Eigen-structure problem optimization for multirate, multi-input multi-output systems applied to a roll rate autopilot

The high degree of freedom in the sampling of multirate systems results in creating an Eigen closed-loop system (Fujimoto and Hori, 2002; Tomizuka, 2004). It should also be noted that uninterrupted application of the eigen-structure method and its stringent conditions to multirate systems will result in the high gain of elements in feedback matrices, which produces unacceptable control inputs and states. Finally, these unacceptable control inputs will lead to inactivity of the multirate control system (direct eigen-structure methods will result in higher or lower sampling rates compared to other allocation feedback methods) (Patton et al., 1995; Tomizuka, 2004).

It can be maintained; by choosing a suitable solution for a closed-loop multirate system, a significant reduction in the control effort and switching properties with high amplitude will be produced, while sensitivity to the effects of internal sampling is negligible (Patton et al., 1995; Tomizuka, 2004; Liu and Chu, 2018).

The first design method (Patel and Patton, 1990) limits the solution in a particular form, requires additional constraints on the direct placement method, which has been termed the special constrained method. The above constraint is assumed to be equivalent to the single-rate system state feedback matrix. The proposed solution attempts to estimate the single-rate matrix and thus decreases the control inputs. The advantage of this method is its simplicity. Extra constraints also require a very simple simplification of the direct value problem. It is also worth mentioning that in this method, the minimization will not be possible simultaneously on the required control effort and the application of the specified model structure (Cimino and Pagilla, 2008; Liu and Chu, 2018). So, an optimal solution should seek to reach an acceptable compromise between these two goals.

The second design method is based on the optimized solutions of multirate systems, and in this research, two optimized special structure placement methods have been proposed (Patel et al., 1993). These methods attempt to minimize the high amplitude of transient switching and control efforts while reducing the sensitivity to the effects of internal sampling.

In this study, two optimized eigen-structure allocation methods are described, and then they are modified and expressed for use in a multirate system. Finally, the mentioned method will be shown with a demonstrative example.

In section 2, constrained eigen-structure method is explained. In section 3, complementary multirate multi-input multi-output (MIMO) system measures for optimized eigen-structure method is proposed. In section 4, a numerical example is solved by the proposed method and finally the simulation results are shown.

Based on the results, by reducing the second norm of feedback gain matrix to 99%, and finding the optimum point by a few steps in the proposed method, the efficiency of the method is proven. It is obvious that this method can be applied to any system with the problem of having high value of the second norm of the gain feedback matrix.

