Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, 1996, pp. 237-246A9638726, AIAA Paper 96-4009

Tuning PID controllers using error-integral criteria and numericaloptimization

Masood SahraianUnited Defense, Santa Clara, CA

Srinivas KodiyalamLockheed Martin Advanced Technology Center, Palo Alto, CA

AIAA, NASA, and ISSMO, Symposium on Multidisciplinary Analysis and Optimization,

6th, Bellevue, WA, Sept. 4-6, 1996, Technical Papers. Pt. 1 (A96-38701 10-31), Reston,

VA, American Institute of Aeronautics and Astronautics, 1996, p. 237-246

A method for tuning proportional integral derivative (PID) controllers based on an error measure and numericaloptimization is presented. A state-space realization of the controller and the plant is used. A response surfaceapproximation-based optimization approach is used to minimize the number of detailed controller responseanalyses. A heuristic algorithm is used to select designs that are close to D-optimal designs to construct the initiallinear approximation. The tuning methodology is directly applicable to multiinput, multioutput control systems,and is shown to obtain improved results compared to the traditional tuning techniques, such as the Ziegler-Nicholsmethod. Realistic applications are provided to illustrate the method outlined. (Author)

Page 1

A96-38726

AIAA-96-4009-CP

Tuning PID Controllers Using Error-IntegralCriteria and Numerical Optimization

Masood Sahraian*United Defense LP, Santa Clara, CA

Srinivas Kodiyalam"Lockheed Martin Advanced Technology Center, Polo Alto, CA

Abstract

A method for tuning PID controllers based onan error measure and numerical optimization ispresented. A state-space realization of thecontroller and the plant is used. A responsesurface approximation based optimizationapproach is used to minimize the number ofdetailed controller response analyses.

A heuristic algorithm is used to select designsthat are close to D-optimal designs to constructthe initial linear approximation. The tuningmethodology is directly applicable to multi-input, multi-output control systems and isshown to obtain improved results compared tothe traditional tuning techniques such asZiegler-Nichols method. Realistic applicationsare provided to illustrate the method outlinedin this paper.

1. Introduction

Proportional, integral, and derivative (PID)control modes are linear control actions whichare implemented in the majority of commercialcontrollers because of their relative ease ofimplementation and adequate performance. APID control law is also known as a three modecontroller; the user can modify the dynamicproperties of this controller by adjusting thethree proportional, integral, and derivativeparameters. A judicial selection of thecontroller parameters such that the controlsystem performance requirements are met isknown as PID tuning.

There are many techniques for tuning PIDcontrollers, among them (1) open-looptechniques, and (2) closed-loop techniques [1].In the open-loop method the process ischaracterized by its response to a unit step inthe controller output. Normally a very simplefirst order system plus a time delay is used as amodel of the process and the process responseis fitted to this model. The controller settingsfor proportional, integral, and derivative gainsare then calculated as a function of processcharacteristics. In a closed-loop method knownas the Ziegler-Nichols method, an ultimateperiod of oscillations for the system response isdetermined by increasing the proportional gainwhile keeping integral and derivative gains attheir minimum values. The controller settingsfor proportional, integral, and derivative gainsare determined such that the process responseproduces a quarter decay ratio (the differencebetween each successive peak in the responseand the steady state value is one fourth of theprevious difference). The closed-loop methodslike open-loop methods rely on a simplemathematical model of the process in the tuningprocedure.

In another closed-loop technique known as theerror-integral method a more systematicprocedure is devised to find the controllersettings based on minimizing the error in theresponse of the controller [2]. A variety oferror-integral measures such as integratedsquare error (ISE) or integrated absolute error(IAE) are used. We refer to Figure 1 to describethis method. In this schematic a simple process(first order system plus time delay) is controlled

Senior Analysis Engineer; Engineering Analysis, Simulation, and Test Department.Senior Staff Scientist, Structures Department, Senior Member AIAA.

Copyright © 1996 by the American Institute of Aeronautics and 237Astronautics, Inc. All rights reserved.

by a PID controller. The controller computesthe control signal U(s) based on its error inputE(s). In error-integral method of tuning acontroller, an error criterion such as thefollowing is defined:

(1-1)

where L is a suitable function of the error e(t)such as a quadratic function (ISE measure).Note that the error e(t) is a function of the PIDparameters, namely Kp ,T, , and IA • Theseparameters are found by setting the Jacobian of<1> equal to zero. It is assumed that the functionL(e(t)) is zero only if the error is identically zeroand that it possesses a minimum value.

