APPENDIX A
APPENDIX A
Project Publications
PROJECT PUBLICATIONS (Supported Wholly or in Part by NASA Grant) 1. R.R. Zakrzewski and R.R. Mohler, "On Nonlinear Model Algorithm Controller Design," Proceedings, IFIP Conf. Sys. Modeling and Optimiz., Zurich, 1991. (See Appendix B) 2. R.R. Mohler, V. Rajkumar, and R.R. Zakrzewski, "Nonlinear Time-Series Based Adaptive Control Applications," Proceedings, IEEE Conf. Decision & Control, Brighton, 1991.
3. R.R. Mohler, Nonlinear Systems: Vol. 2 Applications to Bilinear Control, Prentice Hall, Englewood Cliffs, NJ, 1991.
4. R.R. Mohler, V. Rajkumar, and R.R. Zakrzewski, "On Discrete Nonlinear Self-Tuning Control," Proceedings, Korean Control Conf., Seoul, 1991.
5. R.R. Mohler, R. Zakrzewski, S. Cho, and C. Koo, "New Results on Nonlinear Adaptive High Alpha Control," Comcon 3, Victoria, 1991.
6. J.E. Kurek, "Analysis of Nonlinear Stability Using Robust Stability Analysis for Linear Systems," submitted, 1992.
APPENDIX B
APPENDIX B
On Nonlinear Model Algorithmic Controller Design
Published in Proc. IFIP Conf. Sys. Modeling and Optimization, Zurich, 1991.
ON NONLINEAR MODEL ALGORITHMIC CONTROLLER DESIGN R.R. Zakrzewski and R.R. Mohler Oregon State University Department of Electrical and Computer Engineering Corvallis, OR 97331 USA 1. INTRODUCTION Two nonlinear algorithmic controllers, MAC, are studied here. One uses a block-canceling Volterra approximation, and the other MAC consists of solving an approximating polynomial time series instate and control. Both methods synthesize discrete control sequences and are applied successfully to the control of a simple nonlinear longitudinal aircraft model for large variations in angle of attack.
The Volterra-series approach used here was introduced by Modyaev and Averina [1], and a form of inverse generating control according to an assumed structure is presented by Harris [2]. This work formed the basis for the methods used here. The high angle-of-attack aircraft model derived by Stalford, et al. [3] was the plant simulated for the MAC application. In many traditional design studies, a sequence of linearized perturbation models are derived for different equilibrium flight conditions with linear controllers appropriately derived. Linear adaptive control can be derived according to nonlinear gain scheduling of the control law. A highly successful version of such control, which includes proportional plus integral plus filter (PIF) terms, is presented by Ostroff [4,5]. However, such designs usually require a large number of set-point design computations, and may have stability problems for large fast changes in angle of attack and/or mach number.
For generation of the nonlinear control, a nonlinear time-series based model reference is used. In order to identify such model, experimental data was collected for angle of attack (a) and pitch rate (q) subject to random steps of control (stabilator, 5). To capture such phenomena as limit cycles in the data the steps were rather long (40 s). There were 64 such steps with time discretization of 0.1 s resulting in 25,600 points in a state plane for 64 values of control.
For a least-squares simulated data fit, the following approximation was surprisingly accurate: 2 3 «(k+1) = Pi««(k) + P « (k) + P « (k) + 2a 3a 2 3 P q(k)«(k) * P «q(k)o (k) + P q(k)o (k) * 5a 6 7a 2 3 P u(k)a(k) + P u(k)o (k) + p u(k)a (k) + p 9a 10a 11a 12a 2 3 = P a(k) + a (k) + a (k) + 1q P2q P3q 2 3 p q(k) + P q(k)o(k) + p q(k)a (k) + p q(k)a (k) + 4q 5q 6q 7q 2 3 Paqqu(k) + P9qU(k)a(k) + p u(k)a (k) + p u(k)o (k) + p 10q 11q 12q Even limit cycles are accurately rendered by this model, as well as the stable zone behavior, although large discrepancies occur when the control values are close to the stable/unstable zones border.
