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Helicopter translational rate command using individual channel analysis and design
Control Engineering Practice 6 (1998) 15—23
Helicopter translational rate command using individual channel analysis and design G.J.W. Dudgeon!, J.J. Gribble",* ! Formerly PhD student at the University of Glasgow; now with Rotorcraft Group, Flight Management and Control Department, DERA, Bedford MK41 6AE, England, UK " Lecturer, Centre for Systems and Control, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow, G12 8LT, Scotland, UK Received July 1997; in revised form November 1997
Abstract Translational rate command (TRC) in helicopter flight involves controlling the earth-referenced forward and side velocities, using the helicopter’s cyclic control. Handling qualities requirements for TRC systems suggest that the design should comprise an inner attitude command-attitude hold (ACAH) loop, with TRC control as an outer loop. This forms a non-square system, and a method to determine the implications of this structure for stability robustness is proposed here. Control synthesis was performed using the method of individual channel analysis and design. This technique is a neo-classical, frequency-domain control analysis and design method for multivariable systems. The control law obtained is very simple, and the handling qualities, assessed by non-linear simulation, are predicted to be at ‘Level 1’ (satisfactory without improvement). The conditions for stability robustness are satisfied at frequencies of significance for the handling qualities. ( 1998 Elsevier Science ¸td. All rights reserved. Keywords: Aerospace control; helicopter control; helicopter dynamics; robustness
Notation h 0 h 14 h 1# h 0T hQ h, / p, q, r u % v %
Main rotor collective pitch angle Main rotor longitudinal cyclic pitch angle Main rotor lateral cyclic pitch angle Tail rotor collective pitch angle Height rate (ft/s) Euler angles (rad) Body axes angular velocities (rad/s) Earth-referenced forward velocity (ft/s) Earth-referenced side velocity (ft/s)
1. Introduction Helicopters have distinctive characteristics that make them uniquely suitable for a wide range of military purposes. However, this same versatility means that the achievement of good handling qualities presents a signifi-
* Corresponding author. Fax: (0) 141 330 6004; e-mail: J. [email protected]. 0967-0661/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved PII S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 0 1 4 - 8
cant challenge to the designer, not only because of the range of piloting tasks, but also because of the complex dynamics of the vehicle (Key, 1988). By the early 1980s it had long been recognised within the US aerospace community that the existing handling qualities were inadequate, and efforts were directed towards the development of a new specification. A draft version of the new requirements appeared in 1988, and was used as part of the procurement package for the US Army Light Helicopter Family at that time. The requirements were adopted as a new ‘MIL-SPEC’ by the US Army Aviation and Troop Command in 1989, and since then appear to have become the de facto international standard for work in this area. The publication in 1989 of the revised helicopter handling qualities requirements ADS-33C (now upgraded to ADS-33D, (Anon., 1994)) has provided a focus for much research into the helicopter flight-control problem, by both academic and industrial workers. The highly coupled nature of rotorcraft dynamics has been thought to preclude the use of ‘one-loop-at-a-time’ control design methods, based on classical single-input single-output (SISO) techniques, and much of the recent helicopter
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G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
control research has therefore concentrated on modern synthesis techniques such as H (Walker and Postleth= waite, 1996), eigenstructure assignment (Manness and Murray-Smith, 1992), and linear quadratic Gaussian/ loop transfer recovery (Gribble, 1993). However, such techniques are regarded as lacking visibility and physical insight, being driven more by theoretical considerations than by engineering practicalities. Also, the order of the resulting control laws, which depend on the order of the models used for design, can be relatively high. This paper is concerned with the application of individual channel analysis and design (ICAD) (Leithead and O’Reilly, 1992), to a model of a typical single main rotor combat rotorcraft in hover, in order to obtain multivariable control laws that meet the specifications contained within ADS33D. ICAD clarifies the conditions under which classical frequency-domain-based analysis, of the Nyquist-Bode type, can be applied to multivariable systems, with some assurance that robustness problems associated with loop interaction will not occur. ICAD is particularly suitable for the helicopter flight-control problem because the performance requirements given in ADS-33D are given in terms of the response of specific outputs to specific reference inputs, which is the focus of the ICAD approach. Furthermore, many of the most demanding requirements are expressed in the frequency domain. It will be helpful to introduce a few concepts from ADS-33D. The acceptability of the flying qualities of the aircraft for a given ‘‘mission task element’’ (MTE) is rated on a three-point scale of three different ‘‘levels’’. A rating of Level 1 is the best, and means that the aircraft is ‘‘satisfactory without improvement’’ for that MTE. The document defines various ‘‘response types’’; the response type required is a function of the MTE. Of particular significance for this paper are the ‘‘translational rate command’’ (TRC) and ‘‘attitude command attitude hold’’ (ACAH) response types. In control engineering terms the TRC response type implies that the aircraft should track forward and lateral velocity commands supplied by the pilot. Similarly, the ACAH response type requires the tracking of pitch angle and roll angle commands. As the horizontal force components required for the TRC response on the one hand, and the pitching and rolling moments required by the ACAH response on the other hand, are both produced by the same physical mechanism (essentially by tilting the thrust vector produced by the main rotors) there is a conflict between these two response types. The dynamic performance requirements for a given response type are a function of the ‘‘Usable Cue Environment’’ (UCE) where UCE"1 means that visibility is good and UCE"3 means that visibility is poor. The response type of most relevance to this paper is the translational rate command (TRC), which relates to the hover and low-speed flight regime, and is necessary for Level 1 handling qualities in a degraded visual cue envi-
ronment (UCE"3). The specific mission tasks for which the TRC is required include precision hover, slung load carrying, sonar dunking and mine sweeping. Very little work on the TRC problem has been reported in the literature. Manness and Murray-Smith (1992) used an eigenstructure-assignment technique with full state feedback. These authors encountered difficulties which they attributed to non-minimum-phase zeros, and obtained TRC responses that were not as smooth as those of an attitude-command-attitude-hold system designed using the same technique. Young and Lin (1993) used the H method. The responses of this system leave room for = improvement, although this is likely to be a consequence of the authors having insufficient insight into the dynamics being controlled and not a consequence of using the H method. In particular, the velocities cannot easily be = controlled if the attitude loops are not stabilised (Hoh, 1988). The objective of this paper is to present a design for a TRC system, executed within the ICAD framework and philosophy, which satisfies the ADS-33D performance requirements and possesses an adequate level of robustness. Its specific contributions are the recognition that the performance requirements and plant dynamics imply that the control architecture should comprise two nested sets of feedback loops. Furthermore, because the standard version of ICAD provides robustness conditions only for a single set of feedback loops, the paper also develops techniques to assess the stability robustness of the nested structure. A brief description of the ICAD method is presented in Section 2. The requirements from ADS-33D for the TRC response type are stated in Section 3. These requirements lead naturally to a control-system topology in which high-bandwidth attitude-command-attitude-hold (ACAH) loops are enclosed within TRC feedback loops of lower bandwidth. The robustness analysis of the ACAH system is presented in Section 4.1. The analysis and design issues of the TRC controller are described in detail in Section 4.2. Simulation results based on a detailed non-linear helicopter model are presented in Section 5, and the main body of the paper is concluded in Section 6.
