The cell-mapping method, due to its global optimality, has been applied to solve multi-objective optimization problems (MOPs) and optimal control problems. However, the curse of dimensionality limits its application in high-dimensional systems. In this paper, the multi-parameter sensitivity analysis is investigated to reduce the parameter space dimension, which broadens the application of cell mapping to MOPs in high-dimensional parameter space. A post-processing algorithm for MOPs is introduced to help choose proper control parameters from the Pareto set. The proposed scheme is applied successfully in the control parameter optimization of an adaptive nonsingular terminal sliding-mode control for an antenna servo system on a disturbed carrier. Moreover, as the existing global optimal tracking control with an adjoining cell-mapping method may generate tracking-phase differences, an optimal-sliding-mode combined-control strategy is proposed. By using the combined-control strategy, the azimuth and pitch angles of the antenna system are controlled to catch up to a target trajectory with the minimum cost function and to keep high-precision tracking after that.

Cell mapping is a highly efficient numerical method for the global dynamical characteristic analysis of nonlinear dynamical systems. With the cell-mapping method, a state space region is discretized into a set of cells, and the dynamical behaviors described by an infinite number of point-to-point mappings can be represented by a finite number of cell-to-cell mappings. Then, long-term global dynamical characteristics, including attractors, domains of attraction, equilibrium states, etc., can be obtained by studying short-term cell-mapping relationships. The simple cell mapping (SCM) was first proposed by Hsu (1980). After that, the cell-mapping method was developed by a lot of scholars (Dellnitz and Hohmann, 1997; Sun and Luo, 2012). Due to its powerful global analysis ability, cell mapping has been extended to control optimization fields including multi-objective optimization problems (MOPs) (Fernández et al., 2016) and optimal controls (Martínez-Marín and Zufiria, 1999).

One application of the cell-mapping method in control optimization fields is the solving of MOPs. In a control process, multiple performance indexes are usually considered to evaluate the control performance, such as overshoot, peak time, or steady-state error. The design of control parameters to meet multiple conflicting performance objective functions simultaneously as far as possible yields the multi-objective optimization problem. The traditional single-objective optimization problem has only one unique solution. But the MOP has a set of solutions, which is called the Pareto set. The corresponding optimization objective function values are named the Pareto front. Many numerical algorithms for solving MOPs have been developed, such as the immune algorithm (Chen et al., 2019), particle swarm optimization (Peng et al., 2019; Zhang et al., 2022) and the genetic algorithm (Du et al., 2016). These algorithms are mainly based on bionics. Recently, the cell-mapping method was developed as an effective numerical method to solve MOPs. The cell-mapping method to deal with MOPs is one kind of set-oriented method with subdivision technology. It is global in nature and allows one to approximate the entire set of global Pareto points. Compared with other optimization methods, it can guarantee global optimality well to a great extent and can obtain the Pareto set with one single run of the algorithm. Besides, it is applicable to a wide range of optimization problems and is characterized by an excellent robustness. The SCM was firstly used for multi-objective optimal proportion integration differentiation (PID) control for a nonlinear system with a time delay (Xiong et al., 2013). Then a hybrid algorithm consisting of gradient-based and gradient-free laws for MOPs was presented (Xiong et al., 2014), and a hybrid method consisting of the genetic algorithm and SCM was proposed (Naranjani et al., 2014). A post-processing strategy to select control parameters from the Pareto set is shown by Qin and Sun (2017). The algorithm first sets an ideal point and then circles the ideal point with a radius to narrow the selective area of optimal solutions. As mappings from different pre-image cells to their image cells need to be constructed in the cell-mapping method, it is easy to imagine that the calculation task will grow rapidly when the dimension of an analyzed dynamical system increases. Because the construction of cell-to-cell mappings are naturally parallelizable, the graph process unit (GPU) parallel technology was introduced to the global analysis with the cell-mapping method for high-dimensional nonlinear dynamical systems (Xiong et al., 2015). The MOP design of a sliding-mode control with a parallel simple cell-mapping method was studied (Qin et al., 2017). The multi-objective optimal motion control of a twin-rotor model helicopter based on the parallel simple cell-mapping method was presented (Qin et al., 2020). The cell-mapping method is an interesting alternative to the classical mathematical programming method. It has been applied successfully to lowly and moderately dimensional MOPs. However, the curse of dimensionality still exists even though the GPU parallel technology is adopted. That is the main limitation of the cell-mapping method in its application to MOPs for more control parameters.

