Clearance of Flight Control Laws for Time-Varying Parameters

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(1)Clearance of Fligth Control Laws for Time-Varying Parameters. Jorgen Hansson, Anders Helmersson, Ragnar Wallin, Karin Stahl-Gunnarsson, Fredrik Karlsson and Anders Hansson Division of Automatic Control Department of Electrical Engineering Linkopings universitet, SE-581 83 Linkoping, Sweden WWW: http://www.control.isy.liu.se E-mail: hansson@isy.liu.se 24th November 2003. OMATIC CONTROL AU T. CO MMU. MS NICATION SYSTE. LINKÖPING. Report no.: LiTH-ISY-R-2551 Submitted to 16th IFAC Symposium on Automatic Control in Aerospace Technical reports from the Control & Communication group in Linkoping are available at http://www.control.isy.liu.se/publications..

(2) Abstract In this article exponential stability of the closed loop for the Saab AB VEGAS model controlled by a gain-scheduled linear fractional transformation controller is investigated for time-varying Mach-number. The analysis is based on parameter-dependent Lyapunov-functions which are obtained by investigating feasibility of linear matrix inequalities.. Keywords: Stability Analysis, Lyapunov Stability, Linear ParameterVarying Systems, Linear Fractional Transformation, Gain-Scheduling..

(3) Clearance of Flight Control Laws for Time-Varying Parameters Jörgen Hansson† , Anders Helmersson† , Ragnar Wallin† , Karin St˚ahl-Gunnarsson‡, Fredrik Karlsson‡ and Anders Hansson† † Division. of Automatic Control Department of Electrical Engineering Linköping University SE-581 83 Linköping, Sweden E-mail: hansson@isy.liu.se, Phone: +46 13 281681, Fax: +46 13 282622 ‡ Saab. AB SE-581 88 Linköping, Sweden Abstract In this article exponential stability of the closed loop for the Saab AB VEGAS model controlled by a gain-scheduled linear fractional transformation controller is investigated for time-varying Mach-number. The analysis is based on parameter-dependent Lyapunov-functions which are obtained by investigating feasibility of linear matrix inequalities. Key Words: Stability Analysis, Lyapunov Stability, Linear Parameter-Varying Systems, Linear Fractional Transformation, Gain-Scheduling.. 1 Introduction Clearance of flight control laws is a current research topic as described in e.g. [3]. Traditionally this is a very time-consuming process, and there is much to gain from improved analysis tools which do not require tedious simulations on desktop computers or simulators involving test pilots. Flight control systems are often designed as gain-scheduled controllers, [17]. Scheduling variables are typically velocity, altitude and angle of attack. Traditionally, the design of gain-scheduled controllers is carried out by first linearizing the system to be controlled at a discrete number of operational points parameterized by the scheduling variables. Then linear controllers are designed for each of these operational points, and the overall controller is obtained by interpolation. Even if each linear controller is stable for its linearized system model, there is no guarantee that the overall control scheme is stable when the parameters that the scheduling is based on starts to vary, [15]. However, if bounds on the rate of variation of the parameter is imposed it is possible to show that stability is recovered, [10, 13, 9, 14, 12]. Common to these approaches is that they all assume that the linearized system is exponentially stable, uniformly in the parameter. Under a rate-constraint on the parameter they then typically show that the nonlinear system is uniformly ultimately bounded locally. Recently much attention has been given to analyzing the Linear Parameter Varying (LPV) system, i.e. the linearized system for time-varying parameters, e.g. [4, 1, 7]. It has been shown that computationally attractive schemes can be derived to construct parameter-dependent Lyapunov-functions which prove exponential stability for the LPV system. In [5] it is shown how these results can be used to prove local stability for the underlying nonlinear system, and moreover larger stability regions can be obtained by this new approach as compared to previous approaches. It is the scope of this paper to investigate stability of the LPV model of the short period mode dynamics of the Saab VEGAS model, [2] controlled by the Linear Fractional Transformation (LFT) controller described in [16]..