The above description follows the method givenin [2] and assumes a very simple process andrelies on a procedure which is not suitable forfast, reliable, and automatic tuning of PIDcontrollers. Note that the function <& dependson the controller parameters and that thisfunction is not normally known in a closed formsince that would require having available aninverse Laplace transform of E(s) which is notreadily available. Therefore even for the simplesystem of Figure 1 treated in [2], the minimumof equation (1.1) would have to be solved usinga numerical procedure.

In what follows a method is described based onstate-space methods and numericaloptimization techniques. Numericaloptimization techniques have long been used inintegrated control and structural design ofspace systems [3]. The PID tuning methodproposed in this paper does not makesimplifying assumptions on the type of theprocess and is meant for implementation in aComputer-Aided Control Engineering (CAGE)environment such as Matlab* or Matrix-X" .

2. Formulation of the Problem

Figure 2 shows a general structure for thecontrol system. It is assumed that the plant P(s)has the following state-space realization:

x = Fy=Hx

Gu(t) (2.1)(2.2)

where x is a n x 1 state vector, u is the controlvector, and y is the output vector. Without lossof generality we assume a single input, singleoutput (SISO) system in equations (2.1) and(2.2). In that case u and y are both scalarfunctions of time. The treatment in this paper is,however, general and with simplemodifications may be applied to a multi-input,multi-output (MIMO) system. The modelrepresenting the plant can be either a linearfrequency domain model or a general nonlinearstate-space model normally known as asimulation model. In both cases it is possible totransform the model of the plant into a linear,time-invariant form given by (2.1) and (2,2). Ifthe system we start with is a general nonlinearsimulation model, a linearization about asuitable operating point can be performed totransform the system to the linear, time-invariant form given above. All thetransformation operations described in thisparagraph can be performed using CAGE tools.

Referring to Figure 2 again, the controller D(s) isassumed to be a PID controller given by

U(s)E(s)

(2.3)

where

u(t) = Kp [e(t) + Td e(t)+—^e(t)dt ] (2.4)i /

The error input to the controller is given by

(2.5)

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where yrec (or r(t)) is a reference signal that theoutput of the plant y(t) has to follow, normally aconstant value.

238

We use the following strategy to tune thecontroller parameters as given in (2.4): thecontrol law (2.4) is used in the plant system(2.1). The error output of the plant e(t) is thenused to minimize a performance index given by(1.1). The form of the function L(e(t)) dependson the performance specifications of the controlsystem but it is customary to assume aquadratic function of e(t). The following sectiondescribes the solution method.

3. Solution Method

In this section we first present the details of theequations arising from implementing the PIDcontroller (2.4) in the state-space system (2.1-2).Then the optimization technique to solve for thetuning parameters is presented.

3.1 Details of the Equations

Before using (2.4) in the state-space system weneed to introduce an additional state toproperly represent the integral part of thecontroller. Let

x, = e(t)

which implies

x, =\e(t)dti

Now the augmented state vector is

(3.1.0)

(3.1.1)

(3.1.2)

where x is a n x 1 vector and XT is a scalar sincewe have only a single-output system in thiscase. For a multi-output system Xj would be acolumn vector and the appropriate changeshave to be made. The method described below,however , applies just the same. Theaugmented state-space system is then given bycombining (3.1.0) and (2.1) to get

(3.1.3)

where

F-H

G

Ha=[H 0]

(3.1.4)

(3.1.5)

(3.1.6)

(3.1.7)

(3.1.8)

We now re-write the PID control law (2.4) byusing the error equation (2.5) and its first timederivative. Note that

yref M-Hx = yref (0 - Haxa (3.1.9)

x,=G2xa (3.1.10)

G 2=[o i x n 1] (3.1.11)

After substituting (3.1.9) and (2.5) into (2.4) weget

(3.1.12)

The state-space equations (3.1.3) is re-written bysubstituting for u from (3.1.12)

(3.1.13)

J(Gl+KpGa)yref+KpT<iJGJref

where

239

r — / • f j , f T / ^ i j \ 1 C3. 1 td.\J — \I -r A I j\j ti ) ^.J.l.l'*^

It is assumed that the matrix inside parenthesisin (3.1.14) is nonsingular.

Note that what we have done so far is torepresent the PID controller in a form similar tostate feedback form in equation (3.1.12). Weshould note, however, that this has been an ad-hoc procedure and that: (a) the control law aspresented in (3.1.12) does not necessarily haveany desirable stability property, and (b) thestate-space system (3.1.13) does not necessarilyhave any desirable controllability-observabilityproperties

To summarize this section: the PID control law(2.4) was substituted into the modified state-space equations (3.1.3) to arrive at the closed-loop equations (3.1.13). In the following sectionthis state-space system is used along with theoptimality index (1.1) to find tuning parametersthat will optimize the controller (2.4) in thesense defined by (1.1).

