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4-dimensional Control of a Remotely Piloted Vehicle

R. D. Linehan: K. J. Burnham and D. J. G. James+

Keywords: Mulitvariable flight control, 4- dimensional control, remotely piloted vehicle.

1 Abstract

The problem of accurate tracking to a non-zero reference input is addressed in the context of mul- tivariable flight control. There exists a require- ment of zero steady state error in order to perform specific autopilot commands such as accurate 4- dimensional control: thus enabling the aircraft to be accurately positioned in time and space. The novel algorithm of optimal dyadic pole placement (ODP) is presented as the solution to this ser- vomechanism problem [l]. In addition, in order to propose a pragmatic solution, a twc-stage con- trol strategy is suggested which is shown to offer robust performance which may be implemented through a gain scheduling methodology. The two- stage control strategy achieves zero steady state error through the application of LQR to obtain an approximate track to the reference signal fol- lowed by ODP to effect the negation of the steady state error. The effectiveness of this approach is demonstrated via simulation.

2 Introduction

The aim of this research is to upgrade the control strategies used within the autopilot onboard the Raven 2 remotely piloted vehicle (RPV). In par- ticular there are two main objectives: firstly the development of a control strategy to effect rapid gust rejection and secondly the developement of a strategy to enable 4-dimensional control of the air- craft. A pre-requiste of the latter is the tracking to a profile with zero error: this excludes the inde-

'Currently with Lucas Aerosapce tBoth with Control Theory & Applications Centre,

Coventry University

pendent use of optimal methods which by design minimize the tracking error, but never completely negate it. Even a very small error in velocity can rapidly grow over time into a very large error in position or altitude. The principle objective there- fore is the development of a control strategy which tracks to a non-zero reference veloicty profile with zero error, while displaying gust rejection proper- ties.

The aircraft under consideration is a small, short range, unmanned aircraft, used to procure tacti- cal military surveillance [2]. A basic function of the autopilot is the ability to perform loiter and dash manoeuvres, whereby the controIler tracks to a desired velocity profile. The ability to accu- rately control the velocity of the aircraft enables the positioning of the aircraft in time and space. Such 4-dimensional tracking is of significant op- erational advantage in terms of tactical military surveillance.

3 Mat hemat ical Modelling and Simulation

The mathematical model of the Raven 2 RPV has been derived from first principles through de- termination of the aerodynamic stability deriva- tives [3] and presentation of the differential equa- tions of motion in state space form.

For the purposes of mathematical modelling the aircraft is assumed to be rigid so that its motion can be considered to have six degrees of freedom. However, for the purposes of controller design, it is assumed that there is no interaction between longitudinal and lateral motions, a factor only sig- nificant during rapid rolling manoeuvres.

Due to limitations on space, only the longitudinal equations of motion and subsequent results are detailed. The longitudinal equations of motion

UKACC International Conference on CONTROL '96,2-5 September 1996, Conference Publication No. 427 0 IEE 1996

771

are represented as:

X = A x + B u

y = C x + D u

where:

presented in this paper, it is assumed that full state feedback is available, although the method- olgy has been successfully demonstrated using a realistic state estimator [I].

4 Problem Formulation

-0.6510 -1.5886 29.6714 -0.5110 The novel algorithm of optimal dyadic pole place- 0.7352 -1.6216 -6.4495 0.0273 ment is based upon the dyadic pole placement al-

gorithm proposed by Young [5]. In brief, the al- gorithm may be outlined as follows:

Stage One: Apply an initial feedback matrix K T ~ of the form:

1 -0.0719 0.3226 -0.0005 -9.7966

A = [ 0 0 1 0

-0.0068 0.2524

Kl-1 = O:KI1 (1) [ ' 1 = [ -2912567 -0.i204 1

where K T ~ is of full rank. The choice of the matrix Krl is left to the designer, but is nominally chosen as the (pxp) identity matrix I,,, where p is the number of control inputs. Application of initial feedback, removes cyclicity of the state coefficient matrix.