(4) The remaining part of the paper is organized as follows. In Section 2 the LFT model of the closed loop system is described, in Section 3 stability for the LFT model in terms of parameter-dependent Lyapunovfunctions is analyzed, in Section 4 stability results for the closed loop of VEGAS are presented, and finally in Section 5 some concluding remarks and suggestions for future research are given.. 2 Model In this section the LPV model for the short period mode dynamics will be presented. It has been derived using so called “frozen” linearization, see [8, Chapter 5.7] and [6] for details. Linearization around a nominal solution is much more involved since it would require to find the nominal solution. However, if the parameters vary slowly, the model obtained from frozen linearization will not be too conservative for investigation of stability of the underlying nonlinear system, [5]. Frozen linearization is the standard approach when modeling flight control systems controlled by gain-scheduled controllers. The linearized equations for the short period mode dynamics are given by . . ∆α˙ ∆q˙. =. Zα Mα. Zq Mq. . . ∆α ∆q. +. Zδe Mδe. Zδc Mδc. . ∆δe ∆δc. . (1). Here all ∆-variables are deviations from the linearization points, where α is angle of attack, q is pitch rate, δc is the deflection of the canards, and δe is the deflection of the evelons. The elements of the system matrices can for constant altitude and angle of attack be approximated as linear functions of the Mach number M in the interval 0.2 to 0.5: Zα = 0:1060 2:7155M; Mα = 0:3063 + 11:2584M; =. 0:0027. 1:0344M; 65:0593M;. = 11:2501. Zδc Mδc. −0.6. =. 2:1775M. 0:0059 0:0574M 2:4748 + 20:5557M. 2 Val. data Approx. 1.5. −1. Zq. Zα. =. Val. data Approx.. −0.8. 1. −1.2 0.5. −1.4 −1.6 0.2. 0.3. M. 0.4. 0 0.2. 0.5. 0.3. M. 0.4. 0.5. −0.4 10. Val. data Approx.. 8. Val. data Approx. −0.6. 6. Mq. Mα. Zδe Mδe. Zq = 1 Mq = 0:0389. 4. −0.8. 2 −1. 0 −2 0.2. 0.3. M. 0.4. 0.5. −1.2 0.2. 0.3. M. 0.4. 0.5. Figure 1: Comparison of linear fit and actual data for the A-matrix..

(5) −0.1. 0 Val. data Approx.. Zδe. −0.3. −0.02. −0.4. −0.03. −0.5. −0.04. −0.6 0.2. 0.3. M. 0.4. −0.05 0.2. 0.5. 5. 0.3. M. 0.4. M. 0.4. 0.5. 8 Val. data Approx.. 0. 7. −5. 6. Mδc. Mδe. Val. data Approx.. −0.01. Zδc. −0.2. −10. 5. −15. 4. −20. 3. −25. Val. data Approx.. 2. −30 0.2. 0.3. M. 0.4. 1 0.2. 0.5. 0.3. 0.5. Figure 2: Comparison of linear fit and actual data for the B-matrix. Since the errors in this model are very small, as is seen in figures 1–2, we will consider them to be negligible in the analysis that follows. However, the analysis can easily be extended to consider the true data by incorporating bounds on the difference between the linear fit and the true data. We can write (1) on the form x˙ = A(∆)x + B(∆)u. (2). where . A(∆) = . x=. Zα Mα. . . . Zq ; Mq. B (∆ ) = . ∆α ; ∆q. u=. Zδe Mδe ∆δe ∆δc. . Zδc Mδc. . and where ∆ = M 0:35 is the deviation from nominal Mach 0.35. Do not confuse this usage of ∆ with the previous usage. Notice that the linearized equations defines an LPV system where the parameter is ∆, i.e. the equations in (2) are linear for each fixed value of ∆. Moreover the system matrices depend linearly on ∆: . . A(∆) B(∆). =. . A0. B0. . +. . A1. . B1 ∆. It is straightforward to see that we can describe the LPV system as an LFT system . x˙ z. =. A0 A1. 2 3. B0 B1. x I 4 5 u 0 w. where w = ∆z. This system is controlled with another LFT system, see [16, 6] for details: 2 3. 2. Aξ ξ˙ 4u5 = 4Cu Cζ ζ. Bx Dux Dζx. 32 3. Bω ξ Duω 5 4 x 5 Dζω ω. It turns out that the interconnections of the controller with the LFT system for the model is also an LFT system 2 3 2 32 3 x˙ A0 + B0 Dux B0Cu I B0 Duw x 6ξ˙ 7 6 7 6 B A 0 B ξ x ω 76 7 ξ 6 7=6 7 4 z 5 4A1 + B1 Dux B1Cu 0 B1 Duω 5 4w5 Dζx Cξ 0 Dζω ω ζ where. . w ω. =∆. z ζ.

