APPLICATIONS OF INVERSE SIMULATION WITHIN THE MODEL ...

APPLICATIONS OF INVERSE SIMULATION WITHIN THE MODEL-FOLLOWING CONTROL STRUCTURE Linghai Lu, D.J. Murray-Smith, and E.W. McGookin Dept. of Electronics and Electrical Engineering University of Glasgow, Glasgow G12 8LT, UK

Abstract: This paper focuses on a detailed description and application of inverse simulation, which has been extensively investigated in the aircraft field over the past decade, to replace model inversion in the traditional model-following structure. Unlike most currently available approaches for model inversion, inverse simulation provides an alternative more feasible and causal way to design a feedforward controller for a nonminimum-phase system. The results from inverse simulation, combined with an H∞ feedback controller, have been applied successfully for an eighth-order linear helicopter model and show the validity and effectiveness of the approach. Copyright © 2002 USTRATH Keywords: Automatic control, feedforward control, inverse dynamic problem, multiinput/multi-output systems, tracking systems.

1. INTRODUCTION Since the introduction of regulator theory for linear systems (Fancis and Wonham, 1976) and then further application to the nonlinear regulator field (Isidori and Byrnes, 1990), output tracking has undergone a significant development. Because of defects existing in the regulator approach, and the finite-time transition between stationary setpoints (Graichen et al., 2005), model-inversion based techniques shown in Fig. 1 became an active research topic in the 1990s, especially in the aircraft field (Moghaddam and Moosavi, 2005; Mickle et al., 2004; Zou and Devasia, 2004).

Fig. 1 Model-following control design scheme (Graichen et al., 2005) with plant G. Kff is a model-inversion based feedforward controller (FFC) and Kfb is the feedback controller (FBC)

These techniques have also been applied to other fields such as flexible structures and manipulators (Wang and Chen, 2002), electronic device manufacturing (Rusli et al., 2005), and marine system control (Loo et al., 2005) etc. This model-following structure, shown in Fig. 1, gives precise tracking of a particular output trajectory by producing the feedforward input and state reference trajectories. It is a form of two-degree-offreedom control scheme. The feedforward controller (FFC) plays a vital role in providing precision tracking and the feedback controller (FBC) provides robust stability against uncertainties caused by exogenous disturbances and signal noise. The FBC can be designed by traditional control algorithms such as PID (Visioli, 2004), the H∞ algorithm (Che and Chen, 2001), and the LQ algorithm (Al-Hiddabi and McClamroch, 2002). There are also a number of methods available for the FFC design (Devasia et al., 1996; Campa et al., 2004; Graichen et al., 2005). Among them, nonlinear inversion-based techniques (Devasia et al., 1996) are the most widely accepted and have undergone significant development in the last decade. However,

the complexity of the developed approaches for model inversion, especially methods for the casual inversion of a nonminimum-phase system, means that this application is not easy to be implemented for helicopter and ship models, which involve large numbers of ordinary differential equations. In addition, they also require strictly relative order and thus the approach depends upon the model considered. Such conditions cannot always be satisfied in practical situations (Ramakrishna et al., 2001). Finally, their derivation also involves complicated local coordinate transformations. Inverse simulation is a methodology that can allow one to determine the control law that enables a vehicle with given aerodynamic characteristics to follow a prescribed flight path. Work reported by Avanzini (2004) has proved the possibility of inverse simulation being used to provide a reference input for the controlled model. There he simplified the original helicopter model with a two time-scale method at first and then applied traditional inverse simulation. Further investigations (Lu et al., 2006) have been carried out. This paper focuses on application of the results for a practical helicopter example. In addition, the control effort issue will also be discussed within the context of the modelfollowing structure.

simulation in the case of nonminimum-phase systems has been investigated. It was shown for the first time that inverse simulation can also be used to solve the nonminimum-phase problem for linear systems in a natural and more straightforward way compared with the model inversion approach. This development depends upon redistribution of zeros within the process of inverse simulation and provides a link between the linear inverse system and its discrete counterpart in a mathematical sense. There are some sampling-rate intervals or critical sampling points, for which all the zeros satisfy a condition that their magnitudes in the z-plane are less than one or are close to unity. Under such conditions, inverse simulation may still provide good convergence. Lu et al. (2006) point out that inverse simulation is in fact an approximation to the traditional discretisation process. This method has now been applied successfully to an eighth-order linear helicopter to obtained bounded inputs for prescribed manoeuvres. 3.

