PI(D) tuning for Flight Control Systems via Incremental Nonlinear Dynamic Inversion

PI(D) tuning for Fligh t Con trol Systems via Incremen tal Nonlinear Dynamic In v ersion P aul Acquatella B. ∗ , 1 Wim v an Ek eren ∗ , ∗∗ , 2 Qi Ping Ch u ∗∗ , 3 ∗ DLR, German A er osp ac e Center Institute of System Dynamics and Contr ol D-82234 Ob erpfaffenho fen, Germany ∗∗ Delft University of T e chnolo gy, F aculty of A er osp ac e Engine ering 2629HS Delft, The Netherlands Abstract: Previous r esults rep orted in the rob otics literature show the rela tio nship betw een time-delay c ontr ol (TDC) and pr op ortional-inte gr al-derivative c ont r ol (PID). In this pap er, we show that incremental no nlinear dynamic inv ersion (INDI) — more familiar in the aero s pace communit y — are in fact equiv alent to TDC. This leads to a meaning ful and systematic metho d for PI(D)-co n trol tuning of robust nonlinea r flight co n trol sy stems via INDI. W e considered a reformulation of the plant dyna mics inv ersion which r e mov es effector ble nding mo dels from the resulting control law, resulting in robust mo del-free control laws like PI(D)-control. Keywor ds: ae rospace, tracking, application of nonlinear analysis and design 1. INTRODUCTION Ensuring stabilit y and p erformance in betw een op era tio nal po int s of widely-used gain-scheduled linear PID controllers motiv ates the use o f nonlinear dynamic in version (NDI) for flight con trol systems. NDI cancels o ut nonlinearities in the mo de l via state feedba ck, and then linea r control can be subse q uent ly designed to c lo se the sys tems’ outer -lo op, hence eliminating the need o f linear izing a nd desig ning different controllers for sev era l op erational p oints as in gain-scheduling. In this pap er w e consider nonlinear fligh t con trol str ategies based on incremen tal nonlinear dyna mic inv ersion (INDI). Using sensor a nd actuator measurements for feedback al- lows the design of an incremental control action which, in combination with nonlinear dyna mic in version, stabilizes the p artly -linearized nonlinear system incr emental ly . With this result, dep endency on exact knowledge o f the s ystem dynamics is grea tly reduced, ov ercoming this ma jor ro- bustness is sue from conv ent iona l nonlinea r dynamic in ver- sion. INDI has been considered a sensor-bas ed approach bec ause senso r measurements w ere meant to replac e a large part o f the vehicle mo del. Theoretical dev elopment of incremen ts of nonlinear con- trol action date back from the late nineties and started with activities concerning ‘implicit dynamic inv ersion’ for inv ersion-based flig ht control (Smith (1 9 98); B acon and Ostroff (2000 )), where the a rchitectures consider ed in this pap er were fir stly descr ib ed. Other designatio ns for these developmen ts fo und in the literature are ‘modified NDI’ 1 Researc h Engineer, Spa ce Systems Dynamics Departmen t. paul.acq uatella@dlr.d e . 2 Graduate Studen t, Control & Operations Department. w.vaneke ren@student.t udelft.nl . 3 Asso ciate Professor, Cont rol & Op erations D epartmen t. q.p.chu@ tudelft.nl . and ‘simplified NDI’, but the designation ‘incremental NDI’, in tro duced in (Chen and Zha ng (2008 )), is con- sidered to describ e the metho dolog y a nd nature of these t yp e of co n trol laws b etter (Chen and Zhang (2 008); Chu (2010); Siebe r ling et al. (20 10)). INDI has be en elab ora ted and applied theo retically in the past decade for adv anced flight cont ro l and space applications (Sieber ling e t a l. (2010); Smith (1998); Bacon and Ostro ff (2000 ); B acon et al. (2 000, 2001 ); Acquatella B. et al. (2012); Simpl ´ ıcio et a l. (201 3)). Mor e recently , this technique has b een ap- plied also in practice for quadroto rs and ada ptive co n trol (Smeur et a l. (2016a ,b)). In this pa per , we pr esent thre e main contributions in the context of nonlinear flight control system design. 