Another application of the cell-mapping method in control optimization fields is the solving of optimal control problems. Optimal controls have been widely applied in many engineering fields (Lu et al., 2019). However, the analytic optimal control solutions are usually difficult to find for complex nonlinear systems, especially when the state space region and control inputs are constrained. The cell-mapping method provides an efficient numerical way to solve optimal control problems for complex nonlinear dynamical systems. More importantly, the cell-mapping method searches optimal control solutions in the whole state space region, which can guarantee the global optimality better compared with other methods. The SCM was firstly introduced to solve optimal control problems by Hsu (1985). The solving of fixed-final-state (Crespo and Sun, 2000) and fixed-final-time (Crespo and Sun, 2003) optimal control problems with SCM was studied. To decrease discretization error, the adjoining cell-mapping method was investigated to solve optimal control problems by Zufiria and Martínez-Marín (2003). The optimal control with the adjoining cell mapping performs closed-loop feedback control, which makes it applicable in real physical systems. As for optimal tracking control problems, even numerical solutions are difficult to obtain. The SCM was used for optimal control of tracking moving targets with a bounded state space region by Crespo and Sun (2001), but it was limited to only low-dimension single-input–single-output systems. Recently, a subdivision strategy of adjoining cell mapping was proposed to deal with fixed-final-state global optimal control problems for multi-input–multi-output (MIMO) systems (Cheng and Jiang, 2021), and it was extended to solve the global optimal tracking control for MIMO systems (Tian et al., 2023). However, the low steady-state tracking accuracy and the existence of phase differences between the target trajectory and the real trajectory limit the application of adjoining cell mapping to a wider range of optimal control problems.

In brief, on the one hand, the cell-mapping method is still limited by enormous calculations when dealing with the MOP in high-dimensional parameter space. On the other hand, the global optimal control with the cell-mapping method is currently mainly focused on low-latitude single-input–single-output dynamical systems. Even the subdivision strategy for adjoining cell mapping has been proposed to solve fixed-final-state global optimal controls and global optimal tracking controls for MIMO systems; further research should still be conducted. Therefore, in this paper, the MOP with the cell-mapping method in high-dimensional parameter space is studied, and the global optimal tracking control with the cell-mapping method for nonlinear MIMO systems is investigated. The proposed approaches are applied in the control optimization for a MIMO antenna servo system on a disturbed carrier.

This paper is organized as follows. The modeling and adaptive nonsingular terminal-sliding mode control (ANTSMC) design for the antenna servo system are described in Sect. 2. In Sect. 3, the control parameters' multi-objective optimization for ANTSMC based on multi-parameter sensitivity analysis and simple cell mapping is illustrated, and a post-processing algorithm for MOPs is introduced. In Sect. 4, an optimal-sliding-mode combined-control (OSCC) strategy is proposed and applied in the global optimal tracking control for the antenna servo system. Finally, Sect. 5 concludes the paper. The main contributions of this paper are as follows:

The multi-parameter sensitivity analysis is adopted to reduce the dimension of parameter space, which broadens the application of the cell-mapping method to MOPs in high-dimensional parameter space.

A post-processing algorithm for MOPs is introduced, which can help to select proper control parameters from the Pareto set.

An optimal-sliding-mode combined-control strategy is proposed, which offers a widely applicable and efficient numerical way to solve optimal tracking control problems for nonlinear MIMO systems in engineering.

The antenna system on a mobile carrier has the advantage of real-time communication without the limitation of geographical conditions. It has already been widely applied in the fields of battlefield communications, emergency communication, rescue operations and television relay. An antenna system on a mobile carrier consists of many components, including an antenna, electrical motors, inertial navigation devices and so on. As the carrier is always under external, large disturbance, it is difficult to control the antenna attitude quickly and accurately by means of the classical PID strategy. Consequently, the dynamical modeling is necessary for the control design. The sketch map of an antenna system on a mobile carrier is shown in Fig. 1. Wire-cable vibration isolation equipment

The sketch map of the antenna system.