MODEL-FOLLOWING STRUCTURE DESIGN

The whole control structure scheme for the current application is shown as follows:

The paper is structured as follows. Section 2 recalls the main results presented in a recent paper by Lu et al. (2006) and then the FFC is designed using inverse simulation. Section 3 discusses design of the FBC by the H∞ methodology as well as presenting background information relating to inverse simulation. Section 4 presents numerical results obtained from application of this combined feedforward and feedback approach. 2.

REVIEW OF INVERSE SIMULATION AND MODEL INVERSION

In the last decade, a number of approaches for designing the inversion-based FFC are available (Graichen et al., 2005; Devasia et al., 1996). However, because of their complexity and issues associated with feasibility for specific practical applications, some authors began to look for new approaches to replace these inversion methods. One approach due to Avanzini (2004), has involved inverse simulation methods but did not address, in any detail, the issue of the suitability of inverse simulation in the case of nonminimum-phase systems. Lu et al. (2006) have extended the research and explored the applicability of the inverse simulation to nonminimum-phase systems in detail. Their main results are summarised as follows. The results from Lu et al. (2006) point out that, provided a suitable value of sampling-interval is used, inverse simulation can be applied instead of model inversion for a minimum-phase system. This proposition has been illustrated by an application involving a nonlinear HS125 fixed-wing aircraft model. In addition, the feasibility of inverse

Fig.2 The control structure scheme with disturbance and noise. Kff is inverse simulation and Kfb is a controller designed using an H∞ approach. In Fig. 2, ur represents the inputs calculated from the inverse simulation process, no represents external measurement noise, Rr stands for the reference state variables, and Cm is a transformation matrix which transforms the reference state variables into the desired outputs. W1, W2, and W3 represent the weighting functions. The variables Z1 and Z2, represent the tracking-error signals reshaped by the relevant weighting function and the reshaped control outputs u, respectively. The variable Y is the system output signal. In this paper, the helicopter model considered is the same as that used in (Lu et al., 2006). This model is a linearised nonminimum-phase system representing a twin-engined multi-purpose military helicopter in the hover condition. Details of the state variables, output variables, and parameters are as defined by (Skogestad and Postlethwaite, 1996).

As shown in Fig. 2, in this paper, inverse simulation is used to replace the traditional model inversion for the FFC design. The selected algorithm for inverse simulation is the most widely adopted technique based on the integration method of Hess et al., (1991). In addition, as well as input time histories, inverse simulation provides reference state trajectories for the feedback channels. This is a little different from the scheme shown in Fig. 1 and the system of Fig. 2 follows concepts presented by Devasia et al. (1996). Finally, for the process of inverse simulation, the model adopted is the originally linearised nominal model without disturbance. The four channels of heave velocity ( H& ), roll rate (P), pitch rate (Q), and heading rate (ψ& ) are selected to be the outputs. The inputs are still the original four control channels (main rotor collective pitch, main rotor longitudinal and lateral cyclic pitches and tail rotor collective pitch). The four desired manoeuvres as well as other parameters are as defined by Walker and Postlethwaite, (1996). 3.2 The H∞ feedback controller design Since the pioneering work of Zames (e.g. Zames, 1981), the H∞ algorithm has been a dominant method of design in the multivariable control field and been intensively investigated in aircraft applications. H∞ control has been recognized to be good at simultaneously meeting the high-precision tracking performance requirements, attenuating the control energy saturation, and improving stability robustness (Yang et al., 2005). The latest work relating to this control algorithm in the helicopter field application can be found in (Luo et al. 2003; Postlethwaite et al. 2005).