1) W e revisit the NDI/INDI control laws and we establish the equiv a lence betw een INDI and time-delay con trol (TDC). 2) Ba sed on previous r esults rep or ted in the rob otics liter- ature showing the relationship b etw een discr ete formula- tions o f TDC and prop or tional-integral-deriv ative control (PID), we show that a n equiv alent PI(D) controller with gains < K , T i , ( T d ) > tuned via INDI/TDC is more mean- ingful and sys tematic than heuristic metho ds , since o ne considers desired erro r dynamics g iven by Hurwitz g ains < k P , ( k D ) > . Subsequently , tuning the remaining effector blending gain is muc h less cum b ersome than designing a whole set of g ains iteratively . 3) W e a ls o consider a r eformulation of the plant dyna mics inv ersion as it is done in TDC which re mov es the effector blending model (control deriv atives) fro m the resulting control law. This has not be en the case so far in the rep orted INDI co nt ro llers, ca using r obustness pr oblems bec ause of their uncertainties. Moreover, this a llows to consider the intro duced term as a scheduling v ariable which is only dire c tly related to the prop ortiona l g a in K . 2. FLIGHT VEHICLE MODELING W e are interested in E uler’s equation of motion r epresent- ing flight vehicles’ angular velocity dynamics I ˙ ω + ω × I ω = M B (1) where M B ∈ R 3 is the external mo ment vector in bo dy axes, ω ∈ R 3 is the ang ula r velocity vector, and I ∈ R 3 × 3 the iner tia matrix o f the rigid bo dy a ssuming symmetry ab out the plane x − z of the b o dy . F urthermore, we will be interested in the time history o f the angula r velo city v ector , hence the dy na mics of the rotational motion of a vehicle (1) can b e rewr itten as the following set of differential equations ˙ ω = I − 1  M B − ω × I ω  (2) where ω = " p q r # , M B = " L M N # = S Q " b C l c C m b C n # , I = " I xx 0 I xz 0 I y y 0 I xz 0 I z z # , with p , q , r , the b o dy roll, pitch, and yaw rates, resp ec- tively; L , M , N , the roll, pitch, a nd ya w moments, resp ec- tively; S the wing surface ar ea, Q the dynamic pre ssure, b the wing spa n, c the mean aer o dynamic chord, and C l , C m , C n the moment co efficients for roll, pitch, and yaw, resp ectively . F urthermore, let M B be the s um of mo men ts partially generated b y the aer o dynamics o f the air frame M a and mo men ts generated by control surface deflections M c , and we describe M B linearly in the deflection angles δ assuming the co nt ro l deriv atives to b e linea r a s in Sieb er- ling et a l. (2010 ) with ( M c ) δ = ∂ ∂ δ M c ; therefore M B = M a + M c = M a + ( M c ) δ δ (3) where M a = " L a M a N a # , M c = " L c M c N c # , δ = " δ a δ e δ r # and δ c orresp onding to the control inputs: a ileron, elev a - tor, a nd rudder deflection angles, r esp ectively . Hence the dynamics (2) can be rewritten as ˙ ω = f ( ω ) + G ( ω ) δ (4) with f ( ω ) = I − 1  M a − ω × I ω  , G ( ω ) = I − 1 ( M c ) δ . F or pra ctical implementations, we consider first-order ac- tuator dynamics represented by the follo wing tra nsfer function δ δ c = G a ( s ) = K a τ a s + 1 , (5) and furthermore, we do not consider these actuator dy- namics in the control design pro ce s s as it is usually the case for dynamic inversion-based control. F or that reason, we a s sume that these actuator s are sufficiently fast in the control-bandwidth sense, meaning that 1 /τ a is higher than the control system closed-lo op bandwidth. 