This method reduces the amplitude of control and state signal of multirate MIMO system by reducing the matrix gain designed by feedback matrices. If a straightforward approach is used to create a smooth control level, the constrained gain matrix of a multirate MIMO system will meet the desired targets. The problem with this method is to select the priority constraints for calculating the matrix efficiency. The simple and preliminary method of trial and error may take some acceptable work, but it will require a great deal of time to design (Patel et al., 1993; Patton et al., 1995; Piou and Sobel, 1995; Lixin et al., 2020). Singular pencil matrix of an open loop and closed loop, single rate discrete system is shown in (1) and (2) (Zamani and Brian, 2012). ({A · B · C · D} shows discrete system matrices and index c stands for close loop): (1) F(z)=[zI−ABC0]=[F1(z)F2(z)] F\left( z \right) = \left[ {\matrix{ {zI - A} & B \cr C & 0 \cr } } \right] = \left[ {\matrix{ {{F_1}\left( z \right)} \cr {{F_2}\left( z \right)} \cr } } \right] (2) Fc(z)=[zI−AcBcCc0]=[Fc1(z)Fc2(z)] {F_c}\left( z \right) = \left[ {\matrix{ {zI - {A_c}} & {{B_c}} \cr {{C_c}} & 0 \cr } } \right] = \left[ {\matrix{ {{F_{c1}}\left( z \right)} \cr {{F_{c2}}\left( z \right)} \cr } } \right] F(z) and F_c(z) are strictly equivalent if two constant, nonsingular matrices of the form (3) and (4) exists in such a way that (5) can be written as (3) L=[L11L120L22] L = \left[ {\matrix{ {{L_{11}}} & {{L_{12}}} \cr 0 & {{L_{22}}} \cr } } \right] (4) R=[R110R21R22] R = \left[ {\matrix{ {{R_{11}}} & 0 \cr {{R_{21}}} & {{R_{22}}} \cr } } \right] (5) F(z)=LFc(z)R F\left( z \right) = L{F_c}\left( z \right)R For a state feedback control, only F₁(s) and F_c1(s) are of interest. By considering appropriate feedback system elements only, equation (5) changes to (6) (index c and m stands for close loop and multirate, respectively): (6) F1(z)=L11Fc1(z)[R110R21R22] {F_1}\left( z \right) = {L_{11}}{F_{c1}}\left( z \right)\left[ {\matrix{ {{R_{11}}} & 0 \cr {{R_{21}}} & {{R_{22}}} \cr } } \right] A suitable form for L and R is (7): (7) L11=Tc⋅R=[Tc−10kTc−1G]det G≠0 {L_{11}} = {T_c} \cdot R = \left[ {\matrix{ {T_c^{ - 1}} & 0 \cr {kT_c^{ - 1}} & G \cr } } \right]\det \,G \ne 0 In (7), k is an arbitrary feedback matrix and T_c is the transformation matrix for canonical system. If the singular pencil matrix of the closed loop multirate system is defined as (8), then F(z) and F_m(z) are related by (9) in which S_L and S_R can be decomposed in the form of (10): (8) Fm(z)=[zI−AmBmCm0]=[Fm1(z)Fm2(z)] {F_m}\left( z \right) = \left[ {\matrix{ {zI - {A_m}} & {{B_m}} \cr {{C_m}} & 0 \cr } } \right] = \left[ {\matrix{ {{F_{m1}}\left( z \right)} \cr {{F_{m2}}\left( z \right)} \cr } } \right] (9) F(z)=SLFm(z)SR F\left( z \right) = {S_L}{F_m}\left( z \right){S_R} (10) SL=LP⋅SR=ΦR {S_L} = LP \cdot {S_R} = \Phi R By considering the feedback properties for the matrices L and R, P and Φ matrices condense the multirate properties (Zamani and Brian, 2012). The main difference between these matrices is that under R and L the controllability and observability is invariant respectively. Thus, for holding with feedback in multirate case, by choosing suitable matrices for the open loop multirate system the controllability and observability properties can be maintained in the form of (11) (Wang et al., 2004; Zamani and Brian, 2012). (11) P=[I0II]⋅Φ=[I00Φ22] P = \left[ {\matrix{ I & 0 \cr I & I \cr } } \right] \cdot \Phi = \left[ {\matrix{ I & 0 \cr 0 & {{\Phi _{22}}} \cr } } \right] To consider an appropriate constraint, we can establish equivalence between the desirable multirate closed loop system and the corresponding single-rate closed loop system (Wang et al., 2004). The relationship between an open loop single-rate system and a closed-loop multirate MIMO system will be as follows (Tangirala et al., 1999; Wang et al., 2004): (12) [ZI−AB]=Tc[ZI−Am Bm][Tc−10Φ22KTc−1Φ22G] \left[ {ZI - AB} \right] = {T_c}\left[ {ZI - {A_m}\,{B_m}} \right]\left[ {\matrix{ {T_c^{ - 1}} & 0 \cr {{\Phi _{22}}KT_c^{ - 1}} & {{\Phi _{22}}G} \cr } } \right] In (12), G is the single rate close loop system and T_c is the transformation matrix for canonical system and K is the gain matrix.

Relationship (12) exists if relation (13) is established: (13) Am=Tc−1ATc+BmΦ22K {A_m} = T_c^{ - 1}A{T_c} + {B_m}{\Phi _{22}}K Equation (13) expresses the equivalent of a closed-loop multirate matrix based on a single-rate feedback matrix. The matrix K_m = Φ₂₂K can be used as a smoothing parameter to constrain the problem of locating the eigen multirate structure. This limits the usefulness of the multirate feedback matrix utilization elements, which results in minimizing the control effort required for a multirate MIMO system. To include the direct method of matrix yield constraint, the eigenvalue problem for the selection of special vectors V_i has been modified to give (14). (V_i(V_ai · V_di) is the right eigen vector associated with the λ_i (ith eigenvalue). Indexes a and d stands for achieved and desired, respectively): (14) minki‖Mλiki−kmVdi‖2 \mathop {min }\limits_{{k_i}} {\left\| {{M_{\lambda i}}{k_i} - {k_m}{V_{di}}} \right\|_2} where M_λi is the ith eigenvalue of the block diagonal matrix in canonical system.