As the sliding-mode control possesses good robustness against external disturbance, it is adopted to control the antenna attitude angles for this servo system. The control objective is to make the azimuth

To improve the control performance of ANTSMC, control parameter optimization is necessary. The designed control parameter vector is

The value ranges of ANTSMC parameters.

The overshoot, peak time and steady-state error are usually used to characterize the performance of the closed-loop feedback control. The MOP in this paper can be described as

The cell-mapping method to deal with MOPs can find the global and fine structure of the Pareto set through one single run of the program. Although the global optimality can be guaranteed well, the computational time required increases dramatically when the dimension of the design parameter space increases. Consequently, it is poorly suited to dealing with MOPs in high-dimension parameter space due to the curse of dimensionality. In this paper, we consider the use of sensitivity analysis technology to realize the dimensionality reduction of the parameter space effectively.

There are 12 control parameters in the ANTSMC design for the antenna system. Direct optimization of the 12 control parameters will consume too much computation, which is not cost-effective and is even unfeasible. In fact, these parameters may have different influences on the change of the objective function values. When these parameters change simultaneously, some may play relatively important roles in the system output response compared with the others. Consequently, it is easy to imagine that the relatively important control parameters can be optimized first and the others after that. In this way, the dimensionality of the parameter space is reduced, which greatly economizes the computational cost, and the approximate MOP solutions can be obtained with sufficient accuracy.

This brings us to the next problem of how to assess the influence of each control parameter on the objective functions of the decision maker. The main purpose of parameter sensitivity analysis is to evaluate the importance of each input parameter to the system output. There are two main methods for sensitivity analysis: single-parameter sensitivity analysis and multi-parameter sensitivity analysis. Among them, the multi-parameter sensitivity analysis method allows the situation in which multiple parameters change simultaneously, which has global significance. The Sobol method (Sobol and Kucherenko, 2009) is a classical and sophisticated and the most widely used multi-parameter sensitivity analysis method. It is a sort of Monte Carlo method based on variance and has the advantages of fast convergence and good stability. It also has good applicability in the situation of obvious multiple-parameter interactions in the strongly nonlinear systems. Moreover, the first-order, second-order and high-order global sensitivity coefficients can be obtained simultaneously. The Sobol method divides the total variance into the independent variance and the interaction variance between different parameters. A function

To avoid a sampling-centralization phenomenon, Latin hypercube sampling (LHS) is adopted, which can better cover the parameter space region with a small number of samples. LHS is widely used in the probability statistics of complex systems with random input characteristics. In multi-parameter sensitivity analysis, LHS is an important and commonly used parameter sample acquisition method. The main idea of LHS is to make uniform equal-probability stratification for each dimension of the parameter space and then to select samples from each layer randomly. In order to improve computation speed, two-time independent samples (Saltelli, 2002) are executed. The generated vector samples are scrambled and rearranged subsequently.

To execute parameter sensitivity analysis of ANTSMC for the antenna servo system, 10 000 groups of control parameters are sampled, and two independent samples are taken. In Fig. 2, the sensitivity coefficient of

Sensitivity coefficient convergence curve of

Parameter sensitivity analysis results and importance evaluation. The sensitivity coefficients playing important roles are highlighted in bold.

To make the sensitivity analysis results more visible, it is assumed that, if the sensitivity coefficient of a parameter for an objective function is greater than 5 %, the parameter is considered to be important to this objective function; if the sensitivity coefficient is less than 1 %, the parameter is considered to negligible in relation to this objective function; if the sensitivity coefficient is between 1 % and 5 %, the parameter is relatively minorly important to this objective function. According to multi-parameter sensitivity results, the importance of each parameter to different objective functions is evaluated and also shown in Table 2. The symbols I, M and N indicate important, minorly important and negligible, respectively. Table 2 shows that there are four parameters which have no important impact on all objective functions, namely

The cell-mapping method to solve MOPs can obtain the Pareto optimal set through one single run of the algorithm and can ensure the global optimization well. With the cell-mapping method, a region in the parameter space is discretized into uniform cells. An infinite number of points in the parameter space can be represented by a finite number of cells. In simple cell mapping (SCM), the characteristics of a cell are represented by the characteristics of its center point. The Pareto optimal set is obtained by constructing cell-to-cell mappings in the parameter space and extracting the periodic cells.