3. FLIGHT CO NTR OL LA W DESIGN 3.1 Nonline ar Dynamic Inversion Let us define the control par ameter to b e the angular velocities, hence the o utput is simply y = ω . W e then consider an error vector defined as e = y d − y where y d denotes the smo oth des ired output vector (at least o ne time differentiable). Nonlinear dynamic in version (NDI) is desig ned to linearize and decouple the rotational dynamics in order to o btain an explicit desired closed lo o p dynamics to b e follow ed. Int ro ducing the vir tual control input ν = ˙ ω des , if the matrix G ( ω ) is non-singular (i.e., invertible) in the domain of interest for a ll ω , the no nlinear dynamic in version control co nsists in the follo wing input transformatio n (Slotine and L i (1990); C hu (2010)) δ = G ( ω ) − 1  ν − f ( ω )  (6) which cancels all the nonlinearities, a nd a simple input- output linear relationship between the output y and the new input ν is o bta ined as ˙ y = ν (7) Apart fro m b eing linear, an in teresting result fro m this relationship is that it is als o decoupled since the input ν i only affects the output y i . F r om this fact, the input trans- formation (6 ) is called a de c oupling c ontr ol law , and the resulting linear system (7) is called the single-inte gra tor form. This single-integrator form (7) can b e render ed ex- po nent ially stable with ν = ˙ y d + k P e (8) where ˙ y d is the feedforward term for tra cking tasks, a nd k P ∈ R 3 × 3 a constan t dia gonal matr ix, whos e i − th diagonal element s k P i are chosen so tha t the po lynomials s + k P i ( i = p, q , r ) (9 ) may beco me Hur witz, i.e., k P i < 0 . T his r e s ults in the exp onentially stable and decoupled desir e d error dynamics ˙ e + k P e = 0 (10) which implies that e ( t ) → 0. F rom this t ypical tracking problem it can b e seen that the entire control s y stem will hav e tw o control lo o ps (Ch u (2 010); Sieb erling et al. (2010)): the inner linea rization lo op (6), and the outer control lo op (8). This resulting NDI control law dep ends o n accurate knowledge of the aer o dynamic momen ts, hence it is susceptible to model uncertainties contained in b oth M a and M c . In NDI c o nt ro l design, we consider outputs with relative degrees of o ne (rates), mea ning a first-o rder system to be con trolled, see Fig. 1. Ex tensions of input-output lin- earization for systems inv olving higher r e lative degrees are done via fe e db ack line arization (Slotine and Li (1990 ); Chu (2010)). 3.2 Incr emental Nonline ar Dynamic Inversion The concept of incr ement al nonlinear dynamic in version (INDI) amounts to the applicatio n of NDI to a s ystem expressed in an incremental form. This improves the ro- bustness of the closed-lo o p system a s co mpared with con- ven tional ND I since dependency on the accurate knowl- edge of the plant dynamics is reduced. Unlik e NDI, this P S f r a g r e p la c e m e n t s Reference T ra jectory Po sition Con trol Flight Path Angle and Airsp eed Control Attitude Con trol Rate Control X, Y , Z V , ψ , γ µ, α, β