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 Y=f(X1X2,…,Xn) can be divided into a set of functions with increased dimensions, and the first-order sensitivity coefficient of a parameter Xi can be expressed as 9Si=VE(Y|Xi)VY, where EY|Xi is the conditional expectation, and V is the variance. The total sensitivity coefficient of Xi is denoted as 10STi=EV(Y|X∼i)VY=1-VE(Y|X∼i)VY where X∼i is all the parameters in X except Xi.

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 β2 for the objective function Oq2 is presented. When the number of samples is over about 2000, the sensitivity coefficient of ε2 holds steady at approximately 0.15. The sensitivity coefficients of other control parameters for different objective functions follow the similar regular pattern. The sensitivity analysis result of each control parameter for different objective functions is shown in Table 2. It can be seen from the table that, among all the control parameters, β2, ε2 and c22 have a relatively large influence on Oq2, while β2, β3, ε2, ε3 and c32 have a relatively large influence on Oq3. The control parameters related to q2 also have an influence on Oq3, which is due to the existence of couplings. β2, γ2 and c22 have a greater impact on peak time Tpq2, while β3, γ3 and c32 have a greater impact on Tpq3. Eq2 is mainly affected by β2 and γ2, while Eq3 is mainly affected by β3 and γ3, which is consistent with the fact that the steady-state performance of the sliding-mode control only depends on the parameters of the sliding-mode surface.

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 k2, k3, c21 and c31. In fact, the maximal one among the sensitivity coefficients of the four control parameters for one objective function is merely 3.16 %. This means that the 4 control parameters have quite a small influence on all the objective functions compared with the other 8 parameters when the 12 parameters change simultaneously. Consequently, it can be easily deduced that the MOP described by Eq. (6) can be simplified approximately as the optimization of eight relatively important parameters firstly and the other four parameters next. In this way, the dimension of the parameter space decreases from 12 to 8, which makes the MOP with the cell-mapping method achievable.

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 zkn represent the cell corresponding to its center point parameter vector kn. The free gradient law determines the image cell of a cell zkn by comparing the objective function values of the cell with those its neighborhood cells. If there exists a neighborhood cell zki whose objective function value satisfies the conditions Fki≤Fkn and Fki≠Fkn, zki is regarded as the image cell of the pre-image cell zkn. If there exists more than one neighborhood whose objective function values satisfy the aforementioned condition, the maximally dominant cell in the neighborhood is treated as the image cell. The mapping relationship can be expressed as 11zkn=Cszki. If there exists no image cell of zkn, zkn is recognized as a periodic cell under the concept of MOP with SCM, and the corresponding control parameter group is considered to be a Pareto optimal candidate. All the information of the Pareto optimal set is contained in the one-step cell mappings. After all the periodic cells are found, the dominance relationship checking is instituted to avoid the local Pareto optimal solutions, which can ensure the global optimality. Each periodic cell is compared with other periodic cells, and only those periodic cells which are not dominated by any other periodic cell constitute the Pareto optimal set.

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 58 relatively coarse cells. The objective function values corresponding to every center point vector of these cells are firstly calculated by GPU parallel technology, and the constraint condition described by Eq. (8) is checked. If the constraint condition is satisfied, the corresponding cell is considered to be processed next to find its image cell. If the constraint condition is not satisfied, the corresponding cell is discarded. After the mapping relationship judgment, 153 coarse periodic cells are found whose center vectors represent the Pareto optimal set candidates.

Then each coarse periodic cell is further discretized as 58 smaller cells. Consequently, the final partition scale of the parameter space region is 258. The calculation of objective function values, the constraint condition judgment and the search for periodic cells are executed again. The final Pareto optimal set contains 1012 fine cells after the dominance relationship checking. Figure 3a and b show the projections of the Pareto set on 3D sub-spaces (β2, γ2, ε2) and (ε3, β3, c32). The projections of the Pareto front on 2D planes (Tpq2, Eq2) and (Oq3, Tpq3) are exhibited in Fig. 3c and d, respectively. The Pareto front displays the contradictory relationship between different objective functions well. The parallel calculation is instituted on a GPU device (NVIDIA GeForce GTX 1080Ti graphics card), which has 3584 cores with a frequency of 1480 MHz.

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 Fi, the difference between the maximum value Fimax⁡ and the minimum value Fimin⁡ is recognized first. Then it is divided equally by a fixed positive integer N: 12hi=(Fimax⁡-Fimin⁡)/N,i=1,2,…,nF, where nF is the total number of the objective function. Subsequently, the weight coefficient ηi is designed for every objective function according to the willingness of the decision maker, and a distance ri is set as 13rij=Fimin⁡+jηihi,i=1,2,…,nF,j=1,2,…,N. The region Fimin⁡,rij is used to pick the Pareto front and the corresponding Pareto optimal solutions.