2. ADAPTIVE CONTROL APPROACHES 2.1 Nonlinear Volterra-Based Control Here, as in [6], the Volterra series serves as a conceptual starting point for a nonlinear time series base control. Continuous time controllers based on Volterra j series were systematically developed in [7] with formulae for the controller's kernels given those of the plant and of the desired feedback system. In particular, the problem of so-called exact feedback linearization was solved here. However, those formulae may be of limited practical value because of the properties of Volterra series under feedback. The problem is that even finite (e.g., second order) Volterra series of the open loop results in infinite Volterra series of the closed loop. This makes it necessary for the controller to include theoretically an infinite number of compensating terms even for a quadratic system. The same problem for the discrete time systems was treated in [1] with multidimensional Z transforms to derive the set of formulae equivalent to those for so-called exact feedback linearization [8].
However, they also provided a very elegant transformation of which results in a controller requiring only as many Volterra terms as there are in the assumed plant.
One attractive feature of this controller is that its structure makes it possible to utilize it not only with models represented in the form of Volterra series, but in fact with any model with easily divided linear and nonlinear parts of the dynamic equations such as (2) above.
The following algorithm results: a) according to the linear part of the plant, calculate the linear control u (k) L b) calculate the predicted value of the output at the moment k y(k) = L(y(k-1) ..... y(k-M) u(k-1),...,u(k-M)) f ) ..... y(k-M),u(k-1) ..... u(k-M) c) solve the "linearizing" control equation for x(k) such that tf(y(k),y(k-1) ..... y(k-lvU1),u (k)-x(k),u(k-1) ..... u(k-IVM)) = L =L(x(k),x(k-1) ..... x(k-M+1),y(k),y(k-1) ..... y( 3) calculate the control by u(k) = U (k) - x(k) L This algorithm becomes a sort of prediction controller which tries to estimate the effects of the previous controls knowing the previous values of outputs and then to adjust the current value of control so that the nonlinear part of predicted output is canceled.
This discrete time nonlinear a control algorithm is generated according to an off- line identification of model (1) with a nonlinear aircraft simulation based on [3].
Also, a linear controller was designed according to the linear parts of (l)-(3).
The design was performed to obtain the closed loop model reference behavior of the form G(z) = 0.05/(z - 1.6z + 0.65) In order not to cancel the zero of the plant, the observer polynomial (z-0.7) was introduced. The algorithm for the control value u(k) is as follows. First the estimate of the output at moment k is calculated from (1) with k replaced by k-1.
Then it can be shown that the control becomes P8aUL(k) 2 p 3 2 3 (k) - " K° * 3«* * Ps"^ * Peqq« * P?.^ * Pi2.)
U (4) 2 3 (Ps« + Pg«« + Pio«« + Pna" ) with a(k) and q(k) designating estimates taken from (1). It is seen that if there are no nonlinearities in the model the control reduces to a regular linear controller u = U .
L Simulations were run to test the controller performance especially in the unstable range of angle of attack. The system is successfully stabilized and the transients are very smooth and without significant overshoots for the nonlinear control as demonstrated by Figure la. By different choice of the reference model it is possible to obtain much faster, but at the same time much more "nervous" transients. The elevator control is also relatively smooth and within the range corresponding to the terminal equilibria. As can be seen from Figure Ib, the similar linear control is unstable.
2.2 On-Line Adaptive MAC Algorithm .1 Model algorithmic control (MAC), described for example in [2], consists of solving the model equation for the value of control necessary to obtain required value of output. Usually this desired output trajectory is generated form the setpoint by means of a reference model. In case this model is linear, the algorithm in essence becomes a linearizing one.
Here, the controlled output is assumed to be the angle of attack such that the reference equation becomes: with 2 3 2 3 2 3 T = [fi, a , a , q, qa, qa , qa , U, ua, Ua , Ua , l] (k) T 500 - P. *(k-l) * («(k-l) - a^k-t)) T q(k) = p 4>(k-1) + (q(k-1) - (k-1)) q qmod The controller is assumed to know the values of angle of attack and of pitch rate at the moment k-1. Then it estimates their current values ec(k) and q(k) taking into consideration previous prediction errors and based on them it calculates the control required to achieve a at the moment k+1. The value of control is found as: ref Pl2tt " r U(k) = + 2 3 Ps. P9«« * Pio«« + Pn«« (6) .
where o = (k 1) - (a(k-1) - a (k-1)) r are( + mod and a = a(k), q = q(k) as described above.
