Blending Methodology of Linear Parameter Varying Control Synthesis of F-16 Aircraft System

Transcription

1 NASA/CR111 ICASE Report No. 1 Blending Methodology of Linear Parameter Varying Control Synthesis of F1 Aircraft System JongYeob Shin ICASE, Hampton, Virginia Gary J. Balas University of Minnesota, Minneapolis, Minnesota Alpay M. Kaya University of California, Berkeley, California ICASE NASA Langley Research Center Hampton, Virginia Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Hampton, Virginia Prepared for Langley Research Center under Contract NAS19 October 1

2 BLENDING METHODOLOGY OF LINEAR PARAMETER VARYING CONTROL SYNTHESIS OF F1 AIRCRAFT SYSTEM JONGYEOB SHIN Λ, GARY J. BALAS y, AND ALPAY M. KAYA z Abstract. This paper presents the design of a linear parameter varying (LPV) controller for the F1 longitudinal axes over the entire flight envelope using a blending methodology which lets an LPV controller preserve performance level over each parameter subspace and reduces computational costs for synthesizing an LPV controller. Three blending LPV controller synthesis methodologies are applied to control F1 longitudinal axes. Using a function substitution method, a quasilpv model of the F1 longitudinal axes is constructed from the nonlinear equations of motion over the entire flight envelope, including nontrim regions, to facilitate synthesis of LPV controllers for the F1 aircraft. The nonlinear simulations of the blended LPV controller show that the desired performance and robustness objectives are achieved across all altitude variations. Key words. LPV control synthesis, F1 longitudinal axes Subject classification. Guidance and Control Nomenclature. m, I yy : Mass (slug), Inertial moments (slug ft ) V, q : Velocity (ft sec 1 ), Dynamic pressure (psi), q : Angle of attack (rad), Pitch rate (rad sec 1 ), fl : Pitch angle (rad), Flight path angle (rad) i e, T : Elevator deflection (rad), Thrust (lb) μc, S : Chord length (ft), Reference area (ft ) X ac, X cg : Aerodynamic center position (ft), Center of gravity position (ft) C X, C Z : X and Z force aerodynamic coeicients C Xq, C Zq : Aerodynamic stability derivatives C mo : Pitch moment aerodynamic coeicient Cm q : Pitch moment aerodynamic stability derivative 1. Introduction. Extensive researches have focused over the last ten years on developing analysis and synthesis techniques for gainscheduled controllers for linear parameter varying (LPV) systems [1, 1, 1,,, 15]. In Ref. [1, 1], conditions are given which guarantee stability, robustness, and performance properties of the global gainscheduled designs. Recent theoretical developments have produced methods for synthesizing gainscheduling controllers for LPV systems, which guarantee a level of robust stability and performance across scheduling parameter spaces [,, 15]. In Ref. [], LPV control synthesis methods have also been developed using parameterdependent Lyapunov functions to lead to a less conservative result. This gainscheduling approach has been successfully applied to synthesize controllers for pitchaxis Λ ICASE, MS 1C, NASA Langley Research Center, Hampton, VA 81. This research was supportd by National Aeronautics and Space Administration under NASA Contract No. NAS19 while the first author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 81. y Professor, Department of Aerospace Engineering and Mechanics, University of Minnesota, 11 Union St. SE, Minneapolis, MN z Graduate Research Assistant, Department ofmechanical Engineering, University of California, Berkeley, CA 9. 1

3 missile autopilots [, 11], F1 aircraft lateraldirectional axis during powered approach [, 5], and turbofan engines [8]. One of the potential diiculties in practical uses of the LPV synthesis methodology with parameterdependent Lyapunov functions is that the complexity of the linear matrix inequality (LMI) optimization problem increases exponentially with the number of scheduling parameters and the number of the grid points over the scheduling parameter spaces. One approach to facilitate practical use of LPV synthesis methodology, denoted as the blending approach, has been discussed in Ref. [15, 1]. This approach to control design partitions the full parameter space into overlapping small subspaces. An LPV controller is synthesized for each small region. These regional controllers are blended into a single LPV controller for the entire parameter space. In this paper, this blending approach is applied to control F1 longitudinal axes over the entire flight envelope. To synthesize an LPV controller for an F1 aircraft, an LPV model of the aircraft dynamics is required. Conventional approaches to generate an LPV model of an aircraft are based on Jacobian linearizations at trim points or a change of state coordinates [11] to reduce the nonlinearity of aircraft dynamics. The LPV models constructed by both approaches can present aircraft dynamics at trim conditions. However, the models can not represent aircraft dynamics at nontrim conditions. Instead of using Jacobian linearization or state transformation, the nonlinear terms of aircraft dynamics can be substituted for other functions in quasilpv form [18, 19]. This function substitution approach can be applied for both trim and nontrim conditions. The approach has been used in generating a quasilpv model of a generic missile [18, 19]. In this paper, a quasilpv model of F1 longitudinal axes is provided over the entire flight envelope including nontrim regions, using a function substitution approach. In Section, a brief summary of conventional LPV controller synthesis used in this paper is presented to emphasize the complexity of the LMI optimizations. In Section, three blending LPV control synthesis methodologies are presented. Development of a quasilpv model of F1 longitudinal axes is presented in Section. In Section 5, formulation of the LPV control problem and the blending of two LPV controllers of the F1 aircraft are presented. Nonlinear simulations of the closedloop system with the blended controller are presented in Section and this paper concludes with a brief summary in Section.. LPV Control Synthesis. In this section, a conventional LPV control synthesis using parameterdependent Lyapunov functions[] is briefly described. Consider a generalized linear openloop system as functions of parameters ρ(t) P. For a compact subset P ρr s, the parameter variation set denotes the set of all piecewise continuous functions mapping R (time) into P with a finite number of discontinuities in any interval, where s is the number of parameters. An LPV openloop system can be written as _x(t) e(t) 5 = y(t) A(ρ(t)) B 1 (ρ(t)) B (ρ(t)) C 1 (ρ(t)) D 1 (ρ(t)) C (ρ(t)) D 1 (ρ(t)) 5 x(t) d(t) 5 ; (.1) u(t) where y(t), e(t), d(t) and u(t) are measurements, errors, disturbances, and control signals. Hereafter, ρ denotes ρ(t). The induced L norm of d to e is defined as jjejj sup : ρp;dl ;jjdjj = jjdjj In a conventional LPV synthesis methodology, suppose there is an LPV output feedback controller K(ρ) which stabilizes the closedloop system exponentially and makes the induced L norm of d to e less than fl.

