Compare of Transient Quality in Automatic Control Systems with Classic PID Algorithm and Optimal Regulator

Currently, about 90–95% of generic controllers use the PID algorithm to generate control actions, while 64% of the PID controllers are used in single-circuit automatic control systems. Most of industries (power industry among them) use hundreds of automatic control systems. The quality of their work is the basis of economic efficiency of technical processes, ensuring safety, reliability, durability and environmental friendliness of both technological equipment and automation equipment. There are different modifications of PID-controller structure implementation. In practice the ideal PID controller with a filter and the classic PID regulator (serial connection of the ideal PI controller and the real PD regulator as the direct action elements) are widely used. The problem of choosing a rational structure and a method of parametric optimization of PID controllers, which provide the best direct indicatives of the quality in the development of the main effects in single-circuit automatic control systems, becomes urgent. However, only for the classical PID controllers, which are widely used at present, there are more than three hundred methods for adjusting the three parameters of the optimal dynamic adjustment, as well as the ballast time constant. This results in arising a problem of substantiation of the best structure and method of parametric optimization of classical PID regulators. As a basic option, one of the simplest and most obvious one, viz. the method of automated adjustment of the controller in the Simulink MatLab environment had been chosen, which was compared with the method of full compensation in general for objects with a transfer function in the form of an inertial link with a conditional delay. Two variants of control action realization on the basis of the structural scheme of the optimal regulator developed by the Belarusian national technical University were also offered. In contrast with the classic PID controller, the optimal controller has one parameter of dynamic adjustment setting. The results of simulation of transients at basic perturbations confirmed that the best direct indicatives of the quality are provided with an optimal regulator, which makes it possible to recommend it for wide implementation instead of the classic PID controllers.


Introduction
Adaptive control systems design is one of most effective method to upgrade regulation quality of process variables. Adaptive control systems must consider plant's dynamic behaviours for wide range of load variation and dynamics of disturbances. They must use combined control principle in response to deviation and disturbance [1].
Long list of papers verifies this problem's relevance and importance. These scientific papers deal with PID controllers' adjustment and their realization [2][3][4][5][6][7][8][9][10]. This type of controller is the most difficult for adjustment among continuous controllers. PID controllers are used to regulate plants those are described differential equations of higher order. So transfer functions of plants can't be approximated dynamic elements of first order with time delay, because they can't give significant improvement of PID controller control quality [5]. Problem of adaptation automatic process regulator settings is reputed relevance Сравнение качества переходных процессов систем автоматического управления… too, because dynamic behaviours of plant are varied in wide range of load variation [6].
Within the order of 90-95 % under service regulators are using PID algorithm [6]. Also 64 % PID controllers are used in single loop automatic control systems and 36 % are used in multi loop systems. Thus problem of design and parametric optimization method for PID controllers becomes relevance. Solution to this problem lets to get best regulation costs in single loop automatic control systems to the different disturbances.