The results of the simulations are seen in Figures 2a,b. The-reference trajectory was chosen to be l/z -1.6z+0.65). The actual output of the plant is seen to follow the reference very closely, even though the region of operation was that of the most severe nonlinearities. The control action is also remarkably smooth.
The discrete time nonlinear state space model (1) describes the behavior of the complex nonlinear plant quite accurately in the entire region of operation. In practice, however, such a global model is rather difficult to fit, and consequently one should look for local approximations, depending on the current operating conditions.
In such a situation, on-line adaptive control seems to offer an ideal solution.
The algorithm discussed in the previous section can be made adaptive, or self- tuning, by incorporating on-line identification of the parameters. A recursive least squares (RLS) algorithm was implemented in the following form taken from [8]:
-- - -
p(k) e(k) Qflc-1) = _L_ ( _2) - Q(k-2)4>(k- Q(k T -1) -Kt>(k-1) Q(k-2)4>(k-1)J T = y(k) - p <t>(k-1) w where y may denote a or q and p may stand for p. or p , respectively. The forgetting q factor X was introduced to enable the algorithm to change the estimates of parameters with the change of operating conditions. To avoid the unlimited growth of covariance matrix Q at the steady state when the input is not persistently exciting the variable forgetting factor policy was implemented: 1 - e do) e(k) where e(k) is the current prediction error, e(k) is the average prediction error form last 10 samples and e is equal to 0.01. As an additional precaution the trace of the covariance matrix Q was monitored and Q was reset to diagonal matrix whenever the threshold value was exceeded. Starting values of parameters were taken to be as in (1).
Figure 2 displays the simulation results for a reference model specified as l/(z -1.8z+0.82). Remarkably exact following of the reference trajectory may be observed, although, surprisingly enough, the performance is slightly worse than in the nonadaptive case. Most probably this is due to the fact that prediction error now changes much more quickly because of the ongoing identification process. Thus, approximating the term (y(k+l)-y (k+l)) by (yCk-lJ-y^k-l)) may worsen the niod behavior of the system as two values of y,,,,,, no longer correspond to the same parameter vector. Since the on-line identification process assures (at least in principle) that the prediction error should asymptotically converge to zero it is possible that the correction terms in i(k), q(k), and in control equation (5) ought to be omitted.
The performance of the adaptive nonlinear MAC controller was compared to the linear one, which uses the same control strategy but with a strictly linear model being identified and used for the calculation of the control action. Clear difference between the performance of linear and nonlinear controller can be seen from Figure 3. particularly in control action at the setpoint o = 15°. The linear identifier has obvious difficulties with fitting the parameters of a linear model to the behavior of the plant which is highly nonlinear in this region. As a result, the control starts oscillating for a while. Also, it was seen that the nonlinear algorithm results in control plots that are more smooth, although they still contain one-pulse spikes. To eliminate these spikes weighting of the increments of control can be introduced into the algorithm with little performance deformation.
4. CONCLUSIONS The nonlinear control applications to high angle-of-attack aircraft, as reported here, is of a preliminary nature. However, the analysis does suggest that nonlinear adaptive control can be quite effective to stabilize large rapid maneuvers in angle of attack. Of the comparisons made, the on-linear, nonlinear-time-series and adaptation performed the best and was quite superior to a similar linear MAC.
5. ACKNOWLEDGEMENT The research reported here is supported by NASA Grant No. NAG-1-1081 with supplemental support from NSF Grant No. ECS8913773.
REFERENCES [1] A.D. Modyaev, A.D. Averina, "Analysis and synthesis of discrete control systems based on multidimensional z transforms," in Philosophy of Nonlinear Systems (B. Naumov, ed.), Mir Publishers/CRC Press, 1990.
[2] K. Harris, "Properties of nonlinear model algorithmic control," Proceedings of 24th Conference on Decision and Control, Ft Lauderdale, 1985, vol. 1, pp. 663- 665.
