Learning Control: Some Basic Design Approaches

Copyright © IFAC Algorithms and Architectures for Real-Time Control, Vilamoura, Portugal, 1997

LEARNING CONTROL: SOME BASIC DESIGN APPROACHES

Ferreiro Garcia, R., Pardo Martinez, X.c., Vidal Paz, J., & Coego Botana, J.

Dept. Electronica e Sistemas. Universidade da Coruna. E.S.Marina Civil. Paseo de Ronda, 51, 15011 A Coruna. Spain. E-maiL·{jerreiro. pardo}@Sol.des.fi.udc.es

Abstract: A method to identify by dynamic process mapping and adjust dynamically a PID controller is presented. Such method comprises two main issues: (a) Identifying the process under a reduced and linear math-model, restricted to its actual operating point by learning from real-time operation experience using a deterministic or fuzzy associative memory (DAM or FAM) as an innovation procedure. (b) Control design, which is being associated with a frequency technique for achieving both, phase and magnitude for finally find the PID parameters as control application . Proposed design strategies has been applied for validation by simulation of the dynamic controlled system, based on a mathematical model of process in response to external forces or load disturbances and internal couplings of a disturbed nonlinear tank level process. Keywords: Adaptive Control , FAM Mapping, Model Reduction, Learning Control , Identification.

adjust dynamically the regulator so as to get stability and robustness under changes in the environment (load, system parameters, set point). The main impediment to achieving control system specifications is that if high static gain is necessary for load disturbance rejection requirements, then, closed loop stability is generally affected and viceversa. The PID algorithm, if properly adjusted is characterized by its capability to support large nonlinearities parameter and load changes. Fortunately, if dynamically adjusted its perfonnance is much better, which is the case analyzed in this paper (F erreiro and Vidal, 1995).

1. INTRODUCTION A properly tuning method for PID control algorithms is one of the well accepted ways to save control energy as well achieve both, stability and performance robustness under hard environmental situations like parameter and load variations in most common industrial processes. PID controllers, though limited in their capabilities, are being applied successfully to a surprisingly variety of industrial processes, mainly non linear chemical processes, liquid flow systems, aerospace and marine vehicles, and even gas turbine engines.

Perfonnance specifications may require a control algorithm with more than three degrees of freedom . A way to increase the capabilities of the PID algorithm is by modifying dynamically the PID parameters according the results of the knowledge achieved by gathering real time information from input/output process data and stored by FAM or DAM process mapping mode. In figure I it is shown the general configuration of a dynamic adjusted learning control algorithm

In control system design task, the design specifications are basic, which can be loosely divided into two categories: robust perfonnance specifications and control law specifications. Once restricted control law specifications to the PID algorithm, the robust performance specifications describe how the closed loop system would perfonn if some parts of the system were changed or perturbed. Perturbations are mainly due to parameter variations and load disturbances, which affect closed loop system stability and performance. The objective is to

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Parameter identification is a powerful tool to get the updated model of a system with some restrictions derived of the rapid parameters variation from the internal model or external model such the disturbances. The fact of avoiding modelling errors in control design, guaranties that the control algorithm will achieve the desired performance. Identification may be realized by means of a learning algorithm properly used. Under the same idea, adaptive control is possible provided that the most difficult task of adaptive control is parameter identification. An interesting feature of learning-dam or fam algorithms is that they provide a generic class of nonlinear functions . This is used in the following ways when modelling nonlinear dynamical systems (Astrom and McA voy., 1992). Consider, for example, the linear model defined by the phase state space variable method

Fig. I. Configuration of a learning control system Second section describes the identification and learning design procedure based on the FAM mapping algorithm, third section illustrates the control design by frequency techniques and section forth shows the arrangement of several strategies on applications.

dx dt y

Ax+Bu

= ex

(1 )

(2)

Expression (1) and (2) can be replaced with

2. PROCESS IDENTIFICATION The necessity for applying learning control arises in situations where a system must operate in conditions of uncertainty, and when the available a priori information is so limits that it is impossible or impractical to design in advance a system that has fixed properties and also performs sufficiently well.

where m < n and the function f is a set of input variables to the hypercube which defines the fuzzy associative memory (F AM) and will be used to get the system model by the application of the proposed real time training-learning algorithm, which accumulate the knowledge on a FAM system, and the Dnx is the n order derivative of variable x.

The principal benefits of learning control, given the present state of its technological development, derive from the ability of learning systems to automatically synthesize mapping that can be used advantageously within a control system architecture. Examples of such mapping include a controller mapping that relates measured and desired plant outputs to an appropriate set of control actions or a model parameter mapping that relates the plant operating condition to an accurate set of model parameters (identification) (Waiter Let al. 1992). In general, this mapping may represent dynamic functions.

