Abstract
This paper is concerned with static output tracking control of nonlinear systems with one-sided Lipschitz condition. The dimensions of system model and reference model may be different. A static output feedback controller is designed to guarantee that the system output asymptotically tracks the reference output with disturbance rejection level. A new sufficient condition is derived to obtain the static output feedback gain by linear matrix inequality (LMI), and no equality constraints can be needed. Finally, an example is given to illustrate the effectiveness of the proposed method.
1. Introduction
Tracking control has been a hot point due to its wide applications. The main objective of tracking control is to minimize the error between the state (or output) of the plant and the state (or output) of a given reference model. So it involves two related problems, that is, state feedback tracking controller design [1–3] and output feedback tracking controller design [4–9]. Among them, the latter is paid much attention because of its attractive features such as low overheads of implementing control, the reliability of control systems, and widely practical applications where measurement of all the state variables is not possible. Furthermore, since the static output feedback case needs much lower costs than an observer-based approach, a few meaningful results about static output feedback tracking control have been presented [10–12].
From above results, it has been shown that the solution of the Riccati equation or LMI depends strongly on the Lipschitz constant, but when the Lipschitz constant becomes large, most of the existing results are infeasible. To enlarge the domain of attraction of nonlinear systems, the one-sided Lipschitz condition is proposed [13, 14, 14–21]. The one-sided Lipschitz constant is significantly smaller than the Lipschitz constant, which makes it much more suitable for estimating the influence of nonlinear part. One-sided Lipschitz condition is shown to be an extension of the Lipschitz condition and is less conservative. Recently, the problem of tracking control of nonlinear systems with one-sided Lipschitz condition has been presented [22]. In [22], the stabilization and signal tracking control for one-sided Lipschitz nonlinear differential inclusions are considered; a nonlinear state feedback tracking controller is designed. However, to the authors’ knowledge, there are very few studies concerning static output tracking controller design of nonlinear systems with one-sided Lipschitz condition. These motivate our study.
This paper considers static output tracking control of nonlinear systems with one-sided Lipschitz condition. The dimensions of system model and reference model may be different. A design procedure of static output feedback controller is proposed to guarantee that the system output asymptotically tracks the reference output with disturbance rejection level. A new sufficient condition is obtained by LMI, and no equality constraints can be needed. These will reduce the difficulty in solving output feedback gain. Finally, an example is given to illustrate the effectiveness of the proposed method.
Notations. and denote, respectively, the spaces of -dimensional real numbers and real matrices. Let be a real symmetric matrix; means is positive definite. is the inner product in ; that is, given , , then , where is the transpose of the column vector . is an identity matrix with dimension . refers to either the Euclidean vector norm or the induced matrix 2-norm. represents the omitted symmetric element of a matrix.
2. Problem Formulation
Consider the following nonlinear system: where is the state vector and is the control input. is the system output. represents a nonlinear function that is continuous with respect to and . are matrices with compatible dimensions and is controllable. is of full rank. is a bounded disturbance.
The following concepts about Lipschitz property, the one-sided Lipschitz property, and quadratic inner-boundedness property for the nonlinear function are introduced to further our study.
Definition 1 (see [22]). The nonlinear function is said to be locally Lipschitz in a region including the origin with respect to , if there exists a constant satisfying
Definition 2 (see [22]). The nonlinear function is said to be one-sided Lipschitz, if there exists a constant such that where is called the one-sided Lipschitz constant.
From Definitions 1 and 2, Lipschitz constant must be positive; however, one-sided Lipschitz constant can be positive, zero, or even negative. It is true that any Lipschitz function is also one-sided Lipschitz, not vice versa [15–17, 22].
Definition 3 (see [22]). The nonlinear function is called quadratic inner-bounded in the origin , if there exists constants such that with .
From the definition, the Lipschitz function is quadratically inner-bounded with and , but the converse is not true. Note that is not necessarily positive. In fact, if is restricted to be positive, then it can be shown that must be Lipschitz.
Consider a reference model as follows [4]: where is the reference state, is a bounded reference input, and is the reference output. are matrices with compatible dimensions and is a specific asymptotically stable matrix.
Let . From (1) and (5), we have where and ; and satisfy
Remark 4. In (7), the values of can be obtained by the method in [8]. By using Kronecker product of matrices and the operation, (7) can be equivalent to where If and only if , one of the solutions is where denotes the Moore-Penrose inverse of .
For (6), we design the following static output feedback controller where is from (7) and is a static output feedback gain and determined later.
From (5), (6), and (11), the closed-loop system can be written as where and .
Note that the output matrix is of full rank; without loss of generality, we assume that , is nonsingular.
The objective of this paper is to design a static output feedback controller (11) such that the closed-loop system (12) is asymptotically stable with an -norm bound ; that is, the following conditions are achieved simultaneously.(i)The closed-loop system (12) with is asymptotically stable.(ii)The closed-loop system (12) has a given disturbance rejection level. It is to make, under the zero-valued initial condition, the following inequality holds: for any nonzero , where is a prescribed scalar.
3. Main Results
In this section, a design algorithm is proposed to obtain the output feedback gain via LMI and the sufficient condition includes no equality constraints.