[3] H. Stalford, W.T. Baumann, F.E. Garrett, T.L. Herdman, "Accurate modeling of nonlinear systems using Volterra series submodels," Proceedings of the 1987 American Control Conference, Minneapolis, 1987, Vol. 2, pp. 886-891.
[4] A. Ostroff, "Application of variable-gain output feedback for high-alpha control," AIAA Guidance, Nav. & Control Conf., Boston, 1989.
[5] A. Ostroff,"Superagility application of a variable-gain output feedback control design methodology," NASA High Angle of Attack Tech. Conf., Hampton, VA, 1990.
[6] H. Wakamatsu, "Model reference nonlinear adaptive control system using nonlinear autoregressive moving average model derived from Volterra series and its application to control of respiration," Proceedings of IF AC 10th Triennial World Congress, Munich, 1987, Vol. 10, pp. 191-196.
[7] S.A. Al-Baiyat, "Nonlinear feedback synthesis: a Volterra approach," Ph.D.
dissertation, Department of Electrical Engineering, University of Notre Dame, 1986.
[8] G.C. Goodwin, K.S. Sin, Adaptive Filtering,. Prediction and Control. Prentice Hall, 1984.
Figure la: Step response with non- Figure Ib: Step response with linear linear controller vs. nominal response controller It II • 14 !• 10 It 14 I* Figure 2a: Nonlinear adaptive MAC Figure 2b: Nonlinear adaptive MAC (with reference trajectory)
i T
V j -10
- I I Z
K i
-IS -14 Figure 3a: Linear adaptive MAC (with Figure 3b: Linear adaptive MAC reference trajectory)
APPENDIX C
APPENDIX C Analysis of Nonlinear System Stability Using Robust Stability Analysis for Linear Systems ANALYSIS OF NONLINEAR SYSTEM STABILITY USING ROBUST STABILITY ANALYSIS APPROACH FOR LINEAR SYSTEMS Jerzy E. Kurek Instytut Automatyki Przemyslowej, Politechnika Warszawska ilnst. of Industrial Automatics, Warsaw University of Technology) ul. Chodkiewicza 8, 02-525 Warszawa, Poland Fax (22) 29-2962, Tel (22) 49-9616, Tlx 813307 pwpl During the research author was with Department of Electrical and Computer Engineering Oregon State University, Corvallis, Oregon 97331-3211, U.S.A.
Summary The stability analysis of an airplane using its nonlinear model is presented. The analysis is based on the robust stability analysis approach for linear systems. Then, based on analysis, a small static feedback gain is designed such that the robustness of the closed-loop nonlinear system stability is significantly improved.
Acknowledgment This research was in part supported by NASA Grant No. NAG-1-1081.
1. INTRODUCTION The stability is one of the most important issues in the cont system design. Recently there has been observed a great inter in the methodology of robust stability analysis and design robust control systems for linear dynamic systems [6] .
objective of this paper is to investigate the applicability this approach for nonlinear dynamic system such as an aircr flight in high angle of attack/sideslip flight. The unsta control system can result in the plane crash.
There is considered stability of nonlinear, simplified howev model of the airplane. The organization of the paper is follows. In section 2, the model of the plane is present Stability of the aircraft is considered in section 3. Final concluding remarks are given.
2. THE AIRCRAFT MODEL.
Model of an airplane is highly nonlinear, [4,5]. There usually, however, used simplified models for control sys design, e.g. [1,3,9]. In this paper we consider very simple mo given in [8]: x=A(x)x+Bu+D a where x= is a state vector, a is the angle of attack degrees, q is the pitch ratio in degrees per second and u is elevator control in degrees, 9.168c(«) f-1.83361 _f-5. 4732961 z R n A= a U ~L-8.5950j' ~[ 2.865000J -5.73 and c (a) is a nonlinear function. This function can, however, approximated as follows: -0.072815870 for 0° s a * 14.74° 0.088470922-2.3774/a for 14.74° < a s 17.40° C (a) = z 0.033099050-1.4068/a for 17.40° < a s 18.87° -0.016633734-0.4743/a for 18.87° < a s 28.00° It is easy to find that -0.072818087 < c (a) s -0.048161261 for 14.74° < a £ 17.40 ° ° Z -0.047751524 < c (a) * -0.041453149 for 17.40° < a s 18.87 -0.041768869 < c (a) s -0.033573019 for 18.87° < a s 28.00° This model approximates model taken from measured wind tunnel values of the T-2C airplane [7] . It is known that numerical values of c (a) and b,, are uncertain.