Including nonlinear constraint (15): (15) ‖Nλiki−Vdi‖≤α \left\| {{N_{\lambda i}}{k_i} - {V_{di}}} \right\| \le \alpha In (15), α is the scalar contraction factor and N_λi is the summation of input sample multiplicities for λ_i.

With regard to conditions (12) and (13), the straightforward method will result in obtaining desirable special vectors with the aim of reducing excessive output oscillations (Sato et al., 2016). This will be achieved by estimating the matrix yield of the single-rate feedback matrix (Tangirala et al., 1999; Wang et al., 2004; Sato et al., 2016). The problem is solved if it is only min‖M_λik_i – k_mV_di‖₂ ≤ α. In this case, we will have ‖NλiMλi−1kmVdi−Vdi‖2≤γ {\left\| {{N_{\lambda i}}M_{\lambda i}^{ - 1}{k_m}{V_{di}} - {V_{di}}} \right\|_2} \le \gamma or ‖NλiMλi−1kmVdi−Vdi‖2>γ {\left\| {{N_{\lambda i}}M_{\lambda i}^{ - 1}{k_m}{V_{di}} - {V_{di}}} \right\|_2} > \gamma (γ is the gain margin). Consequently, the solution obtained in relation (16) will be truth: (16) (MλiTkmMλi+ΓNλiTNλi)ki=(MλiTkmVdi−ΓNλiVai) \left( {M_{\lambda i}^T{k_m}{M_{\lambda i}} + \Gamma N_{\lambda i}^T{N_{\lambda i}}} \right){k_i} = \left( {M_{\lambda i}^T{k_m}{V_{di}} - \Gamma {N_{\lambda i}}{V_{ai}}} \right) Of course, the choice of Γ which is a weighting factor, depends on the designer and it will be a difficult task. An appropriate way to select Γ and at the same time establish the constraint (15) is combining both conditions (12) and (13). These limitations will limit our problem to one of the following two solutions: (17) minki‖ρλiki−ζVdi‖2ρλi=Nλi−Mλi and ζ=(In−Φ22K) \matrix{ {\mathop {min }\limits_{{k_i}} {{\left\| {{\rho _{\lambda i}}{k_i} - \zeta {V_{di}}} \right\|}_2}} \cr {{\rho _{\lambda i}} = {N_{\lambda i}} - {M_{\lambda i}}\,{\rm{and}}\,\zeta = \left( {{I_n} - {\Phi _{22}}K} \right)} \cr } The matrix ζ is determined by the single-rate feedback matrix. To simplify, suppose λ_i ∈ R. If ρ_λi ∈ R^{n × n}, equation (17) will have a single solution that will yield the columns of ρ_λi. Then any special closed-loop vector can be obtained from equation (18): (18) Vi=Nλiρλi−IζVdi {V_i} = {N_{\lambda i}}{\rho _{\lambda i}} - {I_\zeta }{V_{di}} where l_ζ is the identity matrix of dimension ζ. For λ_i ∈ c, a solution will be obtained by sharing the space of real and imaginary components of eigenvalues. As a result, the feedback will be able to estimate a single rate feedback. The multirate feedback designed by this method has the same sensitivity characteristics as the restricted single-rate system. For this reason, it is advisable for systems with high interaction modes to be design based on a non-sensitive single rate feedback matrix (Wang et al., 2004; Sato et al., 2016). Common pole assignment techniques can usually be used to produce a smooth feedback matrix for systems that are weakly coupled. In order to determine the usefulness of this method to reduce the matrix elements of the multirate MIMO system, the feedback matrices ‖k‖₂ will be compared with the feedback matrices produced by the direct method (Lee and Lee, 2007).