3.2 Tuning by Optimization

The closed-loop system (3.1.13) matrices aredependent on the three tuning parameterssymbolically presented as

1 =(3.2.1)

The system equations (3.1.13) are re-written inthe following form

*a=A(|)jt .+*(| ,0 (3-2'2)

y=H.x, (3.2.3)

where

KK,TdJG.ynt

(3.2.4)

(3.2.5)

The optimization problem is formulated in thefollowing way: Find the set of design variables,I, such that

(3-2.6)

is minimized, with the side constraints,

e^.!<rwhere,

e(t) = yref(t)-Haxa (3.2.7)

The choice of the function L depends on theperformance requirements and control system.A quadratic function is normally used, i.e.,

(3.2.8)

determines the optimal tuningThe solutionparameters.

4. Optimization based on Response Surfaces

The optimization problem defined by equation(3.2.6-3.2.8) is solved using numericaloptimization techniques [4] along with aresponse surface methodology [5] forapproximating the controller responses as afunction of the tuning parameters, %. Aflowchart of the tuning optimization procedureis shown in Figure 3. The use of responsesurfaces for approximation of the controllerresponses is primarily to minimize the numberof detailed solutions of the state-spaceequations during optimization. Theimplementation of response surface approach issimilar to that described in Reference [6].

A combination of experiment design andregression analysis methods are used with thisimplementation of response surfacemethodology. The layout of the responsesurface is defined using a second orderapproximation of the form:

240

Using a least squares procedure, the polynomialcoefficients, 0., are determined. The first step inthe response surface construction is to generateand analyze (ndv +1) designs for a linearapproximation. Here, ndv corresponds to thenumber of design variables, £. A heuristicalgorithm for generating experimental designsusing Fedorov's exchange algorithm [7], is usedfor generating (ndv +1) designs that are close tobeing D-optimal designs. This algorithmattempts to maximize the determinant of XT Xfor the (ndv+1) selected design points, from aset of candidate design points. Matrix X isreferred to as a design matrix of candidatepoints from which the design points for thelinear approximation would be selected. For thedeterminant evaluation, X consists of only thoserows of the original matrix of candidate pointsthat correspond to the selected points.

If the linear model does not meet therequirements, and as each of the additionaldesigns are analyzed, the response surfaceapproximation is sequentially refined to a fullquadratic approximation. When more numberof designs than what is required for a quadraticapproximation is available, the best (ndv+1 +(ndv+l)*ndv/2) are retained for constructingthe response surface.

Finally, the approximate optimization problemis solved using the BFGS method programmedin ADS optimizer [8].

5. Applications

In this section two applications are presentedwhich illustrate the method. The firstapplication provides a proof of concept. Thesecond application provides a slightly differentformulation of the problem where an optimalplant parameter is optimized along with thePID tuning parameters.

5.1 First Application

In this section the tuning method presentedabove is applied to the following plant [1].

P(s)= 1 (5-D

This transfer function corresponds to thefollowing state-space realization of equations(2.1) and (2.2).

-0.9999 -394.85900 -10 3.2069e-06

394.8590-3.2069* -06

-1

= [0 394.8590 394.8590f

H = 0 0]

The structure of the control system is as inFigure 2. In this figure D(s) is the transferfunction of the PID controller as in equation(2.4).

The search for an optimal design started with abaseline PID design parameters as in Table 1.The baseline design results in a stable system.The dynamic performance of this design,however, is very poor as can be seen in Figure4. This figure shows the closed-loop responseof system (3.2.2) to a unit disturbance and azero reference input. This design results in verylarge settling time and overshoot giving rise toa relative large performance measure for 3> (avalue of 1.1977). The PID parameters and thevalue of performance measure 4> for the finaloptimal design is shown in Table 1. This tablealso shows the controller parameters ascomputed by Ziegler-Nichols and Shinskeymethods [1].

The closed-loop response based on the secondoptimizer iteration is also shown in Figure 4.Despite the fact that the optimizer started witha linear approximation the solution is improvedconsiderably only after two iterations. Anoptimization iteration history is shown inFigure 6. The closed-loop response for the finaloptimal design and for Ziegler-Nichols andShinskey tuning methods are shown in Figure 5.The ISE measure for these three solutions alongwith the time to reach steady state (fj and themaximum overshoot (ymax) are shown in Table 2.Finally the closed-loop poles (eigenvalues ofmatrix A in equation (3.2.4)) for these designsare presented in Table 3. The control systemperformance based on the optimal design is

241

better than that of the Ziegler-Nichols method.The latter method has the best results of manydifferent traditional tuning methods asdiscussed in [1].

