H ∞ Robust Tracking Control of Stochastic TS Fuzzy Systems with Poisson Jumps

A robust adaptive H∞ tracking control design for nonlinear stochastic systems with both Brownian motion and Poisson jumps is proposed, which is based on Takagi–Sugeno (T-S) type fuzzy techniques. Because the state of to-be-controlled systems cannot be known exactly, to overcome this difficulty, the state estimation systems and error estimation systems are introduced to obtain an augmented system. By using the fuzzy systems to approximate the nonlinear systems, an adaptive fuzzy control is employed to achieve the desiredH∞ tracking performance for stochastic systemswith exogenous disturbance. A simulation example is presented to illustrate the tracking performance of the proposed design method.


Introduction
Adaptive control theory is a powerful methodology which has been widely applied to design feedback control for systems with parametric uncertainties or for plants with unknown structure or changing operating conditions [1,2].Because there are uncertainties in the systems, the laws of adaptive control design schemes no more depend on the fixed description of input-output relationships but are related to the estimation of the system's state or output and those estimating errors [3][4][5][6].
The robust  ∞ tracking control theory is a powerful methodology to design feedback controllers where the system parametric uncertainties are seen as the exogenous disturbance [7][8][9].This methodology is widely applied in networked control systems [10], mobile robots [11], etc.The objective of the robust  ∞ tracking control design is to construct an adaptive controller which guarantees the  ∞ tracking control performance.Based on the system's structure, the robust  ∞ tracking control designing methods include linear  ∞ control and nonlinear  ∞ control.Linear  ∞ control design problems are based on solving a kind of algebra Riccati inequalities which can be solved by the LMI's method.And the nonlinear  ∞ control design involves solving a nonlinear Hamilton-Jacobi inequality(HJI) [12].However, it is very difficult to solve the Hamilton-Jacobi equalities or inequalities [13,14].In practice, to overcome this difficulty, the fuzzy methods have been applied to the  ∞ robust control design of nonlinear systems where the Takagi-Sugeno (T-S) type fuzzy is widely used [15][16][17].
Stochastic systems are applied in economics [18], biology [19], and natural science to describe the randomness in models [20][21][22][23].Based on the distribution properties of the inserted random variables, stochastic systems include Itôtype systems driven by Brownian motion [24,25], systems driven by Markovian jumps [26], systems driven by Poisson jumps or Lévy process [27], and their compound forms [28][29][30].The linear stochastic  ∞ theory has been developed since 1990s via the linear matrix inequalities (LMI) approach [31].The nonlinear stochastic  ∞ problems are solved by means of Hamilton-Jacobi equations [25].In practice, there exist sudden shifts in the systems.In order to describe such phenomenon, Poisson jumps are inserted in the model, so the stochastic systems are driven not only by Brownian motion but also by Poisson jumps or Lévy process [27,28].The main complication in the  ∞ tracking control design problem studied here is due to the presence of both deterministic, stochastic perturbations and Poisson jumps terms in the system.Nonlinear  ∞ tracking theory and fuzzy control design are combined together to construct the adaptive fuzzybased controller which guarantees the  ∞ tracking control performance.
This paper is organized as follows: In Section 2, some lemmas about stochastic differential equations with Poisson jumps are reviewed, which will be used in the latter theoretical analysis and deductions.In Section 3, the theories of  ∞ tracking control are extended to the case of linear stochastic systems with Poisson jumps.In Section 4, the  ∞ tracking control design methods based on the Takagi-Sugeno type fuzzy techniques are applied to the nonlinear stochastic systems with Poisson jumps, and the  ∞ tracking controller is obtained.In Section 5, the air-to-air missile pursuit systems are presented to show the effectiveness of the proposed method.
Notation.For convenience, we adopt the following notations: R × : the set of all real  ×  matrices.R  : the set of all −dimensional real vectors. > 0( ≥ 0): the positive definite(semidefinite) matrix .  : the transpose of matrix .: the identity matrix with proper order.E[]: the expectation of random variable .||: the Euclidean norm of vector  ∈ R   .

Preliminaries
Let (Ω, F, F, Pr) be a complete probability space where F = {F  :  ≥ 0} is a filtration generated by Brownian motion (Wiener process) {(),  ≥ 0} and Poisson process {(),  ≥ 0} which are two mutually independent stochastic processes: Let where N denotes the totality of −null sets.Then the filtration F = {F  } ≥0 .We firstly review some basic theories of stochastic differential equations driven by both Brownian motion and Poisson jumps: Under the conditions of Lemma 1, we then review the Itô's formula of (2) (see [32] ( The following lemma is a kind of martingale inequality; see Proposition 7.15 in [33].