p, q , r Fig. 1. F our lo op nonlinear fligh t cont ro l design. W e are fo cused o n nonlinear dynamic inv ersio n of the ra te co n trol loo p (grey b ox) in the following. Image credits: Sonneveldt (201 0 ). control design technique is implicit in the sense that de- sired clo sed-lo op dynamics do not reside in some explicit mo del to b e follow ed but result when the feedback lo o ps are closed (Bacon and O stroff (2000); Bacon et a l. (2000)). T o o btain a n incremental form of system dynamics, we consider a first-o rder T aylor se ries e x pansion of ˙ ω (Smith (1998); Ba con and Ostroff (200 0); Bacon et al. (2000, 2001); Sieb erling et a l. (2010); Acquatella B. et al. (2012, 2013)), not in the geometric s ense, but with res pec t to a suffiently smal l time-delay λ a s ˙ ω = ˙ ω 0 + ∂ ∂ ω  f ( ω ) + G ( ω ) δ      ω = ω 0 δ = δ 0 ( ω − ω 0 ) + ∂ ∂ δ  G ( ω ) δ      ω = ω 0 δ = δ 0 ( δ − δ 0 ) + O (∆ ω 2 , ∆ δ 2 ) ∼ = ˙ ω 0 + f 0 ( ω − ω 0 ) + G 0 ( δ − δ 0 ) with ˙ ω 0 ≡ f ( ω 0 ) + G ( ω 0 ) δ 0 = ˙ ω ( t − λ ) (11a) where ω 0 = ω ( t − λ ) and δ 0 = δ ( t − λ ) are the time- delay ed s ignals of the curr e n t state ω and control δ , re- sp ectively . This means an approximate linear ization abo ut the λ − delay ed s ig nals is p er formed incr emental ly . F or such sufficiently small time-delay λ so that f ( ω ) do es not v ary significant ly during λ , w e a ssume the following approximation to hold ǫ I N D I ( t ) ≡ f ( ω ( t − λ )) − f ( ω ( t )) ∼ = 0 (12) which leads to ∆ ˙ ω ∼ = G 0 · ∆ δ (13) Here, ∆ ˙ ω = ˙ ω − ˙ ω 0 = ˙ ω − ˙ ω ( t − λ ) re presents the incremental accele ration, and ∆ δ = δ − δ 0 represents the so-called incremental c o nt ro l input. F or the obtained approximation ˙ ω ∼ = ˙ ω 0 + G 0 ( δ − δ 0 ), NDI is a pplied to obtain a relation b etw een the incr emental con trol input and the output of the system δ = δ 0 + G − 1 0  ν − ˙ ω 0  (14) Note that the deflection angle δ 0 that corr esp onds to ˙ ω 0 is taken fr om the output of the actuators, and it has bee n assumed that a commanded control is achieved s u f- ficiently fast according to the assumptions of the a ctuator dynamics in (5). The tota l control command a lo ng with the obtained linearizing control ∆ δ can b e rewr itten a s δ ( t ) = δ ( t − λ ) + G − 1 0 h ν − ˙ ω ( t − λ ) i . (15) The dep endency of the closed-lo op sys tem o n accur ate knowledge of the a irframe mo del in f ( ω ) is largely de- creased, improving ro bustness agains t mo del uncertain- ties contained therein. Therefor e, this implicit control law design is more depe nden t on a ccurate measurements or accurate estimates of ˙ ω 0 , the a ngular acceleratio n, and δ 0 , the deflection a ngles, resp ectively . Remark 1 : By using the mea sured ˙ ω ( t − λ ) and δ ( t − λ ) incrementally we practically obtain a robust, mo del-free controller with the self-scheduling prop erties of NDI. Notice, how ever, that typical INDI control la ws ar e nev - ertheless also dep ending on effector blending mo dels r e- flected in G 0 , which makes this implicit controller suscep- tible to uncertainties in these terms. Instead, consider the following transformation as in (Chang and Jung (2009)) ˙ ω = H + ¯ g · δ (16) with H ( t ) = f ( ω ) + ( G ( ω ) − ¯ g ) δ, and with the following (but not limited) options for ¯ g (Chang and J ung (2009)), whe r e n = 3 in our case ¯ g 1 = k G ·I n = k G     1 0 · · · 0 0 1 . . . . . . 0 1     , ¯ g 2 =     k G 1 0 · · · 0 0 k G 2 . . . . . . 