As stated in the preceding sections, the value of ‖k‖₂ is measured to estimate the amount of control effort required by the feedback matrix of a multirate MIMO system (Lee and Lee, 2007). Measuring the value of ‖k‖₂ is a common method for detecting the magnitude of matrix gain amplitudes in continuous and single rate discrete systems. For the multirate MIMO state feedback problem, the gain matrix contains various elements with different domains (Pedro and Pedro, 2016). It must also be noted that as the input increases, the extent of gain associated with a particular input would increase (Zamani and Brian, 2012).

For example, if the gain matrix k₂ is defined as ɛ₁ = 2 and ɛ₁ = 1 for the inputs u₁ and u₂, then the elements associated with the input u₁ (the first two rows of k₂) will be substantially greater than the elements associated with the single-rate input. Therefore, any design method that aims to reduce the control effort for the feedback of a multirate MIMO system should aim to minimize the range of gain elements associated with each multirate input (Sheng et al., 2005). An appropriate cost function for achieving this goal is given in (19): (19) JG=∑i=1m∑j=1nγ(i⋅j){k(i⋅j)}2 {J_G} = \sum\nolimits_{i = 1}^m {\sum\nolimits_{j = 1}^n {\gamma \left( {i \cdot j} \right){{\left\{ {k\left( {i \cdot j} \right)} \right\}}^2}} } In (19), matrix γ has the domain m × n.

This demonstrates the necessity of putting each element in the feedback gain matrix by assigning a non-negative weight to each ij position.

In J_G the larger weighting of the position ij in γ, leads to greater ij element in k. If γ is chosen such that all the weights of the matrix elements in the rows of the multirate inputs are much greater than those of the same elements of the single-rate inputs, the accuracy of the material stated about the feedback design in multirate systems will be achieved using the eigen-structure method.

In order to illustrate the variable nature of MIMO systems and control inputs, a method for calculating the gain matrix is presented (Cimino and Pagilla, 2009). The unstable behavior of these systems occurs at the moments of internal sampling, which is a logical consequence of the changes in the rows of gain matrix elements associated with each input (Word et al., 2007).

It is considerable that if the difference between the elements of each row of gain matrix is minimized, the system behavioral changes at internal sampling points will reduce (Cimino and Pagilla, 2009).

If, like the relation (19), only the rows of the multirate inputs are considered, the appropriate cost function will be written as: (20) JD1=∑p=1m∑i=XΨ∑j=1nα(i,j){k(i,j)−k(i+1⋅j)}2 {J_{D1}} = \sum\nolimits_{p = 1}^m {\sum\nolimits_{i = {\rm{X}}}^\Psi {\sum\nolimits_{j = 1}^n \alpha } } \left( {i,j} \right){\left\{ {k\left( {i,j} \right) - k\left( {i + 1 \cdot j} \right)} \right\}^2} In relation (20) X=∑q=1p−1nq+1 X = \sum\nolimits_{q = 1}^{p - 1} {{n_q} + 1} and {Ψ=X for nq=1Ψ=X+nq−1 for nq>1 \left\{ {\matrix{ {\Psi = X\,\,for\,{n_q} = 1} \cr {\Psi = X + {n_q} - 1\,for\,{n_q} > 1} \cr } } \right. . The matrix α is a matrix of non-negative assigned weights with dimensions similar to the yield matrix k which contains all elements equal to zero in all Ψ rows for each m input and n_q is the sample rate multiplicity. If the weighting elements are selected such that the position of elements of the highly varying rows is linked to large values in the α matrix, the objectives in feedback design will be achieved. By Considering k(k) as the largest and smallest numbers in k, the state changes and controllability can be expressed as: (21) ki(kmi)i=1.....m {k_i}\left( {{k_{mi}}} \right)i = 1.....m In relation (21) k_mi represents rows n_i matrix k that related to i. As a result, it is an appropriate cost function to represent the behavior and changes of state and control signal as: (22) JDnew=∑i=1mki(kmi) {J_{Dnew}} = \sum\nolimits_{i = 1}^m {{k_i}\left( {{k_{mi}}} \right)} The minimal value of J_Dnew will be achieved by minimizing system state changes and controlling behavior.

