1. Introduction
In the last three decades, there has been extensive interest and quantities of papers in switched systems, which are regarded to be good models representing practical systems; refer to [
1,
2,
3,
4,
5,
6,
7] and the references therein. It is well-known that there are three basic problems in the area of switched systems and control, and the third one (most challenging problem) is to design a stabilizing switching law (strategy) for the case where each single subsystem is not stable as desired. When the switched linear systems are composed of unstable LTI deterministic subsystems, there are a few existing results. In [
8,
9], it is shown that if there exists a stable convex combination of subsystem matrices, then there exists a state-dependent switching rule quadratically stabilizing the switched system. For switched continuous-time and discrete-time linear systems with polytopic uncertainties, quadratic stabilizability via state-dependent switching has been discussed [
10]. Ref. [
11] investigates quadratic stability/stabilization of a class of switched nonlinear systems by using a nonlinear programming (Karush–Kuhn–Tucker condition) approach. Ref. [
12] extends the discussion and results in [
8,
9] to achieve quadratic stability for switched linear systems with norm-bounded uncertainties, and a state-dependent switching law has been proposed for quadratic stabilization.
Recently, the results in [
8,
9] were extended in Ref. [
13] to quadratic stabilization of switched linear systems where norm-bounded uncertainties exist in the subsystems. In that context, a convex combination of subsystems is defined including the subsystem matrices and the matrices denoting uncertainties, and is represented by a condition of
norm. It was shown in [
13] that if we manage to obtain such a convex combination of subsystems, which is Hurwitz, and the
norm is smaller than the specified value, we can design a state-dependent switching law such that the switched system is quadratically stable, even though each subsystem is not. The convex combination approach was further discussed for switched affine systems in [
14], and later extended to quadratic stabilization of switched uncertain stochastic systems (SUSS) by state- and output-dependent switching laws in [
15,
16].
Encouraged by these existing contribution, we here aim to study the convex combination approach developed in [
8,
9,
13,
15,
16] for global quadratic
performance in probability (quadratic stability and
gain
in probability: abbreviated as GQ
-P) for SUSSs, which consist of a finite number of linear stochastic subsystems where there are norm-bounded uncertainties. As in the above literature, we challenge the third basic problem, i.e., we consider the situation that for a specified positive scalar
, there is no subsystem that achieves GQ
-P. For our control problem, we define a
new convex combination of subsystems that incorporates the norm-bounded uncertainties,
gain, and stochastic disturbance attenuation in an integrated manner. This is a major extension to the existing convex combination approach. If we can obtain such a convex combination of subsystems that achieves GQ
-P, then we propose a switching law using the Lyapunov matrix obtained by the convex combination system matrices, and prove the SUSS achieves GQ
-P under the switching law. When it is difficult to find such an appropriate convex combination, we proceed to consider designing the state feedback controller for each subsystem so that the convex combination approach may be applied for the closed-loop subsystems.
The outline of this manuscript is as follows. Some preliminaries are first recalled in
Section 2 for general and linear stochastic control systems, quadratic stability, and quadratic
performance in probability; then, the control problem in this paper is formulated. Next,
Section 3 introduces the new convex combination of subsystems and proposes a state-dependent switching law for the SUSS under consideration. It is shown that if we can obtain a convex combination of subsystems that achieves GQ
-P, then a state-dependent switching law can be designed such that the SUSS achieves GQ
-P. A numerical example is provided to show effectiveness of the proposed method. In
Section 4, the simultaneous design of state feedback controllers and state-dependent switching laws is studied so that more flexibility is earned for the control system, and the application to a type of DC–DC boost converters is dealt with. Finally,
Section 5 concludes the paper.
Notations | Descriptions |
| n-dimensional Euclidean space |
| Identity matrix of size |
| transpose of A |
| trace of square matrix A |
| |
| W is symmetric and positive (negative) definite |
| W is symmetric and non-negative (non-positive) definite |
| expectation value of a random variable |
A is Hurwitz | all eigenvalues of A have negative real parts |
SUSS | Switched uncertain stochastic systems |
GQS-P (GAS-P) | globally quadratically (asymptotically) stable in probability |
GQ-P | global quadratic performance in probability |
2. Preliminary Results and Problem Formulation
We first recall some stability results concerning stochastic control systems. For a more detailed description, refer to, for example, Ref. [
17]. At the end of this section, we explain the control problem considered in this paper.
