Abstract
A robust adaptive 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 desired tracking performance for stochastic systems with exogenous disturbance. A simulation example is presented to illustrate the tracking performance of the proposed design method.
1. 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 [3–6].
The robust tracking control theory is a powerful methodology to design feedback controllers where the system parametric uncertainties are seen as the exogenous disturbance [7–9]. 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 [15–17].
Stochastic systems are applied in economics [18], biology [19], and natural science to describe the randomness in models [20–23]. Based on the distribution properties of the inserted random variables, stochastic systems include Itô-type 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 [28–30]. 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 fuzzy-based 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: : the set of all real matrices. : the set of all dimensional real vectors. : the positive definite(semidefinite) matrix . : the transpose of matrix . : the identity matrix with proper order. : the expectation of random variable . : the Euclidean norm of vector .
2. Preliminaries
Let be a complete probability space where is a filtration generated by Brownian motion (Wiener process) and Poisson process which are two mutually independent stochastic processes:(i) is a dimensional standard Brownian motion with and ;(ii) is a Poisson jump process with rate and .
Letwhere denotes the totality of null sets. Then the filtration . We firstly review some basic theories of stochastic differential equations driven by both Brownian motion and Poisson jumps:
Lemma 1. Let be valued functions on and, for some positive constant , satisfy the Lipschitz conditionand linear growth conditionfor all , . Then the stochastic differential equation (2) has a unique adapted solution with right continuation and left limitation.
Under the conditions of Lemma 1, we then review the Itô’s formula of (2) (see [32], Chapter 4, Rule 4.24).
Lemma 2. Let be twice continuously differentiable in and once in ; is the solution of stochastic differential equation (2). Then
The following lemma is a kind of martingale inequality; see Proposition 7.15 in [33].
Lemma 3. For a finite , let be a submartingale (or martingale, or supermartingale) with respect to filtrations . Then for any , there exists
Remark 4. Because Brownian motion is a martingale with respect to filtration , and Poisson process is a submartingale, by Lemma 3, for every finite , there exist and or equivalently, and Therefore, when is large enough, and have upper bound with probability and , respectively. Furthermore, and when .
3. Robust Tracking Control of Linear Systems
Consider the linear control system with the following forms:where , , , , and are the system’s coefficient matrices; is the state, is the measurement, and are the exogenous disturbances; and is the standard dimensional Wiener process and is the Poisson process with Poisson intensity .
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 is the desired reference state to be tracked by , is a specified asymptotically stable matrix, and is a bounded reference input at the steady state, . 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, is the terminal time of control, and . The physical meaning of (14) is that the effect of any on tracking error must be attenuated below a desired level from the viewpoint of energy. The following observer is proposed to deal with the state estimation of linear system (11):Denote the estimation errors asCombining (11) and (15), we getSuppose the controller has the form asThe performance (14) considering control effort is revised aswhere is a positive definite matrix. Let and substitute (18) into (11); then system (11) and (17) can be augmented as the following formand 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 is suggested. The matrices of and include the to-be-designed matrices and . If the initial state , and the estimation error are also considered, the performance of (19) can be described as follows.
Theorem 5. Suppose constitute the solution of the following Riccati matrix inequality:where Then the tracking control performance in (24) is guaranteed for a prescribed .
Proof. Let , . By Itô formula, we get Taking expectation on the both sides, we obtain Thus, we have Completing the square for , we have where and . Together with (24), inequality (24) is proved. This ends the proof.
In order to find a solution for matrix inequality (24), suppose has the following block formand thenwhere and Because and we take and . Then
Proposition 6. Suppose , , and are the solution of the following block matrix inequality:andwhere Then with form of (31), and are the solutions of (24) and (25). Moreover, the tracking control performance in (24) is guaranteed for a prescribed .
Proof. Applying the well-known Schur definiteness criterion to symmetric block matrix , together with and , 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.
Remark 7. In equation (32) and (37), if there exists such that , this implies i.e., Thus, the following systemis stable. Furthermore, if there exists such that , this implies Therefore, system (43) is asymptotically stable; i.e., is stable. In this paper, the estimation error system (17) can be seen as the extension of (43) where (17) presents the differences between and . Furthermore, in (38), ; i.e., and if , then is stable. However, in general, ; the matrix is not necessarily stable.
4. Nonlinear Systems and Corresponding Fuzzy Systems
For nonlinear system as in the following formsthe reference model is suggested as follows:The observer is proposed to deal with the state estimation of system (46)Let and denote , . Design the control with form ofand thenwhere For a positive function , an auxiliary positive function is suggested. The tracking performance considering the initial condition is given by
Theorem 8. Suppose , , constitute the solution of the following Hamilton-Jacobi inequality:where (variable is omitted) and is a fixed positive definite matrix withThen the tracking control performance in (53) is guaranteed for a prescribed .