2. Outline of individual channel analysis and design This section gives a brief review of ICAD to establish the necessary equations for the understanding of this paper. For further details the reader should consult (Leithead and O’Reilly, 1992). An m]m multi-input multioutput (MIMO), linear, time-invariant plant can be modeled by a transfer function (TF) matrix G. A diagonal control matrix K is in the forward path immediately before G, and a feedback loop is closed around GK. (Non-diagonal feedback control can be accommodated within the ICAD framework, but was not found to be
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
necessary in this application, and is considered no further here.) In ICAD, attention is focused on opening one loop at a time, while the other loops remain closed. Without loss of generality, suppose that loop 1 is opened between output 1 of the plant and the input to the (1, 1) element of the diagonal controller K. (This can always be done by renumbering the plant input—output pairs.) Let G be partitioned as
C
D
g G 11 12 (1) G G 21 22 where g is a scalar, G is a 1](m!1) row vector, 11 12 G is a (m!1)]1 column vector and G is a 21 22 (m!1)](m!1) matrix. (All other matrices and vectors are also appropriately partitioned.) A block diagram of the configuration is given in Fig. 1. The TF describing y /r (Channel 1) is given by, 1 1 (2) C ,(g !G H G~1G ) k 1 11 12 2 22 21 1 where, G"
H ,(I #K G )~1K G . 2 2 2 22 2 22 Eq. (2) can be re-written as,
(3)
C "k g (1!c ) 1 1 11 1 where,
(4)
(5) c "g~1G H G~1G 11 12 2 22 21 1 c is known as the multivariable structure function 1 (MSF) of Channel 1. The effects of coupling from r to 2 y can be represented, without loss of information, as an 1 additive disturbance at the output of channel 1. The other Channels of the system can be derived in the same manner. Within the context of ICAD the ‘structure’ of a plant relates to the number of right half-plane poles (RHPPs) (the pole structure) and right half-plane zeros (RHPZs) (the zero structure). ICAD places great emphasis on the importance of the structure of the channels and other SISO transfer functions that arise in the decomposition of the MIMO system. For example, it can be seen from Eq. (4) that the zeros of Channel 1 include the solutions of
Fig. 1. Block diagram of ICAD configuration.
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c (s)"1. It is easy to extend the standard Nyquist cri1 terion to show (Leithead and O’Reilly, 1991) that the number of RHPZs of Channel 1 is related to the number of RHPPs of c (s), and the number of encirclements of the 1 #1 point by c ( ju). Therefore, sufficient conditions for 1 the number of RHPZs of Channel 1 to remain unchanged by plant variation are that both the number of RHPPs of c (s) and the number of encirclements of the #1 point by 1 c ( ju) remain unchanged. 1 The frequency response of Channel C can be used to 1 analyse the transient response and reference tracking of y for the nominal system in exactly the same way as in 1 a classical, single-loop system. ICAD recognises that the c s (i"1, 2 , m) have implications for the robustness of * the closed-loop plant. Referring to Eq. (4), if the Nyquist plot of c is close to the (#1, 0) point, then plant variation 1 may cause c to traverse the (#1, 0) point. If this happens 1 (and the RHPP structure of c does not change) then the 1 zero structure of (1!c ) will change, hence changing the 1 RHPZ structure of C . If an RHPZ is introduced into 1 C at a frequency where there is sufficient gain for a pole 1 of C /(1#C ) to be attracted to it and move into the 1 1 RHP, then the system will become unstable. A RHPZ may also be introduced into (1!c ) if the structure of 1 c changes. Furthermore, if c ( ju) is close to #1 at some 1 I frequency, then small changes in c ( ju) can produce very I large changes in C ( ju), which the channel stability mar* gins may be insufficient to ‘cover’. In summary, a closedloop system may be regarded as possessing stability robustness if: f the C s have adequate gain and phase margins with i respect to the !1 critical point, f the number of RHPPs of the c s is invariant and i f the c s are far from the (#1, 0) point at frequencies of i importance, i.e. frequencies at which the magnitudes of the corresponding channel frequency responses are either much bigger than, or comparable to, 1. An approximation to the MSFs can be obtained by assuming that the bandwidths on the other loops are infinite. For example, in the case of Channel 1, the approximate MSF, CK , is obtained as 1 (6) CK "g~1G G~1G . 11 12 22 21 1 (This is equivalent to using the constrained variable method (Tischler, 1987) to obtain an approximation to the Channel 1 transfer function.) The approximate MSFs are a function of the plant only, and are used to assess the inherent coupling between different input-output pairs of the plant. In particular, it can be determined whether the coupling is strong or weak and, if strong, whether or not it is benign. The importance of the approximate MSFs is that they can be used before detailed control design starts, in order to assess potential structural and robustness problems, and their use in this way is a unique feature of the ICAD approach.
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G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
In the 2]2 case, there is only one approximate MSF, denoted simply as c and given by g g c, 12 21 . g g 11 22
(7)
Also, for the 2]2 case, c and c can be written as ch 1 2 2 and ch respectively where 1 kg h , i ii i"1, 2. i 1#k g i ii
Fig. 2. Block diagram of TRC system.
(8)
For full details of ICAD, including theorems which establish the existence of stabilising controllers, please refer to (Leithead and O’Reilly, 1992).
longer be of the ACAH type. However, for convenience, we will continue to refer to ‘‘the inner ACAH loop’’.)
4. Individual channel analysis and design 3. Design considerations The natural starting point of the analysis and design process is to consult the specifications in order to determine the form of the required controller. ADS-33D (Anon., 1994) states that (i) The translational rate response to step cockpit pitch (roll) control position or force inputs shall have an equivalent rise time no less than 2.5 sec and no greater than 5 sec. (ii) The pitch and roll attitude shall not exhibit objectionable overshoots in response to a step cockpit controller input. Statement (i) indicates that the !3dB bandwidth of the translational rate responses should be approximately 0.4 rad/s for the shortest permissible rise of 2.5 seconds. (This follows from the well known approximate inverse relationship between bandwidth and rise time.) Statement (ii) suggests that the pitch and roll attitudes are required to be controlled in some manner. However, the primary plant inputs for the control of pitch and roll are the same as for the forward and lateral velocities respectively and therefore we cannot hope to control the velocities and attitudes independently. The proposed control structure thus consists of an inner and an outer loop. The inner loop will be ACAH augmentation, and following (Dudgeon et al., 1995), hQ , h#q, /#0.2p and r are fed back to h , h , h and h respectively via the inner loop 0 14 1# 0T controller K. The outer loop constitutes TRC augmentation in which u and v are fed back to the outer loop % % controller K@ which generates pitch and roll channel reference signals for the inner loop. The control structure is shown in block diagram form in Fig. 2, from which it can be seen that the helicopter model is a 6]4 TF matrix, which will be referred to as GH. The inner ACAH loop has 4 inputs and 4 outputs, and the outer TRC loop has 2 inputs and 2 outputs. (With the velocity loops closed the attitude responses of the overall system will no
4.1. ACAH system The ACAH controller is of the same form of the controller described in Dudgeon et al. (1995). Fig. 3 shows the c ’s (i"1.4) of the system. The scaling of the axes is i linear between !2 and 2 and logarithmic above 2 and below !2. The $ symbol on the axes should be interpreted as ‘10 to the power of’. (Liceaga-Castro et al., 1995) have suggested that, in the absence of specific knowledge of plant uncertainty bounds, one should regard the system as potentially lacking in stability robustness if the MSFs approach to within a distance of 0.2 of the #1 critical point. The non-robust region of radius 0.2, centred at the (1, 0) point, is marked in Fig. 3. It can be seen that c is within this 2 region (at frequencies below 0.005 rad/s). Plant uncertainty could cause c to traverse the (1, 0) point and 2 introduce a RHPZ into C , thus causing a potential 2 stability problem. However, as this sensitivity is at frequencies below 0.005 rad/s it is not a problem in practice because any low frequency unstable modes that might occur could easily be stabilized by the pilot with no significant increase in workload. This last statement will seem rather cavalier to readers unfamiliar with (Anon., 1994) and merits further explanation. None of the requirements of this document refer to a time-scale greater than 30 seconds, and most of the requirements cite much shorter time-scales. For example, the attitude command response type is defined with respect to the behaviour of the appropriate step response over a period of 12 seconds. Therefore, unstable modes with a frequency less than about 0.03 rad/s have no handling qualities significance. The gain and phase margins of the channels are shown in Table 1. Because the gain and phase margins are adequate and the c s do not approach the (1, 0) point at frequencies of t interest, the design is regarded as possessing stability robustness. It is noted here that the ACAH augmentation
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
19
identity matrix, then 0 E"
1 0
!GH(1 2 4, 124)KE or
0 0 (I#GH(124, 124)K)E"
1 0
.