Let us start with the general stochastic system
where
is the state,
is an
r-dimensional normalized Wiener process defined on an appropriate probability space, and
is a stochastic differential of
.
is the vector field,
is the diffusion rate matrix function, and both functions are locally Lipschitz satisfying
,
. Similar to the Lyapunov stability theory [
18] for deterministic systems, the following theorem provides the Lyapunov stability condition for the stochastic system (
1).
Lemma 1 ([
19])
. If there exist a function , two class functions and , and a class function , satisfyingthen, the equilibrium of (1) is globally asymptotically stable in probability (GAS-P). If the function
in Lemma 1 is obtained having the form
, where
, we say the equilibrium of the system (or simply the system) is globally quadratically stable in probability (GQS-P). Actually, when
f and
g in (
1) are linear with respect to
x, i.e., taking the form of
where
are constant matrices, we can consider a candidate quadratic Lyapunov function
with
for it. Translating Lemma 1 with this
and
, we obtain the following result.
Lemma 2. If there exists a matrix satisfying the linear matrix inequality (LMI) [20]then, the equilibrium of (2) is GQS-P. Now, we deal with the case of involving uncertainties in the stochastic system (
2) as
where
are constant matrices,
denotes the norm-bounded uncertainty and assumes
without losing generality. According to Lemma 2, the equilibrium
of (
4) is GQS-P if there exists a matrix
satisfying
for any
within the norm bound.
The next well-known lemma is used to analyze the matrix inequality (
5).
Lemma 3 ([
21])
. Assume that and are constant matrices. Then,holds for any satisfying . Using the above lemma and the Schur complement lemma for the matrix inequality (
5), we obtain the following result.
Lemma 4. If there exists a matrix satisfying the LMIthen, the equilibrium of (4) is GQS-P. Next, we consider the following uncertain stochastic system, which corresponds to the system (
4) with disturbance input and controlled output.
Here, is the disturbance input; is the controlled output; and are constant matrices with proper dimension.
Definition 1. The system (8) is said to achieve global quadratic
performance
in probability (GQ
-P)
if it is GQS-P, and moreover, when ,holds for any time and any disturbance input satisfying . Lemma 5. ([
19])
. If there exists a matrix satisfying the LMIor equivalently,then, the system (8) achieves GQ-P. With the above preparation, we now proceed to describe our control problem in detail. Consider the switched uncertain stochastic system (SUSS)
where
is the state,
is the disturbance input,
is the controlled output, and
and
are the same as in (
1). The switching law (signal)
determines the index number of the active subsystem at every time instant, where
is the index set. Thus, there are
subsystems that may be activated, and the dynamics of the
i-th subsystem is represented by
where
,
,
,
,
are constant matrices and
denotes the norm-bounded uncertainty as in (
4). It is assumed, as in the literature, that there is no jump in state
x at the switching instants.
The control problem is formulated as follows:
For given , design a state-dependent switching law such that the SUSS (12) achieves GQ-P.If there is one subsystem in (
13) achieving GQ
-P, we can choose to activate that subsystem for all time (without any switching) and, certainly, the switched system has the same performance. If there are more than two subsystems in (
13) achieving GQ
-P, we may discuss the average dwell time approach [
22,
23,
24] and multiple/piecewise Lyapunov functions approach [
25,
26,
27]. Since we are here challenging the third basic problem in switched systems and control, we assume the following throughout this paper.
Assumption 1. There is NO single subsystem in (13) achieving GQ-P in the sense of Lemma 5. Alternatively, there is NOT any subsystem such that there exists satisfying the LMIor equivalently, 4. State Feedback Controller Design
We focused our attention on design condition and the switching law in the previous section. When Assumption 2 does not hold and feedback control is available, we shall incorporate state feedback controller design together with the switching law in this section.
4.1. Controller Design
Introducing control inputs into the switched system (
12), we have
where
is the control input and
is the constant input matrix.
In the case of state feedback, the design issue is to propose
with a constant feedback gain
K, such that Assumption 2 holds for the closed-loop system
Then, the discussion in the previous section is valid if we replace
with
, where
. In other words, we are considering the following
convex combination system of the subsystems in (
37)
Using the matrix inequality (
19) with
replaced by
, we obtain the design condition
It is noted that in addition to the coupling between
’s and
P, the term
includes the matrix product
in the above matrix inequality. To make (
40) more trackable, we use the Schur complement lemma for (
40) to reach
Multiplying the first row and column of (
41) by
, we obtain
where
. Therefore, if the matrix inequality (
42) is feasible with the variables
,
M, and
’s, the state feedback gain is computed by
, and the matrix
will be used in the switching law defined later.
