Automatica 37 (2001) 1971}1978
Brief Paper Comparison of nonlinear control design techniques on a model of the Caltech ducted fanଝ Jie Yu, Ali Jadbabaie*, James Primbs, Yun Huang Control and Dynamical Systems, MS 107-81, California Institute of Technology, Pasadena, CA 91125, USA Received 8 February 2000; revised 3 November 2000; received in "nal form 17 May 2001
Abstract
In this paper we compare di!erent nonlinear control design methods by applying them to the planar model of a ducted fan engine. The methods used range from Jacobian linearization of the nonlinear plant and designing an LQR controller, to using model predictive control and linear parameter varying methods. The controller design can be divided into two steps. The "rst step requires the derivation of a control Lyapunov function (CLF), while the second involves using an existing CLF to generate a controller. The main premise of this paper is that by combining the best of these two phases, it is possible to "nd controllers that achieve superior performance when compared to those that apply each phase independently. All of the results are compared to the optimal solution which is approximated by solving a trajectory optimization problem with a su$ciently large time horizon. 2001 Published by Elsevier Science Ltd.
Keywords: Nonlinear control design; Optimal control; LPV methods; Model predictive control; Control Lyapunov functions
1. Introduction of this paper is to continue this comparison further, and add some other design methods to the existing ensemble. For more than three decades, nonlinear stabilization The methods are applied to a ducted fan, a research setup has been a major part of control engineering research. developed at Caltech as a testbed for nonlinear control Despite the di!erences among nonlinear control design design. In this paper, we report the simulation results techniques, they can be divided into two major groups. performed using various nonlinear methods on a simpli- The "rst group is comprised of those methods that gener- "ed model of the fan, and experimental results will be ate a control Lyapunov function (CLF), while the second presented in our future papers. group takes an existing CLF and uses it to generate In order to be able to focus only on the di$culties a stabilizing controller. associated with nonlinearity, issues such as structured Motivated by the notion of combining the advantages uncertainty and unmodeled dynamics are not considered of these two viewpoints into the design methodology, at this point, and a quadratic performance index is se- a workshop was organized at the American Control lected. The studied methods include: Jacobian lineariz- Conference in June 1997. Several nonlinear design tech- ation, Frozen Riccati equations (FRE), linear parameter niques were applied to simple toy examples generated by varying methods (LPV), control using global lineariz- the converse HJB method (Doyle, Primbs, Shapiro, ation, and "nally model predictive (receding horizon) & NevistecH , 1996), and they were compared to the opti- control (MPC), including hybrid approaches such as mal solution which was known in advance. The purpose model predictive control combined with the CLF ob- tained using LPV. This paper is organized as follows: In Section 2, we de- scribe the state space model for the ducted fan engine. Sec- ଝA conference version of this paper was presented in IFAC world tion 3 deals with the methods that obtain a control Congress, Beijing, China, 1999. This paper was recommended for publi- Lyapunov function. In Section 4, we describe methods cation in revised form by Associate editor Daizhan Cheng under the direction of Editor Hassan Khalil. that utilize the CLF to generate a stabilizing controller. A * Corresponding author. comparison of di!erent methods is performed in Section 5, E-mail address: [email protected] (A. Jadbabaie). and "nally our conclusions are presented in Section 6.