Theorem 5. Given a constant . Suppose that the function in the system (12) satisfies conditions (2) and (3) with constants , and . The system (12) is asymptotically stable with an -norm bound , if there exist matrices , and scalars , such that the following matrix inequality is true where , , , , , and .
Proof. Consider the following Lyapunov functional candidate:
Calculating the time derivative of along the trajectory of system (12), we have
On the other hand, we have
where , are free-weighting matrices. In the following, can be shortened as for simplicity.
From (3), . Therefore, for any positive scalar , we have
From (4), similarly, for any positive scalar , we have
Then, adding (17) and the left-sided terms in (18) and (19) to time derivative of yields
where .
From (14), we can have
Under the zero initial conditions, we have
That is, .
In fact, when , in the same way, we easily obtain
where and
where .
Noting that (14) implies , we have
This implies that the system (12) with is asymptotically stable. Then the proof is completed.
For (14), it is a nonlinear matrix inequality, there are no effective algorithms for solving , , , , , and scalars , simultaneously. The following theorem gives an approach to solve this problem.
Theorem 6. Given a constant . Suppose that the function in the system (12) satisfies conditions (2) and (3) with constants , and . and are priori selected tuning parameters. The system (12) is asymptotically stable with an -norm bound , if there exist matrix , general matrix with , and scalars , such that the following matrix inequality is true:where , , and , . Furthermore, .
Proof. The solution of (14) can be obtained by the following two-step procedures.
In the first step, can be given.
In the second step, for (14), choose and denote , . Left- and right-multiplying both sides of (14) by and , respectively, we obtain
where , .
By considering the structure of , we have
Define
Then we have
Substituting (30) into (27) and using Schur’s Lemma yield (26).
Because is given in advance, (26) is converted to a LMI. So we easily solve it by Matlab LMI toolbox [23]. The proof is completed.
Remark 7. Theorem 6 provides a new solving method about static output feedback gain by LMI, when the parameters are given in advance. Moreover, no equality constraints are needed, so the difficulty in solving it can be reduced. It should be noted that the special form of matrix will bring some conservatism; however, thanks to this form, the output feedback gain can be easily obtained [24].
The procedure of the proposed static output tracking control of nonlinear systems with one-sided Lipschitz condition is summarized as follows.
Step 1. Find the solutions of and from (10). If there exists no solution, then the reference model must be modified.
Step 2. Observe the structure of in (12): if , is nonsingular, then the procedure directly goes to the next step, or there exists a state transformation for (12) to make satisfy this structure, where is a nonsingular matrix.
Step 3. Choose the parameters , , , and in Theorem 6.
Step 4. Calculate the matrix and in by solving LMI (26).
Step 5. Obtain the static output feedback gain .
4. Discussion
Theorem 6 gives a new approach to solve the static output feedback gain; if we use traditional method including the equality constraints, the following theorem can be given.
Theorem 8. Given a constant . Suppose that the function in the system (12) satisfies conditions (2) and (3) with constants , and . The system (12) is asymptotically stable with an -norm bound , if there exist matrices , and scalars , such that the following matrix inequality is true:where , , , and . Furthermore, .
Proof. The proofs can be easily obtained similar to the arguments in Theorem 6; the details are omitted.
Remark 9. Noting that (31) can be converted to a LMI on condition that the parameters are also given in advance, the equality condition can be equivalently converted to trace . Introduce the condition and Schur’s complement gives Hence, the static output tracking problem can be changed to a problem of finding a global solution of the following minimization problem:
Remark 10. From Theorems 6 and 8, the sufficient conditions in Theorem 8 are more complex, which lead to a large amount of calculation. So the static output feedback gain can be conveniently obtained by the method in Theorem 6.
5. Example
Consider the systems (1) and (5) with , , , , , , , and .
is given by
The above system model can be used to describe the motion of a moving object [22].
Let According to [22], the quadratically inner-bounded property of is verified in , . As the system is globally one-sided Lipschitz, that is, , . Note that the region can be made arbitrarily large by choosing appropriate values for and .
We choose and ; then .
The initial value is and .
By solving (10), and can be obtained as
Given , and , then solving LMI (26) yields
Furthermore, noting that , we obtain
Then the controller is designed as
The simulation results are shown in Figures 1, 2, and 3.
Figure 1 shows the time response of the error . The curves about system output and reference output are shown in Figure 2. Figure 3 shows the control input response curve. From the simulation results, it is concluded that the proposed method is effective.
Remark 11. By Theorem 8, we also can obtain the static output feedback gain, but both conditions (31) and (33) are calculated simultaneously. So the method by Theorem 6 is relatively simple.
6. Conclusion
In this paper, the problem of static output tracking control of nonlinear systems with one-sided Lipschitz condition has been investigated. A static output feedback controller is designed to guarantee that the system output asymptotically tracks the reference output with disturbance rejection level. A sufficient condition is derived to obtain the static output feedback gain by linear matrix inequality (LMI), and no equality constraints can be needed. It should be noted that the special form of matrix will bring some conservatism, so further work is to study how to reduce it.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful for the supports of the National Natural Science Foundation of China under Grants U1404610, 61374077, and 61203047, fundamental research project (142300410293, 142102210564) in the Science and Technology Department of Henan province, the Science and Technology Research Key Project (14A413001) in the Education Department of Henan province, and innovation ability cultivation fund (2014ZCX015) in Henan University of Science and Technology.