0 k G n     . Applying nonlinear dynamic in version (NDI) to (16) re- sults in a n e xpression for the co nt rol input of the vehicle as δ ( t ) = ¯ g − 1  ν ( t ) − H ( t )  . (1 7 ) Considering H 0 = ˙ ω 0 − ¯ g · δ 0 , the incr emental c o unt erpa rt of (17) results in a control law that is neither dep ending on the a irframe mo del nor the effector blending moments δ ( t ) = δ ( t − λ ) + ¯ g − 1 h ν − ˙ ω ( t − λ ) i . (18) Remark 2 : The self-scheduling pro p e rties of INDI in (15) due to the ter m G 0 are now lo s t, suggesting that ¯ g should be an scheduling v aria ble . 3.3 Time Delay Contr ol and Pr op ortional Inte gr al c ontr ol Time delay c ontr ol (TDC) (Chang and Jung (200 9)) de- parts fro m the usua l dynamic inv ersion input tra nsforma- tion o f (16) δ ( t ) = ¯ g − 1  ν ( t ) − ¯ H ( t )  (19) where ¯ H denotes an estimation of H , b eing the nominal case when ¯ H = H which res ults in p erfect inv ersion. Instead o f ha ving an e stimate, the TDC takes the following assumption (Chang and Jung (2009)) analogo us to (12) ǫ T D C ( t ) ≡ H ( t − λ ) − H ( t ) ∼ = 0 . (20) This relationship is used together with (16 ) to obtain wha t is ca lled time-delay estimation (TDE) as the following ¯ H = H ( t − λ ) = ˙ ω ( t − λ ) − ¯ g · δ ( t − λ ) (21) In addition, ǫ ( t ) is called TDE err or at time t . Com bining the equations we obtain the following TDC law δ ( t ) = δ ( t − λ ) + ¯ g − 1  ν − ˙ ω ( t − λ )  (22) which is in fact e quivalent to the INDI con trol law obtained in (18). Appropr iate selection of ¯ g must e nsure stabilit y according to (Chang and Jung (20 09)), and ideally , this term should be tuned accor ding to the b est estimate of the true effector blending moment ˆ g ( ˜ ω ) for measured angular velocities ˜ ω . So far we hav e consider ed deriv ations in co nt inuous-time. F or pra ctical implementations of these controllers and for the matters of up coming discussions, sa mpled-time formulations inv olving c ontin uous and discrete qua n tities as in (Chang and Jung (2009 )) are mor e co nv enien t and restated here. F or that, consider ing that the s ma llest λ one can consider is the eq uiv alent of the sampling p erio d t s of the o n-b oard co mputer. The sampled fo r mulation of (22) may be expressed as δ ( k ) = δ ( k − 1) + ¯ g − 1  ν ( k − 1 ) − ˙ ω ( k − 1)  (23) where it has been neces sary to consider ν at s ample k − 1 for causality reas ons. Replacing the s ampled virtual cont ro l ν according to (8 ) we hav e δ ( k ) = δ ( k − 1 ) + ¯ g − 1  ˙ e ( k − 1) + k P e ( k − 1)  (24) and we can co nsider the following finite difference appr ox- imation of the er ror deriv a tives as angular accele rations are not dir ectly measured ˙ e ( k ) = [ e ( k ) − e ( k − 1)] /t s . (25) Consider now the standar d pr op ortional-inte gr al (PI) con- trol δ ( t ) = K  e ( t ) + T − 1 I Z t 0 e ( σ )d σ  + δ DC , (26) where K ∈ R 3 × 3 denotes a diagona l pr op ortional ga in matrix, T I ∈ R 3 × 3 a co nstant diagona l matrix repres e nt- ing a reset or integral time, a nd δ DC ∈ R 3 denotes a constant vector represe nting a tr im-bias, which acts as a trim setting and is computed by ev aluating the initial conditions. The discrete form of the PI is given b y δ ( k ) = K  e ( k − 1) + T − 1 I k − 1 X i =0 t s e ( i )  + δ DC (27) When substracting tw o co nsecutive terms of this discr e te formulation, we can remov e the integral sum and achieve the so-called PI c ontroller in incremental form δ ( k ) = δ ( k − 