NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM

Transcription

1 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM D.G. Dimogianopoulos, J.D. Hios and S.D. Fassois DEPARTMENT OF MECHANICAL & AERONAUTICAL ENGINEERING GR PATRAS, GREECE sms 14th Mediterranean Conference on Control Automation, June 28-30, 2006, Ancona, Italy. Research supported by the European Commission [STREP project No on Innovative Future Air Transport System (IFATS)].

2 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 1 TALK OUTLINE 1. INTRODUCTION & AIM OF THE WORK 2. SYSTEM MODELLING & CONTROLLER DESIGN 3. STABILITY ANALYSIS: OUTLINE OF PROOF 4. AIRCRAFT SYSTEM CONTROL RESULTS 5. CONCLUDING REMARKS

3 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 2 1. INTRODUCTION & AIMS OF THE WORK The General Problem Singularities in control law design for widely-used classes of nonlinear systems, lack of integral action. Problem Characteristics Systems with complex, operating-point-dependent dynamics may be efficiently represented only by specifically designed nonlinear input-output representations (eg. CCP- NARMAX: Constant Coefficient Pooled - Nonlinear AutoRegressive Moving Average with exogenous excitation). These nonlinear representations rarely facilitate control design based upon traditional techniques [(Generalized) Minimum Variance - (G)MV]. Typical (unwanted) characteristics: Controller action ill-suited to the system dynamics (poor high frequency performance), control signal impossible to compute (singularities) at some instants, switching needed.

4 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 3 Problem Significance Simple, well-suited to system dynamics, nonlinear control design avoiding singularities in the control law computation and featuring integral action. Literature Survey Nonlinear plants controlled by PI(D) designs [Chen, Huang 2004], PI(D) with inverse precompesation [Petridis, Stenton 2003] or loop-shaping [Glass, Franchek 1999]. (G)MV introduced more than 30 yrs ago [Astrom 1971] and successfully used in Linear Time Invariant (LTI) modelled industrial applications. Comments on GMV design for NAR(MA)X plants in [Goodwin, Sin 1984], [Sales, Billings 1990], NARX in [Bittanti, Pirrodi 1997] which mentions the difficulty of solving the control law equation.

5 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 4 MV also combined with Neural Network system representations [Gao et al. 2000], [Zhu, Gao 2004]. GMV for general nonlinear plants [Grimble 2005] with integral action. Aim of the Work Design of a GMV controller based upon a CCP-NARMAX representation and applied to a fast dynamics system (aircraft). Modify the designed controller to obtain: i) Feasible control law values at all times (no singularities). ii) Integral control action.

6 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 5 2. SYSTEM MODELLING & CONTROLLER DESIGN A) System Modelling. The system to be modelled is the Input- Output (IO) relationship between the pilot stick input and the the pitch rate response of a 6 Degrees-Of-Freedom nonlinear aircraft simulator (with characteristics shown in Fig. 1): Figure 1: The aircraft simulator with its inputs outputs and disturbances.

7 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 6 Stochastic Single Input Single Output (SISO) CCP-NARMAX representation (Pooled Nonlinear AutoRegressive Moving Average with exogenous excitation and Constant Coefficients): y j [t] : model output y j [t] = L i=0 θ i p i,j [t] + e j [t] E{e j [t] e s [t τ]} = γ e [j, s]δ[τ] e j [t] NID (0, σe(j)) 2 e j [t] : prediction errors or residuals [zero mean uncorrelated with variance σ 2 e(j)]. u j [t] : model input (not explicitly presented in the previous equations) p i,j [t] : regressors θ j : model coefficients (parameters) NID: Normal Independently Distibuted operating point (flight) j (1) y j [t] = φ T j [t] θ + e j [t] (2) with φ j [t] = [p 0,j [t]... p L,j [t]] T, E{ } = statistical expectation, NID(.,.) = Normally Independently Distributed (with the indicated mean), and variance δ[τ] = Kronecker delta (δ[τ] = 1 when τ = 0 and δ[τ] = 0 when τ 0) and γ e [j, s] = the cross covariance. Lower case/capital bold face symbols designate column vector/matrix quantities, respectively

8 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 7 Assumption 1 : The model is affine in the most recent control variable (also in [Bittanti, Piroddi 1997]). y j [t] = L i=0 θ i p i,j [t] + e j [t] = A l (q 1 )y j [t 1] + G u [t 1]u j [t k] + G F [t 1] + e j [t] A l (q 1 ) : linear polynomial operator G u (q 1 ) : general form operator terms multiplying the most recent control variable u[t k] ([t 1] designates the most recent signal involved) G F (q 1 ) : all remaining terms Assumption 2 : The model has no transmission zeros equal to 1 (also in [Bittanti, Piroddi 1997], useful for control law integral action ).