The final cost function should be capable of measuring the proximity of the right eigenvalues selected from the acceptable space to the arbitrary set if it is the best of all the scenarios mentioned above.

If the ith right eigen vector is shown with V_id, the appropriate measure of the probability of the nearest ith right eigen vector V_i to V_id is expressed by: (23) J=∑i=1nΓ(i)[V¯i−V¯id]T[Vi−Vid] J = \sum\nolimits_{i = 1}^n {\Gamma \left( i \right){{\left[ {{{\bar V}_i} - {{\bar V}_{id}}} \right]}^T}\left[ {{V_i} - {V_{id}}} \right]} In relation (23), β(i) represents the ith non-negative weighting factors and the superscript ‘-’ at the top of the vector V indicates its complex conjugate.

The elements of the rows of matrix Γ show the emphasis on the positioning of the desired right eigen vectors. The low value of Γ(i) also indicates a low priority in the substitution of ith eigen vector (and vice versa). Finally, the low value of J_m indicates the accuracy of the precedence in the right eigen vectors.

In this example, we will discuss the method described for adjusting the parameters of the PI controller in the multirate roll autopilot system.

To better understand the design process, a single-rate system will be designed, followed by a brief explanation of the multirate design.

Finally, the accuracy of the design and the presented algorithm will be shown in the simulation results.

In Figures 1 and 2, q represents the rate of roll angle changes, q_a is the signal required by the autopilot for roll motion.

Figure 1

Figure 2

Also, in the equations ζ denotes the angle of high curvature of the winner.

In this example, the transverse open loop dynamics for a flying vehicle at a speed of about 40 m/s is given by: (36) q−qaζ=−14(s+0.1)(s+3.1)(s+0.01±j0.0698)(s+1.2±j3.499) {{q - {q_a}} \over \zeta } = {{ - 14\left( {s + 0.1} \right)\left( {s + 3.1} \right)} \over {\left( {s + 0.01 \pm j0.0698} \right)\left( {s + 1.2 \pm j3.499} \right)}} To simplify the Following process of designing, the ratio between k_p : k_i is considered 1:10.

Choosing k_i = 2.6 and k_p = 0.26 will result in uniform responses.

For a single-rate design, states and vectors are presented as: (37) x=[uwqθ]=[Progressive speedVertical speedTransverse movement rateTransverse Angle Change Rate]⋅u=[ζ] x = \left[ {\matrix{ u \cr w \cr q \cr \theta \cr } } \right] = \left[ {\matrix{ {Progressive\,speed} \cr {Vertical\,speed} \cr {Transverse\,movement\,rate} \cr {Transverse\,Angle\,Change\,Rate} \cr } } \right] \cdot u = \left[ \zeta \right] Then the state space of the transfer function is defined as standard form of (38) and we write the compensator description as (index “com” stands for compensated system parameters and index “oc” shows the open loop system parameters): (38) {x˙(t)=Ax(t)+Bu(t)y(t)=Cx(t)+Du(t) \left\{ {\matrix{ {\dot x\left( t \right) = Ax\left( t \right) + Bu\left( t \right)} \cr {y\left( t \right) = Cx\left( t \right) + Du\left( t \right)} \cr } } \right. (39) {x˙com(t)=Acomxcom(t)+Bcomucom(t)ycom(t)=Ccomxcom(t)+Dcomucom(t) \left\{ {\matrix{ {{{\dot x}_{com}}\left( t \right) = {A_{com}}{x_{com}}\left( t \right) + {B_{com}}{u_{com}}\left( t \right)} \cr {{y_{com}}\left( t \right) = {C_{com}}{x_{com}}\left( t \right) + {D_{com}}{u_{com}}\left( t \right)} \cr } } \right. In (39), A_com = [0] and B_com = [1] and C_com = [k_i] and D_com = [k_p].