(6) Hence the overall closed loop system can be described as . x˙¯ z¯. =. A¯ C¯. B¯ D¯. . x¯ w¯. where w¯ = ∆¯z. Notice that the closed loop system is an LPV system given by x˙¯ = A¯ (∆)x¯ where A¯ (∆) = A¯ + B¯ (I. ∆D¯ ) 1 ∆C¯ is no longer necessarily linear but linear fractional in ∆.. 3 Stability Analysis Stability of LPV systems can be investigated using e.g. Lyapunov functions. It is well-known that exponential stability of an LPV system, e.g.[8] is equivalent to the existence of a symmetric positive definite matrix P(t ) which satisfies the Linear Matrix Inequality (LMI) P˙ (t ) + P(t )A(∆(t )) + AT (∆(t ))P(t ) < 0. (3). where we have now dropped the bar-notation from the previous section for the closed loop system. We will be interested in stability for ∆(t ) 2 D = f∆ : j∆j γ; j∆˙ j ρg, where γ and ρ are some fixed numbers. In general it is difficult to find P(t ) and often one look for P independent of t. However, this can be too conservative, i.e. there might be P(t ) depending on t which satisfies the LMI but no constant P. It has been suggested by several authors to consider P depending on ∆(t ), e.g. [7] and the references therein, and here we will consider the special case when P depends on ∆ as . P(∆) =. I N (∆). T. . . I P0 ; N (∆ ). N (∆) = (I. ∆D) 1∆C. ˙ and let where P0 is a real symmetric matrix. Let ∇ = ∆I ∆I ∆I, . A=. A 0. 2. . B ; 0. . B= 3. 0 I. 2. C D 6CA CB7 7 6 6C D7 7; 6 C =6 I 7 7 6 0 4 0 05 0 0. . 0 0. 0 6D 6 60 D =6 60 6 4I 0. 3. 0 07 7 07 7 07 7 I5 I. Assume that A(0) is Hurwitz. Then a necessary and sufficient condition for P(∆) to satisfy (3) and to be positive definite is that there exist real symmetric matrices P0 and Θ which satisfy . and. P0 A + A T P0 B T P0. P0 B 0. . T. I ∇. +. . C. . Θ. I ∇. D. T. . Θ C. D. . <. 0. 0. for all ∇ such that ∆ 2 D, [7]. Notice that we do not have to require that P0 is positive definite. The above condition is not easy to verify since there are infinitely many constraints related to ∇. However, if we add the assumption that Θ2 < 0, where Θ2 is the 2,2-block of Θ it turns out that it is sufficient to consider only the vertices of D, [7]..