0 The last two groups of equations can be combined and written in partitioned matrix form as: [0]
Fig. 3. Nyquist plot of ACAH c ’s. i Table 1 Gain and phase margins for Channels of ACAH system TF
GM (dB)
PM (deg)
k g (1!c ) 1 11 1 k g (1!c ) 2 22 2 k g (1!c ) 3 33 3 k g (1!c ) 4 44 4
30.09 13.08 12.56 18.82
51.79 44.41 44.48 50.43
is Level 1 for small signal considerations (Dudgeon et al., 1995). The same system has been the subject of extensive simulation studies for moderate- and large-amplitude manoeuvres, for which the effects of non-linearities are expected to be significant (Dudgeon et al., 1997). The controller was tested over a range of flight conditions without gain scheduling and performed well, building confidence in both the controller and the design methodology. 4.2. TRC system Referring to Fig. 2, the ‘‘open-loop plant’’ for the TRC design is the 2]2 sub-system of the closed-loop ACAH system from inputs 2 and 3 to outputs 5 and 6. This 2]2 system will be referred to as G@ (the prime is used to denote the outer system). Since the derivation of the general expression for the elements of G@ is a little messy, the discussion here will be confined to the derivation of element g@ of G@. The remaining three elements can be 11 derived in a similar way. Element g@ of G@ is the Laplace transform of output 11 5 of the ACAH closed-loop system when input 2 is set equal to 1. If K denotes the ACAH controller, then g@ "GH(5, 12 4)KE or g@ !GH(5, 124)KE"0. 11 11 The error signal E is the difference between the unit signal injected at input 2 and the feedback from the first four outputs from GH. If I denotes the 4-by-4
C
1 !GH(5, 1 2 4)K
DC D
0 I#GH(1 2 4, 1 2 4)
g@ 11 " E
0 1 0 0
where 0 denotes the 4-by-1 null matrix. By using Cramer’s rule, the solution for g@ can be written as the 11 quotient of two determinants:
K
g@ " 11
!GH(5, 1 2 4)K I(1, 1 2 4)#GH(1, 1 2 4)K I(3 2 4, 1 2 4)#GH(3 2 4, 1 2 4)K DI#GH(1 2 4, 1 2 4)KD
DGK D 11 " . DI#GH(1 2 4, 1 2 4)KD
K (9)
The conditions for stability robustness summarised towards the end of Section 2 require the individual TF elements of the open-loop system to be structurally invariant at frequencies of importance. It is customary to assume this to be true without question for the physical plant transfer function GH, and therefore it is not an issue in the design of the inner ACAH system. However, it cannot automatically be assumed to be true for the effective plant transfer function, G@, ‘‘seen’’ by the TRC loops: the individual TF elements of the open-loop TRC system G@ must be checked to ensure that interaction with the ACAH loop has not caused their zeros to become structurally sensitive to variations in the elements of GH. (The poles of G@ may be assumed to be insensitive because the ACAH design has already been found to be robust.) To calculate the sensitivity of the zero structure of DGK D to relative changes in a particular TF element g, ij DGK #DGK D"DGK D#dgC (10) ij ij ij DDGK D"dgC (11) ij DDGK D gC dg dg ij " "¸ (12) ij DGK D DGK D g g ij ij
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G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
where C is the cofactor of g. If Eq. (12) has a magnitude less than 1, then the zero structure of DGK D will remain ij invariant. In this case, it is not necessary to check the sensitivity of the zeros of elements g@ and g@ . To see 21 12 this, consider Fig. 4 which shows the Nyquist plot of the MSF, c@, of the 2-input 2-output effective plant G@ computed using Eq. (6). c@ is shown up to only 2 rad/s so one can clearly see the low frequency behaviour. It is seen that c@ does not encircle the (1, 0) point at low frequency and, once control action is introduced, c@h@ and c@h@ will 2 1 have magnitudes comfortably less than 1 at all frequencies due to the high frequency roll-off of h@ and h@ . This 2 1 means that the zero structure of elements g@ and g@ will 21 12 not affect the structure of (1!c@h@ ) and (1!c@h@ ) and so 2 1 the sensitivities of the zeros of g@ and g@ need not be 21 12 determined. Figs. 5 and 6 show the sensitivities of the zero structure of elements g@ and g@ to changes in the 22 11 individual elements of GK and GK (i.e. ¸ and ¸ cal11 22 11 22 culated from Eq. (10)). Each of these figures has sixteen curves, one for each of the sixteen elements of the 4]4 plant controlled by the four loops of the ACAH system. The salient feature of each of these figures is the greatest of the sensitivity functions at any given frequency. We anticipate potential robustness problems where the greatest sensitivity function has a magnitude much greater than one. It is seen that there is a sensitivity problem at low frequencies, below about 0.02 rad/s, but the same argument as used in section 4.1 shows that this sensitivity is of no practical significance in this particular application. In addition, high sensitivity is seen at approximately 5 rad/s on Fig. 5 and moderate sensitivity at approximately 15 rad/s on Fig. 6. Again, these sensitivities have no practical importance because with TRC channel bandwidths of about 0.4 rad/s and good channel loop-shapes, the channel magnitudes will be much less than 1 at these frequencies.
Fig. 4. Nyquist plot of c@.