4 The controller K(ρ) can be written as _xk u = A k (ρ) C k (ρ) B k (ρ) D k (ρ) x k y : (.) An LPV controller K(ρ) can be constructed from solutions of X(ρ) and Y (ρ) of the following optimization problem []: subject to where X(ρ) ^AT (ρ)+ ^A(ρ)X(ρ) ~A(ρ)Y (ρ)+y (ρ) ~ A T (ρ)+ min fl (.) X;Y R n (μν i ) B (ρ)b T (ρ) 11(ρ) T fl 1 B 1 (ρ) i C 11 (ρ)x(ρ) I ne 1 fl 1 B T 1 (ρ) I (μν i ) C T (ρ)c (ρ) Y (ρ)b 11 (ρ) fl 1 C T 1 i B T 11(ρ)Y (ρ) I nd 1 5 < ; (.) fl 1 C 1 (ρ) I ne X(ρ) fl 1 I n ; (.) fl 1 I n Y (ρ) X(ρ) > ; Y (ρ) > ; 5 < ; (.5) ^A(ρ) A(ρ) B (ρ)c 1 (ρ); ~ A(ρ) A(ρ) B1 (ρ)c (ρ); (.) and n is the number of states of the generalized openloop system. The detailed definitions of matrices C 11 (ρ), C 1 (ρ), and B 11 (ρ) can be found in Ref. []. Note that and μν i is included in the LMIs. The parameter rate _ρ is bounded as ν i» _ρ i» μν i. μν i indicates that every combination of ν i A method to construct an LPV controller K(ρ) from the solutions X(ρ) and Y (ρ) is taken from Ref. []. In this paper, an LPV controller is constructed as []: A k (ρ) =A(ρ)+B (ρ)f (ρ)+q 1 (ρ)y (ρ)l(ρ)c (ρ) fl Q 1 (ρ)m(ρ; _ρ); (.8) B k (ρ) = Q 1 (ρ)y (ρ)l(ρ); (.9) C k (ρ) =F (ρ); D k (ρ) =; where matrices Q(ρ), F (ρ), L(ρ), and M(ρ; _ρ) are defined as Q(ρ) =Y (ρ) fl X 1 (ρ); F (ρ) = [B T (ρ)x 1 (ρ)+d T 1(ρ)C 1 (ρ)]; L(ρ) = [Y 1 (ρ)c T (ρ)+b 1 (ρ)d T 1(ρ)]; M(ρ; _ρ) =H(ρ; _ρ)+fl Q(ρ)[ Q 1 (ρ)y (ρ)l(ρ)d 1 (ρ) B 1 (ρ)]b T 1 (ρ)x 1 (ρ): (.1) (.11)

5 Matrix H(ρ; _ρ) is defined as H(ρ; _ρ) = [X 1 (ρ)a F (ρ)+a F (ρ) T X 1 (ρ)+ _ρ i + C T F (ρ)c F (ρ)+fl X 1 (ρ)b 1 (ρ)b 1 (ρ) T X 1 (ρ)] with A F (ρ) =A(ρ)+B (ρ)f (ρ) and C F (ρ) =C 1 (ρ)+d 1 (ρ)f (ρ). The closedloop system with the controller K(ρ) is exponentially stable and the induced L norm is less than fl. The proof can be found in Ref. []. To make the optimization problem of equation (.) computationally tractable, the scheduling parameters ρ are discretized into grid points. Thus, an infinite number of constraints are represented by a finite number of LMI constraints. Also, X(ρ) and Y (ρ) are represented by a finite number of basis functions h i (ρ) and g i (ρ): X(ρ) = XN x h i (ρ)x i ; Y (ρ) = N X y g i (ρ)y i ; where h i (ρ) and g i (ρ) are continuously dierentiable functions. The LMIs of equations (.)(.) are solved for all grid points of the scheduling parameters simultaneously. The size of the optimization problem is proportional to s+1 N g where s and N g are the number of scheduling parameters and total number of grid points over the scheduling parameter space. The number of decision variables are (N x + N y ) n(n+1), where N x, N y and n are the order of basis functions of X and Y, and the state order. Thus, computational time to solve the optimization problem of equation (.) is dependent on the number of grid points of scheduling parameters, the state order of a generalized openloop system, and basis function orders. The conventional LPV synthesis methodology may require expensive computational cost (computational time and computer memory) when number of scheduling parameters increases. In the next section, a blending method to reduce the computational cost to synthesize an LPV controller and to preserve performance level over parameter subspaces is presented.. LPV Controller Blending Approach. Instead of designing a single LPV controller for the entire parameter set in a conventional LPV synthesis, LPV controllers can be synthesized for parameter subsets, which are overlapped with each other. Then, these LPV controllers are blended into a single LPV controller over the entire parameter set. Thus, the performance of the closedloop system with the blended controller is preserved when parameter trajectories travel over the overlapped parameter subsets. Consider the scheduling parameter vector ρ Pwhich consists of subvectors ρ s P s and ρ d P d. The parameter subset P s can be partitioned into two subsets with the following conditions: P s = P s1 P s ; (.1) P s = P s1 [P s : (.) Suppose that there exist LPV controllers K 1 and K constructed from parameterdependent Lyapunov functions of a parameter subvector ρ d, over each parameter subset P si [P d, i =1;. Thus, X i and Y i are functions of ρ d (not ρ s ) over each parameter subset P si [P d ;i = 1;. Also, the controller K i can stabilize the closedloop system and the inducedl norm of the closedloop system is less than fl i over each parameter subset P si [P d. When a scheduling parameter subvector ρ s is in the intersection subset P s, LMI solution matrices X i (ρ d ) and Y i (ρ d ) are combined into X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ), respectively. There are three methods to

