Abstract: This study presents a bilinear control scheme for a class of bilinear system. A controller is proposed to globally stabilize the bilinear system. Then, the stabilization condition is derived to guarantee the stabilizability of the control system in terms of Linear Matrix Inequalities (LMIs). Finally, a numerical example is utilized to demonstrate the feasibility and effectiveness of the proposed control scheme.
INTRODUCTION
Bilinear systems and controls have been successfully applied to a wide variety of fields in recent years (Mohler, 1973, 1991; Elliott, 1999; Chen and Chen, 2008; Kim et al., 2005; Li et al., 2008). Two distinguished merits should be pointed here for selecting bilinear models to describe nonlinear models. One is that bilinear systems are an adequate approximation than linear models for some real-world systems, including engineering applications in nuclear, thermal and chemical processes and many other non-engineering applications in biology, socioeconomics and immunology (Mohler, 1991, 1973). The other is that many real physic processes can be appropriately modeled as bilinear systems when linear models are inadequate, for example, the population of biological specie (Mohler, 1991). For these two reasons, bilinear system is therefore very essential to design its controller, to explore the stability (Xiang-Shun and Hua-Jing, 2009) and to improve performance by applying various control techniques (Li et al., 2008; Zhang and Li, 2010; Tang et al., 2005; Sun, 2007).
The main contributions of this study are (1) designing a bilinear controller for the bilinear system and (2) describing the stabilization conditions for the bilinear system via LMI.
SYSTEM DESCRIPTION AND CONTROLLER DESIGN
Here, we will introduce the bilinear system and then develop the controller. Firstly, consider a class of bilinear system is described as below:
| (1) |
where, x∈Rnx1 is state, u∈R is control input, A∈Rnxn, B∈Rnx1 and N∈Rnxn.
The controller for the bilinear system Eq. 1 is formulated as follows:
| (2) |
where,
is a vector to be determined and K∈R1xn is a scalar to be assigned.
The control objective is to design a controller Eq. 2 to stabilize the bilinear system Eq. 1 Substituting Eq. 2 into Eq. 1, one can get the closed-loop system:
| (3) |
The main result on the asymptotic stability of the bilinear controlled system is propounded in the next section. Before discussing the proof, we first give the following results which will be used in the proof of our main results.
Lemma 1: Lo and Lin (2006), Chen et al. (2008) and Xu and Lam (2005): Given any matrices A, B and ε with appropriate dimensions such that ε>0 we have:
| (4) |
RESULTS
Theorem 1: If there exist a symmetric and positive definite matrix P, a scalar ρ, some vectors K and some scalars εi = 1,2,3, such that the following LMI (Eq. 4) is satisfied, then the bilinear system (Eq. 1) is globally asymptotically stable via the feedback controller (Eq. 3):
| (5) |
where, Q = P-1 and * denotes the transposed elements in the symmetric positions.
Proof: Consider the Lyapunov function candidate:
| (6) |
where, P is a constant, symmetric and positive definite matrix. Clearly, V(x(t)) is positive definite and radially unbounded. The time derivative of V(x(t)) becomes:
| (7) |
Substituting Eq. 3 into 4, we get:
| (8) |
Consider the equation from the Eq. 8:
| (9) |
Multiplying the above equation on the left and right by P-1 and defining a new variable Q = P-1, we can rewrite Eq. 9 as:
| (10) |
According to Lemma 1, the above equation can be rewritten as:
| (11) |
Assuming that Eq. 11 is negative, then applying the Schur complement to Eq. 11 results in:
| (12) |
which is the LMI (Eq. 5). Thus
Example: Here, the proposed method is used to design a controller for a class of bilinear system. The bilinear system is described as follows:
| (13) |
where,
Let, ρ = 0.3 and choose the controller gain matrix as K = [-0.5, -1]. Applying ρ and all these matrices to inequalities Eq. 4 in Theorem 1 and utilizing the LMI tool box (Tanaka and Wang, 2001), one can figure out the common positive-definite matrices,
| Fig. 1: | State responses of x1 under three different initial conditions (x(0)) = [3.1 -2]T: solid line, x(0) = [-0.8 -1.3]T: dash line, x(0) = [1.5 1.5]T: dot line |
| Fig. 2: | State responses of x2 under three different initial conditions (x(0)) = [3.1 -2]T: solid line, x(0) = [-0.8 -1.3]T: dash line, x(0) = [1.5 1.5]T: dot line) |
The simulations are performed under three different initial conditions x(0) = [3.1 -2]T, x(0) = [-0.8 -1.3]T, and x(0) = [1.5 1.5]T, respectively. Figure 1 shows state responses of x1 and Fig. 2 shows state responses of x2. From these two Fig. 1 and 2, one can find that the states x1 and x2 converge to the equilibrium state within 1 sec.
| Fig. 3: | Control input u(t) for three different initial conditions (x(0)) = [3.1 -2]T: solid line, x(0) = [-0.8 -1.3]T: dash line, x(0) = [1.5 1.5]T: dot line) |
Control input u(t) for three different initial conditions is shown in Fig. 3. From these simulation results, the proposed control scheme for bilinear system is effective and feasible.
CONCLUSIONS
In this study, we have developed a novel controller for a class of bilinear system. Considering to stabilize this kind of problem and to guarantee the stability of the controlled system, the sufficient condition has been derived via LMI in detail. Finally, a numerical simulation has been adopted to demonstrate the feasibility and effectiveness of the proposed schemes.
ACKNOWLEDGMENT
This study was supported by the National Science Council of Taiwan, Republic of China under Grant NSC-98-2221-E-027-120.