With the post-processing algorithm, the Pareto front obtained by solving the MOP of the ANTSMC parameters is processed. N is assigned as 100, and ηi as (0.8,0.8,0.4,0.4,0.3,0.3), which corresponds to (Oq2,Oq3,Tpq2,Tpq3,Eq2,Eq3). After the post-processing algorithm, the top 22 % of the Pareto optimal solutions are picked up from the original 1012 ones. Figure 4a and b show the projections of the Pareto set on 3D sub-spaces (β2, γ2, ε2) and (ε3, β3, c32) after post-processing, while Fig. 4c and d show the projections of the Pareto front on 2D planes (Tpq2, Eq2) and (Oq3, Tpq3) after post-processing. The objective functions fall in the following range: 140≤Oq2≤0.0014%,0≤Oq3≤0.021%,0.376≤Tpq2≤0.416,0.305≤Tpq3≤0.369,3.682×10-11≤Eq2≤1.684×10-10,0.796×10-9≤Eq3≤1.672×10-9. These remaining control parameter groups after the post-processing algorithm possess relatively balanced and close performance under the setting of the weight coefficient ηi. We choose one group from them arbitrarily to carry out the time domain simulation, namely β2=β3=0.06, γ2=1.62, γ3=1.66, ε2=3.4, ε3=6.2, c22=490 and c32=470. The other four control parameters are optimized with the same manner and are set as c21=ε32=5, k2=36 and k3=16. To verify the effectiveness of parameter optimization, the feedback control is instituted with two different groups of control parameters. They are the optimized parameters above and the original unoptimized parameters which are adjusted artificially (Cheng and Jiang, 2019): k2=k3=ε3=5, ε2=2, β2=β3=0.5, γ2=γ3=1.5, c21=c31=5 and c22=c32=500. Figure 5 shows the time history curves and the steady-state error curves of q2 and q3 under the two groups of control parameters. By using the optimized parameters, the peak time Tpq2 decreases from 1.497 to 0.386 s, while Tpq3 decreases from 1.377 to 0.343 s. The steady-state error index Eq2 decreases from 3.282×10-9 to 5.721×10-11, while Eq3 decreases from 3.913×10-8 to 9.415×10-10. The control performance was improved significantly by the optimization design.

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 U=u1,u2,…,uNu, where Nu is the total discretization number. Let zi(n) represents the nth image cell of a previous image cell zi(0); the one-step adjoining cell mapping can then be described as 15zi(n)=Cazi(0),uj. After the mapping database is constructed, the discrete optimal control table (DOCT) can be constructed by a back-stepping search strategy. As a result, every cell is assigned an optimal control input and an optimal cost function value. Those cells with the initial maximum index function value are recognized as uncontrollable cells.

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 zΦ represent the fine cell under the subdivision scale in which the current target state is located, while zΔ is the cell in which the current target state is located. The OSCC can be described as 16uOSCC=uOTCACM,zΔ≠zΦuANTSMC,zΔ=zΦ. Note that the OSCC strategy combines the global optimality of the optimal control with cell mapping and the small steady-state tracking error of the sliding-mode control. With OSCC, a controlled system can be driven to catch up a target trajectory with the minimum cost function value and keep tracking for the target trajectory with a high accuracy after that.

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 M denote the database storing the information of all adjoining cell-mapping pairs described by Eq. (15), and let J denote the database storing the incremental cost function value Jijk from a pre-image cell zi in relation to its kth adjoining image cell under a control input uj. If a cell can reach the target cell where the target trajectory is located in after the n-step adjoining cell mapping, it is denoted as an n-step controllable cell. The initial optimal control input ui∗ and the optimal cumulative cost function Ji∗ are assigned considerably large values. Those cells whose ui∗ and Ji∗ are never updated throughout are recognized as uncontrollable cells. The general procedure of OSCC is shown in Algorithm 1.

Recalling the system in Eq. (1), the control target is set to drive the azimuth angle q2 and pitch angle q3 to track a sinusoidal motion with an amplitude of 30 degrees and a period of 5 s: q20=q30=30∘2πt/5. The bounded state space region for q2,q˙2,q3,q˙3 is set as -180∘,180∘×-6,6×-90∘,90∘×-3,3 with the initial state -60∘,0,60∘,0. The external-control torque inputs M2 and M3 are constrained in -25,25. A quadratic index function which integrates control error and external energy consumption is set to estimate the optimal control performance: 17J=∫0Tea2e22+b2M22+a3e32+bcM32dt, where a2=a3=1, b2=0.01, and b3=0.025. The initial coarse discretization scale of the state space region is 314, while the fine discretization scale is 54. The external-control inputs M2 and M3 are uniformly dispersed with 50 levels each, which generates 2500 different control torque input combinations for (M2,M3). The construction of the cell-mapping database is instituted on the GPU device (NVIDIA GeForce GTX 1080Ti graphics card).

The feedback control evolution trajectories of q2 and q3 in the time domain under OSCC are shown in Fig. 6a and b. The dashed green and solid red lines represent the target trajectory and the real system response trajectory, respectively. It can be easily seen that both q2 and q3 catch up the target trajectories with the optimal performance index function and perform remarkable tracking with high control accuracy after that. The projections of the phase space evolution trajectory on 2D planes q2,q˙2 and q3,q˙3 are presented in Fig. 6c and d, respectively. After catching up the target ellipse trajectories in the phase space, the two practical trajectories coincide well with them. The control moment inputs M2 and M3 in the time domain under OSCC are shown in Fig. 7.

For comparison, the feedback control evolution trajectories of q2 and q3 in the time domain under OTCACM (Tian et al., 2023) are exhibited in Fig. 8a and b. It is obvious that the tracking for q20 almost fails due to the existence of serious phase lag, while the tracking for q30 is effective. This happens as the cost function index described by Eq. (17) implies as small an input energy consumption as possible. Compared with the pitch turntable, the azimuth turntable possesses a relatively large moment of inertia. Therefore, the azimuth motor cannot drive the azimuth turntable to catch up the target trajectory timeously under the same control input level as the pitch motor. The projections of the phase space evolution trajectory on 2D planes q2,q˙2 and q3,q˙3 are presented in Fig. 8c and d, respectively. There exists a large deviation between the practical trajectory and the target elliptical trajectory on the q2,q˙2 plane. The chattering phenomenon is obvious in the tracking for q30, which is due to the fact that the target trajectory changes relatively slowly with time. The phenomena of phase lag and chattering are consistent with the previous analysis.

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 J and the steady errors e2 and e3 are shown in Table 3. The control accuracy increases obviously under OSCC compared with that under TCSCM, while the performance index function value decreases obviously under OSCC compared with that under ANTSMC.