6 calculate X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ), which will be explained later. The blended matrices X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ) should be feasible solutions of the LMI constraints of equations (.)(.5) over the parameter subset P s [P d. When the parameter subvector ρ s is in the parameter subset P si P s, X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ) should be equal to X i (ρ d ) and Y i (ρ d ), respectively, and the partial derivatives should be equal respectively. There are three blending methods for X b and Y b to satisfy the feasibility condition and the boundary conditions. For method I, matrices X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ) can be written as: X b (ρ s ;ρ d )= Y b (ρ s ;ρ d )= X X b xi (ρ s )X i (ρ d ); (.) b yi (ρ s )Y i (ρ d ); (.) where blending functions b xi (ρ s ) and b yi (ρ s ) are dierentiable scalar functions. To satisfy the boundary conditions of X b and Y b, blending functions b xi (ρ s ) and b yi (ρ s ) are defined as: b xi (ρ s )= b yi (ρ s xi (ρ s s =; 8 < : 1; ρ s P si P s ; ; ρ s P s P si ; 8 < 1; ρ s P si P s ; : ; ρ s P s P si ; (.5) yi (ρ s s =; ρ s P si P s : (.) Suppose that the blending functions b xi (ρ s ) and b yi (ρ s ) satisfy the following conditions: X X b xi (ρ s )=1;» b xi (ρ s )» 1; ρ s P s ; (.8) b yi (ρ s )=1;» b yi (ρ s )» 1; ρ s P s ; (.9) and fl is chosen as max(fl 1 ;fl ). Then, the following equations are satisfied, since the LMIs of equations (.)(.5) are convex with respect to X and Y. Hereafter, ρ dependence is omitted for convenience. X b ^AT + ^AXb ~AY b + Y b ~ A T + (μν i (μν i X X b i ) B B T + X b C T 11C 11 X b + fl B 1 B T 1 < ; (.1) b i ) C T C + Y b B 11 B T 11 Y b + fl C T 1 C 1 < : (.11) When the derivatives of blending x i (ρ s ) y i (ρ s ) are small enough to satisfy the s inequalities, μ 8 s < μ : μν i 8 s < : μν i X X X j x j (ρ s ) 1 9 i ; 1 Y j y j (ρ s i 9 = ; A <(MX ); (.1) A <(MY ); (.1) 5

7 it is obviously noted that the blended X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ) can be feasible solutions of LMIs of equations (.)(.) over the parameter subset P s, where matrices M X and M Y are defined as M X X b ^AT + ^AXb M Y ~ AYb + Y b ~ A T + (μν i (μν i X X b i ) B B T + X b C T 11 C 11X b + fl B 1 B T 1 ; b i ) C T C + Y b B 11 B T 11Y b + fl C T 1 C 1 : Here, μ and represent the maximum and the minimum singular values, respectively. I is: The procedure of designing an LPV controller over the entire parameter set using this blending method 1. Partition the entire parameter set into two subsets which have the overlapped subset.. Solve the LMI optimization of equation (.) for X i (ρ d ) and Y i (ρ d )over each parameter subset.. Define blending scalar functions b xi (ρ s ) and b yi (ρ s ) which satisfy the boundary conditions of equations (.5)(.) and the derivative conditions of equations (.1)(.1). Note that the derivative conditions are suicient for X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ) to be feasible solutions of the LMI constraints of equations (.)(.5) over the parameter subset P s [P d. A controller designer chooses candidate blending functions until the feasibility conditions of equations (.)(.5) are satisfied.. Construct an LPV controller over the entire parameter set P s [P d using equations (.8)(.11), from the calculated X b and Y b of equations (.)(.). For method II, an alternative way to calculate the blended matrices X(ρ s ;ρ d ) and Y (ρ s ;ρ d ) as feasible solutions of the LMIs of equations (.)(.) is adding extra LMIs in conventional LPV synthesis with candidate blending functions []. The extra LMIs are: II is: X ~A b xj X j ^AT + ^A X b yj Y j + X X b xj X j + C 11 X b yj Y j ~ A T + B T (μν i X b xj X j ) B B T X b xj X j C T 11 fl 1 B 1 b xj X j I ne 1 fl 1 B T 1 I nd (μν i X b yj Y j ) C T C X b yj Y j B 11 fl 1 C T 1 b yj Y j I nd 1 fl 1 C 1 I ne 5 5 < ; (.1) < : (.15) The procedure of designing an LPV controller over the entire parameter set using this blending method 1. Partition the entire parameter set into two subsets which have the overlapped subset.. Solve the LMI optimization of equation (.) for the solution matrices X 1 (ρ d ) and Y 1 (ρ d )over one of parameter subsets, which is denoted by P s1 [P d.. Define blending scalar functions b xi (ρ s ) and b yi (ρ s ) which satisfy the boundary conditions of equations (.5)(.).