0005-1098/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S 0 0 0 5 -1 0 9 8 ( 0 1 ) 0 0 1 4 9 -2 1972 J. Yu et al. / Automatica 37 (2001) 1971}1978
2. Caltech ducted fan model These equations are referred to as the planar ducted fan equations. The following quadratic cost function was The Caltech ducted fan is a small #ight control experi- used for comparison of di!erent design techniques: ment whose dynamics are representative of either a Harier in hover mode or a thrust vectored aircraft such as F18- J" (x 2(t)Qx (t)#u2(t)Ru(t)) dt, HARV or X-31 in forward #ight (Murray, 1998). This system has been used for a number of studies and papers. where x "[xx yy Q]2, R"I, and Q is chosen to be In particular, a comparison of several linear and nonlinear a diagonal matrix with the following diagonal terms: controllers was performed in Kantner, Bodenheimer, Ben- " dotti, and Murray (1995), Bodenheimer, Bendotti, and Q diag[10 1 10 1 1 1]. (3) Kantner (1996) and van Nieuwstadt and Murray (1996). In Associated with this optimal control problem is the cor- this section we describe the simple planar model of the fan responding value function,de"ned as shown in Fig. 1 which ignores the stand dynamics. This model is useful for initial controller design and serves as
3.4. Linear parameter varying (LPV ) methods
In this technique, the following so-called quasi-LPV representation of a nonlinear input-a$ne system is used to design a state feedback controller x "A((x))x#B((x))u. (13) Assume the underlying parameter varies in the allow- able set FJ ": +3C(R>,RK):3P, ()) P G G ) " , G( ), i 1,2, m , (14) where PLRK is a compact set. If there exists a positive de"nite X() such that the following inequality is Fig. 2. The FRE controller. satis"ed:
*X ! K #A X #X A2 !B B2 X C2 G G( )* ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) G (0 (15) C X !I ( ) ( )
At each frozen state the Riccati equation is solved, and for all 3P where C()"Q ((x)), then the closed-loop then the resulting state fedback controller is feedback to system is stable with the state feedback the system, that is, the state feedback nonlinear control "! 2 \ law is obtained by solving the following: u(x) B ( (x))X ( (x))x. 0"A(x)2P(x)#P(x)A(x)!P(x)g(x)g2(x)P(x)#Q, Moreover, an upper bound on the optimal value function 0100 0 0 00 d g sin cos sin 0 ! 00 ! 0 ! m m m 0 001 0 0 00u x " x# (12) d g(cos!1) sin cos u 000! 0 m m m 0 000 0 1 00 r 0 000 0 0 0 j Results of this approach are shown in Fig. 2. J. Yu et al. / Automatica 37 (2001) 1971}1978 1975 variation of Sontag's original formula. < f#((< f )#q(x)(< gg2<2) ! V V V V 2<2 2 2 g V, < " Vg 0, "* * where 2 2 2 and determining by solving the HJB equation point- + f ()," 1, , 3 !1 2, 5 G wise with (x) # Simulation of the closed-loop system is shown in Fig. 3. 4.2. MPC CLF: This section deals with combining the CLF obtained using methods described in the previous section with 4. CLF-based control schemes a model predictive control (MPC) strategy. First, how- ever we review some results on MPC. So far, we have discussed several methods for generat- ing a CLF. Each of the above mentioned methods 4.2.1. Model predictive control (MPC) have their own technique for generating a controller. In model predictive control (cf. GarcmH a, Prett, However, once a CLF is obtained, there are number & Morari, 1989; Mayne & Michalska, 1990), the current of alternative methods that can be used to imple- control at state x and time t is obtained by determining ment a controller purely from the knowledge of the on-line the optimal control u( over the interval [t, t#¹] CLF. In what follows, some of the important ones are respecting the following objective: described. R>2 inf (q(x( ))#u2( )u( )) d #(x(t#¹)) 4.1. Sontag's formula S R s.t. x "f (x)#g(x)u. Perhaps the most important formula for producing a stabilizing controller based on the existence of a CLF The resulting control trajectory is implemented until was introduced in by Sontag (1989) and has come to be a new state update occurs, usually at pre-speci"ed samp- ¹ known as Sontag's formula. Below, we present a slight ling intervals of time . Repeating these calculations 1976 J. Yu et al. / Automatica 37 (2001) 1971}1978 yields a feedback control law. This type of implementa- tion is commonly referred to as receding or moving horizon. As is evident from this sort of control scheme, obtaining a reduced value of the performance index is of utmost importance. Model predictive control has traditionally been plagued by stability questions. For nonlinear systems, these have been addressed in various ways, including the use of end constraints (which require that the state be identically zero at the end of the horizon ¹), and by various other methods (Mayne & Michalska, 1990; Michalska & Mayne, 1993). Unfortunately, these do not always result in satisfactory solutions when considered in the framework of practical implementation. Often such Fig. 4. Optimal. additional constraints introduce a number