Proof. Applying Itô’s formula to , there exist where Keeping in mind that is positive and applying (54), we prove that This ends the proof.
The nonlinear system (46), (47), (48), (50), and (51) can be described by T-S fuzzy model as follows.
Model Rule IF is , and and is , THENCorresponding reference model is suggested as follows:where are the premise variables, is the fuzzy set, and is the number of model rules.
The following observer is proposed to deal with the state estimation of fuzzy system (61)Then the controller has the form as
Let be the grade of membership of in ; then the membership function of for the ’th rule iswith . The standard membership function is defined byThen the fuzzy system is inferred asandwith estimation errorsThen the controller has the form asand
5. 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.Here is the system state which describes the trajectory of missile, concretely; is the relative distance between the missile and target; see Figure 1; is the radial relative velocity; is the tangential relative velocity; see Figure 2; and is the input control with and being the missiles acceleration components along the and axes. denotes the continuous noise of the system such as the air friction caused by air resistance and the inner thermal noise caused by missile operating system. Poisson process denotes the sudden discrete noise of the system such as directional trimming, or deceptive behavior of the missile to avoid being shot down by shooting out gas from two side jets. is the measurement and is the fighting time which denotes the missile’s effective working time, and .
Take (sec), (m), (m/s), (rad/s), , , , , and the density of Poisson jump is 0.02; i.e., , , and Let the trajectory of the target’s motion which is embedded in a reference model as follows:where which implies the steady state . Let be the reference trajectories with Choose the fuzzy variable and the fuzzy rules as follows:
Rule 1 ( about ).
Rule 2 ( about ).
Rule 3 ( about ).
Rule 4 ( about ).
Rule 5 ( about ).
Rule 6 ( about ).
Rule 7 ( about ).
Remark 9. We find that, in the fuzzy rules, the coefficients of have the form with where . For every rule , , , and are solved by the following steps:
Step 1. Solve such that subject toThe above optimal could be solved by increasing until cannot be found in LMIs (86) and (87).
Step 2. Fix solved by Step 1. Suppose has the form of where . Solve such thatwhere is determined by (37), . Under the conditions of , , for a fixed , can be solved by increasing until matrix inequality (89) can be true.
Step 3. Fix and which are determined by Steps 1 and 2. The positive definite matrix can be obtained by solving the LMIs (38) in which and are fixed matrices.
The weighted function is given by where , , , , . Figure 3 shows the profiles of , .
Figure 4 shows the sample trajectories of the original system (72) and the corresponding fuzzy system (67) with and . 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 , .
By (70), the tracking fuzzy control is obtained byHere , . Figure 5 illustrates the sample trajectories of fuzzy control for system (72) with Poisson process. It shows that the values of control change violently when Poisson jumps occur.
Figures 6, 7, and 8 show the sample trajectories of the components of the state , the reference , and the estimation state . These show that, under the proposed tracking fuzzy control (93), the state and its estimation of fuzzy system (67) with coefficient matrices provided by Rules 1–7 corresponding to (72) take values around the reference trajectory . Figure 9 shows the estimation error which illustrates the performance difference between the state estimation and the state . The four figures confirm that the control defined by (93) can achieve the desired performance (53) with , where is defined by (71).
Now, we compare our results with that in [35]. In [35], the guidance law uses the information for the target acceleration bound, and the guidance command is derived based on the nonlinear planar engagement kinematics. In our guidance law design, we consider the nonlinear model of the system with fuzzy method. Some comparisons are given as follows:
In the control algorithm of [35], the sliding mode control scheme is used to obtain the guidance law. In our control design, an adaptive fuzzy scheme is employed to cancel the exogenous disturbance in the stochastic system with Brownian motion and Poisson jumps.
The controller obtained in [35] is based on solving a nonlinear function . In our paper, the robust tracking controller is given with fuzzy forms in (70) which is only based on the locally linear approximation.
The controller in [35] depends on the sliding surface and the state. Our controller in (70) depends on the difference between the state and the reference trajectory and the estimation errors .
6. 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 61573227), the Research Fund for the Taishan Scholar Project of Shandong Province of China, the Natural Science Foundation of Shandong Province of China (No. ZR2016FM48, ZR2015FM014), SDUST Research Fund (No. 2015TDJH105), and the Ministry of Science and Technology of Taiwan under the contract No. MOST-106-2221-E-007-010-MY2.