Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 103 (2017) 426 – 431
XIIth International Symposium «Intelligent Systems», INTELS’16, 5-7 October 2016, Moscow, Russia
Algorithm selection for the predictive model in control systems with incompletely observable control coordinate A.A. Kobzev*, N.A. Novikova, A.V. Lekareva Vladimir State University named after Alexander and Nikolay Stoletovs, 87, Gorky Street, Vladimir, 600000, Russia
Abstract In this report we examine predictive control of the systems with incompletely observable control coordinate. The scientific and methodical approach to algorithm selection for the predictive model is represented. The article contains the analysis of prediction methods based on the static processing of object state data. We present research results concerning the precision of methods of algorithm specification based on interpolation polynomials and ordinary least squares approximation of data. © 2017 Published The Authors. Published Elsevier B.V.access article under the CC BY-NC-ND license © 2017 by Elsevier B.V.by This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium «Intelligent Systems». Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” Keywords: predictive control; system with incompletely observable control coordinate; extrapolation; Newton polynomial; ordinary least squares method
1. Introduction Dynamic object control theory using predictive models (model predictive control) is one of the perspective approaches towards analysis and synthesis of automatic control systems (ACS) which is based on mathematical optimization methods. The main feature of this approach, which determine its successful application in construction and usage of control systems, is relative simplicity of the main scheme of feedback formation conjoined with high adaptive properties. The latter fact allows one to control multidimensional and multilinked objects of a complex structure including nonlinearities and to account uncertainties in specifying objects and disturbances1.
* Corresponding author. E-mail address:
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1877-0509 © 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” doi:10.1016/j.procs.2017.01.009
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2. Predictive control in systems with incompletely observable control coordinate Among the applications of model predictive control, we should explicitly notice its efficiency in synthesizing systems of dynamic object control with so-called «free blocks» in their circuits. They with regards to their properties cannot be included in a main feedback loop, therefore the dynamic properties of «free blocks» will not be accounted for, and it will negatively reflect on the ACS performance quality. This class includes such objects as mechanoprocessing equipment (point of contact of the tool with the workpiece), industrial robots (working surface of the tool in grasp), movement tracking systems, weapon objects on tracked and wheeled vehicles, mobile robots with attached equipment etc. In these systems feedback sensors regularly either are built in the actuating motor, or are gyroscopic and installed in direct vicinity of trunnion axis. Direct installation of the sensors to achieve data on the control coordinate and its derivatives is practically impossible2,3. One can achieve significant improvement of performance quality of aforementioned ACSs by introduction of the parallel predictive model, in which closure is performed directly on the control coordinate 2. Figure 1 shows the block diagram of an ACS with parallel predictive model. On the diagram three blocks are highlighted: the kinematic transmission actuator described by the transfer functions W1 p , W2 p ; the control object, part of which is covered by the main feedback, with the transfer function W31 p ; the control object, part of which is not covered by the main feedback, with the transfer function W32 p . Also present on the block diagram: V1 p , V2 p — transfer functions for an actuator by disturbance for
the parts of control object covered and not covered my main feedback respectively, g , f – control and disturbance signals, X 1 , X 0 - control coordinate and its intermittent value, covered by the main feedback loop of the ACS; G – system error value. Prediction of the external impact makes it possible to estimate the reaction of the system for some time forward (prediction step). According to these estimates the correction signal is computed (as a function of the difference between errors of predictive model and system), which allows to minimize the error of the ACS. Moreover, bootstrapping methods are applied to form the additional component4,5.
Fig. 1. Block diagram of ACS with predictive model
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3. Scientific and methodical approach to prediction algorithm selection Selection of the method for predicting the functions in dynamic systems requires estimation of the best approximation for the minimal number of measurements6. The scientific and methodical approach to algorithm selection for the predictive model includes the following components: x Analysis of characteristic modes of control object functioning for the informed choice of approximation function type. While choosing the prediction function one must consider the frequency spectra of control and disturbance signals, eigenfrequencies of bypass of tracking actuators, the duration of impulse characteristic of an actuator, the precision of tracking and/or the stabilisation, the frequency properties of blocks not covered by the main feedback. x Selection of an approximation method. Prediction can be performed using interpolation methods, with the task of “forward interpolation” - extrapolation or approximation of data received from sensors. There is a multitude of computationally different (i.e. in computational complexity) methods for function extrapolation, the most widely known being polynomial methods – canonical, Lagrange and Newton. In practice, however, Newton’s interpolation formula with use of finite differences (1) possesses a number of benefits compared to a canonical polynomial and Lagrange’s formula. The Newton polynomial can be written as follows: Pn ( x )
y n q ' y n 1
q ( q 1) 2!
' 2 y n 2 ...
q ( q 1)...( q n 1) n!