8 . Solve the LMI optimization problem of equation (.) with the extra LMI constraints of equations (.1)(.15) for the solution matrices X (ρ d ) and Y (ρ d ) over the other parameter subset P s [P d. 5. Construct an LPV controller over the entire parameter set using equations (.8)(.11) with the X b and Y b calculated using equations (.)(.). The LMI optimization over the second parameter subset P s [P d is related with the solution matrices X 1 and Y 1 over the first parameter subset P s1 [P d. Thus, defining the order of parameter subsets may aect the designed LPV controller over the entire parameter set. Both methods I and II require appropriate blending functions to blend solution matrices over the parameter subset P s. Note that an LPV controller provided by the blending methods I and II changes dependently on which blending functions are selected. It is unknown how this aects the closedloop performance of the designed LPV controller. For method III, blending matrix functions are calculated to minimize the induced L norm fl over the parameter subset P s [P d. The blended solution matrices X b and Y b are rewritten as: X b (ρ s ;ρ d )= 1 [X b 1 (ρ s )X 1 (ρ d )+X 1 (ρ d )X b1 (ρ s )] + 1 [X b (ρ s )X (ρ d )+X (ρ d )X b (ρ s )]; (.1) Y b (ρ s ;ρ d )= 1 [Y b 1 (ρ s )Y 1 (ρ d ) + Y 1 (ρ d )Y b1 (ρ s )] + 1 [Y b (ρ s )Y (ρ d ) + Y (ρ d )Y b (ρ s )]; (.1) where blending matrix functions X b1 (ρ s ), X b (ρ s ), Y b1 (ρ s ), and Y b (ρ s ) are dierentiable symmetric matrix functions bounded over the parameter subset P s. To present the blending matrix functions, basis functions g xj (ρ s ), h xj (ρ s ), g yj (ρ s ), and h yj (ρ s ) for X b1 (ρ s ), X b (ρ s ), Y b1 (ρ s ), and Y b (ρ s ) are introduced, respectively. To satisfy the boundary condition of blended matrices X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ), X b1 (ρ s ), X b (ρ s ), Y b1 (ρ s ), and Y b (ρ s ) are defined as: where X b1 (ρ s )=g x (ρ s )I + Y b1 (ρ s )=g y (ρ s )I + NX xg NX yg g xj (ρ s )X b1j ; X b (ρ s )=h x (ρ s )I + g yj (ρ s )Y b1j ; Y b (ρ s )=h y (ρ s )I + NX xh NX yh h xj (ρ s )X bj ; (.18) h yj (ρ s )Y bj ; (.19) g x (ρ s )=1; h xj (ρ s )=; h x (ρ s )=1; g xj (ρ s )=; g y (ρ s )=1; h yj (ρ s )=; h y (ρ s )=1; g yj (ρ s xj (ρ s x j (ρ s ) =; ρ s P s1 P s ; xj (ρ s ) x j (ρ s ) =; s P s P s ; yj (ρ s ) y j (ρ s ) =; s P s1 P s ; yj (ρ s ) y j (ρ s ) =; s P s P s : si (.) Here, the basis functions are dierentiable over the parameter subset P s and X b1j, X bj, Y b1j, and Y bj are unknown constant matrices in R n n. The unknown constant matrices can be determined solving the following LMI optimization: min fl; (.1) X b 1 ;X b j ;Y b j 1 ;Y b j R n n j

9 X b ^AT + ^AXb ~AY b + Y b ~ A T b (μν i ) B B T X b C T 11 fl 1 B i C 11 X b I ne 1 fl 1 B T 1 I nd (μν i ) C T C Y b B 11 fl 1 C T 1 B T 11 Y b I nd 1 fl 1 C 1 I ne X b fl 1 I n 5 5 < ; (.) < ; (.) ; X b > ; Y b > ; (.) where X b = g x X 1 + h x X +:5 Y b = g y Y 1 + h y Y +:5 8 < NX xg : 8 < NX yg : g xj [X 1 X b1j + X b1j X 1 ]+ g yj [ Y 1 Y b1j + Y b1j Y 1 ]+ NX xh k=1 N yh X k=1 9 = h xk [X X bk + X bk X ] ; ; 9 = h yk [Y Y bk + Y bk Y ] ; : (.5) III is: The procedure of designing an LPV controller over the entire parameter set using this blending method 1. Partition the entire parameter set into two subsets which have the overlapped subset.. Solve the LMI optimization of equation (.) for X i (ρ) and Y i (ρ) over each parameter subset.. Define basis functions for blending matrix functions in equations (.18) and (.19), which satisfy the boundary conditions of equation (.) over each parameter subset.. Solve the LMI optimization of equation (.1) over the parameter subset P s [P d. 5. Construct solution matrices X and Y over the entire parameter set using the solution matrices X 1, X, Y 1, Y, X b, and Y b.. Construct an LPV controller over the entire parameter set using equations (.8)(.11) based on solution matrices X and Y generated in step 5. In this section, there were described three methods to calculate the blended solution matrices X b (ρ s ;ρ d ) and Y b (ρ s ;ρ d ) which are feasible solutions of LMIs of equations (.)(.5) over the parameter subset P s [P d. Since all three methods keep X 1 and Y 1 over the parameter subset P s1 P s, the blended LPV controllers generated by the three methods are equal to the regionally designed LPV controller K 1 over the parameter subset P s1 P s. Both methods I and III can, also, keep the regional LPV controller K over the parameter subset P s P s. The disadvantage of method I is that choosing blending functions is in an adhoc manner. However, method I is the fastest to synthesize an LPV controller over the entire parameter set among the three blending methods. In method II, the blending solution matrices X b and Y b are guaranteed to be feasible solutions over the parameter subset P s. This method II requires solving the LMI optimization with the extra LMI constraints. In method III, optimal blending matrix functions are calculated to minimize the induced L norm fl over the parameter set P s, based on basis functions defined by a controller designer. It is unknown how the basis functions aect the blending matrix functions and the LPV control design. 8