(7) 4 Application to VEGAS In this section we will analyze the stability of the LFT model of the closed loop system for different values ˙ Computations have been made using YALMIP, [11] of γ and ρ, i.e. limits on the absolute values of ∆ and ∆. and the semidefinite programming solver SDPT3, [18]. 0.2. 0.18. 0.16. 0.14. γ. 0.12. 0.1. 0.08. 0.06. 0.04. 0.02. 0. 0. 10. 20. 30. ρ. 40. 50. 60. 70. Figure 3: Trade-off curve showing the largest possible values of γ and ρ such that exponential stability can be proven using the analysis technique of the previous section. Above the vertical line the LPV model is not valid. 0.2. 0.15. 0.1. ∆. 0.05. 0. −0.05. −0.1. −0.15. −0.2 −80. −60. −40. −20. ∆˙. 0. 20. 40. 60. 80. Figure 4: The closed loop remains exponentially stable for values of ∆ and ∆˙ inside any of the boxes. Figure 3 shows the largest possible values of γ and ρ such that exponential stability can be proven using the analysis technique presented in the previous section. Above the vertical line the LPV model is not valid, i.e. M is not in the interval 0.2 to 0.5. In Figure 4 are shown boxes such that if the values ∆(t ) and ∆˙ (t ) for each t is inside a fixed box, then the closed loop system is exponentially stable. Notice that it is not guaranteed that the closed loop system is exponentially stable if the values are inside the union of the boxes.. 5 Conclusions In this article exponential stability of the closed loop for the linearized VEGAS model controlled by a gainscheduled LFT controller has been proven for time-varying Mach-number. In the range of Mach 0.2 to 0.5 stability is guaranteed for any reasonable acceleration. However, the stability results do only carry over to the nonlinear system for much smaller values of the acceleration, since the linearized model obtained by frozen linearization is only a good approximation of the nonlinear system for small values of the acceleration. The reason for considering a relatively narrow range of Mach number is that it is then possible to fit linear models for how the system matrices depend on Mach number. For larger ranges it may be necessary to fit.

(8) rational models which is much more involved. Moreover the closed loop model will be of higher dimension making the computations for investigating feasibility of the LMI much more time-consuming and possibly difficult from a numerical point of view with respect to conditioning of the problem. Special purpose codes for the LMIs considered in this article that potentially may speed up computations are presented in e.g. [20, 19]. LFT modeling in general is still a challenging research area especially when it comes to selection of orders for rational functions of several variables. Future work will be devoted to studying this and also at modeling larger ranges of the flight envelop. Stability of the closed loop nonlinear system in the spirit of [5] is also of interest to investigate, especially applications to large scale examples such as aircrafts.. Acknowledgments The authors gratefully acknowledge financial support from the Swedish Research Council under contract No. 271-2000-770.. References [1] P. Apkarian and H. D. Tuan. Parameterized LMIs in control theory. SIAM J. Control and Optimization, 38(4):1241–1264, 2000. [2] H. Backström. Report on the usage of the generic aerodata model. Technical report, Saab AB, 1997. 2nd edition. [3] C. Fieldings, A. Varga, S. Bennani, and M. Seiler. Advanced Techniques for Clearance of Flight Control Laws, volume 283 of Lecture Notes in Control and Information Sciences. Springer Verlag, New York, 2002. [4] P. Gahinet, P. Apkarian, and M. Chilali. Affine paremeter-dependent Lyapunov functions and real parameteric uncertainty. IEEE Transactions on Automatic Control, 41(3):437–442, 1996. [5] A. Hansson, A. Helmersson, and T. Glad. Stability analysis of nonlinear systems using frozen stationary linearization. Technical Report LiTH-ISY-R-2512, Department of Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden, Apr 2003. [6] J. Hansson. Using linear fractional transformations for clearance of flight control laws. Master’s thesis, Department of Electrical Engineering, Linköping University, 2003. [7] T. Iwasaki and G. Shibata. LPV systemanalysis using quadratic separator for uncertain implicit systems. IEEE Transactions on Automatic Control, AC-46:1195–1208, 2001. [8] H. K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, 1996. [9] H. K. Khalil and P. V. Kokotovic. On stability properties of nonlinear systems with slowly varying inputs. IEEE Transactions on Automatic Control, AC-36:229, 1991. [10] D. A. Lawrence and W. J. Rugh. On a stability theorem for nonlinear systems with slowly varying inputs. IEEE Transactions on Automatic Control, AC-35:860–864, 1990. [11] Johan Löfberg. Yalmip: A matlab interface to sp, maxdet and socp. Technical Report LiTH-ISY-R2328, Department of Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden, Jan 2001. [12] W. J. Rugh. Analytical framework for gain scheduling. IEEE Control Systems Magazine, 11(1):79–89, 1991. [13] J. S. Shamma and M. Athans. Anaysis of gain scheduled control for nonlinear plants. IEEE Transactions on Automatic Control, AC-35(8):898–907, 1990. [14] J. S. Shamma and M. Athans. Guaranteed properties of gain scheduled control for linear parametervarying plants. Automatica, 27(3):559–564, 1991..

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