Z & Our purpose is to consider stability of system (1) in the range of angle of attack 0°^a^28°, and to find a static feedback which can, eventually, improve the stability of the plane in this range.
3. STABILITY ANALYSIS.
Consider linear time- invariant system x=Ax (4) n where xeR is a state vector. Then, assuming that the system is asymptotically stable one can define the following notions, [2].
Definition 1 .
A connected set fl in the system parameters -space (parameters of T matrix A) is a robust time invariant stable (RTIS) set for system (4) iff AeQ... and every time -invariant system x=4x ( 5 ) .
is asymptotically stable for sten.-.
Definition 2_.
A connected set £1. in the system parameters-space is a robust t: varying stable (RTVS) set for system (4) iff AeO^ and every tlr varying system (5) is asymptotically stable for Then, consider four linear models instead of (2), respectively: -0.072815870 for 0° s a s 14.74° -0.048161261 for 14.74° < a s 17.40° c (a) = -0.041453149 for 17.40° < a s 18.87° z -0.016633734 for 18.87° < a s 28.00° It is easy to find that all models are asymptotically stable, are, however, interested in the set of (k-,k~) such that all ' J_ ^ linear closed loop systems will be stable with the follow: feedback u=Kx, K=[k- k] An appropriate region n, can be easily calculated based algorithm 2 proposed in [2]. This is, however, only the sec order system and one can simply obtain analytical formulas for RTIS region in this case. The characteristic polynomial for 2nd order system has the following form s +as+b=0 It is known that all roots of this polynomial are in the left h plane, i.e. a system is asymptotically stable stable, iff a>0 b>0. Based on this, the RTIS region a., for 'stable' feedback ga was calculated. The region, is presented on fig. 1, a dashed ] represents RTIS region for model PI, 0°sosl4.74°, a dotted moc P2 for 14.74°«xil7 .4°, a dash-dotted model P3 for 17 .40°<asl8.8 and a continuous line model P4 for 18.87°<as28°. It -is easy to that the system without feedback, i.e. k-.=k =0, sign + on plane (k-,k~), is very close to the stability region boundary, can improve stability assuming appropriate K from Q (k^k-).
Next, RTVS sets Qy were calculated for these models, according to the algorithm given in [2], for uncertain parameters a . . . , and a--, in A. They are presented on fig. 2. All four models are inside the RTVS region calculated for the model P4 . Moreover, since all time varying (nonlinear) a =9 .168c (a) is smaller than nominal values 1;L z used in linear models it means that the whole nonlinear system (1) is asymptotically stable for 0°so^28°. However, there is a very small upper bound for a-., in this model, namely +Aa < 0.0227 1:L This can cause that with small system uncertainty the system can be unstable. The vertexes of RTVS quadrilateral O^ on the plane (Aa ,Aa- ) are as follows: 11 1 .1. X ^ J_ V = { (-176.5,0), (0.0226,0), (0, -0.2274) (0,0.2944) } yo In order to improve system stability feedback gain matrix K was chosen from Q . Intuitively, it seems that a good gain is "a small T one - a high gain can result in a lack of system controllability because of saturation of the control input, and such that the stability margin with respect to K will be rather large.
Thus, the good choice seems to be K-=[0 0.2]. For this gain one obtains significant improvement of RTVS set. This set is shown on fig. 3. In this case also all models are inside the n . . . - calculated for the model P4 . However, an upper bound for a,, is more than 8 times greater: < 1835 °- Also range for uncertain parameter a--, is almost 6 times larger.