Stage Two: Apply a second stage dyadic feedback gain matrix, of the form:

, U = [ ; ; ] x = [ i ]

The longitudinal state vector x is defined in terms of horizontal veloicty U , vertical velocity w, pitch rate q and pitch angle 8 . The elevator and thrust control inputs U are denoted by the symbols 6~ and ST respectively. r '1

The elevator control input 6~ is limited by the upper and lower saturation values of +0.5236 -+

-0.3665 rad respectively, with a servo-controlled maximum slew rate of 1 rads-I. An idealised thurst control input is formulated such that the lower saturation limit is defined by the ratio of stall speed (21 ms-l) to maximum speed (50 ms-l) and the maximum saturation limit is de- fined as unity. This represents a realistic, veloc- ity dependent thrust variation from 3% to %% thrust.

The state space equations of motion have been de- termined at the following datum flight condition: horizontal velocity = 30 ms-', altitude = 330 m (equivalent air density = 1.1896 kgm-3) and air- craft mass = 80 kg. However, the complete op- erational envelope of the aircraft is considered as this has a significant effect on the open loop flight dynamics of the aircraft.

in order to place the closed-loop poles in their de- sired locations. The second stage feedback ma- trix is determined directly from the algorithm pre- sented by Young:

KTZ = yk;f

-1 yT BT (3)

ko =

where p = [PI pz ...pnlT is the vector composed

~~

from the coefficients of the open-loop characteristic polynomial, I I X - AI

r = [q r2 ... r,IT is the vector composed of the desired closed-loop characteristic polynomial coefficients

The equations of motion are solved within the MATLAB/SIMULINK environment [4]. In gen- eral separate multivariable control strategies are

-1 1 0 0 0

applied both of to which the longitudinal are described and by lateral two-inputlfour- motions; s = [ 7 1 0 ::: I] output systems. For the purposes of the results Pn-1 Pn-2 . . . Pl

772

The resulting complete control law is given by:

4.1 Problems associated with dyadic pole placement

The dyadic pole placement algorithm derived by Young allows a considerable amount of design flex- ibility in terms of the four design parameters [5]:

1. dyadic vector y

2. initial feedback matrix K T ~

3. desired closed-loop pole locations

4. position of lag poles.

However there are a number of problems associ- ated with the application of this strategy. Firstly, due to the necessity of placing lag poles along the real axis, this methodology is more suited to the application of longitudinal flight control than to lateral [l]. This is due to the presence of the spi- ral convergence/divergence mode which is char- acterised by a real root near the origin. Conse- quently, the stability of this lateral mode is very sensitive to the application of lag poles. It is rec- ommended that this methodology is only applied to the longitudinal motion, characterised by com- plex modes of oscillation. However, this restric- tion is of no consequence given the current ap- plication of tracking to forward velocity: a state variable within longitudinal motion.

Secondly the dyadic methodology requires the stipulation of the position of the desired closed- loop pole locations. However for multivariable systems, this is not straightforward, as classi- cal techniques only describe the closed-loop pc- sitions of SISO systems through the use of root locus techniques. A root loci of a multivariable system is only of limited value. Traditionally,

pole locations are assigned through application of flying qualities specifications [6 ] ; however these specifications only result in a range of applicable pole positions and not definitive locations. This highlights the problem of specifying pole locations

in aerospace applicatbns, the desired closed-loop

which are not actually achievable in the closed- loop. As a result it is possible to expend control effort in an attempt to obtain, unobtainable pole positions.

Application of the dyadic methodology is further conplicated due to the inter-dependency of the three design parameters: dyadic vector y, ini- tial feedback matrix K T ~ and lag pole position. Closed-loop systems are very sensitive to the po- sition of lag poles, with speed of response in- creasing significantly with respect to pole location along the real axis. This has a strong influence on the resultant demanded control action. It has been found that with the application of dyadic pole placement and the requirement of zero steady state error, it is common that the demanded con- trol action exceeds the saturation limits: conse- quently a level of detuning is required [l]. The designer can attempt to restrict the speed and magnitude of the demanded control action by tun- ing the dyadic vector and initial feedback ma- trix. However, variations in y and K T ~ affect the achieved position of the lag poles. The design then becomes an iterative process, as the designer continually adjusts y, K T ~ and the position of the lag poles in an attempt to fine tune the system re- sponse. If this iterative process is combined with an attempt to achieve unobtainable closed-loop pole locations then the design may be compro- mised.