Fig. 5. Sensitivity plot for g . 11
Fig. 6. Sensitivity plot for g . 22
In order to determine the form of the controllers required to stabilize the plant with TRC augmentation, it is necessary to perform a structural analysis of G@. Table 2 shows the structure of g@ , g@ and c@. 11 22 The forward velocity channel is assigned as Channel 1, and the side velocity channel as Channel 2. The TF of Channel 2 is (13) C@ "k@ g@ (1!c@h@ ). 1 2 22 2 If k@ is designed such that the number of RHPPs of 1 h@ is the same as the number of RHPPs of k@ g@ , then c@h@ 1 1 11 1 has no RHPPs; this is done by ensuring that k@ g@ does 1 11 not encircle the (!1, 0) point. Because c@h@ will have 1 small magnitude and does not encircle the (1, 0) point, then (1!c@h@ ) has no RHPPs or RHPZs, making 1 C@ stable and minimum phase, and 2 (14) (1!c@h@ )+1. 1 In order to stabilise C@ /(1#C@ ) (and hence, the plant) it 2 2 is necessary for k@ to cause no encirclements of the 2
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
(!1, 0) point of C@ , and hence no encirclements of the 2 (!1, 0) point of k@ g@ . 2 22 Because the level of coupling between the Channels is very low and there are no structural difficulties, the design of the controllers can effectively be performed on the diagonal elements g@ and g@ . The design was per22 11 formed using classical loop shaping techniques and the controllers are shown below. (15) k@ "!8.25(s#0.1)/(s(s#10)) 1 (16) k@ "5.93(s#0.14)(s#0.5)/(s(s#0.29)(s#10)) 2 The Bode plots of Channels 1 and 2 are shown in Fig. 7. The gain and phase margins of the Channels are shown in Table 3 and are seen to be good. The c@h@s (i"1, 2) are i shown in Fig. 8 and are seen to be nowhere near the critical (1, 0) point. Because the ACAH design is robust, and the interaction of the ACAH augmentation on the outer loop elements does not cause sensitivity problems, then the complete design can be regarded as possessing stability robustness.
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actuators, coning and flapping modes and engine dynamics. The step responses are shown in Figs. 9—11. For clarity, only off-axis responses of handling qualities importance are shown. Simple pre-filters were required to shape the responses, but due to space restrictions, they are not listed here. The rise times (calculated as the time to reach 0.632 of the steady state values) are approximately 4.0 secs and 4.3 secs for the u and v responses % % respectively, and both responses are seen to have qualitative first order appearances. The pitch and roll overshoots on the u and v responses are not regarded % % as being objectionable. The responses meet Level 1 requirements. Table 3 Gain and phase margins for TRC design TF
GM (dB)
PM (deg)
k@ g@ (1!c@h ) 1 11 2 k@ g@ (1!c@h ) 2 22 2
12.32 14.82
51.90 50.81
5. Non-linear simulation results The system was simulated using the non-linear helicopter model HELISIM: the model includes rate limited Table 2 Structural assessment TF
RHPZs
RHPPs
g@ 11 g@ 22 c@
0.002$5.309j none 0.003 0.119$0.838j 0.049$0.867j
none none 0.002$5.309j
Fig. 8. Nyquist plot of c@h@ (solid) and c@h@ (dotted). 2 1
Fig. 7. Bode plot of u Channel (solid line) and v Channel (dashed line). % %
Fig. 9. Height rate step response of (i) 15 ft/s and (ii) 30 ft/s.
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G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
Fig. 10. Forward velocity (u ) step response of (i) 15 ft/s and (ii) 30 ft/s. %
Fig. 11. Side velocity (v ) step response of (i) 15 ft/s and (ii) 30 ft/s. %
6. Conclusion TRC augmentation was designed for a model of a typical combat rotorcraft in hover (Smith, 1984). The control architecture involved an inner ACAH and an outer TRC loop. The robustness implications of this architecture were assessed by determining the sensitivity of the outer loop open-loop system due to plant variation in the inner loop. Conditions sufficient for the complete system to be robust are summarized as follows. (i) The ACAH system should be robust. (ii) The sensitivity of the outer open-loop structure due to the inner ACAH loop should be low. (iii) The outer TRC loop should be robust. All three conditions were met, and the system is regarded as possessing stability robustness. The design was also found, using non-linear simulation, to meet Level 1 requirements using only low-order, diagonal, control.
The greatest advantage of ICAD for this application derives from the nature of the specifications, which refer to the responses of specific outputs to specific inputs in SISO terms. ICAD easily copes with this kind of requirement. The channel loop shapes relate directly to on-axis responses. Off-axis responses due to cross-coupling can also be dealt with directly, although this paper has not dwelt on this issue. Other (implementation-related) advantages are that the feedback controllers are diagonal, and the order of the individual controller transfer function elements is low in relation to the order of the plant. The most characteristic feature of ICAD is its identification and use of the multivariable structure functions. Their role in the design of the ACAH system is described in detail in Dudgeon et al. (1995). Here it is sufficient to note that Fig. 3 shows that the coupling in the ACAH system is strong but essentially benign, apart from the low-frequency behaviour of c which is of no practical 2 importance for this application. Fig. 4, with the accompanying structural analysis and knowledge of the anticipated TRC bandwidths, makes it clear that the ACAH system has essentially decoupled the forward and lateral velocities (as one would have hoped), and that the design of the TC controllers can safely proceed solely on the basis of the diagonal elements of G@. In considering the robustness of the design it is convenient to distinguish between performance robustness and stability robustness. Competently designed ICAD controllers achieve performance robustness in the same way as controllers in SISO systems that have been competently designed using classical techniques: the channel gains are high at low frequencies, with a rapid roll-off of 20—30dB per decade in the crossover region, combined with adequate channel gain and phase margins. Stability robustness is assessed in a way that may be regarded as a direct extension of tried-and-tested classical techniques, by measuring the closeness of the approach of certain SISO frequency responses to the appropriate critical points. In particular, the behaviour of the multivariable structure functions in relation to the #1 critical point allows one to assess the effects of loop interaction on stability robustness. It should be appreciated that, in this case, knowledge of the actual plant model error bounds is extremely limited, which would make it difficult to perform formal robust control design in a meaningful way. The results in this paper show that ICAD is a powerful tool for the analysis and design of MIMO helicopter control systems.
Acknowledgements The authors wish to thank DERA (Bedford) for supplying the HELISTAB and HELISIM packages. The authors also wish to thank Professor John O’Reilly of the
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
University of Glasgow for many stimulating discussions on the theory of ICAD. The results reported in this paper formed part of the first-named author’s PhD work at the University of Glasgow. The views expressed in this paper are those of the authors.
