l 1-Induced State-Feedback Controller Design for Positive Fuzzy Systems

The problem of l 1 -induced state-feedback controller design is investigated for positive Takagi-Sugeno (T-S) fuzzy systems with the use of linear Lyapunov function. First, a novel performance characterization is established to guarantee the asymptotic stability of the closed-loop system with l 1 -induced performance. Then, the sufficient conditions are presented to design the required fuzzy controllers and iterative convex optimization approaches are developed to solve the conditions. Finally, one example is presented to show the effectiveness of the derived theoretical results.


Introduction
In the real world, positive systems exist in many branches of science and technology such as industrial engineering and ecology [1,2].In many applications, the state variables of positive systems are used to denote the concentrations of material, thus taking nonnegative values.Many previous approaches used for general systems are no longer applicable to positive systems because of the special structures of positive systems.Positive systems are special systems and they are defined on cones instead of linear spaces.Therefore, some new problems of positive systems appear and they have been studied extensively [3][4][5][6][7][8][9].For example, controllability and reachability for positive systems have been studied in [10,11].The design of state-feedback controllers guaranteeing the closed-loop system to be asymptotically stable has been studied by the linear programming approach in [12].The positive state-space representation for a given transfer function has been proposed in [13].For nonnegative and time-delay compartmental dynamic systems, stability has been thoroughly studied in [14][15][16][17].Research on 2D positive systems has appeared in [18].Moreover, the problem of controller design for positive systems has been investigated by the linear matrix inequality (LMI) approach in [19].
In addition, it is noted that existing research has been conducted mainly for positive linear systems and there have been few results on positive nonlinear systems.The reason is that nonlinearity is difficult to tackle and many methods derived for positive linear systems cannot be directly used for positive nonlinear systems.It is well known that T-S fuzzy model can approximate a nonlinear system arbitrarily well over a compact domain.Such a modeling approach provides an efficient method to tackle some problems for nonlinear systems [20][21][22].With the modeling approach, some research methods applicable to linear systems can be used for nonlinear systems.Consequently, many important results on fuzzy systems have appeared; for example, a novel approach was proposed in [23] for stability analysis and stabilization of discrete-time T-S fuzzy delay systems.The problem of stability analysis and stabilization for 2D discrete fuzzy systems was investigated in [24].As a result of the novel idea of delay partitioning technique, the proposed stability conditions in [23] were much less conservative than most of the existing results.A fuzzy filter design approach was established for fuzzy stochastic systems in [25], and the proposed approach can be used for fault detection problem due to its strong robustness.In recent years, many researchers have focused their interest on positive T-S fuzzy systems [26].In detail, sufficient conditions of asymptotic stability and stabilization for discrete-time positive T-S systems were proposed in [26].In [27], the problem of stability and constrained control was addressed for discrete-time positive T-S fuzzy systems with time-varying delays.
Recently, in comparison with the existing results derived with the quadratic Lyapunov function, a novel approach is derived for the synthesis problem of positive systems by using a linear Lyapunov function.The reason why we use the novel approach is that the variables of positive systems are nonnegative and therefore a linear Lyapunov function becomes valid.Many researchers have proposed some novel results based on the linear Lyapunov function [28][29][30][31][32]. Compared with quadratic Lyapunov function based results, the new results in terms of linear programming are more amenable to analysis and computation.Moreover, the applications of the so-called linear Lyapunov functions in the analysis of positive linear systems naturally lead to a variety of results based on a linear setting, which stimulates the use of  1 -gain as a performance index for positive linear systems.It is noted that some frequently used costs such as  ∞ norm are based on the  2 signal space and these costs are not very natural to describe some of the features of practical physical systems.By contrast, 1-norm gives the sum of the values of the components and it can provide a more natural description for positive systems.It is more appropriate, for example, if the values denote the number of animals in a species or the amount of material.Based on the above discussion, it is noted that the synthesis problems for positive T-S fuzzy systems have not been fully investigated, especially with ℓ 1 -induced performance and linear Lyapunov functions.This motivates our study.
In this paper, we investigate the state-feedback controller design problem for positive T-S fuzzy systems with the use of linear Lyapunov function.The main contributions of the paper are as follows: (1) an ℓ 1 -induced performance index is explicitly presented for positive fuzzy systems and analytically characterized under a linear Lyapunov function framework; (2) the desired state-feedback controllers are derived with which the asymptotic stability of the closed-loop system is guaranteed and the proposed performance is satisfied; (3) iterative convex optimization approach is developed to solve the conditions.
The rest of this paper is organized as follows.In Section 2, some important preliminaries about positive T-S fuzzy systems are introduced.In Section 3, the state-feedback controller is designed for positive T-S fuzzy systems.One example is provided in Section 4 to show the applicability of the theoretical results.The results are finally concluded in Section 5.