As a result, the novel dyadic pole placement al- gorithm is proposed which is a hybrid design based on the linear quadratic regulator and the dyadic pole placement algorithm. This novel ap- proach uses the benefits and simplicity of the LQR methodology to determine the required closed loop pole positions. These optimum achievable closed loop pole positions are then used as start- ing point for a modified dyadic application which negates the tracking error inherent in pure opti- mum techniques.

5 Optimal Dyadic Pole Placement

A design methodology for optimal dyadic pole placement (ODP) is necessary due to the con- flicting requirements and compromise inherent in mulitvariable controller design. The opti- mal dyadic pole placement methodology allows

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the straightforward application of an optimal ap- proach incorporating zero steady state tracking error - hence enabling 4-dimensional control. The following methodology is proposed [l]:

tune the closed-loop system using the LQR strategy in order to obtain achievable opti- mal pole locations for the closed-loop system. This can be performed iteratively through variation of the weighting matrices Q and R until optimal closed loop performance is achieved within the control input constraints.

amend the system to a two stage dyadic form by applying initial feedback to remove non- cyclicity and close the resultant loop using the dyadic pole placement tracking algorithm

specify the desired closed-loop pole locations as those obtained from LQR

specify the desired lag poles by placing one pole at the origin, affecting a pole/zero can- cellation and retain the second pole location as a sliding pole which acts as an additional tuning parameter

if necessary simplify the design procedure by specifying the initial feedback matrix in terms of the dyadic vector yij , such that

It has been found that the definition of the initial feedback matrix in terms of the dyadic vector ensures that as the system is tuned or detuned, these two parameters work in unison rather than in opposition to each other [7].

tune the closed-loop system by sliding the lag pole position along the x-axis in order to obtain the system dynamics defined by the optimal poles. Because the system has been amended from that tuned using LQR, the op- timal dyadic pole placement algorithm does not necessarily achieve the optimal pole lo- cations, but careful tuning of the dyadic pa- rameters will result in optimal locations.

6 Results

Application of the novel ODP procedure allows optimal closed loop performance to be obtained in addition to the benefits of zero steaty state track- ing error. The ODP algorithm has been success- fully applied to the Raven 2 RPV - the aircraft has been simulated tracking to the demanded velocity profile with zero steady state error. This is per- formed within the non-hear rate and saturation limits of the actuators onboard the aircraft.

Initially, the algorithm is assessed at one trim flight condition and manually tuned at that po- sition. However, in practice the flight envelope of the aircraft varies considerably during any one mission. In particular, flight envelope variations in aircraft velocity, altitude and mass effect the values of the aerodynamic stability derivatives contained within the state coefficient and driving matrices. These variations effect the natural sta- bility modes of the aircraft and as a result, the loop gains required to perform specific autopilot commands.

Due to hardware restictions onboard the Raven 2 RPV, it is not feasible to propose an adap- tive, self-tuning strategy to be implemented as the controller design, consequently a gain sched- uled methodology is proposed [7]. This approach is simply applied through prior determination of the loop gains required at specific stages of the flight envelope. It is suggested that the three factors most effecting flight dynamics and ulti- mately loop gains are: veloicty, altitude (air den- sity) and mass. In practice these parameters may be measured directly (velocity) or inferred (air density via altitude measurement and mass via estimation of fuel consumption). Based on these three parameters the feedback gain matrix may be calculated through linear interpolation of the pre-determined gain matrices. This apporach has been succesfully demonstrated via simulation, us- ing fuzzy logic to perform the interpolation [7].

The application of optimal dyadic pole placement via a gain scheduling methodology however results in an additional problem. It has been found that the ODP strategy is sensitive to variations in flight envelope, in some circumstances resulting in sat- urated control action; although this did not effect the stability of the aircraft, only speed of response. Although, it should be noted that this problem may be reduced or eliminated through determina- tion of the feedback gains at more points through-

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out the flight envelope.