References Anon., 1994. Handling qualities requirements for military rotorcraft (ADS-33D). United States Army Aviation Systems and Troop Command, St. Louis, MO., Directorate for Engineering. Dudgeon, G.J.W., Gribble, J.J., O’Reilly, J., 1995. The use of individual channel analysis and design to meet helicopter handling qualities requirements. Proceeding of the 21st ERF 3, Paper VII-3. Dudgeon, G.J.W., Gribble, J.J., O’Reilly, J., 1997. Individual channel analysis and helicopter flight control in moderate- and large-amplitude manoeuvres. Control Eng. Practice, 5(1), 33—38. Gribble, J.J., 1993. Linear-quadratic gaussian loop transfer recovery design for a helicopter in low-speed flight. J. Guid. Cont. and Dynamics 16(4), 754—761. Hoh, R.G., 1988. Dynamic requirements in the new handling qualities specification for US military rotorcraft. The Royal Aeronautical
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Society Helicopter Handling Qualities and Control Conference, Conference papers. Key, D.L., 1988. A new handling qualities specification for US military rotorcraft. The Royal Aeronautical Society Helicopter Handling Qualities and Control Conference, Conference papers. Leithead, W.E. O’Reilly, J., 1991. Performance issues in the individual channel design of 2-input 2-output systems. Int. J. Control 54, 47—82. Leithead, W.E. O’Reilly, J., 1992. m-Input m-output feedback control issues by individual channel design. Part 1. Structural Issues. Int. J. Control 56, 1347—1397. Liceaga-Castro, J., Verde, C., O’Reilly, J., Leithead, W.E., 1995. Helicopter flight control using individual channel design. IEE proc.Control Theory Appl. 142, No. 1. Manness, M.A. Murray-Smith, D.J., 1992. Aspects of multivariable flight control law design for helicopters using eigenstructure assignment. J. American. Heli. Soc. 37(3), 18—32. Smith, J., 1984. An analysis of helicopter flight mechanics part I — Users guide to the software package HELISTAB. RAE, TM FS(B) 569. Tischler, M.B., 1987. Digital control of highly augmented combat rotorcraft. NASA TM 88346, USAAVSCOM TR 87-A-5. Walker, D., J. Postlethwaite, I., 1996. Advanced helicopter flight control using two-degree-of-freedom H infinity optimization. J. Guid. Cont. and Dynamics 19(2), 461—468. Young, J.S., Lin, C.E., 1993. Refined H optimal approach to rotorcraft = flight control. J. Guid., Cont. and Dynamics 16(2).
Helicopter translational rate command using individual channel analysis and design G.J.W. Dudgeon!, J.J. Gribble",* ! Formerly PhD student at the University of Glasgow; now with Rotorcraft Group, Flight Management and Control Department, DERA, Bedford MK41 6AE, England, UK " Lecturer, Centre for Systems and Control, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow, G12 8LT, Scotland, UK Received July 1997; in revised form November 1997
Abstract Translational rate command (TRC) in helicopter flight involves controlling the earth-referenced forward and side velocities, using the helicopter’s cyclic control. Handling qualities requirements for TRC systems suggest that the design should comprise an inner attitude command-attitude hold (ACAH) loop, with TRC control as an outer loop. This forms a non-square system, and a method to determine the implications of this structure for stability robustness is proposed here. Control synthesis was performed using the method of individual channel analysis and design. This technique is a neo-classical, frequency-domain control analysis and design method for multivariable systems. The control law obtained is very simple, and the handling qualities, assessed by non-linear simulation, are predicted to be at ‘Level 1’ (satisfactory without improvement). The conditions for stability robustness are satisfied at frequencies of significance for the handling qualities. ( 1998 Elsevier Science ¸td. All rights reserved. Keywords: Aerospace control; helicopter control; helicopter dynamics; robustness
Notation h 0 h 14 h 1# h 0T hQ h, / p, q, r u % v %
Main rotor collective pitch angle Main rotor longitudinal cyclic pitch angle Main rotor lateral cyclic pitch angle Tail rotor collective pitch angle Height rate (ft/s) Euler angles (rad) Body axes angular velocities (rad/s) Earth-referenced forward velocity (ft/s) Earth-referenced side velocity (ft/s)
1. Introduction Helicopters have distinctive characteristics that make them uniquely suitable for a wide range of military purposes. However, this same versatility means that the achievement of good handling qualities presents a signifi-
* Corresponding author. Fax: (0) 141 330 6004; e-mail: J. [email protected]. 0967-0661/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved PII S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 0 1 4 - 8
cant challenge to the designer, not only because of the range of piloting tasks, but also because of the complex dynamics of the vehicle (Key, 1988). By the early 1980s it had long been recognised within the US aerospace community that the existing handling qualities were inadequate, and efforts were directed towards the development of a new specification. A draft version of the new requirements appeared in 1988, and was used as part of the procurement package for the US Army Light Helicopter Family at that time. The requirements were adopted as a new ‘MIL-SPEC’ by the US Army Aviation and Troop Command in 1989, and since then appear to have become the de facto international standard for work in this area. The publication in 1989 of the revised helicopter handling qualities requirements ADS-33C (now upgraded to ADS-33D, (Anon., 1994)) has provided a focus for much research into the helicopter flight-control problem, by both academic and industrial workers. The highly coupled nature of rotorcraft dynamics has been thought to preclude the use of ‘one-loop-at-a-time’ control design methods, based on classical single-input single-output (SISO) techniques, and much of the recent helicopter
16
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
control research has therefore concentrated on modern synthesis techniques such as H (Walker and Postleth= waite, 1996), eigenstructure assignment (Manness and Murray-Smith, 1992), and linear quadratic Gaussian/ loop transfer recovery (Gribble, 1993). However, such techniques are regarded as lacking visibility and physical insight, being driven more by theoretical considerations than by engineering practicalities. Also, the order of the resulting control laws, which depend on the order of the models used for design, can be relatively high. This paper is concerned with the application of individual channel analysis and design (ICAD) (Leithead and O’Reilly, 1992), to a model of a typical single main rotor combat rotorcraft in hover, in order to obtain multivariable control laws that meet the specifications contained within ADS33D. ICAD clarifies the conditions under which classical frequency-domain-based analysis, of the Nyquist-Bode type, can be applied to multivariable systems, with some assurance that robustness problems associated with loop interaction will not occur. ICAD is particularly suitable for the helicopter flight-control problem because the performance requirements given in ADS-33D are given in terms of the response of specific outputs to specific reference inputs, which is the focus of the ICAD approach. Furthermore, many of the most demanding requirements are expressed in the frequency domain. It will be helpful to introduce a few concepts from ADS-33D. The acceptability of the flying qualities of the aircraft for a given ‘‘mission task element’’ (MTE) is rated on a three-point scale of three different ‘‘levels’’. A rating of Level 1 is the best, and means that the aircraft is ‘‘satisfactory without improvement’’ for that MTE. The document defines various ‘‘response types’’; the response type required is a function of the MTE. Of particular significance for this paper are the ‘‘translational rate command’’ (TRC) and ‘‘attitude command attitude hold’’ (ACAH) response types. In control engineering terms the TRC response type implies that the aircraft should track forward and lateral velocity commands supplied by the pilot. Similarly, the ACAH response type requires the tracking of pitch angle and roll angle commands. As the horizontal force components required for the TRC response on the one hand, and the pitching and rolling moments required by the ACAH response on the other hand, are both produced by the same physical mechanism (essentially by tilting the thrust vector produced by the main rotors) there is a conflict between these two response types. The dynamic performance requirements for a given response type are a function of the ‘‘Usable Cue Environment’’ (UCE) where UCE"1 means that visibility is good and UCE"3 means that visibility is poor. The response type of most relevance to this paper is the translational rate command (TRC), which relates to the hover and low-speed flight regime, and is necessary for Level 1 handling qualities in a degraded visual cue envi-
ronment (UCE"3). The specific mission tasks for which the TRC is required include precision hover, slung load carrying, sonar dunking and mine sweeping. Very little work on the TRC problem has been reported in the literature. Manness and Murray-Smith (1992) used an eigenstructure-assignment technique with full state feedback. These authors encountered difficulties which they attributed to non-minimum-phase zeros, and obtained TRC responses that were not as smooth as those of an attitude-command-attitude-hold system designed using the same technique. Young and Lin (1993) used the H method. The responses of this system leave room for = improvement, although this is likely to be a consequence of the authors having insufficient insight into the dynamics being controlled and not a consequence of using the H method. In particular, the velocities cannot easily be = controlled if the attitude loops are not stabilised (Hoh, 1988). The objective of this paper is to present a design for a TRC system, executed within the ICAD framework and philosophy, which satisfies the ADS-33D performance requirements and possesses an adequate level of robustness. Its specific contributions are the recognition that the performance requirements and plant dynamics imply that the control architecture should comprise two nested sets of feedback loops. Furthermore, because the standard version of ICAD provides robustness conditions only for a single set of feedback loops, the paper also develops techniques to assess the stability robustness of the nested structure. A brief description of the ICAD method is presented in Section 2. The requirements from ADS-33D for the TRC response type are stated in Section 3. These requirements lead naturally to a control-system topology in which high-bandwidth attitude-command-attitude-hold (ACAH) loops are enclosed within TRC feedback loops of lower bandwidth. The robustness analysis of the ACAH system is presented in Section 4.1. The analysis and design issues of the TRC controller are described in detail in Section 4.2. Simulation results based on a detailed non-linear helicopter model are presented in Section 5, and the main body of the paper is concluded in Section 6.