' n y0 ,
(1)
where q x x0 / h is a coefficient determined by a number of steps, ' n y0 ' n Pn x0 ' n 1 y1 ' n 1 y is a n-th degree finite difference of a polynomial Pn x in x0 , h is an interpolation step. When it is required to increase the degree of an interpolation polynomial by 1, adding yet another node, in Lagrange’s formula (where every term itself is a n-th degree polynomial) we must add not only another term, but also calculate each term in the expression anew. Meanwhile an addition of the new node in Newton’s formula (1) is reflected by an addition of another term, without recalculations. This significantly lowers the calculation delays in real time prediction tasks. However, no matter the method for interpolation polynomial construction, on the set of formerly specified points we will get the same result (to the precision of an rounding errors), because the specified n+1 points belong to exactly one n-th degree polynomial7. For the approximation of the functions specified in equidistant nodes there are many formulae for calculating the polynomials based on differences8, the unified approach was developed for finding the interpolation formulae, node selection and error estimation. However, the majority of said methods (Gauss, Stirling, Bessel etc.) are used for interpolation in the middle of an interval and are of little use in an extrapolation task. Employment of different methods for function approximation allows us to compromise between the precision of the presentation of extrapolated data, computational speed and memory usage. However, to achieve the required degree of extrapolation precision (smoothness of the results) one should employ any method necessary. x Estimation of source data quantity for prediction with the required precision – a number of nodes and their fixation (equidistant or with arbitrary distances). x Selection of an extrapolation step based on the theoretical data, modeling results with respect to the previous statements and prediction errors. 4. Research results and conclusions We offer two ways of prediction algorithm specification in control systems with incompletely observable control coordinate based on the statistical processing of control object state data, represented in the form of functional dependencies: 1. Prediction using interpolation polynomials. 2. Function approximation for prediction using ordinary least squares (OLS) method on the data. In particular, during the prediction algorithm selection we have considered Newton polynomial, cubic spline interpolation and OLS approximation.
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Let’s analyze the functioning of a prediction module during extrapolation of the functions y f x , implementing the movement trajectory on a plane. With the constant horizontal component of the movement speed v the coordinate x changes according to the law x vt , where t is time. In that case modeling the movement on a plane can be replaced with an equivalent modeling in time: y
A sin vt ;
y
A sin vt
(2)
A 2
sin 2vt .
(3)
Figure 2 shows absolute errors of prediction of the functions (2)-(3), achieved using the Newton polynomial, cubic spline interpolation and OLS approximation. Errors were evaluated using the expression E y ( x ) Pn ( x ) ,where y x is the true value of the function, Pn x is the result of an extrapolation and n is a degree of interpolation polynomial. The precision analysis for the function prediction using Newton polynomial while modifying their parameters (amplitude, velocity) shows that: x changing the amplitude causes proportionate change in the absolute error value, without modifying the character of the dependency E x ; x increasing the interpolation step twofold (on the interval of h d 0.02T , where T is the base harmonic period) leads to the 40 times increase in an absolute error value with different values of n . With h t 0.06T maximum absolute error exceeds 10% of the base harmonic amplitude and further increase makes extrapolation impossible; x the degree of Newton polynomial (which is determined by a number of interpolation nodes used for prediction) causes sufficient influence on the precision of extrapolation with interpolation step h not exceeding 2% of the base harmonic period. In that case increase of the polynomial degree n by one maximum absolute error decreases 10 times of the function of type (2) and 5 times for the function of type (3). x increasing the interpolation step causes decrease of the influence of polynomial degree (and therefore the source data quantity) towards the prediction precision. When h t 0.04T precision does not depend from n at all. Similar evaluations were performed for the spline extrapolation. Results show that extrapolation using the Newton polynomial with n t 4 maximum error is less than the spline interpolation (reaching 6 times with small h ). The increase of the step h during spline interpolation the error increases, although slower than while using Newton’s formula. With h t 0.025T errors are practically the same. Figure 3 demonstrates the dependency of maximum absolute error value from the base harmonic amplitude (the velocity of change of the function during the interpolation step) for Newton polynomials of third and fourth degree and spline extrapolation. Analysis of prediction calculations using OLS approximation has shown the results similar to the Newton polynomial, while performing significantly more calculations and being more computationally complex. Generalization of the data achieved during simulation of predictive model algorithm has shown the advantage of prediction using Newton polynomial. High precision of computation with relatively low complexity of an algorithm makes it the preferred candidate in synthesizing ACS with model predictive control.
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Absolute error of prediction of trajectories. (a)-(b) absolute error of prediction using Newton polynomial (n=4); (c)-(d) absolute error of prediction using cubic splines; (e)-(f) absolute error of prediction using OLS approximation (with the number of nodes k=5)
A.A. Kobzev et al. / Procedia Computer Science 103 (2017) 426 – 431
Fig. 3. Maximum absolute error of trajectory (3) prediction during the modification of an amplitude A. For Newton interpolation polynomial: 1 – n=3, 2 – n=4, 3 – spline interpolation
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