The motion of the control surfaces is constrained in various ways

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(1)Stabilizability under actuator constraints: an application to aircraft control. S. T. Glad Department of Electrical Engineering Linkoping University S-581 83 Linkoping, Sweden email: torkel@isy.liu.se Presented at MTNS96. 1 Introduction In modern aircraft design it is often found to be advantageous to make the aircraft unstable. It is then necessary to use the control surfaces to stabilize the system. The motion of the control surfaces is constrained in various ways. There are maximum deections that can be used and also maximum rates for moving the surfaces. These constraints mean that it is not possible to stabilize the system for arbitrary initial conditions of the state variables. In the aircraft design it is then necessary to estimate the region of the state space in which it is possible to stabilize the system. For the control of the longitudinal motion of aircraft there is usually one unstable eigenvalue. This is the situation that is considered in this paper. A simplied version of the problem was considered in 4]. The problem of controlling a discrete time linear system under state and control constraints is considered in 5]. The combination of instability and saturating controls is analyzed in 7] using a describing function approach. A general theory for the problem of remaining in a set under the e ect of control constraints is given in 2], 3]. Other relevant references are 6] 1],8]. In this paper the problem is considered in continuous time to stay closer to the physical equations. Also an explicit characterization of the stabilizable region is sought. The mathematical formulation of the problem is considered in the next section.. 2 Problem formulation The system is assumed to be described by linear dynamics x_ = Ax + Bz Research supported by FFV and TFR. 1. (1).

(2) In the aircraft application x will typically consist of angle of attack, rate of rotation, angle of attitude and possibly states describing the control surface actuators. The z variables describe the control surfaces or possibly commands to the servo mechanisms driving the surfaces. The crucial point is that they are limited in both rate of change and in amplitude.. zi zi zi ui z_i ui i = 1 : : : m. (2). The limits are assumed to satisfy. zi < 0 < zi ui < 0 < ui i = 1 : : : m. (3). It is now natural to regard the derivative of z as an input.. u = z_. (4). The system description is thus. x_ = Ax + Bz z_ = u z z z u u u Denition 1 The set of points (x z) for which there exists a control. (5) (6) (7) (8). satisfying (5){(8), and taking the state to the origin asymptotically, is called the stabilizable region of the system (5){(8). 2.1 Variable changes and scaling Now assume that the system matrix has a real unstable eigenvalue > 0 with corresponding left eigenvector v:. vT A = vT (9) Making a variable change = Tx, with T being a nonsingular matrix with v as its rst row, then gives a system description of the form.. T _ = A~ A~0 + vB~ B z 21 22 21 z_ = u. (10) (11). where A~21 , A~22 and B~21 are matrices of appropriate dimensions. With the scaling T T 1 = vT x = t i = v B i zi i = v B2i ui i = 1 : : : m (12). 2.

(3) where Bi denotes the i:th column of B , the dynamics inuencing 1 is described by. 1 = 1 + 1 + + m =. 0. 0. (13) (14). where 0 denotes di erentiation with respect to the scaled time . The bounds on the variables are. . where. ( vT Bi zi. i = vT B i zi . ( vT Bi zi. if vT Bi > 0 if vT Bi < 0. T i = vT B i zi if vT Bi > 0 if v Bi < 0 . (15) (16). ( vT Bi ui. ( vT Bi ui if vT Bi > 0 2 if vT Bi > 0 2. i = vT Bi ui = . (17) i T v B2i ui if v T Bi < 0 if vT Bi < 0 2 (We consider only problems where vT Bi 6= 0, i = 1 : : : m. The general case is. reduced to this by considering fewer control variables.) We note the following. Proposition 1 Assume that A has one real eigenvalue strictly greater than zero, and that the remaining eigenvalues have real parts strictly less than zero. Then the stabilizable region is characterized only in terms of the variables 1 and 1 ,.., m . Proof. To bring the state of (10) to the origin it is necessary to bring the state of (13), (14) to the origin. Conversely, if there is a control on t0 t1 ] which brings the state of (13), (14) to the origin, then that control, continued with. = 0 for t1 will bring the state of (10) to the origin asymptotically, since A~22 has all its eigenvalues strictly in the left half plane. The rst estimate of the stabilizable set is given by Proposition 2 The stabilizable region is contained in the set. ;. X. i < 1 < ;. X. i. (18). Proof. Follows immediately from (13), (15).. We can also construct the following set on the border of the -region. Lemma 1 The set = 1 < 1 1 3. (19).

(4) with. m. X X X 1 = ; i 1 = ; i ; e 0 i (ei ; 1) ;. (20). i=1. i = i =(; i ) > 0 0 = max i i. belongs to the stabilizable region. A similar set can be dened analogously for = .. Proof. For an initial value = and 1 satisfying 1 < 1 < 1 applying the control = 0 brings the state to the point 1 = 1 after nite time. Applying the control. (. i ( ) = 0 0 < 0 ; i. i 0 ; i 0 from that initial condition then results in. 1 (0 ) = 0 (0 ) = 0 Dene the functions.