2. Outline of individual channel analysis and design This section gives a brief review of ICAD to establish the necessary equations for the understanding of this paper. For further details the reader should consult (Leithead and O’Reilly, 1992). An m]m multi-input multioutput (MIMO), linear, time-invariant plant can be modeled by a transfer function (TF) matrix G. A diagonal control matrix K is in the forward path immediately before G, and a feedback loop is closed around GK. (Non-diagonal feedback control can be accommodated within the ICAD framework, but was not found to be
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
necessary in this application, and is considered no further here.) In ICAD, attention is focused on opening one loop at a time, while the other loops remain closed. Without loss of generality, suppose that loop 1 is opened between output 1 of the plant and the input to the (1, 1) element of the diagonal controller K. (This can always be done by renumbering the plant input—output pairs.) Let G be partitioned as
C
D
g G 11 12 (1) G G 21 22 where g is a scalar, G is a 1](m!1) row vector, 11 12 G is a (m!1)]1 column vector and G is a 21 22 (m!1)](m!1) matrix. (All other matrices and vectors are also appropriately partitioned.) A block diagram of the configuration is given in Fig. 1. The TF describing y /r (Channel 1) is given by, 1 1 (2) C ,(g !G H G~1G ) k 1 11 12 2 22 21 1 where, G"
H ,(I #K G )~1K G . 2 2 2 22 2 22 Eq. (2) can be re-written as,
(3)
C "k g (1!c ) 1 1 11 1 where,
(4)
(5) c "g~1G H G~1G 11 12 2 22 21 1 c is known as the multivariable structure function 1 (MSF) of Channel 1. The effects of coupling from r to 2 y can be represented, without loss of information, as an 1 additive disturbance at the output of channel 1. The other Channels of the system can be derived in the same manner. Within the context of ICAD the ‘structure’ of a plant relates to the number of right half-plane poles (RHPPs) (the pole structure) and right half-plane zeros (RHPZs) (the zero structure). ICAD places great emphasis on the importance of the structure of the channels and other SISO transfer functions that arise in the decomposition of the MIMO system. For example, it can be seen from Eq. (4) that the zeros of Channel 1 include the solutions of
Fig. 1. Block diagram of ICAD configuration.
17
c (s)"1. It is easy to extend the standard Nyquist cri1 terion to show (Leithead and O’Reilly, 1991) that the number of RHPZs of Channel 1 is related to the number of RHPPs of c (s), and the number of encirclements of the 1 #1 point by c ( ju). Therefore, sufficient conditions for 1 the number of RHPZs of Channel 1 to remain unchanged by plant variation are that both the number of RHPPs of c (s) and the number of encirclements of the #1 point by 1 c ( ju) remain unchanged. 1 The frequency response of Channel C can be used to 1 analyse the transient response and reference tracking of y for the nominal system in exactly the same way as in 1 a classical, single-loop system. ICAD recognises that the c s (i"1, 2 , m) have implications for the robustness of * the closed-loop plant. Referring to Eq. (4), if the Nyquist plot of c is close to the (#1, 0) point, then plant variation 1 may cause c to traverse the (#1, 0) point. If this happens 1 (and the RHPP structure of c does not change) then the 1 zero structure of (1!c ) will change, hence changing the 1 RHPZ structure of C . If an RHPZ is introduced into 1 C at a frequency where there is sufficient gain for a pole 1 of C /(1#C ) to be attracted to it and move into the 1 1 RHP, then the system will become unstable. A RHPZ may also be introduced into (1!c ) if the structure of 1 c changes. Furthermore, if c ( ju) is close to #1 at some 1 I frequency, then small changes in c ( ju) can produce very I large changes in C ( ju), which the channel stability mar* gins may be insufficient to ‘cover’. In summary, a closedloop system may be regarded as possessing stability robustness if: f the C s have adequate gain and phase margins with i respect to the !1 critical point, f the number of RHPPs of the c s is invariant and i f the c s are far from the (#1, 0) point at frequencies of i importance, i.e. frequencies at which the magnitudes of the corresponding channel frequency responses are either much bigger than, or comparable to, 1. An approximation to the MSFs can be obtained by assuming that the bandwidths on the other loops are infinite. For example, in the case of Channel 1, the approximate MSF, CK , is obtained as 1 (6) CK "g~1G G~1G . 11 12 22 21 1 (This is equivalent to using the constrained variable method (Tischler, 1987) to obtain an approximation to the Channel 1 transfer function.) The approximate MSFs are a function of the plant only, and are used to assess the inherent coupling between different input-output pairs of the plant. In particular, it can be determined whether the coupling is strong or weak and, if strong, whether or not it is benign. The importance of the approximate MSFs is that they can be used before detailed control design starts, in order to assess potential structural and robustness problems, and their use in this way is a unique feature of the ICAD approach.
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G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
In the 2]2 case, there is only one approximate MSF, denoted simply as c and given by g g c, 12 21 . g g 11 22
(7)
Also, for the 2]2 case, c and c can be written as ch 1 2 2 and ch respectively where 1 kg h , i ii i"1, 2. i 1#k g i ii
Fig. 2. Block diagram of TRC system.
(8)
For full details of ICAD, including theorems which establish the existence of stabilising controllers, please refer to (Leithead and O’Reilly, 1992).
longer be of the ACAH type. However, for convenience, we will continue to refer to ‘‘the inner ACAH loop’’.)