𝐻 ∞ Robust Tracking Control of Linear Systems
Consider the linear control system with the following forms: where and  ∈ R   × V are the system's coefficient matrices; () ∈ R  is the state, () ∈ R   is the measurement, () ∈ R   and V() ∈ R  V are the exogenous disturbances; and  1 () is the standard 1−dimensional Wiener process and  1 () is the Poisson process with Poisson intensity  > 0. Now, we review the basic theory of stochastic differential equations driven by both martingale and Poisson jumps.In order to design the tracking control of system (11), a reference model is suggested as follows: where   () ∈ R  is the desired reference state to be tracked by (),   ∈ R × is a specified asymptotically stable matrix, and () is a bounded reference input at the steady state,   () = − −1  ().In practice, () and   are given by user or designer to specify the transient and the steady state of reference signal   () to be tracked.
Denote the tracking errors as For the tracking error   (), we consider the  ∞ robust tracking performance as follows: Here  is a positive definite matrix,   > 0 is the terminal time of control, and w() = [V()  ()  ()  ]  .The physical meaning of ( 14) is that the effect of any w on tracking error   () must be attenuated below a desired level  > 0 from the viewpoint of energy.The following observer is proposed to deal with the state estimation of linear system (11): Denote the estimation errors as Combining ( 11) and ( 15), we get Suppose the controller () has the form as The performance ( 14) considering control effort is revised as where  is a positive definite matrix.Let x = [  ,   ,    ]  and substitute ( 18) into (11); then system ( 11) and ( 17) can be augmented as the following form and the  ∞ performance of (19) equals where In order to achieve the  ∞ tracking performance (19) with a prescribed attenuation level , an auxiliary symmetric positive definite matrix P is suggested.The matrices of Ã and Ẽ include the to-be-designed matrices  and .If the initial state (0),   (0) and the estimation error (0) are also considered, the performance of ( 19) can be described as follows. ∫ Theorem 5. Suppose P ∈ S 3 + (R),  ∈ R   × ,  ∈ R ×  constitute the solution of the following Riccati matrix inequality: where Then the  ∞ tracking control performance in (24) Taking expectation on the both sides, we obtain Thus, we have Completing the square for w, we have where M =  2  − Ẽ  (24), inequality (24) is proved.This ends the proof.
In order to find a solution for matrix inequality (24), suppose P has the following block form and then where we take  = and where ( Then P with form of (31),  =  2  −1 11   and  = − −1    22 are the solutions of ( 24) and (25).Moreover, the  ∞ tracking control performance in (24) is guaranteed for a prescribed  2 .
Proof.Applying the well-known Schur definiteness criterion to symmetric block matrix , together with  =  2  −1 11   and  = − −1    22 , it is easy to see that (38) is equivalent to (24).As far as the  ∞ tracking control performance in ( 24) is guaranteed, it can be directly obtained by Theorem 5.

Nonlinear Systems and Corresponding
Fuzzy Systems the reference model is suggested as follows: The observer is proposed to deal with the state estimation of system (46)  x () = [ ( x ()) +  ( x ())  () Let and denote x = [  ,   ,    ]  , w = [V  ,   ,   ]  .Design the control  with form of and then where For a positive function l( x), an auxiliary positive function Ṽ is suggested.The  ∞ tracking performance considering the initial condition is given by (53) Theorem 8. Suppose Ṽ : R 3 → R + , ( x), Ł( x) constitute the solution of the following Hamilton-Jacobi inequality: where (variable x is omitted) and  ∈ S + (R) is a fixed positive definite matrix with Then the  ∞ tracking control performance in ( 53) is guaranteed for a prescribed  2 > 0.
Proof.Applying Itô's formula to Ṽ( x()), there exist where Keeping in mind that Ṽ is positive and applying (54), we prove that This ends the proof.

Model Rule
Corresponding reference model is suggested as follows: where () = ((), . . .,   ()) are the premise variables,   is the fuzzy set, and  is the number of model rules.

(63)
Then the controller () has the form as Let M (  ()) be the grade of membership of   () in   ; then the membership function of () for the 'th rule is with ∑  =1   (()) > 0. The standard membership function is defined by Then the fuzzy system is inferred as and (71)

Simulation Examples
As the application of tracking system, the air-to-air missile pursuit systems are discussed, which are used to shot down escaping target such as aircraft, fighter, or missile [34,35].The relative dynamic motion between a homing-missile and the escaping target is described by the following system driven by both Wiener process and Poisson process.
Let () be the reference trajectories with Choose the fuzzy variable  =  3 / 1 and the fuzzy rules as follows: Rule 1 ( about −0.45).
(1) = [ [ [ 0 0 0 0 0.17 0 Remark 9. We find that, in the fuzzy rules, the coefficients of  () have the form with where  > 0. For every rule , The above optimal  * 11 could be solved by increasing  until  11 cannot be found in LMIs (86) and (87).
Figure 4 shows the sample trajectories of the original system (72) and the corresponding fuzzy system (67) with  = 0 and V = 0.This illustrates that the T-S fuzzy method can be used to approximate the stochastic nonlinear systems via the fuzzy systems with proper coefficients given by Rule ,  = 1, 2, . . ., 7.

Conclusion
The  ∞ tracking control technique combining with adaptive control algorithm and T-S fuzzy method has been studied to design the  ∞ tracking control to achieve the desired performance for nonlinear stochastic systems driven by both Brownian motion and Poisson jumps.By using adaptive fuzzy control algorithms, the corresponding adaptive fuzzy control laws are derived, which have been employed to treat the  ∞ design problem.A real-world example is used to show the effectiveness of the proposed method.Simulation results illustrate that the proposed adaptive T-S fuzzy control algorithms are practically useful to achieve the  ∞ tracking performance.

Figure 2 :
Figure 2: The geometric relationship between the target and missile.
11,  22 , and  33 are solved by the following steps:   11 +  11  −  2    ≤ −, 3. Fix  11 and  22 which are determined by Steps 1 and 2. The positive definite matrix  33 > 0 can be obtained by solving the LMIs (38) in which  11 and  22 are fixed matrices.