Control input saturation is in fact due to the requirement of the dyadic methodology of zero steady state error in contrast with LQR where the error tends asymptotically to zero. In an attempt to limit control action and to render the control strategy more robust, the tracking commands may be applied in two stages: firstly, LQR is applied as a base control action in order to achieve an ap- proximate track to the reference signal with min- imal control effort. Once steady state is achieved, characterised by a steady state error, the ODP strategy is applied to negate the reference error. Clearly, the required demanded control action to perform the negation of the steady state error is significantly reduced due to the base LQR control action.

Figure 1 demonstrates the implementation of the ODP strategy to negate steady state error while tracking to a 48 ms-’ velocity reference. In this situation, LQR has been implemented as a base control action and achieves steady state at 47.7 ms-l. The presence of this small error eliminates the possibility of performing 4-dimensional con- trol. Although the throttle control action does reach the saturation limits, its should be noted that the maximum aircraft velocity is 50 ms-’, therefore it is un-necessary to prevent throttle sat- uration. The magnitudes of the resulting varia-

tions on board the aircraft, it is proposed that the strategy is implemented via gain scheduling, In order to propose a robust solution to the tracking problem, the tracking control is applied through a two-stage approach: firstly LQR tracks to the in- put signal, obtaining an approximate track char- acterised by a steady state error then the ODP strategy is adopted to negate this error. Although initially the LQR tracking strategy may appear to be a suitable compromise between control ef- fort and tracking accuracy, it should be noted that any steady state error, no matter how small, grows with time, and eliminates the possibility of 4-dimensional control.

References

[l] Linehan R. D. Modelling Simulation and Con- trol of a Remotely Piloted Vehicle. PhD the- sis, Control Theory & Applications Centre, Coventry University, August 1995.

[a] Hudson S. Future roles for the Raven surveil- lance RPV. In Proc. Tenth International Con- ference on Remotely Piloted Vehicles, Bristol, UK, March 1993.

[3] ESDU International plc. Validated Engineer- ing Data Index, 1991.

tions in pitch rate and vertical velocity are Small.

The application of the ODP strategy completely negates the tracking error and the aircraft is shown flying at 48 ms-l. The achievement of the desired velocity profile allows the aircraft to be accurately positioned in time and space. The per- fomance of this approach has also been succesfully demonstrated in the presence of gusts simulated through a Dryden gust model [l].

[4] The Math Works Inc. MATLAB for Wzndows User’s Guzde, December 1991.

[5] Young p. C. and Willems J . c. An ap- proach to the linear multivariable servomech- anism problem. Internatzonal Journal o f Con- trol, 15(5):961-979, 1972.

[6] Department of &fense. Military SPecificatZon- flying qualities of p i lo ted airplanes, 1980. MIL- F-8785(ASG).

[7] Linehan R. D., Burnham K. J . , Hudson S. M., James D. J . G., and King P. J . Opti-

7 Conclusions mal control strategy for a surveillance RPV. In Proc. Colloquzum on the Control and Guzd- ance of Remotely Operated Vehicles. Digest The novel algorithm of optimal dyadic pole place- No: 1995/124, The Institution of Electrical ment has been successfully applied, via simula-

tion, to the tracking problem of the Raven 2 re- ‘lace, London, UK, June 1995. motely piloted vehicle. Tracking to an external

signal with zero steady state error allows effective 4-dimensional tracking of the aircraft.

Due to the sensitivity of the ODP strategy to vari- ations in flight envelope, and the hardware limita-

0.3 t /

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Time (s)

48 I h e E v

U g 47.9 - - 8 3 47.8 - -

7

L a c

- 47.7

3 -1 oo 10 20 30 40

Time Is\

x -5' Y

0 .-

Time (s)

0 A

E -0.02 22 '$3

3 -0.04

w

.g -0.06 > -0.08

I I

10 20 30 40 Time (s)

Time Is\

I

Time (s)

Figure 1: Elimination of the steady state error upon application of the ODP control strategy