4. Individual channel analysis and design 3. Design considerations The natural starting point of the analysis and design process is to consult the specifications in order to determine the form of the required controller. ADS-33D (Anon., 1994) states that (i) The translational rate response to step cockpit pitch (roll) control position or force inputs shall have an equivalent rise time no less than 2.5 sec and no greater than 5 sec. (ii) The pitch and roll attitude shall not exhibit objectionable overshoots in response to a step cockpit controller input. Statement (i) indicates that the !3dB bandwidth of the translational rate responses should be approximately 0.4 rad/s for the shortest permissible rise of 2.5 seconds. (This follows from the well known approximate inverse relationship between bandwidth and rise time.) Statement (ii) suggests that the pitch and roll attitudes are required to be controlled in some manner. However, the primary plant inputs for the control of pitch and roll are the same as for the forward and lateral velocities respectively and therefore we cannot hope to control the velocities and attitudes independently. The proposed control structure thus consists of an inner and an outer loop. The inner loop will be ACAH augmentation, and following (Dudgeon et al., 1995), hQ , h#q, /#0.2p and r are fed back to h , h , h and h respectively via the inner loop 0 14 1# 0T controller K. The outer loop constitutes TRC augmentation in which u and v are fed back to the outer loop % % controller K@ which generates pitch and roll channel reference signals for the inner loop. The control structure is shown in block diagram form in Fig. 2, from which it can be seen that the helicopter model is a 6]4 TF matrix, which will be referred to as GH. The inner ACAH loop has 4 inputs and 4 outputs, and the outer TRC loop has 2 inputs and 2 outputs. (With the velocity loops closed the attitude responses of the overall system will no
4.1. ACAH system The ACAH controller is of the same form of the controller described in Dudgeon et al. (1995). Fig. 3 shows the c ’s (i"1.4) of the system. The scaling of the axes is i linear between !2 and 2 and logarithmic above 2 and below !2. The $ symbol on the axes should be interpreted as ‘10 to the power of’. (Liceaga-Castro et al., 1995) have suggested that, in the absence of specific knowledge of plant uncertainty bounds, one should regard the system as potentially lacking in stability robustness if the MSFs approach to within a distance of 0.2 of the #1 critical point. The non-robust region of radius 0.2, centred at the (1, 0) point, is marked in Fig. 3. It can be seen that c is within this 2 region (at frequencies below 0.005 rad/s). Plant uncertainty could cause c to traverse the (1, 0) point and 2 introduce a RHPZ into C , thus causing a potential 2 stability problem. However, as this sensitivity is at frequencies below 0.005 rad/s it is not a problem in practice because any low frequency unstable modes that might occur could easily be stabilized by the pilot with no significant increase in workload. This last statement will seem rather cavalier to readers unfamiliar with (Anon., 1994) and merits further explanation. None of the requirements of this document refer to a time-scale greater than 30 seconds, and most of the requirements cite much shorter time-scales. For example, the attitude command response type is defined with respect to the behaviour of the appropriate step response over a period of 12 seconds. Therefore, unstable modes with a frequency less than about 0.03 rad/s have no handling qualities significance. The gain and phase margins of the channels are shown in Table 1. Because the gain and phase margins are adequate and the c s do not approach the (1, 0) point at frequencies of t interest, the design is regarded as possessing stability robustness. It is noted here that the ACAH augmentation
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
19
identity matrix, then 0 E"
1 0
!GH(1 2 4, 124)KE or
0 0 (I#GH(124, 124)K)E"
1 0
.
0 The last two groups of equations can be combined and written in partitioned matrix form as: [0]
Fig. 3. Nyquist plot of ACAH c ’s. i Table 1 Gain and phase margins for Channels of ACAH system TF
GM (dB)
PM (deg)
k g (1!c ) 1 11 1 k g (1!c ) 2 22 2 k g (1!c ) 3 33 3 k g (1!c ) 4 44 4
30.09 13.08 12.56 18.82
51.79 44.41 44.48 50.43
is Level 1 for small signal considerations (Dudgeon et al., 1995). The same system has been the subject of extensive simulation studies for moderate- and large-amplitude manoeuvres, for which the effects of non-linearities are expected to be significant (Dudgeon et al., 1997). The controller was tested over a range of flight conditions without gain scheduling and performed well, building confidence in both the controller and the design methodology. 4.2. TRC system Referring to Fig. 2, the ‘‘open-loop plant’’ for the TRC design is the 2]2 sub-system of the closed-loop ACAH system from inputs 2 and 3 to outputs 5 and 6. This 2]2 system will be referred to as G@ (the prime is used to denote the outer system). Since the derivation of the general expression for the elements of G@ is a little messy, the discussion here will be confined to the derivation of element g@ of G@. The remaining three elements can be 11 derived in a similar way. Element g@ of G@ is the Laplace transform of output 11 5 of the ACAH closed-loop system when input 2 is set equal to 1. If K denotes the ACAH controller, then g@ "GH(5, 12 4)KE or g@ !GH(5, 124)KE"0. 11 11 The error signal E is the difference between the unit signal injected at input 2 and the feedback from the first four outputs from GH. If I denotes the 4-by-4
C
1 !GH(5, 1 2 4)K
DC D
0 I#GH(1 2 4, 1 2 4)
g@ 11 " E
0 1 0 0
where 0 denotes the 4-by-1 null matrix. By using Cramer’s rule, the solution for g@ can be written as the 11 quotient of two determinants:
K
g@ " 11
!GH(5, 1 2 4)K I(1, 1 2 4)#GH(1, 1 2 4)K I(3 2 4, 1 2 4)#GH(3 2 4, 1 2 4)K DI#GH(1 2 4, 1 2 4)KD
DGK D 11 " . DI#GH(1 2 4, 1 2 4)KD
K (9)
The conditions for stability robustness summarised towards the end of Section 2 require the individual TF elements of the open-loop system to be structurally invariant at frequencies of importance. It is customary to assume this to be true without question for the physical plant transfer function GH, and therefore it is not an issue in the design of the inner ACAH system. However, it cannot automatically be assumed to be true for the effective plant transfer function, G@, ‘‘seen’’ by the TRC loops: the individual TF elements of the open-loop TRC system G@ must be checked to ensure that interaction with the ACAH loop has not caused their zeros to become structurally sensitive to variations in the elements of GH. (The poles of G@ may be assumed to be insensitive because the ACAH design has already been found to be robust.) To calculate the sensitivity of the zero structure of DGK D to relative changes in a particular TF element g, ij DGK #DGK D"DGK D#dgC (10) ij ij ij DDGK D"dgC (11) ij DDGK D gC dg dg ij " "¸ (12) ij DGK D DGK D g g ij ij
20
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
where C is the cofactor of g. If Eq. (12) has a magnitude less than 1, then the zero structure of DGK D will remain ij invariant. In this case, it is not necessary to check the sensitivity of the zeros of elements g@ and g@ . To see 21 12 this, consider Fig. 4 which shows the Nyquist plot of the MSF, c@, of the 2-input 2-output effective plant G@ computed using Eq. (6). c@ is shown up to only 2 rad/s so one can clearly see the low frequency behaviour. It is seen that c@ does not encircle the (1, 0) point at low frequency and, once control action is introduced, c@h@ and c@h@ will 2 1 have magnitudes comfortably less than 1 at all frequencies due to the high frequency roll-off of h@ and h@ . This 2 1 means that the zero structure of elements g@ and g@ will 21 12 not affect the structure of (1!c@h@ ) and (1!c@h@ ) and so 2 1 the sensitivities of the zeros of g@ and g@ need not be 21 12 determined. Figs. 5 and 6 show the sensitivities of the zero structure of elements g@ and g@ to changes in the 22 11 individual elements of GK and GK (i.e. ¸ and ¸ cal11 22 11 22 culated from Eq. (10)). Each of these figures has sixteen curves, one for each of the sixteen elements of the 4]4 plant controlled by the four loops of the ACAH system. The salient feature of each of these figures is the greatest of the sensitivity functions at any given frequency. We anticipate potential robustness problems where the greatest sensitivity function has a magnitude much greater than one. It is seen that there is a sensitivity problem at low frequencies, below about 0.02 rad/s, but the same argument as used in section 4.1 shows that this sensitivity is of no practical significance in this particular application. In addition, high sensitivity is seen at approximately 5 rad/s on Fig. 5 and moderate sensitivity at approximately 15 rad/s on Fig. 6. Again, these sensitivities have no practical importance because with TRC channel bandwidths of about 0.4 rad/s and good channel loop-shapes, the channel magnitudes will be much less than 1 at these frequencies.
