Examination of a repetitive process control system

In the present work, the application of repetitive filters in the robust process control systems is examined. The functionality of the proposed system and the improved performance, robust performance and filtering properties has been proven.


Introduction.
By increasing the performance requirements of the modern technological process control systems, there is considerable interest in the classes of non-stationary control systems in the conditions of a priori uncertainty and unmodulated dynamics, some of which are the following known classes of systems: robust systems; systems with parametric compensation; systems with interference absorption; systems using fractal algorithms; adaptive systems; systems with reduced sensitivity; systems with variable structure; extreme systems; repetitive systems; intelligent control systems (using neural networks, fuzzy multi-theory, genetic algorithms, chaos theory, etc.). There are differences between them in structural and parametric terms. They are predominantly governed by the class of control process, the type of control and interfering performance, and the controlled objective. In the case where the parametric disturbances in the processes are dominant and there are periodic external disturbances whose nominal parameters are known in advance but which fluctuate operationally around the nominal ones and one of the main requirements for the performance of the system is the performance and the robust performance of the system, application of robust repetitive systems is recommended as an effective. In this context, the aim of the current work is to study the properties and applicability of the repetitive filters in robust systems and systems with robust properties. [1].

Robust systems and systems with robust properties.
In literature are known systems with internal model MM R [1][2][3], systems with interference absorption DAS [1], and gain scheduled control system GSC [1], as well as the methods and the algorithms for their synthesis.  The free parameter method solves the task of analytical synthesis of robust systems MM R (figure 1) with an internal model for process control in the conditions of a priori uncertainty, under the criteria: (1): robust stability and robust performance of the system for a predetermined in the synthesis parameter set  , satisfying the requirements (1.a); local criteria for quality LCQ (1.b)process with minimal integral squared error, and initial conditions for synthesisa priori know * G ,  ,  , where:  The method of the equilibrium partial absorption equation [2] solves the task for analytical synthesis of systems (figure 2), absorbing the influence of wave structured disturbances on the adjustable magnetite, for controlled process under a priori uncertainty, under the criteria (6): minimum Euclidean norm (6.a) of the members of the equilibrium balanced absorption equation on the adjustable magnitude; equality stock sustainability (6.b) (or optimal module); LCQ (6.c) (process with reordering set const   ) etc., and initial conditions for synthesis: a priori known * G ,  ,  , a representative trend of the system error in operating (simulation) conditions. The control u (6) of the systems (figure 3) has two additive constituents: r u -on the basis controller * R , realizing the aim of control of the respective LCQ and d u -to absorb the impact of interference  . The matrix E , F и C (7) reflect the dependence of the vector y from the controlled vector u , the disturbance  and the condition of the system x . The "nominal" system corresponding to the absorption system is a hypothetical parametric, non-intrusive linear system synthesized at the given LCQ (6.c) to the nominal model * G , determined with the "nominal" controller * R (8).
Corresponding to the disturbance absorbing system (figure 2) "the classical" parametric disturbed a.
b. in real operational conditions (or with adequate computer simulation). o y , u are signal external interference aggregated on the assignment channel o y and/or the loading u , and  are aggregated internal additive or multiplicative parametric and/or structural disturbances with a wave structure in process control. Analytical design of the absorber A in the structure (figure 2) of the system is according (9), where the fluent is determined by a known methodology [1], and the optimum gain value A k is determined based on (10) and initial conditions for synthesis: a priori known  , type of regulator, where:

Repetitive control and ML filters.
The repetitive control is effectively applied in systems for controlled technological objects characterized by periodic signal interference [4,5]. In the structure of the repetitive regulator, the internal model in the ML (Memory Loop) [6][7][8][9][10][11]   p M L (figure 4 and figure 6) is used, which consists of a periodic signal generator allowing compensation of periodic disturbances with a predetermined and fixed value of the period p T . The base ML has the ability to "memorize" the frequency of "cut-off" and effectively to counteract it, through its specific structure as a dynamic system. There are dependency bridges for it (14)-(16): Figure 4. Base ML filter Figure 5. Improved ML filter In the literature is well-known and improved [9] ML (figure 5). It differs from the basic one [6][7][8][9][10][11] because it uses a model of delay, but in a different structure. Unlike basic one (17), the improved MLfilter (19) is a sustainable dynamic system.
The values of the controller setting parameters in the controlled technological process systems are subject to the aim and the criteria presented to the system in its synthesis. They are not a function of the respective ML-filter, which is added to the controller. In this sense, the design of the ML-filter in the control systems is autonomous and is not related to the synthesis of the controller. For designing ML-filters with memory is used "rejecting module bandpass filter criterion". The essence of this method is a sequential frequency correction. The design filter criterion is defined by (20).