Figure 2 illustrates the identification task in which input and output data from the process is processed under a training learning phase, and is being accumulated into a fuzzy associative memory . With such information, it is solved the identification problem by computation. Once system model is parameter updated, then state variables can be determined as a function of real time inputs and output to the process. In the same way an adaptive algorithm is to be applied if required.

Although there is some differences between adaptive and leaning control, we see that both adaptive and learning control systems can be based on parameter adjustment algorithms, and that both make use of experimental information gained through closed-loop interactions with the plant. Clear differences exist between both methods: a control system that treats every distinct operating state as a novel one is limited to adaptive operation, whereas a system that correlates past experiences with past situations, and that can recall and exploit those past experiences, is capable of learning. In this work we will treat parameter estimation on the basis of a learning algorithm.

L - ._ _~

Reduced Model Computation

Fig. 2. System model procedure.

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searching by

learning

2.1 Solving the model parameter estimation task. Learning A1goritlun about Dynamic Behaviour

This sub-section deals with the problem of parameter identification, and fmally followed by the state variable determination (F erreiro and vidal, 1995). The problem of real time dynamic parameter identification via learning procedure requires a learning algorithm and a supporting tool to store the knowledge acquired. The input output data is processed in closed-loop and stored into a DAM rule system. Figure 2 shows the approach restricted to a first order general non linear system where the assumed linearized dynamic model is

V'2

Vu

Vl2

Va

V14

V21

V'n

V'u

V'13

V'14

V2Z

V'21

V'12

V'!3

V ' 24

V23

V ' 31

V'31

V'33

V'34

V24

V'41

V'42

V'43

V'44

(5)

Fig. 3. VO variables & FAM as stored knowledge. the output of learning system is the higher order derivative of the system output variable V2' in a first order system, that is or

2.2. Identification by learning techniques

The task of TRAININGILEARNING is a previous part of the identification procedure and is carried out in real time control (F erreiro, R., 1994). In this case it is restricted to a two input variables, and the steps to be performed are described below as,

(6)

a) Find the universe of discourse of any input and output variables.

Once defined the universe of discourse of variables and the number of membership-functions for each variable, the learning algorithm is to be applied under a real time training procedure. The data collected is stored as dynamic knowledge, and displayed into a FAM rule system as shown in figure 2.

b) Initialize the DAM by filling it in the training learning process with the nommal values at the middle of its range. c) Begin .an inner loop until time for arithmetic mean exprres

A set of FAM or DAM system equations may be formulated in order to solve for parameters estimation. When system must be identified around its operating point selection of Input DAM variables would be as close as possible in order to get the plant parameters at this point. An approach is,

d) Read I/O values e) Compute the values of membership-function for each inRut variable. Store the result of adding actual to past (lata as, UTU,k) = UTU-l , k-l) +UU,k)

(11)

NU. k) = NU - 1 .k - 1) + 1

(12)

(7)

where j and k are the membership-function values which belong to input variables.

(8)

f) Come back on inner loop until the time for evaluations is reached.

Figure 3 shows the input/output variables as data to enter the DAM system. Such knowledge from a FAM or DAM rule system may be represented in matrix form as

v

= [VJK

g) Compute arithmetic mean under an hypercube rule-base and update DAM by filtering in low-pass mode as,

(9) UToU,k)

and the plant parameters at the calculated operating point are directly achieved as,

=d · [

UTO U. k) + UTU. k)] NU. k) .d51

(13)

h) Apply expression (10) to achieve the plant parameters.

(10)

i) Initialize UT(j.k) and N(j,k) with zero values each.

where K is the vector of plant parameters achieved from the data of the F AM rule system.

j) Come back on outer loop. repeating training/ learnmg process loop.

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parameter identification task. This variables are the manipulated variable denoted as VID load disturbances as N, and controlled variable as Y

3. PID CONTROL DESIGN In the frequency domain design procedure, the process open-loop frequency response is necessary . Such response is straightforward achieved from the linear model given by (5) which under frequency domain is G(jwo) and defmed in terms of its magnitude and phase angle (phillips and Nagle, 1984).