10 . QuasiLPV Model of F1 Longitudinal Axes. In this section, a quasilpv model of F1 longitudinal axes is presented. The full nonlinear equations of an F1A aircraft are taken from Ref. [1]. The nonlinear equations of F1 longitudinal axes [1] are _V = qs cos m [C X(; i e )+ μc V C qs sin X q ()q]+ m [C Z(; i e )+ μc V C Z q ()q] g sin( )+T cos ; (.1) g cos( ) _ = V + qs _q = _ = q: sin mv cos fi T +[1+ qsμc mv (C Z q () cos C Xq () sin )]q [C Z (; i e ) cos C X (; i e ) sin ]; V t (.) qsμc I yy V (μcc m q ()+ C Zq ())q + qsμc (C mo (; i e )+ I yy μc C Z()); (.) Velocity (V, ft/sec), angle of attack (, rad), pitch rate (q, rad/sec), and pitch angle (, rad) are the state variables and thrust (T, lb) and elevator deflection (i e, rad) are the control variables. The aerodynamic coeicients are lookup tables based on windtunnel data from NASALangley windtunnel tests on an F1 aircraft scaled model [1]. The aerodynamic data were valid for a speed range of 1» V» 9 ft/sec, an angle of attack range of 1 i»» 5 i, and an altitude range of 5» h» ft. These three parameters are scheduling parameters in the quasilpv model derived for the F1 longitudinal axes. Note that V and are both scheduling parameters and states and h is a scheduling parameter which enters implicitly into the nonlinear dynamics. Unfortunately, the control variable i e does not enter ainely in equations (.1)(.). To derive a quasi LPV model of F1 longitudinal axes, it is necessary that all controls be in aine form. This is achieved by transforming (i e,t)into synthetic inputs u s =[u 1 u ] T. For details of the transformation, readers are referred to Ref. [1]. For the F1 quasilpv model, cos and sin are linearized about a trim value o. After tedious algebraic manipulations, equations (.1)(.) are rewritten as (.) _x = A(V; ; h)x + M(V; ; h)u s + f(v; ; h); (.5) x [V q o ] T ; where the elements of matrices A R, M R, and f R 1 are written in Appendix A. Using the function substitution method [18, 19], the nonlinear function f(v; ; h) can be decomposed into quasilinear functions G(V; ; h)[v V o o ] T where G is in R. Thus, a quasilpv model of F1 aircraft longitudinal axes is provided. The details of function substitution are written in Appendix B. To compare simulation results of the nonlinear and the quasilpv model of the F1 aircraft dynamics, several time sets of inputs T and i e are predefined. In this paper, one example of time simulations is presented for space limitation. For example, inputs are set as T = 8 < i e = i eo ; (.) : T o lb;» t» 1; 11» t T o lb; 1» t» 11 sec; where i eo and T o is a trim value. The time simulation results in Figure.1 show that the time responses (.) 9

11 velocity (ft/sec) 5 5 Linear model Nonlinear model θ (deg) α (deg) Linear model Nonlinear model q (deg/sec) Fig..1. Nonlinear vs. quasilpv model simulations of the quasilpv and nonlinear models are almost matched to each other. Note that the quasilpv model provided by the function substitution method may change depending on which one trim point is selected. It is unknown how this aects the quasilpv model or the LPV control design. 5. LPV Control Problem Formulation Control Design Objectives and Weighting Functions. A formulation of the LPV control synthesis of the F1 longitudinal axes is presented in this section. The primary control objective for the F1 longitudinal flight controller is to track velocity and flight path angle commands within 1 ft/sec and. i error range in steadystate conditions. Velocity and flight path angle tracking problems can be formulated as model matching problems in the LPV control synthesis. In this paper, we consider the F1 aircraft as an unmanned aircraft. The ideal 1

12 transfer function from the flight path angle command to the flight path angle measurement is modeled as :(s +1:5) a second order system, with.8 rad/sec bandwidth and a right hand zero at 1.5 rad/sec. s +1:s +: For the velocity tracking problem, the ideal transfer function from the velocity command to the velocity measurement is modeled as the second order system :1 with. rad/sec natural frequency s +:8s +:1 and critical damping. W mu1 V h? + h+ W mu s W act r? h+ Act F1? + h K Sen s s q?? h+ h+ W Vcmd W flcmd fl? h+ v cmd fl cmd W pv + h W pfl W n noises T idv T idfl Fig Interconnection structure for the model matching problems. A block diagram of the interconnection structure for synthesizing an LPV controller for the F1 longitudinal axes is shown in Figure 5.1. The velocity, angle rate and angle sensors are modeled as the firstorder low pass filters 5 s +5, s +, and 1 s +1 (see Ref. [1, 1]). In the LPV controller synthesis models, the sensor models shown in Figure 5.1 are approximated as: 1 Sen = diag([1; 1; ]); (5.1) s +1 since the interesting frequency range for LPV control synthesis is less than 1 rad/sec. The reduced order sensor models help to reduce the overall state order of an LPV controller, since the state order of an LPV controller is same as that of the augmented openloop system. The elevator actuator is modeled as the firstorder lag filter s + and its rate limit is defined as ± deg/sec in Ref. [1, 1]. Since the F1 is a fighter aircraft, we estimated the engine model as the firstorder lag filter s + which allows fast responses in engine dynamics. The thrust rate limit is taken to be 1 lb/sec. In the block diagram in Figure 5.1, the actuator rate and actuator models are: s Act = diag([ s + ; s + ; s s + ; ]): (5.) s + 11