W1 (s) = diag{2* s + 5 ,0.5* s + 5 ,0.5* s + 5 , s + 0.005 s + 0.005 s + 0.005 s + 10 s s 0.5 * ,0.5 * ,0.5 * } s + 0.001 s + 0.001 s + 0.001

W2 ( s ) = diag{0.01* s + 0.001 , 0.01* s + 0.001 , s + 10 s + 10 s + 0.001 s + 0.001 0.01* ,0.01* } s + 10 s + 10 W3 ( s ) = diag{1, 0.1, 0.1, 1, 1, 1}

The reasons for selecting this W3 are based on that only the channels H& , P, Q, andψ& , are selected to be outputs. The selected channels are weighted by setting their values to be 1 and the others are forced to be unimportant by setting their values to 0.1. With these weighting function values, the final calculated cost-function value (γ) is 2.1, which is within the usually acceptable range of 0 to 4 (Skogestad and Postlethwaite, 1996). It should be noted that this might be not the optimal selection of weighting functions and an algorithm for selection of the optimal weighting functions can be found in (Yang et al., 2005). W4 is the unit matrix with compatible dimensions. Fig. 3 and Fig. 4 show the good performance of the designed H∞ controller for the weighting functions selected. inv(I+GK) 20 0 -20 mag (dB)

3.1 The feedforward controller design

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The following are the chosen values for the weighting functions W1(s), W2(s), and W3(s):

Fig.3 The plot of (I+GK)-1 GK.inv(I+GK) 10 0 -10 -20 mag (dB)

In the application of the H design algorithm, all goals are achieved by selecting different kinds of weighting functions since their formulae represent the performance specifications. Hence, the key point of H∞ control design is the choice of proper weighting functions. In Fig. 2, the weighting function W1(s) is used to shape the tracking error between the output signals and the reference input signals. Usually this is selected to be a low pass filter with a bandwidth equal to that of disturbance (Skogestad and Postlethwaite, 1996) since the disturbance is usually a low frequency signal. W2(s) is chosen to be a high pass filter with a crossover frequency approximately equal to that of the desired closed-loop bandwidth. Through this choice, saturation problem in the actuators could be avoided. W3(s) is the weighting function on the reference input and the values of its elements individually depend on the priority given to each input for the application being considered.

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Fig.4 The plot of GK(I+GK)-1 4.

SIMULATION RESULTS

In this section, three groups of manoeuvres are considered. The first group is taken from the standard heave axis response (Walker and Postlethwaite, 1996) based on the latest version of ADS-33E-PRF (Anonymous, 2000). The first group aims to check the validity of proposed method in

over model inversion. The results in the channels H& , P, and Q are nearly the same. However, in the channelψ& , the control structure with the FFC is slight better than the one without the FFC.

which the FFC is designed using inverse simulation. Compared with the first, the other two groups are more demanding and facilitate the investigation of the influence of the FFC on the tracking performance.

Fig. 6 shows the comparison of control efforts from these different approaches. As shown in Fig. 6, the collective control, the longitudinal cyclic control, and the lateral cyclic control are nearly same for these two control structures. The effort in the tail rotor channel with the FFC is slightly smaller than that without FFC. In addition, control inputs with FFC from inverse simulation are bounded regardless of the nonminimum-phase characteristics of the vehicle.

4.1 Manoeuvres from the ADS-33E specification

Other investigations with increasing noise levels have also been carried out. The results are similar to those shown in Fig. 5 and Fig 6. This robustness shown against measurement noise proves the effectiveness of the designed H∞ controller again. Furthermore, all results from the simulations show the validity of proposed approach in terms of the proposed replacement of model inversion by inverse simulation.

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10 P,rad/s

Heave Rate,ft/s

Since the available helicopter model is linearised around the hover situation, the desired vertical rate response is defined as having the qualitative appearance of a first-order lag with an additional pure delay, as shown in Eq. (1). The other three channels P, Q, andψ& are set to be zero in terms of their desired responses. 10 H& ( s ) = e − 0.162⋅s (1) 0.8225 ⋅ s + 1

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4.2 Demanding-manoeuvre applications

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Fig.5 Tracking-performance comparison of standard manoeuvres with minor measurement noise. ----Ideal values; −*−*− No FFC; ---------- With FFC (Δt =0.01s).

g ( s) =

θlS,deg

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ω n2 s 2 + 2ζω n s + ω n2

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where ωn=3.79 and ζ=0

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In this part, two groups of more demanding manoeuvres are selected to compare the tracking performance with or without the FFC. One (group A) involves use of the step response of a standard second-order transfer function, as shown in Eq. (2), for the heading rate. The other three output channels are set to be zero. In the other (group B), the roll rate requirement involves the step response of the transfer of Eq. (2) and other three output channels are zero.