Finally, by combining (38) and (39), the description of the offset loop is written as: (40) {x˙(t)=Aocx(t)+Bocu(t)y(t)=Cocx(t)+Docu(t) \left\{ {\matrix{ {\dot x\left( t \right) = {A_{oc}}x\left( t \right) + {B_{oc}}u\left( t \right)} \cr {y\left( t \right) = {C_{oc}}x\left( t \right) + {D_{oc}}u\left( t \right)} \cr } } \right. In (40) the parameters re defined as: (41) Aoc=[Acom0BCcomA].Boc=[BcomBDcom]⋅Coc=[DCcom C]⋅Doc=[Dcom D] \matrix{ {{A_{oc}}} \hfill & { = \left[ {\matrix{ {{A_{com}}} & 0 \cr {B{C_{com}}} & A \cr } } \right].{B_{oc}} = \left[ {\matrix{ {{B_{com}}} \cr {B{D_{com}}} \cr } } \right] \cdot {C_{oc}}} \hfill \cr {} \hfill & { = \left[ {D{C_{com}}\,C} \right] \cdot {D_{oc}} = \left[ {{D_{com}}\,D} \right]} \hfill \cr } Finally, the state space equation of the closed loop system of roll changes will be written as: (42) Gc(s)=[AcBcCcDc] {G_c}\left( s \right) = \left[ {\matrix{ {{A_c}} & {{B_c}} \cr {{C_c}} & {{D_c}} \cr } } \right] In (42) the parameters re defined as: (43) Ac=[AcomBcomCBCcomA+BDcomC]⋅Bc=[BCcomBDcom]⋅Cc=[DCcom C]⋅Dc=[DcomD] \matrix{ {{A_c}} \hfill & { = \left[ {\matrix{ {{A_{com}}} & {{B_{com}}C} \cr {B{C_{com}}} & {A + B{D_{com}}C} \cr } } \right] \cdot {B_c} = \left[ {\matrix{ {B{C_{com}}} \cr {B{D_{com}}} \cr } } \right] \cdot {C_c}} \hfill \cr {} \hfill & { = \left[ {D{C_{com}}\,C} \right] \cdot {D_c} = \left[ {{D_{com}}D} \right]} \hfill \cr } In order to avoid the unstable effects of discretization the closed loop system with negative feedback, we first consider the open loop discrete system and then compute the closed loop.

As a result, the single-rate discrete system of the open loop is expressed in the manner that its closed-loop response corresponds to the desired system (44): ΦD=[ΦMcom0λMCMcomΦM]⋅λD=[λMcomλMDMcom], {\Phi _D} = \left[ {\matrix{ {{\Phi _{{M_{com}}}}} & 0 \cr {{\lambda _M}{C_{{M_{com}}}}} & {{\Phi _M}} \cr } } \right] \cdot {\lambda _D} = \left[ {\matrix{ {{\lambda _{{M_{com}}}}} \cr {{\lambda _M}{D_{{M_{com}}}}} \cr } } \right], (44) CD=[DMCMcom CM]⋅DD=[DMcom DM] {C_D} = \left[ {{D_M}{C_{{M_{com}}}}\,\,\,{C_M}} \right] \cdot {D_D} = \left[ {{D_{{M_{com}}}}\,\,{D_M}} \right] where {Φ_Mcom · λ_M · C_Mcom · D_Mcom} and {Φ_M · λ_M · C_M · D_M} represent discrete compensator and flying vehicle dynamics for the sampling interval T = 0.1.

Relationships (45) and (46) must be established for the exact matching of the resulting system and the desired one, with respect to (44): (45) AMcom=Acom & BMcom=Bcom, {A_{{M_{com}}}} = {A_{com}}\,\&\, {B_{{M_{com}}}} = {B_{com}}, (46) λMCMcom=λCcom & λMDMcom=λDcom {\lambda _M}{C_{{M_{com}}}} = \lambda {C_{com}}\,\&\, {\lambda _M}{D_{{M_{com}}}} = \lambda {D_{com}} The right-hand side of (45) and (46) are derived from relation (44).

The two expressions of (45) are directly related and need not be solved, and the two expressions (46) are true if λ_M ∈ R^{n × m} and λC_com ∈ R^{n × m} and λD_com ∈ R^{n × m}, will be established.

As a result, the single rate system will only result in an approximate solution for the compensator matrices C_Mcom and D_Mcom and the final solution will depend on λ_M.