Fig. 4. General control configuration So that, next step concerns the computation of values with which the regulator must contribute by phase angle and magnitude for a predefined phase margin and operating crossover frequency specified as design criteria. With such data, proportional gain is achieved deterministic ally from expression (I). Assuming the Ziegler & Nichols relation Kt, between the controller parameters Ti and Td which is closely to the value 4, then, PID design is carried out as foHows : The controHer parameters that satisfy the performance specifications are given as (Ferreiro, R. , 1994)) cooe

Kp = IGUw c )1 =

1_ = Td . wc __._ Tt · wc

cosA

-,;r

(14)

tane

(15)

Figure 5 illustrates some structures of industrial applications for DAM synthesis where a priori knowledge permits to establish a frequency model based design technique by restricting model to a low or first order linear one.

y

Vrn

Y"::1

For a given process G(jw), the selection of a crossover frequency Wc and phase margin Pm univocally defmes the proportional gain KpNevertheless, Td and Ti are not univocally determined, as per last expressions. In order to satisfy the expression in Td and Ti, it will be used the Ziegler-Nichols criteria subjected to some dynamic changes as rule-base conclusions Kt, to satisfy Ziegler-Nichols flexible criteria as shown below.

Td

=

Kt.tanA+.AK?tan(A)2+ 4 . Kt ) 2 . Wc .Kt

Ti = Kt· Td ;

« =2) 5 Kt 5 ( =8»

I y--.J PID I __ Kp -..I System 11 Design I" ::

~ DAM

Vm _ N

DAM System

~

PID Design

~ Kp

::

Fig. 5. Frequency model based design from DAM synthesis methods DAM system synthesis is only possible when no load disturbances affect the process, nor measuring noise, or noise is conveniently filtered. Such topic is shown at figure 5(a). When such is not the case, then, Figure 5(b) shows an approach to the process identification by taking into account the possibility of existence of load disturbances. Such DAM system may be reduced by processing input data as in figure S(c) where the difference between the manipulated variable and system load is considered as a variable. This is particularly useful when some a priori knowledge about the process exist, which means an advantage in reducing the model degrees of freedom.

(16)

(17)

4. SOME LEARNING CONTROL DESIGN STRATEGIES Real-time data gathering is the most important phase in learning control. Such data must be captured simultaneously, that means, a set of input/output data will be collected and stored every cycle, at the end of the same sampling period. Another topic in learning control is that several strategies in data gathering will affect performance.

Two main strategies about system control design are to be considered: a) on-line PID design b) off-line PID design On-line PID design requires a systematic DAM updating task and consequent PID design every time evidence of data changes are appreciated into DAM.

Figure 4 shows the general system control configuration from which data is to be gathered under several methods. In this scheme it is shown the most important variables that are relevant in model

Off-line PID design requires also a systematic DAM updating task in parallel computation with control

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application but only a data-base in which a set of PID's are stored will be modified when significant changes are observed in DAM data. The closed loop control selects the proper PID as function of the relevant information, that are set point, load disturbances, or set point and the difference between manipulated variable and load disturbances. Some combinations might be associated in order to fit the control strategies to every particular process. Future research to extend design alternatives to high order and nonlinear processes is in progress.

5. CONCLUDING REMARKS Some strategies based in model identification by a learning procedure has been shown. Some experience is also achieved from previous works in such topics which justify the following assumptions. In Industrial processes, generally non linear and most of them well defined by a first order model (into or near its operating point) an acceptable PID controller may be achieved. In any case a default controller must be previously defined under the possibility of such method might shut down. With this design method an additional DAM containing a set of PID's as function of set point and load disturbance values might be applied being useful.

6. REFERENCES Astrom, KJ. and McAvoy, T. (1992). Handbook of Intelligent Control. Neural, Fuzzy, and Adaptive Approaches. (David A. White, Donald A.Sofge). Chap. I, pp. 3-21. Van Nostrand Reinhold. New York. Baker, W.L. and Farrell, lA. (1992). Handbook of Intelligent Control. Neural, Fuzzy, and Adaptive Approaches. (David A. White, Donald A. Sofge). Chap. 2, pp. 35-61. Van Nostrand Reinhold. New York. F erreiro Garcia, R. (1994) Associative memories as lerning basis applied to the roll control of an aircraft. 5th Int. Symp. AMST'94. Application of Multivariable System Techniques. University of Bradford. West Yorkshire, U.K . Ferreiro Garcia, R. , and Vidal Paz, l , (1995) Learning Task Applied to Identification of a Marine vehicle. IF AC. Artificial Intelligent in Real Time Control, pp. 197 -201. Elseviere Science Ltd. London. Phillip, C.L. and Nagle, H.T., (1984). Digital Control Systems. Analysis and Design . Prentice- Hall, Inc. Englewood Cliffs, New York, U.SA

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