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Fig. 7 to Fig 9 show results from simulations with minor measurement noise. Results from investigations with increasing noise level are similar to these as well as control inputs from group B.

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Fig.6 Control effort comparison of standard manoeuvres with minor measurement noise. −*−*− No FFC; ---------- With FFC (Δt =0.01s) Figure 5 shows that for the simulations of standard manoeuvres the systems with and without the FFC provide almost the same tracking performance with disturbances and minor measurement noise. The simulation process is causal since no predefined information is required in the current application. This is one of the strengths of the proposed method

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Fig.7 Tracking-performance comparison of demanding manoeuvres (A) with minor measurement noise. ----- Ideal values; −*−*− No FFC; ---------- With FFC (Δt =0.001s)

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the proposed approach for nonminimum-phase systems for various kinds of manoeuvre.

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Fig.8 Control effort comparison of demanding manoeuvres (A) with minor measurement noise. −*−*− No FFC; ---------- With FFC (Δt =0.001s)

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The model-following structure based on inverse simulation for the FFC combined with an effective H∞ controller as the FBC has been successfully designed. The results from an eighth-order linearised nonminimum-phase helicopter model have shown the validity and effectiveness of the approach. The implemented structure always achieves good tracking performance regardless of the manoeuvre adopted. The bounded results from the causal simulation process also prove the feasibility of inverse simulation for replacement of more complex modelinversion techniques in such situations. In addition, control efforts of the new approach are nearly the same as the old one without the FFC. 6.

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CONCLUSIONS

ACKNOWLEDGEMENT

Linghai Lu gratefully acknowledges the award of a Glasgow University Scholarship and an Overseas Research Studentship from the British Government. E. W. McGookin and D. J. Murray-Smith acknowledge support from the UK Engineering & Physical Societies Research Council through grant GR/ S91024/01.

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Fig.9 Tracking-performance comparison of demanding manoeuvres (B) with minor measurement noise. ----- Ideal values; −*−*− No FFC; ---------- With FFC (Δt =0.001s) The tracking performance shown in Fig. 7 and Fig. 9 are quite different from the ones shown in Fig. 5 for the less demanding manoeuvres. The structure with the FFC always provides good results, which are far better than the results without the FFC. This is shown particularly well in Fig. 9, where the structure without the FFC achieves good tracking only in the heave rate channel. The results of the other three channels are unsatisfactory. In addition, further investigations suggest that the use of smaller discretisation intervals will lead to more accurate tracking with the FFC. However, the same phenomenon cannot be found for the structure without the FFC. The different performance from these two types of manoeuvre is probably due to the one of the advantages of the FFC that it can relate the system response to commands by providing a 'directcontrol ' channel. In addition, it is known that, generally, the smaller the discretisation interval, the more accurate will be the control inputs obtained from inverse simulation (Hess et al, 1991). In terms of the control effort comparison, results are broadly similar for both structures, as shown in Fig. 8. These results demonstrate the stability of the algorithm for inverse simulation and the validity of

REFERENCES Al-Hiddabi, S. A. and N. H. McClamroch (2002). Tracking and Maneuver Regulation Control for Nonlinear Nonminimum Phase Systems: Application to Flight Control. IEEE Transactions on Control Systems Technology, 10(6), 780-792. Anonymous (2000), Handling Qualities Requirements for Military Rotorcraft, US Army Aviation and Missile Command, ADS-33EPRF, 2000. Avanzini, G. (2004), Two Time-Scale Approach to Inverse Simulation: Algorithm Development and Application. Personal Communication, Turin Polytechnic Inst. Campa, G., M. L. Fravolini, M. R. Napolitano, M. G. Perhinschi and M. Battipede (2004). A Stochastically Optimal Feedforward and Feedback Technique for Flight Control Systems of High Performance Aircrafts. Proceedings of the 2004 American Control Conference (AAC), 1713-1718. Che, J. and D. Chen (2001). Automatic Landing Control Using H Infinity Control and Stable Inversion. Proceedings of the IEEE Conference on Decision and Control, 1, 241-246. Devasia, S., D. Chen and B. Paden (1996). Nonlinear Inversion-Based Output Tracking. IEEE Transactions on Automatic Control, 41(7), 930-942. Fancis, B. A. and W. M. Wonham (1976). The Internal Model Principle of Control Theory. Automatica, 12(5), 457-465.