9 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 8 Model identification & parameter estimation If N recorded data for the j-th operating point (flight) are available: y j [1]. y j [N] = φ T j [1]. φ T j [N] θ + e j [1]. e j [N] = y j = Φ j θ + e j (3) If data from M operating points (flights) are available: y 1. y M = Φ 1. Φ M θ + e 1. e M = ȳ = Φ θ + ē (4)

10 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 9 Structure selection ( = which terms p ij should be included in Φ) and parameter estimation (computation of θ) is made by the orthogonal algorithm [Korenberg et al. 1988]and a 2-Stage Least Squares (2SLS) method: 1 st Stage: Use all data (from M flights) with the orthogonal algorithm to obtain an initial CCP-NARX model (no regressors involving e[t] terms) and an initial residual sequence ē[t]. 2 nd Stage: Use all data (from M flights) AND the above residual sequence with the orthogonal algorithm to obtain the CCP-NARMAX structure (with regressors involving prediction error terms). Parameter estimation is performed by Ordinary (or Weighted) Least Squares (OLS or WLS) applied to (4 ).

11 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 10 General comments for the CCP-NARMAX repreresentation ALL data records are used as ONE entity for structure selection and parameter estimation: Ideally suited to systems with multiple operating points (aircrafts). Knowledge on dependency between multiple operating points may be easily incorporated in the model building phase [second equation of set (1): cross-correlated residuals are permitted]. Modelling Results M = 99 flights of N = 5001 samples (data recorded at 100 Hz) obtained through the 6 DOF nonlinear simulator. Input signal u[t] is the pilot stick input, while output signal y[t] is the pitch rate. Identified model CCP-NARMAX(15,15,15,1) (that is, model orders n y = n u = n e = 15, and delay k = 1 - see Table in the following page) and L = 45 terms.

12 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 11 pitch rate (deg/sec) prediction error x actual pitch rate one step ahead prediction time (sec) (a) (b) p 1 = y[t 1] p 24 = u[t 8] p 2 = y[t 2] p 25 = u[t 9] p 3 = y[t 3] p 26 = u[t 10] p 4 = y[t 5] p 27 = u[t 11] p 5 = y[t 8] p 28 = u[t 14] p 6 = y[t 9] p 29 = u[t 13] p 7 = y[t 13] p 30 = u[t 7] p 8 = y[t 15] p 31 = e[t 2]e[t 4] p 9 = y[t 1]e[t 2] p 32 = e[t 1]e[t 5] p 10 = u[t 1]u[t 10] p 33 = e[t 7]e[t 10] p 11 = u[t 1]u[t 13] p 34 = e[t 2]e[t 7] p 12 = u[t 5]u[t 14] p 35 = e[t 1]e[t 15] p 13 = u[t 3]u[t 12] p 36 = e[t 4]e[t 5] p 14 = u[t 8]u[t 15] p 37 = e[t 8]e[t 14] p 15 = u[t 11]u[t 14] p 38 = e[t 15] 2 p 16 = u[t 15] 2 p 39 = e[t 2] p 17 = u[t 15]e[t 2] p 40 = e[t 1] p 18 = u[t 14]e[t 1] p 41 = e[t 4] p 19 = u[t 14]e[t 15] p 42 = e[t 5] p 20 = u[t 3]e[t 15] p 43 = e[t 8] p 21 = u[t 14]e[t 2] p 44 = e[t 13] p 22 = u[t 6] p 45 = e[t 9] p 23 = u[t 15] Figure 2: CCP-NARMAX(15,15,15,1) performance: (a) actual pitch rate versus one-step-ahead prediction; (b) one-step-ahead prediction errors. The CCP-NARMAX may be written in the affine form as: with A l (q 1 ) y j [t] = A l y j [t 1] + G u [t 10]u[t 1] + G F [t 1] + e j [t] (5) = θ 1 + θ 2 q 1 + θ 3 q 2 + θ 4 q 4 + θ 5 q 7 + θ 6 q 8 + θ 7 q 12 + θ 8 q 14 G u [t 10] = θ 10 u[t 10] + θ 11 u[t 13] (6)