1 ) + K · t s  ˙ e ( k − 1) + T − 1 I · e ( k − 1)  (28) F ollowing the same steps , and for co mpleteness, w e a lso present the PID extension by simply consider ing the extra deriv a tive term ¨ e δ ( k ) = δ ( k − 1) + K · t s  T D ¨ e ( k − 1 ) + ˙ e ( k − 1) + T − 1 I · e ( k − 1)  , where T D ∈ R 3 × 3 denotes a c onstant diago nal matr ix representing deriv ative time. 3.4 Equivalenc e of IND I/TDC/PI(D) Having in mind the found the equiv a lence b etw een INDI and TDC, a nd comparing ter ms from (24) with (28), we hav e the following relationships as o riginally found in (Chang and Jung (2 009)) w hich are the rela tio nship betw een the discrete form ulations o f TDC and PI in incremental form K = ( ¯ g · t s ) − 1 , T I = k − 1 P (29) Whenever the system under co nsideration is of second- order controller canonica l form, we will hav e error dynam- ics of the form ¨ e + k D ˙ e + k P e = 0, and considering the newly intro duced deriv ative g ain k D related to ¨ e we hav e K = k D · ( ¯ g · t s ) − 1 , T I = k D · k − 1 P , T D = k − 1 D (30) This s uggests not only that an equiv alent discr ete P I(D) controller with gains < K , T i , ( T d ) > can b e obtained via INDI/TDC, but doing so is more meaningful a nd systematic than heuristic methods. This is beca us e we beg in the design fr om desir ed erro r dynamics given by Hurwitz gains < k P , ( k D ) > a nd what follows is finding the remaining effector blending ga in ¯ g either analytically whenever G is w ell kno wn, with a pro per estimate ˆ G , or b y tuning acco r ding to closed-lo op requirements. As alrea dy men tioned, details on a sufficient condition for closed- lo op stability under discrete TDC, and therefore applicable to its equiv alen t INDI, can b e found in (Chang and J ung (2009)) a nd the references therein. In essence, this pro cedure is more efficient a nd muc h less cum b erso me tha n des igning a whole set of ga ins iteratively . Moreov er, for flight control s ystems, the se lf-s cheduling prop erties of inv ersion-ba sed controllers have suggested su- per ior a dv antages with r esp ect to PID controls since these m ust be gain- s cheduled acco rding to the flight en velope v ar iations. The r elationships here outlined sugg ests that PID-scheduling sha ll b e done at the pro p o rtional gain K via the effector blending ga in ¯ g , and not ov er the who le set o f gains < K , T i , ( T d ) > . 4. LO NGITUDINAL FLIGHT CONTROL SIMULA TIO N In this s ection, robust PI tuning via INDI is demonstra ted with a simple y et s ignificant example co nsisting of the tracking control des ign fo r a long itudinal launcher vehicle mo del. The second-order no nlinear model is obtained fro m (Sonneveldt (2010 ); Kim e t al. (2004 )), and it cons ists on longitudinal dynamic equa tions r epresentativ e of a v ehicle trav eling a t an altitude of appr oximately 60 0 0 meter s , with aero dynamic coefficients represented as third order po lynomials in a ngle of attack α and Mach num ber M . The nonlinear equatio ns of motion in the pitch plane are given by ˙ α = q + ¯ qS mV T  C z ( α, M ) + b z ( M ) δ  , (31a) ˙ q = ¯ q S d I y y  C m ( α, M ) + b m ( M ) δ  , (31b) where C z ( α, M ) = ϕ z 1 ( α ) + ϕ z 2 ( α ) M , C m ( α, M ) = ϕ m 1 ( α ) + ϕ m 2 ( α ) M , b z ( M ) = 1 . 6238 M − 6 . 7 240 , b m ( M ) = 12 . 039 3 M − 4 8 . 2246 , and ϕ z 1 ( α ) = − 288 . 7 α 3 + 50 . 32 α | α | − 23 . 89 α, ϕ z 2 ( α ) = − 13 . 53 α | α | + 4 . 185 α, ϕ m 1 ( α ) = 3 03 . 