Abstract
This paper considers the parametric control to the Lorenz system by state feedback. Based on the solutions of the generalized Sylvester matrix equation (GSE), the unified explicit parametric expression of the state feedback gain matrix is proposed. The closed loop of the Lorenz system can be transformed into an arbitrary constant matrix with the desired eigenstructure (eigenvalues and eigenvectors). The freedom provided by the parametric control can be fully used to find a controller to satisfy the robustness criteria. A numerical simulation is developed to illustrate the effectiveness of the proposed approach.
1. Introduction
Lorenz system, first studied by Edward Lorenz, is a simplified mathematical model for atmospheric convection [1], which can display a phenomenon called chaos. Chaotic phenomena can be detected in many physical systems such as mechanical systems, electrical systems, and thermal systems [2]. Many scholars pay close attention to it and get a variety of positive achievements. Kim et al. put forward a robust control approach to regulate and synchronize the generalized Lorenz system based on the backstepping method, while the nonlinear and uncertain items can be estimated and canceled [3]. By using the Lyapunov function and Barbalat’s lemma, Liu et al. consider the problem of synchronization and antisynchronization in the Lorenz system and apply the result to secure communication with uncertain parameters [4]. Wu et al. propose an adaptive control controller composed of a wavelet network and a proportional controller containing adaptive gain; with this controller, the coupled Lorenz system is globally stable [5]. Yu et al. present a novel approach through switched control and superheteroclinic loops to linearize two symmetrical equilibria and also give the circuit implementation [6]. Zhang et al. design a hybrid controller for the Lorenz system with a piecewise linear memristor and provide criteria to maintain that the trivial solutions are exponentially stable in the mean square [7]. Simultaneously, hidden attractors of the classical Lorenz system are also discussed [8, 9]. More results of the Lorenz system can be found in the literature [10, 11].
Robust control is an approach to design controller that explicitly deals with uncertainty. The most commonly used method in the design and synthesis of robust controllers is based on linear matrix inequalities (LMIs) (see [12–14] and the references therein). Lindemann and Dimarogonas consider the robust model predictive control that the robust optimization problem is transformed into a convex quadratic program [15]. Liu et al. investigate the robust formation of tracking control for multiple quadrotors by linear quadratic regulation and robust compensation theory [16]. Cristofaro proposes a robust sliding-mode controller to globally stabilize the tracking error system with disturbances [17]. There also are several methods to deal with robust control and optimization [18–21]. However, the goal of existed methods is to obtain a controller that satisfies some specific requirements, such as robustness. Once the design requirements change, the design needs to be repeated.
Parametric approach, creatively proposed by Duan [22, 23], solves the controller design of quasi-linear systems via state feedback and output feedback, which develops novel research areas. Furthermore, Gu et al. extend a parametric approach to second-order and high-order quasi-linear systems [24–29] and chaotic systems [30]. In this paper, based on the solutions of a class of GSE [31, 32], a whole set of parametric state feedback controllers is established and the closed-loop system is transformed into the desired eigenstructure. The parametric approach provides a group of arbitrary parameters that represent the degrees of design freedom, to implement state feedback and robust optimization.
The major contributions of this paper are summarized in two aspects. On the one hand, the complete parameterization form of state feedback is established such that closed-loop system has an expected eigenstructure, which results in that the chaos phenomenon has been eliminated effectively. On the other hand, the degrees of design freedom in arbitrary parameters are fully used to realize the robust optimization such that robustness can be improved obviously.
The remainder of this paper is organized as follows. Section 2 analyzes the dynamical behaviors of the Lorenz system. Section 3 presents the problem statement of parametric design for Lorenz system via state feedback and provides some preparations. In Section 4, the generally parameterized form of state feedback is established in two cases, and robust optimization is realized by using the degrees of freedom in arbitrary parameters. In Section 5, a numerical example is presented to prove the parametric approach is effective and feasible. Section 6 concludes the proposed results.
Notation: we propose some notations which are used in this paper. Rank, det, and eig represent the rank, determinant, and all eigenvalues of matrix , respectively. deg denotes the degree of polynomial matrix ; diag indicates the diagonal matrix with diagonal elements , . max and min represent the maximum and minimum.
2. Analysis of the Lorenz System
Consider a class of generalized Lorenz systems as follows:where , , are state variables and , , are positive real constants. Let , and the system (1) can be equivalently written aswithand is a smooth nonlinear vector field satisfying
The divergence of the system (1) is given asand with the rate of contraction . Inequality (3) demonstrates the existence of chaotic attractor of the system (1).
With , and , two-dimensional phase portraits of the system (1) are shown in Figure 1.
(a)
(b)
(c)
3. Problem Statement and Preliminaries
3.1. Problem Statement
Consider the following controlled Lorenz system:where is the input vector and and are coefficient matrices satisfying Assumption 1.
Assumption 1.
For the system (6), using the control inputwe can obtain the closed-loop systemwithwhere is the feedback gain matrix to be determined.
The parametric control of Lorenz system (6) via state feedback (7) can be stated as follows.
Problem 1. (PCLS). Let the controlled Lorenz system (6) satisfy Assumption 1. Given an arbitrary matrix find for the system, a state feedback controller in the form of (7) and a nonsingular matrix , satisfying
Remark 1. The above problem requirement (10) is equivalent to find a state feedback controller (7) such that the closed-loop system (8) is similar to a linear time-invariant form with the desired eigenstructure; that is, is similar to an arbitrary matrix by the proposed parametric approach. Because the closed-loop system (8) is stable [33, 34], is chosen as a Hurwitz matrix, that is,
3.2. Preliminaries
When Assumption 1 is met, there exists the following right coprime factorization (RCF):
Denote , andThen,
4. Main Results
4.1. Parametric Control of Lorenz System
With the above discussion, we propose the following Theorem 1 regarding Problem 1 (PCLS).