Fig. 4. Nyquist plot of c@.
Fig. 5. Sensitivity plot for g . 11
Fig. 6. Sensitivity plot for g . 22
In order to determine the form of the controllers required to stabilize the plant with TRC augmentation, it is necessary to perform a structural analysis of G@. Table 2 shows the structure of g@ , g@ and c@. 11 22 The forward velocity channel is assigned as Channel 1, and the side velocity channel as Channel 2. The TF of Channel 2 is (13) C@ "k@ g@ (1!c@h@ ). 1 2 22 2 If k@ is designed such that the number of RHPPs of 1 h@ is the same as the number of RHPPs of k@ g@ , then c@h@ 1 1 11 1 has no RHPPs; this is done by ensuring that k@ g@ does 1 11 not encircle the (!1, 0) point. Because c@h@ will have 1 small magnitude and does not encircle the (1, 0) point, then (1!c@h@ ) has no RHPPs or RHPZs, making 1 C@ stable and minimum phase, and 2 (14) (1!c@h@ )+1. 1 In order to stabilise C@ /(1#C@ ) (and hence, the plant) it 2 2 is necessary for k@ to cause no encirclements of the 2
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
(!1, 0) point of C@ , and hence no encirclements of the 2 (!1, 0) point of k@ g@ . 2 22 Because the level of coupling between the Channels is very low and there are no structural difficulties, the design of the controllers can effectively be performed on the diagonal elements g@ and g@ . The design was per22 11 formed using classical loop shaping techniques and the controllers are shown below. (15) k@ "!8.25(s#0.1)/(s(s#10)) 1 (16) k@ "5.93(s#0.14)(s#0.5)/(s(s#0.29)(s#10)) 2 The Bode plots of Channels 1 and 2 are shown in Fig. 7. The gain and phase margins of the Channels are shown in Table 3 and are seen to be good. The c@h@s (i"1, 2) are i shown in Fig. 8 and are seen to be nowhere near the critical (1, 0) point. Because the ACAH design is robust, and the interaction of the ACAH augmentation on the outer loop elements does not cause sensitivity problems, then the complete design can be regarded as possessing stability robustness.
21
actuators, coning and flapping modes and engine dynamics. The step responses are shown in Figs. 9—11. For clarity, only off-axis responses of handling qualities importance are shown. Simple pre-filters were required to shape the responses, but due to space restrictions, they are not listed here. The rise times (calculated as the time to reach 0.632 of the steady state values) are approximately 4.0 secs and 4.3 secs for the u and v responses % % respectively, and both responses are seen to have qualitative first order appearances. The pitch and roll overshoots on the u and v responses are not regarded % % as being objectionable. The responses meet Level 1 requirements. Table 3 Gain and phase margins for TRC design TF
GM (dB)
PM (deg)
k@ g@ (1!c@h ) 1 11 2 k@ g@ (1!c@h ) 2 22 2
12.32 14.82
51.90 50.81
5. Non-linear simulation results The system was simulated using the non-linear helicopter model HELISIM: the model includes rate limited Table 2 Structural assessment TF
RHPZs
RHPPs
g@ 11 g@ 22 c@
0.002$5.309j none 0.003 0.119$0.838j 0.049$0.867j
none none 0.002$5.309j
Fig. 8. Nyquist plot of c@h@ (solid) and c@h@ (dotted). 2 1
Fig. 7. Bode plot of u Channel (solid line) and v Channel (dashed line). % %
Fig. 9. Height rate step response of (i) 15 ft/s and (ii) 30 ft/s.
22
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
Fig. 10. Forward velocity (u ) step response of (i) 15 ft/s and (ii) 30 ft/s. %
Fig. 11. Side velocity (v ) step response of (i) 15 ft/s and (ii) 30 ft/s. %
6. Conclusion TRC augmentation was designed for a model of a typical combat rotorcraft in hover (Smith, 1984). The control architecture involved an inner ACAH and an outer TRC loop. The robustness implications of this architecture were assessed by determining the sensitivity of the outer loop open-loop system due to plant variation in the inner loop. Conditions sufficient for the complete system to be robust are summarized as follows. (i) The ACAH system should be robust. (ii) The sensitivity of the outer open-loop structure due to the inner ACAH loop should be low. (iii) The outer TRC loop should be robust. All three conditions were met, and the system is regarded as possessing stability robustness. The design was also found, using non-linear simulation, to meet Level 1 requirements using only low-order, diagonal, control.
The greatest advantage of ICAD for this application derives from the nature of the specifications, which refer to the responses of specific outputs to specific inputs in SISO terms. ICAD easily copes with this kind of requirement. The channel loop shapes relate directly to on-axis responses. Off-axis responses due to cross-coupling can also be dealt with directly, although this paper has not dwelt on this issue. Other (implementation-related) advantages are that the feedback controllers are diagonal, and the order of the individual controller transfer function elements is low in relation to the order of the plant. The most characteristic feature of ICAD is its identification and use of the multivariable structure functions. Their role in the design of the ACAH system is described in detail in Dudgeon et al. (1995). Here it is sufficient to note that Fig. 3 shows that the coupling in the ACAH system is strong but essentially benign, apart from the low-frequency behaviour of c which is of no practical 2 importance for this application. Fig. 4, with the accompanying structural analysis and knowledge of the anticipated TRC bandwidths, makes it clear that the ACAH system has essentially decoupled the forward and lateral velocities (as one would have hoped), and that the design of the TC controllers can safely proceed solely on the basis of the diagonal elements of G@. In considering the robustness of the design it is convenient to distinguish between performance robustness and stability robustness. Competently designed ICAD controllers achieve performance robustness in the same way as controllers in SISO systems that have been competently designed using classical techniques: the channel gains are high at low frequencies, with a rapid roll-off of 20—30dB per decade in the crossover region, combined with adequate channel gain and phase margins. Stability robustness is assessed in a way that may be regarded as a direct extension of tried-and-tested classical techniques, by measuring the closeness of the approach of certain SISO frequency responses to the appropriate critical points. In particular, the behaviour of the multivariable structure functions in relation to the #1 critical point allows one to assess the effects of loop interaction on stability robustness. It should be appreciated that, in this case, knowledge of the actual plant model error bounds is extremely limited, which would make it difficult to perform formal robust control design in a meaningful way. The results in this paper show that ICAD is a powerful tool for the analysis and design of MIMO helicopter control systems.
Acknowledgements The authors wish to thank DERA (Bedford) for supplying the HELISTAB and HELISIM packages. The authors also wish to thank Professor John O’Reilly of the
G.J.W. Dudgeon, J.J. Gribble / Control Engineering Practice 6 (1998) 15—23
University of Glasgow for many stimulating discussions on the theory of ICAD. The results reported in this paper formed part of the first-named author’s PhD work at the University of Glasgow. The views expressed in this paper are those of the authors.
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