1 α 3 − 246 . 3 α | α | − 3 7 . 56 α, ϕ m 2 ( α ) = 7 1 . 51 α | α | + 10 . 01 α. These approximations a re v alid for the flight env elop e of − 10 ◦ ≤ α ≤ 10 ◦ and 1 . 8 ≤ M ≤ 2 . 6. T o facilitate the control desig n, the no nlinear longitudinal mo del is rewritten in the more general state-space form as ˙ x 1 = x 2 + f 1 ( x 1 ) + g 1 u (32a) ˙ x 2 = f 2 ( x 1 ) + g 2 u (32b) where: x 1 = α, x 2 = q g 1 = C 1 b z , g 2 = C 2 b m and f 1 ( x 1 ) = C 1  ϕ z 1 ( x 1 ) + ϕ z 2 ( x 1 ) M  , C 1 = ¯ q S mV T , f 2 ( x 1 ) = C 2  ϕ m 1 ( x 1 ) + ϕ m 2 ( x 1 ) M  , C 2 = ¯ q S d I y y . The control ob jective considere d her e is to desig n a P I autopilot via INDI that tracks a smo oth command refer- ence y r with the pitch rate x 2 . It is a ssummed tha t the aero dynamic force and mo ment functions are accurately known and the Mach num ber M is treated as a parameter av ailable fo r measurement. Moreover, for this second- o rder system in non-low er triangular for m due to g 1 u and f 2 ( x 1 ), pitc h rate c o ntrol using INDI is p ossible due to the time- scale separa tion principle (Chu (201 0); Sieber ling et al. (2010)). With r esp ect to actua tor dynamics mo deled as in (5), we consider K a = 1, and τ a = 1 e − 2 . 4.1 Pitch r ate c ontr ol design First, introduce the rate-tr acking er ror z 2 = x 2 − x 2 ref (33) the z 2 − dynamics satisfy the following error ˙ z 2 = ˙ x 2 − ˙ x 2 ref (34) for which we design the fo llowing exp onentially stable desir e d err or dynamics ˙ z 2 + k P 2 z 2 = 0 , k P 2 = 50 rad/s . (35) According to the results previo usly outlined, the incremen- tal nonlinear dyna mic inv ersion control law design follows from consider ing the a ppr oximate dynamics ar ound the current refere nce state fo r the dy namic equation of the pitc h ra te as in (13 ) ˙ q ∼ = ˙ q 0 + ¯ g · ∆ δ (36) assuming that pitch ac c e leration is av a ilable fo r measure- men t and the s c a lar ¯ g to b e a factor of the accurately known estimate of g 2 ¯ g = k G · ˆ g 2 , k G = 1 . This is rewritten in our formulation as ˙ x 2 ∼ = ˙ x 2 0 + ¯ g · ∆ u (37) where recalling that ˙ x 2 0 is an incremental instance b efore ˙ x 2 , and therefor e the incremental nonlinea r dynamic in- version law is hence obtained as u = u 0 + ¯ g − 1  ν − ˙ x 2 0  , (38) with ν = − k P 2 z 2 + ˙ x 2 ref , (39) or more co mpactly u = u 0 + ¯ g − 1  − k P 2 z 2 − ˙ x 2 0 + ˙ x 2 ref  (40) This r esults a s des ired, in the following z 2 − dynamics ˙ z 2 = ˙ x 2 0 + ¯ g · ∆ u − ˙ x 2 ref . (41) Notice that we a re replacing the a ccurate knowledge of f 2 by a measurement (or an estimate) a s f 2 ∼ = ˙ x 2 0 , which will result in a co ntrol law which is no t entirely dep endent on a mo de l, hence mor e r obust. W e now conside r these contin uous-time formulations in sampled-time. T o that end, we re place the small λ with the sampling p erio d t s so that t k = k · t s is the k − th sampling instant at time k , and therefore u ( k ) = u ( k − 1 )+ ¯ g − 1  − k P 2 z 2 ( k − 1) − ˙ x 2 ( k − 1) + ˙ x 2 ref ( k − 1)  , (42) where due to caus ality relationships we need to consider the independent v ariables at the s ame sampling time k − 1. Referring back to the derived relationship b etw een INDI and PI control, the equiv ale nt PI control in incremen tal form is u ( k ) = u ( k − 1) + K · t s  ˙ z 2 ( k − 1) + T − 1 I z 2 ( k − 1)  , (43) with K = ( ¯ g · t s ) − 1 , T I = k − 1 P 2 (44) The na ture of the des ired er r or dynamics (prop ortional) gain k P 2 is