Theorem 1. and are given by (14) and satisfy RCF (12). Problem 1 (PCLS) has a general solution if and only if there exists an arbitrary parameter matrix such thatwhere
Furthermore, the feedback gain matrix is solved bywhere
Proof. Equation (10) is equivalent toLetthen equation (19) becomes the GSE:Using (14), we haveSubstituting the above equations into (12) and equating the coefficients of on both sides, we obtainBased on equations (16), (18), and (23), we haveThis proves that the matrices in (16) and in (18) satisfy the GSE (21). Finally, combining equations (16) and (18) with (20), the feedback gain matrix can be obtained as equation (17).
The proof is completed.
In application, we choose the matrix in the diagonal form ofwhere . Based on Theorem 1, we can derive the following Corollary 1 to Problem 1 (PCLS).
Corollary 1. and are given by (9) and satisfy RCF (8). Problem 1 (PCLS) has a general solution if and only if there exists a group of parameter vectors , such that condition (15) is satisfied, where matrix can be given by columns
Furthermore, the feedback gain matrix is solved bywhere matrix can be given by columns
4.2. Robust Optimization
Theorem 1 and Corollary 1 show that the matrix is an arbitrary parameter which can provide the degrees of freedom to analyse and design problems. In this subsection, robust optimization is considered such that regional pole assignment and robustness criteria are investigated.
4.2.1. Regional Pole Assignment
We aim to locate the eigenvalues , in an admissible set to satisfy control requirements of practical applications. Actually, eigenvalues can be considered within a small interval around the expected locations to reduce the difficulty and improve the flexibility when implementing controller. In this case, the complex pole is defined aswithwhere and are real and imaginary parts of complex pole and , and , are the upper and lower bound of and , respectively. For real pole, equation (29) can be simplified asEquation (29) shows the arbitrary parameter plays an important role in determining the location of eigenvalues, which becomes the decision factor in robust optimization.
4.2.2. Robustness Criteria
Thus, this paper fully employs the freedom to find a feedback controller to satisfy the robustness criteria. For the closed-loop system (8), a bounded disturbance is considered aswhere is the disturbance vector and is the disturbance matrix. In this subsection, a multiobjective robustness criterion is proposed to improve the robustness of the closed-loop system (8).
First, let us introduce the robustness degree [35], a simple and useful method to measure the ability of system for overcoming external disturbances, modeling errors, and other uncertainties, defined aswhere .
Second, we consider the disturbance attenuation which investigates the effects of external disturbances to closed-loop systems, and the objective is expressed in terms of the -norm of the following function [36]:
Third, in order to maintain the performance robustness and stability robustness when parameter uncertainty exists, a general way is to minimize the sensitivity function for closed-loop eigenvalues, and the overall eigenvalue sensitivity is defined as follows:
Combining equations (33)–(35), the multiobjective robustness criteria are established to represent the robustness aswhere , and are the weight coefficients satisfyingwhich means that weight coefficients , and can be updated in real time. Then, we formulate the robust optimization into a multiobjective problem as
Remark 2. Based on the above results, a guideline for robust parametric control of Lorenz system (1) via state feedback is proposed as follows: Step 1: determine a Hurwitz matrix with an expected eigenstructure. Step 2: obtain the based on (12). Step 3: seek arbitrary parameter by robust optimization problem (38). Step 4: calculate the coefficient matrix of state feedback (17).
5. An Example
5.1. System Description
Consider the controlled Lorenz system (6) with , and and we have
Letbased on RCF (8), and and are obtained as
We takeand a bounded disturbance
5.2. Nonoptimized Solution
Using the randn function in Matlab® to generate an arbitrary parameter matrix asand ; according to Theorem 1 or Corollary 1 with (41), we have
By using the feedback controller (46), the closed-loop system is
5.3. Robust Solution
Consider the robust optimization problem (38). Choosing the initial value in (44), the optimal can be obtained as by the fminsearch function in MATLAB Optimization Toolbox:and ; according to Theorem 1 or Corollary 1 with (41), we have
By using the feedback controller (50), the closed-loop system is
5.4. Simulation Comparison
Choose the initial value as , and the simulation results are shown in Figures 2–6.
Figures 2 and 3 show that robust state feedback yields a better transient performance than that of nonoptimized one. In Figure 4, although the amplitude of the robust solution is higher than nonoptimized one when the disturbance existed, it reduces the frequency of oscillation and arrives at stability rapidly. From Figures 5 and 6, we see that the robust solution consumes less energy.
Let and represent the optimized and nonoptimized indices, and we have
, which means the robustness of the closed-loop system is improved effectively through the robust optimization based on degrees of freedom provided by the proposed parametric approach.
6. Conclusion
A parametric approach, inspired by the solutions of GSE, is provided aiming at the Lorenz system in this study. The proposed parametric approach establishes a more unified parametric expression of state feedback concerning matrices and and then transforms the closed-loop system into a linear constant form with the desired eigenstructure. Meanwhile, a robust optimization scheme is also considered based on degrees of design freedom given by the parametric approach. In this scheme, these arbitrary parameters to be optimized are with no physical significances; thus, the optimization interval can be greatly extended, which is a benefit to obtain an approximately globally optimal solution.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Grant nos. 61690210 and 61690212).