of problems related to the feasibility of the on-line optimization. Hence, while receding horizon control exploits the ability to compute on-line, this also introduces many di$culties control trajectory is implemented. The second constraint ranging from stability to feasibility of the on-line optim- (21) obviates the need for a terminal weight, using the ization. Nevertheless, as computer power increases, re- information from the CLF instead. ceding horizon control and on-line computation become This formulation of MPC avoids the stability prob- increasingly attractive. lems of standard MPC by incorporating the stability properties of the CLF. This allows the horizon to be 4.2.2. MPC extensions of CLF formulas varied on-line and the optimization to be preempted Freeman and KokotovicH (1995), proposed to use before the optimizing solution is found, all without loss of a CLF in conjunction with the following optimization to stability. When no horizon is present, it reduces to the determine a control law. pointwise min-norm problem, which also serves as a feas- ible trajectory for the MPC problem when a horizon is Pointwise Min-Norm: present. In order to test this hybrid strategy on the planar model of the ducted fan, we used the CLF obtained by minimize u2u (17) the Quasi-LPV method described in Section 3.4. Simula- # )! subject to Table 1 increasing importance will be whether on-line techniques Values of the cost function J using di!erent methods for the ducted fan such as MPC#CLF can realistically be implemented in example a real time environment such as the Caltech ducted fan Method Cost experiment. This will be a topic of ongoing interest as this research continues. ¹ " ¹ " MPC w/o CLF ( F 0.3, 0.05) Unstable Jacobian linearization#LQR Unstable FRE 2801 Global linearization#LQR using LMIs 2423 6. Conclusions Quasi-LPV 1795 CLF from LPV#Sontag 1761 The purpose of this paper was to perform a compari- # ¹ " ¹ " CLF from LPV MPC ( F 0.1 s, 0.05) 1572 son between several existing nonlinear design methods # ¹ " ¹ " CLF from LPV MPC ( F 0.3 s, 0.05) 1451 on the planar model of a ducted fan. The design schemes &Optimal' 1362 were divided into two groups. The "rst dealt with di!er- ent methods of generating control Lyapunov functions (CLFs). Despite the fact that the CLF generating methods have their standard method of generating a con- initial condition, although with a rather poor cost. Simu- troller, it was shown via simulation that if the CLF lation results are supplied in Fig. 2. methodology is combined with an MPC framework, the Next, we "nd that the Global Linearization techniques hybrid method works better than any of the individual provide a guarantee of stability and reduce the cost even ones. Simulation results were performed for a speci"c further. When even more o!-line computation is thrown initial condition which incorporated the true nonlinear at the problem as in LPV techniques, the cost reduces nature of the problem. Further research in this direction even further. The result of the LPV simulation is given in is under progress, and our goal is to do a similar com- Fig. 3. Note that in LPV and FRE techniques, a design parison with more sophisticated models and eventually choice is involved in the selection of a state dependent with the Caltech ducted fan experiment. representation. Although no systematic procedure was used (nor does one exist), the results obtained here were the best of the state dependent representations that were References tested. The results from LPV on down were all qualitatively Artstein, Z. (1983). Stabilization with relaxed controls. Nonlinear similar to the optimal trajectory. Applying Sontag's for- Analysis, 7(11), 1163}1173. mula with the aid of the CLF from LPV results in a cost Bodenheimer, B., Bendotti, P., & Kantner, M. (1996). Linear param- and trajectory very similar to that obtained from the eter-varying control of a ducted fan engine. International Journal on Robust and Nonlinear Control, 6, 1023}1044. standard LPV implementation. When this approach was Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear extended to an on-line MPC implementation, a signi"- matrix inequalities in system and control theory Studies in applied cant drop in the cost was observed. As the horizon was mathematics, Vol. 15. Philadelphia, PA: SIAM. increased from ¹ "0.1 to 0.3, the cost steadily de- Cloutier, J. R., D'Souza, C. N., & Mracek, C. P. (1996). Nonlinear F H creased, providing the lowest cost observed in any of regulation and nonlinear control via the state-dependent Riccati equation technique. Proceedings of the xrst international the simulations. Due to the similarity in results, only the conference on nonlinear problems in aviation and aerospace, optimal trajectory is supplied in Fig. 4 for reference. Daytona Beach, FL. The simulation results indicate that the hybrid tech- Doyle, J. C., Primbs, J., Shapiro, B., & NevisticH , V. (1996). Non- niques outperform the `CLF-onlyaa