stability (JISpS) and stochastic Lyapunov stability criterion are put forward. Moreover, the martingale process caused by the stochastic Markovian jumps is converted into Wiener noise. By using backstepping technology and stochastic small-gain theorem, a switching controller is proposed which ensures JISpS in probability of the jump nonlinear system. And system output can be regulated in probability to a small neighborhood of the equilibrium point with preset precision. The result presented in this paper also stands for the general nonjump system.
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This work has been funded by BK21 research project: Switching Control of Systems with Structure Uncertainty and Noise.
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Zhu, J., Park, J., Lee, KS. et al. Switching Controller Design for a Class of Markovian Jump Nonlinear Systems Using Stochastic Small-Gain Theorem. Adv Differ Equ 2009, 896218 (2009). https://doi.org/10.1155/2009/896218
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DOI: https://doi.org/10.1155/2009/896218