Graichen, K., V. Hagenmeyer and M. Zeitz (2005). A New Approach to Inversion-Based Feedforward Control Design for Nonlinear Systems. Automatica, 41(12), 2033-2041. Hess, R. A., C. Gao and S. H. Wang (1991). Generalized Technique for Inverse Simulation Applied to Aircraft Maneuvers. Journal of Guidance, Control, and Dynamics, 14 (5), 920926. Isidori, A. and C. I. Byrnes (1990). Output Regulation of Nonlinear Systems. IEEE Transactions on Automatic Control, 35(2), 3544-3548. Loo, M., E. W. McGookin and D. J. Murray-Smith (2005). Application of Inverse Model Control to IFAC Benchmark Models. 4th International Conference on Advanced Engineering Design. Lu, L., D. J. Murray-Smith and E. W. McGookin (2006). Relationships between Model Inversion and Inverse Simulation Techniques. Accepted for publication, 5th MATHMOD 2006, Vienna. Luo, C. C., R. F. Liu, C. D. Yang and Y. H. Chang (2003). Helicopter H Infinity Control Design with Robust Flying Quality. Aerospace Science and Technology, 7(2), 159-169. Mickle, M. C., R. Huang and J. J. Zhu (2004). Unstable, Nonminimum Phase, Nonlinear Tracking by Trajectory Linearization Control. Proceedings of the IEEE Conference on Decision and Control, 812-818. Moghaddam, M. M. and S. F. Moosavi (2005). Robust Maneuvering Control Design of an Aircraft via Dynamic Inversion and MSynthesis. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 11-18. Postlethwaite, I., E. Prempain, E. Turkoglu, M. C. Turner, K. Ellis and A. W. Gubbels (2005). Design and Flight Testing of Various H Infinity Controllers for the Bell 205 Helicopter. Control Engineering Practice, 13(3), 383-398. Ramakrishna, V., L. R. Hunt and G. Meyer (2001). Parameter Variations, Relative Degree, and Stable Inversion. Automatica, 37 (6), 871-880. Rusli, E., T. O. Drews, D. L. Ma, R. C. Alkire and R. D. Braatz (2005). Robust Nonlinear FeedbackFeedforward Control of A Coupled Kinetic Monte Carlo-Finite Difference Simulation. Proceedings of the 2005 American Control Conference, 2548-2553. Skogestad, S. and I. Postlethwaite (1996). In: Multivariable Feedback Control: Analysis and Design. JOHN WILEY & SONS, Chichester. Visioli, A. (2004). A New Design for a PID Plus Feedforward Controller. Journal of Process Control, 14(4), 457-463. Walker, D. J. and Postlethwaite, I. (1996). Advanced Helicopter Flight Control Using Two-Degreeof-Freedom H Infinity Optimization. Journal of Guidance, Control, and Dynamics, 19(2), 461-468. Wang, X. and D. Chen (2002). Tip Trajectory Tracking for a One-link Flexible Manipulator Using Causal Inversion. Proceedings of the

IEEE Conference on Control Applications, 507512. Yang, C. D., C. C. Luo, S. J. Liu and Y. H. Chang (2005). Applications of Genetic Taguchi Algorithm in Flight Control Designs. Journal of Aerospace Engineering, 18(4), 232-241. Zames, G. (1981). Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses. IEEE Transactions on Automatic Control, 26(2), 301-320. Zou, Q. and S. Devasia (2004). Preview-Based Inversion of Nonlinear Nonminimum-phase Systems: VTOL Example. Proceedings of the 43rd IEEE Conference on Decision and Control, 4350-4356.