(5) ( ) =. m X i=1. i +. m X i 1 ; e(i i)=i. (21). ;. i=1. ( ) = 1 +

(6) ( ). (22). where and are constants. The function has the following useful property Proposition 3 Along solutions of (5){(8) one has. m ( )= d ( ( ) ( ) ) = ( ( ) ( ) ) + X (. ;. ) 1;e i i i i i d i=1 (23) ;. Proof. Straightforward calculation. We can now show. 4.

(7) Lemma 2 If the system (5),(6) at points satisfying (7) is controlled with an. satisfying (8), then. d (24) d ( ( ) ( ) ) ( ( ) ( ) ) d (25) d ( ( ) ( ) ) ( ( ) ( ) ) In (24) equality is obtained precisely when, for each i, either i = i or i = i . In (25) equality is obtained precisely when, for each i, either i = i or i = i .. Proof. From (23) we get. m ( )= d ( ( ) ( ) ) = ( ( ) ( ) ) + X (. ;. ) 1;e i i i i i d i=1 ;. Since, from the denitions of and . i ; i 0 i ; i 0 it follows that the terms in the sum are all less than or equal to zero. Equality is obtained when each term is zero, which occurs when either i = i or 1 = e(i i )=i . A similar calculation can be made for ( ( ) ( ) ). ;. 3 The main result The basic stabilizability result can now be stated Theorem 1 For a system where A has one real eigenvalue > 0 and where the remaining eigenvalues have negative real parts the stabilizable region is given by the set. ;

(8) (z ) < 1 < ;

(9) (z ). (26). Proof. From Lemma 2 it follows that (( ) ( ) ). 0 for all t > 0 if ( (0) (0) ) 0. Since (0 0 ) > 0 it follows that no point ( ) with ( z^) 0 can belong to the stabilizable set. A similar argument holds for ( ) 0. This shows that the stabilizable region must be contained in (26). To show that the stabilizable region is all of (26) consider a point satisfying those conditions together with. 1 + 1 + + m < 0 Using a control with = 0 will keep constant while 1 decreases until. ( ) = > 0 5.

(10) is satised. Using, for each i, i = i until i = i and then i = 0, will eventually bring the state to the point = while satisfying. d d ( ) = ( ) It follows that a point is reached where > 0. With small enough it will. belong to the set dened in (19), which from Lemma 1 is known to belong to the stabilizable set. A similar calculation can be made for the case. 1 + 1 + + m 0. 4 Graphical description of the stabilizable region Consider a system with two inputs and. ; = = 1 ; = = 1 The stabilizable set i shown in gure 1. The set described in Lemma 1 is part of the vertical line to the right at the back of the gure ( 1 = 1, 2 = 1). ratio 1:1. 2 1.5 1. vTx. 0.5 0 1 −0.5 0.5. −1 0. −1.5. −0.5. −2 −1. −0.5. 0. 0.5. 1. −1. z2. z1. Figure 1: Stabilizable set. 1 and 2 are on the horisontal exes, on the vertical axis. 6.

(11) 5 Conclusions The stabilizable set of a linear system with constraints on the amplitudes and rates of actuator variables can be computed explicitly for the case of one unstable eigenvalue. This covers the typical description of unstable aircraft dynamics in the longitudinal mode. Of course it would be interesting to extend the results to the case of several unstable eigenvalues. In the aircraft control case this would cover also oscillatory instabilities, for instance in the lateral mode, and combined instabilities in the longitudinal and lateral motion.. References 1] Georges Bitsoris and Marina Vassilaki. Constrained regulation of linear systems. Automatica, 31(2):223{227, 1995. 2] Arie Feuer and Michael Heymann. Admissible sets in linear feedback systems with bounded controls. Int. J. Control, 23(3):381{392, 1976. 3] Arie Feuer and Michael Heymann. !-invariance in control systems with bounded controls. Journal of Mathematical Analysis and Applications, 53:266{276, 1976. 4] S.T. Glad. Stabilizable regions for unstable systems with rate and amplitude bounds on the control. pages 489{492, Tahoe City, CA, USA, 1995. IFAC Symposium on Nonlinear Control Systems Design, NOLCOS'95. 5] Per-Olof Gutman and Michael Cwikel. An algorithm to nd maximal state constraint sets for discrete-time linear dynamical systems with bounded control and states. IEEE Transactions on Automatic Control, AC-32(3):251{ 254, March 1987. 6] Jin-Hoon Kim and Zeungnam Bien. Robust stability of uncertain linear systems with saturating actuators. IEEE Transactions on Automatic Control, 39(1):202{207, 1994. 7] M. M. Seron, G. C. Goodwin, and S. F. Graebe. Control system design issues for unstable linear systems with saturated inputs. IEE Proc. Control Theory Appl., 142(4):335{344, 1995. 8] G. F. Wredenhagen and P. R. Belanger. Piecewise-linear lq control for systems with input constraints. Automatica, 30(3):403{416, 1994.. 7.

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