techniques. While linear games: examples and counterexamples. Proceedings of the this is just a numerical observation, theoretical justi"ca- 35th IEEE conference on decision and control, Kobe, Japan (pp. 3915}3920). tion can also be given; see Primbs (1998) and Jadbabaie Freeman, R., & KokotovicH , P. (1995). Optimal nonlinear controllers for (2000) for more details. feedback linearizable systems. Proceedings of the American control To summarize, in general the following trends were conference, Seattle, Washington, DC (pp. 2722}2726). observed. While not uniformly true, the more detailed Freeman, R., & Primbs, J. (1996). Control Lyapunov functions: New and sophisticated techniques, which generally involve ideas from an old source. Proceedings of the 35th IEEE conference on decision and control, Kobe, Japan (pp. 3926}3931). extensive o!-line analysis, tended to outperform the GarcmHa, C. E., Prett, D. M., & Morari, M. (1989). Model predictive simpler, less theoretically sound techniques. Extensive control: theory and practice*a survey. Automatica, 25(3), 335}348. computation was also found to be extremely bene"cial, Jadbabaie, A. (2000). Receding horizon control of nonlinear systems: especially when employed in an on-line manner, but only a control Lyapunov function approach. Ph.D. thesis, California Insti- when used under the guidance of a solid theoretical tute of Technology, Pasadena, CA. Kantner, M., Bodenheimer, B., Bendotti, P., & Murray, R. M. (1995). framework. This accounts for the di!erence bet- An experimental comparison of controllers for a vectored thrust, ween the near optimal MPC#CLF controllers and ducted fan engine. Proceedings of the American control conference, the unstable standard MPC approach. A question of Seattle (pp. 1956}1961). 1978 J. Yu et al. / Automatica 37 (2001) 1971}1978 Khalil, H. K. (1996). Nonlinear systems (2nd ed.). Upper Saddle River, Wu, F., Hua Xin Yang, ., Packard, A., & Becker, G. (1996). Induced ¸ NJ: Prentice-Hall. -Norm control for lpv systems with bounded parameter vari- KristicH , M., Kanellakopoulos, I., & KokotovicH , P. (1995). Nonlinear and ation. International Journal of Nonlinear and Robust Control, 6, adaptive control design. New York: Wiley. 983}998. Lukes, D. (1969). Optimal regulation of nonlinear dynamical systems. SIAM Journal of Control, Vol. 7, pp. 75}100. Lur'e, A. I., & Postnikov, V. N. (1944). On the theory of stability Ali Jadbabaie was born in Tehran, Iran in of control systems. Applied Mathematics and Mechanics, 8(3), 1972. He received the B.S. degree (with 246}248. high honors) from Sharif University of Mayne, D. Q., & Michalska, H. (1990). Receding horizon control of Technology in 1995, the M.S. degree from University of New Mexico in 1997, both in nonlinear systems. IEEE Transactions on Automatic Control, 35(7), electrical engineering, and the Ph.D. de- 814}824. gree in control and dynamical systems Michalska, H., & Mayne, D. Q. (1993). Robust receding horizon control from California Institute of Technology of constrained nonlinear systems. IEEE Transactions on Automatic (Caltech) in 2000. Since October 2000, he Control, 38(11), 1623}1633. has been a post-doctoral scholar in control Murray, R. M. (1998). Modeling of th caltech ducted fan, class and dynamical systems at Caltech where notes for cds 111. Technical Report, California Institute of Techno- his research interests include optimiza- logy, Control and Dynamical Systems, 107-81, Pasadena, CA tion-based control of nonlinear systems with applications to unmanned 91125. aerial vehicles, optimization-based design of analog CMOS circuits, and robust control theory. van Nieuwstadt, M., & Murray, R. M. (1996). Real time trajectory generation for di!erentially #at systems. Proceedings of the IFAC world congress. James Primbs received B.S. degrees in Primbs, J. A. (1998). Nonlinear optimal control: a receding horizon electrical engineering and mathematics approach. Ph.D. Thesis, California Institute of Technology, at the University of California, Davis in Pasadena, CA. 1994, the M.S. degree in electrical engine- Primbs, J. A., NevisticH , V., & Doyle, J. C. (2000). A receding horizon ering at Stanford University in 1995 generalization of point-wise min-norm controllers. IEEE ¹ransac- and the Ph.D. degree in control and tionson Automatic Control , 45(8), 898}909. dynamical systems from the California In- stitute of Technology in 1999. He is cur- Sontag, E. D. (1989). A universal construction of Artstein's theorem rently a post-doctoral researcher in the on nonlinear stabilization. Systems and Control Letters, 13(12), control and dynamical systems depart- 117}123. ment at Caltech. His research interests in- Tsiotras, P., Corless, M., & Rotea, M. A. (1998). An l disturbance clude dynamic optimization, "nancial attenuation solution to the nonlinear benchmark problem. Interna- engineering, and the modeling of complex business, economic, and tional Journal on Robust and Nonlinear Control, 8, 311}330. "nancial systems.