To summarize the above discussion up to now, we obtain the following theorem.
Theorem 2. If there exist matrices , M and non-negative scalars satisfying such that the matrix inequality (42) holds, then the SUSS (37) together with the state feedback achieves GQ-P under the switching law Proof. If the matrix inequality (
42) holds, we obtain (
40). Since
, by using the proof of Theorem 1, the switched system (
38) achieves GQ
-P under the switching law (44). □
Remark 4. The matrix inequality (41) is equivalent towhich is a convex combination of the matrix inequalities for each subsystem to achieve GQ-P through a state feedback. Therefore, the condition of Theorem 2 requires that a convex combination of subsystems should achieve GQ-P by a switching state feedback, although every single subsystem cannot make it. In this sense, this condition can be regarded as the state feedback version of the design condition in the previous section. 4.2. Application to Boost Converters
We consider the DC–DC boost converter model dealt with in [
28], which is depicted in
Figure 2. For simplicity and easiness to follow, suppose that there is only one transistor–diode switch
S; thus, the system is composed of two subsystems. In the case of more than two switches in the circuit, the number of subsystems will be more than three, but the discussion and the results can be applied in the same manner.
Let the inductor current and the capacitance voltage be the state variables, and combine them into the state vector Suppose that is the control input , is the disturbance input of the input voltage, and the capacitance voltage is the controlled output . Moreover, suppose that there are independent stochastic disturbance with the states and .
Then, when the switch
S is closed, the state space model is
and when the switch
S is open, the state space model is
In the above, and denote the variations of the load resistance and R, which are supposed to be bounded by and , respectively.
Observing the nominal part and the uncertain part in the system matrices of (
46) and (
47), we see that the above switched system takes the form of (
13), where
and
with
. It is noted that the first-order Taylor expansion is used in the above to separate the uncertainty from the term
. Moreover, although we have only assumed the uncertainties in the load resistance
and
R, we can use the same formulation to deal with bounded uncertainties in
L and
C.
To perform the numerical simulation, we need to set up the physical parameters in the above state space models. Here, we assume
,
,
,
, and the uncertainties are
,
. Substituting the above parameter values together with
,
into (
48), we obtain the coefficient matrices
Since both
and
are Hurwitz, the achievable
gain
is essential. First, we try to solve the design condition (
19) with
by adjusting the combination parameters
and
, but there is no feasible solution, which means the desired quadratic stability with
gain
cannot be achieved through switching if there is NO state feedback.
Next, we set
(and thus,
) to solve the design condition (
42) with
It turns out that the condition (
42) is feasible when
, and the solutions are
Then, the feedback gain matrices
K is computed by
To activate the switching laws (
43) and (
44), as stated in Theorem 2, we use
With the initial value
and the disturbance input
, the state trajectories of
in the switched system are depicted in
Figure 3, which present good convergence. Moreover, it is confirmed that the inequality (
9) holds for any
on average when several trials are performed, which implies that the desired quadratic stability and
performance in probability has been achieved.
5. Conclusions
We have dealt with the global quadratic performance analysis problem for a class of switched uncertain linear stochastic systems. Under the assumption that no single subsystem achieves GQ-P but a convex combination of the subsystems can make it, we proposed a state-dependent switching law such that the SUSS achieves the desired GQ-P. We also extended the discussion to the design of switching state feedback controller, together with its application to quadratic stabilization of a boost converter.
It is noted that the convex combination approach proposed in this paper incorporates norm-bounded uncertainties,
gain analysis (attenuation of
disturbance attenuation), and stochastic noise reduction in an integrated manner; thus, it is a major extension to the existing results in the literature. Our future work will consider the applicability and extension of such a convex combination approach to switched positive systems [
29], switched affine systems [
30,
31], switched dynamical output feedback [
32], fault detection observer design [
33], and event-triggered control [
34] for switched and hybrid systems. Furthermore, it is important to apply the proposed design condition and the algorithm for more practical electronic circuits and other real systems.