The objective function value corresponding to the center point of each cell is calculated firstly, and the free gradient law (Qin et al., 2017) is adopted to construct one-step simple cell mapping. Let

To improve calculation speed, the subdivision technology and GPU parallel technology are adopted. Recall the MOP described in Eq. (6). The parameter space region is firstly discretized as

Then each coarse periodic cell is further discretized as

The projections of the Pareto set on 3D sub-spaces

A set of points in the parameter space which represent the Pareto optimal solutions is obtained after the MOP solving. Usually, the multi-objective optimal control design generates hundreds or thousands of Pareto optimal solutions. How the decision maker selects appropriate control parameters from the Pareto set is a post-processing issue. In this paper, a post-processing algorithm for MOPs is introduced. For every objective function

With the post-processing algorithm, the Pareto front obtained by solving the MOP of the ANTSMC parameters is processed.

The projections of the Pareto set on 3D sub-spaces

Numerical simulations of

The optimal control input is designed to drive a dynamical system from an initial state to the predesigned terminal state so that a designed cost function reaches the extreme value (maximum or minimum). In the next section of this paper, the optimal control objective is set to make the cost function reach the minimum value. However, it is usually difficult to solve optimal control problems analytically for complex nonlinear dynamical systems. When the control input is bounded, even numerical optimal control solutions are quite difficult to obtain. That is an important factor that limits the application of optimal controls in engineering.

The cell mapping offers an efficient numerical way to solve optimal control problems. With the cell-mapping method, the continuum state space region is firstly discretized into a finite number of cells. Simultaneously, the bounded control inputs are also uniformly discretized. For each cell, the cell-to-cell mappings are constructed under different control input levels. Based on the constructed mapping database, the global optimal control solutions can be searched out. The cell-mapping method to deal with optimal control problems has universal applicability for linear and nonlinear dynamical systems. Moreover, it can ensure the global optimality of optimal control solutions as the search is instituted in the whole state space region. In addition, all the optimal control solutions of different controllable cells can be obtained, and uncontrollable cells can be recognized with one single run of the program.

The adjoining cell-mapping method, as an improvement of the simple cell-mapping method in dealing with optimal control problems, can construct the mapping database with smaller discretization error and can search the optimal control solution more efficiently. Furthermore, the closed-loop feedback control can be performed, which guarantees robustness in relation to the external disturbance. In the adjoining cell mapping, to deal with optimal control problems, the bounded external-control input is uniformly discretized as

Due to the existence of dimension disasters, the optimal control with the cell-mapping method is mainly applied in two-dimensional single-input–single-output system over quite a long time. The two-level subdivision strategy can make the cell-mapping method feasible to solve fixed-terminal-state optimal control problems for multi-input–multi-output (MIMO) systems. The state space region of interest can first be first discretized into relatively coarse cells, and the mapping database can then be constructed with GPU parallel technology. The optimal control inputs for all the controllable coarse cells can be searched for by means of the back-stepping search strategy, and the uncontrollable cells can be identified. Then the feedback control is instituted from a controllable initial state, and the integral trajectory starting from the initial coarse cell will cross several coarse cells in the state space before it reaches the cell where the target set is located. The region consisting of these continuous cells in the state space is further discretized, and the search is executed again to obtain the fine DOCT.

As for the trajectory-tracking optimal control problem, the target set is not fixed but is instead constantly changing over time. So the cell-mapping method to deal with fixed-terminal-state optimal control problems is no longer applicable. When the cell in which the target set is located changes, it is necessary to construct a new DOCT. Naturally, it is imagined that the target trajectory can be discretized into a series of points, namely, a series of cells in this paper. Then the trajectory-tracking optimal control problem can be transformed into a sequence of fixed-terminal-state optimal control problems, and the two-level subdivision strategy for the adjoining cell-mapping method can still be adopted to obtain optimal control solutions with a high computational efficiency. Based on this idea, a global optimal tracking control strategy with an adjoining cell-mapping method (OTCACM) is introduced (Tian et al., 2023). By using an adaptive criterion to judge the availability of adjoining cell-mapping pairs, the cell-mapping method is extended to solve optimal tracking control problems for MIMO systems for the first time.