Note that in the Ziegler-Nichols method anapproximation of the plant transfer function(equation (5.1)) is required to arrive at a firstorder plus dead-time model as in Figure 1. Thereason for this approximation is that theultimate period and gain which are used tocompute the PID parameters in Ziegler-Nicholsmethod are based on this type of model [2]. Inthe method presented in this paper, however,no such approximations are necessary and theoriginal model of the plant is used to computean optimal PID design.

Although the application presented here is for asingle-input, single-output (SISO) system, themethod is directly applicable to a multi-input,multi-output (MIMO) system such as acascaded control used in many processingplants. In fact we expect the method to be morepowerful in such cases where the traditionalmethods break down.

5.2 Second Application

Our present focus is on applying the presentmethodology to an integrated control-processdesign. In this application the PID tuningparameters and a process gain are optimizedsimultaneously. The process gain canincorporate such effects as the actuatorcharacteristics.

Summary

A methodology for optimal tuning of PIDcontrollers was presented. An integratedsquare error (ISE) measure containing theclosed-loop response error in the controlledvariable was minimized by using responsesurface optimization. The result of thisoptimization is a tuned controller which resultsin superior dynamic performance. Other errormeasures such as integrated absolute error

(IAE) can be used as well. The method wasapplied to a SISO control system. A MIMOcontrol system, however, can be tackled directlyusing the method presented in this paper.

References

1. L. Martins and D.E. Carvalho, DynamicSystems and Automatic Control, 1994.

2. A.M. Lopez, et. al., Tuning Controllers withError Integral Criteria, InstrumentationTechnology, November 1967.

3. A. Messac, et. al., Control-StructureIntegrated Design: a Computational Approach,Proceedings, AIAA/ASME/ASCE/AHS/ASC 31st Structures, Structural Dynamicsand Materials Conference, AIAA,Washington, D.C., April 1991.

4. G.N. Vanderplaats, Numerical OptimizationTechniques for Engineering Design withApplications, McGraw-Hill, New York, USA,

5. J. Krottmaier, Optimizing EngineeringDesigns, McGraw-Hill, Berkshire, UK,1993.

6. H. Thomas, Optimization of StructuresDesigned using Nonlinear FEM Analysis,AIAA/ASME/ASCE/AHS/ASC 37thStructures, Structural Dynamics andMaterials Conference, AIAA, Washington,D.C., April 1996. AIAA CP: 96-1385.

7. N.K. Nguyen and A. J. Miller, A Review ofSome Exchange Algorithms for ConstructingDiscrete D-optimal Designs, Comput.Statistics and Data Analysis, vol. 14, 1992,pp. 489-498.

8. G.N. Vanderplaats, ADS Users Manual,Version 2.01, VMA Engineering, ColoradoSprings, Colorado, 1987.

242

DisturbancePID Controller N(s) Process (Plant)

R(s) xrx E(s) 1 Y(s)

Figure 1: A typical PID Controller for Simplified Plant

Controller Pr ocess (Plant)

/»(*)Y(s)

Figure 2: A Generic Structure for a Control System

243

Initial Controller Parameters / Model

Heuristic algorithm to identify<ndv4l)

D-Dptimal designs, using Fedoiov exchange

Response SurfaceConstruction

Update ControllerParameters

> o p t

Numerical Optimization -ADS

4>

Response SurfaceApproximation

Figure 3: Flow of the Tuning Optimization Procedure

Figure 4: Comparison of Closed-loop Responses

244

0.2

10t

Figure 5: Comparison of Closed-loop Responses

ErrorCriterion '

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Iteration Number

Figure 6: Optimization Iteration history for Error Criterion <t>

245

Design

Ziegler-Nichols

Shinskey

Baseline Design

Optimal Design

K,

4.8000

4.0000

1.600

7.0048

T,

1.8138

1.2334

0.7255

0.9178

r.0.4534

0.2902

0.3628

0.6631

<E>

0.0558

0.1383

1.1977

0.0184

Table 1: PID Parameters for Different Designs

Design

Ziegler-Nichols

Shinskey

Optimal Design

4>

0.0558

0.1383

0.0184

*„

14

60

12

"max

0.19

0.22

0.12

Table 2: Dynamic Performance Results for Different Designs

Design

Ziegler-Nichols

Shinskey

Optimal Design

First Pair

-1.0957+ 0.0205 j

-0.0970+1.2791)

- 0.5278 ± 1.4492 j

Second Pair

- 0.4043 ± 1.4283 j

-1.4030 ± 0.0487 j

- 0.9721 ± 1.5110 j

Table 3: Closed-loop Poles for Different Designs

246