13 The performance weighting functions are chosen based on the desired performance objectives. The performance weighting function of flight path angle W pfl,, is derived based on the performance 1(s=1 + 1) (s=:+1) objective tokeep a fl tracking error less than. i for 1 radian command in steady state flight. Since the bandwidth of the ideal model from the flight path angle command to the flight path angle measurement is.8 rad/sec, the roll o frequency of the weighting function is chosen as. rad/sec to specify the tracking error less than. i at the low frequency region (<. rad/sec). The performance weighting function for the velocity, W pv, s= + 1 s +1, is derived to track velocity commands within 1 ft/sec error range in steady state flight. The unmodeled dynamics are included in the multiplicative uncertainty models, W mu1 and W mu. The uncertainty weighting functions are rolled up in the midfrequency range in order to limit the bandwidth of the LPV controllers. The multiplicative uncertainty weight functions are set as :1(s=:5 + 1) W mu1 = ; s=8 + 1 W mu = :1(s=:+1) : s=5 + 1 The sensor noise models are taken as constant across frequency to reduce the state order of the LPV controllers. The velocity, angle, and angle rate sensor noises are modeled as white noises with amplitudes of.8 ft/sec,.1 i, and. deg/sec, respectively. To solve LMI equations (.)(.), the basis function sets need to be defined for X and Y. There is no analytical method to choose the best basis function set. Most often the basis functions used are power series[], Legendre polynomials[1], or aine functions of scheduling parameters [9]. Here, the basis function set for X(ρ) and Y (ρ) ischosen as the first order power series f1; ρg of the scheduling parameter of velocity to reduce the computation time in the LPV control synthesis. Note that the basis functions of X and Y do not have to be same. 5.. Blending Two Controllers. In this section, synthesizing an LPV controller for the F1 longitudinal axes using the blending approach is demonstrated. To apply the blending approach for control of the F1, the entire parameter set (the flight envelope) is partitioned into two subsets: high and low altitude regions. Parameter subsets are: P 1 f(v; ; h)j1» V» 9 ft/sec; 1» <5 i ; 5» h< ftg; P f(v; ; h)j1» V» 9 ft/sec; 1» <5 i ; 1 <h» ftg; P P 1 P : (5.) To use methods I and II, blending functions are required over the parameter subset P, which are satisfied with the boundary conditions of equations (.5)(.). Blending functions b 1 (h) and b (h) are chosen as: and b 1 (h) = 8 >< >: 1; h» 1 ft; :5[1 + cos( h 1 ß)]; 1 <h< ft; ; ft» h; (5.) b 1 (h)+b (h) =1: (5.5) 1

14 Using the blending functions, the solution matrices X i and Y i are blended across the parameter subset P as: X b (V; ; h) =b 1 (h)x 1 (V;)+b (h)x (V;); (5.) Y b (V; ; h) =b 1 (h)y 1 (V;)+b (h)y (V;): (5.) Using method I, an LPV controller K I is constructed with the solution matrices X b and Y b using equations (.8)(.11) over the entire parameter set P 1 [P. Using the blending functions in equations (5.)(5.5) and the solution matrices X 1 and Y 1 over the parameter subset P 1, the solution matrices X and Y are determined solving the LMI optimization over the parameter set P with the extra LMIs of equations (.1)(.15) described in method II. An LPV controller is constructed based on the calculated blended matrices X b and Y b using equations (.8)(.11) over the entire parameter set. To blend solution matrices X i (V;) and Y i (V;) calculated over each parameter subset, the basis functions for blending matrix functions of equations (.18) and (.19) are required to use method III. In this paper, the basis function sets are chosen as fg ; g 1 g for X b1 and Y b1, and as fh ; h 1 g for X b and Y b, respectively. g (h) = g 1 (h) = 8 >< >: 8 >< >: 1; h» 1 ft; :5[1 + cos( h 1 ß)]; 1 <h< ft; ; ft» h; ; h» 1 ft; :15[1 + cos( h 1 ß)]; 1 <h< ft; ; ft» h; (5.8) (5.9) h (h) =1 g (h); h 1 (h) = g 1 (h): (5.1) Note that it is not necessary to choose the same basis functions for X bi and Y bi. From the solution of the LMI optimization of equation (.1), the matrices X b (V; ; h) and Y b (V; ; h) are calculated using equations (.1) (.19). An LPV controller is constructed from the matrices X b (V; ; h) and Y b (V; ; h) using equations (.8)(.11) over the entire parameter set. For comparison, an LPV controller is constructed with solution matrices X and Y over the entire parameter set using the conventional LPV controller synthesis approach. It takes approximately hours on 9 MHz PIII machine running Linux. Using the blending methods I, II, and III, it takes approximately, 5, and hours on the same machine, respectively. The computation time to synthesize the LPV controller for the F1 longitudinal axes is reduced using the blending approaches.. Nonlinear Simulations. The 1 state nonlinear F1 aircraft dynamics [1] with the synthesized LPV controllers, K 1, K,, K I,, and, are simulated to compare their time simulations in this section. Recall that the LPV controllers K 1 and K are constructed over parameter subsets P 1 and P, respectively. The LPV controller is synthesized over the entire parameter set, using the conventional LPV control synthesis methodology. The LPV controllers K I,, and are the blended controller using the blending methods I, II, and III, respectively. In simulations, the LPV controller is implemented 1

15 1 5 V ideal 1 V (ft/sec) V trim15 h 8 ft γ (deg) V ideal 1 δ e (deg) dδ e /dt (deg/sec) 5 1 T (lb) dt/dt (lb/sec) Fig..1. Time simulations with the LPV controllers around 8 ft altitude. using linear interpolation at the currentvalues of the scheduling parameters between the grid point solutions. The full state models of actuators and sensors are included in the nonlinear F1 aircraft simulations. Using these LPV controllers, the velocity and fl step responses are simulated around 8 ft and ft altitude, respectively. The step input sizes are a 1 ft/sec velocity command and a 5 i fl command at 1 sec. The simulation results in Figure.1 show that all velocity and fl measurements match the ideal responses within ±.5 ft/sec and. i tracking error. It is observed that these LPV controllers achieve the desired performance objectives. The LPV controllers K 1, K I,, and are exactly equal to each other over the parameter subset P 1 P, since the blending methods I, II, and III keep the regional LPV controller K 1 over the parameter subset P 1 P. The step responses of velocity and fl with the LPV controller in Figure.1 represent the step responses with the LPV controllers K 1, K I, and in the simulations around 8 ft altitude. 1