13 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 12 B) Controller Design. i) Standard GMV Control law (for simplicity the index j is dropped): Minimize: {( ) 2 ( 2 } J[t + k] = E P(q 1 )y[t + k] + Q(q )f(u[t])) 1 t (7) with P(q 1 ) = k I(1 aq 1 ) 1 q 1 and f(u[t]) = G u [t 9]u[t], Q = q 0 (fixed number). Consider J[t+k] u[t] = 0, and using the affine system form (5): u[t] = H[t] = P 1 H[t] (k I +q0 2)G u[t 9] ( A l y[t] + G F [t] ) ) + k (8) I(1 a) (e[t] + G 1 q 1 u [t 10]u[t 1] ii) Modified GMV Control law by introducing the design parameter k b : 1 u Mod [t] = (k I + q0 2)G H[t] (9) u[t 9] + k b k b permits to shift the threshold of a rapidly growing u[t] from G u = 0 to a userchosen value (application dependent).

14 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM STABILITY ANALYSIS: OUTLINE OF PROOF Use of u mod [t] instead of u[t] depends on the value of G u [t]. i) Consider critical instant t = t where for the first time G u [t ] < ǫ but G u [t ] 0 with ǫ a small user defined constant. ii) Compute J MV [t + 1] (using u[t ]) and J mod [t + 1] (using u mod [t ]). Note that even if u[t], J MV [t + 1] still finite. It results J Mod [t + 1] J MV [t + 1] = q2 0(q ) { } (k I + q0 2 [t ] + p 2 0E e 2 )2H2 MV [t + 1] [ O( 1 M ) ± O( 1 M ) + Λ2 [t] ] H 2 [t ] + p 2 0E { } e 2 Mod[t + 1] (10) (11) with M a very large number and Λ 2 [t] 1 for G u 0, and H[t] in (8)

15 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 14 Then for k I, q 0 1 (true in practice in order to obtain integral action): J MV [t + 1] < J Mod [t + 1] < (12) Theorem: The CCP-NARMAX modelled system satisfying assumptions (1),(2) under the control action (8) and (9) is stable and the signals involved are, at worst, bounded.

16 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM AIRCRAFT SYSTEM CONTROL RESULTS Consider k b [t] = γ exp( G u [t 9] ), γ > 0 (13) This function permits the transition from (8) to (9) without switching and u mod may be used all the time. In this case γ = Disturbances are due to turbulence levels and the wheel input (the wheel-to-pitch rate relationship has not been included in the SISO CCP-NARMAX model). For comparison purposes: PID designed using procedure in MATLAB R Optimization Toolbox K PID P = , K PID I = , K PID D =

17 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 16 pitch rate (deg/sec) (a) open loop system system with PID control system with MV control pitch rate (deg/sec) (c) open loop system system with PID control system with MV control 1 1 Controled input: Stick (lbs) (b) PID control input MV control input disturbance time (sec) Disurbance: Wheel (lbs) Controled input: Stick (lbs) 10 (d) PID control input MV control input disturbance time (sec) Disurbance: Wheel (lbs) Figure 3: PID versus MV-like control: (a) Pitch rate response (low turbulence); (b) Control input (low turbulence); (c) Pitch rate response (increased turbulence); (d) Control input (increased turbulence).

18 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM 17 Comments: i) The GMV-like controller has less overshoot and regulates the input faster than the PID controller (aircraft stays less time off-course), for low turbulence conditions. The same goes for increased turbulence conditions. ii) The GMV-like controller signal range is bigger than PID s, but control activity is quite smooth (not aggressive as is typical for the GMV controller) iii) The output never reaches zero even in absence of control inputs (wheel or stick) due to internal noise of the aircraft simulator.

19 NONLINEAR INTEGRAL MINIMUM VARIANCE-LIKE CONTROL WITH APPLICATION TO AN AIRCRAFT SYSTEM CONCLUDING REMARKS 1. A minimum variance-like controller with integral action for an affine (in the most recent control variable) CCP-NARMAX representation has been designed. 2. The GMV-like control law has been suitably modified to avoid singularities in the control law calculation. 3. The controller has been applied to an aircraft system to regulate its pitch rate around zero. Comparisons with a classical PID controller shows quicker pitch rate regulation with less overshoot even in the presence of significant disturbances (turbulence, wheel input).