However, it should be noted that the OTCACM may generate phase differences between the target trajectory and the system's real response trajectory. Before the target trajectory is cached up, the current system trajectory and the target trajectory are not in the same cell. Consequently, transformation of the trajectory-tracking optimal control to a sequence of fixed-terminal-state optimal controls is appropriate. However, after the current system trajectory and the target trajectory are in the same cell, real-time tracking for the target trajectory in the time domain is necessary to ensure a high control accuracy. It can be imagined that, on the one hand, if the target trajectory changes relatively slowly with time, the system response trajectory under optimal controls may shuttle back and forth around the target cell, which will result in the chattering phenomenon. The chattering amplitude and tracking error depend on the discretization scale of the state space region, namely, the size of cell; on the other hand, if the target trajectory changes relatively quickly with time, in some situations, such as the optimal control with minimum energy consumption, the resulting control input by OTCACM may be not insufficient to drive the system to catch the target cell quickly, which will result in the obvious phase lag phenomenon.

Based on the above analysis, an optimal and sliding-mode combined-control strategy (OSCC) is proposed. The OSCC employs the OTCACM law before the target trajectory is cached up and the ANTSMC law with optimized control parameters after the target trajectory is cached up. Let

Equation (16) means that the cells in which the current target state and system state are located need to be identified. If they are same, the ANTSMC law is adopted. If not, the OTCACM law is adopted. To detail the general procedure of OSCC, let

The general procedure of OSCC.

Discretize the state space region into

Construct

Find the target cell. Set it as

Find all the mappings in

Update

The coarse cells crossed by the evolution trajectory from the current state to the target cell constitute a new region

Closed-loop feedback control with ANTSMC law

The fine DOCT remains unchanged

Closed-loop feedback control under the fine DOCT

Recalling the system in Eq. (1), the control target is set to drive the azimuth angle

The feedback control evolution trajectories of

The feedback control evolution trajectories in the time domain under OSCC for

The control moment inputs

For comparison, the feedback control evolution trajectories of

The feedback control evolution trajectories in time domain under OTCACM for

The minimum time, minimum energy consumption and minimum quadratic performance index tracking controls are instituted, respectively, for the antenna servo system with three different algorithms, namely, OTCACM, ANTSMC and OSCC. The comparisons of cost function value

Control performance comparison of three control algorithms.

In this paper, the cell-mapping method is applied in the control optimization for an antenna servo system on a disturbed carrier.

To conquer the curse of dimensionality in the cell-mapping method for solving MOPs, the multi-parameter sensitivity analysis is implemented to realize the dimension reduction of the parameter space. A post-processing algorithm is also proposed to provide a reference in order for the decision maker to select proper control parameters from the Pareto set. The ANTSMC control parameters for the antenna servo system are optimized effectively with the proposed scheme. In addition, the OSCC strategy is introduced to overcome the tracking-phase difference phenomenon in the existing OTCACM. The OSCC combines the global optimality of optimal controls with cell mapping and the high tracking accuracy of the sliding-mode control. Simulation results show that, with the OSCC strategy, the antenna attitude angles are driven to catch up the target trajectories successfully with the global optimal cost function index and keep high accuracy tracking after that. The proposed approaches in this paper make the cell-mapping method more practical in the control optimization field, providing widely applicable and effective numerical solutions for control optimization problems for nonlinear dynamical systems in engineering.

Some or all data or code generated or used during the study are available from the corresponding author by request.

Zhui Tian developed the research idea and designed the study. Zhui Tian and Yongdong Cheng performed the analysis and simulations. Zhui Tian discussed the results. Yongdong Cheng prepared the paper. All the authors provided input on the paper for revision before submission.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by the Natural Science Basic Research Plan in Shannxi Province of China (grant no. 2021JQ-183).

This paper was edited by Peng Yan and reviewed by Khubab Ahmed and two anonymous referees.