16 The actuator deflections and their rates of the time simulations are shown in Figure.1. It is noted that the blended LPV controller uses smaller actuator deflections and their rates than the LPV controller does to achieve the performance objectives. 1 5 V ideal 1 V (ft/sec) V trim15 h ft γ (deg) V ideal 1 δ e (deg) dδ e /dt (deg/sec) T (lb) dt/dt (lb/sec) Fig... Time simulations with the LPV controllers around ft altitude. The LPV controllers K, K I, and are exactly equal to each other over the parameter subset P P. The blended LPV controller is dierent from K over the parameter subset P P, since in the method II, the blended controller is constructed solving the LMI optimization of equations (.)(.) with the extra LMI constraints of equations (.1)(.15). The step responses with the LPV controllers,, and are shown in Figure.. It is observed that these LPV controllers achieve the desired performance objectives over the parameter set P P. The simulation results show that the LPV controller uses the largest actuator deflections and their rates to achieve the performance objectives. For comparison, there is simulated one of candidate maneuvers that the F1 aircraft flies from 15 ft 15

17 to ft across the parameter subset P, using and 1 ft/sec velocity step and 1 i fl step commands. Velocity, flight path angle, altitude, and angle of attack time responses with the LPV controllers,,, and K I are shown in Figure.. All the LPV controllers can achieve the performance objectives across the parameter subset P. It is noticed that the velocity tracking performance with the blended controllers is slightly better than the LPV controller designed over the entire parameter set P 1 [P using the conventional LPV controller synthesis methodology. The actuator deflections and their rates with the LPV controllers are shown in Figure.. The simulation results show that the blended controllers use much smaller actuator deflections to track the velocity and fl commands. V (ft/sec) K I 55 V ideal 5 γ (deg) 5 1 K I V ideal 1.5 x h (ft).5 K I alpha (deg) K I Fig... Time simulations with the LPV controllers for the candidate maneuver. Sensor noises are integrated into the F1 aircraft simulations for the same situation of flying the F1 aircraft from 15 ft to ft. Sensor noises for pitch rate, velocity and pitch angle are set as white noises with ±.5 deg/sec, ±.8 ft/sec, and ±.1 i amplitudes. The simulation results are omitted for space 1

18 5 x δ e (deg) 5 T (lb) K I K I δ e /dt (deg/sec) K I dt/dt (lb/sec) 1 8 K I Fig... Actuator time responses with the LPV controllers for the candidate maneuver. limitations. The LPV controllers,,, and K I stabilize the F1 aircraft with the sensor noises and achieve tracking performance objectives over the scheduling parameter variations. The thrust signal and its rate do not exceed their limits: T < 19 lb and j dt dt j < 1 lb/sec. Also, the elevator actuator and its rates do not exceed their limits: ji e j < 5 i and j die dt j < deg/sec. The blended controllers and K I can preserve the performance level of the regional LPV controllers K 1 and K over each parameter subset P 1 and P. Note that method I requires the slow parametervarying blending functions. In this example, the parameter intersection range is wide enough to provide the slowvarying blending functions for method I. When the parameter intersection space is narrow, method III is appropriate to blend two LPV controllers over the parameter intersection space. 1

19 . Summary and Discussion. In this paper, the methodologies of blending LPV controllers are discussed to preserve the performance level over each parameter subset and to reduce computation time to synthesize an LPV controller over the entire parameter set. The blending approaches are to design each LPV controller for the set of a small number of scheduling parameters over the parameter subsets and blend all controllers scheduled on the entire scheduling parameters using the blending functions. The quasilpv model of the F1 longitudinal axes is provided by a function substitution method over the entire flight envelope to facilitate synthesis of an LPV controller. The two LPV controllers of the F1 longitudinal axes are synthesized as functions of velocity and angle of attack at two regions: low and high altitudes, respectively. The two LPV controllers are blended into a single LPV controller as functions of velocity, angle of attack, and altitude over the entire flight envelope, using the three blending approaches. It is noted from the nonlinear time simulations that the blended controllers achieve better performance than the LPV controller constructed using the conventional LPV synthesis methodology. The blended controllers constructed using methods I, II, and III achieve the performance objectives and stabilize the closedloop system of the F1 aircraft with sensor noises. Acknowledgements. The technical monitor was Dr. Christine Belcastro at NASA Langley Research Center. REFERENCES [1] B. Stevens and F. Lewis, Aircraft Control and Simulation, John Wiley and Sons Inc., 199. [] F. Wu, Control of Linear Parameter Varying Systems, PhD thesis, Department ofmechanical Engineering, University of California, Berkeley, [] F. Wu, A. Packard, and G. Balas, LPV Control Design for PitchAxis Missile Autopilots, in Proceedings of the th Conference on Decision and Control, New Orleans, LA, 1995, pp [] G. Balas, A. Packard, J. Renfrow, C. Mullaney, and R. M'Closky, Control of the F1 Aircraft LateralDirectional Axis During Powered Approach, Journal of Guidance, Control, and Dynamics, 1 (1998), pp [5] G. Balas, I. Fialho, A. Packard, J. Renfrow, and C. Mullaney, On the Design of LPV Controllers for the F1 Aircraft LateralDirectional Axis During Powered Approach, in Proceedings of the American Control Conference, Albuquerque, NM, 199, pp [] G. Becker, Quadratic Stability and Performance of Linear Parameter Dependent Systems, PhD thesis, Department of Engineering, University of California, Berkeley, 199. [], ParameterDependent Control of an UnderActuated Mechanical System, in Proceedings of the th IEEE Conference on Decision and Control, New Orieans, LA, 1995, pp [8] G. Wolodkin, G. Balas, and W. Garrard, Application to Parameter Dependent Robust Control Synthesis to Turbofan Engines, in AIAA, Aerospace Sciences Meeting and Exhibit, AIAA989, Reno, NV, [9] G. Wood, Control of ParameterDependent Mechanical Systems, PhD thesis, Department of Engineering, University ofcambridge, [1] J. Primbs, Nonlinear Optimal Control: A Receding Horizon Approach, PhD thesis, Department of Engineering, University of Cambridge, [11] J. Shamma, GainScheduled Missile Autopilot Design Using Linear Parameter Varying Transformations, Journal of Guidance, Control, and Dynamics, 1 (199), pp