Preliminaries
In this section, we introduce notations and several results concerning positive T-S fuzzy systems.
Consider the following fuzzy system described by the th rule.
Next, some useful results are introduced.
Here, the definition of ℓ 1 -induced performance is introduced.We say that a stable positive system (2) has ℓ 1 -induced performance at the level  if, under zero initial conditions, sup where  > 0 is a given scalar.

State-Feedback Controller Synthesis for Positive Fuzzy Systems
In this section, the ℓ 1 -induced state-feedback controller synthesis problem is formulated.Based on the stability and performance conditions, a state-feedback controller is designed for positive fuzzy systems.Finally, an iterative convex optimization approach is developed to solve the conditions accordingly.
Here, we deal with the stabilization problem for the following positive fuzzy systems: where () ∈ R  , () ∈ R  , () ∈ R  , and () ∈ R  are the system state, disturbance input, control input, and controlled output, respectively.
For the positive fuzzy system in (9), we construct the following parallel distributed compensation (PDC) fuzzy controller.
The overall fuzzy controller is represented by Then, the closed-loop system with the PDC fuzzy controller (11) is given by The problem to be addressed in this paper is described as follows.
First, the stability characterization of the closed-loop system (12) is proposed as follows.
In the following, our aim is to derive a numerically tractable means to synthesize a derived controller.It is noted that when matrix   is fixed, (24) turns out to be linear with respect to the other variables.Therefore, a natural way is to fix   and solve ( 24)-( 25) by linear programming.

Algorithm PPSFCD
Step 1. Set  = 1.We use Proposition 5 to solve for  1  such that the positive system ( 9) with ( 10) is positive and asymptotically stable.
Step 2. For fixed    , solve the following optimization problem for    and   .OP1: minimize   subject to the following constraints: Step 3.For fixed    , solve the following optimization problem for    .OP2: minimize   subject to the following constraints: Denote  *  as the solution to the optimization problem.
Remark 7. The parameter  can be optimized iteratively.Notice that  * +1 ≤  *  since the corresponding parameters obtained in Step 3 will be utilized as the initial values in Step 2 to derive a smaller .Since the sequence { *  } is bounded from below, the convergence of the iterative process is naturally guaranteed.

Illustrative Example
In this section, one illustrative example is presented in the following to illustrate the effectiveness of the proposed controller design approach.
With this choice of initial condition, Figures 1 and 2 show the response of open-loop system.It can be seen that the open-loop system is not globally asymptotically stable.Figures 3 and 4 show the state response of the closed-loop system, from which we can see that the state of the closedloop system converges to zero.

Conclusion
In this paper, the state-feedback controller synthesis problem for positive T-S fuzzy systems has been addressed under ℓ 1 performance.Novel performance characterization of the closed-loop system has been established.Based on the novel characterization, sufficient conditions have been developed for the existence of state-feedback controllers.Moreover, iterative convex optimization algorithms have been developed to solve the design conditions.Finally, one example has been presented to demonstrate the effectiveness of the proposed approach.Further research topics include the controller synthesis problem for positive fuzzy systems with timevarying or distributed delays with linear Lyapunov functions.