Indeed, the vertexes of RTVS quadrilateral in this case on the plane (Aa.,,,Aa ) are as follows V = { (-25049,0), (0.1835,0), (0,-3.547), (0,3.211) } vl Then, it was considered feedback gain K =[0.2 0] . This ga however, seems to be worse situated in the RTIS set Q_ than considering the stability region with respect to K. Neverthele also in this case one obtains improvement of robust stability the closed loop system. The appropriate RTVS set £1, is shown fig. 4. In this case an upper bound for a^.. is as follows < 0 .0613 Similarly, range for perturbation in a--, is larger than for K are as The vertexes of RTVS set Q follows V = { (-65.15,0), (0.0613,0), (0, -0.5770), (0,1.3641) } V2 It should be noted that all RTVS sets were calculated un< assumption Q=I in algorithm 3 [2] .
From the above analysis follows that relatively small sta linear feedback gain K=[0 0.2] significantly improves stabil.
of the system. It should be emphsized that every nonlinear/tij varying system (1) with a^ and a from the obtained RTVS set will be asymptotically stable. This way we have designed a robu stable nonlinear closed-loop system.
4. CONCLUDING REMARKS.
A robust-stable nonlinear controlsystem has been designed. It shown that small linear static feedback gain can significa
j
improve stability of the airplane. The feedback gain seems to ( so small that it should not constrain control signal during pi maneuvering. This should also results in better a controllabil of the plane.
/ The stability analysis and feedback gain synthesis were done us methods designed for linear systems [2]. This, approach can also used for more complicated nonlinear systems. For instar assuming as a base model for the airplane, the linear 9th order model given in [1,9]. This model is unstable, but, as it was shown in [2], one can deal also with unstable models using the same approach.
Presented results also show the power of the approach proposed in [2] .
REFERENCES.
[ 1] S.Grag, D.L.Mattern and R.E.Bullard, "Integrated flight/ propulsion control system design on a centralized approach", J. Guidance Control and Dynamics, vol. 14, 1991.
[ 2] J.E.Kurek, "Robust stability region for linear systems - part I: continuous-time systems", IEEE Trans. Automat. Control, submitted.
[ 3] R.R.Mohler et. al., Nonlinear stability and control study of highly maneuverable high performance aircraft, OSU-ECE report NASA 91-01, Oregon State University, 1991.
[ 4] R.R.Mohler, Nonlinear systems v. 1: Dynamics and control, Prentice-Hall, Englewood Cliffs, 1991.
[ 5] A.J.Ostroff, "Application of variable-gain output feedback for high-alpha control", AIAA Guidance Navigation and Control Conf., Boston, paper No. 89-3576, 1989.
[ 6] D.D.Siljak, "Parameter space methods for robust control design: a guided tour", IEEE Trans. Automat. Control, vol.
AC-34, pp. 674-688, 1989.
[ 7] H.Stalford, "Application of the estimation-before-modeling (EBM) system identification method to the high angle of attack/sideslip flight of the T-2C jet trainer aircraft", Naval Air development Center Report NADC-76097-30, vol. Ill, 1979.
[ 8] H.Stalford, W.T.Baumann, F.E.Garret, T.L.Herdman, "Accurate modeling of nonlinear systems using Volterra series submodels", Proc. 1987 American Control Conference, Minneapolis, vol. 2, pp. 886-891, 1987.
[9] T.Troudet, S.Garg, W.C.Merrill, "Neural network application to aircraft control system design", Proc. AIAA Guidance, Navigation and Control Conference, New Orleans, Loiusiana, vol. 3, pp. 993-1009, 1991.
Fig. 1. The RTIS region on plane (kl. k2)- for PI. P2. P3. P4 1.2 - 0.8 0.6 CM 0.4 0.2 -0.2 " -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 kl Fig. 2. The HTVS region for PI. P2. P3, P4 with K-[0. 0] I CM -4 rt) T'J -6 ; I.'/' Hi I- •10 "- -2 -1 -5 -4 -3 all Fig. 3. The RTVS region for PI. P2. P3. P4 with K-[0.0.2] 0 - -2 CM -4 ID -6 -a -10 -2 -1 -5 -4 -3 all Fig. 4. The RTVS region for PI. P2. P3. P4 with K-[0.2. 0] O r -2r cu -4 - CO -6 -a - -10 -5 -3 -2 -1 all