Description of simulation model
Plant's transfer function is a second-order relaxation circuit with delay time. Parameters of this transfer function are specified with the help of plant's transfer function experiment diagram for controlling action channel [9] 11,2 e 1,6e where k -plant's transfer function coefficient; Т, σ -larger and lesser transfer function time constants, s; τ с -delay time for controlling action channel, s. Plant's transfer function for external disturbance channel where k ext -transfer function coefficient for external disturbance channel; Т exttransfer function time constant for external disturbance channel, s. Widely used transfer function of classic PID controller is written as where k с -transfer function coefficient of controller; Т i , Т d , Т b -integration, derivative and ballast time constants, s.
There are a lot of different adjustment methods for PID controllers at this moment [2]. Automatized controller adjustment with the help of Simulink MatLab API is one of the most simplest methods. Process of adjustment and optimization is written [7]. Settings of optimal dynamic adjustment for PID controller after automatized controller adjustment are: k с = 2,789, Т i = 0,0246 s, Т d = 38,67 s and Т b = 0,4 s (first variant). These settings are chosen of minimum integral of the squared error (ISE). Automatic control system has minimum overshoot and minimum time, when controlled variable get to the controller's dead band (±2 %), under the given settings. This method can't help to calculate controller settings, which let controlled variable be changed without overshoot to the controlled variable step input.
Full compensation method in general terms will be the second variant of PID controller dynamic adjustment settings [9]. Derivative time is equal to delay time for controlling action channel under this method. Ballast time constant is calculated as mean value Т b = Т d /N, where N = 10 [2]. Time constants are equal for the second variant of PID controller dynamic adjustment settings: Т d = τ c = 11,2 s; Transfer function coefficient is calculated as where ξ -damping coefficient (equal 1), that help to rectify overshoot to the controlled variable step input. PID controller structure can be made with the help of optimal regulator transfer function to the controlled variable step input [10]. Optimal regulator let to operate input step without overshoot ( fig. 2).
Optimal regulator transfer function under input step can be found with the help of equations (8) and (1) [ where T cl -one and only one calculated dynamic adjustment setting of optimal regulator, which help to calculate regulation costs of automatic control system to the controlled variable step input. Filter transfer function with the help of equations (1) and (10) is equal The numerical value of T cl is calculated with the help of golden ratio number sequence (third variant) [10] T cl1 = 0.618τ c = 0.618 ⋅ 11.2 = 6.92 s. (12) The numerical value of T cl must be increased to make maximum control action equals to automatized controller adjustment method with the help of Simulink MatLab API (fourth variant) T cl2 = 0.725τ c = 0.725 ⋅ 11.2 = 8.12 s. (13) The tab. 1 gives dynamic adjustment settings for all four methods. Optimal regulator (T cl1 = 6.92 s) ----6.92 4 Optimal regulator (T cl2 = 8.12 s) ----8.12     The tab. 2 gives regulation costs of the automatic control systems to the controlled variable step input (x sp ), internal disturbance (f 1 ) and external disturbance (f 2 ). When regulation costs of four compared controllers (PID controllers and optimal regulators) were analyzed, it was found that fourth variant has the best regulation costs to the controlled variable step input, but third variant marginally better than fourth variant to the internal and external disturbances.

Results of transient simulation
1. It has been suggested three variants of PID and optimal controller adjustment (number 2-4), which were compared with automatized adjusted controller (Simulink MatLab API) to the input step in automatic control system.
2. ACS automatized adjustment with the help of Simulink MatLab API didn't let to adjust controller in such a way that controlled variable varies monotonically without overshoot to input step.
3. If PID controller has non-free behavior (only controller coefficient and time constants are adjusted), then controller adjustment with the help of full compensation method in general terms (second variant) has some advantages compared to automatized adjustment (first variant). There are: no overshoot; time, when controlled variable get to the controller's dead band, is by a 1.1 % less and maximum control action variation is by a 8.3 % less to input step. But maximum dynamic controlled variable variation is by a 38.7 % larger and stabilization time is by a 38.2 % larger than first variant to the internal disturbance (f 1 ). And maximum dynamic controlled variable variation is by a 3.9 % larger and stabilization time is by a 4.1 % larger than first variant to the external disturbance (f 2 ).
4. If PID controller has free behavior (controller structure can be changed), then it is appropriate to use optimal controller transfer function for controller adjustment. Simulation results of transients show significant improvement of control quality to the controlled variable step input. Stabilization time is by a 40.4 % less (third variant) and by a 32.6 % less (fourth variant) than automatized controller adjustment (first variant). But maximum control action variation is by a 37.2 % larger for third variant and by a 1.0 % less for fourth variant. As well these variants have no overshoot. 5. First, third and fourth variants have virtually the same regulation costs to the internal disturbance f 1 . Stabilization time is by a 2.9 % larger (third variant) and by a 5.9 % larger (fourth variant) than first variant. But maximum control action variation is by a 10.0 % less for third variant and by a 2.8 % less for fourth variant.
6. Use of optimal regulator let to improve regulation costs to the external disturbance f 2 . Stabilization time is by a 25.5 % less (third variant) and by a 20.4 % less (fourth variant) than first variant. But maximum control action variation is by a 2.3 % less for third variant and by a 0.8 % less for fourth variant.
7. Controller design with the help of optimal regulator transfer function let to improve greatly transient regulation costs to the input step and disturbances and let to simplify adjustment process too, because of optimal regulator has one and only one dynamic adjustment setting. Curves of control action variation in open-loop automatic control system are in close agreement for optimal and PID controllers.