Compensator gains are defined as (47) CMcom=(λM)†λCcom & DMcom=(λM)†λDcom {C_{{M_{com}}}} = {\left( {{\lambda _M}} \right)^{\rm{\dagger }}}\lambda {C_{com}}\,{\rm{\& }}\,{D_{{M_{com}}}} = {\left( {{\lambda _M}} \right)^{\rm{\dagger }}}\lambda {D_{com}} († represents the inversion by Moore–Penrose method).

By applying this matching method, we can determine the amount of PI compensator gain for the roll autopilot loop: (48) ki=2.5 & kp=0.35 {k_i} = 2.5\,{\rm{\& }}\,{k_p} = 0.35 In the above multirate system, it has both q – q_a error and control signals for updating at a fast nT {n \over T} rate.

The expressed single-rate design process illustrates the discrete compensator dynamics for the Tn1 {T \over {{n_1}}} interval repeated for the multirate mode.

This has yielded a bunch of gains as in Table 1.

As it is clear, during the main T interval, only the proportional gain needs to be adjusted (the update rate for the first system is n/T and for the second system is 1/T).

In order to optimize the applied procedure, it should be noted that the feedback matrix k is very effective in determining how it performs and its norm is equal to 143.801 (‖k‖ = 143.801).

According to the literature, there is a choice for the weighting matrix γ for optimized feedback with the lowest second norm (49): (49) γ=[1001001001001001001001001001001001001001001001111111111] \gamma = \left[ {\matrix{ {100} & {100} & {100} & {100} & {100} \cr {100} & {100} & {100} & {100} & {100} \cr {100} & {100} & {100} & {100} & {100} \cr 1 & 1 & 1 & 1 & 1 \cr 1 & 1 & 1 & 1 & 1 \cr } } \right] The difference in the first three rows with the last two rows is due to the achievement of greater values for the elements associated with u₁ and τ than u₂ and ζ.

After performing several times, the optimization algorithm presented with different initial steps s₀, in all cases in the first 7 to 8 steps ‖k‖₂ decreases rapidly and significantly and this decrease does not lead to a decrease in system sensitivity.

The matrix k which results in the lowest ‖k‖₂ is: (50) k=[−0.060.020.5−0.060.010.17−0.04−0.01−0.270.09−0.01−0.003−0.250.29−0.070.040.35−0.221.240.260.07−0.39−1.23−0.2−0.25] k = \left[ {\matrix{ { - 0.06} & {0.02} & {0.5} & { - 0.06} & {0.01} \cr {0.17} & { - 0.04} & { - 0.01} & { - 0.27} & {0.09} \cr { - 0.01} & { - 0.003} & { - 0.25} & {0.29} & { - 0.07} \cr {0.04} & {0.35} & { - 0.22} & {1.24} & {0.26} \cr {0.07} & { - 0.39} & { - 1.23} & { - 0.2} & { - 0.25} \cr } } \right] The function of the lateral subsystem mentioned for k expressed in (16) is given in Table 2.

The optimized simulation results for the two different maneuvers are shown in Figure 2(a) and (b), and as it can be seen, the output is very well managed to track the input.

And the autopilot CRE error (Li et al., 2019) for inputs of Figure 3 is shown in Figure 4, respectively.

Figure 3

Figure 4

The following results can be expressed with respect to Table 2 and the simulation:

Using the designed method, norm of the feedback Matrix is about 1% of the original matrix.

A novel approach by using partial derivatives to a gain optimization process to optimize the nonlinearly constrained eigen-structure method for multirate MIMO systems is proposed in this paper. The proposed method works by applying optimal solutions to the special structure method of a multirate system based on the gradient search around the initial eigen-structure assignment, and is able to minimize all elements of the gain matrix, while maintaining model specifications according to the initial design. For the purpose of computation, a weighting matrix is used with the aim of affecting the value by which all elements of the gain matrix are minimized. Also, the use of gain modification method to minimize the gain elements in enhancing the lateral stability of the mentioned system has reduced the ‖K‖₂ to 99%. As a result, due to the linear time response of the low-form stable system, we have seen the absorption of all switched behavior in response to the desired input.