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). ({
Relationship (12) exists if relation (13) is established:
Including nonlinear constraint (15):
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
As stated in the preceding sections, the value of ‖
For example, if the gain matrix
This demonstrates the necessity of putting each element in the feedback gain matrix by assigning a non-negative weight to each
In
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:
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
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(
This section of the article describes a method for obtaining a set of right eigen vectors in such a way that the gain elements of the original design are minimized. For multirate MIMO systems, the increased range of acceptable eigen vector space allows further modifications to the feedback matrix.
In the optimal control method, we need to select an appropriate cost function that represents the order in which the (
According to this point, the cost function
The cost function can be rewritten by combining equations (23) and (24) and (25):
We then define
In relation (32),
For some multi-input, multi-output multirate issues, the search process can include very large times without compromising the final design's sensitivity.
Finally, a method for updating gain based on successive halves of the search path is presented.
Using the internal search step
Calculate (
Calculate
If
Updating
Continue step 2
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,
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:
Choosing
For a single-rate design, states and vectors are presented as:
Finally, by combining (38) and (39), the description of the offset loop is written as:
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):
Relationships (45) and (46) must be established for the exact matching of the resulting system and the desired one, with respect to (44):
The two expressions of (45) are directly related and need not be solved, and the two expressions (46) are true if
As a result, the single rate system will only result in an approximate solution for the compensator matrices
Compensator gains are defined as
By applying this matching method, we can determine the amount of PI compensator gain for the roll autopilot loop:
The expressed single-rate design process illustrates the discrete compensator dynamics for the
This has yielded a bunch of gains as in Table 1.
Calculated gains.
2.5 | 0.39 |
2.5 | 0.48 |
2.5 | 0.41 |
2.5 | 0.506 |
As it is clear, during the main
In order to optimize the applied procedure, it should be noted that the feedback matrix
According to the literature, there is a choice for the weighting matrix γ for optimized feedback with the lowest second norm (49):
After performing several times, the optimization algorithm presented with different initial steps
The matrix
Performance parameters after optimization
4.69 | |
9.501 | |
1.429 | ‖ |
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.
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.
Because the output optimally detects the input, this will allow for a better correlation between the roll and yaw movements of the flying vehicle.
Reduce the need for rudder control in the designed method than the original system.
Based on the above results, it can be said that the modified design requires much less activity and control level and the control signal amplitude.
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 ‖
Calculated gains.
2.5 | 0.39 |
2.5 | 0.48 |
2.5 | 0.41 |
2.5 | 0.506 |
Performance parameters after optimization
4.69 | |
9.501 | |
1.429 | ‖ |