therefore of an in tegral control actio n, whereas the effector blending gain ¯ g act as pro p o rtional cont ro l. Having designed for desired err or dyna mics, and for a given sampling time t s , tuning a pitch rate co ntroller is only a matter of selecting a prop er effecto r blending ga in ¯ g a ccording to p erfor mance requirements. Remark 3 : Notice at this point that having the PI control in incremental form intro duces a finite difference of the error state, which is the equiv a lent co un terpar t of what has be e n cons ide r ed the ac celeration or s tate deriv ative ˙ x 2 0 in INDI c ontrollers. Remark 4 : Notice also that desig ning the PI control gains via INDI is hig hly b eneficial, s inc e only the effector blending ga in is the tuning v aria ble . T his strong ly suggests that r obust a daptive cont ro l can be achiev ed by scheduling this v ariable online dur ing flight and not over the whole set o f gains. Sim ulation results for the INDI/PI control are presented in Figure 2, considering s mo oth rate doublets for a nominal longitudinal dynamics mo del a t Mach 2. F or b oth con- trollers, the same zero-mea n Gaussian white-noise with standard dev iation s d q = 1 e − 3 rad/s is added to the rates to simulate no is y measure ments. The designed INDI gains of k P 2 = 50 rad/s and k G = 1 are mapped to PI gains resulting in K = 100 ˆ g − 1 2 and T I = 0 . 02 s, b oth controllers showing iden tical clo sed-lo op r esp onse as exp ected. With this example, it is demonstrated ho w a self-sc heduled PI can b e tuned via INDI by departing from desired er ror dynamics with the g ain k P 2 , and consider ing an accura te effector blending mo del e stimate ¯ g = ˆ g 2 . q RE F q I N D I q P I q I N D I − q P I α I N D I α I N D I − α P I δ I N D I δ I N D I − δ P I P S f r a g r e p la c e m e n t s δ (deg) time (s) time (s) q (deg/s) α (deg) 0 5 1 0 1 5 2 0 0 5 10 15 20 0 5 10 15 20 − 6 − 4 − 2 0 2 4 6 − 6 − 4 − 2 0 2 4 6 − 6 − 4 − 2 0 2 4 6 Fig. 2. INDI/PI nominal tra cking control simulation o f the flight mo del (31) for k P 2 = 50 rad/s and k G = 1 5. CO NCLUSIONS This paper presented a meaningful and sys tema tic metho d for PI(D) tuning of r obust nonlinear flig ht control systems based o n results previously r ep orted in the ro b o tics lit- erature (Chang and Jung (2009)) re garding the relation- ship b etw een time-delay c ontr ol (TDC) and pr op ortional- inte gr al-derivative c ontr ol (PID). The method was demon- strated in the context of an example for the pitch rate tracking of a conv entional longitudinal nonlinear flig ht mo del, sho wing the same trac king perfor mance under nominal conditions. Being incremental nonlinear dynamic inversion (INDI) equiv a lent to TDC clearly suggests that impo sing de- sir e d err or dynamics , as usual f or INDI control laws, and then mapping these in to an equiv alent incremental PI(D)-controller tog ether with control deriv atives lea ds to a meaningful and sy s tematic PI(D) gain tuning method, which is very difficult to do heuristically . W e co nsidered a refor mulation o f the plan t dynamics in- version which reduces knowledge of the effector blending mo del (cont ro l deriv atives) from the resulting cont ro l law, reducing feedback co nt ro l dependency o n accura te knowl- edge o f b oth the aircraft a nd effector blending mo dels, hence re s ulting in ro bust and mo del-free co nt ro l laws like the PI(D) control. 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