20 [1] J. Shamma and M. Athans, Analysis of Gain Scheduled Control for Nonlinear Plants, IEEE Transactions on Automatic Control, 5 (199), pp [1], Guaranteed Properties of Gain Scheduled Control for Linear ParameterVarying Plants, Automatica, 5 (1991), pp [1] JY. Shin, Worstcase Analysis and Linear Parameter Varying Control of Aerospace System, PhD thesis, University of Minnesota,. [15] L. Lee, Identification and Robust Control of Linear ParameterVarying Systems, PhD thesis, Department ofmechanical Engineering, University of California, Berkeley, 199. [1] L. Nguyen, M. Ogburn, W. Gilbert, K. Kibler, P. Brown, and P. Deal, Simulator Study of Stall/PostStall Characteristics of a Fighter Airplane with Relaxed Longitudinal Static Stability, tech. report, NASA Scientific and Technical Information Branch, 199. NASA Technical Paper 158. [1] P. Apkarian and R. Adams, Advanced GainScheduling Techniques for Uncertain Systems, IEEE Transactions on Control Systems Technology, (1998), pp. 1. [18] W. Tan, Applications of Linear ParameterVarying Control Theory, Master's thesis, Department of Mechanical Engineering, University of California at Berkeley, 199. [19] W. Tan, A. Packard, and G. Balas, QuasiLPV Modeling and LPV Control of a Generic Missile, in Proceedings of the American Control Conference, Chicago, IL,, pp Appendix A. Nonlinear Equations of F1 Longitudinal Axes. From nonlinear equations (.1) (.), the nonlinear equations of the F1 longitudinal axes are rewritten as follows: _V q _ 5 = A 11 (V;) A 1 (V; ; h) A 1 (V;) A (V; ; h) V q o 5 + M(V; ; h)u s + f(v;); (A.1) where A 11 = A 1 = ; A 1 (1; 1) = qsμc mv (C X q () cos + C Zq () sin ); A 1 (1; ) = g( cos cos o sin sin o ); A 1 (; 1) = 1 + qsμc mv ( C X q () sin + C Zq () cos ); A 1 (; ) = g V (sin cos o cos sin o ); f = A = qsμc VI (μcc mq ()+ C Zq ()) yy 5 ; 1 qs m C Z () sin + g( cos sin o + sin cos o ) qs mv C Z() cos + g V (sin sin o + cos cos o ) qs I C Z () yy 5 : 19

21 The gain matrix M is constructed as lookup tables and u s are synthetic inputs. The term of M(V; ; h)u s can represent the following terms: M(V; ; h)u s (V; ; i e ;T) ß qs m [ μ CZe i e sin + C X (; i e ) cos ]+ qs mv [ CZe μ i e cos C X (; i e ) sin ] qsμc I yy C M (; i e )+ qs I yy CZe μ i e cos m T sin mv T The detailed methods to determine M and synthetic inputs u s are available in Ref. [1]. The synthetic inputs can vary in the range of 1» u 1» 1 and» u» 1. The units of u 1 and u are 5 i and 19 lb, respectively. Appendix B. Decomposition. Set the state variables in a quasilpv model of the F1 aircraft dynamics as V V o, o, q q o and o, where V o, o, q o and o represent a trim point. Then, equation (A.1) is rewritten as _ο1 _ο = F (z)+ A 11 (z) A 1 (z) A 1 (z) A (z) ο 1 + ο B 1 (z) B (z) ~u; 5 : (B.1) where F (z) =f(z)+ A 11 (z) A 1 (z) A 1 (z) A (z) z o + w o B 1 (z) B (z) u o ; ο 1 = z z o ; ο = w w o ; ~u = u s u o ; (B.) and z =[V ] T ; w =[q ] T : (B.) To provide a quasilpv model of F1 aircraft dynamics from equation (B.1), the term F (z) should be decomposed into linear parameter varying functions written as F (z) =F (z o + ο 1 )= g z (z o + ο 1 ) g w (z o + ο 1 ) ο 1 ; (B.) where g z R and g w R. There are an infinite number of possible solutions of g z and g w to satisfy equation (B.). To determine functions g z and g w, more constraints are required. In this paper, the variation of g z and g w over the entire flight envelope is minimized. With these constraints, an optimization problem is formulated to determine g z and g w. For example, to determine the first row ofg z, an optimization problem is follows: min l; (B.5) g z 11 R;gz 1 R subject to F 1 (z o + ο 1 )=[g z11 (z o + ο 1 ) g z1 (z o + ο 1 )]ο 1 (B.) g z11 (z o + ο 1 1 j»l; g z1 (z o + ο 1 )

22 where F 1 is the first rowoff. To make the optimization problem of equation (B.5) computationally tractable, the continuous constraints are evaluated at grid points over the parameter set. Thus, the matrix g z can be determined at every grid point ofο 1. Using solutions of the optimization for g z and g w, a quasilpv model of F1 longitudinal axes is written as: _V q _ 5 = A 11 (V;)+g z (V;) A 1 (V; ; h) A 1 (V;)+g w (V;) A (V; ; h) V V o o q o 5 + M(V t;;h)(u s u so ): (B.) 1